Proceedings.
International Congress of Mathematicians.

Page  [unnumbered] BIBLIOGRAPHIC RECORD TARGET Graduate Library University of Michigan Preservation Office Storage Number: AAG4063 UL FMT S RT a BL s T/C DT 07/15/88 R/DT 07/15/88 CC STAT mm E/L 1 010:: a 52001808//r62 035/1:: a (RLIN)MIUG83-S20809 035/2:: a (CaOTULAS)159944565 040:: aCU IcCU IdNvU IdMiU 041:: a engfregeritarus 050/1:0: a QA1 b.I8 082/1:: a 510.631 111:2: a International Congress of Mathematicians. 245:00: 1 a Proceedings. 260:: a [v.p.] 300/1:: lav. b ill. c24-31cm. 310:: [ a Quadrennial 362/1:0: a 1897 -500/1:: [ a Some volumes have title in translation: Actes, Atti, Comptes rendus (varies slightly), Trudy, Verhandlungen. 515/2:: a No congresses held 1916,1937-1949. 515/3:: I a Congresses held 1897-1912 called lst-5th; those for 1920-1958 designated new series, lst-8th. 546/4:: | a Contributions in English, French, German, Italian and Russian. Scanned by Imagenes Digitales Nogales, AZ On behalf of Preservation Division The University of Michigan Libraries Date work Began: Camera Operator:

Page  [unnumbered] PROCEEDINGS OF THE FIFTH INTERNATIONAL CONGRESS OF MATHEMATICIANS

Page  [unnumbered] CAMBRIDGE UNIVERSITY PRESS kLonuon: FETTER LANE, E.C. C. F. CLAY, MANAGER I ~laFp 0 ebinburgbj: 1oo, PRINCES STREET Berlin: A. ASHER AND CO. tL4ig: F. A. BROCKHAUS jlctb Lork: G. P. PUTNAM'S SONS 13ambap an-b Calcutta: MACMILLAN AND CO., LTD. All 'rights reserved

Page  [unnumbered] PROCEEDINGS OF THE FIFTH INTERNATIONAL CONGRESS OF MATHEMATICIANS (Cambridge, 22-2 8 August 1912) EDITED BY THE GENERAL SECRETARIES OF THE CONGRESS E. W. HOBSON SADLEIRIAN PROFESSOR OF PURE MATHEMATICS IN THE UNIVERSITY OF CAMBRIDGE AND A. E. H. LOVE SEDLEIAN PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF OXFORD VOL. II. COMMUNICATIONS TO SECTIONS II-IV Cambridge: at the University Press 1913

Page  [unnumbered] cambribte: PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS

Page  [unnumbered] COMMUNICATIONS PAGE SECTION II (GEOMETRY).......... 7 SECTIONS III (a) AND III (b) (ASTRONOMY AND STATISTICS)... 61 SECTION III (a) (MECHANICS, PHYSICAL MATHEMATICS, ASTRONOMY). 175 SECTION III(b) (ECONOMICS, ACTUARIAL SCIENCE, STATISTICS).. 341 SECTIONS IV(a) AND IV(b) (PHILOSOPHY, HISTORY AND DIDACTICS) 447 SECTION IV (a) (PHILOSOPHY AND HISTORY).... 459 SECTION IV (b) (DIDACTICS)........ 543

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Page  [unnumbered] SECTION II (GEOMETRY)

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Page  9 SUR LA NOTION DE " CLASSE " DE TRANSFORMATIONS D'UNE MULTIPLICITE PAR L. E. J. BRoUWEIP,. Soet I t /'des mnultiplicite's ferme'es: nous dirons de deux represenaions univoques et continues de /. sur tk' qu.'elles appartiennent 'a la mn'me clatsse, Si. nolS pouvons arriver' de l'une 'a lautre an moyen d'une modification continu. Le representations dle la me'me classe posse'dent tontes le me'me degre' (voir mon travail " Ueber Abbil'dung von Mannigfaltigkeiten," Af!athein. Annalen, Bd. 71), mais ce qui. est remarquablc, c'est qu'en tin grand nomnbre de cas linverse de cc the'ore'me est encre ra, popie'te dont j'indiquerai brie'vement la demonstration pour.eca u jk et ~t' sont toutes les deux des sphe'res. Nous comuiencerons par de'montrer la proprie'te pour certaines categories speciales de representations, (]oul nous nous e'Reverons par e'tapes aux representations les plus gene~rales. Conside"rons d'abord les repre'sentations riernanm jennes, c'est-a'-dire les repre'sentations qui an sens de F'analysis situs sont identiques aux representations des surfaces de Riemann de genre zero sur le plan complexe. D'une part M. Klein a demiontr6 que toutes les repre'sentations de degre' r 'a ramifications sirnples peuivent e'tre transforme'es d'une nmani~re continue les unes en les antres en faisant mnouvoir convenablement, les points de ramification (proprie'te' de'coulant dii the'ore'me de Ltiroth-Clebsch sur l'arrangemnent canonique des coupures de ramification), d'autre part on pent, pour les representations ii ramifications multiples, se'parer ces ramifications en un nombre 'equivalent de ramifications simples, de sorte que pour les repre~sentations riemanniennes notre the'ore'me se trouve d4'niontre'. Conside'rous ensuite les 7-epre'sentations cctnoniqutes, pour lesquelles ni - 1 courbes simples fermn'es, situe'es dans /z et sans points communs, sont repre'sente'es chacune en un seul point (le tk, tandis que les n regions de'termininees par elles sont repre'sente'es biunivoquement, et cela on toutes positivement, ou bien toutes nc'fgativement. Une representation canonique pouvant e~tre transforin~e, au moyen d'une modification continue arbitra-irement petite, en une representation riemannidnne, notre the'oreme est encore de~m-ontr4' pour les representations canoniques. Passons atix r-epre"Se'ntations normales, ne satisfaisant qu'aux conditions suivantes: (1) elles sont sans contraction cyclique, c'est-a'-dire il n'existe sur at aucune courbe simple ferme~e represents'e en un seul point de /'; (2) elles sont simnplicia les; (3) elle's

Page  10 10 10 ~~~~~~L. E. J. BROUWER sont sans plis*, c'est-ah-dire de chaque element de 1a, dont l'image est biunivoque, l'indicatrice de l'image a le iue'me signe. Pour les representations normales nous de'montrons notre the'ore'me en faisant subir aux arcs de courbe de u' images d'e6lements de iz, une sorte d'extension continue les rendant images biunivoques et transforinant la representation normale en une representation riemannienne. Envisageons enfin les r-epre'sentations simpliciales sans plis, pour lesquelles il existe sur a un1 nombre fini d'ensembles polygonaux d'un seul tenant, repre'sente's chacun en un seul point de lk' et divisant ~k en un nombre fini de regions, pour chacune desquelles la representation est ou normale on nulle part dense. Pour ces dernie'res representations nous de'montrons notre the'ore'me en re'duisant d'abord les ensembles polygonaux 'a des courbes siruples ferme'es sans points communs-ce qui nous donne une representation simpliciale sans plus "h' contraction simple "-et re'duisant ensuite la representation de chacune des regions partielles, dont l'image n'est pas nulle part dense, successivement 'a une representation riemannienne et 'a une representation canonique, ce qui nous donne une telle representation simpliciale sans plis 'a contraction simple, pour laquelle les regions partielles de p se divisent en trois sortes, repre'sente'es sur /' successivement avec le degre' 0, avec le degre' + 1, et avec le degre' - 1. Or, an moyen de nouvelles transformations continues de la representation, on peut d'une part remplacer lensemible d'une region de 2d et une region adjacente de 3m sorte par une seule re'gion de resorte, d'autre part faire absorber une region de Ire sorte par une region adjacente de 2de ou de 3me sorte, proce'de qui finit par nous donner une representation canonique. Maintenant les r-epre'sentations gene'rales ne pre'sentent plus de difflculte', puisqu'en 'commendant par les rendre simpliciales et de'truisant ensuite les plis morceau par miorceau, on peut les transformer d'une manie're continue en representations simpliciales sans plus. *Par un "1pis " nous entendons ici une suite finie d'e'l~ments de A, dont chaque 616ment, excepts le dernier, poss~de en commun avec son suivant un " cote' de contact," tons ces c6te's de contact e'tant repr6sente's sur un meme arc de courbe de A', et le premier et le dernier element de la suite 6tant repre'sente's avec des indicatrices oppos6es.

Page  11 ON THE EXTENSION OF A THEOREM OF W. STAHL BY F. MORLEY. ~ 1. Counter-curves. Given d + 1 binary forms f)...fa all of order m, an involution I" is defined by (if) -ofo +. + dfd = 0. The natural geometric representation of I" is the sections of a rational curve pa of order m in a space Sd by spaces Sd_. All forms b, also of order mn, which are apolar to all forms f define the counterinvolution (or conjugate involution) (X() -= 0o0 +... + X,-d-1 (m-d — = 0, which is represented by a curve Pm-,_l, the counter-curve of p. A rational curve carries an infinity of involutions; for example in the case of a planar curve all algebraic curves of given order on the flexes will cut out an involution, and all algebraic curves of given class on the stationary lines will cut out an involution by means of the common lines. Particular interest attaches, in the case of planar curves, to involutions given by curves on the double lines. For the curve p^, the curve of lowest class on the double lines is of class 2 (m - 3); the common lines of this and p give an 14(1 3); and my point is that this is the dual form of the counter-curve. In other words, the counter-involution (or in Stahl's phrase fundamental involution) on a curve p^ is given by a series of theorems, the first of which was given by Stahl (Crelle, Vol. 101), as follows: For the quartic pa4, any conic on the double lines has with p 4 common lines; these give a set of the counter-involution. For the quintic p," any range of quartics on the double lines contains 5 which touch p; the parameters of contact are in the counter-involution. For the sextic p.2 any web of sextics on the double lines contains 6 which osculate p; the 6 parameters are in the counter-involution. And so on. I shall give the proof in the case of p,5, by a method which applies to all cases.

Page  12 12 F. MORiLEY ~ 2. Algebraic theory of the planar rational quintic*. The argument does not require explicit expressions, but only the degrees of the coefficients in these expressions. Thus the form (cf) which, when zero, gives the p25 is sufficiently indicated by A t.......................................(1), where A denotes a constant, 4 the coordinates of a line in the plane, t a parameter. Regarding this as a binary quintic we have for two such quintics AVt5, Aqrt5 the combinative transvectants A'2x8, A2T4, A2..............(2. 1, 2. 2, 2. 3); the first being the line-equation or point-section of the curve. Making x and ~ incident in (1) and (2) we get A3t3r6, A3t5r4, A3t5............(3. 1, 3. 2, 3. 3), where in (3. 1) we have removed from an A3t5T8 the factor (tr)2. The remaining factor A3t3T6 expresses the incidence of point t and line 7, and may be called the incidence form. The coefficients in (3) are linear in the coordinates of p25, that is in the 3-rowed determinants formed from the constants A. The incidence form is a special case of Gordan's involution form A3 t................................... (4), which expresses that the points tj, t2, t3 are collinear. By developing Gordan's form Professor Coble (On symmetrical binary forms and involutions, Am. Jour. Vols. xxxi. and xxxII., in particular Vol. xxxII. p. 359) obtained the 3 forms in r alone, which are linear in the coordinates. These linear forms are: A'r9,, 3r A...............(5. 1, 5. 2, 5. 3), or say (fr)9, (Tr)5, (hz)3, the first being the stationaryform which gives the triple roots of (1). If, after Study, we write the equation of p2 symbolically: (ar) (at)5, then f is i abc /3y ] ya | ay / (at)3 (/t)3 (yt)3, g is abc \f3y \\rya a/3 fi \ry 2 (at)3 (/3t) (yt), his abc / y ya a 3 y i (at) l a 2 (3t)2; and in general to write the linear forms for a p4 we write all covariants of degree 3 of a form (ax)n-2; for a p" all covariants of degree 4 of a form (ax)'-3, and so on. These linear forms are fundamental. All covariant line curves of p25 are invariants of the binary forms (1) and (5). All covariant point curves are invariants of (2) and (5). All covariant connexes are invariants of (1), (2) and (5). Covariants of the binary systems (1) and (5) yield covariant linear systems of curves. And so on. * Compare W. Gross, Annalen, Vol. xxxII.

Page  13 ON THE EXTENSION OF A THEOREM OF W. STAHL 13 ~ 3. Quartics on the double lines of a rational planar quintic. Consider the web of curves of class 4 on the 12 double lines of P25. They have with p25 32 common lines; these are the double lines counting twice and 8 variable lines. Two of these determine the rest, that is we have an involution 128, which is pictured by a curve, say of lines, r8. When the quartic touches a stationary line, this line counts as two common lines; there are then 9 lines of r28 which count twice as tangents, i.e. r28 has nine stationary lines. It is then, as a point curve, of order 5, say p,5; and it has the same stationary form (ft)9 aS p2. Hence its incidence form is (ft) (fr)..., and is linearly built from j; g, h. Denote it by A 1:3 tl r,)....................................(6). The lines T, T, of p and p cut out the sections A3tr6T tTr2 and A1,tl3r t1" 1. These are apolar if A T6r- 8 = 0....................................(7). Since no form of degree 2 and order 16 can be built from f, g, h, (7) vanishes if = =,. It is then I A6 7 77 7................................ (8). Now (7) shows that eight tangent-sections of p are apolar to a tangent-section 7 of p. Either then all sections of p are apolar to all those of p, in which case the curves are counter-curves, or else the eight tangent-sections are all on a point x. Looking now at the curve p.5 only, the form A6T7r17 gives for every 7, a point x on the line 71, and the values of r are the parameters of the other lines from x. When T1 is a stationary parameter, T= T1. Hence when T= T], the form A6T7r,7 contains the factor f The other factor must be g. Thus as Tr varies the locus of x crosses each stationary line once, and therefore is a line; and it meets the curve where g = 0. This line g is the line (2. 3), whose equation is A2x = 0. Hence, either the form (8) vanishes identically or it is the condition that the lines r and T1 meet on the line g. This condition is obtained by making two tangentsections of p25 apolar. Hence when the curves p2* and p5 are projectively distinct, they are counter-curves. It is proved then that given the forms f, g, h, there are two and but two incidence forms, linear in the coordinates; these defining the curve itself and the counter-curve. ~ 4. The self-counter quintic. The curves p and p cease to be distinct when the involution set up by the quartics is also the involution set up by lines of p25 on a point; that is when the quartics can break up into a cubic on the double lines and an arbitrary point. Hence the double lines of a self-counter quintic are on a cubic.

Page  14 14 F. MORLEY This interesting special curve has to satisfy 3 conditions; any two line-sections are apolar, so that the line A2x = 0 and therefore its section (gr)5 = 0 become arbitrary. ~ 5. Quintics on the double lines. Passing now to quintics on the double lines, we can write after Cayley (Works, Vol. ii. p. 469) a covariant of (1) of degree 5 and order 9 which vanishes when (1) has two double roots. Hence we have special quintics on the double lines A5 5t9, and the apolarity-relation of this with an arbitrary form (at)9 gives the general such quintic, a A5 55. The common lines of this and p are aA15740 and removing the parameters of the double lines, an A12724, we have for the variable common lines a j r16 This can only be the Jacobian of (at)9 and (fr)9. We have then in the common lines a picture of the Jacobian of the form f and an arbitrary form a, forming an involution I816. In the breaking up of the quintic into a quartic and a point we have the breaking up of the 8'16 into two counter-involutions 1 28and I28. And this raises in a new form an old question: how many curves p,5 are there when the stationary form (fr)9 is given? This is equivalent to the question: how many planes in S5 meet 9 given planes?-which is raised in Meyer's Apolaritdt. Professor Coble solves this problem of enumeration as follows. Consider first the inter-relations of the linear formsf, g, h. ~ 6. The linear forms. Writing 1 012 etc. for the 3-rowed determinants formed from the coefficients of three line-sections written without binomial coefficients we have for the stationary form =I1012 T.+3 0131T8+{61014 +3 0231}T7 21 1 + {1 0151 + 8 10241 + 1125 1 T6 + 15 0251+ 61034 + 31124 1} r5 + {1510351+61125 + 31134 }4 + 1010451 + 81135 + 1234 1 r3 + {6 145 1 + 3235 } T 2+ 3 245 T + 345 1...........................(9. 1); and for g and h as calculated by Mr Tracey g = 12 0141 - 023 1} T5 + {10 1015 -1123 1} 4 + {10 1 025 - 2 124 1} " + {10 0351- 2 1:34 } T2 + [10 0451- 234 I T+2 1451 - 235...................(9. 2),

Page  15 ON THE EXTENSION OF A THEOREM OF W. STAHL 15 = t20 1 015 -51 024 +2 123 1} T3 + {15 1025 i - 15 034 +3 1124} iT2 -{15 035 -15 125 +31134}'T - {20 045 - 5 j135 + 2 234 }.....................(9. 3). Thus the 20 coefficients of f, g, h are 20 linearly independent combinations of the 20 linearly independent coordinates 1 012 i etc. of the curve. Any relation among these coefficients must be due to a syzygy connecting the comitants off, g, h. 'Call X + p + v - 3 the weight of i X\v |. Then the coefficients of f have their natural weight, those of g and h have their natural weight increased by 2 and 3 respectively. The weights and orders of the comitants of the second degree are: Weight Order Forms 0 18 ff 2 14 Ifft2, fg 3 12 Ifg i, fh 4 10 tffi4, fh I, gg, Ifg 2......(10). 5 8 ifh I2,.fg 3, gh 6 6 6 Iff", Ifhl, Igg2", hh, fg 4, Ighl 7 4 Ifg, 5 gh12 8 2 iffi8, iggs4, hhl2 Now the linearly independent relations of the second degree in the coordinates are known; there are 30 of the type 012 1034j+l0231 041++1 031 024, and one of the type 1012 345 +... There is then one of weight 4 i012 0341+..., indicating a syzygy of weight 4 and order 10; two of weight 5, 1012i 035 +... and 1102 111341 +..., indicating a syzygy of weight 5 and order 8; and so on. In any syzygy g must occur throughout to an even or an odd degree, since we pass from a curve to the counter-curve by changing the sign of g. Thus the second syzygy must be either Ifh 2=0 or \fg3 =gh, and it is seen from the leading terms that the former is not the case. We have then fg = gh.................................... (11), where the numerical factors are selected as in (9). The equation (11) defines g as an irrational covariant off.

Page  16 16 F. MORLEY ~ 7. The enumeration. Inserting a factor of proportionality h. we get from h,. Ifg 13 = gh 9 bilinear forms in g0... g, and h,, h0... h. Take now 10 bilinear forms in 6 g's and 5 h's (ag) (/3h), and first suppose these factors ag, /3h not symbolic. The condition for a common solution is the product of ( 16) determinants in the coefficients a, and of ( 1) determinants in /. It contains the coefficients a of each form to the degree (9) and / to the same degree (4). Hence when the bilinear forms are not degenerate the invariant which expresses that they have a common solution is of degree (9) in the coefficients of each form. Since this invariant would be obtained by substituting in the tenth form the solutions of the other 9, and multiplying, it is the number of such solutions. There are then () forms g when f is given, and therefore there are 126 pairs of counter-quintics.with a given stationary form. Or again there are 126 pairs of planes meeting 9 given planes in a space 5. In starting with a p2' we have isolated one of these 126 pairs of solutions. The other 125 pairs are then represented by pairs of nets of quintics on the double lines; for the involution I,16 breaks up for each solution into two independent involutions 12s and I28. ~ 8. Case of the p'4. The explicit form for the conies on the double lines of a quartic is easy to obtain. The line-section being A t4, its Hessian is A2 2t4. If the counter-involution be (a)4 + X (bT)4, the Hessian is apolar to this if aA2 2 + XbA2 O = 0...........................(12), a range of conics. Each line of the conic X gives a section whose Hessian is apolar to the quartic X. Hence a common line of the range gives a section whose Hessian is apolar to all quartics X, i.e. whose Hessian is also a section. But such a section is given by a double line. Hence (12) is the conics on the double lines. Eliminating X we have A52t4.

Page  17 ON THE EXTENSION OF A THEOREM OF W. STAHL 17 In terms of the linear forms (fr)6 and (gr)2 of the quartic this is Hf 3 + Hg, where H is the Hessian of a section. In this case of the quartic, cubics on the double lines can break up in 5 ways, once into an arbitrary point and a conic on the double lines and four times into a point on a double line and a conic on the other three. Thus the 5 binary involutions with the given Jacobian f are the counter-involution of the p24 and those of the tangents to p24 from a point on the double line. Similarly for p::5 the stationary form (fr)8 is, we know, the Jacobian of 14 pencils of binary quintics. One of these pencils is the counter-involution of the curve; the rest refer to the 13 lines, each of which is in 4 planes of the curve. From a point on such a line we can draw 5 other planes, and thus we have 13 pencils of binary quintics, obviously having the given Jacobian. 5I. C. II.

Page  18 CERTAIN CONTINUOUS DEFORMATIONS OF SURFACES APPLICABLE TO THE QUADRICS BY LUTHER PFAHLER EISENHART. Bianchi* has established an important transformation of surfaces applicable to the quadrics, such that, if S be one of these surfaces and S a transform, the lines joining corresponding points M and M on S and S form a congruence for which S and S are the focal surfaces, and the asymptotic lines on these surfaces correspond. A congruence possessing the latter property is called a W-congruence. It is a general theorem concerning W-congruences that either focal surface admits of an infinitesimal deformation such that the direction of deformation at any point is parallel to the normal to the other focal surface at the corresponding point. Hence the Bianchi transformations of surfaces applicable to the quadrics carry with them certain infinitesimal deformations of these surfaces. It is the purpose of this paper to show that with the aid of these transformations one may derive from a surface S, applicable to a quadric Q, a suite of continuous deforms of S. Such a suite we call a system (Q). Although there is an essential difference in the form of the equations according as Q is a paraboloid or a central quadric, the results are analogous in all cases. Unless otherwise stated the following equations refer to the case where Q is the real hyperbolic paraboloid - - - = 2z and the transformation is real. P q We assume that the parametric curves u = const., v = const. on S correspond to the generators on Q. In this case the equations of a transform S are of the form ax ax x = x ~ I' +m ~ m.(1) - +- +.............................(1), where 1 and m are the two ratiost U V w m=( V.. 2), W ' m W.......................... 2), U, V, W being algebraic functions of u, v, of an essential constant i which fixes the confocal quadric QK determining the transformation, and of a function X which must satisfy the system+ ax v -H + (2DU+ + U (D U + aD"Tr)...(3), au IcH 2/cJ/H av =cH 2 ci/H * Lezioni di Geometria Differenziale, Vol. II. Pisa (1909). Hereafter a reference to this volume will be of the form B. p. -. + B. p. 13. + B. p. 81.

Page  19 DEFORMATIONS OF SURFACES APPLICABLE TO THE QUADR[CS 19 where 4H= EG- F2; E, F, G and D, D), D" being the fundamental functions for S; and e is + 1 or - 1 according to the system of generators used on Q, to determine the transformation. In accordance with the general theorem previously referred to, an infinitesimal deformation of S is given by x,=x+a,, y,=y+ (7l, z,=z+ac...................... (4), where a is an infinitesimal constant such that powers higher than the first are negligible, and I, v, o are of the form eT (Fl + ) a + (El + Fr) x + 2 - AXj.........(5), the functions X3, Y3, Z3 being the direction-cosines of the normal to S; A a deterrinate function; and T a function to be determined by the well-known conditions of infinitesimal deformation, namely ax ax ax a~ ax a~ axaa E.t =0, =s0, -v+ -- =o 0............ (6). au all dv av al aav av au( The resulting equations are reducible by means of (3) to aT alogH dlogm eV/H A D -- + - - -e + _ - =0 9t atu du 1\/p 2 m? aT alogH dlogl e VH-AD". 7 + - __ + - - ~ = 0..................(7), v + v Adv /v 2 1 i aT aT dl dm e/H I — r-m a+ + p -a AD' = 0 aU av du dv,/pq d a a ax d a a ax where r = + =e +_ d? dau X' ax i' dv =v ax av' Equations (7) are readily shown to be consistent by means of the Gauss and Codazzi equations for S. If we are interested not in a discrete infinitesimal deformation of S, but in a continuous deformation, we must look upon the functions appearing in the above expressions as involving a third parameter, say w. The preceding results are true in this case also, but there are further conditions to be satisfied. Now in equation (5) 3x: must be replaced by -, and the conditions of integrability aw ' a ax\ a (ax) a (ax) a ax- 9altaw)~ w W a a~w[)v............. ). must be satisfied. If we put /=E X 1, = /X........................(9), ax _.(9), du av conditions (8) are reducible to axlr(FXV x) B..'(1 a- = eT(FX, - /EGX2) + X,, =eT (EGX, -FX2)+ G-X (10) where B and are determinate functions of, v,, T D, D D". where B and C are determinate functions of u, v, X, T, D, D', D". 2 —2

Page  20 20 LUTHER PFAHLER EISENHART From (10) follows ax, -= (FX1- / -GX2) - (EG X1 - FX2) )......(11). aw 4, H -\ 4 By means of the Gauss equations for S we can find the expressions for ax, ax, aX2 ax,. ax3 ax, au' v; au' av a'u a. (12). 9ue ' 8v? au? au ' a^5 8 v.................. The necessary and sufficient condition that a system (Q) exists is that the integrability conditions of the expressions (10), (11), (12) be satisfied. These lead to a system of five independent partial differential equations in two dependent variables X and T, and three independent variables u, v, w. Two of these equations are of the first order and the other three of the second order. These equations are such that one may readily establish the existence of sets of integrals. When a set of values of X and T satisfying these equations is known, the functions D, D', D" follow directly and the corresponding system.(Q) is defined intrinsically. When one has a system (Q), the transformation (1) applied to each surface of the system leads to a family of surfaces S, each of which possesses the property that the deformation of a member of the first system is in the direction of the normal to the corresponding member of the second system at the corresponding point. We call the second the conjugate system to the original system (Q) and we show that The system conjugate to a given system (Q) is itself a system (Q) whose conjugate system is the given system. In order to establish this result it is necessary to give an analytical proof of the fact that the transformation (1) is reciprocal*. Of particular interest is the case in which all of the surfaces w= const. of a system (Q) are ruled. For this case we have D=0, D'= 2x, D"= 2 /B (v, w), where b is to be determined. If we put eT _ Hm ' the set of equations determining the system reduces to ap 2 a 1 air 1 av ax Vf — a......(13), A- + - o- + o-F(v, X) 0; = 0, - = - a1 a =- V.....(13) aw v v' K V av Vv 3v K where F is a determinate function. The surfaces w = const. of the conjugate system are likewise ruled. We have in this case a very interesting type of rectilinear congruence, consisting of a family of applicable ruled surfaces. When the locus of the lines of striction of these surfaces is taken as the surface of reference of the congruence, the Kummer functions assume a very simple form. When the quadric Q is a hyperboloid of revolution, the lines of striction on the * Bianchi has given a geometrical proof of this fact for the case when S is a ruled deform of Q; B. p. 36.

Page  21 DEFORMATIONS OF SURFACES APPLICABLE TO THE QUADRICS 21 applicable ruled surfaces are curves of Bertrand. For a system (Q) of these ruled surfaces the intrinsic equations of these curves of Bertrand are FI II ' 4 s\11) 11**(1, 2 c. sec4 (- 2, -2 = sec4 2...........(14), p 2aL' \2a/j' '-I'r 2 2a where b(s, w) satisfies an equation similar to (13). The corresponding lines of striction of the conjugate system are likewise curves of Bertrand. Regarding these systems we have the theorem: As 0 in equations (14) varies with w, a given Bertrand curve of the family is deformed into another curve of the family, the deformation at each point being in the direction of the principal normal to the corresponding Bertrand curve of the conjugate system. When in particular the quadric Q is the imaginary sphere x2 + y2 + z2 + 1 = 0, the equations define a system of isogonal deforms of pseudospherical surfaces of the type studied by Bianchi*. If in (2) and (3) we replace K by K1, the corresponding equation (1) defines a transformation of S into a surface S,. When this transformation is applied to the surfaces of the original system (Q), one gets a suite of surfaces S, which is not in general a system (Q). When, however, these surfaces form such a system, we have thus a transformation of a system (Q) into another of the same kind. The system conjugate to a given system arises from a particular transformation of this sort. Regarding general transformations of systems (Q), we announce the theorem: In order that the transforms of a system (Q) form a system (Q), it is necessary and sufficient that XI satisfy in addition to equations analogous to (3) also an equation of the form ax, - aw where I is a function whose form varies with the character of the quadric Q. We have established this theorem, from another point of view, for the paraboloids, and owing to the analogy between all the other results we feel sure that it is true also for the central quadricst. However, we have been unable to complete the proof in the general case by the methods of this paper. * Sopra una classe di deformazioni continue delle superficie pseudosferiche, Annali di Matematica, Tome xvIII. Ser. III. pp. 1-67. t In fact, Bianchi (I.c.) has established this theorem for systems (Q) of pseudospherical surfaces.

Page  22 RECENTI PROGLESSI NELLA GEOMETRIA PROIETTIVA DIFFERENZIALE DEGLI IPERSPAZI Di ENRICo BoMPIANI. Dopo le classiche ricerche di Monge e della sua scuola, ove accanto a proprieta' inetriche differenziali di una superficie dello spazio ordinario vengono anlche studiate proprieth di carattere proiettivo, passa un lungo periodo di tempo, dovuto forse al rapido ed esuberante sviluppo della geometria metrica nella Francia stessa, prima che 1' indagine geometrica volga di nuovo i suoi sforzi a quel. ramo di geometria che s' indica col nomne piuttosto moderno di proijettivo-differenziale. Si raccolgono in esso quei risultati di geometria differenziale che appartengono (secondo un concetto del Klein) al gruppo proiettivo (invarianti cioe' per una trasformazione di questo gruppo). Scarsa eredita' ha lasciato a noi il secolo passato in questo campo; la teoria delle tangenti coniugate di Dupin per le superficie dello spazio ordinario ne costituisce certo la parte piii' preziosa. La proiettivita' di Chasles fra punti e piani tangenti lungo una generatrice di una rigata appartiene pure a questo gruppo. Alla ricerca delle proiettivita' di Chasles degeneri si riduce ii problema della distribuzione in sviluppabili delle rette di una congruenza, e a questo equivale a sua volta la determinazione di un doppio sistemna coniugato di linee sopra una superficie. PhiU recenti (1882) sono alcuni risultati del Koenigs sul lnodo di comportarsi di una congruenza o di un complesso lineare rispetto ad una congruenza o ad un complesso qualsiasi nell' intorno di una generatrice. Qunesti teoremi, dati da Koenigs come analoghi al teorema di Meusnier (di natura mietrica, non proiettiva) non sembrano facilmyente ricollegarsi agli altri ricordati, ed occorre infatti salire agli iperspazi perche' vengano in piena luce le diverse analogie *. In Italia si veniva intanto sviluppando la geometria proiettiva degli iperspazi;, nel 1886 ii Prof. Del Pezzo ottenne per via sintetica i primi resultati relativi a spazi pluritangeiti a varieth immerse in un ambiente qualsiasi. Poco dopo, nel 1888, il Prof. Segre estese ai sistemi 00nl di rette di S,, le proprieth focali di una congruenza di S,. * Tralascio di ricordare, perch6 non interviene nel seguito, una ricerca del Darboux (1880) "1sur le contact des courbes et des surfaces."

Page  23 GEOMETRIA PROIETTIVA DIFFERENZIALE DEGLI IPERSPAZI 2 23 Un ventennio, trascorse inoperoso, sopra questi risultati, fatta eccezione per le ricerche del Wilczynski (relative allo spazio ordinario) che avro' poi occasione di ricordare. In una serie di lavori pubblicati dal 1906 al 1910 il Prof. Segre ha ripreso e condotto ad un elevato grado di perfezione questo ramo di geornetria iperspaziale *. Ii contributo che ad esso recano, sotto 1' impulso, del Segre, giovani geometri in Italia e fuori puo6 percio, caratterizzarsi veramente italiano. lo mi propongo di dare uno sguardo all' opera compiuta, cercando di porre in luce le idee fondamentali, e di completarne in qualche punto lo svolgimento. Due tipi di problemi si pongono in quest' ordine di ricerche. Un primo problema, che puo" dirsi ristretto, ha per oggetto lo studio della costruzione dell' intorno di un punto di una varietah in un ambiente qualsiasi. Ii secondo, problema esteso, ricerca caratteri die interessano la costruzione di tutta la varieth (per es. 1' esser costituita da spazi lineari). Vediamo come si pongano e Si risolvano le quistioni del 10 tipo. Partiamo dallo spazio ordinario. All' intorno del 10 ordine di un punto di una superficie Si sostituisce il piano tangente: le proprietih proiettivo-differenziali del 20 ordine risultano dal cosderare le intersezioni di esso con i piani infinitamente vicini. Per porre la questione analoga nel modo pii\ generale ci conviene introdurre una nuova locuzione. Consideriamo sopra una varieta' Vm,, le curve passanti per un suo punto ed aventi ivi uno stesso S~, osculatore t; gli Sh (h > v) osculatori a queste curve nel punto comune, se non riempiono 1' ambiente, appartengono ad un determinato spazio che dir6 li-osculatore alla varieta' secondo lo S, fissato, o di indici h, v. Se lo S, e, il punto stesso (v = 0) lo spazio definito (ove esista) si dira' semplicemente h-osculatore o, se si vuo le, per avere una locuzione pii\ simile all' ordinaria e a quella adotat da De Pezoh-tangente alla Vr. nel punto+. Cosi' definiti questi spazi rimane a studiare la legge di distribuzione degli spazi osculatori in essi contenuti. Fissato uno spazio Sh-1 osculatore (v = h - 1) gli Sh osculatori ivi a curve della varieth descrivono intorno ad esso un sistema lineare che riempie lo spazio h-osculatore alla Vm, secondo lo Sh, Ma le cose si presentano diversamente quando lo S,. osculatore fissato abbia dimensione minore di h - 1. Alcuni dei teoremi relativi sono stati studiati dal Prof. Segre e la ricerca pub essere agevolmente continuata servendosi degli stessi metodi da lui usati. * Di questi lavori citerb, come piU' importanti, i due seguenti: "Su una, classe di superficie degli iperspazi legate colle equazioni lineari alle derivate parziali di 20 ordine," Atti Ace. Sceneze di Torino, VOL. XLII, 1906-07. " 1Preliminari di una teoria delle varieth luoghi di spazi, " Rendic. Cir. Jifatem. Palernio, t. xxx, 1910. Gli altri si trovano negli Atti della B. Ace. delle Scienze di Torino, nel periodo di tempo ricordato. Tralascio nel seguito le indicazioni particolari relative alle memorie del Prof. Segre. t Intendo con ci6 di fissare anche gli spazi osculatori di dimensione inferiore, quando essi non vengano determinati dallo S,.; cfr. " ISopra hicune estensioni dei teoremi di Meusnier e di Eulero " (in corso di stampa negli A tti dell' Accad. d. S'cienze di Torino, 1913). + Cf. P. Del Pezzo " Sugli spazi tangenti ad una superficie o ad una varietfi immersa in uno spazio di PMi dimensioni," Rend. Ace. di scienze Fis. MJat. di Napoli, fasc. 80, 1886.

Page  24 24 ENRICO BOMPIANI Ad assicurarne 1' interesse basterah ricordare che, seguendo le idee di Pluecker e di Klein, in questa geometria pub' ritenersi inclusa la geometria proiettiva di un qualsiasi elemento generatore dello spazio, e la stessa geometria metrica di un ambiente qualsiasi. E basta infatti una semplice interpretazione di linguaggio per ritrovare i. teoremi ricordati di Koenigs sui sistemi di rette in S,,, alcuni recenti teoremi di C. L. E. Mooret sni sistemi di rette in 54 o di cerchi in 83, e infine un' estesissima generalizzazione dei teoremi di Meusnier e di Eulero per una qualsiasi varieth immersa in un ambiente di dirnensione arbitraria. La nozione di coniugio potra, estendersi in diverso modo (in relazione alla dimensione della varieth e dell' ambiente) imponendo speciali condizioni d' incidenza a spazi osculatori a curve uscenti dai punti successivi di una curva assegnata. Le direzioni die danno luogo all' incidenza voluta possono dirsi coniuigate alle tangenti alla curva data negli stessi punti, ma la corrispondenza non ha in generale carattere involutorio. Volendo generalizzare la nozione di asintotica, si cercherah se esistono sulla varieth curv, o ingenrale, varieth di dimensione inferiore, tali che alcuni loro spazi osculatori siano legati da speciali condizioni d' appartenenza con spazi di dimensione conveniente, osculatori alla varieth data. La determinazione di. queste curve sopra una varietah costituisce un problema del 20 tipo. Per portare un esempio, sopra una superficie generale di 84 si pub definire un sistema 00 2di curve tali che lo S., osculatore ad una di esse in un punto riesca ivi tangente alla superficie (mentre non esistono in generale sulla superficie asintotiche nel senso ordinario). Interpretando questo risultato nello spazio ordinario si ottengono teoremi noti sulle congruenze di rette e di sfere4. A queste proprietii di carattere generale ne andranno sostituite altre per quelle varieth tali che gli intorni di tutti i loro punti siano costruiti in modo particolare. Pub darsi infatti die alcuni degli spazi h-osculatori sopra definiti siano di dimensione minore di quella prevista dal caso generale. Q uesto fatto si esprime analiticamente ponendo dei legami lineari omogenei fra le coordinate proiettive dei punti die definiscono quegli spazi h-osculatori; cioe scrivendo die le coordinate proiettive omogenee di un punto della varieth e le loro derivate fino a quelle di un certo ordine sono fra loro legate linearmente; o, piiit in breve, che le coordinate di un punto della varieth sono soluzioni di un sistema di equzioi lledeivate parziali, lineari ed omogenee. Alle superficie che rappresentano un' equazione di Laplace ha dedicato un' importante memoria il Prof. Segre. Dall' esistenza di un doppio sistema coniugato di linee sulla superficie si deduce, come fa il Darboux per lo spazio ordiuario, la trasformazione di Laplace per 1' equazione ricordata. Servendosi di questa imagine geometrica riesce facile * Sur les propri~t~s infinite'simales de l'espace regl6,," Annales de 1'L~cole norinale, 1882 (2), t. 11. t Infinitesimal properties of lines in 84, with applications to circles in S3," Proceedings of the Amierican Academny of Arts and sciences, vol. XLVi, no 5, 1911. Cf. la nmia nota " Sopra una trasformazione classica di Sophus Lie,' Atti R1. Accad. dei Lincei, vol. xxi, serie 5, 1,912, f. 11.

Page  25 GEOMETRIA PROIETTIVA DIFFERENZIALE DEGLI IPERSPAZI 2 25 assegnare dei criteri perch' 1' equazione data sia integrabile col. metodo di Laplace *. Come giah accennava il Prof. Segre nel suo lavoro, le ricerche ivi contenute sono suscettibili di diverse estensioni. In luogo di una sola equazione del 20 ordine a 2 variabili si puo6 per es. considerare un sistema di piui ecjuazioni del 20 ordine a kc variabili indipendenti; ii Terracini, in una memoria sulle Vk die rappresentano piut di (Ic- 1) equazioni di Laplace linearmente indipendenti t, assegna loro una pro2 prieth (non caratteristica per6') che le riavvicina alle sviluppabili dello spazio ordinario. Ii caso Ic = 3 era giah stato ampiamente trattato in una memoria del Sig. C. H. Sisam$+ Un' altra estensione delle ricerche del Segre si ha invece considerando una o piu equazioni di ordine superiore al secondo. In particolare, 1' equazione a due variabili, di qualsiasi ordine, a caratteristica completa, ha un' imagine geometrica del tutto simile a quella deli' equazione di Laplace: si pub costruire per essa una successione di trasformate e assegnare condizioni sufficienti affinche' la sua integrazione sia ricondotta a quella di equazioni a derivate ordinarie. Come Si e' visto il principio del mietodo per lo studio di proprieth proiettivodifferenziali consiste nel sostituire all' intorno di un punto uno spazio lineare di dimensione conveniente. Si presenta quindi come particolarmente interessante, sotto un doppio punto di vista, lo studio delle variethi luoghi di spazil: sia per la costruzione della teoria generale, sia per vedere come si semplifichino per esse le proprieth osservate per una varieth~ qualsiasi. I fondamenti di una teoria delle varietht luoghi di spazi sono stati posti dal Prof. Segre nella Memoria citata del 1910. In essa sono trattate le questioni esseuziali relative agli spazi tangenti, agli spazil caratteristici, alla nozione di sviluppabilitah e alle sue diverse estensioni nonche' molte delle questioni generali prima accennate per una varieth qualsiasi. I metodi svolti in questa memoria sono serviti di modello nelle ricerche successive che gia' abbiamo avuto occasione di ricordare. Un indirizzo completamente differente per la geometria proiettivo-differenziale nello spazio ordinario veniva sviluppato dal Wilc zynski in una serie di Memorie la cui pubblicazione incomincia nel 1901: ne formano oggetto lo studio delle curve piane e sghembe, delle rigate, delle superficie curve, delle congruenze di. rette e dei sistemi di curve piane. Ii Wilczynski, mettendosi dal punto di vista gia' adottato dall' Halphen nella sua Tesi e nei lavori successivi (1875-1881) per la rappresentazione di curve piane e sghembe, si e' servito di un sistema di equazioni differeuziali lineari opportunamente scelte per rappresentare gli enti ricordati. Dalle proprietht del sistema scelto risulta che ii corrisponidenite ente geometrico e' definito a meno di una trasformazione * Cf. la mia memoria sull' equazione di Laplace (Rendiconti del Circ. Mat. di Palermo, t. xxxiv, fasc. iii, 1912). ~Circolo Matematico di Palermo, t. xxxiii, 1912. "On Three Spreads satisfying four or more homogeneous linear partial differential equations of the second order," American Journal of Mathematics, vol. xxxii, no 2, 1911.

Page  26 26 26 ~~~~~~EN111CO BOMPIANI proiettiva: gli invarianti del sistema sono dunque atti a definire le proprietiL proiettive dell' ente in esame. Ed ii Wilczynski e' riuscito infatti a dare forma invariantiva alla teoria delle rigate, delle superficie, e delle congruenze di rette, arricehendola in molti punti di nuovi risultati*. Nulla di piii naturale che tentarne 1' estensione agli iperspazi. E 1' estensione e' certo ricca d' interesse, perche& se da un lato la rappresentazione analitica fornisce un nuovo mezzo per lo studio degli enti geometrici, dali' altro ii modello geometrico sussidia deli' intuizione ad esso propria la ricerca analitica. Ne valga ad esempio la memoria " Sugli invarianti differenziali proiettivi delle curve in un iperspazio "t in cui ii Prof. Berzolari ha esteso i resultati deli' Haiphen. Consideriamo le superficie rigate di uno spazio qualsiasiSn Servendosi della nozione generale di spazio h-osculatore ad una varietiL in un pinto, si riesce a stabilire per queste superficie una teoria delle curve tracciate su di esse e definite da proprieth proiettive, del tutto analoghe alle asintotiche sulle rigate di S,. Insieme a questo sistema di curve, che potremmo chiamare quasi-asintotiche, vien definito uin altro sisterna di curve associate ad esse, die in generale non apparteugono alla rigata: se vi appartengono coincidono con le curve prima definite. Qunesto accade per es. nello spazio ordinario. Se la rigata e' algebrica rientrano in questa classe generale di curve quasi-asintotiche le direttrici mninime giah studiate dal Prof. Segre$++ Dali' esistenza di queste curve si pu6' dedurre una classificazione pr-oiettiva delle rigate iin un iperspazio; la natura proiettiva dell' intorno di una generatrice generica puO caratterizzarsi mediante alcuni numeri interi che dir6 indici di sviluppabilita& dei quali il piii grande rappresenta il massimo numero di generatrici consecutive linearmente indipendenti. Una rigata di S,, si. diraL del tipo generale quando i suoi indici di sviluppabilith' hanno i massimi valori possibili. Una rigata qualsiasi si pub rappresentare per mezzo di un sistema di 2 equazioni differenziali liineari ed omogenee e ad esse pub riportarsi la nozione degli indici di sviluppabilita'. Quneste considerazioni possono estendersi in direzioni diverse. Possono considerarsi sistemi semplicernente infiniti di spazi' lineari Sk1; una loro rappresentazione si otterrah servendosi di un sistema di k equazioni differenziali lineari omogenee. Oppure potranno considerarsi sistemi piii volte infiniti di rette o di spazi qualsiasi rappresentati da sistemi di equazioni alle derivate parziali. Se, per limitarci a quanto ' e detto sopra, abbiamo in v ista un sistema pih volte infinito di rette, il problema analogo a quello della ricerca delle sviluppabili in una congruenza di 53 consisteraL nel ricercare entro il sistema le rigate aventi i minimi indici di sviluppabilitah * Per i lavori del Wilczynski anteriori al 1906 (che si riferiscono in modo particolare ai sistemi di rette) si puo vedere ii suo libro Projective differential geometry of curves and ruled surfaces (Teubner, 1906). Gli altri si trovano in Transactions of the American Mathensatical Society nel periodo indicato. + Annali di Mceatematica, s. ii, t. xxvi (1897). + "Sulle rigate razionali in uno spazio lineare qualunque " (Atti dell' Accad. d. Scienze di Torino, vol. xix, 1884), ed. altre note stillo stesso argomento (ibid., 1885 e 1886).

Page  27 GEOMETRIA PROIETTIVA DIFFERENZJALE DEGLI IPERSPAZI 27 possibili. Queste sono sviluppabili nel senso ordinario solo, in generale, per i sistemi '~ di Sn. Non e' forse illecito sperare cie le considerazioni geometriche di Fano e di Enriques sulle varieth algebriche che ammettono infinite trasformazioni proiettive *, alle quali si debbono progressi notevoli nella teoria della equazione differenziale lineare le cui soluzioni sono legate da relazioni algebriche, porteranno anche ainto nel campo piii vasto al quale ho cercato ora d' accennaret *Cf. per es. 1' esposizione del Fano n1ei Math. Ann. Bd 53 (1899), p. 493. tNell' intervallo di tempo trascorso fra la lettura di questa comunicazione e la stampa degli Atti (Febbraio 1913) sono apparsi i seguenti lavori sull' argomento che c' interessa: A. Ranum: "On the Projective Differential Geometry of N-dimensional Spreads generated by oo Flats" (Ann. di Matem. pura ed applicata, s. iii, t. xix, p. 205). E. Cairo: " Sopra un sistema 2; di supe~ficie P di S."j (Periodico di Matematica, anno xxvii, fase. vi; anno xxviii, fasc. iii). E. Artom: " Ricerche proiettive sulle linee tracciate in una superficie immersa in uno spazio a pih4 dinmensioni " (Periodico di Matematica, anno XXVIII, fasc. ii).

Page  28 THE GENERAL THEORY OF MOVING AXES BY ERIC H. NEVILLE. For more than fifty years mathematicians both pure and applied have recognized in a moving frame of mutually perpendicular axes one of the most effective weapons in their arsenal. Geometers use it, Darboux's great work being the most astounding example of its power; in attacking particular problems in rigid dynamics it is indispensable; and if the applied mathematician when he concerns himself with curvilinear coordinates all but confines his attention to such systems as are triply orthogonal, that is chiefly, I think, because the axis-frame which corresponds to such a system is of the kind to which he is accustomed. It has been my good fortune to succeed in obtaining a general theory of moving axes, where the frame is subject to no conditions whatever of orthogonality or rigidity, to mount, as it were, the old weapon in such a way that it can be trained immediately on many objects hitherto not within its range, and brought to bear directly on many points which could be attacked in the past only after a more or less tedious detour. In the time now available it might be possible to give the steps of the general theory, but I could then mention not even one of the numerous applications. But if I confine myself to a mere statement of results, I may succeed in shewing you something of the power of the method and of the variety of uses to which I believe we shall some day see it put. In dealing with oblique axes, it is of the greatest importance always to be ready to use both the direction-cosines and the direction-ratios of any line, and in the case of any vector to deal not only with its three components but also with its three projections, the projection of a vector along an axis being of course the numerical magnitude of the vector multiplied by the cosine of the angle between the vector and the axis. We shall use throughout this paper X, p, v for the angles between the axes of reference, A, B, C for the angles between the planes of reference, and T for the sine of the solid angle of the frame, the magnitude, that is, which may be defined indifferently as sin / sin v sin A, as sin v sin X sin B, or as sin X sin p sin C; also we shall denote the components of a typical vector by x, y, z, and its projections by p, q, r, and when the vector changes continuously and there exists another vector which is the rate of change of the first, we shall denote the components of the derived vector by Dx/Dt, Dy/Dt, Dz/Dt, and its projections by Dp/Dt, Dq/Dt, Dr/Dt. The simplest change possible to an oblique frame of reference occurs when two of the axes remain fixed in direction and the third rotates round one of these two, tracing out part of a right circular cone with that fixed line as axis. When the angle

Page  29 THE GENERAL THEORY OF MOVING AXES 29 of rotation is infinitesimal we may call such a motion an elementary spin, and we realise at once that there are six such spins possible to the frame, that the frame may be brought from its initial position to any adjacent position by one and only one set of six elementary spins, and that the measures of these spins will not depend, when first order magnitudes only are considered, on the order in which the spins take place. The first spin, that of the y-axis round the x-axis, is measured by the angle through which the zy-plane has revolved in the positive direction (that is, for this spin, towards the xz-plane), and is denoted by 02, the other five spins similarly measured being denoted by 03, 03, 1b, *1, #2; and it is a matter of no great difficulty to ascertain the effect of an elementary spin on the angles of the frame and on the components and projections of a fixed vector. When the frame changes continuously we have, instead of the six spins described above, six rates of spin which we denote by 02', 03', 03 01, #1', 2 2/; and from the results relating to elementary spins-results which are of course only approximate-we can write down at once the following accurate and fundamental expressions, (cos X) T (02-, D, (lx sin X t d - ~/) - + (zo/ - X-1,') cot C + (xO1' - yOe') cot B, D~t - t 2 T.y lDI, dp T (y - lDt d t) each typical of a set of three. It is the last of these which shews how vital it is to make use of the projections as well as of the components of our vectors, and which also leads us to expect that the increase of difficulty of manipulation as we pass from orthogonal frames to oblique ones will prove not in the slightest degree comparable with the increase of freedom; but even in the expressions for the components of the rate of change we find no unfamiliar coefficients, but merely those which occur in the equations connecting components with projections, for we have always three equations typified by sin2 X Tx = p --- - q cot C- r cot B. The general theory requires for its completion the study of the features which arise when the motion of the frame depends on more variables than one. The investigation, following the lines of work by Kirchhoff and Combescure on moving frames of the simplest kind, presents no difficulty, but, anxious to say something of the applications of the abstract theory, I cannot spare time even to state so much of the notation as would enable me to write down typical results in this part of the subject. The most evident, and it may be the most important, application of the theory now outlined is to the theory of curvilinear coordinates in space. Taking for coordinates u, v, w, let us write the square of the line-element in the form ds2 = (L, M, N, P, Q, R3du, dv, dw)2, and let us also put L=U2, M=V2, N=W2

Page  30 30 ERIC H. NEVILLE to prevent the appearance of radicals in what follows, and again L = 11, P = 23= S32, and so on, the double-affix notation enabling us often to include a number of investigations in one piece of work, and enabling us also to write for brevity the expression for the square of the line-element in the form ds 2=(Sjdu, dv, dw)2. Then at each point of space there is associated with the coordinate system a reference frame, one axis being the tangent to the curve along which v and w are constant and having its positive direction so chosen that for a small movement in the positive direction ds = Udu, and the other axes being similarly determined. If we suppose any curve drawn in space, and the origin of a moving frame to describe this curve while at every point the axes are those of the reference frame at that point, then the six rates of spin of the moving frame are linear functions of u', v', w', the rates of change of the coordinates. It transpires that all the coefficients in the expressions for these rates of spin can be found in terms of L, P, and so on, and their first derivatives, by means of equations which occur in the work on frames whose motion depends on several variables, and we have only to substitute in the expressions for the components and the projections of the rate of change of a given vector the values so obtained for the rates of spin, to obtain the values of such magnitudes as Dx dx Dp dp. D d Dt d- and -D d-t The expression so found for Dt dt is necessarily lineoR Dt Dtdt Dt dt s linear in the two sets of variables x, y, z and u', v', w', and we might expect to have to deal with its nine coefficients, and so in all with twenty-seven combinations of the first derivatives of the coefficients such as L and P; but inspection reveals that Dx dx though there is no symmetry in the expression for Dt d-' the expression for 1 Dt d (x U) is symmetrical in the two sets of variables x/U, y/V, z/W and U JLb dt u', v', w, and contains therefore only six coefficients. We write All fbr the coefficient of a', and A'3 or A32, since it proves convenient to retain both forms, for the coefficient of Y w' + v', and so on, and we may write compactly I Dx d (xlU) ') UDt dt + AyU)VI jjUI VI)) Similarly six magnitudes Br' are introduced as coefficients in the expression for Dt, and six magnitudes Cr' as coefficients in the expression for W Dt and so we have in all eighteen functions of the eighteen first derivatives of the original coefficients S". These eighteen functions are in fact independent linear functions of the derivatives, and the expression of the derivatives in terms of them proves simple enough, for if we denote differentiation with respect to the variables u, v, w by suffixes 1, 2, 3, we find the single formula Sirs = SrlAst + SI2B.st + Sr3Cst + Ss Art + Sl2Brt + SS3C,t for r, s, t = 1, 2, 3, to cover every case. Already we have in outline all that is essential to enable us to write out in terms of curvilinear coordinates of any kind the majority of the equations of applied

Page  31 THE GENERAL THEORY OF MOVING AXES 31 mathematics, and to deal with many problems relating to curves and surfaces; but a word must be said of the expression of rates of rates of change, not only because rates of rates of change are often needed, but also because the analysis itself is related to results of the greatest importance. When we come to discuss rates of rates of change, we find three equations of the form 1 TDhx d(xU) ( A1d (x/U U) d(y/V) d(zlW)Y, v,' U Dt 2 dt' dt dt dt + ( VU' jt' v" ~ +, y j ',, )' each containing only sixteen coefficients although we might have feared to find forty-five. Of the sixteen coefficients six are already familiar, and thus the three equations together introduce us to thirty functions of the second-order derivatives of L, P, and so on, functions which we denote by Xt, Y'.t, Zst, r, s, t having the values 1, 2, 3, and the affixes being permutable. In terms of these thirty functions all thirty-six of the second-order derivatives are expressible, and so we come across the six relations which Cayley discovered must exist between these derivatives if the quadratic form (Sdtt, dv, dw)' is to represent the square of the line-element in Euclidean space. These relations come to us however in a form far more compact than that which Cayley gives, for the two typical equations as we find them are M33- 2P23 + N2, = 2 (S$A23, B23, C023) -2 (SRAA2, B22, C22A33, B33, 33) and -L23-P~+Qi2+~R3=2(S.AI, B11, CIIA23, B23, 0C2)-2(SVA12, B12, C12313 B13, C13). In order to illustrate both the facility with which curvilinear coordinates can be manipulated and the value of the kinematical point of view in differential geometry, let me mention a result which has not, as far as I know, hitherto been given even in terms of Cartesian coordinates. Associated with a family of surfaces in a region of space there is at every point a frame which we may call the fundamental frame, formed by the tangents to the two lines of curvature of the member of the family passing through that point and the normal to this surface. This frame is completely rectangular and consequently its rotation as we pass along an orthogonal trajectory of the family requires only three rates of spin, i, j, and k, for its description. Of these, i and j, the angular velocities about the tangents to the lines of curvature, though simple enough in form, shall not detain us now; but to k, the angular velocity of the fundamental frame about the normal as we pass from surface to surface along the orthogonal trajectory, I wish to draw your attention. For k is a magnitude of the greatest importance in the study of the family, being that which will vanish when and only when the family is a Lame family. Hence having found the analytical expression for k, namely G3 G,-GAll-G2B"1-G3C11, 1 -,A-i A11- 1-)2B-13, 20>1, 0, 0, L 2J3IJI GG-GA~-G2B -G3C~,. -A22-I)A _-r2B '-r3C22, 0, 2>2, 0, M G33-GIA -G2B -GC33, 33-()lA33 —2BB3- C33 O, O 0, 2,)3 N | G=-G~A3-GB3-G~, ~-(I3A 323-~ B23-q) C23 0, ) q, P G3-GIA'-G2B -G3C3I, I3 -(iA31)-A 31-)Bi- C31,, ), (), Q G3-G 3A1-G,B-1- G3 C1, Dp1_- eDA32- q1,B2 — 31 C_12, 31~, q)3, 0, R

Page  32 32 ERIC H. NEVILLE where ( (au, v, w)= constant is the equation of the family, I is the difference between the principal curvatures of the surface, suffixes denote differentiation, Jdudvdw is the element of volume bounded by six reference surfaces, and 1/G is dJP/dn, the rate of change of the defining function of the family as we pass along the orthogonal trajectory, so that analytically J2- L, R, Q, J L, R, Q, (, R M, P G2, M, P, ) Q, P, N Q, P, N, )3 (q), (,3, e, 0 while the magnitudes Arl, B"8, C"r are those we have met earlier in this paper, we canl write down Darboux's famous equation, not merely expressed in terms of coordinates the most general, but also with its meaning, its necessity, and its sufficiency, made evident. Scant as is the best justice I can do to my subject here, I can at least mention one other and most valuable development, that of its application to the theory of a single surface. In dealing with a single surface, striated by two families of curves of reference, the frame which we have to use has two of its axes tangents to curves of reference, and the third, the normal to the surface, is at right angles to the other two. The theory of such a frame is of course only a special case of the general theory, but these frames, which may be called birectals, are of so frequent occurrence that the results relating to them are of particular interest; and indeed it was the desire to have these frames available for use in differential geometry that prompted the whole research of which I am speaking, the value of the quite general theory becoming evident only as the work proceeded. In a moving birectal the constancy of the right angles involves that 02 and 03 are equal, and also that 43 and )01 are equal, and we therefore drop the suffixes from these symbols and speak of only four spins, 0, ), #1, and,.2 The simplicity of the whole geometrical theory will be best appreciated from the mere statement that for a curve on the surface, dividing the angle between the reference curves into angles a, /3, if the rates of spin of the birectal already described, as the origin moves along the curve, are 0', 4', 1',, then the normal curvature C,,, the geodesic torsion so, and the geodesic curvature Cg, are* given by the expressions, = 0' sin a - 4' sin 3, s = 0' cos a + ' cos /3, Kg = + 1, =- + #2, expressions which simplify not at all when the curves of reference happen to be orthogonal. But this is not all. The analysis connected with coordinates on a surface can be deduced from the theory of the birectal just as the analysis connected with * In Darboux's notation these three magnitudes are respectively cos w 1 dw sin s p T ds p

Page  33 THE GENERAL THEORY OF MOVING AXES 33. coordinates in space is obtained from the general theory, and if we write the square of the line-element in the two forms ds2 = Edua + 2Fdudv + Gdv2 = U2d't2 + 2UVcos w datdv + V2dv, we are led to introduce six combinations of the first derivatives of E, F, G, and also to retain three other magnitudes describing the motion of the frame. These nine magnitudes are identical with certain of those which Forsyth has found it necessary to use in his purely algebraic work, and, adopting a notation differing as little as possible from his, we denote the last three by L, M, N, and of the first six, three by Frs and three by Ais, r, s having the values 1, 2 and being permutable. Then the fundamental formula) are 1 Dx _ d(x/U) 'x Y, IT - U Dt -t ( + F - U,) -z (L, M, 1VG G -F ' EG-F ', v 1 Dy d(y/V) x -, E - A' V F -t dt 4 u, TX-~^EGF ---F EG - I'2 Dz dz L, )~ x, / Dt =d + LM, N,-' -u' ') from the last of which it is soon seen that the magnitudes L, M, N are the coefficients in the quadratic form representing the normal curvature of the surface, that is, are multiples by 1/UVsinw of those magnitudes introduced by Gauss and denoted in Darboux's treatise by D, D', D". I cannot hope to give you even a faint indication of the range of application of these last formula. Let it suffice to say that from them the equation of Gauss and the equations of Codazzi and Mainardi follow immediately, that they enable the actual analytical expressions to be given for two functions of direction discovered by Laguerre and Darboux respectively, that the determination froila them of the invariants connected with a family of curves on the surface is immediate, and that they have led to a geometrical interpretation of the equations of Codazzi and Mainardi which is to the best of my belief new, and which I should like to give here. Let D = constant represent any family of curves on a surface, let df/ds, df/dg denote the rates of change of any scalar function f along the curve of the family and along the orthogonal trajectory respectively, and let the normal curvature, the geodesic torsion, and the geodesic curvature, of the curve of the family be Kc, Sg, Kg and the mean curvature of the surface itself be H. Then the two functions d (KIt, -1 H) d ds -l 2%%,g2,, ds + 2 (K, - )H) K, ds ds are virtually those which Laguerre and Darboux respectively discovered to be functions only of direction on the surface; and the two functions d(K n-AH) + d {log (dI/dg)} d (K, H) d {log (d(P/dg)} Idg + ds dg ' - ds in which the logarithmic derivative will be recognized to be the geodesic curvature of the orthogonal trajectory of the family, are analogous functions of direction which make their appearance as soon as we come to deal with a family of curves. As far M. C. II. 3

Page  34 t34 ERIC H. NEVILLE as I know, these four functions, which we denote for the moment by Ls, Ds, Lg, and Dy, might well be supposed, until the actual expressions for them are forthcoming, to be distinct; but on evaluating them we find them to be connected by the two simple relations 1 dH I dH Ls~D W, D - dg' L +r D)- 2 ds ' L)-L = 2 dg ' which constitute the required geometrical interpretation. I should have liked to say something of the relation of the analysis deduced from the theory of moving axes to the process of covariant differentiation, or to write down in terms of curvilinear coordinates some of the most important equations of mathematical physics, but my real desire was to arouse your interest in a subject which I believe to be worthy of some attention, and if in spite of the limits within which I have had to confine myself I have succeeded in that, I am more than satisfied.

Page  35 UEBER DIE TEJLTJNG DES IRAUMES, DURCII 6 EBENEN UND DIE SECHSELACHE VON M. BRUCKNER. In dem auf dem Kongresse in Romi gehaltenen Vortrage hatte ich die Methoden der Ableitung der aussergewdhnlichen Polyeder angegeben und sie durch die Bestimmung der Sechsflache erlautert, wobei ich am Schiusse auf eine spa~tere ausftthrliche Darstellung hinwies. Bei der Ausftihrung des damals skizzierten Programms zeigte sich, dass die Untersuchungen einen vorher nicht vermuteten Umfang annahrnen, so dass es mehrjdihriger Arbeit bedurfte, das hier vorliegende Manuskript fertigzustellen. Ueberdies nalimen die Ergebnisse htichst merkwtirdige, zum Teil von der frtiheren abweichende Gestalt an. Gestatten Sie mir, m.ll., von den Endresultaten Ihnen hente einen kurzen Ueberblick zu geben, wobei ich im Interesse der Kiarheit auf die einleitenden Betraclitungen niiher eingehen mtichte. Zu der gewttnschten Ableitung der aussergew6hnlichen Polyeder im Mbbiusschen Sinne waren zwei Wege vorgezeichnet. Entweder man benutzt die erweiterten Fundamentalkonstruktionen Eberhards, das Abschneiden von Ecken, Kanten und Kantenpaaren, um die n, + 1-Flache aus den als bekannt vorausgesetzten it-Flachen zu erhalten, oder man setzt die Polyeder direkt aus den sie bildenden Zellen zusammen, in die der Raum durch die betreffende Zahi von Ebenen geteilt wird. Die Anwendung beider Methoden zur Auffindung der Fiinfflache ist leiclit, aber an sich von geringem Jnteresse, da einseitige Polyeder und zweiseitige von h~iherem Zusammenhange nocli nicht existieren. Doch bildet emn eingehendes Studinin der Raumfigur aus 5 Ebenen die Voraussetzung ffur die Mtbglichkeit, die kom-pliziertein Figuren aus 6 Ebenen tibersehen zu kdnnen. Es telilen bekanntlich n Ebenen in aligemeinster Lage den proj ektiven Raum in it + 1) -2)Zellen, von denen 1. 2.83 (nm-1) (n -2) (n- 3) 1. 23 - -gesehiossen sind, die iibrigen, transgredienten aus 2 Teilen bestehen, die im Unendlichen zusammenhaingen. Fur = 5 ergibt sich nur eine einzige Figur mit 4 endlichen und 11 transgredienten Zellen, wie auch die 5 Ebenen nu Raume gelegen sein mbgen. Die Linearfiguren, die in jeder Ebene durch die iibrigen erzeugt werden, sind volistandige Vierseite des einen nur vorhandenen Typus und alle 5 mdglichen transgredienten konvexen Fiinifiache sind in diesem einen Gebilde vorhanden. Es ist vorteilhaft, emn ffir allemal das volistandige Fiinfflach in einer bestimmten Form in der Vorstellung festzuhalten. Bezeichnet man das endliche Tetraeder mit [F das Pentaeder mit P und die transgredienten Tetraeder

Page  36 36 36 M~~~~~~~A. BRU~CKNER und Pentaeder mit Ti~ und Pi~, wo die Indices i und j die Zahien der Ecken auf den beiden im Unendlichen zusammienhdngenden Teilen anzeigen, so besteht der durch 5 Ebenen geteilte Raum aus 2T, 2P, 2T,+,, 1T2~,, 2P,+,, 2P4~2, 2P/4+2, lP3+3 und 1P'3+,3, wobei die Striehelung anzeigt, dass die betreffenden Zellen mit den vorhergehenden allomorph sind. Ftigt man nun eine sechste Ebene hinzu, so entstehen laut obiger Formel 10 gesehiossene und 16 transgrediente Zellen, unter denen sich nun auch die beiden von einander versehiedenen Sechsflaehe H' und H2 befinden, von denen das erste als Grenzflachen 2 FUnfecke, 2 Viereeke und 2 Dreiecke, das zweite aber 6 Viereeke hat. Da die analytisehe Behandlung des Problems zeigt, dass von den 26 Zellen des durch 6 Ebenen geteilten Raumles stets 6 Tetraeder, 12 Pentaeder, 6 Hexaeder H' und 2 Hexaeder H2 sein miissen, es aber, wie die Untersuehung zeigt, 20 versehiedene transgrediente konvexe Seclisflache H'j+j nnd 6 dergleichen H 2j~j gibt, so kdnnen nielit samtliehe transgrediente Seehsflache ill einer vorliegenden Sechsteilung des Raurnes zugleich auftreten. Es sind die verschiedenen Teilungen des Raumes dureli 6 Ebenen demnach durch den Charakter der transgredienten Zellen ebensowohi wie durch die Art und Anordnung der endlichen Zellen zu unlterscheiden. In j eder der 6 Ebenen besteht nun die durch die tibrigen funf erzeugte Linearfigur aus einem Fiinfeck, 5 Vierecken nnd 5 Drelecken, von denen 6 gesehiossene Poly-gone, die iibrigen transgredient sind, und es existieren 6 von einander unterschiedene Teilungen der Ebene durch 5 Gerade, wenn man beachtet, wie sich die transgredienten und geschiossenen Zellen unter die verschiedenen Polygone verteilen. Die Einftthrung einer sechsten- Ebene in das vollstandige Fiinfflach ergibt demnach in jener eine der 6 Linearfiguren, deren einziges Etinfeck nur als Sehnitt eines geschiossenen oder transgredienten Pentaeders P bezw. Pij~ entstanden sein kann. Um demnach alle von einander verschiedenen Sechsteilungen des Raumles zu erhalten, sind sdmtliche mdglichen Fiinfeekschnitte der Pentaeder des volistanldigen Ftinifiaches. auszufilhren. Dies zeigte sich auf 152 Weisen in6glich, aber die erhaltenen Sechsteilungen des Raumies sind zurni grossen Teile. isom-orph und es ist eine eingehende Untersuchung ndtig, urn die Auzahi der von einander versehiedenen Typen festzustellen. Das Resultat ist: Es existieren 42 verschiedene Teilungen des Raumles dureh 6 Ebenen, die nach der Art der endlichen Zellen zunaehst in 9 Ordnungen zerfallen, von denen jede mehrere Gruppen enthalt, die durch die Besehaffenheit und Verteilung der transgredienten Zellen untersehieden sind, wie die folgende Uebersicht zeigt: Die endlichen Zellen Zahi der Gruppen 3HI, 41P 3 T2 3//i, 3P), 4T 4 3H/i, 2P5 5 T 3 2H', 5P, 3 T 4 211' 4P, I1T 6 2H1'. 3P, 5 T2 1//1, 1H2, 5P, 3T 1 1H', 1H', 4P, 4T 3 I H2, 6P, 3 T 1 3

Page  37 UEBER DIE TEILUNG DES RAUMES 3 37 Versehiedene Kontrollen lassen, abgesehen von der Eindeutigkeit der Konstruktionsmethode die Behauptung berechtigt erscheinen, dass keine Figur tibersehen ist. Es lassen sich diese 42 Seebsfiache noch in verschiedener anderer Weise gruppieren, sei es, dass man ihre Schnittfiguren in der unendlicliweiten Ebene untersucht, wonach sie in 4 Kiassen zu bringen sind, da 6 Gerade in einer Ebene vier von einander dnrch die Anzahl der i-eckigen Zellen untersehiedene Teilungen ergeben, sei es, dass man die volistandigen Siebenflache betrachtet, die durch Hinznfiigung der unendlicliweiten Ebene zu den sechs im Endlichen liegenden entstehen, wonach die Sechsflache in 12 Kiassen zerfallen wllrden. Der zweite Hauptteil meiner Untersuchungen begreift die Auffindung der endlichen d. h. geschlossenen aussergewt~hnlichen Sechsflache. Wie die Betraclitung der von 5 Geraden gebildeten Linearfigur zeigt, kdnnen Achtecke als Grenzfldchen von Seclisfiachen nicht auftreten, wohi aber Sechsecke, bei denen 2 Kanten in einer Schnittgeraden uend Siebenecke, bei denen 2 Paare Kanten in 2 Schnittgeraden zu liegen kommen. Demnach sind ftir die zweiseitigen aligerneinen Sechsflache vorn Gesehieclite p = 0 und p = 1 die Anzahlen f ar die Flachen f', f~, f5, fi, f, die Uzisungen der Gleichungen:,3f3+2f4+f.5= 12......(1) bezw. 3f3~ 2f4 +.f=f......(2). Ftir die Sechsflache vom. Geschlecht p =0 verweise ich auf die in den Akten des Romkongresses S. 294 gemachten Angaben. Die daselbst fUr die Ableitung dieser Gebilde erkhirten Fundamentalkonstruktionen sind ffir die Secbsflache vom Geschlecht p = 1 durch die gleichfalls dort scion erlauterte Methode der Kombination der Zellen des volistandigen Sechsflaches za ersetzen. Nattirlich ist diese zweite Methode die aligeineinere, die sdrntliche emn- nnd zweiseitigen Sechsflache abzuleiten gestattet. Man hat nur die 10 geschlossenen Zellen der durch 6 Ebenen gebildeten Raumteilung zu 2, 3, 4,...,1 9, 10 in jeder zulassigen Weise zusammenzft~igen, was mit Beriicksichtigung der Zahi 42 der volistandigen Figuren emn zwar mtihsames, aber eine bestimmte endliche.Anzahl von Sechsflachen ergebendes Verfaliren ist, das die Geduld insofern auf eine harte Probe steilt, als sich nattirlich 'jedes Sechsflach wiederholt vorfindet. Ich habe mich schliesslich, wegen der ungeheuren Anzahl existierender Sechsflache, begniigt, von alien Ltisungen der obigen Gleichungen typische Vertreter zu konstruieren. Ftir p = 1 existieren die folgenden Ltisungen der Gleichung (2), wobei die Zahl der Kanten stets 18, die der Ecken 12 ist: (a.) f7= 4, f4= 2, (6); (i3) f7= 2, f6= 2, f,= 2, (7); (,y) f6= 6, (2)* Diese Polyeder fuir p = 1 haben zum Teil die Gestalt eines Ringesjt. Einseitige Sechsflache treten ftir die Zusammenhangszahl a- = 1 und a- = 2 auf (vergi. hierzu Dehn, Math. Encyldl. iiui, S. 202) und es sind die Zahlen ffur die i-kantigen Grenzflachen an die Gleichungen gebunden: 31f3+ 2f4-i-f6+; fUr a-= 1.(e =10,kl,= 15).......(3) bezw. ~ f+f+5f; ftir o-=2.(e=12, k=18).......(4). *Die eingeklammerten Zahien geben hier wvie im folgenden die Anzahl der kolnstruierten und im Manuskript beschriebenen Sechsflache. t Es wurde die Art der Ableitung im Vortrage hier an der Figur eiues volistdndigen Sechsflaches erliiutert; einie B~eihe Modelle wurde vorgelegt und ein zweiseitiges vom Geschlecht p =1 ausfiihrlich besprochen.

Page  38 38 M. BROCKNER Die erste Gleichung besitzt die konstruierbaren Ldsungen: (ca) f-7= 3, /3=3'0, (7); (/3) f-I= 1, f = 2, f = 2, f3= 1, (29) (,y) f = 2, f, -2, f4= 2, (31); (8) f = 6, (9). Da die Gleichung (4) mit der frtther unter (2) fuir die zweiseitigen Polyeder fiur p = 1 aufgeftihrten iibereinstimrn-t, haben die einseitig'en Sechsflache ftir a- =2 die gleichen Grenzfhichen wie oben angegeben und es bedurfte in jedem Falle rioch hier wie dort der Untersuchung, ob das Mtibiussche Kantengesetz gtiltig war oder nicht, urn die betr. Seehsflache richtig einuzuordnen. Es sind 33 soiche einseitige Sechsflache ftir a- = 2 aufgefuinden worden*. Die Zahi aller besebriebenen zwei- unid einseitigen ailgemeinen Sechsflache betragt 228 bezw. 109. Auf die aus den ailgemeinen sich ergebenden singuldren volistaindigen Seehsflache, so wie auf die in ihnen enthaltenen singularen gesehiossenen Polyeder, mdchte ich heute nicht eingehen, doch ist auch hier die Untersuchung abgeschlossenf und ich hoffe bald die ansfitirliche Darstellung an anderer Stelle zu verdifentlichen. * Ein solches wurde im Modell im Ansehiusse an die Zeichnung der vollstiindigen Figur erl.Rutert. t Es wurden 286 zweiseitige und 98 einseitige singuliire Sechsfla~che beschrieben, wvomit freilich deren Anzahi. sicher noch nicht ersch6pft ist.

Page  39 SURI LEQUIYALENT ANALYTIQUE DU PROBLEME DES PIHINCIPES DE LA GEOMETRIE METRIQUE PAR C. ST~]~PHANOS. L'ensemble des formules de la geometrie analytique Enclidienne ou. non Euclidienne dans un espace 'a m dimensions est comple'tement determine' lorscju'on donne 1'expression de la longueur ds de 1ele'6ment line'aire joignant deux points infiniment voisins (xi, x2, *.. xm,) et (xi + dx,,... xm, + dxrn,) de cet espace. II en est ainsi non seulement lorsqu'on fait usage de coordonne'es projectives (caracterise'es par ce fait que les plans sont represente's par des equations line'aires), mais aussi quand on emploie des coordonne'es (curvilignes) quelconques. L'expression de ds ds =f (xi, x2,... xm, dxi,... dxm,) =f(x, dx) en coordonne's quelconques, e'tant la transforme'e de l'expression de lFd~ment line'aire Euclidien on non Euclidien en coordonne'es projectives, doit satisfaire 'a un certain nombre de conditions ne'cessaires et sauffisantes pour qn'il puisse par un changement convenable de variables prendre la forme de 1ele'6ment line'aire (Euclidien on. non Euclidien) en coordonne'es proqjectives. Le carr' de f(x, dx) doit ainsi e're une forme entie're et homoge'ne dn second degre6 par rapport aux diffi~rentielles dxj, dX2,... dxm., sonmise 'aun certain syste'me d'equations diff6rentielles aux de'rive'es partielles. Ces conditions ne'cessaires et suffisantes auxquelles doit satisfaire l'expression de la longucur de 1ele'Mment line'aire en coordonne'es quelconques peuvent e'tre conside'rees comme e'~quivalent analytique de l'ensemble des principes propres 'a la geometrie Euclidienne on non Euclidienne consid6re'e comime examinant les proprie'tes des figures de l'espace qni restent invariables par nn certain gronpe de transformations (de'placemients) de l'espace Euclidien on non Euclidien. La consideration de ces conditions ne'cessaires et suffisantes est indispensable lorsqu'on vent traiter en coordonne~es qnelconques des proble'mes de geometrie diff~rentielle, de ine'canique et de physique mathe'myatiqne. Les formunles anxqnelles on est conduit ainsi out la proprie'te de conserver leur caracte're d'invariance par rapport 'a lensemble de toutes les transformations ponctuelles de l'espace a fin dimensions. Un grand noinbre de questions inte'ressantes, pouvant conduire 'a des re'sultats bien remarquables, sont lie'es 'a cette maniere de conside'rer le proble'me des principes de la geometrie.

Page  40 ON RATIONAL RIGHT-ANGLED TRIANGLES BY ARTEMAS MARTIN. -" and are you such fools, To Square for this?"-Titus Andronicus, ii, 1. In a former paper the writer gave three methods of proof of the celebrated proposition that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the legs (sides including the right angle). Other proofs are presented here which may be of interest. IX. Let ABC be any right-angled triangle (Fig. 1), B the right angle, 0 the centre of the inscribed circle. Put OD, OE, OF = r radius of inscribed circle, AC=a, AB=b, BC —c. / Then E area of triangle ABC= -bc= r (a+ b +c)... (1). 1 ___ From the figure we have B D C Fig. 1. BF+ BD = b + c- = 2r...............(2), since AE= AF, DC= FC, and BD = BE = r. Hence frolm (2) we have = I (b + c - a).................................( ). Substituting this value in (1) and reducing we get a2 = 2 + c2....................................... (4). This proof was communicated in May, 1891, by Dr L. A. Bauer, then of the U.S. Coast and Geodetic Survey, now of the Carnegie Institution of Washington. Another method of the same proof. Let CD= CF = m, AE = AF =n, OD =OE=OF=r, in Fig. 1: then AC= a = m+ n, AB = b = + r, BC=c= + m r. Now (me + r) (n + r) = r (2m? + 2n + 2r).......................(1), since each expression is double the area of the triangle ABC. Therefore nn = mr +?r + r2, and 2m', = 2mr + 2 r' + 2r2 "..............................(2).

Page  41 ON RATIONAL RIGHT-ANGLED TRIANGLES 41 Adding rn2 + n2 to each member of (2), (n +, n)2 (n + r)2 + (( + r)2, or a2= b2 + c2. This method was communicated by Lucius Brown, Hudson, Mass. The following is not new, but is of interest because of its simplicity. Let ABC (Fig. 2) be any right-angled triangle (right-angled at B), DEAC the square on the hypotenuse A C. Produce BC to H, making CH=AB; produce BA to F, making AF=BC; draw EH F D G perpendicular to BH, and produce HE to G, making EG = CH; E draw FG through D; then FD=AB= CH=GE, and DG=EH=BC= =F. A - / Let AC = a, AB=b, BC =c; then B C H Fig. 2. BH =BF=FG=GH=b+c; oDEAC = c&, A ABC= ADFA = AGED = CHE= ~bc. oFGBH = oDEAC + 4 ABC; therefore (b + c)2 = a2 + 2bc, or b2+ 2bc + c2 = a2 + 2bc. Dropping 2bc from each side we have left b2 + c2 = a2. X. Every prime number of the form 4m + 1 is the hypotenuse of a prime rational right-angled triangle. The hypotenuse of a prime rational right-angled triangle is either a prime number of the form 4m + 1 or the product of two or more such primes. A number that is the product of n different prime numbers of the form 4m + 1 is the common hypotenuse of 2n-1 different prime rational right-angled triangles. Every number which contains a prime factor of the form 4m + 1 is the hypotenuse of a rational (but not prime) right-angled triangle. The hypotenuse of any rational right-angled triangle is n times a prime number of the form 4m + 1, or n times the product of two or more such primes, where n may be any whole number. The triangle will be prime when n is a prime of the form 4m + 1. A prime number of the form 4n+ 3 can not be the hypotenuse of a rational right-angled triangle. A number which does not contain a prime factor of the form 4m + 1 can not be the hypotenuse of a rational right-angled triangle. XI. In Miscellaneous Notes and Queries, edited by the late S. C. Gould, Manchester, N.H., Vol. IX, No. 6 (June, 1892), p. 141, in a paper on "Rational Right-Angled Triangles," the late Benjamin F. Burleson of Oneida Castle, N.Y., in his solution of Problem 4, "Write out all the rational right-angled triangles there

Page  42 42 ARTEMAS MARTIN are whose hypotenuses do not exceed 100," stated that there are 55 such triangles; but he inadvertently included in his second series of triangles three that are in the first series, so that three triangles (10, 24, 26), (14, 48, 50), (54, 72, 90), were counted twice. Deducting 3 from 55 we have left 52, which is the number of rational rightangled triangles whose hypotenuses do not exceed 100. See Mathematical Magazine, Vol. II, No. 12 (Sept. 1910), pp. 319-20; Miscellaneous Notes and Queries, Vol. x, No. 3 (Sept. 1892), p. 230; Educational Times Reprint, Vol. xxxix (London, 1883), pp. 37-8, Quest. 6966. The prime rational right-angled triangles whose hypotenuses are less than 3000 are given in the appended table for reference, and are arranged in the order of the magnitude of the hypotenuses for convenience of use in this paper. I have extended this table, in manuscript, so as to include all prime rational right-angled triangles whose hypotenuses are less than 10000. For triangles whose hypotenuses exceed 3000, consult the table on pp. 301-8 of No. 12, Vol. II, of the Mathematical Magazine, beginning with p. 304. I have extended this table in manuscript to p= 101, q = 100. There are 158 prime rational right-angled triangles whose hypotenuses are less than 1000; 161 whose hypotenuses are between 1000 and 2000; 158 whose hypotenuses are between 2000 and 3000; or 477 whose hypotenuses are less than 3000. I have checked the number of these triangles in the appended table with the table by Edward Sang (Transactions of the Royal Society of Edinburgh, Vol. xIII, 1864, pp. 757-58), and both agree to the extent of Sang's table, except that he omitted the following two triangles (576, 943, 1105), (744, 817, 1105), at the end of his table. XII. To find rational right-angled triangles whose sides are whole numbers and hypotenuses consecutive whole numbers. Examples. 1. To find pairs or couples of right-angled triangles whose hypotenuses are consecutive integers. Let u, v, w; x, y, w + 1 denote the sides of a pair of such triangles, and, to get a general solution, we must solve the equations u2 + v2 = 2.................................... (1), x2+ y2= ( + 1).................................(2); but it is much easier to find the triangles wanted by trial from the appended table with the aid of Table I in Barlow's Tables (London, 1814), when the hypotenuses are not greater than 3000; when the hypotenuses are greater than 3000, use the table of right-angled triangles in the Mathematical Magazine on pp. 301-8, beginning with p. 304. If we multiply the sides of the first triangle in the table at the end of this paper by 5, we have a triangle whose sides are 15, 20, 25. Multiplying the sides of the second triangle in the table by 2, we get the triangle whose sides are 10, 24, 26;

Page  43 ON RATIONAL RIGHT-ANGLED TRIANGLES 43 and we have two triangles whose hypotenuses differ by unity or are consecutive numbers. In a similar manner we find from the first and fifth triangles in the table the pair 20, 21, 29; 18, 24, 30. From the first and third triangles we get the pair 16, 30, 34; 21, 28, 35. The following pairs, and 15, 36, 39; 24, 32, 40. 48. 55, 73; 24, 70, 74. 24, 143, 145; 96, 110, 146. 252, 844, 900; 451, 780, 901. many others, are easily found: 14, 48, 50; 20, 48, 24, 45, 51. 28, 45, 52; 53. 104; 105. 60, 80, 100; 20, 99, 101. 119, 120, 169; 102, 136, 170. 696, 697, 985; 680, 714, 986. 40, 96, 63, 84, 65, 156, 80, 150, 169; 170. 324, 945, 999; 352, 936, 1000. 2. To find groups of three rational right-angled triangles whose hypotenuses are consecutive numbers. Multiplying the second triangle in the appended table by 3, the first triangle by 8, we have, with the seventh triangle, 15, 36, 39; 24, 32, 40; 9, 40, 41; a group of three triangles whose hypotenuses are consecutive numbers. In a similar manner the following groups of three such triangles have been found, and many other groups may be found with hypotenuses not exceeding 1000. 48, 55, 24, 70, 45, 60, 69, 80, 45, 52, 120, 105, 92, 84, 108, 165, 126, 140, 73; 74; 75. 115; 116; 117. 173; 174; 175. 60, 80, 100; 20, 99, 101; 48, 90, 102. 40, 96, 104; 63, 84, 105; 56, 90, 106. 81, 108, 64, 120, 88, 105, 135; 136; 137. 193; 194; 195. 48, 140, 51, 140, 90, 120, 148; 149; 150. 232; 233; 234. 95, 130, 117, 168, 144, 156, 160, 105, 90, 168, 208, 216,

Page  44 44 ARTEMAS MARTIN 44, 147, 54, 240, 196, 240, 244; 245; 246. 102, 280, 298: 115, 276, 299; 180, 240, 300. 155, 372, 403; 80, 396, 404; 243, 324, 405. 84, 245, 259; 100, 240, 260; 180, 189, 261. 140, 336, 364; 240, 275, 365; 66, 360, 366. 280, 294, 406; 132, 385, 407; 192, 360, 408. 128, 105, 176, 81, 222, 196, 240, 252, 210, 360, 296, 315, 272; 273; 274. 369; 370; 371. 120, 391, 409; 168, 374, 410; 264, 315, 411. 3. To find groups of four rational right-angled triangles whose hypotenuses are consecutive numbers less than 1000. In the School Messenger, edited by G. H. Harvill, Vol. II, No. 6 (June, 1885), p. 218, the late B. F. Burleson, in a paper on "Complementary Squares" (rightangled triangles), said: " With hypotenuses less than 1000 I have found 21 sets of series, 4 in each, in which the hypotenuses differ by unity"; but he did not give the triangles. The present writer has found 45 such sets or groups of four triangles which are exhibited below. It is to be regretted that Mr Burleson did not publish the groups he found so that it could be seen whether any of them are missing here. 30, 40, 50; 108, 144, 180; 24, 20, 28, 120, 144, 132, 45, 48, 45, 182, 165, 176, 51; 52; 53. 218; 219; 220; 19, 70, 33, 160, 105, 90, 180, 168, 180, 168, 208, 206, 181; 182; 183. 232; 233; 234; 40, 140, 96, 45, 44, 147, 54, 95, 120, 25, 170, 198, 202; 147, 203; 180, 204; 200, 205. 21, 220, 221. 120, 225, 235. 240, 196, 240, 228, 288, 312, 264, 244; 245; 246; 247. 128, 105, 176, 240, 252, 210, 272; 273; 274; 136, 200, 195, 255, 210, 216, 289; 290; 291; 312; 313: 314; 77, 264, 275. 192, 220, 292. 189, 252, 315. 75, 168, 220, 192, 308, 270, 231, 256, 317; 318; 319; 320. 175, 288, 337; 240, 130, 312, 338; 180, 45, 336, 339; 210, 160, 300, 340. 135, 252, 299, 280, 324, 348; 349; 350; 351. 260, 288, 388; 189, 340, 389; 150, 360, 390: 184, 345, 391. 155, 372, 403; 132, 385, 407; 80, 243, 280, 396, 324, 294, 404; 405; 406. 192, 120, 168, 360, 391, 374, 408; 409; 410.

Page  45 ON RATIONAL RIGHT-ANGLED TRIANGLES 4 45 280, 126, 99, 60, 168, 156, 312, 279, 215, 336, 264, 320, 354, 84, 192,:368, 150, 381, 336, 245, 52, 90, 455, 104, 480, 344, 132, 108, 324, 351, 80, 498, 540, 498, 345' 320. 351, 432, 440, 448, 490, 495, 416, 440, 516, 448, 495, 462, 472, 585, 560, 465, 616, 508, 540, 588, 675, 672, 504, 672, 550, 645, 720, 725, 768, 720, 798, 605, 629, 664, 756, 768, 449; 450; 451; 452. 518; 519; 520; 521. 559;560; 561; 562. 590; 591; 592; 593. 634; 635; 636; 637. 677; 678; 679; 680. 730; 731; 732; 733. 800; 801; 802; 803. 829; 830; 831; 832. 168, 120, ~216, 276, 425, 442, 405, 368, 57, 540, 256, 480, 100, 455, 210, 504, 48, 272, 2 85", 400, 420, 110, 235, 288, 315, 360, 393, 144, 480, 153, 360, 315, 481, 462, 96, 380, 310, 207, 160, 280, 504, 580, 58, 480, 575, 51 1, 504, 420, 441, 600, 564, 540, 572, 546, 524, 640, 504, 680, 598, 624, 600, 616, 765, 672, 744, 780, 792, 759, 672, 609, 840, 693, 457; 458; 459; 460. 543; 544; 545;546. 577; 578; 579; 580. 609; 610; 611; 612. 653; 654;~ 655; 656. 696; 697; 698; 699. 769; 770; 771; 772. 806; 807; 808; 809. 840; 841; 842; 843. 108, 340, 190, 287, 352, 99, 154, 380, 390, 408, 384, 225, 273, 440, 175, 50, 385, 350, 189, 260, 196, 260, 270, 228, 217, 520, 252, 378, 315, 180, 421, 528, 600, 335, 480, 585, 480, 357, 456, 396, 420, 540, 528, 399, 432, 495, 440, 540, 560, 576, 600, 624, 552, 576, 648, 624, 672, 651, 648, 665, 744, 576, 735, 640, 756, 800, 700, 630, 630, 804, 728, 648, 492; 493; 494; 495. 548; 549; 550; 551. 582; 583; 584; 585. 623'; 624; 625; 626. 673;674: 675; 676. 700; 701; 702; 703. 775; 776; 777; 778. 819;820;821; 822. 870; 871; 872; 873.

Page  46 46 ARTEMAS MARTIN 345, 828, 897; 560, 702, 898; 620, 651, 899; 252, 864, 900. 75, 936, 939; 564, 752, 940; 580, 741, 941; 510, 792, 942. 365, 876, 949; 266, 912, 950; 225, 924, 951; 448, 840, 952. These groups of four triangles are distinct and independent; no group contains a triangle found in any other group. 4. To find groups of five numbers less than 1000. I have found the following are less than 1000. triangles whose hypotenuses are consecutive whole 15 groups of five such triangles whose hypotenuses 120, 144, 182, 165, 218; 219; 136, 255, 289; 155, 200, 210, 290; 80, 195, 216, 291; 243, 372, 396, 324, 403; 404; 405; I L32. 176, 220; 21, 220, 221; 72, 210, 222. 192, 68, 220, 285, 292; 293. 280, 294, 406; 132, 385, 407. 168, 156, 312, 490, 495, 416, 518; 519; 520; 390, 432, 582; 385, 408, 495, 583; 350, 384, 440, 584; 189, 552, 576, 648, 673; 674; 675; 279, 440, 521; 225, 540, 585; 260, 624, 676; 360, 378, 522. 136. 570, 586. 52, 675, 677. 480, 504, 696; 481, 153, 680, 697; 462, 360, 598, 698; 96, 315, 624, 699; 380, 196, 672, 700. 195, 483, 644, 805; 240, 310, 744, 806; 315, 207, 780, 807; 180, 160, 792, 808; 421, 280, 759, 809. 528, 600, 616, 765, 672, 748, 782, 756, 800, 700, 630, 769; 770; 771; 772; 773. 818; 819; 820; 821; 822. 217, 520, 252, 378, 171, 540, 498, 345, 320, 392, 744, 576, 735, 640, 760, 629, 664, 756, 768, 735, 775; 776; 777; 778; 779. 829; 830; 831; 832; 833. 345, 560, 620, 828, 702, 651, 897; 898; 899; 75, 936, 939; 564, 752, 940; 365, 876, 949; 266, 912, 950; 252, 864, 900; 451, 780, 901. 580, 510, 207, 741, 792, 920, 941; 942; 943. 225, 448, 615, 924, 840, 728, 951; 952; 953. 5. To find groups of six numbers less than 1000. triangles whose hypotenuses are consecutive whole

Page  47 ON RATIONAL RIGHT-ANGLED TRIANGLES 47 I have found 155, 80, 243, 280, 132, 192, 217, 520, 252, 378, 171, 396, the following 372, 403; 396, 404; 324, 405; 294, 406; 385, 407; 360, 408. seven groups of 385, 350, 189, 260, 52, 90, 483, 310, 207, 160, 280, 486, 552, 576, 648, 624, 675, 672, 644, 744, 780, 792, 759, 648, 673; 674; 675; 676; 677; 678. six such triangles: 417, 556, 695; 480, 504, 696; 153, 680, 697; 360, 598, 698; 315, 624, 699; 196, 672, 700. 744, 576, 735, 640, 760, 672, 775; 776; 777; 778; 779; 780. 805; 806; 807; 808; 809; 810. 345, 560, 620, 252, 451, 198, 828, 702, 651, 864, 780, 880, 897; 898; 899; 900; 901; 902. 365, 876, 949; 225, 924, 951; 266, 912, 950; 448, 840, 952; 615, 728, 953; 504, 810, 954. 6. To find groups of seven right-angled triangles whose hypotenuses are consecutive whole numbers less than 1000. I have found the following four groups of seven such right-angled triangles: 155, 80, 243, 280, 132, 192, 120, 417, 480, 153, 360, 315, 196, 260, 372, 396, 324, 294, 385, 360, 391, 556, 504, 680, 598, 624, 672, 651, 403; 404; 405; 406; 407; 408; 409. 695; 696; 697; 698; 699; 700; 701. 385, 350, 189, 260, 52, 90, 455, 365, 266, 225, 448, 615, 504, 573, 552, 576, 648, 624, 675, 672, 504, 876, 912, 924, 840, 728, 810, 764, 673; 674; 675; 676; 677; 678; 679. 949; 950; 951; 952; 953; 954; 955. 7. To find groups of eight right-angled triangles whose hypotenuses are consecutive whole numbers. I have found the following seven groups of eight such triangles. The hypotenuses of the first three are less than 1000. 155, 372, 403; 385, 552, 673; 417, 556, 695; 80, 396, 404; 350, 576, 674; 480, 504, 696;

Page  48 48 ARTEMAS MARTIN 243, 324, 405; 189, 648, 675; 153, 680, 697; 280, 294, 406; 260, 624, 676; 360, 598, 698; 132, 385, 407; 52, 675, 677; 315, 624, 699; 192, 360, 408: 90, 672, 678; 196, 672, 700; 120, 391, 409; 455, 504, 679; 260, 651, 701; 168, 374, 410. 104, 672, 680. 270, 648, 702. 528, 1025, 1153; 715, 1428, 1597; 1008, 1344, 1680; 96, 1150, 1154: 752, 1410, 1598; 720, 1519, 1681 693, 924, 1155; 351, 1560, 1599; 82, 1680, 1682; 644, 960, 1156: 448, 1536, 1600; 792, 1485, 1683; 765, 868, 1157; 80, 1599, 1601; 116, 1680, 1684: 570, 1008, 1158. 702, 1440, 1602; 164, 1677, 1685; 209, 1140, 1159; 420, 1547, 1603; 960, 1386, 1686; 800, 840, 1160. 160, 1596, 1604. 840, 1463, 1687. 1240, 1302, 1798: 224, 1785, 1799; 504, 1728, 1800; 649, 1680, 1801; 952, 1530, 1802; 720, 1653, 1803; 396, 1760, 1804; 1083, 1444, 1805. The late B. F. Burleson gave the first and second of the foregoing groups of eight right-angled triangles whose hypotenuses are consecutive numbers in the late S. C. Gould's Notes and Queries, Vol. II, No. 12 (December, 1886), p. 201, and remarked: "These are the only 2 sets of 8 triangles in a series in which the hypotenuses are in regular sequence ever obtained. They were found by Charles Kriele, of Pennsylvania, when aged, infirm, and nearly blind, and sent by him to the writer of this article." Mr Burleson gave these two groups in Harvill's School Messenger, Vol. II, No. 6 (June, 1885), p. 219, saying he had found "2 sets of series, 8 in each, in which the hypotenuses differ only by unity," and made no mention there of Kriele. I have added five other groups of 8 triangles, so seven such groups are now known. I have also added another triangle to the first, third and fifth groups, making three groups of nine triangles whose hypotenuses are consecutive whole numbers, which are exhibited below: 155, 372, 403; 417, 556, 695; 715, 1428, 1597: 80, 396, 404; 480, 504, 696; 752, 1410, 1598; 243, 324, 405; 153, 680, 697; 351, 1560, 1599; 280, 294, 406; 360, 598, 698; 448, 1536. 1600;

Page  49 ON RATIONAL RIGHT-ANGLED TRIANGLES 49 132, 385, 407; 315, 624, 699; 80, 1599, 1601; 192, 360, 408; 196, 672, 700; 702, 1440, 1602; 120, 391, 409; 260, 651, 701; 420, 1547, 1603; 168, 374, 410; 270, 648, 702; 160, 1596, 1604; 264, 315, 411. 420, 665, 703. 963, 1284, 1605. I have added another triangle to the last of the groups above and thus have one group of 10 rational right-angled triangles whose hypotenuses are consecutive whole numbers which is given below: 715, 1428, 1597; 702, 1440, 1602; 752, 1410, 1598; 420, 1547, 1603; 351, 1560, 1599; 160, 1596, 1604; 448, 1536, 1600; 963, 1284, 1605; 80, 1599, 1601; 1056, 1210, 1606. Since what precedes was written, Mr B. O. M. De Beck of Cincinnati, Ohio, has sent me, from notes he made from Barlow's Tables more than 60 years ago, eight groups of 10 triangles whose hypotenuses are consecutive numbers; eight groups of 11 such triangles; six groups of 12 triangles; two groups of 13 triangles, and one group of 14 triangles, the hypotenuses of all the groups being between 1000 and 10000. He also sent me later the following groups with hypotenuses between 10000 and 20000: thirteen groups of 10 triangles; nine groups of 11 triangles; ten groups of 12 triangles; three groups of 13 triangles; one group of 14 triangles, and one group of 15 triangles. Mr De Beck also contributed one group of 15 triangles with hypotenuses from 85478 to 85492, inclusive. Space considerations prevent me from giving here more than one of each of the different groups whose hypotenuses are between 1000 and 10000. I give also the group of 15 triangles whose hypotenuses are between 10000 and 20000. 1247, 2140, 3103; 1440, 1650, 2190; 1692, 2256, 2820; 2080, 2304, 3104; 175, 2184, 2191; 1085, 2604, 2821; 1863, 2484, 3105; 1408, 1680, 2192; 1328, 2490, 2822; 990, 2944, 3106; 1032, 1935, 2193; 1740, 2223, 2823; 1195, 2868, 3107; 1170, 1856, 2194; 1800, 2176, 2824; 1008, 2940, 3108; 1317, 1756, 2195; 1695, 2260, 2825; 1309, 2820, 3109; 396, 2160, 2196; 1530, 2376, 2826; 1866, 2488, 3110; 845, 2028, 2197; 352, 2805, 2827; 561, 3060, 3111; 1190, 1848, 2198; 560, 2772, 2828; 1512, 2720, 3112. 324, 2175, 2199; 621, 2760, 2829; 1320, 1760, 2200. 1698, 2264, 2830; 969, 2660, 2831. 4 M. C. II.

Page  50 50 ARTEMAS MARTIN 3630, 2376, 2848, 1635, 3354, 3633, 3744, 3465, 3142, 3984, 3636, 4180, 2030, 4840, 5565, 5340, 5828, 5040, 4844, 4760, 4968, 4600, 4565, 4848, 4389, 5712, 6050; 6051; 6052; 6053; 6054; 6055; 6056; 6057; 6058; 6059; 6060; 6061; 6062. 5535, 2080, 5796, 5148, 131, 490, 5700, 3584, 5151, 4536, 3565, 1140, 2520, 5154, 6552, 8322, 6325, 6864, 8580, 8568, 6417, 7800, 6868, 7290, 7812, 8512, 8211, 6872, 8577; 8578; 8579; 8580; 8581; 8582; 8583; 8584; 8585; 8586; 8587: 8588; 8589; 8590. 2460, 3960, 3479, 6840, 5848, 4780, 8379, 1650, 1240, 4032, 4495, 6384, 8100, 5236, 2355, 12177, 11776, 11928, 110374, 10965, 11472, 9180, 12320, 12369, 11760, 11592, 10670, 9435, 11280, 12212, 12423: 12424; 12425; 12426; 12427; 12428; 12429; 12430; 12431; 12432; 12433; 12434; 12435; 12436; 12437. XIII. In each of the following nine groups of three right-angled triangles, the three hypotenuses of each group are the sides of a right-angled triangle. 21, 72, 75; 40, 96. 104; 56, 105, 119; 60, 80, 100; 72, 135, 153; 72, 96, 120; 35, 120, 125. 57, 176, 185. 65, 156, 169. 64, 120, 136; 60, 252, 273; 55, 300, 305. 240, 320, 400; 264, 495, 561; 111, 680, 689. 60, 63, 87; 160, 384, 416; 243, 324, 425. 480, 504, 696; 455, 528, 697; 473, 864, 985. 60, 144, 156; 460, 483, 667; 440, 525, 685. 387, 516, 645; 560, 588, 812; 315, 988, 1037. Such groups of right-angled triangles could be obtained by solving the following simultaneous equations, but it is easier to find them from the table of right-angled triangles, with the aid of Barlow's Table I, as I have done. Let p, q, r be the legs and hypotenuse of the first triangle; s, t, u the legs and hypotenuse of the second triangle; v, w, x the legs and hypotenuse of the third triangle; then we shall have p2 + q2 = r, s2 + t = U2, V2 + W2 = X2, r2 + u2 = X2.

Page  51 ON RATIONAL RIGHT-ANGLED TRIANGLES,51 XIV. In the following six groups of three rational right-angled triangles, the hypotenuses of each group form a rational scalene triangle. 5, 12, 13; 15, 36, 39; 8, 15, 17; 24, 32, 40; 9, 40, 41; 16, 63, 65; 27, 36, 45. 14, 48, 50. 48, 64, 80. 15, 112, 113; 27, 120, 123; 88, 105, 137; 24, 143, 145; 48, 140, 148; 96, 110, 146; 130, 144, 194. 21, 220, 221. 140, 225, 265. In each of the groups below the three hypotenuses of each group are the sides of a rational scalene triangle whose sides are consecutive numbers. 24, 45, 51: 95, 168, 193; 360, 627, 723; 20, 48, 52; 130, 144, 194; 76, 720, 724; 28, 45, 53. 117, 156, 195. 120, 715, 725. 1776, 2035, 2701; 5040, 8733, 10083; 1330, 2352, 2702; 280, 10080, 10084; 1428, 2295, 2703. 3960, 9275, 10085. XV. As stated on a preceding page, a number which is the product of n different prime factors of the form 4m + 1 is the hypotenuse of 2n-1 (incorrectly given 2'1 in Barlow's Theory of Numbers, p. 177) different prime right-angled triangles. See Matteson's Diophantine Problems with Solutions (Washington, 1888), p. 10. Examples. 1. The least two primes of the form 4m + 1 are 5 and 13, and their product = 65 = 82 + 12 = 72 + 42, is the hypotenuse of each of the two prime triangles 16, 63, 65; 33, 56, 65. There are also two other triangles, not prime triangles, with hypotenuses = 65, obtained as follows: (3, 4, 5) x 13 =39, 52, 65; (5, 12, 13) x 5 = 25, 60, 65. The product of 5 and 17 = 85 = 92 + 22 = 72 + 62, and therefore 85 is the common hypotenuse of the two prime triangles 13, 84, 85; 36, 77, 85. The other two triangles in this case are (3, 4, 5)x 17 = 51, 68, 85; (8, 15, 17)x 5=40, 75, 85. The product of 13 and 17 = 221 = 142 + 52 112 + 102, and 221 is the common hypotenuse of the two prime triangles 21, 220, 221; 140, 171, 221. The other two triangles with hypotenuses = 221 are (5, 12, 13) x 17 = 85, 204, 221; (8, 15, 17) x 13 = 104, 195, 221. 4-2

Page  52 52 ARTEMAS MARTIN And, generally, (a2 + b2) (c2 + d2) = (ac + bd)2 + (ad - be2 = (ad + bc)2 + (ac - bd)2. There are 36 couples or pairs of prime right-angled triangles having a common hypotenuse less than 1000, 41 such pairs of triangles whose hypotenuses are between 1000 and 2000, 39 pairs whose hypotenuses are between 2000 and 3000; or, 116 pairs with hypotenuses less than 3000. See table of prime rational right-angled triangles at end of paper. 2. The product of three prime numbers of the form 4m + 1 is the common hypotenuse of 2' = 4 prime right-angled triangles. The product of the least three primes of this form is 5 x 13 x 17= 1105 = 332+4 =32292=31122=242 + 232, which is, therefore, the hypotenuse of each of the following four prime right-angled triangles, viz.: 47, 1104, 1105; 264, 1073, 1105; 576, 943, 1105: 744, 817, 1105. The last two of these triangles are omitted at the end of Edward Sang's table of right-angled triangles referred to on a preceding page. There are nine other triangles having 1105 for hypotenuse which are not prime triangles, but are multiples of prime triangles. They are given below with the method of obtaining them. (3, 4, 5) x 13 x 17 = 663, 884, 1105; (5, 12, 13) x 5 x 17= 425, 1020, 1105; (8, 15, 17) x 5 x 13 = 520, 975, 1105; (16, 63, 65) x 17 = 272, 1071, 1105; (33, 56, 65) x 17 = 561, 952, 1105; (36, 77, 85) x 13 = 468, 1001, 1105; (13, 84, 85) x 13 = 169, 1092, 1105; (21, 220, 221)x 5 105, 1100, 1105; (140, 171, 221)x 5 = 700, 885, 1105. In general, (a2 + b2) (c2 + (12) (e2 + f2) = [e (ad + be) + f (ac - bd)]2 + [e (ac - bd) -f(ad + bc)]2, = [e (ac - bd) + f (ad + bc)]2 + [e (ad + b) - f (ac - bd)]2, = [e (ad - be) + f (ac + bd)]2 + [e (ac + bd) - f(ad - bc)]2, = [e (ac + bd) + f(ad - bc)]2 + [e (ad - be) -f (ac + bd)]2. Take a = 2, b= 1; c =3, d=2; e = 4, f=; then 5 x 13 x 17 = 1105 =322+ 92 =242+ 232=312 +122 =332+ 42, as found before. 5 x 13 x 29 = 1885 = 432+62 =422+11 = 382 +21 = 342+ 272;

Page  53 ON RATIONAL RIGHT-ANGLED TRIANGLES 53 hence 1885 is the common hypotenuse of the four prime right-angled triangles 427, 1836, 1885; 516, 1813, 1885: 924, 1643, 1885; 1003, 1596, 1885. There are also nine other right-angled triangles having 1885 for a common hypotenuse that are not prime triangles, which are given below: 221, 1872, 1885; 675, 1760, 1885; 957, 1624, 1885; 312, 1859, 1885; 725, 1740, 1885; 1131, 1508, 1885; 464, 1787, 1885; 760, 1725, 1885; 1300, 1365, 1885. There are only two groups of four prime rational right-angled triangles having a common hypotenuse less than 2000, and only three groups of such triangles with hypotenuses between 2000 and 3000, viz.: 5 x 13 x 37 = 2405 5 x 17 x 29245, 5 x 13 x 41 = 2665; 2405 = 492 + 22= 472 + 14 = 462 + 17 = 382 + 312; 2465 = 492 + 82 = 472 + 16 = 442 + 232 = 412 + 282; 2665 = 512 + 82 = 482 + 192= 442 + 272= 372 + 362. Hence the three groups of 4 prime triangles are 196, 2397, 2405; 784, 2337, 2465; 73, 2664, 2665; 483, 2356, 2405; 897, 2296, 2465; 816, 2537, 2665; 1316, 2013, 2405; 1407, 2024, 2465; 1207, 2376, 2665; 1564, 1827, 2405. 1504, 1953, 2465. 1824, 1943, 2665. The following curious theorem is given without proof in Matteson's Diophantine Problems with Solutions, p. 11. If a number N is the product of 1, 2, 3, 4, 5,... or n different prime factors of the form 4m +1, the terms of the following series, 1,, 42, 133, 404, 1215, 3646, 10937, 3280s, etc., will represent the number of ways N2 can be the sum of two squares, where the subscripts, 1, 2, 3, 4,... n, denote the number of prime factors composing N. Any term of this series is equal to three times the preceding term, + 1. Thus the square of a prime number of the form 4m + 1 is the sum of two squares in one way only; the square of a number which is the product of two different primes of the form 4m + 1 is the sum of two squares in 1 x 3 + 1 = 4 different ways; the square of a number which is the product of three different primes of the form 4m + 1 is the sum of two squares in 4 x 3 + 1 = 13 different ways; the square of a number which is the product of four different primes of the form 4m + 1 is the sum of two squares in 13 x 3 + 1 = 40 different ways, etc. If we put u, for the nth term in the above series we have un+11 = 3Ut, + 1.

Page  54 54 ARTEMAS MARTIN The solution of this Finite-Difference equation gives n =1 (3- 1). See Boole's Calculus of Finite Differences, second edition (London, 1872), p. 165. Hence if a number is the product of n different prime numbers of the form 4m + 1, it is the common hypotenuse of 1 (3n - 1) different rational integral-sided right-angled triangles, 2"-1 of which are prime triangles and i (3 - 2" - 1) are not prime triangles. Examples. 1. If n = 1, then (31 - 1)= 1, and a prime number of the form 4m + 1 can be the hypotenuse of but one rational right-angled triangle. 2. If n = 2, then (32 - 1) = 4, and any number which is the product of two different primes of the form 4m + 1 is the common hypotenuse of four different rational right-angled triangles. 3. If n = 3, then - (3 - 1) = 13, and any number which is the product of three different primes of the form 4,n + 1 is the common hypotenuse of 13 different rational right-angled triangles. 4. If n = 4, then ~ (34 - 1) = 40, and any number which is the product of four different primes of the form 4m + 1 is the common hypotenuse of 40 different rational right-angled triangles. 5. If, = 5, then I (35 - 1) = 121, and any number which is the product of five different primes of the form 4m + 1 is the common hypotenuse of 121 different rational right-angled triangles. And so on, to any value of n. Find all the rational right-angled triangles whose hypotenuses are 32045. Solttion. 32045 = 5 x 13 x 17 x 29 = 1792 _ 22= 1782 + 192 = 173' + 462= 1662' + 672 = 163 + 742 = 1572 + 862 = 1442 + 1092 = 1312 + 122'. Hence the eight prime right-angled triangles whose hypotenuses are 32045 are as follows: 716, 32037, 32045; 15916, 27813, 32045' 2277, 31964, 32045; 17253, 27004, 32045; 6764, 31323, 32045; 21093, 24124, 32045; 8283, 30956, 32045; 22244, 23067, 32045. The other thirty-two triangles are found as follows: (3, 4, 5) x 13 x 17 x 29 = 19227, 25636, 32045; (5, 12, 13) x 5 x 17 x 29 = 12325, 29580, 32045; (8, 15, 17) x 5 x 13 x 29 = 15080, 28275, 32045; (20, 21, 29) x 5 x 13 x 17 = 22100, 23205, 32045; (16, 63, 65) x 17 x 29 = 7888, 31059, 32045; (33, 56, 65) x 17 x 29 = 16269, 27608, 32045; (13, 84, 85) x 13 x 29 = 4901, 31668, 32045:

Page  55 ON RATIONAL RIGHT-ANGLED TRIANGLES 55 (36, 77, 85) x 13 x 29 = 13572, 29029, 32045; (17, 144, 145) x 13 x 17 = 3757, 31824, 32045; (24, 143, 145) x 13 x 17 = 5304, 31603, 32045; (21, 220, 221) x5x29= 3045, 31900, 32045; (140, 171, 221) x 5 x 29= 20300, 24795, 32045; (135, 352, 377) x 5 x 17 = 11475, 29920, 32045; (152, 345, 377) x 5 x 17 = 12920, 29325, 32045; (132, 425, 493) x 5 x13 = 8580, 30875, 32045; (155, 468, 493) x 5 x 13 = 10075, 30420, 32045; (47, 1104, 1105) x 29 = 1363, 32016, 32045; (264, 1073, 1105) x 29 = 7656, 31117, 32045; (576, 943, 1105) x 29 = 16704, 27347, 32045; (744, 817, 1105) x 29 - 21576, 23693, 32045; (427, 1836, 1885) x 17 = 7259, 31212, 32045; (516, 1813, 1885) x 17= 8772, 30821, 32045; (924, 1643, 1885) x 17 = 15708, 27931, 32045; (1003, 1596, 1885) x 17 = 17051, 27132, 32045; (784, 2337, 2465) x 13 = 10192, 30381, 32045; (897, 2296, 2465) x 13 = 11661, 29848, 32045; (1407, 2024, 2465) x 13 = 18291, 26312, 32045; (1504, 1953, 2465) x 13 = 19552, 25389, 32045; (480, 6391, 6409)x 5 = 2400, 31955, 32045; (791, 6360, 6409)x 5 = 3955, 31800, 32045; (3959, 5040, 6409)x 5 =19795, 25200, 32045; (4200, 4841, 6409) x 5 = 21000, 24205, 32045. See Mlatteson's Diophantine Problems with Solutions, p. 10; also, Miscellaneous Notes and Queries, Vol. x, No. 3 (Sept., 1892), p. 225. XVI. As stated elsewhere, a prime number of the form 4m + 1 is the sum of two squares in one way only, and therefore is the hypotenuse of only one rational right-angled triangle; but the square of such a prime is the sum of two squares in two ways, and therefore is the common hypotenuse of two different right-angled triangles; the cube of such a prime is the sum of two squares in three ways, and so is the common hypotenuse of three different right-angled triangles; and, generally, the nth power of a prime number of the form 4m + 1 is the sum of two squares in n ways, and is the common hypotenuse of n right-angled triangles, but only one of the triangles is prime. Examples. 1. 52 = 32 + 42, and there is one triangle whose hypotenuse is 5. '5- 7 - 24' -15- + 2).

Page  56 56 ARTEMAS MARTIN and 25 is the common hypotenuse of the two triangles 7, 24, 25; 15, 20, 25. 1252 = (5::)2 = 442 + 1172 = 352 + 1202 = 752 + 1002, and we have the three triangles 44, 117, 125; 35, 120, 125; 75, 100, 125. (54)2 = 6252 = 33(2) + 527 = 220 + 5852 =175 + 600 = 3752 + 500; hence we have the four triangles 175, 600, 625; 220, 585, 625; 336, 527, 625; 375, 500, 625. 2. The next prime of the form 4m + 1 being 13, we have 132 =52 +122, and there is only one triangle whose hypotenuse is 1.3. 1692 = 1192 + 1202 = 652 + 1562, and the two triangles with hypotenuse 169 are 65, 156, 169; 119, 120, 169. 133 = 21 97, and 21972 = 8282 + 20352 = 15472 +15602= 845 + 20282; therefore the three triangles having hypotenuse = 2197 are 828, 2035, 2197; 845, 2028, 2197; 1547, 1560, 2197. 134 = 28561 is beyond the limit of the table of right-angled triangles given at the end of this paper, but the triangles can be computed by the aid of Barlow's Tables, before-lmentioned, Table 1, p. 3. I find 285612 = 23!92 + 285602 = 107642 + 264552 = 201112 + 202802 = 108952 + 263642, and therefore the four triangles are 239, 28560, 28561; 10764, 26455, 28561; 10895, 26364, 28561; 20111, 20280, 28561.

Page  57 ON RATIONAL RIGHT-ANGLED TRIANGLES 57 Table of Prime Rational Right-Angled Triangles whose Hypotenuses are less than 3000. 3 5 8 7 20 12 9 28 11 16 33 48 13 36 39 65 20 60 15 44 88 17 24 51 85 119 52 19 57 104 95 28 84 133 21 140 60 105 120 32 23 96 69 115 160 161 68 136 207 25 75 36 204 175 1.80 225 27 76 252 135 4 12 15 24 21 35 40 45 60 63 56 55 84 77 80 72 99 91 112 117 105 144 143 140 132 120 165 180 176 153 168 195 187 156 220 171 221 208 209 255 264 247 260 252 231 240 285 273 224 312 308 323 253 288 299 272 364 357 275 352 5 13 17 25 29 37 41 53 61 65 65 73 85 85 89 97 101 109 113 125 137 145 145 149 157 169 173 181 185 185 193 197 205 205 221 221 229 233 241 257 265 265 269 277 28] 289 293 305 305 313 317 325 325 337 349 353 365 365 373 377 152 345 189 340 228 325 40 399 120 391 29 420 87 416 297 304 145 408 84 437 203 396 280 351 1.68 425 261 380 31 480 319 360 44 483 93 476 132 475 155 468 217 456 336 377 220 459 279 440 92 525 308 435 341 420 33 544 184 513 165 532 276 493 396 403 231 520 48 575 368 465 240 551 35 612 105 608 336 527 100 621 429 460 200 609 31.5 572 300 589 385 552 52 675 37 684 156 667 111 680 400 561 185 672 455 528 260 651 259 660 333 644 364 627 108 725 216 713 407 624 468 595 377 389 397 401 409 421 425 425 433 445 445 449 45 7 461 481 481 485 485 493 493 505 505 509 521 533 533 541 545 545 557 565 565 569 577 593 601 613 617 625 629 629 641 653 661] 673 677 685 685 689 689 697 697 701 709 725 725 733 745 745 757 39 481 195 56 273 168 432 555 280 429 540 41 116 123 205 2:32 287 504 348 369 60 451 464 616 43 533 129 215 580 301 420 615 124 387 248 473 696 372 559 45 660 64 496 192 31]5 645 320 620 7 31 448 495 132 585 47 264 576 744 141 235 329 760 600 748 783 736 775 665 572 759 700 629 840 837 836 828 825 816 703 805 800 899 780 777 663 924 756 920 912 741 900 851 728 957 884 945 864 697 925 840 1012 779 1023 897 1015 988 812 999 861 780 975 952 1085 928 1104 1073 943 817 1100 1092 1080 761 769 773 785 785 793 793 797 809 821 829 841 845 845 853 857 865 865 877 881 901 901 905 905 925 925 929 937 941 949 949 953 965 965 977 985 985 997 1009 1013 1021 1025 1025 1033 1037 1037 1049 1061 1069 1073 1073 1093 1097 1105 1105 1105 1105 1109 1117 1129 423 1064 1145 704 903 1145 528 1025 1153 68 1155 1157 765 868 1157 204 1147 1165 517 1044 1165 340 1131 1181 611 1020 1189 660 989 1189 832 855 1193 49 1200 1201 147 1196 1205 476 1107 1205 245 1188 1213 705 992 1217 140 1221 1229 612 1075 1237 280 1209 1241 441 1160 1241 799 960 1249 420 1189 1261 539 1140 1261 748 1035 1277 637 1116 1285 893 924 1285 560 1161 1289 72 1295 1297 51 1300 1301 255 1288 1313 735 1088 1313:360 1271 1321 357 1276 1325 884 987 1325 504 1247 1345 833 1056 1345 561 1240 1361 840 1081 1369 148 1365 1373 931 1020 1381 296 1353 1385 663 1216 1385 53 1404 1405 444 1333 1405 159 1400 1409 265 1392 1417 792 1175 1417 371 1380 1429 592 1305 1433 76 1443 1445 477 1364 1445 228 1435 1453 583 1344 1465 936 1127 1465 380 1419 1469 740 1269 1469 969 1120 1481 689 1320 1489 532 1395 1493 55 1512 1513

Page  58 58 ARTEMAS MARTIN Table of Prime Rational Right-Angled Triangles whose Hypotenuses are less than 3000 (cont.) 888 165 795 156 684 312 385 901 495 836 1036 624 1007 715 80 240 988 780 57 1113 285 399 560 935 1140 720 164 627 1045 328 741 492 1092 880 1155 59 177 656 295 84 413 969 1040 1248 531 820 420 649 767 984 172 885 61 183 344 305 1148 427 516 924 1225 1508 1292 1517 1363 1505 1-488 1260 1472 1323 1173 1457 1224 1428 1599 1591 1275 1421 1624 1184 1612 1600 1551 1368 1219 1519 1677 1564 1332 1665 1540 1645 1325 1749 1292 1740 1736 1617 1728 1763 1716 1480 1431 1265 1700 1581 1739 1680 1656 1537 1845 1628 1860 1856 1833 1848 1485 1836 1813 1643 1513 1517 1517 1525 1525 1537 '5t37 1 549 1553 1565 15;'65 1585 1585 1597 1601 1609 1613 1 62 1 1625 1625 1637 1649 1649 1657 1 669 1681 1685 1685 1.693 1697 1 709 17 17 1717 1 721 1733 1741 1745 1753 1 765 1.765 1769 1769 1 777 1781 1 781 1789 1H01 1825 1825 1853 1853 1861 1865 1865 1873 1877 1885 1885 1885 1003 1311 549 688 671 1121 1092 88 264 793 860 440 1239 915 63 616 1032 315 1037 1357 792 180 360 1159 693 1204 1428 819 1281 720 1144 1376 65 92 195 276 1403 1071 455 460 585 1320 644 1197 7i15 15125 828 188 1260 376 1496 1012 6 7 564 201 1449 335 1105 1596 1360 1820 17 85 1.800 1560 1595 1935 1425 1927 1749 1911 1520 1748 1984 1887 1705 1972 1716 1476 1855 2021. 2009 1680 1924 1653 1475 1900 1640 1961 1767 1593 2112 2115 2188 2107 1596 1840 2088 2091 2072 1 711 2067 1804 2052 1548 2035 2205 1829 2193 1647 1995 2244 2173 2240 1720 2232 1968 1885 1889 1901 1913 1921 1921 1933 1937 1937 1945 1945 1949 1961 1961 1973 1985 1985 1993 1997 2005 2005 2017 2029 2041 2041 2045 2045 2053 2069 2081 2089 2105 2105 2113 2117 2117 2125 2125 2129 2137 2141 2153 2161 2165 2165 21 73 2173 2197 2213 2221 2225 222,5 2237 2245 2245 2249 2249 2257 2257 469 752 1440 603 1196 1235 15r75 96 737 940 480 871 1365 1380 672 1128 1005 1495 69 1139 345 196 483 1316 1564 392 1056 1273 588 759 784 897 1407 1504 1248 1748 100 980 300 1541 71 213 1173 355 1692 497 1176 700 1 67 75 639 1311 781 900 1632 204 923 1809 408 1100 2220 2145 1769 2204 1947 1932 1672 2303 2184 2109 2279 2160 1-892 1891 2255 2065 2132 1848 2380 2100 2368 2397 2356 2013 1827 2 385 2183 2064 2365 2320 2337 2296 2024 1953 2135 1755 2499 2301 2491 1980 2520 2516 2236 2508 1885 2496 2257 2451 1932 2480 2200 2460 2419 2015 2597 2436 1880 2585 2379 2269 2273 2281 2285 2285 2293 2297 2305 2305 2309 2329 2329 2333 2341 2353 2353 2357 2377 2381 2389 2393 2405 2405 2405 2405 2417 2425 2425 2437 2441 2465 2465 2465 2465 24773 2477 250]2501 2509 2509 2521 2525 2525 2533 2533 2545 2545 2549 2557 2561 2561 2581 2581 2593 2605 2605 2609 261 7 2621 1065 1568 73 816 1207 1824 219 1300 365 511 1725 '1020 1349 104 657 312 803 1 764 520 1491 949 728 1095 936 1633 1700 1428 1960 1241 75 212 424 1144 1775 525 636 1387 1632 1900 825 848 -1533 -1917' 1-060 108 167 9 510 2059 1.272 7 7 756 231 385 1825 1425 17468 2408 2633 2145 2657 2664 2665 2537 2 66,5 2376 26650 1943 2665 2660 2669 2331 2669 2652 2677 2640 268.9 2068 2693 2501 2701 2340 2701 2703 2705 2624 2705 2695 271:3 2604 2725 20771 2725 2679 2729 2300 2~741 2580 2749 2655 2753 2552 2777 2623 2785 2256 2785) 2211 2789 2405 2797 2001 2801 2520 2809 2812 2813 2805 2813 2793 2825 2583 2825 2208 2833 2788 2837 2773 2845 2484 2845 2345 2857 2139 2861 2752 28713 2745 2873 2444 2885 2156 2 88,5 2728 2897i 2709 2909 2915 2917 2479 2929 2400 2929 2891 2941 2100 2941 2665 2953 2668 2957' 2964 2965 28671 2965 2960 2969 2952 2977 2352 29771 2632 2993 2415 2993 I i i I i I - - - -- - -— L.,, - --- --- —.- ---

Page  59 THE CHARACTERS OF PLANE CURVES BY W. ESSON. In a paper published in the Proceedings of the London Mathematical Society, Vol. xxvIII, 1897, I took the view that an n-ic considered as an envelope consists of an m-an, 28 points united in pairs at each of the 8 double points or cuts and 3c points united in triads at each of the K cusps of the n-ic. This view I developed in papers read before the Oxford Mathematical Society at various dates since 1897. I propose to bring this view before the members of the Congress of Mathematicians for their consideration. 1. There are two kinds of joins of a point to an n-ic. (1) The tangents from the point to the curve proper. (2) The joins of the point to the double points or cuts of the curve and to the cusps of the curve. These joins are called tangent-joins, cutjoins and cusp-joins. 2. According to the view taken in this paper the total number of joins of a point to an n-ic is the sum of the number of tangent-joins, of cut-joins and of cuspjoins, and this total number depends only on the degree n of the n-ic. 3. In a pencil of n-ics each member of which is determined by the position of a point on a given curve, there are in general 3 (n - 1)2 positions of the point, each of which determines an n-ic with a double point. As the point on the given curve approaches a position which determines an n-ic with a double point two of the tangent-joins of a given point to the n-ic continuously approach each other and are ultimately united on the join of the given point to the double point. Thus a pair of tangent-joins are changed into a united pair of cut-joins. See fig. 1, I, p. 64. 4. In a pencil of n-ics each member of which has a double point in a given position there are two members each of which has a cusp at the place of the double point. As the point on the given curve approaches a position which determines an n-ic with a cusp one of the tangent-joins of a given point to the n-ic continuously approaches the join of the given point to the cusp and is ultimately united with that join. See fig. 1, II, p. 64.

Page  60 60 W. ESSON 5. There is thus a continuous change effected of tangent-joins into cut-joins and cusp-joins whilst the total number of joins of a given point to the curve remains the same throughout the change. 6. If this t,otal number is M; the number of tangent-joins m; the number of cut-joins 28 and the number of cusp-joins 3/c, then m + 28 + 3c = M, and M is a numb r depending only upon n the degree of the pencil of n-ics. 7. If finally the n-ic takes the form of n straight lines for which mm and K are zeros and 28 = v (n - 1), then m + 28 + 3K = (n- 1). 8. Thus the n-ic considered as an envelope consists of an m-an 28 points uhited in pairs at the double points and 3K points united in triads at the cusps and the total number of joins of a given point to the curve is n (n - 1). 9. The total number of joins to a curve at a double point is six, of which four are tangent-joins, two united on each tangent at the double point, and two are cutjoins each united to one of these tangents. Let the cut be considered as the point A on the branch to which a is the tangent at A and as the point B on the branch to which b is the tangent at B. The join of A to A is the tangent a which counts as two united tangents; the join of A to B is united to the tangent b; similarly for the join of B to B and to A. 10. The total number of joins to a curve at a cusp is six, of which three are tangent-joins and three are cusp-joins all united on the tangent to the curve at the cusp. 11. When the double point A, B becomes a cusp C the tangents a, b and the cut-joins AB, BA are all united on the tangent c at C. As in the case of the joins to the curve of any point one of the four tangents now united on c becomes a cusp-join thus reducing the number of tangent-joins to three and increasing the number of cusp-joins to three. 12. The view that an n-ic considered as an envelope consists of an m-an, 28 points united in pairs at the 8 cuts and 3/c points united in triads at the Kc cusps leads to the following relation between the joins of an n-ic with itself, T 4 28 (m - 4) + 3K (m - 3) + 28 (8 - 1) + c (K - 1) + 68K = T, T being the number of double tangents or tangent-joins of the curve and T the whole number of joins which depends only on n the degree of the n-ic. In addition to the number r of tangent-joins of the curve proper there is to be reckoned: (1) The tangent-joins to the curve of the points united in pairs at each of the 8 cuts. The number of such joins from a cut is m - 4, excluding the tangents at the cut, and the whole number is 28 (m - 4).

Page  61 THE CHARACTERS OF PLANE CURVES 61 (2) The tangent-joins to the curve of the points united in triads at each of the K cusps. The number of such joins from a cusp is m - 3, excluding the tangents at the cusp, and the whole number is 3K (m - 3). (3) The joins of each pair of points at a cut to the pair at another cut. The number of such joins is 28 (8 - 1). (4) The joins of each triad of points at a cusp to the triad at another cusp. The number of such joins is 9 c( (- 1). (5) The joins of each pair of points at a cut to each triad of points at a cusp. The number of such joins is 68Sc. The value of T is obtained from its value when the n-ic becomes n straight lines and is 1n (n - 2) (12 - 9) as will be shown later (~ 16). See fig. 2, p. 65. 13. If the view taken here of the n-ic considered as an envelope is correct it follows by the principle of duality that an m-an with 7 double tangents and t inflexions consists of an n-ic, 2T straight lines united in pairs on each double tangent and 3L straight lines united in triads on each inflexion tangent. The cuts of a straight line with the mn-an are (1) the n cuts with the n-ic, (2) the 2r + 3 cuts with the straight lines united in pairs on the double tangents and in triads on the inflexion tangents. It is not necessary to state in detail the consequences of the dual theory. A. L. Dixon, F.R.S., has given an analytic proof of the foregoing theory. He has shown that when the lineal f(x, y, z) expresses an n-ic with 8 cuts and K cusps the corresponding pointal is Tr 1Ur2. 7rKIs 3. O* m (n, V, ~), a3,,/ representing a cut and a cusp respectively, and m + 28 + 3c = n (n - 1). 14. In the paper to the London Mathematical Society quoted above the characters of a curve in terms of its degree n, class r and deficiency D were stated to be e = 2 (n - 1) -m + 2D, = 2 (m-1)- n + 2D, = (n - 1) (n - 6) + n - 3D, = (m - 1) (m - 6) + - 3D. The number of conditions c which determine the curve is mt + + + 1 - D. 15. Assuming that a straight line is of the first degree and of class zero and is determined by two conditions, it follows that its deficiency is zero. From these data the characters of a straight line are Kc = 0, t = - 3, 8 = 0, r = 4. Similarly the characters of a point are / = - 3, ~ = 0, 8 = 4, T = 0. 16. The further relation, deduced from the characters given above, + 68 + 8K= = 3n(n- 2), implies that a cut absorbs six inflexions and a cusp eight inflexions.

Page  62 62 W. ESSON Consider now the simplest form of an n-ic, viz. n straight lines, which is of class zero, has in (n - 1) cuts and is determined by 2n conditions. The deficiency obtained from 2n = n + 1 - D is D = - (n - 1). The characters are K = 0, =-3n, 8 -= n (n - 1), = 4n. Hence from the relation r + 28 (m - 4) + 28 (8 - 1)= T (~ 12), T= 4n - 4n (n - 1) + n (n - 1) {Ln (n - 1)- 1} = -n(n- 2)(n- 9), the result quoted above (~ 12). 17. Similar results are obtained from the consideration of the simplest form of an m-an, viz. m points. 18. The classification of n-ics based upon their characters should include the cases of n-ics which are composed of curves of lower degrees and have zero or negative deficiency. A table of such curves will commence with rn=n(n-1), D= (n- l)(n-2), /c=O, t=3n(n-2), 8=0, r=.n(n - 2)(V2-9) and end with ni=0, D=-(n-1), K=0, 1=-3n, = - n (n-1), r=4n. 19. A few results are added on the characters of curves which are the loci of cuts of two (m, n) linear pencils. 20. When an (m + n)-ic so described has no cuts except those at the centres of the pencils and no cusps, the class of the curve is 2inn and its deficiency (m - 1) (n- 1), /C- 0, = 3 (2mn - n - n), = -m(m - 1)+ n (n - 1), r = 2 (mn - 1) (mn - 2)- 4 (m - 1) (n - 1), c = mn + 2 (m + n). 21. When I - 1 cuts at the centre of one pencil and n - 1 cuts at the centre of the other pencil are changed into cusps, the class of the curve is 2mn - n - i + 2 and the deficiency is as before (m - 1) (n - 1), Ke = mn + n -2, t = 6 (in - l) (n - 1) + it + n - 2, = 2(in + n - 1) (m + n - -2) - (m - 1) (n- 1), r + l = 1 (2mn - n - n + 1) (2mn - m - mn) - 1) (n - 1), c=(m+ 1)(n+ 1)+1.

Page  63 THE CHARACTERS OF PLANE CURVES 63 22. The simplest form of the (m + n)-ic consists of an (mn + n)-an, (m + n) (mn -1) points united at the centre of one pencil and (m + n) (n - 1) points united at the other centre. The joins of a point to the locus are m + n tangent-joins and (m + n) (m + n - 2) cut and cusp-joins. The deficiency is zero, Kc = m + - n-2, t = m - -2, + K =7 T((m + i - 1) (mi + n - 2), T + -(vt + l n - 1) (mn + n - 2), c = 2 ( + n) + 1. 23. The analytic expression for the simplest form of the (mi + n)-ic is the lineal +e p l of t i- The pointal of this is &(m+n) (n-i) (m+n,) (an-l) {,-,mlnlnm+n _ [_ (m + n)]m+n fm t'n} which exhibits the nature of simplest form of the (m + n)-ic as explained in the preceding paragraph. 24. The shape of the simplest form of the (m + n)-ic is, when m and n are both odd numbers, an oval having tangents at the centres of the pencils meeting the curve in m + 1, n + 1 points respectively and, when one of the numbers mn, n is even and the other odd, a curve with an inflexion at one centre and a cusp at the other centre of the pencils. 25. The following results of a research into the characters of a proper curve of degree n and least class m0 for that degree were communicated to the Oxford Mathematical Society in 1896. 26. The least class n,, of a curve of degree (p - )p is p and of a curve of degree intermediate to (p - l)p and p (p + 1) + 1 is p + 1, except in the case of a curve of degree p (p + 1)- 1 the least class of which is p + 2. 27. The least deficiency Do of a curve of degree,n and of least class mo for that degree is 1 (n + i) - m0 + 1, i being 1 or 0 according as n is odd or even. 28. The least class mO' of a curve of degree n and of no deficiency is a (n + i) + 1, i being 1 or 0 according as n is odd or even. 29. If a is the curve of degree n of least class m0 and of least deficiency Do for that degree and class, and /i the curve of degree n, of no deficiency and of least class mo' for that degree and deficiency, the class mo' of 3 exceeds the class mo of a by Do, the number of cusps K0 of a exceeds the number of cusps Ko' of 3 by 3Do, the number of cuts 0' of 3 exceeds the number of cuts o8 of a by 4Do and the number of conditions which determine /3 exceeds the number of conditions which determine a by 2Do.

Page  64 64L W. ESSjON 30. The following tables give the values of the of degrees from 6 to 12. characters of a and /3 for curves a / *1l ii 8 9 10 11 12 144I) I6D) K4) TO4 Lo CO 13 1 0 9 0 0 9 4 I 4 10 1 1 11 4 1 8 12 2 0 12 4 2 10 16 0 1 12 4 2 16 18 1 0 13 5 2 24 19:3 1 15 4 i 3 28 24 0 0 14 6 4 0 4 6 3 0 11 5 0 8 7 5 1 13 5 0 12 9 6 0 14 9 6 0 18 10 9 1 16 1) 6 0 24 12 10 0 17 11 7 0 32 13 14 1 19 12 7 0 40 15 15 0 20 There is a similar theory of the least degree of a proper curve of the rth class. I P (3) p p X/ (2) q P - e- (i p Q R - 1q p ~~p (1 ) TI q q (1.2) r (1,2,3) 7 r~~~~~~~ Fig. 1. I. P determines p, one of a pencil of 4-ics which are 1.2-ans. ( determines a 4-ic q with a cut. The tangents (1), (2) to p become united on the join (1, 2) to the cut of q. The 12-an now consists of a 10-an and a 2-an, viz, two points united at the cut. II. Q determines q, one of a pencil of 4-ics each consisting of a 10-an and 2-an. 1 determines r with a cusp. The tangent (3) is united to the join (1, 2) to the cut

Page  65 THE CHARACTERS OF PLANE CURVES 65 which is now a cusp. The 10-an and 2-an is changed into a 9-an and 3-an, viz. three points united at the cusp. In the three cases the number of joins of a point to the curve is 12. 3 3 1 Fig. 2. A 4-ic with 2 cuts and 1 cusp is a 5-an and 7-an, viz. 2 points at each cut and 3 points at the cusp. There are 2 tangent-joins of the curve, 4 cut-joins to the curve, 6 cusp-joins to the curve, 4 joins of cut to cut, 12 joins of cuts to cusp, in all 28 joins. M. C. II. 5

Page  66 DIE ZENTI{ALPROJEKTION IN DER ABSOLUTEN GEOMETRIE VON MARCEL GROSSMANN. Bekanntlich wird in der Zentralprojektion die Lage des Zentrums gegen die Bildebene bestirnmt durch den Distanzkreis. Legt mian die euklidische Geometrie zugrunde, so spielt dieser Kreis bei den Konstruktionen eine doppelte iRolle: er nimmnt die Umlegungen des Zentrums mit Ebenen auf, die zur Bildebene normal sind, und er definiert das Orthogonalsysteim des Zentrums, da er konjugiert ist zn dem imaginaren Direktrixkreis des Polarsystems, weiches die Bildebene ans diesem Orthogonalsystemn schneidet. Will man die Methode der Zentralprojektion anwenden auf die beiden rbichteuklidischen Geometrieen, so hat man zn beachten, dass dem Distanzkreis diese doppelte Bedeutung nicht melir znkommt. 1st namlich A OB emn Dnrchmesser des Distanz'kreises, Z das Projektionszentrum, so ist der Winkel AZB kleiner oder grtsser als emn rechter Winkel, je naclidem man die Geometrie von Lobatschefskiy oder von Biemann voraussetzt. Triigt man also in der Ebene AZB in Z einen halben rechtenl Winkel an die Distanz OZ an, so wird dessen zweiter Schenkel die Bildebene ausserhaib oder innerhaib des Distanzkreises schneiden, je nachdem man die erste bezw. zweite der genannten Geornetrien voraussetzt. Das Orthogonalsystem des Zentrums wird also von der Bildebene in einem Polarsystem geschnitten, Plir weiches der reelle Reprdisentant emn dem Distanzkreis konzentrischer Kreis ist, dessen Radius grtisser oder kleiner ist als die Distanz, je naehdem man die Geometric von Lobatschefsky bezw. von Riemann annimmt. Dieser Kreis mdge der Orthogortalicreis des Zentrums genannt werden. Die Geometrie des Masses ist bestimmt, wenn man in einer gegebenen Bildebene den Distanzkreis D und den Orthogonalkreis K eines bestimmnten Pnnktes Z, der nicht in der Bildebene liegt, als gegeben voraussetzt. In Fig. 1 sci der Orthogonalkreis K grizsser als der Distanzkreis D, so dass die dadurch bestimmte Geometric hyperboliseli wird. Dann kann man leicht Geraden bestimmen, die zueinander parallel sind, d. h. Punkte des absoluten Kegelschnittes co der Bildebene konstruieren. Es seien nhmlieh AB und CD zwei zneinander rechtwinklige Dnrchmesser des Distanzkreises. Der llalbstrahl OG schneide den Orthogonalkreis in E. Die Tangenten, die den Distanzkreis in A bezw. D bertihren, schneiden sich in einem cigentlichen Piunkte F, sofern die Distanz d, was imme r mtiglich ist, so gewahlt wird,

Page  67 DIE ZENTRALPROJEKTION IN DER ABSOLUTEN GEOMETRIE 6 67 dass der Orthogonalkreis emn eigentlicher Kreis ist. Denn da die Winkel FOD und EBO je gleich 7r sind, so sind die rechtwinkligen Dreieeke EDO und EOB kongruent,.1 1 ' 4 weil OD = BO = d ist. Also ist auch DE = OE =,a, wenn der Radius des Orthogonalkreises mit a bezeichnet wird. Da nun das Viereek A ODE drei rechte Winkel hat, so schneidet der Orthogonalkreis-nach Lobatschefslcy-die Seite AE in einem Punkte G der Parallelen dureli 0 zu DE, so dass der Schnittpunkt U von OG und DE ein Punkt des absoluten Kegelselinitt-es ist. Der Winkel DOG ist daher der zur Distanz d gehiirigc Paralleiwinkel HI (d). Fig. 1. In der Zentralprojektion der euklidischen Geometrie werden die Geraden uind Ebenen des Raumes durch ihre Spur- und Flucbtelemente bestimmt. Die fundamentale Bedeutung, die der unendlich-fernen Ebene bei dieser Bestimmung zukommt, ist in der Zentralprojektion der nichteuklidischen Geometrie einer Ebene zuzuweisen, deren Auswahl nach Zweekmaissigkeitsgrflnden zu erfolgen hat. Als besonders geeignet empfiehlt sich die absolute Polarebene des Projektionszentrums. Es bildet ftir die Durchfiihrbarkeit der Konstruktionen kein Hindernis, dass diese Ebene in der hyperbolischen Geometrie uneigentlich ist. Ftir den Spezialfall der enklidischen Geometrie geht die genannte Ebene in die unendlich-ferne Ebene fiber. Die Zentralprojektion des Schnittpunktes einer Geraden mit der absoluten Polarebene des Zentrums m~ige der Eixpunkt der Geraden genannt werden. Die Zentralproj ektion der Schnittgeraden einer Ebene mit der absoluten Polarebene des Zentrums mtige die Eixlinie der Ebene genannt werden. Die gemeinsame Normale der Spur- und der Fixlinie einer Ebene geht durch den Mittelpunkt des Distanzkreises (Hauptpunkt). Die bisher getroffenen Festsetzungen ermdglichen die Entwieklung der Konstruktionen der Zentralprojektion nach einheitlieher Methode ftir alle drei Geometrien, d. h. unabhhngig von der besonderen Form des Parallelenpostulates. In diesem Sinne in6ge von einer absolutteni Zentralproqjektion gesprochen werden.

Page  68 68 68 ~~~~~MARCEL GROSSMANN Die descriptiven Aufgaben der absoluten Zentralproj ektion werden ebenso geliist wie in der Zentralprojektion der euklidischen Geometrie. Als Beispiel ftir die Behandlungmretrischer, Aufgaben diene das Normalenp robilem. In Fig. 2 sei der Distanzkreis D grisser angenommen als der zugehiirige Orthogonalkreis K, so dass eine elliptische Mlassbestirnmung) definiert wird, bei weicher eine Unterscheidung von eigentlichen und uneigentlichen Elementen wegfhillt. Eine Gerade g sei gegeben durch ihren Spurpunkt Sq, und ihren Fixpunkt Qg' so dass (Z) 9 ).1.1.11 Rf -X \\ 195 — I -, I I \ I K1 \ I I I I 11 -I I I I I I I I I I I I I I I I, \ I \1 I \\ I Fig. 2. g'-S~Qj ihre Zentralprojektion ist. Alle Ebenen, die zur Geraden g normal sind, bilden emn Ebenenbiischel, dessen Scheitelkante die zur Geraden g im absoluten Polarsystem konjugierte Gerade 1 ist. Die projizierende Ebene von 1, d. h. die Normalebene dureli Z zu g, entspricht im absoluten Polarsystem dem Punkte Qg, ist also normal zum Fixstrahl ZQg'. Man lege daher die Ebene ZOQg' in die Bildebene um, tibertrage die Reehtwinkelinvolution urn (Z) auf den Distanzkreis, bestimmie den Pol J dieser Involution, und konstruiere die Normale (Z) 1? zuim Fi~xstrahl (Z) Q,'. Die Spur der gesnchten projizierenden Ebene geht dnrch 1? normal zu OR und ist die Zentralprojektion der konjugierten Geraden 1.

Page  69 DIE ZENT1RALPR{OJEKrTION IN DERl AI3SOLUTEN GfEOTMETRIE 6 69 Da nun aber die Beziehung zwischen den beiden konjugierten Geraden g und 1 eine gegenseitige ist, so liegt der Fixpunkt Qe' der Geraden I in der Normalen Zn durch den Hauptpunkt 0. Zur Bestimmung der Geraden 1 ist nur noch der Spurpunkt S, zu konstruieren. Die absolute Polarebene dieses Spurpunktes ist aber die normalprojizierende Ebene der Geraden g. Ziir Bestiminung der Spur dieser Ebene, d. h. der Normalproj'ektion der Geraden g, bestimmt man die Normalprojektion des Punktes Q~, indem man in der vorhin bentitzten Umlegung die Normale von o auf (Z) R zieht, die den nmgelegten Fixstrahl (Z) 9)' in der Umlegung (Q9) schneidet. Die Normalprojektion Q,* von Qq liegt in der Normalen von (Q,) auf OQq'. Die Gerade SqQJ* ist die Normalprojektion g* der Geraden g, und der Spurpunkt SI von 1 liegt in der Normalen von 0 auf diese Gerade. Damit ist die Gerade I durch Spur- und Fixpunkt bestimmnt und das Normalenproblem bis auf deskriptive Konstrnktionen erledigt. Die Durchftihrung dieses Beispieles liisst erkennen, dass die metrischen Aufgaben der absoluten Zentralprojektion nur ltisbar sind bei ausgiebiger Verwendung projektiver Methoden. Eine ausfifihrliehere Darstellung der auf die Zentralproj ektion gegriindeten Konstruktionen der absoluten Geometrie gedenke ich in BdIlde zn veriuifentliehen.

Page  70 ON THE CHARACTERISTIC NUMBERS OF THE POLYTOPES e e... e e,- S(n + 1) AND ee.... e e,,_, Mo OF SPACE S. BY P. H. SCHOUTE. Introduction. In my "Analytical treatment of the polytopes regularly derived from the regular polytopes" (Verhandelingen of Amsterdam, Vol. XI, No. 3) I show how it is always possible to deduce the characteristic numbers of any expansion or expansion and contraction form derived from one of the three regular polytopes, simplex, measure polytope and cross polytope, of space Sl, by passing first to the symbol of coordinates. But I am quite well aware of the fact that this way is an indirect one, and that it is to be considered as a desideratum to find a short cut leading from the expansion and contraction symbols to the characteristic numbers immediately. As to the tracing of this short cut I have tried what I could do myself with respect to the case of the measure polytope Me,; but my results are so unsatisfactory that I cannot advise anyone who is in a hurry to be at home in time for dinner to follow that short cut. If ap,,, represents in general the number of limits (l)p of p dimensions of a polytope (P), of n dimensions I find in the case of successive expansions for el Mn el e21Mn ao, = 2 (n + 1)2 a0oi 6 (n + 1)3 a),b = {(I - P) + (n + 1a l)_ a,, = 3n (nt + 1)3 P = 1, 2,..., mn pn = {(n p)- p) + p + ) ( m-) + 1} ( + m ao, = 24 ( + 1)4 an =, 12n (n + 1)4 a.,,, = 2 (2n2- 6n + 7)(ni + 1)4 a = {(a - p)3 + (p - 1) (n -p)2 + ( (p- 2p + 9) ( l- p) + 1} (,I + 1)l,+ p = 3, 4,..., n. el e e ee41 M a,,, = 120 (n + 1)5 a,,, = 60n (n+ 1)5 an = 10 (2n - 7n + 11)(n + 1), a,, = 5 (n3 - 8n2 + 23n - 22) (n + 1)

Page  71 ON THE CHARACTERISTIC NUMBERS OF THE POLYTOPES 71 c,, = {( - p))4 + I (3) - 7) (n - p)3 + -_L (7p2 - 35) + 66) (1 - p)2 + - (p)3- 10p2 + 49p - 36) (n - 1) + 1} (no + 1)~,+ p=4, 5,..., n from which results it is not yet an easy task to deduce the general laws. In the hope that another mathematician may bring this general investigation to a close I will confine myself here to the case that all the expansion operations are applied to the regular cell, and try to find the characteristic numbers of these polytopes in function of the number of dimensions n, though also here the easiest way to success is the indirect one which passes by the symbol of coordinates. As the result of the application of all the expansion operations to the polarly related measure polytope and cross polytope is the same we have to consider two polytopes only, viz. ele2... e_2 en-_ S (n + 1) and ele,... e,_2 en-_ A n. 1. By means of the method indicated in the memoir quoted above we find the results deposed in the two following tables: el e2... en2_ ej,_1, S (l + 1) it a0 a, o a2 a3 a4 a5 a6 a7 a8 2 6 6 1 3 24 36 14 1 4 120 240 150 30 1 -5 7 120 1800 1560 540 62 1 |6 5040 15120 16800 8400 1806 126 1 7 40320 141120 191520 126000 40824 5796 254 1 8 362880 1451520 2328480 1905120 834120 186480 18150 510 1 el e2... en- 2 en_-l n i ai a1 a2 a3 a4 a5 a6 a7 ac 2 8 1 3 48 72 26 1 4 384 768 464 80 1 5 3840 9600 8160 2640 242 1 6 46080 138240 151680 72960 14168 728 1 7 645120 2257920 3037440 1948800 595728 73752 2186 1 8 10321920 41287680 65802240 52899840 22305024 4612608 377504 6560 1 In these tables the unit always indicates the polytope itself.

Page  72 72 P. H. SCHOUTE 2. From the results contained in these tables we can deduce a simple relation between the characteristic numbers placed in a horizontal row and those placed in the immediately preceding one, a relation remembering the principal property of the number triangle of Pascal. So we deduce in both cases the results for n = 4 from those of n = 3, beginning with the unit at the right-hand side, as follows: S(n + 1) Mn 1=1 1=1 30 = 2(14 + 1) 80 = 3.26 + 2.1 150 = 3(36 + 14) 464 = 5.72+ 4.1 240 = 4 (24 + 36) 768 = 7.48 + 6.72 120 = 5( 24) 384 = 8.48 or in general: for S (a + 1)... a+l,n = (a -p) (ap,_l + ap+,ln-_)........................(1), for A1,,... apl,n = 2 (n - p) - 1 ap,,2- + 2 (n - - 1) ap+,,n-1...(2). It is the chief aim of this communication to prove these relations, hitherto of an experimental character. 3. After having communicated the experimental laws (1) and (2) to Mrs A. Boole Stott, entreating her to send me a general geometrical proof of them, I received within a month her considerations dealing with the deduction of the.characteristic numbers for n = 2, 3 and 4 from those for n = 1, 2 and 3 respectively. As her study forms the basis of my general analytic proof I feel myself bound to communicate her results first. I will do so by treating in detail the transition of hexagon to tO in the case of relation (1) and that of octagon to tCO in the case of relation (2). Deduction of tO. If on a hexagon ABC... (Fig. 1) situated in a horizontal plane a we construct a tO-or, if one likes, if we place a tO with a limiting hexagon A B Fig. 1. ABC... on the table-it is at once evident that the vertices of this polyhedron lie in four horizontal planes a, a a2, a3,, and that the last plane which is uppermost contains a limiting hexagon AB3C3... equipollent to ABC..., whilst the intermediate

Page  73 ON THE CHARACTERISTIC NUMBERS OF THE POLYTOPES 73 ones divide the tO into three slices of the same thickness. Now from this diagram we can deduce immediately the following relations number of faces = 2 (1 + 6) number of edges = 3 (6 + 6), number of vertices = 4.6 J as the relation (1) demands. We treat each of these limits for itself. Faces. The original face ABC... presents itself in two parallel positions ABC... and A3B3C..., giving 2 x 1. Moreover between each edge of ABC... and the corresponding edge of A3B3C3... stand two inclined faces, so between AB and A3B3 the hexagon ABBB2AA, and the square A2B2B3A3, between BC and B3C3 the square BCCB1 and the hexagon BCIC2C3B3B2, giving 2 x 6. So we find 2 (1 + 6). Edges. Each edge of ABC... presents itself in three parallel positions, so AB in the positions AB, A2B2, A3B3 and BC in the positions BC, BIC,, B3C3, giving 3 x 6. Moreover from each vertex of ABC... to the corresponding vertex of A3B3C7... leads a broken line consisting of three inclined edges, giving 3 x 6. So we find 3 (6 + 6). Vertices. Each vertex of ABC... presents itself four times, giving 4 x 6. Deduction of tCO. If on an octagon ABC... (Fig. 2) situated in a horizontal,A, A B Fig. 2. plane a we construct a tCO-or if we place a tCO with a limiting octagon ABC... on the table-it is at once evident that the vertices of this polyhedron lie in six horizontal planes a, a,, a2, as, a4, a5, that the last plane which is uppermost contains an octagon A5B5Cs... equipollent to ABC..., whilst the four intermediate ones divide the tCO into five slices, all but the middle one of the same thickness. Now from this diagram we derive the relations number of faces = 2.1 + 3.8 number of edges =4.8+5.8, number of vertices = 6.8 J in accordance with relation (2).

Page  74 74 1'. H. SCHOUTE Faces. The original face ABC... presents itself in two positions ABC... and AB.C,..., giving 2 x 1. Moreover, between each edge of ABC... and the corresponding edge of A5BCs... stand three inclined faces, so between AB and A5B5 the square ABBIA1, the octagon AB,B2B3B4AAA.,A and the square A4B4B5A5, between BC and BAC, the hexagon BCCGC2B2B1, the square BJC2CJB2 and the hexagon B3C3C4C5B5B4, giving 3x8. So we find 2.1+3.8. Edges. Each edge of ABC... presents itself in four parallel positions, so AB in the positions AB, A1B1, A4B4, A5B' and BC in the positions BC, B2,C, BC3, B5C,, giving 4 x 8. Moreover, from each vertex of ABC... to the corresponding vertex of A5,BC... leads a broken line of five inclined edges, giving 5 x 8. So we find 4.8+5.8. Vertices. Each vertex of ABC... presents itself six times, giving 6 x 8. 4. Analytic proof of relation (1). The symbol of coordinates of the polytope ee2... e,_-S(n + 1) of space Sn is (n, n -,..., 1, 0). This symbol shows that the vertices of the polytope lie in the n + 1 parallel spaces x,0, = 0, x, = 1,..., Ix,+ = i, and that these spaces Sn divide the polytope into n slices of the same thickness = 1. In each of these spaces lie n! vertices, so in x,,+ = k the vertices for which moreover (n, n- 1,..., +, k+, k- -2,..., 0) = x, X2,..., ~n. Only for the extreme spaces xn+i = 0 and xn+1 = n these vertices are the vertices of a limiting polytope n(, - I,..., 1) = (n - 1, n - 2,..., 0) - ee2... en-2 S (n). For shortness we indicate the polytope ee2... e,,1_ S (n + 1) itself by the symbol (P)n and its limits situated in the spaces,,+ = 0 and x,+,= n by (P)(1" and (P)('^ 7 —1 6 1 respectively. In connection with the geometrical considerations of Art. 3 we suppose, in order to facilitate the reasoning of the analytic proof of relation (1), that the limits (P)^l1 and ((n) of (P)\ are horizontal, (P)(o) being at the bottom and (P)-)1 at the top, which includes that all the other limits (1)ni_ of (P)n are inclined, and now we prove the following lemma: "In (P)n every limit (1)(0) of (P)(_) is joined to the corresponding limit (1) (') of (P)1_) by a chain of n-p inclined limits (I)p+, with the property that any (l)p+, of this chain is in contact with each of the two immediately adjacent ones by a horizontal limit (l)p equipollent to (1)() and (1)(n, which enables us to consider the n-p limits (1)p+l of this chain and the n-p + 1 limits (1)p formed by (1)(), the n-p-1 intermediate p-dimensional limits of contact and (1)() as the limits (l)p+, and (l)\ of (P),, deduced from the chosen limit (l)(0) of (P)O) "We find all the limits (l)p+I of (P)n by joining together the system of the sets of n -p inclined limits (1)p+l deduced from every limit ()(0) of (P)(O) and the system ( (P),n1, and the system of the sets of i - p horizontal limits (l)p+I deduced from every limit (1)(~) of P(o) " X-pd-1 V f l —i In order to prove this lemma we select any limit (1)(0 of the polytope (P)"1, represented itself by X,1+1 = 0, (XI,..,..., = 1, 2,..., m,

Page  75 ON THE CHIARACTERISTIC NUMBERS OF THE POLYTOPES 7 75 by following Theorem III of our memoir quoted above, i.e. by writing its extended symbol with the it - p + 1 syllables in the form =r+ `0- (""I, X2,..., Xk)= 1, 2,..., -, (Xhk+l, Xk,+2 ' ' +k2) = C~ 1 + 2,..,k~k - wvhere the indices 1I of the coordinates xi have beeni chosen so as to suit this arrangeinent. If this symbol is abridged to alid we maintain tacitly the order of successio'~ 1 1.. x~b of the coordinates, the symbol of the corresponding (1()of (P)("') becomes n (0,., -1e1-1) (ki,, 01 + 1c2 -1)... (k1e+1'2+ +k 11... n-i), and now, in accordance with the first part of the lemma, we have to look out for a limit (1l)~ not entirely situated in x,,1- = 0 of which (1)(O) is a limit. Evidently we p get this (l)p+l by putting the digit 0 inside the brackets of the syllable (1,...,1c), giving us (0, 1,..., Ik1) (k + 1,., 1C ~IC2)... (k_1+ 1i2~...,,_~ 1 ) of which (1l)~l the limit (l)p opposite to (l)(o) is represented by p we represent this (1)p by the symbol (1)(k,) after the value k1, of the horizontal space )p"I )1 = 1-, bearing it. So, in order to pass f 1rom (l)(o) to the first (l). equipollent to it on the way towards (1)(fl) mentioned in the lemma, we have to leap from x,,+1 = 0 to 11 = 1e,, and now it is once more evident that the second limit (1)p~, passing through (1(-)is obtained by putting the new digit k, of x,,~1 inside the brackets of the syllable (k-1 + 1,...,I k-1 + k2), giving US for that new (1),+ (0,... Ik11- 1) (1c1, k1+ 1,., k-1+ k2)... (k-1+1IC2 +... +k — ) and for its (1)p opposite to (1)(l which may be represented by the symbol (1)(kl~I,), Going on in this manner we find the n - p + 1 equipollent horizontal limits )(1$, (k)1) (k)(l+k2)., (k1) +k2+-~ki-j-), (j)(kl+k,2+...+k~tp-1) leading to the n - p inclined limits (1l)~l mentioned in the first part of the lemma; so this part is proved. Any limit (1,+ of (P),,L is either horizontal or inclined. In the first case it lies in the space xn+1 = q, where q represents one of the digits from 0 to n, which proves that it belongs to one of the systems of n - p horizontal limits deduced from a certain limit (l)(o) of (p)()_. In the second case xn+, is contained under the coordinates )p~ 1 lcorresponding to a certain syllable (r, r + 1,.,s - 1, s) of the extended symbol of (1)p+1, and then this (l)p~ admits two horizontal limits (l). which can be got, by

Page  76 76 P-. ~H. SCHOUTE putting either the minimum digit r or the maximum digit s outside the brackets in the form r (r-il,..s-1s), s (r, r+ 1,..s-1 and taking r in the first and s in the second case for the value of x,,m+; so any inclined limit (l)p+, of (P),, belongs to one of the systems of n -p inclined limits deduced from a certain limit (1)(o) of (P)0 Evidently the lemma proved now includes the relation (1); so this relation is proved now too. 5. We now prove the following theorem: "If a,,,, represents the number of the limits (l)p of (P)n we have = 2 (2n - 1), an-,, n= 3 (3m - 2. 221 + 1), an-.-,, n-4(4 _' -3.3n +3.2n" - ), an-,, 2= 5 (5n - 4. 4n ~ 6. 3m, - 4. 2"' + 1), and in general p =(p + 1) ( — 1) (P)k (p-k + 1)..................(I), k=O for the values of 1, 2,..., n of p." We prove by means of (1) that the relation (I) holds for n as soon as it holds for * n -1. But as (I) deals with a__,l instead of a,,,, we transcribe (1) by substituting n - q for p + I in the form an-tj1n = (q + 1) (anq...1, n-i + a21.q, iii), or if we replace y by p a, = (p + 1) (p + a2.12,1). (1'). Now, when (I) holds for n - 1 we have = (P + 1) J( )k (pXk (p-h + 1~' k=O p-i a(n,1)-(p.1), n-1 =p P (_ 1)k (p -1ik (p - k)n-i. k=O In order to facilitate the addition of these two equations we transform the latter by the substitution J = o' - 1, after which we drop the dash of ic' and replace the expression p (p - I)k-i then making its appearance by k ((pk; so the second equation becomes n-i = - ~~ (~J)k J (p) p-J +1T1 T1'he1 by combining the terms with (p)k (p - Jo + 1)n-I of both forms, which compels us to take the term with Jo =0 of the first equation by itself, we find an.-2..l21.1 + ap'- = $ i/t(p + 1) - Ic} (P~ (P - Jo + 1pn1 + (p + 1) (p + 1)22-i - J)k (P)k (P -k0+l)n~+(P l+1 k=1 k=0

Page  77 ON THE CHARACTERISTIC NUMBERS OF THE POLYTOPES 77 So by means of (1') we get anp,n- = (p +1) E (- 1)k (p)k(p — + 1)n k=O which is the equation (I) of the theorem for n. As the relation (I) holds for n = 3 we find by application of the result deduced just now that it holds in general for any positive integer value of n. Remark 1. By means of the general results a0,- = (n + 1)! and a,,,, = n (n + 1)! we find by the way the relations E (-1)k )k (n) - k + IP1 = ~!, k=O n-1 (- 1)k (n -l)k(n-) = 1 ( + l)!. k=O Remark 2. According to the preceding developments the determination of an-p,n is a problem of the theory of permutations and combinations. If we call a row of subsequent integers a group of subsequent numbers this problem can be stated as follows: "Given the n+l numbers 1, 2,..., n +. To find all the possible ways of splitting up these n + 1 numbers into p + 1 groups of subsequent numbers, to determine how many times r the numbers of each of these sets of (p + 1) groups can be distributed differently over n + 1 quantities x,, x2,..., xn+, and to evaluate 2r extended over the different sets of the total system." 6. Analytic proof of relation (2). The symbol of coordinates of the polytope ele2... e-_l M of space S, is [1 + (n - 1)2, 1 +(n- 2) 2,..., 1 + \/2, 1]. This symbol shows that the vertices of the polytope lie in the 2n parallel spaces x = + (1 + k /2), where k assumes one of the values n-1, n- 2,..., 0, and that these spaces Sn^l divide the polytope into 2n - 1 slices, all of the same thickness /2 but the middle one comprised between x = + 1 which admits the thickness 2. In each of these spaces lie 2n- (n - 1)! vertices, so in =1 + k /2 the vertices for which moreover [1 +(n- 1) 2,..., 1+(k+ 1)/2, 1 + (k-1)2,..., 1 +2, 1]=Xl, 2,..., X2, Xn1, Only for the extreme spaces xi = + {1 + (n - 1) \2} these vertices are the vertices of a limiting polytope [1 +(n - 2) /2,..., 1] = ee2... en_2Mn_. For shortness we indicate the ele2... e._l M,, itself by the symbol [M], and its limits situated in the spaces n = + {1 + (n- 1) /2} by [M]+1 and [M]_, respectively. We also here suppose that the spaces x, = constant are horizontal, []f]_l lying at the bottom and [M]+ l at the top. Now we prove the lemma: "In [M]n every limit (1)<+ of [M]+_l is joined to the corresponding limit (1) of [M]_i by a chain of 2 (n- p) + 1 inclined limits (1)p+, with the property that any (l)p+l of this chain is in contact with each of the two immediately adjacent ones by a horizontal limit (1)p equipollent to (1)+ and (1), which enables us to consider the 2 (n - p) + 1 limits (1)p+, of this chain and the 2 (n - p + 1) limits (l)p formed by (1)+,

Page  78 78 P. H. SCHOUTE the 2 (n - p) intermediate p-dimensional limits of contact and (1)p as the limits (l)p+, and (l)p of [M]n deduced from the chosen limit (1)+ of [M]+_." "We find all the limits (1)p+I of [M], by joining together the system of the sets of 2 (n- p) + inclined limits (l)p+i deduced from every limit (l)+ of [M]+and the system of the sets of 2 ( - p) horizontal limits (l)p+ deduced from every limit (1)+. of [M]+ " The space S,,_ represented by x= 0 is a space of symmetry of [M]n. So, if starting from the limit (I)+ situated in x = 1 + (n - 1) /2, we have determined the limits (l)p+I of the chain mentioned in the statement of the lemma which are situated above xn = 0, the mirror images of these (/)p+l with respect to xn = 0 will belong also to the chain, and these two parts of the chain, the upper and the lower one, must be found to be connected by an unpaired limit (l)p+, which is its own mirror image, the space Sp+, containing it being normal to x,, = 0. Nevertheless we call this connecting link between the two parts of the chain inclined, as we use inclined here for "non horizontal" and not for "non perpendicular." So we find that the number of limits (l)p+, forming the chain must be odd, and all we have to do is to show that the number of these (I)p+l lying above x= 0 is n-p. Now according to Theorem XXX of the second part* of my memoir quoted above the extended symbol of the limit (1)+ selected can present itself in two different forms, which have to be considered successively, i.e. we can split up the n digits 1, 1+ V2,..., 1 (n- 1)1/2 either in n-p or in n-p + groups of subsequent digits and place all these groups between round brackets with exception of that group of the n-p + 1 which contains the digit 1, this digit having to be placed between square brackets. So e.g. in the example of the octagon [1 + V2, 1] =., %o in the upper plane x3 = 2 + V2 of the tCO-see the diagram of this polyhedron-we find edges with two different kinds of symbols, i.e. edges (1 + /2, 1) and edges [1] 1 +'2; here AB5 belongs to the second kind, B5C, to the first, and so on alternately. First case (of n-p groups). Writing for short 1 + k, - 1 /2 instead of 1 + (k - 1) /2 we represent the (l)+ selected by the extended symbol (1,.., I + ] -'1 v'2) (1 + k, V/2,..., 1 + i+c,-l V'2)......(1 +, + ++ kn-p-. V2,..., I + n- 2 /2) 1 + n- 1 V/2, where the groups of digits refer successively to the sets of positive coordinates (XI... Xl) (. 1+,~, Xk k2) '+ (X l+ke-a...+kn-p-l+l) "~ ) Xn-l) Xl the subscripts i and the signs of the coordinates xi having been chosen so as to suit this arrangement. Then we get the first (1)p+I of the chain, i.e. the (1)p+, passing through the selected (1)+, by bringing the digit 1+ n- 1/2 inside the brackets of the syllable (1 + ki + k2 +... + kep_- - /2,..., 1+ n- 2 /2) and its second horizontal - This part will appear in the beginning of 1913.

Page  79 ON THE CHARACTERISTIC NUMBERS OF THE POLYTOPES 79 (1)p by taking the smallest digit 11 + A + k2 +... + k-p-1 '/2 placed at the other side as value of xn out of the brackets. So the first new horizontal limit is represented by the symbol (..., 1 + kl,- 1 V2) (1 + kl V2, I..., 1 + / + k-1 2)... (...(1+ 1 - 2+...- +kn-l+1 2,..., 1 -+ - 1 12)1 + k, + - 2+... + Ic- -1 2, the order of succession and the signs of the xi being maintained. Continuing this process by bringing the new value 1 +k + k, +... + kn-p 2 inside and taking 1 + ic, + k2 +... +- k,-p, /2 out of the brackets of the syllable (1 + 1- + t,- + *.+... —2 /2,..., 1 + + k.2 -... + le-p-~1 /2), etc., we find for the values of xc, for (I)+ and the successive limits (1)p of contact lying above x, = 0 + n-1 V2, 1i + k + k2 +c...+ — _l 2/2, 1 + k + k2 +.. + knp_2 /2,.., 1 + l /2, 1. So there are on either side of xn = 0, (I)+ and (1)- included, n - p + 1 horizontal limits (I)p, and therefore in toto 2 (n- p) + 1 limits ()p+l composing the chain, etc. Second case (of n-p + 1 groups). Under the same notation we start here from the extended symbol [1,..., 2](1+c2..., I +k- ] (1,..., l+ +2-1 2)...... (1 + k, + c2 +... + knp /2,..., 1 + - 2 /2) 1 + 2-1 /2. By following closely the same process we get above x, = 0 for xc the n-p values 1 -+ n-1 \/2, 1 + ]2 + k +... + kn-p N/2, 1 + I+ - ]C +... - _p_ /2,..., + k-] V/2, also leading to 2 (n -p) + 1 limits (1)p+,, etc. Remark. The difference between the two cases (with n -p and with, -,p + 1 groups), already indicated above for the edges of an octagon, greatly influences the character of the components (1)p+: of the chain. In the first case of t -p groups these components agree in this with the components of the chain of Art. 4 that they are prismotopes the constituents of which are exclusively polytopes ele,... ek_, S (k + 1) for different values of k; on the contrary in the second case these components are prismotopes one of the constituents of which is always an ee,... ek_, Mk for a certain value of k*. The proof of the second part of the lemma, though it also breaks up into the two different cases distinguished above, can be copied from that given in Art. 4. 7. We now prove the following theorem: "If ap,,, represents the number of the limits (1)p of [M],, we have n-l= 3n - 1, a,n,,, = 5n - 2.3n + 1, a-,_3n = 7n - 3. 5 + 3. 3n - 1, a_, n = 9n - 4. 71 + 6 5 - 4. 31 + 1,............................................. In the two diagrams of tO and tCO the square partakes in all the chains of faces, as the square belongs to both categories.

Page  80 80 P. H. SCHOUTE and in general an-p,= (- 1)' ()k {2 (p - ) }..................(II), k=O for the values 1, 2,..., n of p." We prove by means of (2) that (II) holds for n as soon as it holds for n - 1. But to that end we have to transform (2) into an-p,n = (2p + 1) an-p-1,,_l + 2pan-L,,_n-................. (2') first. Now, when (II) holds for n - 1, we have (2p + 1) ~a.__-P, _n- = (2p + 1) E (_ 1)k (p)k {2 (p - k) + I}fn-1, k=O p-1 2pa(n-1)-(-),n-1 = 2p (- i)k (p - l)k {2 (p - k)- 1}nk=0 In the second of these equations we put = k'- 1, drop afterwards the dash of k', and replace p (p - 1)k- by k (p)k; so we get for the second equation 2pan._,n_1 =- 2 I (- 1)k k (p)k f2 (p - k) + 1}.-1. k=1 Then by combining the terms with (p)k {2 (p - k) + l}n-, and taking the term with k = 0 of the first equation by itself, we find (2p + 1)a _-1,,/-1 + 2pan-p, t-_ = E (- )k {(2p) + 1) - 2kI (p)k {2(p - k) + l}n1 + (2p + 1) (2p + 1)n-l k=l = (-1)k (p)k {2 (p-k)+ 1}n + (2p + 1)' k-l k c0 So by means of (2') we get a,,, n = E (- I)k (p)k {2 (p - k) + 1}12, k=O which is the equation (II) for n. As (II) holds for n = 3 it holds for any positive integer value of n. Remark 1. By means of the general results a,,, = 2n. n! and a,n = 211.n.i! we find by the way the relations (-l)k (n{2 (n - k)+ I}'1 =2n. kc=0 n-1 E (- 1)k(nI -l)k 2 (n - k)- -}1 = 2n-1. n. n! k=O Remark 2. We leave it to the reader to interpret the determination of an-_,, in this case of ee... e_-, M as a problem of the theory of permutations and combinations.

Page  81 CONFORMAL GEOMETRY BY EDWARD KASNER. INTRODUCTION. The connection of conformal transformations with the theory of functions of a complex variable is so important that the geometry based on the group of conformal transformations has been studied almost exclusively as an auxiliary to the theory of functions. Conformal geometry, that is the study of those properties of geometric configurations which are invariant under all conformal transformations, has therefore not yet been developed into a systematic theory, comparable, for example, with projective geometry. The former theory is naturally more difficult than the latter, since it is based on a larger group. It would seem that it deserves study for its own sake, and also as an example of a geometry based on an infinite continuous group, instead of (in Lie's terminology), a finite continuous group. The simplest configurations to investigate in conformal geometry are curves and sets of curves. The curves considered will throughout be assumed to be analytic: we shall confine ourselves in fact to regular real analytic arcs. The conformal transformations are assumed to be regular in a region including the curves considered. A single curve has no invariants; any curve may be transformed conformally into any other curve, in particular into a straight line. The first configuration of real interest is composed of two curves having a common point, that is, a curvilinear angle. The conformal transformations operating on the angle are assumed to be regular in a region including the vertex of the angle. The result of such a transformation is a new curvilinear angle, having the same magnitude as the original. This common magnitude we shall denote by 6: this is the radian measure of the angle formed by the tangent lines at the vertex. It is a differential invariant of the first order since it depends only on the slopes of the curves. If two curvilinear angles are conformally equivalent, they certainly have the same 0, but is this sufficient? If two angles are equal in magnitude, that is have the same 0, will they necessarily be conformally equivalent? This is certainly a fundamental question in conformal geometry. We shall show that the answer is not always in the affirmative. Curvilinear angles exist, which while equal in magnitude, are not conformally equivalent. For example: it is possible to construct a curvilinear angle formed by two analytic arcs, intersecting orthogonally, which can not be transformed conformally into an angle formed by a pair of perpendicular straight lines (the transformation to be of course regular at the vertex). 6 M. C. II.

Page  82 82 EDWARD KASNER This fact indicates the existence of other invariants besides 0. The principal problem of the present paper is to find these absolute invariants of higher order. The invariants considered are differential invariants involving curvatures and derivatives of the curvatures of the sides of the curvilinear angle. Only invariants of finite order are discussed. The existence of invariants of infinite order, that is, those involving all the coefficients in the power series representing the sides of the angle, is not settled. It is shown that if 0 is a rational part of wr (in which case we shall term the angle rational), there exist higher invariants. If the angle is irrational, it is shown that no higher invariants exist. Among the rational angles we have to include the case where 0 is zero: such an angle is termed a horn angle. Each type of horn angle has a unique invariant whose order depends on the degree of contact of the sides. The question as to when two curvilinear angles are conformally equivalent is not yet completely solved. In order that two angles shall be equivalent, it is necessary that not simply 0, but the higher invariants obtained in this paper shall be equal. Whether this is also sufficient depends on the existence of invariants of infinite order. At the end of the paper we consider briefly the corresponding (dual) problem of equilong geometry. CONFORMAL INVARIANTS OF CURVILINEAR ANGLES. THEOREM I. An irrational curvilinear angle has no conformal invariants oj higher order; that is, the magnitude 0 of the angle is its only invariant. To prove this consider two angles of the same magnitude 0, where O/qr is irrational. Without loss of generality we may assume the angles in reduced form, that is, one side a straight line. Thus the angle in the plane z = x + iy is formed say by the axis of reals and the curve y = ax + ax2 +.................................(1); and the angle in the plane Z = X + iY is formed by the axis of reals and the curve Y = A X +A2 X2+..............................(2). By assumption ac = A = tan 0. To shew that there is no other invariant relation involving a finite number of higher coefficients of (1) and (2), we prove that a regular conformal transformation Z = C1 + C22 +................................(3), (where the c's are real and c1 = 0), converting (1) into (2), can be (formally) found. For the determination of the c's, we have an infinite number of equations. The first is an identity in virtue of ac =AI and allows c, to be arbitrary. The second involves cl and c2 and is linear in c2. The Kth involves cl up to cK and is linear in cK. The coefficient of c, can be put into the form sin /c - tan 0 cos c0............................. (4). This cannot vanish for any integer c > 1; for if it did 0 would be a rational part of rt. Hence for arbitrary values of ac, as,..., at and Al, A2,..., An values of C1, C2,..., Cn can be determined: in fact in c 1 ways since cl is arbitrary.

Page  83 CONFORMAL GEOMETRY 83 For arbitrary curves (1) and (2) the series (3) can be formally calculated, the coefficients being rational functions of the parameter cl. Probably for some value of cl the corresponding series (3) will be convergent. This, however, the author has not succeeded in proving. If there are actually cases where (3) is divergent for all values of cl, this would indicate the existence of invariant relations involving all the coefficients of the curves, that is, invariants of infinite order. If the angle is rational, 0i/r=p/q, where the fraction is supposed to be reduced to lowest terms, some of the expressions (4) will vanish, namely those for which K is q + 1 or 2q + 1 or 3q + 1, etc. The series (3) cannot usually be found even formally. Conditions of consistency arise. The discussion of the system of equations for the c's is complicated, but by a certain device we can prove THEOREM II. Every curvilinear angle of rational magnitude has a conformal invariant of higher order. For this purpose, we consider the process of general symmetry, or reflexion, with respect to an analytic curve. The function-theoretic definition of Schwarz may be stated in purely geometric language as follows: two points are symmetric with respect to a given curve provided the pairs of minimal lines determined by the points intersect on the given curve. For an analytic arc a neighbourhood (region) can be found such that each point in the neighbourhood has a unique symmetric point or image in the neighbourhood. The process is covariant under the conformal group. It is fundamental in our geometry. Consider then an angle of magnitude 0 with vertex V and curved sides a and b. The image of a with respect to b is a definite curve c passing through V. The magnitude of the angle b, c is of course 0. Take the image of b with respect to c, etc. If 0 is irrational we shall obtain curves passing through V so that their tangents are everywhere dense. But if 0 is rational we shall after a finite number of steps arrive at a curve having the same direction as a. Hence every rational angle determines a unique horn angle, i.e. angle of magnitude zero, formed by two curves in contact at the vertex. Any invariant of this angle will be an invariant of the original rational angle. It now remains to prove THEOREM III. Every horn angle has one and only one higher conformal invariant. If the order of contact of the two sides of the horn angle is h - 1, so that they have h consecutive points in common, we shall say that the angle is of type h and denote it by HI,. Of course the integer h is invariant under conformal transformation as it is under all contact transformation. The simplest type (ordinary contact) is H,. Consider two angles of type h. Taking one side in each plane to be the axis of reals, and the vertex as origin, the curved sides are of the form y = Ca xh + ah+lxh+l +..............................(5), Y= AhXh + Ah+lX h...........................(6). Expressing the fact that the transformation (3) converts (4) into (5), we are led to a system of equations for the determination of the coefficients cl, c,.... It is sufficient 6- 2

Page  84 84 EDWARD KASNER to note that the Kth equation involves the first K coefficients and is linear with respect to CK, the coefficient of this term being (/c- h) ah...................................(7) By assumption ah does not vanish, therefore expression (7) vanishes when and only when K = h. The first h equations thus involve only c1, C,..., ch/_i: eliminating these we find a relation involving ah,..., a2h-l and Ah,..., A2h-1. There are no other relations since no other eliminations are possible. The order of the unique invariant of a horn angle of type h is 2h - 1. We denote the invariant by Ih-. For a horn angle H, (contact of first order), the invariant in question is 3a3 -22.......................................8). This is, of course, for the reduced form, in which one side is the axis of reals. For the general form where both sides of the angle are curved, we find d7,1 d72 ds( ds, I = ( _ )...............................(9, (7I - elI) where l7, y2 denote the curvatures of the sides (at the vertex), and s1, s2 denote the arc lengths. This is the simplest example of a conformal invariant of higher order. Every horn angle can be reduced conformally (so far as terms of finite order are concerned) to the normal form in which the sides are y = 0 and y = xh + \2h-~l. The constant X is then essential, being equivalent to the invariant I2h_-1 Return now to the discussion of rational angles. Such an angle determines a horn angle by the process of successive symmetry described above. If this angle is of type h, its invariant I^h-_ will be an invariant of the rational angle. This proves Theorem II. In general if the ratio of 0 to vr is p to q (in lowest terms) then the type h equals q + 1 so that the invariant is of order 2q + 1. However, for some rational angles, h may have a greater value. It is necessary to classify rational angles of a given magnitude first according to the values of the arithmetic invariant h, and then according to the values of the geometric (differential) invariant I2h-_. A question remains. Can a rational angle have more than one higher invariant? The related horn angle has only one invariant as stated in Theorem II. The horn angle is determined by the rational angle, but is the converse true? We are led to the following general problem of conformal geometry: given two curves a and b through a point V, find n - 1 curves through V so that the image of a in the first is the second, the image of the first in the second is the third, and so on until finally the image of the penultimate in the last is b. This may be termed the equipartition problem for curvilinear angles. Obviously if the solution exists the magnitudes of the angles formed by the successive curves will all be equal. The simplest case, n = 2, is the problem of bisection of a curvilinear angle: to construct a curve c so that the given curves a and b shall be symmetric with respect to c. Judging from ordinary angles, where the sides are straight lines, we should expect two solutions, an internal and an external bisector. This is, however, not

Page  85 CONFORMAL GEOMETRY 85 always true. We state only the following results. If the given angle is a horn angle for which the sides a and b have contact of exactly the first order, then there will be an internal bisector, but no external. If the sides of the horn angle have contact of exactly the second order, then both internal and external bisectors exist*. Consider now a curvilinear angle of magnitude 7r/2. The image of the first side in the second will be a curve having contact of even order with the first side. For a general right angle the contact will be of second order, that is, the related horn angle is of type 3. Right angles such that the type of the horn angle is 5, 7, 9,... are of greater and greater specialty. In the extreme exceptional case, the order of contact is infinite, that is, the right angle is such that each side is its own image with respect to the other side: this kind of right angle is conformally reducible to a pair of perpendicular straight lines. A right angle of general type has a unique invariant; this is of the fifth order. This follows because the related horn angle, of type 3, uniquely determines the right angle. The invariant in question is the Ih of the related horn angle. If the original right angle is given in reduced form so that one side is y= 0 and the other is x =/3 +/Y33 +..., the invariant is found to be 3. - 322 The exceptional right angles arise when 8, vanishes. Probably for each type of rational angle, there is only one higher invariant. EQUILONG GEOMETRY. This theory is a sort of dual (not however the direct projective dual) of the conformal theory. It is based on the infinite group of equilong transformations introduced by Scheffers in 1905. These are related to the theory of functions of dual numbers u +jv, where j is an imaginary unit whose square is zero, just as the conformal transformations are related to functions of the ordinary complex numbers x + iy, where the square of i equals - 1. Dual numbers are interpreted geometrically by directed straight lines, using Hessian line coordinates; curves are considered as envelopes of straight lines and are also oriented. Any curve may be transformed into any other curve; in particular, any curve may be transformed into a point (of course, a point is considered as the envelope of all the straight lines passing through it). A single curve, therefore, has no invariant. The first configuration to be studied consists of two curves having a common tangent line. This is in fact the dual of the figure discussed in the first part, namely: a curvilinear angle composed of two curves having a common point. The configuration now to be studied has not been given any special name. It possesses an obvious invariant with respect to the equilong group, namely: the * In general if the type of the horn angle is even, no external bisector exists. If the type is odd, an external bisector probably exists.

Page  86 86 EDWARD KASNER distance measured on the common tangent between the two points of contact. The question arises as to whether the configuration has additional invariants, that is, invariants of higher (but finite) order. A special case arises when the distance between the points of contact is zero. In this case the curves have a common tangent and a common point, and therefore form a horn angle. A horn angle arises in both theories since it is in fact a self-dual configuration. The following results are obtained: THEOREM IV. The figure formed by two curves having a common tangent and distinct points of contact has no higher equilong invariant. THEOREM V. If the points of contact coincide, so that the figure is a horn angle, then there will be one, and only one, equilong invariant of higher order. In this, as well as in the previous discussion, the existence of invariants of infinite order is left unsettled. The decision as to whether such invariants exist depends on questions of convergence of certain power series which can be formally constructed. For a horn angle of type h, the unique invariant is of order 2h - 1. It is of course different from the corresponding conformal invariant I2h-i and is here denoted by Jh-l. For the simplest type, h = 2, the equilong invariant is dr, dr2 dO1 dO2 (r - r2)2 where rl, r', denote the radii of curvature and 0,, 02 denote the inclinations of the two curves to any fixed initial line. Of course the values of the radii and the derivatives are taken at the vertex of the angle. CHARACTERIZATION OF THE TWO GROUPS. From the usual points of view conformal transformations and equilong transformations are characterized in entirely different ways. Conformal transformations are singled out from all point transformations by the requirement that the angle between two curves at a common point shall have its magnitude preserved. Equilong transformations, on the other hand, are singled out from all line transformations by requiring that the distance between two curves measured on a common tangent shall be preserved. We shall show that both groups may be characterized in the domain of all contact transformations in terms of their behaviour with respect to horn angles. Every contact transformation turns a horn angle into a horn angle. It is sufficient to consider the simplest type, where the contact is of first order. Conformal transformations leave unaltered a certain invariant of the third order, discussed in the first part and denoted by Is. Equilong transformations leave unaltered the invariant denoted by J,. Both of these quantities are differential expressions of the third order; they are in fact combinations of the curvatures of the sides of the horn angle, and of the rates of variation of the curvatures.

Page  87 CONFORMAL GEOMETRY 87 THEOREM VI. Conformal transformations are the only contact transformations for which the 13 of every horn angle is unaltered. Equilong transformations are the only contact transformations for which the J3 of every horn angle is unaltered. It thus appears that the conformal group may be characterized without mentioning angular magnitude, and the equilong group may be characterized without mentioning distance. We note that the two invariants may be put into the form d201 d202 dS d22 kds- ds2J d2sl d2s2 d o — 2 dO2 J3 ds ds2' kdO, dOd which differ only by the interchange of the letters s and 0. It is to be remembered that there is no (known) automatic principle by means of which we can pass from the results of conformal geometry to those of equilong geometry; we have to deal with a general analogy rather than a strict duality. What is, for example, the analogue of isothermal systems of curves, which are so important in conformal geometry? The resulting systems are defined in terms of Hessian line coordinates by linear differential equations of the first order and their theory is essentially simpler than that of isothermal systems.

Page  88 SUII LES SUJRFACES ISOTHERMIQUES PAR G. TzITZE'ICA. 1.M. Wiiczynski a trouve' une nouvelle me'thode pour e'~tude des propri'te's infinite'simales des courbes, des surfaces et des congruences de droites. En se bornant aux propri~ ~ poetesiiaris comme base une 4quation diff~rentielle i'n'air dans le cas d'une courbe plane on gauche, un syste'me d'e4quations line'aires ordinaires dans le cas des surfaces re'gle'es, et aux de'rive'es partielles dans le cas des surfaces generales et des congruences. D m on ce~ ind endam ent de M. Wilczynski et en partant d'un probl'm particulier, j 'avais trouve un syste'me d'e'quations aux de'rive'es partielles qui de'finissait toutes les surfaces se de'duisant de l'une d'entre cules par des transformations affines. On est conduit d'une manie're naturelle 'a un principe general que l'on pent enoncer de la mnanie're suivante. Etant don~ne'e une classe de figures du plan on de l'espace-courbes, surfaces, congruences-pour 6tudier leurs proprie'te's inflnite'simales qui restent invariables lorsqu'on, applique a' ces figures toutes les transformations d'un, groupe, il faut d'abord trouver un syste'me approprie6 de coordonne'es pour la ditermination d'un 6e'lment quelconque de ces figures, former ensuite une e'quation ou vn syste'me d'e'quations differentielles ve'rifie4 par les coordonne'es d'itn e'lenent variable d'une figure et de toutes les figures transforme'es. Pour e'tudier, par exemple, les proprie'te's infinite'simales affines des courbes planes, on choisit d'abord des coordonne's carte'siennes obliques x et y et si, pour plus de simplicite', on prend parmi les transformations affines celles qui laissent invariable l'origine, on aura comme equation diff~rentielle de'finissant une courbe plane et ses transforme'es une equation de la forme O"~ PO'~q O =0..................(1), p et q e'tant des fonctions donne'es du parame'tre t variable sur la courbe, O' et 9" de'signant les de'rive'es de la fonction inconnue 0 par rapport 'a cc parame'tre. Deux solutions x (t) et y (t) qui forment un syste'me fondamental de (1) de'finissent une courbe integrale. Les invariants et les covariants de le'quation (1) par rapport aux transformations qui ne changent pas les courbes int6grales, 'a savoir celles qui remplacent 0 par const. 0 et t par 0 (t), conduisent aux proprie'tes de ces courbes. 2. Je veuix faire une application du principe general que je viens d'e'noncer i le'tude des surfaces isothermiques, et montrer spe'cialement que l'on obtient par cette voie d'une manie're re'gulie're le'quation aux de'rive'es partielles du quatrie'me ordre trouvee par MM. Rothe et Calapso.

Page  89 SUR LES SURFACES ISOTHERMIQUES 8 89 On sait que toute surface 'a lignes de courbure isothermes garde cette proprie'te apres une transformation conforme de l'espace. Le groupe de transformations que nous conside'rerons sera done le groupe G des transformations conformes de l'espace. Commne les transformnations de cc groupe s'expriment par des transformations line'aires des coordonne'es pentasphe'riques, ce seront ces coordonne'es que nous emploierons dans cette e'tude. Comme enfin les lignes de courbure d'une surface restenlt lignes de courbure sur la surface obtenue par une transformation du groupe G, nous prendrons comme coordonne's curvilignes ut et v sur une surface isothermique ses lignes de courbure. Cela e'tant pose', ii re'sulte que le syste'me ve'rifie' par les coordonne'es pentasphe'riques x1 2., t d'un. point mobile sur une surface isotherrnique doit 6'tre line'aire. Une des 4quations de ce syste'me est connue depuis longtemps, elle est due 'a M. Darboux; c'est une equation do Laplace 'a invariants e'gaux que nous supposerons prise sons la formo re'duite a2O_. O...................(2). Cependant, ii ost clair quo cette equation ne pent suffire pour de'finir une surface isothermique et ses transforme'es. II faut ajouter 'a le'quation (2) un. noinbre d'eSqua~tions suffisant, pour que le syste'me obtenu puisse de'finir seulement cinq solutions line'airement inde'pendantes. A cet effet nous remarquons que si les fonctions xi (i = 1, 2,...,1 5) de it et v sont connues, nous pouvons determiner cinq fonctions a, b, c, d, e, de manie're que I'equation line'airo a:38O D20?rO Mo MJ =a -— + + c +d-+ eO..........(3) admette les xi coinmo solutions particulie'res. En effet, en introduisant ces fonctions 'a la place de 0 dans e'~quation (3), on obtient pour determiner a, b, c, d, e un. syste'me line'aire, qui n'est impossible on inde'termine' que dans le cas oii I'on. aurait aa12 aV2 aa av X 0() le premier membre e'tant un. determinant dont on obtient les cinq lignos, en faisant i= 1, 2.. 5. Or, cette relation signifierait que les xi ve'rifieraient une 6quation de la formo a2 a20 aM aM A — ~ +B +0C + D-+EO=0..........(5), ajt2 av2 am av ce qui est impossible, le systeune forme' par esqatns()t(5ne povat vir plus de quatre solutions line'airement inde'pendantes et dans ce cas la surface serait uno sphe're. On de'montrerait do la me'me maniere que lPon peut determiner cinq autres fonctions a', b', c' d' et e', do manie'ro quo los xi ve'rifient aussi e'~quation a38e,a20 a'o lao aM = a a2 + c'~ + d' ~-+ e'O..........(6). Il ost aise' do voir maintonant quo le syste'me forme' par los equations (2), (3) et (6) admet cinq solutions line'airement inde'pendantos.

Page  90 90 G. TZITZEICA On peut donc dire que une surface isothermique et toutes celles qu'on en d6duit par des transformations conformes de l'espace sont definies par un systeme de la forme de celui forme par les equations (2), (3) et (6). 3. Cependant, il faut remarquer que les coefficients h, a, b, c, d, e, a', b' c', d', e' ne sont pas des fonctions arbitraires de u et v. Ils doivent verifier deux sortes de conditions. D'abord celles qui resultent de la relation bien connue 2 = 12 + x22 +... + 52 = 0.....................(7), ensuite les conditions d'integrabilite. En partant de (7) on en deduit d'abord ax ax x =O, = =~ )............... (8), de plus ' l'aide de (2) on a a tax a / f-I -0 --- -1=0. av \aU/ = a ( = v. Nous laisserons de cote le cas ou l'on aurait (ax\2 ax\2 S a- = 0, aV) = 0. On pourra done changer les variables u et v, sans changer la forme des equations de notre systeme, de maniere que l'on ait 2 \(a} = 1' e(~a=v) = 1...........................(9). On tire alors de (8) a x a2 x ax ax 2.X -=-1, 2x-=-1, 2 =0...............(10) au 2 av2 au av et de lia,x x-=o0, Vz a-=0 au3 ') av~ ou a+ b=0, a'+ b'= 0...........................(11). On aura de 11me111,a ax 3x ax a2x -avu 0a2 U aV =....................... - et de (9) a9 a2x ax a9x( ^ 9 v av- a, o........................(13). a- a,6 2;:~ a a=o De (12) on tire aisement ax a3 ax a3x Y, ---- = h, C- -=h av au3 au av"x d'ou d = c'= h...................................(14). On a aussi 2 au 2 a 0' u) + =0, Ea2) +d'=0.........(15).

Page  91 SUR LES SURFACES ISOTHERMIQUES 91 Enfin de celles-ci on obtient ou2 ad' _2ac +2e, = 2b'd' + "e av am' au av' ah = - bd'- e, ah - a - e av ou Nous avons trouve le premier groupe (I) de relations entre les coefficients. Ce sont les relations (11), (14) et (16). 4. Il nous reste a former les conditions d'integrabilite. Nous ecrirons d'abord que l'on a a /a30\ a2 / a 20 av Y3) au2 KauaV) et en remplacant les derivees donnees par les equations (2), (3) et (6) on obtiendra une equation du second ordre qui doit etre verifide identiquement. On deduit les relations Dc ab ah ac h = +a'b, a+ +bb'+d=, 2 = ah +bc' + av av 9v au av m. ad a2h ah ae 7 bd + e+ =, = a - + be' + ch + (v 'aU2 ada av De mene, en 6crivant que l'on a a a930 = a2 a av au a@v3) - aV2 asuoa)V on trouvera les relations aa' ab' ac' aa'+c' + - =O, h=a'b+ -, a'ce+ e'-= 0, au am......a ad' ah a2h ah ae( t'd + b' + = 2 — = ae + b' + dh + au av' av2 v a u Les relations (17) ct (18) d'integrabilite forment le groupe (II) de relations entre les coefficients de notre systeme. 5. Nous pouvons simplifier ces relations en remarquant d'abord que des relations (17) et (18) on tire h = + ab' + a'b........................... (19). av + a' On pourra done poser a I --, -, - - - l ( a 'u i av' e6tant une nouvelle fonction inconnue. A l'aide de (11) on a aussi 1 a, ' aI, - - ' a =- a-; et alors (19) donne 1 a2q h =- q auav

Page  92 92 G. TZITZEICA Si done on connaissait la fonction b, on connaitrait par la mneme les coefficients a, b, d, h, a', b', c'. II nous reste a determiner les coefficients c, e et d', e'. Or on a les relations )c 2c + a c = h 2c ab + ~2e, -2 -2 - - 2e', = l +u av au > dav ad' _ah 2d' 3 2 ad' 2d' 3-.= 2 =2 2e, = + 2e', du ov a on a' v 4 av dont on tire iinmmediatenment c + d' = 2mb2, m = const. ah ae Prenons maintenant la derniere relation de (17) et remplacons-y a et v par leurs valeurs tirees de (16). On obtient ainsi a2h,, a 2i ad' ab - = - a (a'c + e') + be' + ch - - b -- d' — ah2 av2 av av' o h 8,h a a1 ~bS ou a + =h +. ao - (c + d'), aU2 avd2 2 atU aV ou enfin a2 (1 a a /21 a12 2 a2 a av av2 a a av/ a av (...............(20) qui est precisement l'equation de MM. Rothe et Calapso. Si ) est une integrale de cette equation, c est determine par ac a=4m ah ac ai= -=4m - - 2... ---=2.?a aou av av au ou, a cause de (20), la condition d'integrabilite est verifiee. Une fois les coefficients connus le systeme est determine et completement integrable. Rermarques. 1. Dans l'equation (20) on ne peut avoir n = 0 que dans le cas d'une sphere. 2. Les groupes (I) et (II) de relations sont necessaires pour notre probleme. Nous n'avons pas demontre qu'ils sont aussi suffisants. Je le ferai a une autre occasion. 3. Les proprietes des surfaces isothermiques s'etudieraient en formant les invariants et les covariants du systeme verifie par les xi par rapport aux transformations 'ff= kO, u'= + + ', v'= + v + k", k, k' et k" etant des constantes arbitraires.

Page  93 THE PEDAL LINE OF THE TRIANGLE IN NONEUCLIDEAN GEOMETRY BY DUNCAN M. Y. SOMMERVILLE. 1. Consider a triangle ac8y, and let the equation of the absolute referred to it be, in point coordinates, (A, y, Z)-( -a h g x, y, Z)2=0 h bf g f c and in line coordinates, (V7,r, )2 n A IS G,, -= H B F G F C the capital letters denoting, as usual, the cofactors of the corresponding small letters in the determinant a h g h b f g f The polar of the point (x', y', z') is (AZx,, zx', y, z')= 0, and this is also the condition that the points (x, y, z), (x', y', z') should be a quadrant distant; the pole of the line (a', a', ~') is c(v~,, ~'~ n:',X ', l')= 0o, and this is also the condition that the lines (:, 7, 0), (,', y', b') should be at right angles. The poles of,y7, ya, a/3 are a -(A, H, G), ' - (H, B, F), y'=(G, F, C). The equations of the altitudes aa', /3f/', yy' of the triangle are Gy= Hz, Hz= Fx, Fx = Gy. The triangles ao/y, a',/3y' are in perspective, the centre of perspective being the common orthocentre 0 - (, H ). The axis of perspective LMN is the polar of 0

Page  94 94 DUNCAN M. Y. SOMMERVILLE with respect to the absolute and its coordinates are (,i,, ). We may call this the orthaxis of the triangle since it cuts each side in a point distant a quadrant from the opposite vertex. 2. Take any point P (x,, yy, z1) and let the perpendiculars Pa', P/3', Py', meet the corresponding sides of the triangle a/3y in X, Y, Z. For certain positions of P the points X, Y, Z will be collinear and the line X YZ is called the pedal line of the point P. We shall call the locus of P the pedal or orthocentral locus, and the envelope of XYZ the pedal or orthocentral envelope of the triangle afcj. Reciprocally: take any line p (i, ql, 5i) and let it cut the sides of the triangle a'/'y' in X', Y', Z'. For certain positions of the line p the lines aX', 3 Y', yZ' will be concurrent in a point P. We shall call the envelope of p the orthaxal envelope and the locus of P the orthaxal locus of the triangle a/,y. Evidently the orthaxal locus and envelope of the triangle a,83y are the orthocentral locus and envelope of the triangle a'f'y'. If P is any point on the pedal locus, P (aa'//3',yy') is a pencil in involution, and conversely.

Page  95 THE PEDAL LINE OF THE TRIANGLE IN NON-EUCLIDEAN GEOMETRY 95 Let these rays cut the corresponding pedal line in X1, X, Y1, Y, Z1, Z, and let aX, cut fy in K. Then P (aa', f'V') = (X,X, YZ) = (KX, y3) = P (aa', y) = P (a'a, 3y) which proves the theorem*. Conversely, if P (aa'ff3'iyy') is a pencil in involution and Pa', P3', Py' cut fly, ya, a/3 in X, Y, Z, then X, Y, Z are collinear. P (a', 'y') = P (a'a, /y) = P (aa', y3) = (KX, y/3) = a (PX, 7yP). Now Pa and aP coincide, and Pa', P13', P7' cut aX, ay, a/S in X, Y, Z, therefore X, Y, Z are collinear. In exactly the same way it is proved that if Pa, P'f, Py cut the corresponding sides of the triangle a',/'y' in X', Y', Z', then X', Y', Z' are collinear. Hence if P is any point on the orthocentral locus it is also on the orthaxal locus, and conversely. Hence the orthaxal and orthocentral loci coincide, and so also do the envelopes. 3. To find the equation of the pedal locus. The equation of Pa' is x (Gy - Hz,) + y (Az, - Gx) + z (Hx, - Ay,) = 0, and this cuts f8y in X, whose coordinates are 0: -(Hx - Ay): (Az,- Gx,). The equation of the locus of P is therefore (Cy- Fz) (Az - Gx) (Bx - Hy) = (Fy - Bz) (Gz - C) (Hx - Ay), or A yz (hy + gz) = 2 (fgh - abc) xyz, which represents a cubic curve passiig through the nine points of intersection of the lines ay', la', 7yf' with afi', fy', ya', i.e. through the points a, S, y, a', 18('), Call the last three points U, V, W. Then aU, 3V, TyW are concurrent in the point whose coordinates are (AF, BG, CH). The following points of the cubic which are in the same row or the same column are collinear: V W y' a' M a,8 N Therefore LVW, and similarly MWT and NUV are sets of collinear points. The equation of the cubic may also be written (x y z\ A B C0\ ABC-Babe +f + f gy + y +- h + =- g2h2 xyZ. The curve therefore passes also through L, M, N. * It is otherwise obvious since ap3yXYZ is a complete quadrilateral.

Page  96 96 DUNCAN M. Y. SOMMERVILLE If ABC= Aabc the cubic breaks up into a straight line (the orthaxis), and a conic circumscribing the triangle a/3ry. This condition is also the condition that the triangles aboy, a','"y' should be inscribed in the same conic. This conic, which exists even when the cubic does not degenerate, touches the cubic at a, 83, y. 4. The tangents to the cubic at a, a', U, L meet in a point on the cubic. The following points of the cubic which are in the same row or column are collinear: a a a' a' 3 y L ' By' L N M L N M L therefore the tangents at a, a' and L cut the cubic in a point L', the tangential of L. Again we have a L N 13' a' /3 ' U therefore aL and a' U cut in a point, R say, on the cubic. Similarly /3M and 13' V cut the cubic in S, and y7N and y' W cut the cubic in T. Again S T and S T Bfy L 3' r U MNL V NU therefore the tangent at U meets the cubic also in L'. Since therefore foul real tangents can be drawn from L' the cubic is bipartite, and is of the sixth class; it has nine points of inflexion, of which three are real and collinear. 5. To find the equation of the pedal envelope. Let the coordinates of the line X YZ be A, 7, f. Where this cuts /3y we have x, y, z = 0, - g,, and the equation of the line through this point perpendicular to /3y, i.e. through a', is x (Hr + G) - yAr - zA= O. The three lines will be concurrent if Hv + G~ - AD - A =0, - Be F + H - B~ _C' - C G- + Fq that is, the line-equation of the pedal envelope is Aa^i (Hv + G) = 2 (FGH - ABC), or (cq —.f) (a- g~) (by - ha) = (fq - b ) (g - c) (h a - a), or77 F /(Ha b 5 c+ A( Habc-ABC2 ) ljP & i P G ^ ^^ = 72G

Page  97 THE PEDAL LINE OF THE TRIANGLE IN NON-EUCLIDEAN GEOMETRY 97 which represents a curve of the third class touching the lines /,y, 7a, a,3, 13'y, y'a', a'/f' and the polars of U, V, W with respect to the absolute; it also touches the three altitudes aa', Bf', yy'. By the transformation (,, )=( c It,, yx/,,z), (, y,z)=( A. H G ~,, ~), b f H B F g f c G F C Cy - Fz becomes - A (h -ar), etc., and the point-equation of the pedal locus passes into the line-equation of the pedal envelope, i.e. the pedal envelope is the polar reciprocal of the pedal locus with respect to the absolute*. The envelope is therefore a curve of the sixth degree with nine cusps of which three are real, and the three cuspidal tangents are concurrent. If A BC = A abc and A $= 0, the pedal envelope breaks up into a point (the orthocentre) and a conic touching the sides of the triangle. 6. If A= 0, so that the absolute degenerates to two distinct straight lines through I, and the cofactors A, B, C, F, G, H do not all vanish, the polars of a,,, ly, all pass through I and the triangle a'/,'y' degenerates to the point I. PI is the pedal line of any point P, so that the orthocentral locus disappears and the orthocentral envelope reduces to the point I. The orthaxal locus and envelope are, however, definite. Let the equation of the absolute be (ax + by + cz) (a'x + b'y + c'z) = 0, or shortly, uv = 0. The polar of a (1, 0, O) is av + a'u = 0. Consider the line (I, rj, g) or w = 0; let it cut the polar of a in X' and join aX'. The equation of aX' is 2w u v 2aa'w = a (av + a'u) or =- + - a a and we get similar equations for f1Y' and yZ'. The condition that aX', tYY', yZ' should be concurrent in a point P gives the line-equation of the envelope of w, a a I I 1 1 = V b b' ~ c c which represents a conic touching the sides of the triangle afy and also touching the two lines u, v, i.e. the envelope is a circle inscribed in the given triangle. The envelope also includes the point uv. * This is already obvious since the orthaxal envelope, which is reciprocal to the orthocentral locus, coincides with the orthocentral envelope.,. C. II. 7

Page  98 98 DUNCAN M. Y. SOMMERVILLE 7. To find the locus of P(x', y', z'). Pa is yz'-y'z= 0 and this cuts av + a'u = 0, the polar of a, in x, y, z = (ab' + a'b) y' + (ac' + a'c) z', - 2aa'y', - 2aa'z'. The equation of the locus of P is therefore (ab' + a'b) y + (ca' + c'a) z - 2aa'y - 2aa'z = 0, - 2bb'x (be' + bc) + (ab' + a'b) x - 2bb'z - 2cc'x - 2cc'y (ca' + c'a) x + (be' + bc) y or a'u + av - 2aa'x - 2aa'y - 2aa'z = - 2bb'x b'u + by - 2bb'y - 2bb'z - 2cc'x - 2cc'y c' + cv - 2cc'z which represents a cubic passing through a, 3, y, and through the intersections of /8y (x = 0) and the polar of a (av + a'u = 0), etc. The equation may also be written a'b'cIU' + abcv3 = uv X (a2b'c' + a'2bc) x, which shows that the cubic has a double-point at uv, u and v being the tangents at the double-point, and also C (a2b'c' + a'bc)x = 0 is the line upon which the three points of inflexion lie. When the double-point is a crunode, so that a, v are real, two of the points of inflexion are imaginary; when the double-point is an acnode, with u, v imaginary, the three points of inflexion are all real. 8. The triangle a/3y and its orthaxis form a complete quadrilateral. Take the diagonal triangle as triangle of reference and choose line-coordinates I, V, r so that the coordinates of the sides of the triangle and the orthaxis are (- 1, 1, 1), (, - 1, 1), (1, 1,- 1), (1, 1, 1). The point-coordinates of the vertex a are (0, 1, 1). Let the line-coordinates of I be (ao, b,, Co), and consider the equation a, + bo +Co 0. This represents a conic touching the sides of the diagonal triangle. It also touches the line (_, -,-) which passes through a. The harmonic conjugate of this line with respect to a/3, ay is (b, - Co, - ao, a,) which is the line aL. Therefore this conic touches the sides of the diagonal triangle and the bisectors of the angles of the complete quadrilateral. The equations of the sides of the diagonal triangle in the former coordinates are?X X (ab' + a'b) y + (ca' + c'a) z = 0, /t Y- (b' + b'c) z + (ab' + a'b)x = 0, vZ (ca' + c'a) x + (be' + b'c) y = 0. Taking X (be' + b'c) = f (ca' + c'a) = v (ab' + a'b) we get _ Y: Y Z + Y Z=-X+ Y+Z:X-Y+ ZX+ Y-Z. be' + b'c ca' + c'a ab' + a'b

Page  99 THE PEDAL LINE OF THE TRIANGLE IN NON-EUCLIDEAN GEOMETRY 9 99 Then ax + by + cz = 2abc (X ~ + i'- + Z so that aX b' c'Z (t b c a bi c a 6b c a, b, c,, = aa' (b' c2 b2c'12) bb' (C'2 a2 - c2a'2): cc' (a'2b2 a-2b 12), and Yaocab'. abc' = aa'bb'cc'Y a2 (b'3c2 - b"C'2) = 0. Therefore the conic touches also the lines u and v and is therefore a circle. This corresponds in ordinary geometry to the nine-point circle, and we may therefore call it the mine-line circle. 9. The axis of this circle is the polar of the point I. Let us find its equation. ao bo co The pole of the line (lmm) with respect to + - ~ - = 0 is 4c (c., + bon) +, q (ci + amn) + ' (bol + a,mi) - 0, therefore the polar of (xyz) is 1, m, n = a0 (- aox + boy + coz), bo (aox - boy + coz), co (aox ~ boy - coz). The polar of (a0, b,, c,) is therefore a. (- a02 + b,0 + C02) X- + b. (a02 - b02 + C02) Y + Co0(a02 + b02 - c02) Z = 0. We have to express this in terms of x, y, z, a, b, c, a', b', c'. We find - a02 + b02 ~ c02 - (b2c'2 + b'c2C) (C2a'2 _ C'2a2) (a2b'O a'2b2). Hence we get 1 aa' (b~c'2 - b'"c~) (b~c'2 + b ca'-c) (a~b'" - a'2b2) (bc' + b'c) [(ab' + a'b) y + (ca' + c'a) z] = 0, or, leaving out the symmetrical factor,:x [bb' (c2a'2 ~ C'2a2) (ca' + c'a) (ab' + a'b) + cc' (a2b'" + a'2b2) (ab' + a'b) (ca' + c'a)] = "x (ca' + c'a) (ab' + a'b) [bb' (c2a'2 + c'2a2) + cc' (a2b'2 + a'2b2)] = Cx (ca' + c'a) (ab' + a'b) (bc' + b'c) (a 2b'c' + a '2bc) = 0, or, again leaving out the symmetrical factor, the equation finally reduces to Y (a2b'c' + a'2bc) x = 0, and so the line of imfiexions is the axis of the nine-line circle. 10. The three points of infiexion are mutually equidistant. We may write the equation of the cubic, u and v being the tangents at the double-point and w the line of infiexions, U3 + v3 = 3uVwV, or:3 = it" + v3~ + U- 3uvw = (I t +V + w) (U + &wv + a02W) (It + o) 2V+ WW). 7-2

Page  100 100 DUNCAN M. Y. SOMMERVILLE The last three factors equated to zero give the equations of the inflexional tangents. The lines joining the double-point to the points of inflexion are u + v = O +0, U + v = 0 + 0. The cross-ratios (i, v; u + v, u + wv), (U, v; u + ov, u + o2v), (u, v; u + w2v, u + v) are each equal to o, hence the distance between any pair of points of inflexion is 22i Xi -a 7 2r 2log e =- 3 or -3. 11. When the absolute degenerates to two coincident straight lines we have the case of Euclidean geometry. The orthaxal locus degenerates to the line at infinity, and the orthaxal envelope disappears. For the orthocentral locus and envelope we have the well-known properties of the circumcircle, the nine-point circle and the three-cusped hypocycloid, of which those in the preceding sections are reciprocals. 12. There are other cases in which the pedal locus and envelope degenerate: (1) When a side of the triangle touches the absolute this side forms part of the locus, and the point of contact forms part of the envelope. (2) When a vertex of the triangle lies on the absolute the tangent at this point to the absolute forms part of the locus, and the vertex forms part of the envelope. (3) When a side of the triangle is the polar of the opposite vertex, i.e. when the triangle has two right angles, the locus consists of this side and two lines through the opposite vertex; the envelope consists of the vertex and two points on the opposite side. 13. In a certain case the pedal locus breaks up into a straight line and a circle circumscribing the triangle. We have first the condition ABC = Aabc. The equation of a circle with axis Ix + my + nz = 0 is ax2 + by2 + cz2 + 2fyz + 2gzx + 2hxy = k (lx + my + nz)-. This circumscribes the triangle a/3y if a = kl2, b= k2, c= kn2. The equation then becomes (f- /bc) yz + (g - /ca) zx + (h - \/ab) xy = 0. In order that this may be the same as A B C, yz + zx+ xy =0, we must have afe2= b2 = ch2 = t, say. The equation of the cubic then becomes x,f _ (t -p)(2t +p) I' x p (t + p) where p =fgl.

Page  101 THE PEDAL LINE OF THE TRIANGLE IN NON-EUCLIDEAN GEOMETRY 101 (1) t=p=fgh. The point-equation of the absolute reduces to (= 0o, and the line-equation to f2/ - ghv - (ffi + grq + a2h ) (fa + o2gq + (h) = 0, i.e. the absolute degenerates to two coincident straight lines and a pair of imaginary points, and the geometry is Euclidean. (2) 2t+p=0. In this case the absolute need not degenerate, but the triangle must be equilateral. The equation of the pair of tangents from a to the absolute is Cy - 2Fyz + Bz = 0, or (p + t) h2y2 + 2pghyz + (p + t) g2z2 = 0. The angle a of the triangle is therefore i -p + V2( - (p +2 1 p log = cos2 p- ip (p + t) p+t When the cubic reduces, the angle is cos-1 2 = i cosh-1 2, the side is cosh-l (- 2) = cosh-l 2 + itr, and the vertices of the triangle are all outside the absolute. The pedal envelope reduces to fi th i.e. the orthocentre and the circle inscribed in the triangle. When the absolute and the degenerate cubic are represented by concentric circles on the Euclidean plane this leads to the following theorem of Euclidean geometry *: "If AB'CA'BC' is a regular hexagon inscribed in a circle, and P is any point on the circle, and if PA' cuts BC in X, PB' cuts CA in Y and PC' cuts AB in Z; then X, Y, Z are collinear on a line through the centre of the circle. And also if three parallel lines be drawn through A', B', C' cutting BC, CA, AB in X, Y, Z, then X, Y, Z are collinear and the envelope of the line XYZ is the circle inscribed in the triangle ABC." For A', B', C' are the poles of BC, CA, AB with respect to a concentric circle of radius -, and this circle is the absolute. * Cf. Edu. Times, Sept. 1912 (Question 17308). * Cf. Educ. Times, Sept. 1912 (Question 17308).

Page  102 SURt LES HESSIENNES SUCCESSIVES D'UNE COURBE DU TROISIEME DEGtE" PAR B. HoSTINSKY'. Soit x3 +-II+yZ3+z36mxyz =O...............(1) I'equation d'une courbe du trolsie'me clegre4 en coordonne'es homnogenes x, y, z. Les invariants de la courbe sent S ='fa(I - in), T= 1 - 20m),3 - 8rnt'; 1 T2 la quantite' r = --- est son invariant absoin. A chaque valeur de r, correspondent 12 valeurs (le mi; en general, si r est r"Iel, deux valeuirs de mt (deux courbes) sont re'elles et dix imaginaires. Repre'sentons par H0, le premier menibre de le'quation (1) et e'crivons Hk 6 D xyz) Nous obtenons ainsi une suite de courbes du troisie'me degre4 Ho, Hi, H2,... Hn...............(2), que nous a ppelerons Hessiennes snecessives; chaque courbe de la suite (2) est la ilessienne de la precedente et Fe'quation d'une quelconque de ces courbes aura toujours la, forme (1). Le proble'me que je veux traiter maintenant consiste 'a rechercher les conditions ponr que la suite (2) soit pe'riodique. En d'autres mets: Un entier n e'tant donne6, on cherche nne courbe Ho identique avec II, Supposons que cette condition soit remplie par une courbe H,. Les courbes Ho, Hi, H2...HIconstituent nn cycle d'ordre n; dans la suite (2) pour chaque ic, les courbes Hk et Hk~n seront identiques. La recherche des cycles d'un ordre donne' conduit 'a une equation alge~brique dent il est alse6 indiquer le mode de formation. On s'appuiera snr les formules bien connues de Salmon qui expriment les invariants Sk+i, Tk+, de Hk1en fonction de ceux de Hk et sur e'6quation 108Hk~l = 4Sk2 Hk + TkHk~1.

Page  103 SUR LES HESSIENN ES SUCCESSIVES DIUNE COIJHBE IDU TROISIELME DEGR~I 103 On trouve ainsi Bin = X,. Ho ~ Y, Hi HI X1,1 JfY. ne renfermnant pas les coordonne'es x, y, z. Pour que la courbe H,, soit identique avec, H,, it faut que.n = 0....................(.3). Yf,, est un polyno~me, homogene en S-l et T2; apres avoir divise' le premier -membre de (3) par une puissance convenable de S', on aura une edquation qui ne contiendra que l'invariant absolu. r de la courbe H,. L'e'quation ainsi obtenue sera une equation Abe'lienne. Car si elle admet la racine r"k, elle doit admettre en me~me temps la rac kine rk+,, rk'tant l'invariant absoin, de 11k. Mais on trouve que rki rk (2?k + 3)2 cc qui exprimie la proprie't6 caracte'ristique des equations Abe'liennes. Voici les re'sultats du calcul, dans les cas les plus simples, oii 1'ordre n dii cycle ne de'passe pas 4: I=1. Quatre courbes de'genere'es en triangles. =2. Six courbes harmoniques formant trois cycles du deuxie'm-e ordre; uil seul cycle est reel. C'est 'a Salmon que nous devons cet exemple inte'ressant. n = 3. L'6'quation (3) s'e'crit r2 + 3r + 3 =0. De'signons par r, r' ses racines. Tout cycle du troisie'me ordre est compose' de courbes ayant le iie'me invariant absolu, I (ou. r'). Les 12 courbes de l'iuvariant r- (on r-') se divisent en quatre cycles; il y a done huit cycles du troisie'me ordre ne comprenant que de courbes imaginaires. it= 4. L'e4quation (3) est re'ductible; on trouve (r2 + Gr~+6. (r 4-3r' — 21r - 27r- -9) =0. Ii y a deux espe'ces de cycles du quatrie'me ordre: (a) Soient r, r' les racines du facteur quadratique. Les 24 courbes dont l' invariant est r on. r' forment 6 cycles. Les quatre courbes composant un tel cycle auront resp. les invariants 1r, r' r, r'. Un seul cycle sera r'el. (b) Dh'sign-on's par pp' p11 "' les racines du facteur biquadratique. Les 48 courbes correspondantes forment 12 cycles; les courbes d'un me'me cycle auront resp. les invariants p, p', p", p"'. Deux cycles seront re'els. Si l'ordre nt du cycle de'asse 4, une discussion coinpiete exige un calcul assez long. Toutefois on pent de'montrer le the'ore'me suivant: II n'y a pas de cycle d'ordre n compose6 de courbes re'elles, si n est un nombre impair. La demonstratiou repose sur la remarque suivante: La Hessienne de (1) est represente'e par la meine equation (1) si l'on y remplace mn par &i2 n0

Page  104 104 B3. HOSTINSKY Une discussion de cette dernie're formule nous montre que, si la courbe originale n'a pas d'ovale, la Hessienne en aura un, et si la courbe originale est doue'e d'un ovale, la Hessienne sera sans ovale. Reprenons une suite de ilessiennes successives re'elles. Les courbes d'indicc pair H0, H2, H4... ont par exemple un ovale tandisque les courbes d'indice impair sont sans ovale. Si l'orclre d'un cycle n e'tait un nombre impair, on aurait H0 =H cc qui est impossible. Par consequent l'ordre d'un cycle reel doit e~tre toujours un nombre pair.

Page  105 GEOMETRISCHE MAXIMA UND MINIMA MIT ANWENDUNG AUF DIE OPTIK VON JOHANNEs FINSTERBUSCR. ~.UEBER DIE AUFL6SUNGS-MIETHODEN. 1. Geometrische Maximum- und Minimum-Aufgaben k6nnen entweder synthetiseli oder analytisch behandelt werden. Die synthetische oder reingeometrische Aufkisungsmethode ist durch die Anschaulichkeit und Eleganz ihrer L6sungen bemerkenswert, tragt aber vielfach den Charakter der Zufalligkeit, da sich, urn mit J. Steiner zu reden, CCnicht emn einziges gemeinsames Grundprinzip aufstellen lasst," wdihrend die analytisehe oder rechnerische Behandlung in der Anwendung der iDifferentialreehnung eine umfassende Methode besitzt, wie die erstere nieht ihres Gleichen hat. Dairnit soil aber nicht gesagt sein, dass ffir alle Aufgabengruppen die analytisehe Behandlung mit Wiherer Analysis allen anderen Methoden vorzuziehen sei. Selbst unter den rechnerischen Aufl~isungen eines Problems ist sie es bei weiteml ineht imimer. So ffihirt z. B. die Anwendung des bekannten Doppelsatzes: (Swuinnm-e Produk't a Bei gegebener Pd t von ii positivenm Grissen ist dere? mm~ a { r~ssten}, wven die nt Gr~ssen, alle eirnander gleich sind, tiills die Znrtickfuhrung eines Problems auf ihn gelingt, deshaib kitrzer zum Ziele, weil hierdureh die Anfstellung der zu differenzierenden Fiinktion tiberffiissig wird. Auf den Beweis dieses Doppelsatzes soll hier verziehtet werden *; doch sei erwaihnt, dass or fdr ni = 2 und n. = 3 auch reingeometriseh gefiihrt werden kann. FUr n = 2 steht er in Euklid's Eleinenten im vi. Buche als Satz 27 und bildet naeh Moritz Cantor " das erste Maximum, welehes in der Gesehichte der Mathematik nacligewiesen worden istt. " Ebensowenig m~elite ieh auf die vielen Anwendungein des Doppelsatzes naher eingehen. Nur des Zusammenhanges mit einer unten behandelten Aufgabe wegen, werde mit seiner Hilfe kurz bewiesen, dass unter alien Sehnenvierecken von gegebenem Um~fang das Quadr-at den grd6ssten, fmhait hat. Bezeichnen a, b, c, d die Seiten eines beliebigen Selinenvierecks vom gegebenen Umfange 2s, so folgt aus der bekaunten Hero nisehen Flachenformel F = Vs-)( )( )( ) Arithmetische Beweise finden sich z. B. in: Rudolf Sturm, Maximta und Z11iinihna in der eleinentareit Geonietrie, 1910, S. 1-4. t Moritz Cantor, Geschichte derIll at heinatik, i, S. 252.

Page  106 106 106 ~~~~JOHAN NES FINSTERBUSCH in der die Sumnie der vier Faktoren = 2s, also konstant ist, dass ihr Produkt und daher F zum Maximum wird, weun die Faktoren einander gleich sind, d. h. wenn das Sehnenviereck emn Quadrat ist *. Die synthetischeni Met hoden verdanken ihren Ausbau in erster Linie Jacob Steiner. "Zwei der bedeutendsten Abhandlungen, in denen er die Ergebuisse seiner langjahrigen Untersuehungen ilber Maximumt und Minimtun bei den Figuren;in der Ebene, attf, der Kugeijldche und im Raume ilberhauptt jniedergelegt hat," sind his heute der Ausgangspunkt ahfnlieher, wenn auch niclit ehen zahireicher Abhandlungen geblieben. 2. Tim Folgenden will ich aus mneinen Untersuchungen fiber denselben Gegenstand zunaclist ecinige hekannte Aufgahen aus der Geometrie der Vieleck-e in neuer Weise synthetiseh behandeln. Meine Beweise, die die bisher vorhandenen an Einfaehheit und Anschauilichikeit tibertreffen dtirften, enthalten zuigleich den Schliissel zur Lbsung anderer derartiger Aufgahen, z. B. der Steiner'schen Schliessungs-&itze ilber gespiegelte Lichtstrahl en. Als Hanptaufgahe hehandle ich die Veraligeineinerung des Problems der Minimal-A blenicung, die emn Lichtstrahl beim Durchgang durch eml Prisrna erfih~rt. Ehe ich inich zn den Anfgahen selhst wende, mdehte ich einige knrze Bemnerku~ngen Alber die synthetischen Met hoden, deren ich mich bediene, vorausschicken. Die zu ldsende Autfgabe kann als Thteorem oder als Problemn gesteilt sein. Wahrend die analytische Auflbisungsart in beiden FKillen durch das Yerschwinden des ersten Differentialquotienten die Bedingungsgleiehung des Extremums gewinnt und ferner durch das positive oder negative Vorzeiehen des zweiten Differentialquotienten erkennt, oh emn Minimum oder Maximum vorliegt, hehandelt die synthetisehe Methode beside Falfle verschieden: I. Beiin Beweise eiries T h o r e in s, wenn also die das Extreinum kennzeichnende Eigenschaft sehon bekannt ist, gehen wir von der Figur des Extreinuins aus und veraindern diese derart, dass alle Eigensehaften der Figur, abgesehen von der hesonderen des Extremums, erhalten bleihen. Durch Vergleichnng beider Figuren, deren einzelne entsprechende Gr6ssen um Endliches von einander ahweichen, wird die Figur des Extremums als solche nachgewiesen. Dabei wird zugleich entsehieden, oh cmn Minimum Oder Maximum vorliegt. II. Handelt es sich dagegen um ein Prohieni, soil also erst die noch unhekanute, das Extremum bedingende Eigenschaft gefunden werden, so sind es zwei Wege, die zuin Ziele fiihren: II a. Methode der gleichen, endlich getrennten Werte, wie ich sic nennen m~ichte. Die in der Aufgabe aufgeftihrten Eigenschaften reichen zur vollstiindigen Bestimmung der Figur nieht hin. Wir nehbmen deshalb als noch fehlendes Bestimmungsstiick einen heliehigen Wert der veranderlichen Grdsse, die zum Extremunm werden soil, hinzu. Dadurch geht, wenn emn Extreinum existiert, die hisher unbestimmite Aufgabe in cine immer mnehrdeutig hestirnite fiber. Aus zwei verschiedenen Ldsungen, in geeigneter Weise auf einander bezogen, ist nun abzuleiten, weiche Beziehung * Rudolf Sturm, a. a. 0., S. 25. t Jacob Steiner, J'Verke, Hi, S. 177-308.

Page  107 GEOMETRISCHE MAXIM'vA UND MINIMA MIT ANWENDUNG AUF DIE OPTIK 107 zwischen den Grdssen bestehen muss, wen di edn nlc vesheee Figuren zur Deekung gelangen sollen. Diese Bedingung ist die gesuchte, Eigensehaft des Extremums. Damit ist das Problem zum Theorem geworden. Die Art des Extremums wird nach I entschieden. II b. Methode der gYleichen, zusctrmenfallenden Werte oder Met hode des UnendlichKleinert. Man geht von der Analysis-Figur des Extremums aius und denkt sieh diese nur unendlich wenig verandert. Die Beziehungen beider Figuren drticken sieb lfl einer Gleichung aus, die die unendlich kleine Verainderung der veranderliehen Grdsse enthalt, deren Extremum gesueht wird. Wird diese unendlich kleine Veranderung gleich Null gesetzt, so erhalt man die Bedingungsgleichung des Extremums. Die Art des Extremnums wird auci hier nach I entsehieden. Diese Methode II b, eine geometrische Vorliiufeirin der Differentialreehnung, maeht demnach wie diese von dem Verschwinden der Aenderung einer verainderliehen Grdsse in der Nahe des Extremums Gebrauch. Deeken sich also hierin ihrem Wesen nach beide Methoden, die synthetische und die analytische, so ist jedoch bei remngeometrisehen Prohliemen die erstere der hdheren Analysis weit vorzuziehen, da sie vielmehr als diese geeignet ist, " das eigentliche Wesen oder die wahre Ursache des Maximums und Minimums auzugeben " (Steiner, a. a. 0., ii, 179). Das erseheint ganz natiilich und einleuchtend, wenn man bedenkt, dass bei der synthetischen Methode die Bedingung des Extreinums gleichsam organisch aus der Figur hervorgeht und so ihre wesentlichen Gr~issen direkt zu einander in Beziehung bringt; wahreuid bei der Differentialrechnung die zu Grunde gelegten Grtssen (Koordinaten) meist reeht fremndartig oder nur in losem Zusammienhange zu den wesentlichen Gr~issen der Aufgabe stehen, sodass es oft sehr schwer ist, aus der erhaltenen Bedingungsgleichung cine einfache Eigenschaft des Extremnums herauszulesen. "Die synthetische Methode hat noch in neuester Zeit zu einzelnen schdnen Ergebnissen geftihrt, welehe durch die Anwendung d,~r Differentialrechnung nieht gefunden worden waren " (Fi1e dlIer). Auch von den nun zu behandeinden Aufgaben hatte ich wohl schwerlich so einfache J3eweise und Konstruktionen gefunden, wenn ich inich nicht synthetiseher Methoden bedient hatte. 2. EiNIGE AUFGABEN _UBER VIELECKE. 8. fTsJitzwfklgf Dreieck ABC hat das Dreiecic der H~;heifatsspanlcte DEE den koleinsten Umifa ng. Von diesem Satze hat Sc hwvar z einen schdnen Beweis gegeben, der falschlicherweise Jacob Steiner zugeschrieben worden ist*. Mein ebenfalls reingeometriseher I3eweis hat nicht wvie jener 6 auf einander folgende Umklappungen oder Spiegelungell des Dreiecks ABC niutig, sondern nur je eine um die Seiten AB und AC (Fig. 1). Geht man von einem beliebigen Punkte X der Seite BC aus und sind X, und X, die ihm entsprechenden Spiegelpunkte, so hat jedes beliebige eingeschriebene Dreick X YZ den geknickten Streckeuzug X1 YZX2 zum Umfang, also cinen grdsseren als das Dreick X Y0,Z,, dessen Seite Y0)Z0 auf der Strecke XX2 liegt. Da je zwei Seiten des Hdhenfusspunktdreiecks DEE mit einer Seite des gegebenen *Vergleiche: J. Steiner, Ges. TV., ii, Anmerkung auf Seite 728 und H. A. Schwarz, Ges. W., ii, 349 (nach iR. Stu rm, a. a. 0., Seite 90).

Page  108 108 108 ~~~~JOHANNES FINSTERBUSCHI Dreiecks, gleiche Winkel einschliessen, also symmetriseh Zn ihr liegen, was beim vorigen Dreleck nur an den Ecken Y0, und Z,, der Fall war, so ist dessen Umfang DEFD, gleich der Strecke DID2, wenn D1 nnd D, die Spiegelpnnkte vom THihenfusspunkt D bedeuten. Werden nun die Punkte X1 und X2 auf D1 D, orthogonal projiziert, so folgt aus der Gleiehheit der Streeken DX = DI X, = DJX2 und der Gleichheit der Winkel bei D1 und D, mit den Winkeln a bei D die Gleiehheit ihrer Projektionen X,'D1 = X,'D, auf D1 D,, und also X1'X2' = D D,. 1)a nun die Proj ektion X11X,' kleiner ist als die projizierte Streeke XX, oder der projizierte Streekenzug XI YZX2, so hat das H~ihenfusspnnktdreieck den kleinsten Umfang*. 4. Vorstehendes Theorem kann auch als " Problem " gel6st werden: IDa XS + X2S = Umfang des Dreiecks CBS, also konstant ist, kann unsere Aufgabe auf die einfachere znriiekgefiihrt werden: Welches unter allien Dreiecken XSX,, die in demi Winkel S auir der Spitze und der Schenkelsumme iihereinstimrnen, hat die kleinste Grundlinie XX, I Dreieek XISX, und sein symmetrisehes in Bezug auf die Winkelhalbierende von S stimmen in der Grdsse von X1X2,, dessen Extremum gesueht wird, tiberein. Nach Methode II a rntissen lim Falle des Extremuins beide Dreiecke sich deeken, foiglich ist die Basis D1D, des gleichsehenkligen unter den Dreieeken XjSX, das gesnchte Extremum. Seine- Art wird wie oben nach Methode I bestimmt. 5. tiner allea konvexeut Vieleckent V voit gegebenen Seitea hat das einemn Kreise eingeschriebe'ne oder Sehuvet- Vieleck V0 die gr~sste F/ache. IDa meine Beweismethode ftir das it-Eck genau dieselbe bleibt, wie ftir das Viereek, behandeln wir das letztere. Wir gehen vom Sehnen-Viereck V0, aus (Fig. 2) und zerschneiden es vom, Mittelpunkte M des umgeschriebenen Kreises, aus, durch die 4 nach den Ecken gehenden Radien in gleiehsehenklige Dreieeke Di, deren algebraisehe Summe gleich der Fhiehe des Sehnenviereeks V0 ist. Liegt der Mittelpunkt M innerhalb des Viereeks, so sind alle 4 Dreieeke positiv; nur wenn M1 ausserhalb des Vierecks liegt, ist das gleiehschenklige IDreieek mit der grtissten Viereeksseite als Basis negativ zu nehmen. Werden die Winkel des Sehnenviereeks (als Gelenkviereek aufgefasst) veraindert (Fig. 3), s0 beschreiben die nach den Ecken gezogenen Radien vier Sektoreri S~j, die wir als positiv oder negativ in Rechnung bringen, je naehdem durch sie der betreffende Winkel des Vierecks vergr~issert oder verkleinert wird. Der Mittelpunkt M beschreibt emn Kreisbogenviereck M1M2-MMYJ lifM. Dann folgt fair die neue Figur des Vierecks V "S~Di~ S M+ V. Nun ist aber ~ ~=V,, und:Si = 0, weil die algebraische Summe der Zentriwinkel = 0 sein muss, damit die Winkelsumme des Vierecks, konstant = 4 Rechte bleibt. Also erhalten wir V0= V+ M, womt der Satz bewiesen ist. * Laut Jahresbericht xxxvi-xxxix, 1906-1909 des Vereins f~r Naturkunde zu Zwickau habe ich den Beweis in der Sitzung vom 13./i. 1908 vorgetragen. In Weber and Welistein, Enzykl. d. Elernentarmatlh., iiil, zweite Aufi. 1910 steht ein Beweis, der mit meinem -Uibereinstimmt und nur am Ende die Aehnlichkeit der gleichschenkligen Dreiecke X A X2 und DAD2 heranzieht, urn D1D2 <XX2 nachzuweisen. In der ersten Aufiage iii, 1907, S. 309, stand dieser Beweis noch nicht.

Page  [unnumbered] MAXIMA U. MINIMA MIT ANWENDUNO AUF DIE OPTIK. J/ \ *e / * \ / \ "s / /S

Page  [unnumbered] JOHANNES FINSTERBUSCH. 4,r,,1 C, 6. 7. Lo.2. / 4 I "~ ~ I\ co

Page  109 GEOMETPIRSCIIE MAXIMA UND MINIMA MIT ANWENDUNG AUF DIE OPTIK 109 Die Verwandlung eines konvexen Vielecks in emn Sehnenvieleck von denselben Seiten ist immer m~iglich und eindeutig. Alle Sehnenvielecke von denselben Seiten, aber in versehiedener Reihenfolge, sind flachengleich. 6. ()Unter alie)? konvexen Vielecleen V von gegebenevn Wiinkeli ud {Umfang hat das einern Kreise umgeschrieben e oder Tangeidntieek- V de {gr~ssten Inhalt. Da auch bier die Beweisfifibrung ffir das n-Eck genau dieselbe ist wvie beim Viereck, geben wir wieder die letztere. Wir geben vom Tangentenviereek V0 aus (Fig. 4) und zersehneiden es vomn Mittelpunkte M des eingescbriebenen Kreises aus durch die nach den 4 Berifihrungspunkten gehenden Radien in Teilvierecke, sogenannte rechtwinklige Deltoide Di. Durch (beim Viereck abweebselnde) Verlangerungen oder V~erkiirzungen der Seiten, wodurcb die Winkel des Vierecks erhalten bleiben, (Fig. 5) bescbreiben die 4 Radien Reebtecke lii, die wir als positiv oder negativ in Rechnung stellen, je nachdern es sich urn Verlangerung oder Verkiirzung der Vierecksseiten bandelt. Der Mittelpunkt Al bescbreibt im ailgemeinen ein geradliniges Viereck AM, M M, ME ill. Dann ergibt die neue Figur des Vierecks V YDi ~ IR= V~+M. Im Falle (a) ist nun, da der Umfang erhalten bleibt, die algebraische Summe der Verlangerungen und also auch die algebraisehe Summe der Rechtecke Y~ = 0 urn] folglich V0 = V+ M, womit der Satz (a) bewiesen ist. Im Falle (b) ist, da der Inhalt nicht veraindert wird, Y. D= V0 V, und mithin "SRi= Ml oder:~R > 0. Ist aber die algebraisebe Summe- der Rechtecke grtisser als NTull So ist es auch die algebraische Sumine der VTerlangerungen der Vierecksseiten, womit der Satz (t) bewiesen ist. 7. Nur andeutungsweise werde ffir 2n-Ecke emn Zusammenhang der Aufgabe (a) init den S te in er'scben Schliesswngssitzen A~ber gespiegelte Lichtstra~hleti, gegeben. Werden im Viereck V= AIAAA, die Winkeihalbierenden gezogen und auf ihnen in den je 4 Puukten Ai und Alj Lote errichtet, so scbneiden sich die letzteren in einem Punkte Al, der zugleich der Diagonalschnittpunkt von den Sehnenvierecken NJN2N3N4 und SS2S.3S4ist. Durch Parallelversehiebunglangs derStreckenM13,MMM, M, M, MMA sebliessen sich die Deltoide Di zum Tangentenviereck V =- AOAOA,0A40 zusammen. Das Tangentenviereck V0, und das Viereck V sind also demselben Sebnenviereck S1 S S3 5 eingescbrieben, und man erbalt die S te in er'scben Satze, die bierdurcb eine neue Beleuchtung erfahren. Ich gedenke an anderer Stelle ausftihrlich hierauf zurflckzukommen, da ich hier auf die vielen interessanten Beziebungen, die sich aligemein bei 2n-Ecken und (2vn ~ 1)-Ecken ergeben, nicht eingeben kann. ~ 3. DIE ABLENKUNG DEs LiCRTES IN PRISMEN. 8. Bekanntlicb erfihrt emn Licbtstrahl, der im Hauptschnitt durcb emn Prisma gebt, immer eine Ablenkung von der brechenden Kante des Prismas weg oder nach ihr bin, je nacbdem das Prisma optisch diebter oder diinner als seine Urngrebung, ist,

Page  110 110 110 ~~~~JOHANNES FINSTERBUSCII und der Ablenkiingswinkel ist bei symmetrisehem Durehgang am Kleinsten. iliervon gibt es zahireiche Beweise, die sich auf goniometrische Rechnungen mit oder ohne Anwendung der htiheren Analysis sttitzen. Emn reingeometrischer Beweis, der diesen symmetrischen Durchgang nicht nur als extremen Wert der Ablenkung, sondern auch als Minimum erkennen lehrt, scheint nicht vorhanden zu sein. Ich gebe im Folgenden einen soichen, den ich am 12./7. 1911 gefunden habe, und, zwar behandle ich gleich die ailgemeine, wie ich glaube neue Anfgabe: Das Minirnvnm oder Maximum der A blenk-ung em es Lichtstrahls im Hauptschnitt eines Prismas zzifin deni, dessen Begrenzungseben en, an, zwei verschiedene optische Mittel angrenzen. Als gegebene Grt~ssen betrachten wir die Verhaidtnisse der Lichtgeschwindigkeiten in den dre1 optischen Mittein c,. c, oder die relativen Brechungsexponenten no, n1l, n nnd den brechenden Winkel w des Prismas. Zwischen den Gr6ssen c und n bestehit dann der Zusammenhang corio = elni =C N..................(1). Wir rechnen im Hanptschnitt des Prismas positiv den brechenden Winkel w, wenn dessen Scheitel links von der Lichtstrahlenrichtung liegft, ferner den Einfallswinkel c, und ]3rechnngswinkel /3, beim Uebergang in das Pte Mittel, wenn die Drehung des Strahles in das Einfallslot im Shinne des Uhrzeigers erfolgt, und endlich den Ablenkungswinkel cpu, wenn bei der vten Brechung der Lichtstrahl in diesem Sinne gedreht wird. Dem Snellins-schen ]3rechungsgesetz zufolge erhalten wir dann das Gleichnngssystem: sin ao co Ill sin a,~ el __f2 sin /3 e c1n0, sin /32 C2 P1 uind hieranis fuir (lie, Gesamitablenkung 0 l+ 0:_~L0 /2...............(3). 9. Die Abbildung (Fig. 6) stellt einen solchen Strahlengang ffir den Fall dar, dass das Licht ans Luft in emn Glasprisma eintritt und aus diesem in Wasser austritt. Um die Figur m~iglichst tibersichtlich zu gestalten, ist der Winkel (0 so gross gewdlhlt worden, dass a, und A3 unserer obigen Festsetzung gemiiss negativ werden. Ihre' absoluten Werte seien durch d, und /32 angedentet. Zur Ermittelung des gebrochenen Strahles aus dem einfallenden dient die bekannte Konstruktion von R e u s c h. Sie ist nicht an den beiden Uebergangsstellen 0, und 02 sondern, da parallelen Einfallsstrahlen auch parallele gebrochene Strablen entsprechen, an der brechenden Kante 0 des Prismas ausgeftihirt worden. Wir geben zwei versehiedene Konstruktionen, deren Zusammenhang wiclitig ist. (1) Um -0 seien zunaclist 3 konzentrische Kreise gezogen, deren Radien im Verhhltnis n0, n, n, stehen. \Ton dem Punkte N0 auf Kreis n0, des einfallenden Strahles ausgehend bestimimen die Einfallslote N, No 01 L und N, N2 JL02 die Richtungen des im Prisma verlaufenden Strahles N10 fl 0102 und des anstretenden Strahles N20. (2) Eine zweite ebenso einfache Konstrnktion beginnt mit 3 umi 0 konzentrischen Kreisen, deren Radien sich wie die Lichtgeschwindigkeiten co0: l: c. verhalten. Von dern Punkte Gl' auf Kreis cl des einfallenden Strahies ausgehend,

Page  111 GEOMETRISCHE MAXIMA UND MINIMA MIT ANWENDUNG AUF DIE OPTIK III bestimmnt das Einfallslot 0i'C0 0,L den im Prisma verlaufenden Strahi 00020 und das dureli 02 gehende Lot 0201"' 11LO, auf die Austrittsebene die Richtung C,"'0 des austretenden Strahies. In beiden eben besprochenen, Konstruktionen ist das Brechungsgesetz gewahrt. Sie gelten ffir jeden, das Prisma durehsetzenden Strahiengang. Bei Yergleichung mehrerer Strahiengainge, wollen wir in der Figur 020O festhalten, und alle tibrigen Geraden, also auch die Einfallslote und Begrenznngsebenen des Prismas, die entsprechenden Drehungen ausffihren lassen. 10. Mit Hulfe der Methode II b soil nun an der Hand der Figur cine Bedingung ftir das Minimum -der Ablenkung gesucht werden. Wir nehrnien an, der gezeichnete Strablengang sei der des gesuchten Minimums. Eine oc kleine Aenderung da0o des Einfallswinkels a,, zieht dann oc kleine Aenderungen der Ubrigen Winkel nach sich. Nach 8 (2) ist hierbei du, d/31. Da wir vorn Extremum des Ablenkungswinkels 0 ausgehen, muss dessen oc kleine Aenderung do gegen die tibrigen verschwinden. Es muss also dcp = do, + do,2 = 0 sein. Hieraus folgt auf dem Kreise c, die Gleichheit dler B6gen Wird auf 0o,0' = xO in 00 das Lot 0,D'1 001, und ebenso auf 020"' = X2 In 02 (las Lot 02D"' 1100. errichtet, so schneidet auf dem fiber 02 D' als Durehmesser Cos Oro beschriebenen Kreise, kO der oc kliene Winkel d/31 einen Bogen — d/31 ab und ebenso auf dem jiber 0 "'D"' -- als Durehmesser beschriebenen Kreise k.2 COS /32 der x kleine Winkel da-, einen B~ogen Cos d182 ab. Diese fallen aber mit dlen oc kleinen Biigen c~do, und c1df02 zusammen, da die Kreise ko, und k-2 sich mnit dem Kreise, C ]In Oi' bcziiglich Oi"' beiriihren. Wir erhalten also die Gleichungen co d,8d/1 =eldo, uind =-' i aus denen nach obeni x 0 ___ ___2 _ cos ao COS /32' d. h. die Gleichheit dler Durchmesser der Kreise k, und le. D',= D"'01"' oder auch D'0 = D"'0..........(El) folgt. Dies ist eine gesuchte geometrisehe Bedingung ftir das Extremum der Ablenkung. Fiir jeden anderen Strahlengang haben die Durchmesser der Kreise Ko und k-2 verschiedene Lange. Handelt es sich im Besonderen urn emn Prisma, das an zwei gleiche optische Mittel angrenzt, ist also c2 = CO, so fallen 02 und 0O in einen Punkt zusammen und dieser ist dann emn Schnittpunkt der beiden gleichen Kreise k," und 1c2. Da deren gemeinsame Sehne, hier COO, immer dureli 0 geht, so ist, falls /3' und a, - - -a2 versehiedene Vorzeichen habeni (wie in der Figur), diese die Symmetrieaxe der Figur. Es folgt also a-, = /3, und w =2, = 2/3,. Wenn dagegen /32 und a, gleiche Vorzeichen

Page  112 IT2 112 ~~~~JOHIANNES FINSTERBUSCIT haben, decken sich die beiden gleichen Kreise k0, und 1c2 und die Winkel a, und i3, woraus folgt, dass wo = 0 ist. Dieser Fall seheidet aus. 11. ilaben wir j etzt fuir den ailgemeinen Fall von 3 versehiedenen Mitteln eine geometrische Bedingung (E1) fuir das Extremum gewonnen, so ist noch zn untersuchen, ob wir es mit einem Maximum oder Minimum zu tun haben. Da bei einem Prisma fuir kleine, aber endliche w positive und negative Ablenkungen (P m~iglich sind (im besonderen Falle, von nur 2 versehiedenen Mitteln dagegen nicht), so wollen wvir von einemn Maximum oder Minimum reden, je nachdem der absolute Wert 0(Pm der extrernen Ablenkung gr~isser oder kleiner ist, als die der benachbarten Strahlengiinge. Wir fubhren diesern Nachweis ganz elementar nach Methode I. Wir nehmen an, der durch obige geomnetrisehe Bedingung bestimmnte Strahlengang liege gezeichnet vor. Fiir einen belhebigen anderen Strahlengang bleibe wieder 0,O fest, die Linfalislote und das Prisma sind dann urn einen endlic-hen Winkel gedreht. Dabei schneiden sich die zwei neuen Einfallslote C0A,' und 0QA,"' immer in einem Punkte A. auf einem festen durch 0o und 02 gehenden " o-Kreis." Auf den, gleichen Kreisen k-, und kL werden von ihnen die gleichen Biigen C0/A, = 0"`A, abgeschnitten. Wird nun durch Drehung uim 0 der Kreis k-0 mit dem Kreis k2 zur Deekung gebracht, wodurch alle mit bewegten Geraden umn den Winkel (Pm gedreht werden, so deekt sich CA,' mit 0,'A,'. Es ist Winkel A,'0A,' = CjOCC"'. Demnach ist der neue Ablenkungswinkel AO'0A2.. (P um A,'0A2"` grosser als (P.Ebenso folgt, dass wenn die neuen Lote sich auf der anderen Seite von L~. in B. schneiden, der neue Ablenkuingswinkel B010B211' (Pb um B,' 01,"' grdsser ist als (Pm,. _(a >01fA < (Pb. Es handelt sich also in unserer Figiir um emn positives Minimum (Pm = (Pmin. In gleicher Weise kann in jedemi vorliegenden Falle Lage und Art der extremen Ablenkung best1immt werden. Ehe wir jedoch die Ergebnisse ffir die verschiedenen Falfle zusammenstellen, sollen aus der oben gefundenen Bledingung einige andere abgeleitet, oder auch direkt gefunden werden, die zur Erledigung versehiedener Fragen besonders geeignet sind. 12. Bei jedem befiebigen Stralilengang gelten ftir die 4 durch die Punkte Co, und N, bez. 0, und N, auf die Begrenzungsebenen des Prismas gefaillten Lote der beiden oben gegebenen Konstruktionen: D'O GLo' N1L, N,0 cI'( = /L0, = NL, moo und-0"L"'NL M0 FUr den extremen Stralilengang sind (El), d. h. unserer obigen Bedingung D'O = D"'0 gemass alle Quotienten einander gleich, woraus MOhO = M20 MO 11 folgt; d. h.: (E,) Fur den, extremen Strahiengang schneiden sich die durch No und N, zu den, Begrenzungsebenen des Prismas gezogenen Parallelen in einewm Punicte 11 aaf NO, oder der durch NNN2 bestimmte Kreis co' beriilhrt Kreis nl. Dies lasst sich auch direkt ganz elementar nach Methode II a begrtunden. Entspricht das \Tiereck N/,N, AT,'O einem beliebigen Strahiengange vorn Mittelstrahl

Page  113 GEOMETRISCHE MAXIMA UND MINIMA MIT ANWENDUNG AUF DIE OPTIK 113 N,O, dessen Eintritts- und Austrittsstrahl den Winkel N,,'ON,'= 0b einschliessen, so wird der wo'-Kreis vom Kreise n, nicht nur in N,, sondern auch in einern anderen Punkte N,' geschnitten. Das Viereck NO,'N,'N,' 0 entspricht also ebenfalls einem Strahiengang vom Ablenkungswinkel b.Fiir den extremen Strahiengang muss 4 einen soichen Wert haben, dass die beiden Vierecke auch mit ihren Mitteistrahien (E- IDiagonalen) sich decken, d. h. N, und iNi'missen zusammenfallen, der co'-Kreis also den Kreis n1 bertihren. Aus (E,) folgt ferner: (E,) Beim extrernen Strahlendurch gang werden alle vier Einfallslote von den betreffei? den, Kreisen c oder n in demselben Verhdltnis geschnitten. (E,) Die Sehnenvierecke NONNM, L,JA7,L,0, 0,0O,L und C2OJJCL., sind einander ihnlich. 13. Aus vorstehenden Siitzen kann nun leiclit eine einfache Konstruiction, der Winkcel a,u'nd /3, des extremen Strahienganges gefunden werden: Wir wahien das Sehnenviereck LoNTL20. Da /3 - a, = lei + di= ) also bekannt ist, suchen wir cos /3,:cos a, = NjoL, NL2 zu bestiminen. Dies kann auf verschiedene Art elementar gesehehen. Sehr einfach ist folgende reingeometrische L6sung: Nach (E,) verhiilt sich NLo: NL2 =NLo N2L2. Durch entsprechende Addition und Subtraktion ergibt sich hieraus (NLo )2 (AL2.)2 = (N, Lo +.No Lo) (1,TLo - No Lo):(N, L2 + N2L2)(N L2 - N2L2). Die beiden letzten Glieder Sind nun Produkte von den Abschnitten zweier Sehnen im Kreise nin, von denen die eine durch N,', die andere dureli N, geht. Nach dem Sehnensatze der Planimetrie sind diese Produkte aber gleich den Quadraten der cntsprecheuden halben kleinsten Sehinen im Kreise in,, die dureh N, und N., gehen. Es ist daher das gesuchte VerhhIltnis * Cos /3, Cos at, = N, Lo N. L Vn, 2 _no2 n,/ 2 _- oder: Die von N, auf die Begrenizuntgsebenen des Prismas gefitliten, Lote verhalten sich wie die Sehnen im Kreise nl, die die Kreise no und n,2 beriihren. Dasselbe gilt ftir die Winkelsehenkel von co in allen Vierecken die in (E,) aufgeftihrt worden sind. EMS dieser Vierecke kann nun leiclit konstruiert werden. 14. Werden die in 12 an zweiter oder dritter Stelle tibereinander stehenden Quotienten durch die Winkel a und /3 allein oder durch die Winkel und Radien c bez. it ausgedriickt, so erhd~t man ftir den extremen Strablengang die Formeln Cotg/3, _Cotga un ct, c Cos /3,- 2 cCOS a, cotg a, cotg /3, e Cs d05 a, Cos /82 tang ao tang a, oder ~~~~~~tang /3, tang,-.(2) und Cos a, Cos a,= co= n2...............( c) COS /31 COS,8 /3 10,, In der Diskussion, die dem Vortrag folgte, gab ich auf Anregung von Professor S ch on t e ejine andr e Ableitung mit Hilfe, von Formel (E1,). -24. C. T?, 8

Page  114 114 114 ~~~~JOHANNES FINSTERBUISCH Von diesen beiden Bedingungsleichungen ftir den extremen Strahiengang ist besonders die erste wichtig, da sie nur die vier Winkelgr6ssen enthalt und da aus ihr wichtige Folgerungen gezogen werden kiinnen. (E,) in der Prismenfigur schneiden sich beirn extremen Strahlengang der eintretende und austretende Strahl in einem, Punicte S auf dein Durchmesser OL, wo L der Schnittpunkt der Eirtfallslote ist. Zum Beweise fialle man von S Lote auf die Begrenzungsebenen und wende die Bedingung (E,) an. Da der Quotient der Tangenten zweier spitzer Winkel, ebenso wie der Quotient ihrer Sinus >- 1 ist, je nachdem der Quotient der Winkel selbst es ist, so kann die Gleichung (E,) nur dann bestehen, wenn emn Bruch > 1, der andere < 1 ist. Es muss daher entweder CO > C1 < C2 od er C0 < C1 > C2 sein, d. h.: (E,) Beim Prisma ist emn extremer Durchgang nur dann m~glich, wvenn das Prisina dichter oder di)~nner als seine b eid en umgebenden Mittel ist. 15. Es fragt sich, wie weit, vom Extremum ausgehend, in der Figur die Einfallslote nach beiden Seiten gedreht werden kdnnen, damit noch emn Anstreten des Strahis und nicht eine Totaireflexion an der zweiten Begrenzungsebene erfolge, wie weit also z. B. auf dem o-Kreise A., oder B,,, wandern k~innen. 'St C1 < CO, C2, SO bestimmt je eine von c0 und von c, an den Kreis c, gelegte Tangente diese beiden Grenzlagen. Im ailgemeinen ist keine emn Extremum, sondern nur in dern besonderen Falle, dass das oben betrachtete Extremum mnit einer dieser Grenzlagen zusammenfallt. Wir gehen niclit nhher darauf ein Grenzlagen gibt es auch, wenn C. < C, K C. Oder rC, > C1 > c2 ist, also nach (E8) kein Extremum vorhanden ist. Wahrend irn Sonderfalle C2 =CO, wie einleitend in 8 bemerkt wurde, die Ablenkungswinkel 0 und ihr Extremum 0c/m entweder immer positiv oder immer negatiIv sind, so ist dies im ailgemeinen Falle nicht so einfach (vergi. 11). Aussehiaggebend ist der w-Kreis. Wenn dieser den Kreis c, bertihrt, ist (CO_ - (2mC1?- n.) In, Man erhalt fuir die drei zu unterscheidenden Falle co wo folgende Uebersicht fiber die Vorzeichen der Ablenlcungswinlcel und die Art des Extremumsk, jo nachdem das Prisina optisch dichter oder optisch dibiner, als seine Umgebung ist: L 0 > co)o. Der co-Kreis schneidet den Kreis c, nicht. c0 >0, b<0, Irn, = + Omnj, - Min. (i) c = co)

Page  115 GEOMETRISCHE MAXIMA UND MINIMA MIT ANWENDUNG AUF DIE OPTIK 115 Der w-Kreis bertihrt c, und fidt mit den beiden Kreisen k0, und k, zusammen. ObmO=, ObmO=. Das Prisma ist fuir den extreynen Strahlendurchgang emn " Geradsichtsprisma." III, (0 < coo. Der w-Kreis schneidet den Kreis c1. 0, 0 POm = lfjlW4 'k = + 4Yinax Fur zwei verschiedene Strahlendurchga~nge, die den Schnittpunkten A. und B. des w-Kreises mit c, entsprechen, wird 0b = 0, das Prisma also zum " Geradsichtsprisma." Diese beiden Strahiendureligange sind keine Extrema, sondern bilden nur die Uebergauge von positiver zu negativer Ablenkung. 16. (E,) Der Tangensbedingung (E,) leann emn einfaches geonsetriscites Gewand in Gestalt eines geschlosserten Vierecks H0 G1 H1 G, gegeben werden, dessent Ecken abwvechselnd auf zwei einander in 0 senikrecht schneidenden Geraden liegeln (Fig. 7). Sind fuir einen beliebigen Strahiengang die Winkel HOG1O = a0o, HIGI0 = fl1, H1GO = a1, H2G2O = 132, mithin GI H1G2 =' - ot, so sind H, und Ho verschiedene Punkte, geh6ren die Winkel dagegen dem extremen Strahiengang an, so ist H2- Ho. In letzterem Falle folgt, wenn 11 00 h0, HI 0=- hi, GI 0 = gi, G10 = g2, aus der Identitait h o~ hihi hi) gI g2 g1 g1 obige rpangensbedingting (E2). Die Fusspuvlce A0 B1A1B1, der 4 mvo 0 auf die Seiten, des Vierecics gefdtllte)? Lote a., bl, a1, b2, liege n auf eie Kreise *, wie man leicht erkennt, wn a i rechtwinkligen Dreiecke der Figur durch ihre umgeschriebeneni Kreise ersetzt und die Figur von 0 aus myvers transformiert. Diesem. Satze J. Steiner' s m~zchte ich hinzuffigen, dass A0BI und A1B2 sich auf HO, A0)B, und BAI, sich auf GO schneide'n was ebenfalls durch Inversion leicht bewiesen werden kann. Das Sehnenviereck- B10A1H1j (Fig. 7) ist den in (E,) aufgefilhrten (Fig. 6) dhnlich. Werden die reciproken Werte der Strecken g und h doppelt ausgedrtickt, so folgt sin a0 1I sini,8, a( 9 bi Cos a0 1 _COS13 a0 ht0 b22 4sina 21 sin/32K a, 92 b2 Cos a, cosfl13, a, hil b1i Werden diese Gleichuingen quadriert und die durch Kiammern bezeichneten Paare J. Steiner, Ges. JK7, ii, S. 358, Lehrs. 2.

Page  116 116 116 ~~~~JOHANNES FINSTERBUSCR oder alle vier addiert, so folgen einfache Beziehungen zwischen den Strecken allein. In letzterem Falle ergibt sich I+- + E-b, eine Beziehung, die fuir manche Berechnungen gute Dienste leistet. 17. Zum Schiusse noch einige Bemerkungen u-ber den Verlauf des Lichtstrahles irn Hauptschnitte eines Prismensysterns bei minimaler oder maximaler Ablenkitng. Abgesehen von einigen Sonderfhllen ist es mir his jetzt niclit gelungen, ffir dieses aligemeine Problem von beliebig vielen Brechungen synthetisch zu entsprechend einfachen elementargeometrischen Ergebnissen zu kommen, oder das Problem des Prismnensystems auf das soeben behandelte von drei lichtbrechenden Medien zurtickznfiihren. Ich beschrainke mich daher vorlaufig darauf, mit Hulfe der Differentialrechnung die Herleitung der Bedingung (E) ffir den extremen Strahiengang kiirz anzudeuten. Von den drei verschiedenen Bedingungsgleichnngen, die ich gebe, babe ich nur die Cosinusbedingung (El) in dem Werke: "die Theorie der optischen Instrumente, bearbeitet von wissenschaftlichen Mitarbeitern der optischen Werkstatte von Carl Zeiss, " gefunden*. Die wichtigere Tangensbedingung (Eil) ist dem Bearbeiter des Kapitels liber Prismensysteme entgangen, was um so auff~fl~iger ist, als er diese fair 2 Brechungen, allerdings nur in dem besonderen Falle c2 = c, des gewthnlichen Prismas, angibt. Bei Aufstellung der Gleichungssysteme, deren Differentiation zur Bedinguingsgleichung fUhrt, bin ich m~iglichst dem oben angefiihrten Werke gefolgt. Wir setzen Ic Begrenzungsebenen, k- - 1 Prismen von den brechenden Winkeln W,, und Ic + 1 optisehe Mittel von den Lichtgeschwindigkeiten c1. vorauis. Ueber die, Vorzeichen der Winkel a~ und j3 siehe die in 8 getroffenen Bestimmungren. I )ie einzelnen Ablenkungen seien 4), und die Gesamtablenkurng 4.Dann gelten analog wile in 8 folgende Gleichiingssysteme: sin aV, c,, sin/- 3,, -C(=1, 2,...,Ile........ (1), k k -I Fiir den extremen Wert erhalten wir durcli Differentiation der Gleichungen (1) his (3) und Elimination der c durch (1) die Gleichungssysteme: da - c,-, Cos p3,, __tang avd13, c,, Cos a,,1 tangf,3vvl,..,k. 1) dca. = ( 1l 21...,I ic-i.(2), dI~ da,8k _I 0 oder da,8,............(3'). *a. a. 0., i, Bd. Die Bilderzeugung in optischen Instrurnenten; viii Kap. F. L6we (Czapski), Prismen unid Prismnensy~stevie, S. 422,

Page  117 GEOMETRISCHE MNAXIMA UND MINIMA MIT ANWENDUNG AUF DIE OPTIK 1-17 Die Multiplikation der Gleichungen (1') ergibt nach (2') und (3') als Bedingung ffir den extremen Strahlengang, CZOS ao COS C(i... COS Otk-_ Co -'i~k.(El) COS /3' COS /3,... COS /3k Ck '110 und tang -a0 tang -a.,... tang -ak-i 1. (l tang /31 tang /32..tang /3k -..(E......... (El,,) Ganz analog wie in (E8,) folgt hieraus, class ein Extremumn Shi der Ablenkcung nur danun statt~fiinden kcain, wenn weder ai1le Brechungen aus je einem ditnuereit iu emn dichteres Medium noch umgekehrt erfolgen.. (E1v) Anch hierl hisst sich die Tangensbedingung (E11) durelinemgeschiossenes 21c-Eck darstellen, dessen Seiten abwechselnd sich auf zwei zu einander senkrechten Geraden schneiden. Die vom Schnittpunkt beider auf die Vielecksseiten gefillten Lote gentigen wie in (E,,) der Gleichung: (to a a'k- b1 b21. bk. E)

Page  118 ON BINODES AND NODAL CURVES BY Miss H. P. HUDSON. The first part of this note shews that a binode can be studied by means of two auxiliary non-singular surfaces which approximate to the separate sheets of the given surface. The biplanes are the simplest of such surfaces; and if the binode is a Bi, i.e. if it reduces the class of the surface by i, then the approximation can be carried to the order i- 2. This leads to a very convenient algebraic definition of a Bi. The second part investigates the reduction in class due to a general nodal curve; incidentally, the number of pinch-points is determined. The method of ~ I. might be applied to unodes of all orders; then the two auxiliary surfaces have contact with one another, of a certain order which is the other characteristic of the singularity. The method of ~ II. might be applied to cuspidal, tacnodal, etc. curves, and also to curves of higher multiplicity, determining in each case the reduction in the class of the surface, and the number of its ordinary singularities. REFERENCES. SALMON, Camb. and Dubl. Math. J. ii. p. 72, 1846. CAYLEY, Coll. Math. Papers, vi. p. 123, 1868. ROHN, Math. Ann. xxII. p. 124, 1883. SEGRE, Ann. di Mat. (2) xxv. p. 1, 1896. BASSET, Geometr.y of Surfaces, Cambridge, 1910. I. (Cartesian coordinates.) Theorem. If the origin 0 is an isolated Bi on a surface F, then F= H.K +f,.................................(I) where fi is small of order i near 0, and H, K are surfaces each passing once only through 0. The form of the equation shews that each of these surfaces has contact of order i - 2 with one sheet of F; we may say that to this order of approximation, F is degenerate. Let yz be the biplanes, and let F be expressed in the form F= H. K +f, where H= y + 2, K=z +f,

Page  119 ON BINODES AND NODAL CURVES 119 and I has its greatest possible value, which implies that fi contains a term in x1. If 1 = 3, the form assumed merely expresses that 0 is a binode. We have to prove I = i, the reduction in class. Now the class of F is the number of points at which the tangent planes pass through an arbitrary straight line, i.e. the number of intersections other than 0 of F and the curve F of intersection of PI, P2, the first polars with regard to F of any two points AI (xlyzi1), A2(x2y2z2); and i is the number of intersections of F, F which coincide at 0. a a a a. Then if Ai = i x+ i + y z +Z w -w, (w = w = 1) the equations of F are Pi = HA1K + KA1H + Alf, = 0) P2 - HA,2K + KAH + A2 fi = O and r passes through 0 and touches the edge of the binode. At points of F near 0 we have: AH, AK finite; x small of order 1; y, z small of order 2; H, K, A/f small of order I- 1; and since fi contains a term in x1, both ft and F are small of order I, and exactly 1 intersections of F, F coincide at 0. Therefore I = i. The converse is proved by reversing the argument, so that a Bi may be defined by means of the form (I), which is specially convenient for algebraic transformations. If the Bi is decomposed by Segre's method into adjacent double points, we must suppose them arranged along the curve H = K = 0, which is a generalization of the edge of the binode, just as H, K are generalizations of the biplanes. Note that H is not determinate, but may be replaced by any other surface having contact of order i- 2 with H at 0. II. (Homogeneous coordinates.) If the degree u of F is given, there is an upper limit to i, but its relation to n is unknown. If n = 3, it is 6; if n = 4, it is _ 15. If i exceeds this limit, then 0 ceases to be an isolated binode, and F acquires a nodal curve C through 0. Then r degenerates into C and a residual F', and there are an infinite number of points near 0 common to F, P,, P.,. The argument of ~ I. then shews that there exist non-singular surfaces having contact of any order with one sheet of F at 0, but we cannot in this way determine the reduction in class. Let C be of degree m and rank r, with no singularities except d nodes; then r' is of degree (n- 1)2 —m, and the class of F is the number of intersections of F, ' not lying on C. Now r' meets C in (i) the r points at which the tangent line to C meets the arbitrary straight line A1A2; these are points of contact of PI, P2, and at each of them F' meets F in two coincident points; (ii) the q pinch-points of F, which are points of contact of PI, P2; these are ordinary singularities, and at each of them r' meets F in. three points;

Page  120 120 3IISS H. P. HUDSON (iii) the d double points of C, which are tacnodes of F and points of osculation of P1, P2; through each there passes one branch of F', meeting F in four points; (iv) the points of higher multiplicity of F on C; these are not ordinary singularities, and we only consider t triple points of F at simple points of C; each is a common node of P1, P2, and through it there pass three branches of F', each meeting F in three points. Therefore the reduction in class is mn + 2r + 3q + 4d + 9t and it remains to determine q. First let C be the straight line z = w =. Then the pair of tangent planes at any point (xy00) of C is given by an expression of the form T - (U, V, W z, )2 where U, V, W are functions of x, y of degree n -2. The pinch-points are among the 2 (n- 2) intersections of C with the surface V2- UW; in general they are distinct, and there are no higher singularities. But each of the t triple points of F gives a squared linear factor in V - UW, and in this case q = 2 (n- 2- t). If V2-UW has another squared factor, then a pair of pinch-points coincide; through such a point there passes one branch of r', touching C and meeting F in six coincident points. If Vf - UW is a perfect square, all the pinch-points coincide in pairs. Now this is also the condition that T falls into two factors, each linear in z, w and rational in x, y; then there exist two auxiliary surfaces, given by these factors, each passing once only through C and touching one sheet of F. This last property defines an important class of double curves; it always implies that the pinch-points coincide in pairs, but if C is a general curve, the converse is not true. Next let C be the total intersection of two surfaces 4), T of degrees k, 1. Then F (U, V, WI, r)2 and the pinch-points are among the intersections of C with V2 - UW, in number 2k1 (n - - 1) = 2 {t (n - 2) - r - 2d}. This number does not include the double points of C, for they do not in general lie on V2- UW, but it includes the triple points of F, each counted twice, and the number of pinch-points is q =2 {m (n- 2)- r - 2d-t}........................(II). Finally, let C be a general curve, the partial intersection of <(, P, and let the residual C' (of degree m' etc.) meet C in I points, supposed ordinary points of C and of F. Take any two surfaces F2, F2, of degrees nl, n2, passing through C' and not through C, and having no multiple points on C'. Then the complex surface F. F. F2 has the complex curve of total intersection C + U' as a nodal curve, and F. F,. F2- ( U, V, W V 3, w ).

Page  121 ON BINODES AND NODAL CURVES 121 Now each of the I points is quadruple on F. Fl. F2 and double on V2 - UW, and absorbs four intersections of V2 - UW, C + C'; there are m'n - 21 other intersections of F, C' and m (n, + n2)- 21 of Fl. F2, C, which are triple points on the complex surface; therefore the number of pinch-points of F. F1. F2 on C+ C' is 2 [(m + nz')(n +, + n, + - 2) - (r + r') - 2 (d + d'+ I) - {t + m'n + m(nl + n2) - 41}] - 41. But the pinch-points on C' coincide in pairs at the mn' (ni, + 2 - 2) - r'- 2d' points of contact* of F,, F2 on C'. Therefore the number of pinch-points of F on C is as before q= 2 (n - 2)- r - 2d - t}. Substituting this value for q in II, we have the final formula for the reduction in the class of the surface due to a general nodal curve: m (7n - 12) -4r- 8d + 3t.................... (III). * Salmon, Geom. of Three Dim. 5th ed. p. 358.

Page  122 ON THE CONFORMAL REPRESENTATION OF CONVEX DOMAINS BY E. STUDY. We shall term a monotonic graph any continuous set of points (0, )) in a Cartesian plane such that 1. If 0, - 0 > 0, then 23 - i, - 0. 2. If O2 - O > 0, then 02 - 0 0. 3. The set is reproduced by the translation 0*=0+27r, e*=O+27r. By such a graph, which is a special kind of periodic curve, a sort of functional dependence e (0) is given. To a definite value of 0 there will correspond either a definite value of O or a whole continuum of values, ranging between and including two extremes. These extreme values we may appropriately call ~ (0-0) and E (0 + 0), indicating thereby a process of passing to the limit by which they can be obtained. We shall speak in this case of a vertical segment (0 = const.) contained in the graph. Of course there may be horizontal segments (O = const.) as well. The sum of the lengths (0 +0)- o(0 -0) of all vertical segments contained in any primitive section of the graph to, < 0 < 0, + 27r, 0o - o < )0 + 27r} cannot exceed 27r; therefore the values of 0 belonging to this interval for which e ( - 0) and e (0 + 0) do not coincide, will, at the utmost, form a denumerable set. The primitive section may always be chosen so as not to divide any vertical or horizontal segment in two. It then contains one representative of each set of segments that correspond in the group 0* = +2K~T, H*= +2 2Kr ( =+ 1, 2,...). Finally it is clear that ) ( - 0) + 0 ( + 0) 2 is a monotonic function, such that H( + 27r) = H(0) + 27r, H(0) = H( -0) + H(0 + 0) 2 determined by, and in its turn determining, the graph.

Page  123 ON THE CONFORMAL REPRESENTATION OF CONVEX DOMAINS 123 Graphs derived from one another by means of a translation {0* = 0 + const., H. = e + const.}, and its inverse, we shall consider as not essentially different, or equivalent. Monotonic graphs arise in connection with the conformal representation of convex domains on to one another. Let us especially consider the (direct) mapping w = w (z) of a circular domain, say Izi< 1, on to a second convex domain. The word convex domain we may take in a somewhat broader sense than is customary. We include under this denomination all domains that can be looked upon as limits of sequences of convex polygons, with the exception of the plane itself*. Thus a "convex domain" may extend to infinity; e.g. a parallel strip and a half-plane are convex domains. Leaving aside, however, for the instant, infinite domains as well as all those with angular points or segments of straight lines on their boundaries, we readily see that from the mapping w = w (z) a set of equivalent monotonic graphs can be derived: 0 and e are the angles between two (arbitrary) fixed straight lines and two corresponding tangents of the two boundaries. It must be supposed, of course, that, starting from an initial couple of values (00, 40), 0 and ~ change continuously, when the two tangents turn around the two domains. Now this statement can be generalised so as to include all the limiting cases mentioned. Taking this for granted-the necessary definitions are too long to be reproduced here-we arrive at the conclusion: I. To every mapping w= w (z) of the circular domain lz < 1 on to another convex domain there corresponds a definite set of equivalent monotonic graphs. The same set corresponds, of course, to all the mappings given by the functions W(z)= Aw(z)+B {A, B = const., A 0}. The interest we attribute to this theorem consists in the possibility of its reversion: II. The set of mapping functions W (z) is completely determined by the set of corresponding graphs, which, in its turn, can be prescribed arbitrarily (subject to conditions 1. 2. 3.). In order to get rid of the ambiguity contained in these assertions, we represent the points of the circle, once for all, by the formula =eio. On the other hand, we select from the set of functions W (z) the one whose expression as a power series is w (z) = z +... Then the functional dependence @ (0) or the monotonic function H (0) is determined apart from an additive constant, to which we may attribute an arbitrary value. If now we choose out of a primitive section of the graph two values 01, 02, such that * Of course a more elementary (but less short) definition may be given.

Page  124 124 E. STUDY 62 - 01 > 0, and that the corresponding values i = ~ (08) and ~2 = O (02) are definite, we may look upon the difference AO =, - 1 = H - H, as a mass, positive or zero, situated in an arc of the circle, whose length is A = 02- 1. It follows that the sum of all these masses is 27r, and that their distribution over the boundary ] = 1, determines the function H(0) or the graph (8), apart from an additive constant. Further it follows incidentally that if,0 < 0 < 02, Lim A= 0, then Lim Ae = e (0 + 0) - O (0 - 0). Thus we see that the theorems I and II are equivalent to the following: III. If the function (z) = +... maps the circular domain z < 1 on to another convex domain, it determines a distribution of masses, positive or zero, whose sum is 2rr, over the boundary | 1 = 1 of the circle, and vice versa. Moreover it is possible to determine explicitly the function w (z), if the distribution of masses (or one of the corresponding graphs) is given, and vice versa. Starting from the graph as the given element, we form, by means of a so-called Stieltjes-Integral, the function f() =-.7 l( og( ) {,< I} extending the integration, in the positive sense, over the boundary of the circle, viz., over a primitive interval of the graph. Then w(z)= jdzef(z) o is the function required. If, conversely, w (z) is known, we consider the function (any one branch of the function) (z) = log (dw {z I <l}. i k~dz/ Let the point z, from the interior of the circle, approach the point ' of the boundary on a straight line. Then the real part of C (z) will tend towards a definite value, possibly depending upon the direction of approach. The range of values obtained thereby, including its lower and upper limits, is the range of values of e belonging to the value 0 in one of the required graphs. The two limiting values of the range are e (0 - ) and 0 ( + 0); approaching ~ on a radius, we obtain H(0 = (- 0)+ (0+o) H(0) 2 Thus the study of the mapping of circular domains on to other convex domains is seen to be equivalent to the study of monotonic graphs.

Page  125 ON THE CONFORMAL REPRESENTATION OF CONVEX DOMAINS 125 To further elucidate this we add a few applications. To any horizontal segment in the primitive interval of the graph there corresponds a segment of a straight line, forming part of the boundary of the map. If a part of the graph is analytic but not vertical, the corresponding part of the boundary is analytic too. Hence: If the graph is nowhere analytic, except in its vertical segments, then the circle is a natural boundary to the mapping functions W(2). To any vertical segment of the graph there corresponds an angular point in the boundary of the map, whose internal angle is 7r - { (O + 0) - O (O - 0), provided this difference is positive. If there is no vertical segment in the graph whose length reaclhes or exceeds Vr, then the map remains entirely in the finite domain. If there is, in the primitive section of the graph, one vertical segment of length equal to wr, the map extends to infinity as the interior of a parabola does, viz., in one direction only. If there are two such segments, the map is a parallel strip. If there is a vertical segment whose length exceeds wr and is less than 27r, the map behaves like the interior of one branch of a hyperbola, extending to infinity in an infinity of directions. If there is a vertical segment of length 27r (and consequently one horizontal segment of the same length), the map is a half-plane. Special interest belongs to the case in which the graph is entirely made up of horizontal and vertical segments, a finite (and necessarily even) number being contained in a primitive section of the graph. If we exclude the cases in which the length of a vertical segment reaches or exceeds vr, the interior of the circle is mapped on to the interior of an ordinary (finite) convex polygon. The expression of w(z) reduces to a well-known formula, due to H. A. Schwarz and Christoffel. This formula was, in fact, the author's starting point. The process of passing from convex polygons to the other convex domains suggests also the methods of proof to be applied. Of all the statements given the converse is true. Finally it may be noticed that the "monotonic" graphs may be replaced by "graphs" of a more general description. Instead of our function H(0) any such functions of limited variation (fonctions a variation bornee) as satisfy the conditions H(0 + 27r)= H (0) +27r, H )H() = H(0)+ H( + 0) 2 may be used. Substituting this in the formula for w (z) we still obtain the mapping of the domain z < 1 on to another domain. But the author has not been successful in finding the characteristics of the map in this (much) more general case. For proofs and further developments we refer to the booklet Vorlesungen fiber ausgewdhlte Gegenstdnde der Geometrie, lI. (Leipzig, 1912).

Page  126 TJBER DIE BEGRIFFE " LINJE " UND " FLACHE"~ VON Z. JANISZEWSKI. Mein Referat hat einen ganz negativen Charakter: ich will nur einige kritische Bemerkungen flber den heutigen Stand einer Frage mitteilen und keine Ldsung derselben. Ich stelle mich bier auf rein geomnetrischen (besser: rnengentheoretischern) Boden und betraebte alle geometriscben Gebilde als Punktmengen. Der Einfachbeit halber beschrinke ich mich auf im dreidimensionalen euklidisehen Ranme nirgendsdicbte Kontinua (d. h. Kontinua ohne innere Pnnkte). Die geometrischen Gebilde, mit weichen wir gewtihnlich zu tun haben-die, so koinpliziert sie tins auch erseheinen mdgen, doch immer verhaltnissmassig ejufach sind-verteilen sich ganz nattrlieb in zwei grundverschiedene Arten: in Linien und Flachen. Es kiinnen zwar Gebilde audi auts Linien und Flachen zusammengesetzt sein-wir beschriinken uns aber auf soiche, die in der Umgebung jedes Punktes dieselbe, Dimensionsanahcd haben. Bei dieser Bescbrankung seheint zwischen Linien und Flachen eine Kluft zu bestehen. Die-bis heute offene-Frage, die ich bier an die Spitze stelle, lautet: ist es wirklich so, gilt dies auch von den allgemeinsten geometrisehen Gebilden? Oder giebt es vielleicht soiche Kontinua, die weder Linie, noch Flache genannt werden kdnnen? Ftir die Beantwortung dieser Hauptfrage bedarf man einer ganz ailgemneinev Definition der Linie und der Flache. Ftir den zweidimensionalen Raum ist die Frage bekanntlicb gekist, oder sie ist vielmehr gegenstandslos, da dort jedes nirgendsdi chte Kontinunm eine Linie ist (die Zoretti Ga'ntorsche Linie nennt). Fur den dreidimensionalen Raum aber besitzen wir keine solche allgemeingtiltige Definition, welche aus den nirgendsdiehten Kontinuen die Linien aussonderte. Viele besebranken sich von vornherein (offen oder stillscbweigend) auf spezielle Gebilde (das gilt auch von den abstrakten, tiefen Untersuchungen von Enriques). Die am besten bekannte, Jordanscbe Definition stebt gerade im Widerspruch zu den geometrischen Fordernngen (wie die " Peanosche Kurve " zei~gt) und interessirt uns, also nicht*. * Dabei ist es noch zu betonen, dass man iiberhaupt nicht die uns interessierende Frage als Untersuchung fiber den Funktionsbegriff auffassen darf. Der letztere scheint zu eng zn sein, es sei denn, dass man ganz unregelmnissig mehrdeutige Funktionen betrachten will. Denn, wie will man z. B. die unten angegehene Kontinua als eindeutige Funktionen auffassen?

Page  127 PBER DIE BEGRIFFE " LINIE " UND " FLACHE "12 127 Eine wesentlich aindere Untersuchung ist die von. Fr~chet, die von vornherein die Existenz versehiedener Dirnensionstypen annimrnt; Fr~echet schreibt z. B. dern Kreise einen hiblheren Dimensionstypus zu, als der Strecke. Diese sehr interessante Idee aber schliesst unsere Frage nicht aus. Diese, wtirde in Fre'chet'scher Terminologie nngefahr so lauten: spaltet sich die Mannigfaltigkeit der den nirgendsdichten. Kontinnen entsprechenden Dimensionstypen in zwei Kiassen oder nicht? Es kdnnte jedoch scheinen. als liessen sich einige Aussagen als aligemeingiiltige DefinitIonen der Linie oder der Flache anfstellen. Ich will nun eine Linie konstruiren die emn Gegenbeispiel zn einer soicher vermeinten Definition bildet und die als Vorbild oder Bestandsteil vieler anderer Gegenbeispiele zu anderen Versuchen der Definitionen dient. Man kdnnte nAmlich meinen dass emn Kontinuum eine F~dche ist, wenn. es mit jeder- es schneidenden Ebene, die einer gegebenen Ebene parallel ist, eine Linie gemnein hat. Nun, diese Eigenschaft besitzt folgendermassen konstruirte Linie. Sei 13 eine auf der Strecke PQ nirgends dichte, perfekte Pnnktmenge; wir verbinden jeden Punikt derselben mit einem ausserhaib der Geraden PQ liegenden Punkte A durch geraden Strecken. Die Gesammtheit dieser Strecken bildet ein Kontinuum (5, das offenbar eine (Cantorsehe) Linie ist. In A richten wir eine Senkrechte AB zu der Ebene APQ. Jetzt ordnen wir den Punkten von AB die Punkte von 13 (und also die Strecken des Kontinuums (S) in der von Cantor angegebenen Weise zu: jedem Punkte der Strecke AB entspricht emn od~er zwei Punkte der Menge 13. Wir betrachten nun Q als orthogonale Projektion der zu konstruirenden Linie. In jedem Punkte der Streeke AB riebten wir eine (ev. zwei) Senkrechte parallel und gleich der ihm entsprechenden Strecke des Kontinnuuns (5; ihre G"'esamtheit bildet das gesuchte Kontinnuum Pt. Dass St eine Linie ist kdnnen wir deshalb behaupten, weil es mit einer ebener Cantorsehen Linie p. Q SV elementarverwandt (homeomorph) ist, die wir folgendermassen konstruiren: Wir verbinden dureli gerade Strecken die sich A B entsprechenden Punkte einer Strecke AB und die Punkte einer perfekten Menge Ts, die auf euler zu AB parallelen Strecke PQ liegt und dort nirgends dicht ist. Es ist offenbar, dass dieses Kontiniunm nirgends dicht ist; also ist es eine Linie. Ebenso ersichtlich ist seine Eleruentarverwandschaft mit dem vorher konstruirten Kontintlnu. Sk ist also eine Linie und doch besitzt es die Eigenschaft mit jeder zu APQ parallelen urid es schneidenden Ebene eine Linie (gerade Strecke) gemein zu haben. Die Linien Sk und ~~ sind noch dadurcli merkwtirdig, dass sie emn Kontinnum enthalten (die Strecke A B), von dem jeder Punkt emn Verzweigungspnnkt. ist. Es klingt in der Tat paradox dass die mehrfachen Punkte einer (Cantorschen, nicht Jordanschen) Linie emn Kontinnum bilden ktinnen*. * Im Raume von unendlich vielen Dimensionen wfirde man mit Hfilfe von ~~ eine Linie konstruiren. k6nnen, von der jede?, Punkt ei- Verzweigu-ngspunkt ist.

Page  128 128 128 ~~~~~~Z. JANISZEWSKI Emn ganz analoges Gegenbeispiel hisst sich ftir eine Definition der folgenden Art konstruiren: Die Umgebung eines Punktes A des gegebenen Kontinuum. 2 ist eine Flaqche, wenn jede mit einern geniigend kleinen Radius urn ihn beschriebene Kugel mit 2 eine Linie (z. B. einen Kreis) gemein hat. Dagegen wird diese Definition wahrscheinlich gut, wenn man dieselbe auch fuir eine unendliche Menge zu A benachbarter Punkte voraussetzt, welche von der Machtigkeit des Koutinnums ist. Eine ahunliche Bemerkung gilt von der zuerst versuchten Definition. Ich gehe zu einem zweiten Beispiel tiber. Dieses beweist: es gibt Koitimua, die keinien einfachen Bogen enthalten. Es ist mir in der Tat gelungen emn Kontinuum zu konstruiren, derart, dass emn jedes Teilkontinnuum emn Huufungskontinuum (eontinu de condensation) enthalt *. Man entbhht sie als Grenze einer Folge von Linien, deren erste die Linie y=sinI- ist, und die folgenden enthailt man nach der Methode der Kondensation der Singularitaten indem man tiberall die einfachen Bogen durch die der Linie ysin - elementarverwandte Linien ersetzt. Man muss nur dabei daftir sorgen, dass die Abainderungen die man macht, gentigend schnell gegen Null konvergieren. Es scheint mir diese Linie insofern interessant zn sein, als alle bisherigen (mir bekannten) Beispiele der Topologie aus lauter geraden Strecken konstruirt waren oder werden konnten, und man meinen ktinnte, dies gelte aligemein. Wir konstruiren jetzt auf dieser Linie als Direetrix einen Zylinder. Die so erhaltene Flache enthdlt Icein Abbild eirter Kreisjhiche. Sie zeigt also, dass die Definitionen einer Flhche weiche sich auf solche Abbildung sttitzen, nicht allgemein sind. Im Vorhergehenden habe ich mich bei der Beurteilung der Definitionen auf gewisse Forderiingen gestiitzt, die leicht zu formulieren sind; z. B., dass emn einer Cantorschen ebenen Linie elementarverwandtes Kontinnum selbst Linie heisse, und ebenso, emn in eine endliche Anzahl von Linien zerlegbares Kontinnum; und dass emn Kontinnum, dass das Innere einer Kugel in Gebiete teilt, eine Flache heisse. Sie sind notwendig und unmittelbar durch den intuitiven Sinn der Worte " Linie " und ",Flache " diktiert; sie scheinen aber nicht hinreichend zu sein. Die eventuellen weiteren Forderungen zu formulieren und ihre gegenseitige Unabhaingigkeit zu beweisen scheint mir hier die nachste positive Aufgabe zu sein. Bevor dies getan wird kann man von keiner Definition behaupten, dass sie allgemeingtiiltig ist. Ich schliesse diese, nur negativen Entwickelungen mit der Bemerkung dass ich die so ganz aligemeine Frage nur zur Orientirung behandle. Ich meine dagegen, dass es bei dem jetzigem Stande der Wissensehaft, die Untersuchungen, die sich auf eiufachere Gebilde beschranken, am ehesten hierauf emn Licht werfen kdnnen. *ich habe hier von Herrn Brouwer erfabren, dass ihm dasselbe Beispiel bekannt war.

Page  129 ZUR ANALYSIS SITUS DERi DOPPELMANNIGFALTIGKETTEN UND DERl PROJEKTIVYEN iRXUME VON D~PNEs KONIG. In diesemn Vortrage beabsichtigen wir die projektiven Raume einer beliebigen Dimensionszahl in bezug auf ihre Emn- und Zweiseitigkeit zu untersuchen. Und zwar so, dass die Resultate, so weit als mdglich, mittels der kombinatorischen Methoden der Analysis situs abgeleitet werden. Dabei werden auch gewisse allgemeinere Resultate Uaber sogen. Doppelmannigfaltigkeiten bewiesen. ~ 1. Die von Dehn und Heegaard begrtindete und von Steinitz weiter entwickelte kombinatorische Theorie * der Analysis situs stiitzt sich auf den Begriff der Zellen (Zn) und Sphdren (Sn) (der obere Index bezeichnet stets die Dimensionszahi). Als 0-dimensionale Zelle (ZO) wird der Punkt, als 0-dimensionale Sphare (SI) das Punktepaar bezeichnet, als Z' eine von zwei Punkten (ein'er SO) begrenzte " Strecke "; zwei ZI mit gemeinsamer Grenze bilden em 5I', U. S. W. Im Aligemeinen: emn Zn1 wird dtirch ein S',' begrenzt und zwei Zn mit derselben 5I, —Grenze bilden eineS" Unter der Indicatrix einer S" = (ZO, Z 0), die aiis den Punkten Zo und Zo besteht, versteht man eine der zwei Reihenfolgen dieser zwei Punkte. Ebenso definirt man die zwei Indicatrizen einer ZI, die durch diese So begrenzt wird. Im Ailgemeinen hisst sich die Indicatrix der n-dimensionalen Zellen und Spharen in rekcursiver Weise folgendermassen definiren. ISt Sn-1 die Grenze einer Zn1, so bestimmt die Indicatrix von Sn-1 eine Indicatrix fttr Zn, und umgekehrt. Die Indicatrix ftir Sn = (Znj, 4n) wird angegeben, indem man ftir Zn und Z4 je eine Indicatrix in der Weise bestimmt, dass die gemeinsame Grenze S U-1 von Zn und 4n hierdurch einmal die eine und einmal die andere Indicatrix erhalt.-Die IndicatrixBestimmung der Zellen und Spharen lauft also lim Grunde stets auf irgend eine Angabe tiber die Reihenfolge zweier Elemente hinaus. Nun erweitert man die Bedeutung des Wortes Sphdre, indem man die aus Sn durch interne Tr-ansformation hervorgehende Mannigfaltigkeit ebenfalls als n dimensionale Sphiire bezeichnet. (Durch Sn sollen stets nur die Sphdren im engeren Sinne bezeichnet werden.) Die " interne Transformation " (ftur ihre prazise Erkiarung verweisen wir auf die erwahnten Arbeiten t) bestehit aus einer Reihe *Dehn und Heegaard: "Analysis situs," Encyki. d. Math. B. iii. p. 153 (1907); Steinitz: Sitzungsber. d. Berliner Math. Ges. B. 7, p. 29 (1908).-~ 1 unserer Note enthuilt im Wesentlichen nur Erkhirungen dieser zwei Arbeiten in einer unseren Zwecken angepassten Darstellung. t Dehn-Heegaard, p. 159; Steinitz, p. 32. IL C. I I. 9

Page  130 130 DINES K6NIG von Schritten, die-dureli Einftthrung je eines neuen Z K- ein ZK durch zwei ZK ersetzen. Diese Schritte werden der Reihe nach fttr K = 0, 1, 2,..., n ausgeftihrt. Nun kann also eine Sphare durch eine beliebige Anzahl von Zellen gebildet werden. Auch diese aligenmeinere Spharen n-ter Dimension kdinnen gewisse Z11+ begrenzen.Der Begriff der Indicatrix lasst sich auf diese aligerneinere Spliaren und Zellen unmittelbar tibertragen. Eine enciliche Menge gewisser Zn bildet eine "geschiossene JMannigfaltigkeit" 9ycn, falls jede Zn-' die zur Grenze einer dieser Zn gehdrt, genau zweien dieser Zn angehdrt. Ftir die allgemeine ~)1T' lasst sich der Begriff der internen Transformation ebenso erklqren, wie ffir die Sn1. Wir wollen der Einfachheit halber den Begriff der geschlossenen Mannigfaltigkeit noch dadurch einschrdnken, dass wir annehmen, dass je zwei jener Z", welche die Mannigfaltigkeit bilden, htiestens eine gemeinsame Grenz-Zn-I besitzen *. Die Zn, welehe die Mannigfaltigkeit NJn bilden, so wie die Zn-i, welche die Grenze dieser Zn bilden, u. s. w. his zu den ZO (Punkten) herunter, bilden die iconstituirendent Elernente Von s1Thn. Zwei konstitnirende Elemente Z i und ZiK heissen inzident, wenn Z i konstituirendes Element von Z i~K ist. Eine q)Vn heisst bekanntlich zweiseitig, falls man ihren Zn je eine Indicatrix in der Weise znordnen kann, dass hierdurch jeder konstituirenden Zn-i beidemal versehiedene Indicatrizen zugeordnet seien, d. h. wenn das verallgemeinerte Mdbins'sehe Gesetzt ftir die Zn erftillt werden kann. Ist dies mdglich, so ist es -wenn wir noch 9Jnals zu~sammenhdngend+ voraussetzen-auf zweierlei Weise m~zglieh und durch diese zwei Mdgliehkeiten sind die zwei Indicatrizen einer 9)V definirt, die duirch 'S)Mn und s)Nn bezeichnet werden kdnnen. Dureli die Angabe einer Indicatrix fair ~9NI' ist also jeder ihrer Zn eine Jndicatrix dem. Mtibius'sehen Gesetze entsprechend zugeordnet. Wir werden von zwei mit Indicatrizen versehenen Zn der ~JtN sagen, dass sie " dieselbe " Indicatrix 4und Z4 erhalten haben (in Zeichen: 4 n 4Z), wenn sie zur selben Indicatri'x von 9J1" gehdren. (Dies hat nattirlich nur ffir zweiseitige s7Jtn einen Sinn.) 2. Seien die konstituirenden Elemente der zusammenhangenden geschlossenen und zweiseitigen Mannigfaltigkeit NMn und es sei eine Beziehung zwischen diesen Elementen in der Weise gegeben, dass j eder Z' in umkehrbar eindeutiger Weise emn Z' von derselben Dimension entspricht, und zwar so, dass (1) stets Z' +j 4' ist und (2) inzidenten Elementen wieder solehe entsprechen. Eine solche Beziehung soll als eine _R* Diese Eigenschaft bildet keine wesentliche Einschriinkung, da sie sich durch interne Transformation stets erreichen lisst. tVon M~bius ffur n =2 formulirt: Werke, ii. p. 475. Wir wollen 9)Pn" zusammenhiingend nennen, wenn von jeder ihrer Zn aus jede andere durch n-dimensionale Nachbarzellen sich erreichen lIdsst. Zwei Zn heissen dabei Nachbarzellen, wenn ihre Grenzen eine gemeinsame Z" — besitzen,

Page  131 ZUR ANALYSIS SITUS DER DOPPELMANNIGFALTIGKE1TEN11 131 Beziehung fuir s)Thn bezeichnet werden. Z"~ und ZK heissen " entsprechende" Elem ente. Ist ftir ein Z~ eine Indicatrix ZK (durch das oben angegebene rekursive Vertahren) gegeben und ersetzt man in dieser Indieatrix-Bestimmnng jede Zelle (von delDimension 0, 1, 2....bis zu n) durch die entsprechende, so wird hierdureh ftir Zn eine bestimmte Indicatrix-sie sei Zn -angegeben. Sie soil die der Indicatrix Z n von Z entsprechende Indicatrix von Z,' genannt werden. (A) Ist nun fiar (lie zweiseitige ~)J ejne R-Beziehung fest gegeben, so ist farjedes p entweder stets Z It = Zu oder stets 4n =j Zn, also Zu, = 4'O. Sind namlich zwei Zellen Z4 und Z4 von V'so mit je einer Indicatrix versehen, dass ihre einzige gemeinsame Z11' beidemal verschiedene Indicatrizen hat, so gilt dasselbe ftir die entsprechenden Indicatrizen der entsprechenden zwei n-dimensionalen Zellen, d. h.: dann und nur dann haben 4 und Z4 dieselbe Indicatrix, wenn diess fiiir die entsprechenden Indicatrizen der entsprechenden Zellen der Fall ist. Da nun s))Jn zusammenhangend ist, tibertrdgt sich dieses Resultat unmittelbar auch auf den Fall, wo 4n nnd 4u, beliebige, nicht speziell Nachbarzellen von TVZn sind.,Durch das jetzt bewiesene Resultat zerfallen die mit einer bestimmten R-Beziehung versehenen Mannigfaltigkeiten in zwei Klassen I und II1, je nach dem stets Zu=Z4 oder stets Zn= Zn ist.-Eine bestimmte Indicatrix von g2yn' bestimmt ffir die Zn von 91Tn je eine Indicatrix. Ersetzt man diese Indicatrizen durch diejenigenl, welche der (urspriinglichen) Indicatrix der entsprechenden Zellen entsprechen, so geniigen, wie wir sahen, auch diese nenen Indicatrizen dem Mdbius'schen Gesetz und bestimmen also eine Indicatrix fuir Nn1'. Diese soil die der ursprttnglichen Indicatrix entsprechende Indicatrix von qJIn genannt werden. V9PI gehbirt also zur Kiasse I oder II, je nach dem die entsprechende Indicatrix von DJIn mit der urspriinglichen iibereinstimmt oder nicht. Es soil nun speziell qVPn die Sphare S 1(~ Z2) bedeuten und die gemeinsaine Grenze von Z4 und Z4 sei 81-1 Da unter den. konstituirenden Elementen z, 4,z, 4,... z n1 4n- z, 4 i der Sn nur zwei Zellen der selben Dimension vorkommen, so ist ffir S1, nur eine ]R-Beziehung mdglich, durch weiche die "diametralen" Zeilen ZK und ZK als einander entsprechend gelten. In weiche der zwei Kiassen gehbrt nun Sn? Eine Indicatrix von Sn wird angegeben, wenn man zu Sn~-1 als Elemente von 4u und dann als dem Elemente von 4" die zwei verschiedenen Indicatrizen in der einen oder anderen Weise zuordnet. Aus einer Indicatrix von Sn ergibt sich die entsprechende-in Foige der rekursiven Indicatrix-Bestimmuing)-dadurch, dass man (1) die Indicatrix 9-2

Page  132 132 DPNES K6NTG von Sn-i durch die entsprechende ersetzt und. (2) Zn und. Z n vertauscht. Ist also ftir SI,' die entsprechende Indicatrix mit der urspriinglichen identisch, so ist (da dann nur (2) auszuftihren ist) ftir 5n, die entsprechende von der ursprtinglichen verschieden, und. umgekehrt. Also gehdrt Sn zur Kiasse I oder II, je nach dem Sn-I zur Kiasse II oder I geh~irt. Man sieht aber unmittelbar, dass So = (Z, Z0)zu Kiasse II gehbrt, also gilt folgender Satz: (B) SI' geh~rt in die Klasse I oder II je nach dem n ungerade oder gerade ist. Mann kann leicht einsehn, dass die Zugehtirigkeit zur Kiasse I oder II (ebenso, wie die Emn- oder Zweiseitigkeit) eine in bezug auf interne Transformationen invariante Eigeiischaft der Mannigfaltigkeiten bedeutet *. Diess bedeutet genauer Folgendes. Aiis der mnit einer R-Beziehung versehenen V1 soil dureb eine interne Transformation NJ4 entstehen und es soll ftir 'N eine 1?-Beziehung, BR, von der Beschaffenheit existieren, dass zwei Zellen Zn und Z" von 91 n~ gemass R, nur dann einander entsprechen, wenn dies gemaiss 1? ftir diejenige zwei Zellen Von q1Tn gilt, aus denen 4 ' resp. Z4 entstanden sind. In diesem Falle geh6rt )in in bezug auf RI in dieselbe Klasse I oder II, wie ~)Z in bezug auf 1?. ~ 3. Sei wieder fuir die allgemeine zusammenhaingende und. zweiseitige Mannigfaltigkeit gJZn mit den konstituirenden Elementen eine -R-IBeziehung gegeben, welehe die Zellen Z K und. Z (K= 0, 1, n.. p = 1, 2,..., iK) als einander entspreehende Zellen charakterisirt. Die Zellen in (I) lassen sich also in Paare versehiedener Zellen derselben Dimension zusammenfassen. Sind (Z>, Z) und (Z, Z) zwei solehe Paare und sind. Z' und. Z.zu einander inzident, so gilt das selbe fair Z4 und.Z~ da ja eine R-Beziehung r 01~~~~~~~~~~~~~~ Inzidenzen niemals zersttirt. In diesem Falle (d. h. wenn Z' mit Z I oder mit ZS iuzident ist) werden wir diese zwei Paare selbst inzident nennen. Als Dirnensionszahl des Paares (Z>, 4) soll die Zahl Kc bezeichnet werden. Es kann nun eventuell eine Mannigfaltigkeit sN1 so angegeben werden, dass zwisehen ihren konstituirenden Elementen und. den oben genannten Paaren eine umkehrbar eindeutige Beziehung in der Weise besteht, dass jedem dieser Elemente emn Paar von derselben Dimension entspricht und. inzidenten Elementen inzidente Paare entsprechen. In diesem Falle heisst 9)Jn eine Doppelrnannigfaltigkeit von S))Zn Ist die R-Beziehung fuir qThn gegeben, so ist Rn)Z durch NniTh vollstandig bestimmt, da es bei der Definition der Mannigfaltigkeiten nur auf die Inzidenzen ankommt. Gehdrt nun )Jnzur Klasse I, und. denkt man ihre Z'n dem MdNlbius'schen Gesetze entspreehend mit Indicatrizen versehn, so lassen sich diese Indicatrizen unmittelbar auf die Z11 von s)Nfn tibertragen, so dass also in diesem Falle auch XRrt zweiseitig ist. * Diess ist von. Wichtigkeit, wenn man von der kombinatorischen Theorie zur analytisch-geometrischen Theorie iibergeht, wo die Mannigfaltigkeiten als unendliche stetige Punktmenagen betrachtet werden; da zwei 11Wn, von denen die eine aus der anderen durch interne Transformation hervorgeht, im Sinne der Analysis situs aequnivalente Punktmengen hedenten.

Page  133 ZUR ANALYSIS SITUS DERl DOPPELMANNIGFALTIGKEITEN13 133 Ebenso sieht man, dass auch umgekehrt falls die Zn von t)IPn dem Mdbius'schen Gesetze entsprechendi mit Indicatrizen versehien werden. kdnnen, sN1 unbedingt zur Kiasse I gehdren muss. Man hat also den Satz: (C))I ist zweiseitig oder einseitig, je nach dem 9T u Kas joenI e~t 4. Will man nun die hier gewonnenen Resultate auf den n-dimensionalen projektiven Raum, KRI, anwenden, so kann. man natiirlich analytische Hilfsmittel nicht vermeiden, da Kn, als die Gesamtheit der Verhdltnisse (XI 2... Xn nA selbst analytisch definirt wird. Den bisher kombinatorisch definirten Begriffen entsprechen in bekannter Weise, was bier wohi nicht naher beschrieben werden muss, gewisse analytisch definirbare Gebiete des N-dimensionalen Raumes, d. h. der Gesamtheit der " Pnnkte "(XI, X2,.. -, XjAT, wenn nur N geniigend gross gewAhit wird. Die in der kombinatorischen Theorie definirte Aussage " durch die 1R-Beziehung von g1Th erscheint 9NPI als Doppelmannigfaltigke-it von 91.Th" bedeutet in der analytisch-geometrischen Auffassung der Mannigfaltigkeiten Folgendes: " Man kann. die Punkte von qTn und. VIZ, in vollkommen ein-zweideutiger und stetiger Weise auf einander so beziehen, dass zwei Gebiete (Zellen) von 9Mn, die dureli die R-Beziehung einander entsprechen, demselben Gebiete von VJ1' entsprechen sollen." Diess lasst sich auch nmkehren. Besteht namlich eine ein-zwei deutige und stetige pnnktweise Beziehung zwischen den Pnnktmengen 9Mn und _Nn so kann. man diese zwei Mannigfaltigkeiten aus Zellen in der Weise aufbauen, dass 9)n-'im kombinatorischen Sinne-als Doppelmannigfaltigkceit von ~Y erscheint. Dabei werden die entsprechenden Zellen der " Mannigfaltigkeit " 9NTh aus entsprechenden. Punkten der " Punktmenge" 9Thq gebildet. Die n-dimensionale Sphdre kann man als Punktrnenge am einfachsten durch die Gesamntheit derj enigen Punkte (XI, IX2,..,Xn, Xn~i) des it + 1 -dimensionalen Raumes reprasentiren, fuir die 1 2 2 =1+ ist. Lisst man den zwei diametralen. Punkten (a1I, a2, *. a ICt1) und (- a,-a,.., - a+,) dieser Sphare den einzigen Punkt (a1 a2:...:t~l des n-dirnensionalen proj ektiven Raumies, KR1, entsprechen, so ist hierdureli eine ein-zweidentige und stetige pnnktweise Beziehung ftir KR und SI' gegeben. Werden also die diametralen Elemente der Sphdre als eiinander entsprechend betrachtet, so ist Sit Doppehnannigfaltiglceit f~r W1'. Unsere Satze (B) und (C) ergeben. also anch das Resultat * Der n-dirnensiona le projek-tive Raum ist einseitig fi)r gerades n und zweiseitig fidr ungerades n. * Diesen Satz habe ich schon im Archiv der Mlath. B. 19, p. 214 ausgesprochen. Dort wurde er ausfiihrlich nur fuir i= 2 und 3 bewieseii.

Page  134 ON THE THEORY OF CONNEXES BY D. SINZOV. I propose to give an account of the results I have obtained in the theory of connexes. Published only in Russian, they are not easily accessible to the mathematicians of other countries, and I hope it is not superfluous to present such an account. A. Clebsch, in his remarkable paper, "Ueber eine Fundamentalaufgabe der Invariantentheorie," Gottingen Abh. xvII. 1872, had pointed out a new and vast domain of geometrical study-the investigation of configurations with composed elements (point, line) in the plane, (point, line), (point, plane), (line, plane), (point, line, plane) in the space. He himself published his investigations only on the first of these configurations-on the ternary point-line connex: "Ueber ein neues Grundgebilde der analytischen Geometrie der Ebene" (Gotting. Nachr. 1872; Math. Ann. B. v. 1872). Important contributions to this theory were made by Mr Kyp. Stephanos ("On the conjugate connex," Bull. Darboux, (2) IV. 1880), MM. G. Darboux, L. Autonne and H. Poincare on the algebraic integration of the differential equation of the first order and degree, connected with the connex (m, 1), and by Mr G. Darboux on the theory of Singular Solutions. Further particular cases investigated are those of (1, 1)-bilinear connex, connected with collineation in the plane (A. Clebsch and P. Gordan and others*), of (2, 1) (Godt, Amodeo), and generally (m, 1) (above mentioned, Miillendorf, Voss), (2, 2) (Armenante, Peano). There should be mentioned also the papers of G. Fouret on the systems of curves defined by an algebraical differential equation of the first order (Bull. S. Math. France, II. v, vi, xix). At first inspired by the exposition of F. Lindemann (Clebsch-Lindemann, Vortrtge iib. analytische Geometrie, B. I.) I tried eighteen years ago to extend the results above mentioned to the point-plane space connex (" The theory of connexes in space in connection with the theory of differential equations of the first order," Kasan Univ. 1894-1895, and separately 8vo. 254 pp.), a special case of which-namely (2, 1)-was studied by R. Krause (M. An. 14)t. I have given an account in Bull. Darboux (2) xxII. I cannot at the present time speak about my paper, the matter having been treated afterwards in a masterly manner by L. Autonne in his prize memoir (Mem. cour. de Belgique, 1902) and by M. Stuyvaert, "Recherches relatives * Battaglini, Lazzeri, Voss; on collineation: Pasch, Muth, Rosanes, Kraus, Keller, St. Smith. t A number of papers on the collineations in space and the bilinear space connex (Battaglini, Predella, Pittarelli, Loria, Segre, Richelot, St. Smith, Hirst, Lazzeri, Muth), and the papers of G. Fouret on the implexes are also to be mentioned.

Page  135 ON THE THEORY OF CONNEXES 135 (ix connexes de l'espace," Mem. Belg. Coll. 8vo. t. 61, 1902. I would note only, that on the last pages I give a generalization to n dimensions, i.e. I mark shortly the character of the configurations with element: point, hyperplane in a space of n dimensions, and of the integration problem connected with it. The incompleteness of the theory of the plane (ternary) connex makes it necessary to return to this more simple case. In my further publications I had concentrated my efforts on the theory of conjugate connex and on the influence of certain singularities of the given connex on the class and order of its conjugate. Instead of Clebsch's definition of a singular element, as an element to which corresponds more than one element in the conjugate connex, it seems to me to be more convenient to regard the singular elements as those whose point is a singular point of the corresponding connex-curve, or line is the singular line, or both together-or [which is quite the same] as those for which the equations of the polar pair of the element (X, U) /(xu)=0, SXf/'=0, Uf;'=0.....................(1) are satisfied for every X, or for every U, which belong consequently to all the polar pairs of elements (X, U), whatever may be the point X, if the element is the pointsingular one, i.e. if f/, = 0, (i= l,2,3)..............................(a), or whatever may be the line U, if the element (x, u) is the line-singular one, i.e. if f< = 0, (i= 1, 2,3)............................. (b), or finally for every element (X, U), if both systems (a) and (b) are satisfied simultaneously-in which case the element (x, u) may be called a properly-singular element of the connex. By the convenient modification of the way of finding the order and class of the conjugate connex it is possible to appreciate the influence of certain singularities: 1. A principal point (Hauptpunkt) diminishes the class of the conjugate connex by n, the order remaning unaltered. Reciprocally a principal line diminishes the order of the conjugate connex by m, the class remaining unaltered. 2. A pair of properly-singular elements (l, v) (i.e. o 1 elements forming a pair of curves) diminishes the order of the conjugate connex by iA, the class by v. 3. co 2 properly-singular elements, forming a coincidence, defined as the intersection of two connexes (mu, v), (a/', v') evoke diminutions: Am' = (3m - 2') vv'+ (3n - 1) (,v' + vU') - [(,a + t') vv' + (v + v') (,av' + v')], a.Z' = (3m - 1) (Uv' + v.') + (3n - 2),u' - [(t + i') (uv' + v/L') + (v + v'),aU']. The demonstration of the last theorem is based on the determination of the number of elements of intersection of the four connexes having a coincidence in common (Goettler). It is perhaps noteworthy that in calculating examples to these theorems I have found an example of a connex ICX2Lx32 1 + K32x 'U1 2 4- +cxx23 + Kl'-'t= - 0,

Page  136 136 D. SINZOV whose conjugate configuration is not a connex, but the coincidence 2y3 Y23 _y33 K1 V1 KVV2 K22 3 In following paper (1910) I have extended the results above indicated to the space point-plane connex: a principal point diminishes the class of the conjugate connex by n2, a principal plane diminishes the order by m2, a pair (curve in space, developable surface) (A,, v) of properly-singular elements depresses the order of the conjugate connex by gu, the class by v. Further formulae of reduction for the cases of 0 2, o 3 and co4 properly-singular elements defined as intersection of 4, 3 and 2 connexes, are too complicated to be given here. I pass to another contribution to the theory of connexes,-the introduction of the osculating bilinear connex ikfik" X Uk = 0 (i, = 1,2, 3), (fik' = & ) If we define the singular elements of a coincidence f(x, t1 =0, b (x, u) = 0, as those for which any one or both systems of equations are satisfied: af af af ax, x2 ax. -( ) a, a ****(........................ 1ax, a2 ax3 a/ af af' au, au, au3 aU1 a2 _ aU3 We can easily prove that the osclation-element (x, ) is the properly-singular element WVe can easily prove that the osculation-elemen~t (x, uc) is the properl~y-singular element of the coincidence-intersection of the connex with the osculating bilinear connex, both systems (a) and (b) being satisfied. The analogy with the theorem about the intersection of a surface with its tangential-plane gives an idea how to classify the elements of connex in accordance with the behaviour of the osculating bilinear connex, i.e. according to the properties of the collineation defined by it. The equation of the third degree fil-K f21 f31 l2 f22-K f3 =0, fl/3 2fi f33-K or K3-iK2 + i'K- = 0 defines the values of K, corresponding to the invariant points and lines of the collineation J; X1+ +f2kX. +.3kX3 = K. Xk, Jfl U1 + f2 U2 + Ujf. U= K. j.

Page  137 ON THE THEORY OF CONNEXES 137 A. A 0. I. 1. K, K1 $ K and all real. Hyperbolic element. 2. K1 + K2 $ K3 but two are complex. Elliptic element. 3. K, =-K K 3. Parabolic element. 4. K, K2 = K3. Only one invariant point, and one invariant line. II. One of the invariant points coincides with the point of osculation-element (X = x) [point-singualar element of the principal coincidence]. The same cases are to be distinguished. III. One of the invariant lines coincides with the line of the osculation-element (U = u) [line-singular element of the principal coincidence]. IV. Both the point and line of osculation-element are invariant point and line [properly-singular element of the principal coincidence]. B. A = 0. Collineation is a degenerate one. I. Elements analogous to the inflexion-point of a curve. II. X = x for K = 0. Point-singular element of the connex. III. U = u for K = 0. Line-singular element of the connex. IV. X = x, U = u for K = 0. Properly-singular element of the connex. Very interesting is the case of n = 1 i.e. of connex (m, 1), A1Au + Au2 + A33 = 0. Then the line-singular elements of the principal coincidence are determined by the equations A, (x) A2(x) A3 (x) XI X2 X3 i.e. their points are principal points of the connex. They are named in this case critical points of the equation, and the classification above given coincides with that given by H. Poincare. The same method is applicable to the classification of the elements of the space point-plane connex and of point-hyperplane connex in the space of n dimensions. I adjourn to another occasion the account of my papers on the point-line-plane connexes.

Page  138 ON PAIRS OF FRENETIAN TRIHEDRA By NICHOL~As HATZIDAIKIS. A. Historical notes. The first attempts at a generalization of the formulae of Frenet are those of Ph. Gilbert (cfr Enz. d. m. WV. iii. D. 3, p. 168), of Serret (Galcul Duff. et Int., but only on normials of surfaces) and of Aoust in his various essays on inclined curvature (courbure incline'e) (0.11. v. 57th, pp. 217-9 (1863),-Annali di lMlatematica, Ser. I, v. 6th, pp. 65-87 (1864),-Ser. II, v. 2nd, pp. 39-64 (1868-9), —Ser. II, v. 3rd, pp. 55-69 (1869-70), —Analyse inflnite'siinale des courbes trace'es sur une surface quelconque, Paris, 1869)*. Prof. K. Zorawski, in recent times, has considered, in the Polish journal Prace Matemat~yczno-fiz~yczne (v. xvii, pp. 41-6: lUber Kritmmungseigenschaften der Scharen von Linienelemtenten), such a general trihedron of straight lines with applications to surfaces. Dr Bruno Arndt's inaugural dissertation: Uber die Verallgemeinerung des Kritmmungsbegriffes f~r -Raumkurven deals, also, with similar considerations. Lastly, Prof W. Fr. Meyer has extended the generalized formulae of Frenet to the n-dimensional space (fahresbericht der D. M.- V., v. xix, 1910: Ausdehnung der Frenet'sch~en, Formeln uind verwandter auf den -Rn,) and, more recently, he has published a separate book: Uber die Theorie benachbarter Geraden und einen verallgerneinerten Kri~mmungsbegriff (1911), which, however, mainly deals with the " shortest distances" of two infinitely near trihedra. B. The generalized formulae of Frenet. 1. Let a direction TT' in space of three dimensions be given through its three Cosines A, B, F, functions of an independent variable s. The direction HH' with the A'l B' p' cosins ~ = __ H ____ Z tevidently perpendicular 'to TT', and N/ A ~ VS~A' V\/AS~ the direction BB', common perpendicular to TT' and HH', with the cosines A=e(BZ-FH), M=se(I"-AZ), N=-e(AH-B=*-), ( I=)) According to Prof. W. Fr. Meyer the generalization of the first three formulae of Frenet was known already by Jacobi. (1A t- A' means -etc. (is

Page  139 ON PAIRS OF FRENETIAN TRIHEDRA 139 constitute, together with TT', a generalized trihedron of Frenet. TT' corresponds to the tangent in the case of a customary principal trihedron of a curve and may be called the primitive or departure direction; similarly, we give the names of principal normal direction to HH' and binormal direction to BB'. It is very easy to show how the ordinary formulae of Frenet can be demonstrated for this trihedron. Putting 1 V -SA'2p (>0), we get H Z A B' = r' = A ',,............... If we further put dA = - /SdA 2, dM = - H ScdA2, dN -Z,1 /dA2, the direction ( I1, HI, ZI) is the same as HH' (E, H, Z); indeed, it is SA2 = 1, SAdA = O, SA1 =0, (VSdA2 > 0), and SAA= 0, SAdA + SAdA =0, or - V/SdA2. SA,1 + VSdA2. SA = 0; but SA = 0, consequently SAj1 = 0; then, from the equations SAES =0, SAE = we get 51= =- H H, Z=Z, and therefore ~w~ ZH 1 A=- M'- ' -=- pwhere / —.A). We find, at last, solving the equations 1 1 S '= O0, SAE'= - SA'= SAR Z- S' = (A P' M R' AA A B M N that =H' = - Z'- = - -'R - P R' P R(3). that -=- +, H'=-P+, Z=-P+ R (............3) (1), (2) and (3) constitute the extended formulae of Frenet*. V/SdA2 dZ is the elementary arc of the first spherical indicatrix of TT', or angle of contingence, ds- = its curvature and P its first radius. V/SdA2 _dT is the angle of torsion, ds -P dT 1 = R the torsion and R the second radius. V/Sd=2 = V/d2 + dT2 - dH is the angle ds R d of total curvature, d- = 1p +R the total curvature and K = _-+R the radius of total curvature. All these quantities depend, of course, on the choice of the variable s. If, instead of three directions, we consider three determinate straight lines through the same point, this point describes a curve. p, R, K are then, if we choose the arc of this curve as s, the curvatures of the curve with respect to TT', as Prof. W. F. Meyer defines, or the curvatures of TT' with respect to the curve. The last designation is more general and more exact, the curvature belonging to TT', not r Cfr Serret, Calcul DiTf. et Integral.

Page  140 140 NICHOLAS HATZIDAKIS to the accidental presence of the curve. On the other hand, the first designation is more adapted to the customary conception of curvature, because, if TT' coincides with the tangent of the curve, IHH', BB' become the two other principal directions, 1 1 1 and p, -R the three customary curvatures. C. Relations between the curvatures of two Frenetian trihedra. 2. Let us consider two trihedra of Frenet: (D) [s, TT', HH', BB'] and (D,) [si, TT,', HH,', BB/']; s, being given as function of s (s, = (s)), we can express the cosines and the curvatures of (D1) by means of the corresponding elements of (D) and the relative cosines of T,'T with respect to (D). Let, for this purpose, a, b, c be these relative cosines; we have then Al=Aa+~b+Ac, B1=Ba+Hb+Mc, F=rFa+Zb+Nc...(4), and consequently, if we take (1), (2), (3) into consideration, we get dAl, b = a _ ds, s =A (a _ p + + A) ' ( dBi (S) B(a'-k) + H + r )+M( (M where a' da.) r1,P,, T / b\ r+g7 C\ a _ c b\. = - lc d (s ) = r a-P).........(5); squaring and adding these relations, we get + (S)2 a C b2 C ' f n\ a, af C\ ( = (- - + Sc'2 + 2b - )+2b ( - )= a2+b2 b2 + c2 2ac 2 la b 2 b c ~- + + Sa' +.. = p + - PR Pa +p ab' +R b'c multiplying (5) by A, B, F, or,, H, Z, or A, M, N and adding, we get, since dAl _ 'ds~= p~ etc., x b = a c b,,(S - = a, b,, / \(S) bI, /(S). C + _ P> (s). p=a (-p, P' +)p R f (s) = c' + U V W or x= 2 -U 3VSU2. (7), x = SA'j, t = S31, = SAS1 being the relative cosines of HH1' with respect to (D) and U, V, W abbreviations for a-, b '+a c b * We have given this relation for the first time, in 1906. (Cfr 'ErrerTpis Tro eOvLKovo IIave7rLaTe1,cLov, pp. 349-354: "Generalization of a formula from the theory of the surfaces.") It has been found independently by Dr Arndt in his dissertation (cfr Historical Notes).

Page  141 I ON PAIRS OF FRENETIAN TRIHEDRA The relative cosines of BB1 are consequently b + c+2 ac bW-cV __be'_ - _cb'_ + _ - _ bW-cV R P I = 0. (ba - c)) =. U cU -W ca - ac' - bp + VSU2 )a' b\ a?)( b+ a~ 2+ b2 ca (IV-bU zabb - ba' -{ n =. (a)- bx) = = 0. __- b U 0 VOW2 b/,b2 /,, a c 2 b2 b\2.........(s). And the absolute cosines of (D,) are the following: A, = Aa + 5b + Ac, etc., =- AUU-~AW etc. U(c - Ab) + V(Aa- -Ac) + W(Ab - ~a) nAI~e —, = c -. etc. 3. Let us now calculate the torsion of (D,). Differentiating the formula AU+~V+AW Vsu2 we get ds 4 ~; + = (~) = A UU + A FA + A WI + UA' + E' + WA', wege Xs~ 141 = -) or d-, /.VSU+ (VSU2)'=AU'+=V +ATW+ U ~+ V (- + ) + W(- R). ds, P: Squaring this equation and the similar ones for HI, Z, and adding, we find _J/2,/U W\2 V 2 2 '2 SU2 + (\ 2)2 = 1' + R) ++ ( p(UV'- U'V)+R (VW'-V'W) or 0 (SU2)2=S2.SU'2 -(SUU') +SU2 U- 2W) or =SU ) + + - Tr// +" JW which can be written P K = S(UV' - VU')2 + 4 2 W' '2_ r/U W\ V2 2. 2 +) +- + p(U V- R(V - WV')...(10). It remains to calculate the differences UV'- VU', etc., and to replace by I 1 1 p2 + 2 and by its value (6). The formula thus found expresses the torsion of T T1' by means of a, b, c, but, as this formula is very long, we omit it here.

Page  142 142 NICHOLAS HATZIDAKIS 1 Remark. We can, of course, find an expression for similar to the expression for p (6), which contains 1, m, n instead of a, b, c. But the notion of torsion P2 requires (as well as for the customary torsion of a curve), that is expressed by means of the cosines of the departure direction. An important special case is that of constant cosines a, b, c. The preceding formulae for cosines and curvatures become then shorter, '()2/a C \2 b2 a2+b 2 b2 +c2 2ac P -P K ~ 2 + R2 PR............................. b a c b P_ _ P R R V /P R) i+ 2 K — c4 2K2 P R) + K2......(7'). b2 + c2 ac C a\ a'+b2 ca 1 0 R P M ~~\P R P R 1= b, m=6, n=0-...... (8'). -; 1 (1' 1 (1<\2 j /KrU _W 2 _ V2 i pK.2 -- ~ L (- R) P('))]..... (10). 4. Renmark. (D,) is a Frenetian trihedron with respect to the fixed axes, but evidently not with respect to (D). We can, however, from TT,' as departure direction, construct a third Frenetian trihedron (Dr): T T1', HHr', BrBr with the cosines da bdc db a, ), c; x,. =,etc.; 1,0. d; (82= 1), etc.: da /2tda\2 Sds, / Sds, da Xr, dr a r. dl X,. it is then da, etc. etc.; — ds, Rds- P,.' ds,,. tRc' ds, -,. I /o(da l.2 / dl, where /da2 and sd) are the relative curvatures of T1T1' with respect to (D). (Dr) and (1),) are in general different. They coincide only when P P R R ba c' 7 c =, eqat ins which express that, These conditions are verified, (1) when b = 0, = 0, equations which express that

Page  143 ON PAIRS OF FRENETIAN TRIHEDRA 143 a c TT,' is the direction of the Mozzian axis of (D)*; (2) when + = 0, because this condition is a consequence of b b P R a- c when b < 0, and it verifies also the other equation a c b / b a c P R i P P R b ~ c ~ a' b ' a c The first case, b =, p = p is a special case of (2), for b = 0; we have then a C aa' + cc' = 0, or - =, and the condition a Cc a C +p= 0 becomes p-. R P -P R The condition (2) expresses that H1H,' is perpendicular to the Mozzian axis of (D). D. Relations between the normal and geodesic curvatures of two Frenetian trihedra. 5. The mutual position of the two trihedra can also be defined by the following angles: the angle a- D - TT', T T1', the angle r = HH', SS' and the angle i = HiHf', S', SS' being the direction, which is the common perpendicular to TT' and T T'. It is then very easyt to find three general formulae between the normal curvatures COS _ 1 cos1 _3 1 P -P,,, P -p i, sin v 1 sin _ 1 the geodesic curvatures p p, PI Pi, and the geodesic torsions of the two trihedra 1 dw_ 1 1 dw 1 R s R ds R' R d Rg These formulae are the following +' (s) cos ~ sin \2 Pin P,, Rg c,(s) ds dn g _P,+ ds r........................(11). 4' (s) sin Q cos f RiY Pit Rg They can be shown by trigonometry in the same way as for the curves in my paper (loc. cit.). * Viz. a direction parallel to the momentary axis of the spherical image of (D). t Cfr Nyt Tidsskrift for IMatematik, Copenhagen, v. xiv: N. Hatzidakis, "Om kurveteoretiske Invarianter."

Page  144 ZUM 6-EBENENPIIOBLEM IM P4 VON ROLAND WEITZENB6iCK. Es gibt im 114 " im allgemeinen " 5 Gerade, weiche 6 gegebene Ebenen schneiden. Es wird geometrisch erlautert, wie man diese 5 Geraden erhalten kann, und es wird eine Gleichung ffinften Grades aufgestellt, weiche das Problem 16st. Der Gegenstand des Vortrages wird in einer ausftihrlichen Arbeit in den Wiener- Berich ten. behandelt werden (voraussichtlich Januar 1913).

Page  145 SUR LES EQUATIONS DIFFERENTIELLES DE LA GEOMETRIE PAR JULES DRACH. La the'orie d'inte'gration, que j 'ai de'veloppe'e Sons le nom de the'orie de la ratioiwalite' ou inte'gration logique (et dont on trouvera les points essentiels dans ma Communication au Congre's, Section d'Analyse), donne pour les equations diff~rentielles de la Ge'ome'trie des rnithodes re'guli'e pemett ant de pr 'voir des reductions dans la difficulte' de l'inte'gration, c'est-ah-dire uane re'duction du groupe de rationalite' On rame'ne ainsi des propositions classiques, dues 'a linge'niosite' des ge'ome'tres, a leur source analytique commune et l'on pent, sans difficnlte', en indiquer de nouvelles. J'appelle eqnations diffi~rentielles de la Ge'ome'trie, celles oii interviennent on des fonctions arbitraires ou des fonctions que ion regarde comme donne'es mais dont on ne precise pas la nature transcendante. Le domaine des fonctions que l'on doit regarder comme donne'es on. connues, se compose done des fouctions rationnelles de certains elements qni out kt4 repre'sente's par un signe explicite (j e dirai explicite'es) et cc domaine comprendra touj ours, avec une fonction f (x, y) de deux variables par exemple, toutes ses de'rive'es a, a. Ainsi il est bien entendn qu'on n'explicite pas un signe fonctionrnel d'operation: log it on sin ut ne font pas ne'cessairement partie du domaine qui comprend ut; les seules fonctions de u qui appartiennent touj ours "acc domaine sont les fonctions rationnelles de ut et de ses de'rive'es dont les coefficients sont rationnels par rapport aux autres elements du domaine. L'incertitude oi Nu'on est de la nature des transcendantes donne'es ne permet pas (labontir 'a des conclusions precises sur l'integration, sauf dans le cas ge'ne'ral; ce n')est que si les transcendantes sont de'finies par un syste'me irre'ductible 'a partir des elements rationnels absolus (variables et constantes arbitraires) que la the'orie de l'nte'gration logique pent s'appliquer jusqu'au bout. Ii s'agit donc seulement ici d'une application particulie're et partielle de la the'orie de la ratiovalite': les propositions analytiques ne sont pas distinctes de celles qui interviennent dans cette theorie. M. C. IL 10

Page  146 146 JULES DRACH 146 JULES DRACII~~~~~~~~~~~~~~~~~~~~~~~. Equations du premier ordre. 1. Soit une equation du premier ordre dv = md'a...................(1), oii m est une fonction donnie de u, et v. Supposons que l'on connaisse, pour les courbes ~b(u, V) = const. qui satisfont a' le'quation. (1), une proprijte' ge'orntrique: de quelle utilite' cette connaissance peut-elle 9'tre pour l'inte'gration de (1)? Conside'rons d'abord le cas oii les courbes e'tant trace'es sur une surface (S), d'e'lement line'aire donne' par dS2 = Ed u2 + 2Fdudv + Gdv2, la proprie'te' ge'ometrique en question se conserve dans une deformation continue de (S); elle s'exprime alors par une relation, liRp I(011b(), 12 (j),...,~ fi, 11 (f2),..)=..........(2), entre les invariants de Beltrami de la fonction 0b (c'est-ah-dire les parame'tres diff6 -rentiels A (P), A2 (sb), A (sb, AO4),...) et les invariants absolus du ds2 (on de Minding) de'duits de la courbure totale fl par l'application 'a fl des operateurs A (s), A2 (sb).. Cette relation (2) pent e'tre regarde'e comme une forme re'duite de la relation oii (P est une fonction arbitraire de b Si la relation (3) ne renferme qn'en apparence la fonction CP, c'est-ah-dire si elle garde la me'me forme en 0b, a, a. qnel1 que soit (1? nons ne tirons ancun parti de la connaissance de (2): elie est simplement le'quation aux de'rive'es partielles au qui de'finit le rapport m =. Elle ne renferme donc qne le quotientauess av av de'rive'es. Exernples: La relation AO = 0; la relation = F (fQ, A (~2),.. oiu est la courbure ge'ode'sique de la courbe ~ const. et F une fonction quelconque des invariants absolus; la relation qui exprime que les lignes pour lesquelles dv =mdu sont des asymptotiques sur l'une des de'forme'es de (5). Si la relation (3) renferme PD, 4, (D'. (Oil D=do I*..) lorsqn'on tient compte des formules de transformation: ait au'"' 2 au2 ~ au} et si on l'identifte 'a l'nne de ses riduites (obtenue en particularisant PD), on obtiendra un syste~me diff~rentiel qui definit, an moyen de 0b, l'une des expressions suivantes: I 4) V' V1// 3 (D '\2 wI' P 2 Y'L'

Page  147 SUR LES 1~QUATIONS DIFFE'RENTIELLES DE LA GE~OME'TRIE14 147 Suivant les cas, on de'duira de (3) par une m~thode re,'gulie're, qui consiste essentiellement dans la formation des conditions d'inte'grabilite' de (1) et (3), C'esta-dire, comme on l'a vu en Analyse, dans I'application de l'ope'ration X(f)- +Mav' au v qui donne par exemple: am a24o ~ X av)+ av av2 a2v nne equation dont l'ordre diff~rentiel ne de'passe pas trois et qni deffinit impiitement l'une des expressions: a3p __20 aq a24 aob aV3 3 aV2 v' av2 av 2 ao Toutes les autres equations satisfaites par 0b sont des consequences de cette relation unique. On pent donc ajouter 'a e'~quation: X()ap ma= l'une des equations: a9V aV2 ar' aq a30p 3 a2 I2 ~ av av~ 2 v &V av oii B, K, J, I sont connus en u, v sans inte'gration. Les equations qni deffinissent m sont alors, respectivement: am am a2 M X (R) =O, X (K) +K =0, X(J)~J + ~= 0, av av av2 am a3m X (I)+ 21 + -- =0. av av3 On observera que si les expressions de R, K, J, I sont, par rapport aux de'rive'es de in, d'ordre p, e'~quation aux de'rive'es partielles que doit verifier m est d'ordre an plus e~gal 'a (p +3). Si les elemnents qui figurent dans (1) et dans (2) appartiennent 'a un domaine de rationalite6 [A] bien de'flni, on pourra preciser dans cc domaine la reduction nlte'rieure de l'inte'gration de (1), s'il en existe une qui ne soit pas uniquement celle signale'e plus haut. 1?emarque: Tout cc que nous venons de dire s'applique uniquement aux solutions propres de (2), c'est-a'-dire 'a celles qui ne satisfont "a ancune relation d'ordre inftirieur par rapport aux de'rive'es de f. L'e6quation (2) pent posse'der des 10-2

Page  148 148 148 ~~~~~JULES DRACH solutions particulie'res impropres qui pourront e"tre traite'es par la me'me me'thode mais conduiront 'a des conclusions diff~rent es. Ainsi zA2 6=0 qui de'finit sur une surface les lignes isothermes pent posse'der des solutions de AO4 = F (p); cela arrive sur toutes les surfaces dont 1'e'le'ment line'aire pent s'5'crire ds2 = - (d~ + d#) c as-hdr qui sont applicables sur des surfaces de revolution. Pour les solution propres de ZA2 k 0, la me'thode conduit ha la connaissance de J; pour les autres on a K. Exemples: Si la relation (2) est: P1 oii l'on de'signe par p~, le rayon de courbure ge'ode'sique des courbes f (ut, v) =const., elle donne f sans inte'gration. On a de tne'me sans integration, connaissant senlement leur equation diff6 -rentielle, les lignes qui sur une de'forme'e de (S) sont sur des sphe'res concentriques; pour celles qui sont dans des plans paralie'es on connailtra ao= K en me"me temps av2~ ~ ~ ~~a Iipent exister, pour certaines formes du d82, des cas d'exception qui s'interpre'tent aise'ment. La relation A2 F fl (fl) 1 Pob.. donnera, c'est-hi-dire un multiplicateur de 1'e6quation (1) dA - mdu = 0; les relations A (b; 2) = 0 on z(AO4)= 0 donnent senlenment -, c'est-h\-dire la de'rive'e logarithmique d'un multipliav2 a cateur. De me'me si l'on connalit 1'c'quation diffdrentielle dv -mdu= 0.(1)............... des 'lignes p (it, v) = const. qni permettent de mettre -tn element line'aire donne' sons La forme dS2 = 0q2 + 2 oncnat 6galement a2 o- J sans integration. Ii en est de r~epu e aV2 a lignes 0b (u, v) == coust. telles que ds2 =cos2 a dc02 + sin2 ad#*2. Enfin si la relation (2) est

Page  149 SUR LES E~QUATIONS DIFFIERENTIELLES DE LA G]~OMIftllIE14 149 on pelt obtenir sans int6gration: av3 3 Jv2 et l'inte'gration de (1) se rame'ne, 'a deux cjuadratures pre's, 'a celle de deux equations de Riccati. 2. Il est clair que ce qui permet la the'ori~e precedente c'est l'intervention du groupe ponctuel (P = (P 0) on peut donc appliquer la mithode a' une proprijte' ge'ondtriqiae quelconque du faisceau de courbes 0b (u, v) = const. dorit on suppose connue l'equation ditfrentielle dv= mdu...................(1), cette proprie'te e'tant exprim~e par une relation, de nature quelconque, entre les invariants ge'ome'triques, qui se ramene, en derniere analyse, 'a une relation entre Um v~, _ '- P3ar exemple si l'on suppose que sur la surface x = x(U, v), y = y(u, v), z = z(u, v).(S)....... les conrbes de'finies par (1) sont planes, ou sphe'riques, ou trace'es sur des surfaces alge'briques de degre connn, on les obtient sans integration. Ainsi, soit AQJ) +m O= 0...............(a), dans l'hypothe'se o&" les courbes 0 (ui, v) = const. sont planes on peut, si elles ne sont pas des' droites, de'finir l'inte'grale de (a) par la relation: z = ax + 13y + ry oii a, 13, 'y sout des solutions de A (s6) = 0. Deux d'entre elles sont des fonctions entie'rement de'terrniine'es de la troisie'me. En appliquant l'ope'ration A (f) on obtient les equations A (z) =a A (x) + 8A (y), A, (z) =aA2(X)+ +3A, (y), avec A2 (C)=A(A (A)) et l'ensemble de ces trois relations d~termine a, 13, ry. Deux de ces expressions a et 13 seront d'ailleurs des fonctions de ry, si ry est variable. L'equation aux de'rive'es partielles que doit verifier m est simplement: A (z) A (x) A (y) A2(z) A2(x) A2(y) =0. A3(z) A3(x) A,(y) Le cas d'exception, oii les lignes p=coust. sont des droites, se reconnai't acc que: A2(z)_ A2(x)_ A2(y). A (z) A (x) AQy)' on n'Ia que deux equations distinctes pour trouver a, 13, ry et les droites sont les axes des faisceaux de plans ainsi obteuns. D'ailleurs elles sont 'a priori connues

Page  150 1010 150 ~~~~~JULES DRACH puisqu'en un point de (S) 1'e'quation (1) donne la direction de la tangente 'a la courbe. On connait de me'me aosur les lignes telles que la de'rive' de f suivant la direction conjugue' de la tangente 'a = const. est uine fonction de (sauf si cette fonction est nulle): 1'e'quation (2) est alors DIU O- + F~ a (IV1=FLf) E D D'j+)2F (Dj D _o Ra _ Da On connailt simplernent pour les lignes qui satisfont 'a 'Al(s) 0, A2 e'tant un parame'tre analogue 'a celui de Beltrami mais construit avec une forme diff~rentielle quadratique quelconqu e, par exemple avec Ddu ~2 2D'dudv + D'dv2; etc. On forme sans difficulte' des exemples oiLi la proprie'te ge'ometrique dont nous supposons l'existence n')entraine que la connaissance de a9V3 3 (,V2 i - =I; av \av/ celui signale' plus haut s'appliqne encore lorsque AOb et A~24 sont construits avec une forme quadratique quelconque, de'finie sur (S), au lieu du carr6 de 1ele'kment line'aire. 3. Les raisonnements precedents se transportent, sans y rien changer d'essentiel, aun syste'me complet d'e'quations line'aires aux de'rive'es partielles qui n'adrnet qn'nne solution. On saura, done, par une me'thode re'gulie're, tirer parti de la connaissance d'nne proprikte' ge'ometrique de la famille de surfaces 0b (x, y, z) = const. pour la determination de 4 an moyen d'un syste'me comnplet connul: ax ay az qu'on pent touj ours regarder comme de'finissant les surfaces normales aux courbes de la congruence: d y d x~ Y= Z. On se trouve ici dans le cas o ii, identiquement: X y a Ya-x+ ax-) =0.

Page  151 SUR LES EQUATIONS DIFFE'RENTIELLES DE LA G~OM]ftRIE 151 Si l'on sait, par exemple, que les surfaces (p=const. forment une famille isotherme, on a: On en de'duit la connaissance de -)c'est-ah-dire que (p sera de'fini 'a une transformation line'aire pre~s, par des relations obtenues sans iltegration. Si ces surfaces (p= const. sont des plans ou des sphe'res on des surfaces alge'briques de degre' connu, on me'me des surfaces transcendantes d6tinies par une relation de forme cowatne en x, Y, z: fl (x, y, z) = 0, on les obtient sans integration. Si les surfaces (p=const. sont orthogonales anx surfaces AOb = const., auquel cas z~((, AO)= =0, on pent encore de'dnire de I' (p cest-a'-dire que (p est d'fini aux transformations pre's du groupe line'aire. Enfin (p serait de'fini aux transformations pres du gronpe projectif, si l'on avait A (, A0)-.2 A2 )2 1 xkfl2.4F(x, y, z). 4. La me'thode generale, employee pour e'difier la the'orie de la rationalite', donne aussi un proce'de re'gulier pour diduire de la connaissance d'unie propri~te' geome'trique du re'sectm des courbes (p (in, v) = const., -4'(u, v) = const., difinies respectivernent par les e'quations: au aJv B a* + n a*..........................(2), une reduction du groupe de rationalite' de ces e'qaations. Si l'on connait nne relation R~u~v(p~jo ' a%~** = 0.............(3), d'ordre quelconque par rapport aux de'rive'es de (p et de +f, l'application de l'ope'rateur A (co) qui donne et aussi A (#) + (n -rn) *= 0, a3v A av+ (n m) Ev2 +J 0

Page  152 152 JULES DRACH et par suite n'introduit pas d'e'lements f, autres que ceux qui figurent dans 1?, conduit 'a une relation av Gu I*1~ v ~..........(4), dont (3) et toutes les autres sont des consequences, et dont le premier mernbre est l'un des (juatre invariants: a30p ~ 20 2 ~b j~K,: _=f, - 3=( L'equation: A (F) = A (G)...................(5), pent alors, en tenant compte des equations re'solvantes am a ~~M a?2rnb aM a2m A(Ob)=0, A(K)~K - =0, A(J)+J ~ -=0, A(I)+21 + = =0, a3v a9v D2 av e'tre de'barrasse'e de cJ et de ses de'rive'es. Elle est done une relation connue entre Mais nous savons qu'on. en pent touj ours de'duire une relation analogue donnant explicitement en u, v l'un des quatre invariants: Dra*.~3 3 (7IJ2 Dv a Ii est done loisible de supposer tout de suite que dans R (on dans G) la fonction Jrne -figure qu'au second ordre. Les types possibles de relations entre 0b et 4r peuvent ainsi 9'tre e'tablis a priori et la reduction de l'intq'qration de (1) et de (2) pre'ise'e pour chacun d'eux. On observera qne le'quation (2) est ne'cessairement sp'ciale, et comme on anrait pn permuter le ro'le de et 4r, il en est de me'me de (1). Toute proprie'te ge'ometrique du re'seau ('* qui conduit 'a une relation (3) [c'est-ah-dire qui ne s'exprime pas par une relation entre u, v, m, net leurs de'rive'es] donnera donc, en dernie're analyse, une des relations-types dont nious venous d'e'tablir l'existence. (a) Supposons, par exemple, quill existe une relation oiu +J est connu, c'est-ah-dire que l'inte'gration de (1) se rame'ne 'a celle de (2) et r ciproquement (sans d'ailleurs qu'aucune d'elles s'i~nte'gre expe'licitemnent); on aura dans cc cas l'identite': A DF F aFD aF DE n) __ _ Du v +* Dar v d'oii re'sulte eutre *f,,* It, v une relation du premier ordre, 'a laquelle nous pouvons Dv

Page  153 SUR LES ]QUATIONS DIFF1RENTIELLES DE LA GIOM1TRIE 153 supposer la forme: a, a = K1 (u, v). La fonction F (qui depend de,) doit done verifier identiquement la relation aF aF aF - + m -- +- (m - n) K, = 0, 8u 8v 8a* et cette condition est suffisante. On observe que F peut etre suppose lineaire en X et l'on trouve immediatement F= +/3(u, ), avec la condition: A (/3) + (m - n) K, = 0. En resume si l'equation (2): aku av B (O)= aaf + Wav = 0, possede le multiplicateur K1, c'est-a-dire si an B (K) +K- a=O, a~~ a~ ~vlav l'6quation (1) A (<)= - + m = 0, oi l'on a pris m de maniere a vrifier la condition: -nK, + m + K =0, au \av possede une solution l, liee C r par la relation: = 4, + 3. L'equation (1) possede 6videmment le multiplicateur connu -o- = K + -. av av (b) Si la relation unique, supposee connue, a la forme O=F u, v, 4,,, elle donnera de meme (F)KF FaF aF a _aF ( a F a n a n }. A (F) = + — t-( -n) aV + - ( - n) - - - =0, au av a* 9v + a ( ) aV v av) \av d'ou l'on conclut l'existence d'une relation connue: a2_ a* av2 av ' ou J1 ne depend que de u, v et verifie: an aln B (J1)+ -1 +J = 0. L'identite a- + m - + (m - a) - + (m - n) J a - -= 0 u 9v a 9v 9* / v 9v a a* \9av

Page  154 154 JULES DRACH conduit aisement a l'expression unique: F-= c + a v acceptable sous la condition: A(a) +a {(n-n) J, -v + (m-n)=0 (c 0). On peut encore prendre a arbitrairement et determiner m par cette relation. D'ailleurs l'6quation: av- c+= cv + + aJi av ay/ av av jointe a celle qui donne (, permet d'exprimer expjicitement e et -v au moyen de (P et -a lorsque c f O. Si l'on a c = 0, auquel cas /a * a c = a a\ a* = aya)' av = +a1} av' log ( est donne par une quadrature; une autre quadrature est necessaire pour obtenir 4 quand ( est suppose connu. La correspondance entre les solutions des equations (1) et (2) est evidente. (c) Pour epuiser l'etude des cas oiu ( est explicitement connu quand on a integre (2), il reste a examiner l'hypothese: = F (u,, k, a- ) a v/' La fonction F devra satisfaire identiquement a la relation: aF aF aF a f aF ( a n _r9 M- +n m +4- - (m - n) — + - (- n) - a \ au + n a n a) v a a -\ av- av av \ av aF y n ana2*f a2n a 0 + la2#\tm~n av3 av aV2 av2 av 0 a* a3 3 /a2*2 a 2a~y ou l'on suppose -a 3 2 # -, -3 (=2 I ), av Wv- 2 \ )2 av Ie 6tant connu en u, v et verifiant: an a3 n B (11) + 21 + = 0. av av. La reduction de F a sa forme la plus simple donnera, en observant que F est determine' aux transformations pres qui remplacent t par f () ouif est arbitraire: /(a22 a2 2 av av2 v * L'hypothese O = a, oui p est entier, se ramene a celle-la par l'extraction de la racine piime de a. II lyvrl~o -, av,

Page  155 SUR LES E~QUATIONS DIFFE~RENTIELLES DE LA GI~OM]ftRIE 155 oti /3 y sont des fonctions de u, v seul. qui. s'expriment explicitement au rnoyen de a (u, v) et de ses de'rive'es. Cette dernie're satisfait 'a une condition du troisie'me ordre facile 'a former, qui, jointe 'a la re'solvante A, caract6rise ce cas. En general s'exprine rationnellement en ~,~ et ii existe une relation donnant expliciternent amoede U, V. aV2 anmye e b BRemarqite. II est clair que tout ce que nous venons de dire au sujet de l'inte'gration simultane'e de deux equations du premier ordre, pourra se re'peter, avec des changements e'vidents, pour trois ou un pius grand nombre d'e'quations. Nous avons done la possibilite' par uine mithode re'gulie're d'utiliser la connaissance d'une proprie'te ge'ome'trique du syste'me des courbes a (ui, v) = const., /3 (ui, v) = const., ry (u, v) =const.,. pour l'inte'gration des equations du premier ordre qui de'finissent separement ces courbes. Exemples ge'omitriques. Supposons qu'il s'agisse de mettre un element line'aire conun sous la forme d 82 = a2d0p2 - Modpd + C2 di#2, et que l'on connaisse les equations A 0 a i o-0 *+ nn 0,*('0r) qui de'finissent 0b et ~f.La relation donnera explicitement oa =0 UV) et l'on en de'duira aise'ment les expressions de a2et de V av a9v au moyen de ui et v. Supposons de me~me que les courbes (p= const., *r= const., deffinies par A((p)=0, B(*q)=0, satisfassent 'a la condition A~ (sb, zf) = A;on en de'duira une relation doninant an moyen de '*,2 puis une relation en ui, v, a* a2 ', c'est-ah-dire une expression explicite de A1. On aurait de me~me a2 ~J (u, v), explicitement, en u, V.

Page  156 156 156 ~~~~~JULES DRACH On retrouve aise'ment aussi les propositions classiques en Ge'ome'trie: s'il s'agit de mettre un ds2 sons une forme connue ds2= 2 f (b, g) do d#, on a, avec les equations diff~rentielles des courbes =const., =const., la condition 1 -d'oi' 1'on de'duira explicitement ~b et *'f, sauf 10 si f2 a la forme F(p - '4) (F quelconque, surfaces de revolution) auquel cas on ne corinait que aoet ~, 202 si f2 a la forme av av -auquel ca nn'a que les invariants du groupe qui cosrv __gop *)2 ca n oscv 2'gop 6'tudie' dans ma communication en Analyse: 0 et *f sont deffinis 'a une meme transformation proj ective pre~s. Des remarques analogues peuvent se faire lorsque f~ n'est pas entiewement connu, mais satisfait 'a certaines equations aux drves patels lesrdneaet par exemple, la the'orie de la reduction d'un ds a la forme de Liouville, avec, la discussion correspondante. Equations dua second ordre. 5. La me'thode que nous venous d'exposer pour une on plusieurs equations du premier ordre, s'appliquie quels que soient l'ordre et le nombre des equations diff6 -rentielles ordinaires que lon considere. En Ge'ome'trie, le cas des equations du second ordre (on des equations line'aires aux de~rive'es partielles h trois variables x,,)es particulie'rement important. Le groupe de transformations qui intervient est le groupe general 'a deux variables et le nombre des types de reductions possbe es vIsnd soixante. Si l'on deffinit la congruence des courbes FP ~b(x, y, z) = const., fr(x, yz) =const., par les equations: A()= A()= 0...................(1), ax ay az' on saura par une mithode re'g'alie're, tirer parti, pour l'inte'gration des equations (1), de la connaissance d'une proprie'te ge'oimetrique quelconque des courbes FP et l'on obtiendra touj ours, en definitive, l'expression explicite, au moyen de x, y, z, des invariants diffirentiels en ao, a*q* I...de l'un des types de ay', az'*' ay, az' groupes de transformations en 4, *J. II en sera de me'me si l'on connai't pour un syste'me de deux congruences de courbes 4=a, q =b; b1=a,, b,1 une proprie'te ge'ometrique et si lYon cherche it en tirer parti pour la determination de l'une on de lautre congruence "a partir de ses equations diff~rentielles. Mais le nombre des types de relations possibles, entre /, -4 et leurs de'rive'es d'nne part, 0b1, *,r et leurs de'rive'es d'antre part, types qu'on peut e'tablir a priori, est extre'mement e'leve'.

Page  157 SURLESI~iUAION DIFERENTIELLES DE LA GEOMETRIE15 157 Je me bornerai 'a indiquer des exemples tout 'a fait simples. (a) Supposons que les courbes = const., * =const., deffinies par le syste'me dx=dy _dz..(1) 1 m n................. you les equations A ()0, A (J)=0 avec A (f) = ~ m V+ n Vsoient p)lanfes. On pourra poser: z=ax + 3y + y, a, /3, ry 6tant des fonetions 'a determiner de ~het ~ Si deux de ces fonetions, a et /3 par exemple, sont distinctes en 0 et 'fon pourra prendre ~b=a, *f = /3 et l'on aura en appliquant l'ope'ration A (f) n =a + /3m, A (n)= 3A (m), qui donnent explicitenient a et /3 en x, y, z. La condition qui exprime que les courbes sont planes est:A A A (n) A (in) Si a, /3, y sont fouctions d'un seul argument, et si a n'est pas constant, on pourra prendre 4 = a; a et /3 seront donne's comme plus haut, mais /3 sera une fonction de a, d'ailleurs connue, et ii en sera de me'me de y. On connalitra done explicitement 4; la determination de *( de'pendra d'une equation 'a deux variables, renfermant le parame'tre. En resume' si les plans qui renferment les courbes de la congruence dependent de deux parame'tres, chaque plan ne contient qu'un nonmbre limite6 de courbes qu'on obtient sans integration; si ces plans ne dependent que d'un parame'tre, on a touj ours ces plans sans integration, mais les courbes qui sont dans chacun d'eux sont donne's par uine equation diffrentielle du premier ordre qui peut 'tre quelconque. Cela s"tend au cas oil les courbes de la congruence sont sur des sph'res, des surfaces alge'briques de degre6 connu, ou meme des surfaces transcendantes dont e'6quation ~fl (x, y, z) = 0 a une forme connue en x, Y, Z. (b) Supposons qu'il s'agiisse de determiner les trajectoires orthogonales d'une famille. de surfaces f (x, Z, ) = a, qui soit une farnille de Lame'. En posant: (f g)a _ fa + afag, on a pour les inconnues f,-rles trois conditions: La dernie're, transform4e par la me'thode generale., s'e4crit: a( N ( ~ P a _ _N a __al

Page  158 158 JULES DRACH elle ne garde sa forme, si lCon ne peut avoir AO4 = A*~J = 0, qu 'en prenant (ou permutant (p et p).Les surfaces (p=const., =const se dtri tdn separement par des syste'mes complets 'a une solution. On sait bien en effet que ces surfaces deffinissent les deux syste'mes de lignes de courbure des surfaces f (x, 'y, z) = a~. Si l'on peutt avoir AOb = 0, zA* = 0, les trajectoires sont intersections de developpables isotropes: ceci ne se pre'sente que si f (x, y, z) = a de'finit des plans ou des sphe'res (lignes de courbure inde'termine'es) et 1'on retrouve alors les re'snltats classiques. Iiemarqite. Ces re'sultats, relatifs aux trajectoires orthogonales d'une famille de plans on de sphe'res (et les re'sultats analogues relatifs aux trajectoires orthogonales d'une fa-mille de droites on de cercies dans le plan on sur nne surface connue), penvent aussi s'obtenir par l'application directe de la the'orie de la rationalite6. Si les equations diff~rentielles 'a inte'grer dx dy dz.() 1 ~ ~ ~ ~............... rn sont donne'es en coordonne'es carte'siennes (x, y, z), on pent former explicitement an moyen de x, y, z, m, nL et leurs de'rive'es les expressions des nouvelles variables qui ramenent le syste'me (1) 'a sa forme la plus simple; on pent aussi donner explicitement an moyen des me~mes elements les expressions des invariants, du groupe FP: (P =f (01b,#), P=g9(4w1k) qui caracte'rise la reduction pour le'quation' aF aF aF A(F) = -~m + n-= 0 dont (p et *f sont deux solutions. Par exemple, supposons que e'~quation dy =_ X dx- n(,) soit e'6quation d'une famille de cercles ayant leurs centres sur l'axe ox et proposons nous de trouver leurs trajectoires orthogonales. Si l'on pose: A (c)- =0, en ecrivant 1'e6quation des cercles: on trouve ~~~x 2+ my - 2a =) 0, on trouve x +my -a= 0,

Page  159 SUR LES EQUATIONS DIFFERENTIELLES DE LA GE~OME'TRIE 159 pour determiner a et /3 w=c (a), c'est-h-. dire e'6quation explicite des cercies. La condition que doit verifier m [qui serait de'fini implicitement par les deux equations pr'c'dentes en y posant /3 = co (a), co arbitraire] est simplement: 1 + M2~y ~ +M- =0 L equation qui de~finit les trajectoires orthogonales est: on en connait un multiplicctteur qni donne *4 par la quadrature: d* dx +r-n? dy d*y/ji+rn2

Page  160

Page  161 COMMUNICATION SECTIONS III (a) AND III (b). JOINT MEETING (ASTRONOMY AND STATISTICS) M. C. 1I. 11

Page  162

Page  163 ON THE LAW OF DISTRIBUTION OF ERRORS BY R. A. SAMPSON. I wish to draw attention to certain points in the accepted Theory of Errors, in particular to the foundation of the law by which errors are distributed according to their magnitude. This is given by Gauss's Error Function y = h7r- exp (- h22) and its study is a branch of the theory of probabilities, but following Chrystfal's example (Algebra, Ch. xxxviii.) I shall discard as far as possible the language of probabilities and speak of the conclusions as relating to averages only. If there is but one quantity to determine and that is directly observed by observations of equal weight we accept as the value of the quantity the Arithmetic Mean of the observations, almost of necessity; but no rationally convincing reason has been given for doing so except in the case of no more than two observations, when the Arithmetic Mean does not favour one more than the other, either in method or result. The same cannot be said when the observations number three or more, for then the result favours the middle ones, and therefore the Arithmetic Mean is usually presented either as an axiom or as a practical necessity. But it does not end the difficulty to accept it, because we have to supersede it by a more general rule when the observations give the unknown affected by a variable coefficient, as aix + ni = 0, [i = 1, 2,... s], where ai varies from one observation to another. We cannot deduce any solution of these from the Arithmetic Mean, and therefore we take a rule which includes it as a particular case. Thus if we take a solution x, which, substituted in the members on the left hand of each observation-equation gives residuals Ai or vi = aix + ni, we may adopt as determining x the rule A12 + Ao2 +... + A82 = minimum, which furnishes us with a single equation for x, viz. [aa] x + [an] = 0, where as usual [aa] stands for eai, [i = 1, 2,... s]. The Arithmetic Mean can be deduced from this rule, known as the rule of Least Squares, by taking the case where al = a2 =... = as, but the converse is not true, unless we supplement the Arithmetic Mean by postulating the existence of a law of distribution of errors by which the proportionate frequency of occurrence of any error between the magnitudes A and A + dA may be denoted by ~ (A) dA. Gauss who introduced this idea showed-provided at any rate < (A) may be differentiated-that if the rule of the Arithmetic Mean is not in any case to be 11-2

Page  164 164 R. A. SAMPSON contradicted we must have ( (A) varying as exp (- h/A2), or introducing a factor so as to make J (- d = 1, or ( (A) dA a measure of the proportionate occurrence of the individual error out of the whole, f (A) = herr-2 exp (- h2 A2) (Theoria Motus, ~ 177); and then we find that out of all possible values for the unknown x, leaving residuals Al, A2,... A, in the observations, that one is of most frequent occurrence, as measured by ) (A) >( (A)... < (A,) which makes [AA] a minimum, and this is the rule of Least Squares. It should not escape remark that this robs the simple Arithmetic Mean of its axiomatic character, since if an axiom requires to be generalised, it is the generalised form that is axiomatic; besides, neither the rule of Least Squares nor the existence of a differentiable function b((A) measuring the frequency of occurrence of each error A can be called an axiom. Indeed the Error Function introduces an idea which is entirely novel. In place of treating discrepancies between observation and adopted value as functions of the time or other circumstances of their origin, these features are ignored and they are rearranged merely in order of their magnitude; and the very sweeping conclusion is accepted that for every kind of observation and for every source of error where the word "error" is not even defined except as a residual, the errors arranged in a distribution in order of magnitude follow a law which is identically the same, depending only upon a single parameter controlling the scale. And I would here remark that discussions such as those with which Poincare deals in his Calcul des Probabilite's in which the law is made a function of the quantity measured as well as of the error, are not sufficient, since this quantity is only one of the circumstances which is left out of view. For example in the readings of a divided circle, the error may indeed be a function of the magnitude of the angle measured, but it may also be a function of the temperature, the hour of day, the season of the year, the attention of the observer and so forth, and all of these are equally ignored. The first who put this striking theory to a practical test was Bessel. In the Fundamenta Astronomiae (1818), p. 20, he makes a comparison of the actual and the calculated number of errors within stated limits for 470 determinations of the right ascensions of Sirius and Altair by Bradley and gives the following table: Number of Errors Number of Between by Theory Errors in fact 00s0 and 081 95 94 '1,, 2 89 88 '2,, ' 3 78 78 '3,, 4 64 58 *4 '5 50 51 *5.6 36 36 6,, 7 24 26 *7, 8 15 14 8, 9 9 10 '9,, 10 5 over 1'0 5 8

Page  165 ON THE LAW OF DISTRIBUTION OF ERRORS 165 The comparison has been made in numberless instances since and always shows an agreement of this kind, namely, one that is almost exact but shows a slight systematic preponderance of fact over theory in the occurrence of the largest errors, a feature which Bessel conjectures may be due to such causes as a slip of the pen or a disturbance of the instrument, or the like. The agreement is seldom so perfect as in these observations of Bradley's, but it is generally so good that Bauschinger in the Encyklopd(die der Math. Wiss. (I. 2, p. 774) gives it in his opinion as the best demonstration of the law. But this can hardly be accepted as final. In the first place it is no explanation of the origin of the law to verify its truth, and besides though an error, in its statement, is treated merely as a residual, the law is not verified at all closely in the case of all residuals. The following will serve as an example. In the Greenwich Observations, 1909, there are 504 observations of the nadir point of the Transit Circle made by reflection of the wire in a mercury trough (besides stellar observations). If the residuals of these observations from their mean are taken and analysed according to their distribution in magnitude we get the numbers given below, while the error curve derived for comparison in the usual way indicates the numbers placed alongside them. Greenwich Nadir Observations. Number of Observations Limits of Error - - ___ Actual Calculated 0"0 to 0"`2 79 95 0"'2, 0"'4 96 90 0"4,, 0"6 75 80 0"6, 0'"8 86 68 0"'S, 1"'0 55 54 1"-0, 1"-2 48 41 1"-2 1"-4 20 29 1"-4 1"-6 22 20 1"-6, 1"8 16 12 1"-8 2"-0 3 7 2" 2" 2 3 4 2"2 2"/4 0 2 2"'4 2"'6 1 1 2"/6 " 2/"8 0 1 2"'8,, 3"0 0 0 The result is shown in Fig. 2 and may be compared with those from Bradley's transits in Fig. 1. It will be seen that the nadirs oscillate sharply to and fro across their error curve. The reason doubtless is that there is a strongly marked seasonal term in the position of the nadir, which is brought out when the observations are meaned by months, thus, recording only seconds. January... 55"'14 July...... 53"'68 February... 5"`28 August... 3"-18 March... 5"'29 September... 3"'77 April... 4"-78 October... 3"'86 May... 4"'61 November... 4"'46 June...... 4"`45 December... 4"'56

Page  166 166 R. A. SAMPSON I shall return to this question of oscillation, and now merely remark that its presence makes it an important question to define clearly what class of residuals 80 60 _ _ 50 40.........- - -.30 - 10 -------- -------.... _______ i --- u u01 U02 0-3 U04 0-5 0-6 0.7 Fig. 1. Bradley's Transits and the Error Curve. 0o8 0.9 1.u Fig. 2. Greenwich Nadirs, not cleared of Annual Fluctuation. it is that really satisfies Gauss's law. Gauss does not occupy himself with this question. He simply introduces his function with the words "probabilitas...cuilibet errori A tribuenda exprimetur per functionem ipsius A quam per OA denotabimus," and argues certain properties of it, as that it may be discontinuous,-though he afterwards differentiates it. Laplace first offered a demonstration that the error function possessed a definite form and might be derived from the combination of an unlimited number of small errors individually following any arbitrary laws of distribution. A revised form of this is given by Poisson (Connaissance des Teimps,

Page  167 ON THE LAW OF DISTRIBUTION OF ERRORS 167 1827, p. 273) and this will be examined with interest because the theorem is remarkable, not to say improbable, and Poisson's proof has passed the scrutiny of such carefil judges as Todhunter and Glaisher. The crucial point is the following. Taking the problem "If E is the sum of s errors, e, e1,... es-_ each multiplied by a coefficient, so that E = 7e + 7y,1 +... + y%-1~es, and the law of frequency of occurrence of any value for the error ei is fi (ei), to find the probability that E shall lie between b - c and b + c, where the limits for the maximum error are - a and + a," Poisson obtains the expression for the probability 2 r da P - I ppI2... ps-I Cos (4 + 0, +... + bs-, - ba) sin ca -, where rOw+a r+a pi cos fi =.fi (x) cos y axdx, pi sin ^ = j fi (x) sin 7y axdx, - -a d and it remains to be proved that p tends to a limit for n -- o, which is independent of the forms of all the functions f and of which the variable part is exp (- hI2x). Poisson first proves that every p is less than unity by the following ingenious method: P= - f (cx) os yaxdx + f (x) sin ryaxdx} r+a r+a r+a r+a ~(x) cos 7otdx ~(x') cos,,/x'dx' + I -= fJ__ f) cos ya.dxf_ f(x) cos yax'dx' ~+ f f (x) sin yaxdx f (x') sin yax'dx' +aa -+ a jj= /+f (x) f (x') [cos yax cos yax' + si n a yax'] dxix'. -( J -(a -= f f./'(,f ) /'(xf') cos ya (x - x') xdx,'., - a. -a +a +a <f ( x f (x') dx dx', -a -a +a i.e. < f(x) dx, or <1. Further, if p be developed in ascending powers of a, so that p = - y2lha2... in order that the product R = pp,... p,,_i should converge to a limit other than zero it is necessary either that y2h2, y,2h12,... should decrease regularly in a certain way, or else a2 must be indefinitely small. Poisson gives an example of the former but treats it as exceptional; confining attention to the latter and keeping only the first two terms of each p, we have E log p _ log R = - a2 i17i22 - i ayh ti4 - i,,..... and then if a - 0 but so that a2_yi2hi2 is finite, it appears that a47/Th4,... all -- 0 and we have the form R= exp (- a2_ y/2hi), from which the required result is shown to follow. But it will be remarked that this step draws a conclusion as to the coefficient of a4 in the development of R, while the terms in a4 in the separate factors p have been omitted. The conclusion is therefore unwarranted and there is no proof

Page  168 168 R. A. SAMPSON at all that peculiarities of the functions f efface themselves in the final result. I do not believe that the theorem is true. If the arbitrary distributions fi (x) have any zeros-and this is not excluded by the process of demonstration-I do not see how they can fail to reappear as zeros in the product R. Poincare remarks (Calcul des Probabilites, p. 149), "On a fait une hypothese..., et cette hypothese a ete appelee loi des erreurs. Elle ne s'obtient pas par des deductions rigoureuses; plus d'une demonstration qu'on a voulu en donner est grossiere, entre autres celle qui s'appuie sur l'affirmation que la probabilit6 des ecarts est proportionnelle aux ecarts*. Tout le monde y croit cependant, me disait un jour M. Lippmann, car les experimentateurs s'imaginent que c'est un theoreme de mathematiques, et les mathematiciens que c'est un fait experimental." A large part of Poincare's book is occupied with questions of the justification of this law, but I shall not refer to it further, for while it is very interesting it would lose by summarising and it does not touch upon the point of view which I give below. I think it will be generally felt that the doctrine of the Error Function is so striking, so important, and so well supported by experience that it ought either to be proved or else reduced to a clear axiomatic position. I would therefore offer a few remarks towards its elucidation. These make no pretension to the generality aimed at by the earlier writers, but for that reason they are easier to bring under critical review. The term "accidental error" has come to carry with it an undefined suggestion of a peculiar quality, but there seems no reason to treat an error otherwise than as a disregarded unknown, or more commonly, as a host of small disregarded unknowns. Let us include these unknowns as y, z,... in the equations representing the observations so that we have aix + biy + ciz +... + nzi = 0 [i = 1, 2,... s], equations which are satisfied exactly. If we multiply by ai... and add, the former solution [aiai] x + [aini] = 0 gives the value of x correctly provided only [aibi] = [aic] =... = 0, but substituted in equations in the truncated form aix + ni it leaves residuals Ai or vi=-biy - ciz -.... These are the quantities under discussion. We disregard any individuality they may have and schedule them according to their values and the number of times each value occurs. It should be remarked that not until we reach very complicated cases will the graph of this schedule or distribution consist of anything but a few straight lines parallel to the axis of x. Thus for y= sin t and many others the distribution is represented by j = 1 for - 1 < < 1, and r = () beyond these limits (Fig. 3). Fig. 4 shows y = sin t + sin 2t and its distribution. The distributions for y = log t, y = 1 + t, y= 1/(1 + t3),... are the same, namely a line parallel to the axis of: extending from - oo to + oo. Those for y = et, y = 1 + t2, y = 1/(1 + t2),... consist of a line extending along OF from - oo to 0, and parallel to 0O from 0 to + o. It will thus be seen that it is in making the distribution that the features of the finctions which compose the * Query, read fonction des Rcarts.

Page  169 ON THE LAW OF DISTRIBUTION OF ERRORS 169 residual errors become effaced. It seems probable that Laplace and Poisson were on the wrong lines in attributing complete generality to the law of occurrence of their individual errors. In what follows I shall in all cases assume the graph of the Fig. 3. ' l I..,, ~, '' I I Fig. 4. distribution shows no negative values (which would be meaningless) and zero for x = - c or + oc if not before. converges to Also, I shall deal only with complete distributions, in which every error is represented by its fill proportionate number, giving a graph which is discontinuous only at a finite number of points, and shall postpone any remarks as to the degree to which this departs from actual experience. I regard the superposition of two errors as the disturbance of one distribution by another, in the following way. The error A' is added to the error A. The latter occurs a proportionate number of times q (A) dA, and the former l, (A') dA', where r+J - (A)dA = 1, j-X l, (a')d = al; -ooD hence the combination A + A' will occur the proportionate frequency (A) 1 (A') dAdA'; and if e - A + A', the result of the disturbance of the first distribution by the second will be a distribution in which any error O occurs the proportionate number of times d() f (e - A') f, (A') dA'. I would now remark that if f (A) = hIr- exp (- h2A2), l (') = h'7r- exp (- '2 A'2), the step described reproduces for the distribution of (? a function of the same type;

Page  170 1'70 R. A. SAMPSON for hth'w-r- d& exp {- h1,2( + 2h2o4&- H2A'2 [H2 - hi ~ h'2] = hh'-7r2 exp{ h2J2 ~ h4(OH)2/H2 f d&l 1-( 2m (/H)2 = hh'/H. wr 2 exp {- (hh'6JH)2j, and note that the "measure of precision," hh'/H follows the rule H 2 1 1 hj2 42 42 which is required for agreement with the ordinary theory. We notice too that the step is commutative in order, that is, the result is the same whether A is disturbed by &', or A' by A. It is clear that this reproduction of form is an important fact*; so far as I can determine it belongs with any generality only to the functions exp (x) and exp (X2). There are other cases where there is a partial reproduction of form. Thus if wxe take ~ (n)= r + A2), + A/2), we have I'D ( = 7-2 - I (dOH 1 (2 + 4L tanO1 + (A - ~2ano = 7r'2 dA/ g _ \ + 2?T r2 + 'A'2 + 00\ (H) 2+ 4 H I + (A/ - OH )2 _ '0(0-r~-4 (oo 2+ 4' (")' + LE'~~~~~~~~~~~/"2 '2 which may be considered the, same type; but if we take, (&) a/r-1(a + n'2) we find ~(e,=.,~l+MdA' 1 a2 + y 2 1 + (OH - 4 )2 2a77r-1 l()2 OH4 9 2OH 2(a2 + 1) + (a2 - 1)2' which is a more complicated form. There is an obstacle to proving any definite result in the difficulty of saying what is to be understood by the same form. I must leave this question aside as it is not material to the present discussion to answer it. It will suffice to know that exp (- h2X2), and of course also exp (- h2x2) cos kX +~ y) reproduce themselves. The latter form alone does not represent any real distribution because it contains negative values, but it is important when we have to consider the disturbance of one distribution by another, neither of which follows the law exp (- Mw2 ). We may take as typical the form (A) = hf(I + A)-' exp(- h22)[1 +aexp ikj], where exp ikA is introduced in place of cos Jca to make the subsequent work morc * I learn from Father Willaert, S.J. that this is a known result and that he, and before him M. d'Ocagne, had published it. Cf. Willaert, Ann. Soc. SCieitilique (le Braxelles, t. xxxi, April 1907. d'Ocagne, Bull. Soc. Math. d(e Franice, t. XXIII, 1895.

Page  171 ON THE LAW OF DISTRIBUTION OF ERRORS 171 compact, and the phase-angle 7 is included by reckoning a as complex. We must +x have a! in all cases less than 1, and in order to make f B(A)dA = 1, we have 1 + A = 1 + a exp (- k2/4h). The distribution then represents a simple case of fluctuation about the distribution hr - 2 exp (- h2 A2). Now let this distribution be disturbed by the distribution 01 (AL) = h'rT-' (1 + A')-' exp (- h'2A'2) [1 + a' exp ik'A'], where a' < 1, A' = a' exp (-k/4h'2). Consider in particular the term factored by aa', hh,' w7 (1 + A) (1 + At) exp (- h2A2 - h''A') [... aa' exp i (k/A + k,' ')]. Write A = - A'; hh',.- t+A)(1 + A)' exp {- h2O + 2h2A' - H2" 2} [... aa' exp i ckO + (k' - ) A'l] ( + A) ( + A' ) =7r i 4) i+ A') exp 4- Aexp - (H H -) } [.. aa exp OH + H k + k H-k/ _ h \} Integrating with respect to A' from - o to + so, hh'/H xp h2'2 2 D (k-k')2 (. h' + h'- 1k -, ---- ' -----. exp - ff +. aa exp -, 2 P 2 -7r2(1+A)(1~+A) exp {H )- L + aa'exp - (4H2 Jexp { '-' Thus the expression for the resulting distribution with respect to ~ is —,(G)_ 1ghh'/H fh) 1 Ih \ ^ \ } q) (O)= —,-1 A exp {- 0J- [1+ a oxp - exp if- A -7r2(l +A)(l +A') - - A2).hA^), ( (k2-')2 2^k'+ 12k ) +a' exp j- kH2 exp - h-k' 0 exp I-(- k'2)} expi -- h'k } We notice that it is indifferent which of the two we take as the disturbing distribution. Consider first the terms in a, a' compared with their primitive values. The effect of the disturbance is to make these fluctuating terms in every case recede in amplitude relatively to the non-fluctuating portion and at the same time to lengthen the span or period of the fluctuation. The more rapid the fluctuation originally, the more rapid is its decay. The fluctuating term will be smoothed out until its amplitude is so small and its period so long that it is indistinguishable from the nonfluctuating term. We must also consider the term in aa', especially for the case k = k': in this case the amplitude of this term is Iaa' and the argument kO is reproduced, or, if the fluctuations of the two distributions are of the same period, this period is reproduced in their joint effect. The same would be true if subsequent disturbing distributions had this period, so that it would run unchanged through the whole and not become lengthened like the others. But it would remain a solitary term, and since

Page  172 172 R. A. SAMPSON a I, Ia',... are all less than unity its amplitude would become less and less at every step. Although the fluctuating terms decrease in amplitude they increase in number and it is desirable to examine the convergency of their sum if the steps are indefinitely multiplied. It is evident that the expression will be always positive, since each step consists in an integration of elements each of which is positive. And the sum J (00 6) =1, since the terms within [ ] on p. 171 give respectively I I ( k2 \ 1ph'2k H\ 2, 1 kl'\ ok' H 2' - + T expr 4T2 exp -.2hh' + a IT exp 4H- exp H 2hh' a ( (k-kj') }h (h2kI'+h2k ff} 2 + aa '7r exp- - -H exp {-h' H- 4H2 exp H2 '2hh' +a' exp {= 7 [1 + a exp (- + a ex ( ) + act exp (- 4^-4 )] =72(1 +A)(1 +A'). With regard to individual values, it would seem that infinity is not excluded, since this may occur in the standard curve y=h7r-2 exp(-h2x2) for the special case of h = oo. It is of interest to examine the case when the disturbance is small but with a strongly marked oscillation; that is, when we have h' large compared to h and a' also large, subject to I a' < 1. We find the " measure of precision " becomes hh' 1 ih HW 22h'' and within [ ] the three variable terms are respectively t -(I ) (xp {ik (I - h ) + '- 4) exp { } + aa' (1 4h /2) exp {i ( h'2j ()} The most marked change is in the period of the oscillation factored by a'. The coefficient is not greatly altered, but the lengthening of period will have the effect of smoothing the oscillation away. In place of the oscillation with argument k), we now have two, with coefficients slightly different from and less than a, aa' respectively and arguments each slightly different from kO. The foregoing discussion is confined to the cases where the two distributions which disturb one another contain each only one or at any rate a finite number of fluctuating terms. To meet the cases of some of the distributions shown on p. 169 above, it would be necessary to consider an infinite number of terms, and it is certain that a number of points of difficulty would arise in the discussion, relating generally to questions of convergence. I shall not touch upon these, for as far as they are of

Page  173 ON THE LAW OF DISTRIBUTION OF ERRORS 173 interest they would be better dealt with by someone with more special knowledge of them than I; but if it is allowed to carry forward the conclusions of the previous pages to the general case the points are already made that seem necessary for understanding why in practice Gauss's error function always verifies itself. I would summarise these points as follows. We see that the graph of the error function is in its general run by no means remote from the graphs of such distributions as would most obviously occur; further if one of these distributions be treated as an error function qualified by fluctuations, and is then disturbed by the superposition of a second distribution of similar type, the superposition produces a general smoothing in every case, the non-fluctuating portion in the result being perpetuated and the fluctuating terms being relatively effaced both by spreading out their periods and by diminishing their intensities. If we assume that observed errors are due to a large number of small periodic errors this seems a sufficient explanation of the fact that observed errors do always present themselves as nearly pure illustrations of the graph of e-x2 when distributed in order of their magnitude, whether we can or cannot detect any systematic feature in them by arranging them in order of their origin or otherwise. But if any error A occurs with a relative frequency hwr- exp (- h2 A) dA, and a number of' observations are offered which condition but do not determine a set of errors A,, A2,... among all possible solutions which may occur, as already remarked, that one will in practice occur most frequently which makes A12 + A22 +... a minimum, which is a proof of the rule of Least Squares, which again contains as a particular case the Arithmetic Mean. It thus appears that there is no axiomatic element in the whole theory since the form of the error function follows from the definition of a distribution of superposed errors, and the other rules from the form of the error function. The fact that the Arithmetic Mean seems more primitive and axiomatic than the error function from which I derive it is no objection to this view, since results which are obvious may be deduced as particular cases from all correct theories. It also appears exactly what is the degree of authority possessed by the rule of Least Squares. It must be regarded as one of an indefinite number of legitimate solutions,-or if we separate the solutions by finite steps,-one of a finite number, all of which will in practice regularly occur. Among these solutions the least square solution is the case that will actually occur more frequently than any other individual case. But there is nothing at all to show that it represents the fact or individual case that we want, nor even that it will occur more frequently than some one out of the whole of the discarded possible cases. In the foregoing discussion the observations have been supposed so numerous as to give a continuous graph for the error function, and the successive superposed distributions of errors so numerous as to give a continuous disturbance of the first, and this to happen repeatedly, and we may ask what room there is for these infinitely numerous disturbances in practice where it always appears that a hundred or two observations will determine the error function quite closely. But there is no objection to regard the actual observations as a mere selection, taken at random or on any system not deliberately unrepresentative, from an infinite sequence which it is open to take, and so as generally yielding results that are representative of the whole.

Page  174

Page  175 COMMUNICATIONS SECTION III (a) (MECHANICS, PHYSICAL MATHEMATICS, ASTRONOMY)

Page  176

Page  177 ON DOUBLE LINES IN PERIODOGRAMS BY H. H. TURNER. We are indebted to Professor Schuster for the suggestion that for investigation of the vibrations or periodicities of a material system we should put aside preconceived ideas as to their probable special values and investigate all periods (within certain obvious limits) indifferently. We were not without hints in the past to this effect: thus Euler had specified the period of free oscillation of the earth (considered as a rotating rigid body) as about 10 months: when the coefficients of the terms of this period were found to be insensible it was concluded that there was no sensible free oscillation. Had the period not been assumed, the truth might have been discovered much earlier, viz. that owing to the earth's sensible departure from rigidity the period was much longer than this-about 14 months. But in spite of sporadic hints of this kind, it was reserved for Professor Schuster to formulate and emphasize the general necessity of approaching such problems without prejudices. In the course of much work with his method of the " periodogram," a conviction has gradually been forced upon the writer that the number of "hidden periodicities" may be greater than has been supposed. Putting aside the problems of gravitational astronomy where the existence of numerous periodicities is already well recognized, it may fairly be said that the expectations of investigators have favoured a small number of real periodicities-such as an annual and perhaps semi-annual period in meteorological elements, an eleven year period in sunspots, and so on; perhaps one or two hitherto unknown outside these, but not many. Here again we have not been without hints to the contrary: when Professor Schuster investigated sunspots he found, not one period, but several, even many. But it may be questioned whether this very fact has not raised its own cloud of mistrust from the existing prejudices in favour of a single periodicity, or at least a strictly limited number. It is unnecessary to debate at length whether periodicities are in general few or many: for patient investigation may be trusted to decide the matter. But it may be worth while to call attention to the doubt for the reason that the actual course of the investigation, at any rate in its early stages, may be modified by the attitude of the investigator. To shew this let us consider the extreme cases side by side. First suppose we have only a single noteworthy periodicity, such as the annual term in the rainfall. Calculating the coefficients of terms of all assumed periods, they will all be small except those near twelve months. They will not be zero, owing to the existence of accidental errors: and Professor Schuster has pointed out the analogy M. C. II. 12

Page  178 178 H. H. TURNER between the "accidental" part of the periodogram thus formed, and the faint continuous spectrum which in practice always accompanies a bright line spectrum. He has further used its average value as a standard of comparison for the reality of the "bright lines " themselves, tabulating the chances of "reality" when their intensities have assigned ratios to the average intensity of the accidental terms. And he remarks that we must, in determining this average value, exclude portions of the periodogram disturbed by "diffraction effects"; for example, near the annual term the coefficients will not fall abruptly to their accidental value, but will be enhanced by the neighbourhood of the annual term, in a manner strictly analogous to that in which bright lines are diffracted by a spectroscope of low resolving power. In the periodogram the equivalent of the "resolving power" is the total length of the series of observations, in units of the period under investigation: and even in our most extensive series this is in general small, so that the resolving. power is low. Hence diffraction effects, i.e. spuriously enhanced coefficients, extend to some distance on each side of a definite periodicity. Now consider the other extreme case-instead of a single well-marked periodicity suppose we have a periodogram full of periodicities, like a spectrum full of lines. Owing to the low resolving power each line will be accompanied by diffraction wings which may overlap those of the next line. The continuous or accidental spectrum will disappear, being covered up by the diffraction spectrum which will be much more intense: and no bright line will shew against the bright background with anything approaching the expected contrast. The investigator may be tempted to condemn the indications as too feeble to be worth notice, being actually misled by their very number and strength. To avoid misunderstanding it may be well to repeat that this is only stated as a possibility and not as an ascertained fact. But there are hints that it may be well to keep the possibility in mind: and one practical consequence is that we must devise an alternative, or at any rate a supplement, to Professor Schuster's criterion for comparison with the continuous spectrum. We must be careful that we really get down to this continuous spectrum by removing the diffraction effects, and the quickest way to remove them is to remove their cause, i.e. to subtract any suspected periodicities from the material under investigation. But the nature of the hints referred to, and of a suggestion for an improved criterion, will be best gathered from a couple of examples from recent experience. (1) Azimuths of Greenwich Transit Circle, Periods near 14 months. The Greenwich Transit Circle has been in position since 1851. The azimuth error shews slight variations, partly secular, partly annual, partly hitherto unexplained. Attention was recently called to the latter in connection with seismological questions; and for their more convenient investigation the secular and annual terms were first removed. The residual monthly means were then collected in periods of 14 months, and five consecutive periods grouped together to form 10 groups. The 14 means of each group were analyzed harmonically to obtain the constants of a term of the form A cos (O - a),

Page  179 ON DOUBLE LINES IN PERIODOGRAMS ]179 A being expressed in units of 0"O01, and the unit for 0 being such that 360~ corresponds exactly to 14 months. The quantities found by Fourier analysis are of course s= Asina and c = A cosa. Now from the values of a given for each group as below, it is seen that a is certainly not constant, and hence there is no sensible term of exactly 14 months. But the values of a increase steadily by an average amount not far from 90~, as exemplified in the column /, deduced empirically. Had the differences a - been irregular we might have set them down to accidental causes. But there is an Table I. Greenwich Azimuths. 14 months. Group a / a - A Acos (a -- 3) A sin (a -- ) s - A sin fl c - A cos1 l a,) I 154~ 104 + 50~ 33 +21 +25 - 2 -19 186~ II 231 194 +:37 65 1 +52 +39 -:39 -25 238 III 312 284 +28 231 +20 +11 - 1 + 4 284 IV 348 14 - 26 17 + 15 - 8 -14 0 270 V 60 104 -44 32 +23 -22 +11 +27 22 VI 107 194 -87 21 + -21 +31 +13 68 VII 200 284 -84 12 + 1 -12 +12 -22 152 VI II 63 14 +49 31 +20 + 23 +16 - 2 107 IX 147 104 +43 19 +14 + 13 - 6 - 5 230 X 1 223 194 + 29 36 +31 +17 -13 -10 233 Mean +20 + 7 + 15 + 13 obvious run about them to which we shall presently return. Meanwhile the consistently positive value of A cos (a - /3) shews that we have a periodicity of coefficient approximately 20 (i.e. 0"'20) and period such that the phase increases 90~ in five periods of 14 months, i.e. the period is 14 (I + 5-0-o) = 14-70 months = 447-1 days., 5 x 360 ] The column A sin (a - 3) shews us whether we have hit off, by our preliminary discussion of a alone apart from A, (a) Exactly the right slope: in the present case the earlier numbers are larger than the later. The value /3i = 89~ would have been better than f8 = 90~. Trial and error is probably the quickest way of finding out the best value of /8 when both A and a are taken into account. For the present we shall accept the value 90~ as good enough. (b) Exactly the right epoch. We have adopted an epoch to make the mean value of a -, zero; but when we introduce the coefficients A we upset the compensation. Trial and error will often be the quickest way to put this right: in the present case it is found that we may profitably increase the values of /8 by 21~: and we will denote the corrected value of f8 by /31. We can now compare this calculated periodicity with the observations. The theoretical values of.s and c are Al sin /3 and Al cos,/,: and subtracting them from the observed values we get the Sth and 9th columns of Table I. 12-2

Page  180 180 H. H. TURNER Now these columns may be utilized in two ways:Firstly, if there is nothing systematic about the residuals, they will give an indication of the accidental error. The mean numerical values are + 15 and + 13. The probable error of the mean of 10 may thus be put at + 5: and the coefficient + 20, being four times its probable error, has thus a strong claim to consideration. It is suggested that in this way these residuals provide the supplementary test of the reality of periodicities, the need for which was indicated above. But secondly we may, before regarding them as purely accidental errors, discuss the columns for the existence of a possible neighbouring periodicity. In the present case this has already been suggested by the fourth column of Table I. We may proceed to investigate it by forming a, from the relation tan a2 = (s - A sin /,)/(c - A cos /i), and from the values of a2 given in the 10th column we deduce by a diagrammatic or other empirical method the values /3 increasing uniformly by 47~ as in Table II. Table II. Greenwich Azimuths. 14 months. Group a, a 2- P B Bcos(a, -/2) Bsin (a2-o) s2-A2sin/32 c2-A2cos I a3 f3 I 186 187 - 1 19 +19 - 0 0 +1 - 50~ II I 38 234 + 4 57 +57 + 4 -23 -13 241 214 III 284 281 + 3 4 + 4 +0 +18 0 90 18 IV 270 328 -58 14 + 7 -11 - 4 -17 193 182 V 22 15 + 7 29 +29 + 4 + 6 + 8 37 346 VI 68 62 + 6 34 +34 + 3 +14 + 4 75 150 VII 152 109 +43 25 +18 +17 - 7 -15 192 314 VIII 107 156 -49 16 +10 -12 + 8 +16 27 118 IX 230 203 +27 8 + 7 + 4 + 2 +13 9 282 X 233 250 -17 17 +16 - 5 + 6 - 3 116 86 Mean +20 ~9 +9 It seems to me that the mere inspection of the differences a - /32 is convincing that we are itl the presence of a second periodicity. But we can proceed to test it just as before. Forming B2 = (s - A sin /31)2 + (c - A cos 81)2, the values of B are given in the 5th column of Table II: and resolving the coefficient in the observed direction, we find the mean value (say A2) of B cos (a - /32) to be A2 = + 20 (i.e. + 0"'20): so that this line is equally intense with the former. Forming B sin (a2 - /82) there is no suggestion of sensible error in slope or epoch. Forming s2 - A sin /2 and c., - A cos /2 we see that the mean numerical error has been reduced from + 14 to + 9; so that the probable error of a mean of 10 is now + 3: and the coefficients are thus nearly seven times their probable errors. The residuals still shew traces of progression as exemplified by the columns a3 and /38 where /38 steadily increases by 164~. But this corresponds to a period of 14 + 36 = 1 5ths approx.64 14 1 + ~ Q3 ) - 1-sn months approx. x 360 OU

Page  181 ON DOUBLE LINES IN PERIODOGRAMS 181. and is better dealt with in connection with means for 15 months. We have seen enough to realize the complex nature of the problem, and the manner in which it may be solved in steps. Without repeating the detailed process we may notice a second example of an even more striking character. The azimuths of the Cape Transit Circle from 1856-1904 have been discussed in a manner similar to that above described for Greenwich, removing first the secular and annual terms and then dealing with the residuals. When these are grouped in periods of 12 months, so far from the removal of the annual term leaving residuals of an accidental nature, they shew a well-marked progression. On analysis this is found to be due to the presence of at least three periodicities near 12 months: and the Greenwich observations confirm them from quite independent material. The details will be gathered from Table III. Table III. Azimathal Terms near One Year (qfter removing mean annual periodicity). Period Coefficient Phase in -- l Months Greenwich Cape Gr. Cape G- C 1 176 0"-20 2"6 294~ 106~ 188~ (11-60 0 '19 1 '( 315 145 170 (12-40 0 16 1 0 215 48 167 The 11-76 months period occurs in the Greenwich rainfall (over 80 years' observations) with a coefficient one quarter of the annual. The other pair of terms may be combined to give a variation in the annual term running through its cycle in 30 years; and this corresponds to what Mr Chandler has noticed in the variation of latitude. But the discussion of the origin of these terms cannot be undertaken here. The points to which the attention of mathematicians may appropriately be drawn are Firstly, the evidence for groups of periodicities in certain cases: and their possible existence in others; Secondly, the method of dealing with them above outlined.

Page  182 RELATIONS AMONG FAMILIES OF PERIODIC ORBITS IN THE RESTRICTED PROBLEMS OF THREE BODIES BY F. R. MOULTON. The restricted problem of three bodies is that in which an infinitesimal body moves subject to the attractions of two finite masses revolving in circular orbits. In the present discussion only those orbits will be considered which lie in the plane of motion of the finite masses. The masses of the finite bodies will be denoted by 1 - f and /u, and in the physical problem,/ may have any value from zero to unity. The differential equations which the motion of the infinitesimal body satisfies may be written in the form x" = F(; x, x', y, y') y" = F2 (; x, x', y, y') where the accents denote derivatives. These equations admit the integral '2 + y'2= F(, x, y) -C....................... (2). Only orbits for which C is finite will be considered. The functions F, and F2 are regular except for such values of x and y that the infinitesimal body collides with one of the finite bodies, or when at least one of the equations x' = o, y = o, x = cc, or y = co is satisfied. It follows from the integral (2) that x' = oo or y'= only in case of a collision or when at least one of the equations x = oc, y = oo is satisfied. The general solutions of equations (1) involve four arbitrary constants, one of which is additive to t because F, and F., do not involve t explicitly. One of the constants may be identified with C of the integral (2). The general solutions of (1) may be written in the form x =fi (t-to; ~; c,, c2, C) ) y=f2 (t-to;; c,, c2, C)...... (3 I), where the constants of integration are to, cl, Co, and C. When the initial conditions are such that the right members of equations (1) are regular, f, and f, are regular analytic functions of t - to,, c,, c2, and C in the neighbourhood of their initial values. When the initial conditions belong to a collision, x and y are regular functions of (t - to)] 1,, c, c2, and C in the neighbourhood of their initial values. There is an exception only when the collision is with a body of vanishing mass. If the collision is with A, then f, and f2 are functions of w-s and fI = 0 is an essential singular point.

Page  183 RELATIONS AMONG FAMILIES OF PERIODIC ORBITS 183 In order that the solutions (3) shall be periodic with the period T three equations, 1i(T; /a; cl, c.,, C)=O( ~0 (T;.; C1, C2, C)= O.......................... (4), (T;; c,C2, C) = 0o must be satisfied. The properties of 01, 02, and -h are similar to those of fi and f2. The method of discussion consists in establishing the existence of certain classes of periodic solutions for favorable values of the parameters; in determining the character of the possible changes they may undergo when the arbitrary parameters are varied; and in tracing out the connections among the various families of orbits from considerations of continuity and analysis situs, and from the results obtained by numerical calculations. The demonstrations of the existence and properties of classes of periodic solutions and the possible changes they may undergo can not be given in the limits of a single paper. The proofs of the theorems which follow will be found in a book on periodic orbits which is now being published by the Carnegie Institution of Washington. Moreover, the exigencies of time make it advisable to limit this report to the consideration of those orbits in which the motion of the infinitesimal body is retrograde, that is opposite to that of revolution of the finite bodies. The retrograde orbits are chosen because, in the first place, very little has been published on them, and because, in the second place, they are much simpler than those in which the motion is direct. The theory for the latter encounters the difficulties connected with the consideration of infinite periods. Among the theorems upon which the discussion of the relations among classes of periodic orbits depends the following may be mentioned: 1. There are three families of small periodic orbits about each of the finite masses in which the motion is retrograde (also direct). When the orbits are sufficiently small the members of only one family are real, and they are symmetrical with respect to the line joining the finite bodies. These results are true for all values of, between zero and unity. 2. There are six families of large periodic orbits about both of the finite masses in which the motion is retrograde. With respect to fixed axes the motion is retrograde in three families and direct in the other three. When the orbits are sufficiently large the members of only two families are real, and they are symmetrical with respect to the line joining the finite masses. These results are true for all values of p between zero and unity. 3. There is a single family of small periodic orbits about each of the three straight-line Lagrangian equilibrium points. The motion is retrograde with respect to the equilibrium points, and the orbits are symmetrical with respect to the line joining the finite bodies. These results are true for all values of tt between zero and unity. 4. If, < (1 (- V/i.), or if - < / ( - -), there are two families of small periodic orbits about each of the equilateral triangular equilibrium points. The motion is retrograde with respect to the equilibrium points, and the orbits about one of the equilibrium points are, with respect to the line joining the finite bodies,

Page  184 184 F. R. MOULTON symmetrically opposite to those about the other equilibrium point. For / greater than, but sufficiently near to, I (1 - _/23) there are two corresponding families about each of the equilateral triangular points, but they all have finite dimensions. They are the analytic continuation, with respect to /t, of the orbits for,/ < 1 (1 - V/a). 5. For /u sufficiently small there are closed ejectional orbits from 1 - / of the form indicated in Fig. 1. For 1 - / sufficiently small there are similar orbits of Fig. 1. ejection from /i. In both cases there are also closed orbits of ejection making an arbitrary number of circuits around the finite masses. The closed orbits of ejection are not periodic, for their coordinates, considered as functions of t, have branch-points at which three branches permute when there is a collision. 6. There are two families of periodic orbits related to each closed orbit of ejection of the type considered in 5, and they have the closed orbits of ejection as limiting forms. A single closed orbit of ejection is given in heavy line in Fig. 2, and one member of each of the associated families of periodic orbits in lighter lines. Fig. 2. 7. Real periodic orbits appear, or disappear, with the variation of C or / only in united pairs. (Poincare's Theorem.)

Page  185 RELATIONS AMONG FAMILIES OF PERIODIC ORBITS 185 8. Let an orbit which, for a certain value of C, belongs to two series of orbits (a united pair) be called a double orbit. Such a double orbit is the beginning of Sir George Darwin's Satellites B and C, Acta Mathematica, Vol. xxI. Double orbits can appear, or disappear, with the variation of u only in united pairs. 9. Orbits can unite, with variation of C or /u, only when they coincide throughout. 10. Orbits can acquire loops only by passing through an ejectional form, or by passing through a cusped form at a curve of zero relative velocity. Sir George Darwin's computations have proved that the periodic orbits can have cusps. 11. Only direct orbits, or orbits having direct loops, can have cusps. Cusps can appear or disappear, with the variation of at, only in united pairs. Fig. 3. A number of illustrations of the evolution of certain families of retrograde orbits will now be given. It is to be understood that the series of changes indicated in the diagrams are inferred from the existence of the families of periodic orbits and the limiting cases mentioned in the foregoing theorems, and from the changes periodic orbits can undergo, and that the results are not fully established by direct analysis. Fig. 4.

Page  186 186 F. R. MOULTON A. Evolution of retrograde orbit about,u showing the changes from a small approximately circular orbit (theorem 1) for large values of C to a closed orbit of ejection from 1 - u (theorem 5) for a smaller value of C, and to an orbit of two loops, one enclosing I - / and the other enclosing both finite bodies. The orbits are shown in Fig. 3. B. Evolution of closed orbits of ejection from 1 - as p, varies from zero to unity. The notation is 0 I, < p <... < un - 1 and the corresponding orbits are shown in Fig. 4. C. Evolution of the large retrograde periodic orbits enclosing both finite masses (theorem 2) for small C to an ejectional form (ejection from both finite masses simultaneously when pj = 1 - z = 0'5, the case shown in Fig. 5) for larger values of C, and finally to a direct orbit about both finite bodies consisting of a single loop. The evolution suggested here should be tested by a numerical calculation. Fig. 5 (1-=2=tl). D. Evolution of the oscillating satellites (theorem 3) about the straight line equilibrium points which are not between the finite masses, for decreasing values of C, to a closed orbit of ejection (theorem 5), and then to an orbit having a double loop about the finite body. The smaller loop expands and coincides with the larger when the orbit is a member of the retrograde series having a single loop (theorem 1). The orbits of two loops near this are Poincare's orbits of the deuxieme genre. The diagram, Fig. 6, gives only that series of orbits which passes,A. Fig. 6.

Page  187 RELATIONS AMONG FAMILIES OF PERIODIC ORBITS 187 E. Evolution of the oscillating satellites (theorem 3) about the straight line equilibrium point which is between the finite bodies, for decreasing values of C, to orbits consisting of small loops about each of the finite bodies, and a large loop enclosing both finite bodies. The curves shown in Fig. 7 are for sL = 1 - = 0'5, when the orbit becomes an orbit of ejection for both finite bodies simultaneously. Fig. 7.

Page  188 STABILE ANORDNUNGEN VON ELEKTRONEN IM ATOM VON L. F6PPL. In meiner Gdttinger Dissertation, die ich auf Anregung ineines verehrten Lehrers Hilbert angefertigt habe, habe ich mir die Aufgabe gesteilt, das in der Physik aligemein bekannte Thomson'sche Atommodell einer strengen mathematischen Untersuchung zu unterziehen. J. J. Thomson denkt sich bekanntlich das Atom als eine Kugel, die raumlich homogen mit positiver Elektrizithit geladen ist, wdihrend die negative Elektrizitht in Form von Elektronen auftritt, die sich innerhaib der positiv geladenen Kugel reibnngslos bewegen kdnnen. Diese Elektronen werden einerseits von der positiven Ladung der Kugel nach dem Kugelmittelpunkt mit der quasi-elastischen Kraft K = - - a angezogen, wobei 7' die Anzahl der positiven Ladungseinheiten der Kugel, b der Radius der Kugel und a der Abstand des Elektrons vom Ktugelmittelpunkt bedeuten; andererseits stossen sie sich umgekehrt proportional dem Quadrat ihres Abstandes von einander ab. Unter dem Einflusse dieser beiden Krafte werden die Elektronen in der Kugel gewisse Gleichgewiehtslagen aufsuchen. Die Anfgabe besteht nun darin, diese Gleichgewichtslagen auf Stabilitat zn prtifen. Man kann von vorne herein sofort von jeder beliebigen Anzahl von Elektronen gewisse Gleiehgewichtslagen angeben z.B. diejenigen, bei der alle n Elektronen in gleichen Abstanden auf einem Kreis liegen, dessen Mittelpunkt mit dem Kugelmittelpunkt zusammenfailt. Jedoch wird man sofort zugeben, dass diese Gleiehgewichtslagen fair grdssere Anzahlen n sieher labil sind, indem man das Gefuihl hat, dass die Elektronen tejiweise aus der Ebene heraustreten, um den Raum nin den Kngelmittelpunkt herum mdglichst gleiehmiissig auszuftillen. In der Tat habe ich anch bewiesen, dass unter allen mdgliehen Gleichgewichtslagen innerhaib der Kugel diejenige die stabile ist, bei der die Summe der Quadrate der Abstande von Kugelmittelpunkt mndgliehst klein ist. J. J. Thomson hat in seiner Arbeit in dem Phil. Mag., 1904 die Annahmie gemacht, dass alle Elektronen mit gleicher Winkelgeschwindigkeit um eine in der Kugel feste dureh den Kngelmittelpunkt gehende Achse rotieren. Mit Hilfe der durch die Rotation hervorgerufenen Zentrifugalkraft reduziert J. J. Thomson das ganze Problem auf ein ebenes. Er kommt zu dem interessanten Resultat, dass 1, 2, 3 Elektronen bei beliebiger Rotation, 4 und 5 Elektronen bei gentigend starker Rotation in gleichen Abstanden auf dem Umfang eines Kreises um den Kugelmittelpunkt stabile Anordnungen darstellen; dass dagegen 6 in gleicher Weise

Page  189 STABILE ANORDNUNGEN VON ELEKTRONEN IM ATOM 189 angeordnete Elektronen selbst unter Annabme noch so grosser Rotationsgeschwindigkeit labil sind und erst wieder durch ein siebentes Elektron in Mittelpunkt der Kugel stabilisiert werden. So weit ftthrt J. J. Thomson seine Rechnungen streng durch. Bei grdsseren Anzahl von Elektronen nimmt J. J. Thomson zur Vereinfachung der Rechnung an, class alle Elektronen immer in einer Ebene bleiben. Offenbar ist diese Annabme fuir eine strenge Stabilitaitsuntersuchung unzuldssig, worauf J. J. Thomson auch hinweist. Um diese Vernachiassigung zn vermeiden, babe ich von Rotation vorerst ganz abgesehen und einfach die stabilen Konfigurationen der Rnhe untersucht, wodurch sich die Aufgabe gegentiber der Thomson'schen zu einem dreidimensionalen Problem erweiterte. Im iibrigen babe ich den von J. J. Thomson eingeschlagenen Weg der Stabilitdtsnntersuchung bentitzt; namlich die Methode der kleinen Schwingungen. Ich begann mit der allgemeinen Stabilitatsnntersuchung eines einzelnen Elektronenringes, d.h. alle n Elektronen sind in gleichen Abstanden auf einem Kreis ur den Kugelmittelpunkt angeordnet. Die Rechnung ergab die Thomson'schen Resultate, dass bloss fttr n = 1, 2 und 3 Stabilitat eintritt, wabrend schon bei =4 diejenige Schwingung labil ist, bei der zwei gegentiberliegende Elektronen des Ringes in dem einen Sinn, die beiden anderen ir entgegengesetzten Sinn parallel der Symmetrieachse sich verschieben. Man iiberzeugt sich leicht, dass diese Verriickung im weiteren Verlauf der Bewegung auf die Tetraederanordnnng der vier Elektronen fiihrt. Diese ist aber, wie ich zeigen konnte, stabil. Uber die Hauptschwingungen eines Einzelringes in der Ringebene, die icl genan durchdiskutiert babe, liesse sich noch vieles Interessante sagen; doch verweise ich auf meine Dissertation, wo man sich an Hand von Figuren ein kares Bild machen kann. Mathematisch ist vor allen Dingen interessant wie in der analytischen Behandlung der Stabilitatsfr-age die geometrische Symmetrie der Anordnungen zuim Ausdrnck kommt. Man erhalt naimlich bei Anwendung der Methode der kleinen Schwingungen ftir die Untersuchung der Stabilitat von n Elektronen bekanntlich 3n Gleichungen und die charakteristische Determinante liefert fflr das Quadrat der Frequenzen elne Gleichung vom Grade 3n, deren Ldsung nattirlich bei einigermassen grossem it untiberwindliche Schwierigkeiten machen wiirde, wenn nicht wegen des symmetrischen Baues der Determinante, weiche in unserem Fall eine sogenannte Zirknlante ist, die eine Gleichung vom Grade 3n in n einzelne Gleichungen je vom Grade 3 zerfallen wtirde. Diese ffir die wirkliche zablenmassige Berechnung nattirlich fundamentale Tatsache tritt bei allen unseren Anordnungen wieder auf, sodass wir mit der Untersuchung von Gleichungen dritten Grades auskommen. Doch kehre ich nun zur Beschreibung meiner Resultate zurtick. Haben wir gesehen, dass 4 Elektronen nicht mehr in Ringanordnnng stabil sind, sondern an den Ecken des regularen Tetraeders ibre stabile Lage besitzen, so werden wir bei 5 Elektronen wieder zur Ringanordnung zurtickgeftihrt, aber mit dem Unterschiede gegen frtiher, dass nunmehr zu einem Elektronenring, dessen Mittelpunkt mit der Kugelmittelpunkt zusammenffallt, noch beiderseits in gleichen Abstainden von dem Elektronenring je ein weiteres Elektron hinzutritt-Polelektron, wie ich es nennen will. Die stabilen Anordnungen dieses Typs, den ich in Nachabmung der geo

Page  190 190 L. F6PPL metriscben Gestalt dtircb it = 1 ~ m + 1 bezeichnen will und bei dem die Rechnungen des isolierten Elektronenringes~mit geringen Anderungen tibernommen werden kduinen, sind folgende: 5=1+3+1, 6=1~+4+1, 7=1~5~1. Dabei ist 6=1 ~ +4 ~ 1 die Anordniing (ier Elektronen an den 6 Ecken des regularen Oktaeders. Dagegenr sind (lie Anordnungen 8 = 1 + 6 + 1 und. 9 = 1 ~ 7 + 1, wie ich gezeigt babe, labil. Man wird vermuten, dass 8 Elektronen an den Ecken des Wiirfels stabil sind. Jedoch habe ich zeigen kdnnen, dass die Wtirfelanordnung labil ist und zwar siebit man, wenn man (lie labile Sebwinguingen des Wtirfels verfolgt, dass die 8 Elektronen eine Anordnnng nach zwei paralle-len Ringen von je 4 Elektronen liefern, die gegenseitig versebrankt liegen, so dass ibre Proj ektion auf eine Ebene senkrecbt zur Symmetrieaebse einen Achterring gibt. Auch diesen Anordnungstyp babe ich ailgemein untersucht; aneh sehliesslich unter Hinzunabme von Polelektronen zn beiden Seiten des Doppelringes. TDie langwierigen Zablenrechnungen ergaben folgende stabilen Anordnungen: 8=4+4, 10=1+4~4~1, 12=1+5+5+1 (Jkosaeder), 14=1~6+6+1. Dagegen babe ich gefunden, dass die Anordnungen 16 = 1 + 7 ~ 7 + 1 nnd 18 =1 ~ 8 ~ 8 ~ 1 labil. sind. Eine labile Schwingung dieser letzteren Anordnnng von 18 Elektronen ist diejenige, bei der die 4 Elektronen mit geraden Nummern jeder der beiden Achterringe in der einen Riebtung, die 4 mit ungeraden Nummern in entgegengesetzter Riebtung sicb versebieben. Verfolgt man diese labile Verschiebung in ibrem weiteren Verlauf, so ftibfrt sie aus Symmetriegrtinden auf die folgende Anordnung der 18 Elektronen: 18 = 1 + 4 + 8 +4 ~1I d.b. abgeseben von den beiden Polelektronen treten drei Ringe auf: emn Aebterring und zu beiden Seiten desselben je emn Viererring nnd zwar so, dass die Ringe gegenseitig verschrainkt liegen. Diese Anordnuing babe ich nicht mebr ndiher untergesnecht. Die, Stabilitaitsuntersuchung maclit bier grosse Scbwierigkeiten; selbst die Bestimmung der Gleicbgewicbtslage ist bier keine leiehte Aufgabe. Ich babe nur noch den Fall von 20 Elektronen ins Aiuge gefasst, weil die 20 Elektronen emn besonderes Interesse beansprucben, da sie mdglieberweise an den 20 Ecken des Dodekaeders eine stabile Anordnung finden kiinnen; und zwar babe ich die Dodekaederanordnnng der 20 Elektronen mit derjenigen Anordnung verglicben, die sieb an die oben besprocbenen Anordnungen von 12, 14, 18 Elektronen ungezwungen anreibt; namlieb 20 = 5 + 10 +5a. Es stelite sieb dabei beraus, dass die Dodekaederanordnung emn grtisseres Gesamtpotential besitzt als diese letzere Anordnung und daber labil ist. Es lassen sieb daber von unseren Gesicbtspnnkt aus die 5 regularen Kdrper in zwei Gruppen einteilen: Tetraeder, Oktaeder und Ikosaeder liefern stabile, Wiirfel und Dodekaeder labile Anordnungen. Abgeseben von den drei Anordnungen nach regularen K~irpern babe ieb-nm nun einen zusammenfassenden Uberblick zu geben-gefunden, dass bei allen stabilen Anordnungen eine Acbse ausgezeicbnet ist, indem sie Symmetrieacbse ist. Ausserdem findet man bei allen Anordnungen mit mebr als einemn Ring, dass die Ringe geg~enseltig verscbriinkt oder wie man aueh sagen kann verzabnt liegen. Nimnt.

Page  191 STABILE ANORDNUNGEN VON ELEKTRONEN TM ATOM19 191 man nunl an, dass beide Eigenschaften ailgemein fttr stabile A-nordnungeni Geltung besitzen, so kann man sich auch fuir gr~iisere Anzahlen von Elektronen, deren strenge Stabilitatsuntersuchung wegen zu umstindlicher Rechnungen nicht mehr gelingt, folgendes Bild von den stabilen Anordnungen entwerfen: es werden 1, 2 oder 3 Pole auftreten, nhrnlieh entweder einer irn Kugelmittelpnnkt oder zwei Zn beiden Seiten der Ringsysteme oder sebliesslich gieiehzeitig einer im. Kugelrnittelpunkt und *je emier zu beiden Seiten der Elektronenringe. Abgesehen von diesen Polen werden sieh die Elektronen in Form von- Ringen urn den Kugelmittelpunkt gruppieren. Dabel spielen natiirlieh die Teilbarkeitsverhaltnisse der Zahien eine grosse Rolle. Wegen dieser Eigensehaft lassen sich gewisse Zahien ans der Zahienreihe heransgreifen, bei denen imrieder derselbe Anordnungstyp naeh drei parallelen Ringein auftreten muss. IDiese Zahien w~aehsen abweehselnd urn 24 und 48 Eiriheiten, so dass wir anf diese Weise anch formal das periodisehe System. der Elemlente nachahmen k~innen. Wegen naherer Ausffihrungen hierzu muss, ieh auf meine Dissertation verweisen. Was speziell die letzte.n Ausftihrungen betrifft., so m6ehte ieh bemerken, dass ieh den Boden der strengen Mathematik verlassen habe. Aber immerhin kommt diesen 1)arlegungen doeh etwas melir als nur einern Phantasiegebilde zu, da sie eine niaheliegende Verallgemneinerung der mathemati seh strengen Folgerungen sind, wie ich sic im. Hauptteil meiner Arbeit fiir geringere Anzahlen von Elektronen abgeleitet habe.

Page  192 ON THE PRACTICAL APPLICABILITY OF STOKES' LAW OF RESISTANCE, AND THE MODIFICATIONS OF IT REQUIRED IN CERTAIN CASES BY M. S. SMOLUCHOWSKI. ~ 1. Stokes' law for the resistance of a sphere in a viscous liquid rests, as is well known, on the fundamental assumptions: I. Slowness of motion, so that the inertia terms in the hydrodynamical equations may be neglected, in comparison with the effects of viscosity. II. Complete adhesion without slip, of the liquid to the sphere, this being considered as a rigid body. III. Unboundedness of the liquid and immobility at infinity. In what follows I should like to contribute some remarks on this law with regard to certain cases of practical importance, where the underlying conditions are changed to some extent, which may be of some interest to those who are engaged with research work on subjects connected with Stokes' law. First let us touch briefly the question of slipping, connected with the second of the above assumptions. Stokes' calculation can be generalised, by allowing the liquid to slip along the surface of the sphere, with a velocity proportional to the frictional force in a tangential direction [which in the case of a parallel laminar flow implies the surface condition i/u =, ]. In this case, as Basset has shown, the simple law of Stokes has to be replaced by 1R3+ 2~ F= 6-7r Rc fM + 23..............(1). Thus the minimal value of the resistance, for the case of infinite slip (/3 = 0), is two-thirds of the maximal value for no slip (/ = oo ). Now it is generally assumed, on account of the experimental researches of Poiseuille, Whetham, Couette, Ladenburg and others, that the slip of liquids along solid walls is negligibly small. Mr Arnold's* recent measurements prove, by their exact agreement with Stokes' law, that the coefficient of sliding friction / is certainly greater than 5,000 and probably greater than 50,000. * H. D. Arnold, Phil. Mag. 22, p. 755 (1911).

Page  193 ON THE PRACTICAL APPLICABILITY OF STOKES' LAW OF RESISTANCE 193 ~ 2. On the other side, his experiments, on bubbles of gas moving through liquid, gave the unexpected result that the slip at clean* surfaces between gas and liquid is infinite, as the velocity turned out too great by 50 per cent. Now I think a different explanation of those experiments to be preferable, as in the case of gas bubbles or liquid drops also the interior liquid is subject to circulation. Some time ago I advised Mr Rybczynski in Lemberg to calculate the motion of a viscous sphere through viscous liquid. The calculation is quite easy and the resultt, published January last year, and deduced also half a year later, quite independently of course, by M. Hadamard, is equally simple. It shows that for slow motion the inner liquid retains its spherical shape and that the resistance is F = 67-pRe 3p' + 2pF =, + 2.............................. (2), where u' designates the viscosity of the liquid in the interior of the sphere. Comparison with the above formula shows that the resistance experienced by a gas bubble or liquid drop without slip is the same as the resistance of a solid sphere with a coefficient of surface friction / = —; in fact the velocity and the stream lines of the outer liquid are identical in both cases. It would be interesting to verify the above formula by experiments on liquids with similar values of pu and A'; in the case of Mr Arnold's experiments the viscosity in the interior was negligible in comparison with the viscosity of the outer medium, which had' the same effect as if the surface slip were infinite. So far his results too a explained without the assumption of surface slip. 3. However, there is a case where the existenceof surface slip has been proved beyond doubt, namely in rarefied gases. As is well known, the magnitude of the coefficient of slipping 7= is, according to the.kinetic theory and also to the old experiments of Kundt and Warburg, roughly equal to the mean length of the free path of the gas molecules; therefore the phenomenon plays an important part even at ordinary pressures in the motion of very minute droplets, as in Millikan's experiments. Now unfortunately one cannot use formula (1) for this case, with substitution of the empirical value for 3, except for the case of comparatively small slip. For if the mean length X is comparable with the dimensions of the moving sphere, the ordinary hydrodynamical equations cease to be valid altogether, since the implicit assumption underlying them, that the state of the gas is varying little for distances comparable with X, is impaired. Therefore also the interesting deduction of a corrected formula by Prof. E. Cunningham+ is not to be considered as a demonstration and Messrs Knudsen and * I.e. provided the surface be not contaminated with solid films. t W. Rybczynski, Bull. Acad. d. Sciences Cracovie, 1911, p. 40; J. Hadamard, Comptes Rendus, 152, p. 1735 (1911); 154, p. 109 (1912). + E. Cunningham, Proc. Roy. Soc. 83, p. 357 (1910). M. C. II. 13 I —. 0t

Page  194 194 M. S. SMOLUCHOWSKI S. Weber may be right in trying to get closer approximation by other, purely empirical formulas*. At any rate the formula proposed by Cunningham F=6rtRc I +A - serves remarkably well for interpolation, considering the experiments of those authors and those of Mr McKeehant. It is preferable to write it in the form F=67rtLRc 1+ R, where p is the density of the gas, as mistakes are easily involved by using the mean length of free path X, which is a very indefinite term and really has no precise meaning. For great rarefaction the resistance is proportional to the cross-section of the sphere, and for this case the calculation can be carried out exactly, if the way is known, how the interaction between the surface of the sphere and the gas molecules takes place. If they rebound like elastic bodies, we get in accordance with Cunningham F= 4 /8 Rrp pcV where V is the square root of the mean square of molecular velocity. The numerical coefficient, as calculated from the experiments mentioned above, is considerably larger, it amounts to 1'65 (Knudsen and Weber) or 184 (McKeehan). McKeehan concludes that molecules are reflected from the surface of the sphere only in a normal direction; I think, however, that his theoretical formula is not quite exact and at any rate his conclusion seems to me at variance with fundamental principles of the kinetic theory of gases. I think that the experimental results are explained best by the view, supported also by other researches of this kind, especially those of Knudsen, that a solid surface acts in scattering the impinging molecules irregularly in all directions whether with or without change of mean kinetic energy. We shall not go into these questions now, however, as they belong to the kinetic theory of gases, not to hydrodynamics. ~ 4. Now let us consider what modifications are required in Stokes' law, if the third of the fundamental assumptions is impaired, the liquid being limited by solid walls, or a greater number of similar spherical bodies being contained in it. In this case the linear form of the hydrodynamical equations makes it possible to attain their solution by a method of successive approximations, analogous to the method of images used in the theory of electrostatic potential. It consists in the successive superposition of solutions formed as if the fluid would extend to infinity, but so chosen as to annul the residual motion at the boundaries, with increasing approximation. This method was used first by H. Lorentz in order to determine the influence of an infinite plane wall on the progressive movement of a sphere, and we shall refer * M. Knudsen and S. Weber, Ann. d. Phys. 36, p. 981 (1911). f McKeehan, Physik. Zeitsch. 12, p. 707 (1911).

Page  195 ON THE PRACTICAL APPLICABILITY OF STOKES' LAW OF RESISTANCE 195 to his formulas later on*. He found that the resistance of the sphere is increased by a fraction amounting to R for normal motion, -R for parallel motion, if a 8a 16 a denotes the distance from the wall. Mr Stock in Lemberg has extended the calculation for the second case to the fourth order of approximation, including terms with (X)4. In a somewhat similar way Ladenburg+ calculated the resistance experienced by a sphere, when moving along the axis of an unlimited cylindrical tube, and his result, indicating an increase in comparison with the usual formula of Stokes in the proportion of 1:1 + 24 (where p =radius of the tube), has been verified with very P satisfactory approximation by his own experiments and by those of Mr Arnold. ~ 5. Now let us apply this method to the case where a greater number of similar spheres are in motion, and extend a little further now an investigation which I had begun in a paper published last year~. Imagine a sphere of radius R, moving with the velocity c along the X-axis, its centre being situated at the distance x from the origin. It would produce at the point P (with coordinates,:, r, ) certain current Re velocities u0, v0, w0, of order R, defined by Stokes' equations, if the fluid be unlimited. But if we assume this point P to be the centre of a solid sphere of radius R, we have to superpose a fluid motion uivuw, chosen so as to annul the velocities of the primary motion at the points of this sphere and satisfying the conditions of rest for infinity. This motion may be called the "reflected" motion; it can be found with any degree of approximation, by making use of the solution of the hydrodynamical equations given by Lamb, in form of a development in spherical harmonics. But Re as it is of order c at the surface of the second sphere, which is its origin, it seems probable, a priori, that its magnitude at the first sphere will be of order c (-), and I have verified this as well as the following results by explicit calculation. /R\2 Thus if we confine ourselves to terms of order c (), we can apply a simplified method of evaluating the mutual influence of such spheres, by neglecting the difference between the velocity at the centre of the second sphere and at its surface. That is to say, the sphere P, being at rest, is subjected to frictional forces X= 67rwpRuo, Y= 67ruRvo, Z= 67rwRwo * H. A. Lorentz, Abhandlungen i. th. Physik, I. p. 23 (1906). In Millikan's determinations of the ionic charge the increase of resistance due to the presence of the condenser plates may produce an increase of the order of one-thousandth. + J. Stock, Bull. Acad. d. Sciences Cracovie, 1911, p. 18. + R. Ladenburg, Ann. d. Phys. 23, p. 447 (1907). ~ M. S. Smoluchowski, Bull. Acad. d. Sciences Cracovie, 1911, p. 28. 13-2

Page  196 196 M. S. SMOLUCHOWSKI on account of the motion of the first sphere; on the other side, the moving sphere experiences a reaction by virtue of the presence of the sphere P, such as if this would execute simultaneously the three motions - ut, - v, - w,; the three current systems resulting therefrom, according to the usual formulas of Stokes, produce at the centre of the first sphere nine current components, giving rise to nine components of frictional force, to be calculated each according to Stokes' law of resistance. If both spheres are in simultaneous motion, the mechanical effects are found by superposition of the forces corresponding to the two cases where one of them is moving and the other one at rest. In this way an interesting conclusion is obtained for the case where both spheres are moving in parallel directions with equal velocity: then both are subjected to equal additional forces in the same direction, one component in the direction of motion, tending to diminish the resistance by the amount - rc [1 - - - 0 2 r [ 4 r the other component along the line joining the centres, towards the sphere which is going ahead, of amount 9- R2 cs [1-9 [where 0 is the angle between 2 r L 4 rJ the line of centres and the direction of motion]. Thus two heavy spheres of this kind would sink faster than Stokes' law is indicating and, besides, their path must be deflected from the vertical towards the line of centres by an angle e defined by sin 3 3e = ~ r sin e = F1 3 R] sin 0 cos 0. ~ 6. Analogous methods are applicable to a greater assemblage of spheres. The motion results from superposition of simpler solutions, where one sphere is supposed moving and all the other ones resting. Each of the component solutions comprises the direct action, and for higher approximation also its "reflections." Now if the parallel motion of a cloud of n similar spheres is considered, the resistance of each of them will be diminished by an expression proceeding after powers of R, the first term of which will be of the order of magnitude ucR2 -. We see that these developments would be divergent for an infinite number of spheres. It is evident that for instance an infinite row of spherical particles, arranged at equal distances, would acquire infinite velocity, by virtue of their gravity, as also an infinite cylinder would behave in the same way. This applies a fortiori to two-dimensional infinite assemblages. Stokes' law of resistance will not be true even approximately, and the developnR. ment will cease to be convergent in general, unless - is small, where S denotes a kind of mean distance, comparable with the linear dimensions of the cloud. ~ 7. The same result follows from the following simple reasoning. Imagine a spherical cloud of radius S, containing n spherical particles, each of radius R and density a, suspended in a medium of viscosity /u, of negligible density, for example a cloud of minute drops of water in air. Then currents will take place in the

Page  197 ON THE PRACTICAL APPLICABILITY OF STOKES' LAW OF RESISTANCE 197 spherical cloud and it will attain a certain velocity as a whole, which may be calculated after the formula (2), just as if the cloud would form a homogeneous /JV medium of density a () and of the same viscosity as the outer medium. The 4 nRigomass velocity resulting therefrom, of amount 5 Sg, is superposed upon the displacement of the particles, relative to the moving cloud, taking place with velocity 9 R2g Thus evidently the downward velocity will be much increased, and nR Stokes' law cannot be true even approximately, unless g is small in comparison to unity. This condition shows that Stokes' law can be applied only to particles constituting clouds of exceedingly scarce crowding, and it is easily seen that it would be quite erroneous to apply it to actual fogs or actual clouds in the atmosphere, with diminished transparency [as in this case the aggregate cross-section of the particles nR2r is comparable with the cross-section of the cloud S2r]. As an illustration how nR cautious we must be in this respect, I may mention that the ratio - amounts to 10 and even to 100 for a cubic centimetre cloud as produced by Sir J. J. Thomson and H. A. Wilson, in their experiments on the determination of the ionic charge. ~ 8. What has been said applies of course only to clouds moving in an otherwise unlimited medium. The conditions of motion are quite different for a cloud contained in a closed vessel, as in the experiments just referred to. Prof. E. Cunningham has attempted to evaluate the order of magnitude of the correction to be applied to Stokes' law in this case. His estimate is founded on the supposition that each particle moves approximately in such a way, as if it were contained in a rigid spherical envelope, of radius comparable with half the distance to its next neighbours. Now this supposition does not seem quite evident, although we shall see that it leads to a result of the right order. We can calculate the resultant motion in quite an exact way, if we consider a homogeneous assemblage of equal spherical particles, moving all of them with the same velocity c in the direction of negative X, towards an infinite rigid wall, which we assume to be the plane YZ. In this case we see, by making use of H. A. Lorentz's calculation before alluded to, that a moving sphere x, y, z produces at a point:, situated on the axis of X, a velocity component =- + + Q X + l ]. j (3). 4r| \ r 4 p p2 P4 J... The first part of this expression, containing r= V/(x- )2 + y2 + z is the component of direct motion, according to Stokes; the second part is the component caused by "reflection" at the plane YZ; it contains the distance between the point g and the reflected source p = /(x +: )2 + y2 + '2. The terms with higher powers of - have been neglected, as we confine ourselves to the first approximation. The total current produced in the point e by the motion of all the particles is equal to U =;u, where the summation is to be extended over

Page  198 198 M. S. SMOLUCHOWSKI all their values of x, y, z. Now we might consider it right to replace the sunmmation by an integration, since one particle corresponds to a space A3, if A denotes a sort of mean distance between the particles. In this case the result would be very simple, for we should have U= f udxdydz. The integrals of the separate terms constituting u can be evaluated explicitly if we extend them to a cylinder with YZ as basis, of height h and of radius G. Then we can use the well-known expression for the potential of a disk in points of its axis, and expressions derivable from it by differentiation with respect to:, and by these means we find the unexpected result that the integral current U is zero, if we extend the summation to an infinite value of G. But in reality U is not defined by integration but by summation. Evidently both operations lead to the same result for distant parts of the space, but not for those parts whose distance from the point ~ is comparable with the distances A between two particles. Therefore the resultant current U in points at a great distance (in comparison with A) from the wall will be given by U= 3 Re 1 f-i /s,,, )Af^ x \ U............... (4), where =12 f 1 + ( 2 dxldydz- S 1 + x A2 r\ r2 r r to be extended over a space great in comparison with A, is a purely numerical coefficient. In order to evaluate /3 we must know how the particles are arranged. If we suppose an arrangement in rectangular order, we can get easily an approximate value by explicit calculation and by integrating over a cube of height H, constructed around the point I, which gives Ji (1 + dxdydz =8H2 [log(L + 3)- lo2 - A It is sufficient to take H equal to a small uneven multiple of, as the expression for /8 is rapidly converging with extension of the limits of integration. In this way I have found the approximate value 3 = 3'09, and therefore the resistance for one particle will be F=67w-Rc l + A3 = 6V7rRc 1 + 2-32 j............(5). This formula would apply, of course, also if the particles were arranged in a different way, but then the numerical value of / would be different. Our result agrees to the order of magnitude with Prof. Cunningham's estimate, which led him for the case of an equilateral arrangement to a similar formula, with a coefficient of R included within the limits 3'67 and 4'5. ~ 9. However, the practical application of this formula is rather questionable, as it applies only to a regular arrangement of particles. If they were arranged in

Page  199 ON THE PRACTICAL APPLICABILITY OF STOKES' LAW OF RESISTANCE 199 clusters, the correction might even become negative. It is interesting to note that the average value of /3, for a particle whose position relatively to the other ones is defined by pure accident, would be zero, and that seems quite natural, as the average current of liquid U in the cross-section must be zero. Thus it follows, what we should not have expected at first sight, that Stokes' law applies for the particles of an actual cloud, on an average with no correction whatever, of this order of magnitude. The evaluation of the quadratic terms would be much more complicated of course, as then all possible kinds of single reflections caused by any one sphere have to be taken into account. The general result of our calculation shows at any rate that Stokes' law is undergoing but small corrections if applied to the particles of a uniform cloud filling a closed vessel. But it is important to note that things will change entirely if the cloud is not of quite uniform density or if it does not fill the whole empty space between the walls. Then as a rule convective currents will arise, which in certain cases may be of preponderant influence. Their velocity may be calculated approximately by considering the medium as a homogeneous liquid subjected to certain forces, the intensity of which per unit volume corresponds to the aggregate force acting on the particles contained in it. Consider for instance an electrolyte in an electric field. If it is conducting in accordance with Ohm's law, the average electric density is zero and no currents will take place. But in bad liquid conductors, with deviations from Boyle's law, convective currents may arise, which may influence also materially the apparent value of the conductivity. They have been observed long ago, for instance by Warburg*. Similar movements may be produced in ionised gases, and I think more attention ought to be paid to them than usually is done. In experiments where the saturation current of strong radio-active material is observed between condenser plates wide apartt, these phenomena may be of importance as producing an apparently greater mobility of the ions than under normal conditions. ~ 10. There is another application of the theoretical methods exposed above which may be mentioned. Imagine a two-dimensional infinite assemblage of equal spherical particles, distributed uniformly over the plane x =l, whilst the plane YZ again may be supposed to be a rigid wall. Now let all these particles be moving along the plane in direction Ywith equal velocity c; what motion will be produced in the surrounding liquid, and what will be the resistance experienced by every particle? According to Lorentz again the motion produced by a single sphere moving parallel to a fixed wall is, when higher powers of the ratio i, which we suppose to be a small quantity, are neglected: 3 Rc Fi jY1 ]3 RC li 2(\ 3Rcx(x~ ) 9 RcXy2 (X+) 3 = I 1+ - I + p + - p 4 r L4 p J L P 2 p 3 2 p5 where the first term is the direct current according to Stokes, while the remaining terms represent the current reflected by the wall, just as in the former example. * E. Warburg, Wied. Ann. d. Phys. 54, p. 396 (1895). t Cf. Rutherford, Radio-activity, pp. 35, 84.

Page  200 200 M. S. SMOLUCHOWSKI We might also in this case calculate the resultant current by forming Sv over all values of y and z, and derive therefrom the resistance of a single particle. But we shall confine ourselves to the following remarks. In the extreme case where the particles are so crowded, as nearly to touch one another, a lamellar flow will take place in the liquid between the fixed wall and the cx plane x=1 with a velocity v= I, while on the other side of the plane x =1 the liquid will be dragged along by the sheet of moving particles with the constant velocity c. The frictional force per unit of surface of the plane x = 1 is evidently equal to!-c, therefore the resistance experienced by each particle is F =/cA2 F ' which is much smaller than Stokes' law would indicate, as A is of the order of R but the distance I is supposed to be of higher order. Now consider the other extreme case, where the distances A between the particles are so great that Stokes' law is approximately valid, which requires A to be of order 1. Let us calculate the resultant motion of the liquid for points at infinite distance from the wall (4 = oo ). For such points the summation mentioned above can be replaced by integration; besides we can put- - = 21, 1 - r = 61- r p r& ' r 3 p 3 r.5 and thus we get This iJteg + b +, d = This integral can be transformed by putting y = s sin S, z = s cos b, dydz = sdsdo, and we get finally 6_Rrc A By comparing this with Stokes' law for the resistance F we have =F 1 A25) that means that in both cases the liquid at a great distance from the wall will be dragged along, in a parallel direction to it, with such a velocity as if the force corresponding to unit surface F- were distributed uniformly over the liquid, in a A 2 plane at a distance I from the fixed wall. This result, which can be generalised for a greater number of similar layers, seems natural enough if the distances between the particles are small in comparison with their distance from the wall, so that the assemblage can be considered as if forming a homogeneous medium, but we see it remains true for particles widely apart. Without going into further details, I may only mention that this result has an important bearing on the theory of electric endosmose, which will be explained elsewhere with full details. ~ 11. I may conclude with a brief remark about the influence of the inertia terms in the hydrodynamical equations (assumption I), which have been neglected

Page  201 ON THE PRACTICAL APPLICABILITY OF STOKES' LAW OF RESISTANCE 201 as well in Stokes' original calculations as in the above reasonings. It is well known Rcathat this neglection is justified only if the ratio is small in comparison to unity. But it has been proved by Oseen* in an important paper, commented upon in a very interesting way by H. Lamb, that the solution given by Stokes is defective even if this criterion is fulfilled; for at distances r where rca is large, the inertia terms must be of prevalent influence over viscosity. Oseen himself has given a solution which is different from Stokes' equations for those distant parts of the space and gives better approximation there. However, the resistance of the sphere depends only on the state of movement in its immediate neighbourhood, therefore the resistance law of Stokes is not impaired by those results. The condition of its validity may be defined more exactly by means of the recent experiments of Mr Arnold, which have shown that it holds with very good accuracy (one half per cent.) for spheres moving under influence of gravity, provided their radius is smaller than 0'6 r, where the critical radius r is defined by the relation -- = 1. This means that Rcathe ratio -- must be smaller than (0'6) = 0'22. /F ~ 12. The inertia terms are of greater importance, in the case before alluded to, where the motion of a greater number of similar spheres is considered. For it is legitimate to calculate the forces of reaction between such spheres by using Stokes' equations for slow motion only if they are lying within the space where viscosity is predominant over inertia. Mr Oseen has generalised recentlyt the calculation of the interaction of two spheres given by me by introducing in it his solution of Stokes' problem. The forces exerted on the two spheres come out unequal in this case and are given by much more complicated expressions. They become identical with the first approximation given by me if the distance r between the two spheres rcor satisfies the condition that - is small. Mr Oseen thinks this to be a great restric2Ft tion on the validity of those formulas for experimental purposes, but he omits the factor ao in the above expression. We satisfy ourselves easily that, for instance, in the case of water-drops in air, as in Sir J. J. Thomson's and H. A. Wilson's condensation experiments, the limit of validity for r is of the order of several centimetres; in Perrin's experiments on the applicability of Stokes' law to the particles of emulsions it would amount to hundreds of metres. It is also sufficiently great for direct experiments, when highly viscous liquids are used, as Ladenburg did in his elaborate research. Ordinary hydraulic experiments, with water and spheres of a size to be handled conveniently, are excluded of course when Stokes' law or any of those modifications are in question. One might try to apply Oseen's method of approximate correction for inertia also to the other cases treated above, but it will imply rather cumbersome calculations and, besides, for movements in closed vessels it will be generally of lesser importance than in a liquid extending to infinity. * Oseen, Arkiv f. mat. astr.fysik, 6 (1911); H. Lamb, Phil. Mag. 21, p. 112 (1911). t F. Oseen, Arkiv f. mat. astr.fysik, 7 (1912).

Page  202 THE APPLICATION OF THE METHOD OF W. RITZ TO THE THEORY OF THE TIDES BY A. E. H. LOVE. Four years ago W. Ritz published a memoir* in which he propounded a new method of solving problems of mathematical physics; and in this and a subsequent memoirt he gave examples of the application of the method to obtain approximate numerical solutions of problems which had long defied the efforts of analysts. The salient features of the method may be described as follows:-First the problem is formulated as a problem of Calculus of Variations, that is to say the differential equations of the problem are recognised as conditions required to be satisfied in order to minimize, or render stationary under suitable restrictions, an integral, which may, for example, be the expression for the potential energy of a vibrating system. In the next place the dependent variable, which may, for example, denote a component of displacement in a vibrating system, is taken to be expanded in an infinite series of functions. These functions may be polynomials, or the series may be a Fourier series, or the functions in question may be normal functions of some quite different vibrating system. It matters little what functions are employed provided that an arbitrary function, arbitrary in the sense in which functions that occur in mathematical physics are arbitrary, can be expanded in terms of them. The only necessary restriction affecting the choice of the functions in a series of which the dependent variable is expanded is that, if the variation of the integral is to be subject to special conditions which hold at a boundary, the functions in question must satisfy similar conditions. If, for example, the dependent variable is a displacement, and the boundary is so fixed that the displacement there vanishes, the dependent variable must be expanded in a series of functions which vanish at the boundary. In the practical application of the method the choice of a set of normal functions is conditioned by the circumstance that, unless a suitable set of normal functions is chosen, the integral to be rendered stationary may be intractable. After the formulation of the problem as a problem of Calculus of Variations, and choice of a suitable set of normal functions in a series of which the dependent variable is to be expanded, * W. Ritz, " Ueber eine neue Methode zur LIsung gewisser Variationsprobleme der mathematischen Physik," J.f. Math. (Crelle), Bd 135 (1908). t W. Ritz, "Theorie der Transversalschwingungen einer quadratischen Platte mit freien Rindern," Ann. d. Phys. (4te Folge), Bd 28 (1909). The two memoirs are reprinted in "Gesammelte Werke Walther Ritz, CEuvres publi6es par la Societe Suisse de Physique," Paris, 1911.

Page  203 THE APPLICATION OF THE METHOD OF W. RITZ TO THlE THEORY OF THE TIDES 203 the next step is to assume the dependent variable to be expressed approximately by a terminated series of such functions fitted with arbitrary coefficients. This series is then substituted in the integral, and the integral is thus expressed as a rational integral function, frequently a homogeneous quadratic function, of the coefficients in the terminated series. The coefficients are then adjusted to make this rational integral function stationary subject to the satisfaction of some specified conditions. The approximation to the correct value for the dependent variable improves as the number of terms in the assumed series increases, but it is often quite good when the number of terms is small. The approximate expressions which are thus obtained for the dependent variable, or variables, do not in general satisfy the differential equations of the problem*. They satisfy boundary conditions of the type above described, and the functions to which they are approximations satisfy the differential equations; but the series, when differentiated term by term, do not in general satisfy these differential equations even formally. The security for the series being approximate expressions for functions which do satisfy the differential equations is furnished by the satisfaction of the conditions that arise from the formulation of the problem as a problem in the Calculus of Variations. The suggestion that this method might be applied to the dynamical theory of the tides was made by Poincaret. Taking as the dependent variable the vertical displacement of the surface of the sea at any point of specified latitude and longitude, he determined the form of the integral which must be stationary when the differential equations are satisfied, and pointed out some difficulties that beset the application of the method. One of these difficulties is the possible existence of critical latitudes at which some terms of the integral would be infinite or indeterminate. This particular difficulty may be turned by choosing a different dependent variable. I propose to explain a method of procedure, differing slightly from Poincare's, and shall restrict the investigation to the problem of the free oscillations of a sheet of water of uniform depth covering a rotating globe. The water will be treated as frictionless, and the effects of the self-attraction will be neglected. The problem in the more extended form in which account is taken of the self-attraction having been solved very completely by S. S. Hough, the comparison of the results to be obtained by the present method with those obtained by him makes it possible to proceed with great confidence to problems relating to sheets of water of varying depths enclosed by rigid boundaries. I have gone some way with such applications but the work is as yet unfinished. Let a denote the radius of the rotating globe, h the depth of the water, g the value of gravity at the surface. Let o denote the angular velocity of the rotation. For the sake of brevity the globe will be compared with the earth, but it is to be understood that no forces except an attraction towards the centre, specified by g, are * The idea of leaving the differential equations of the problem out of account when once the integral to be rendered stationary has been found is perhaps the most novel feature of Ritz's method. The method has much in common with approximate methods previously used by Lord Rayleigh. In regard to this matter reference may be made to a paper by Lord Rayleigh in Phil. Mag. (ser. 6), vol. 22 (1911), pp. 225-229. t H. Poincare, Leqons de Mecanique Celeste, t. 3 (1910), pp. 297-303. + S. S. Hough, "The application of harmonic analysis to the dynamical theory of the tides," Parts I and II, London, Phil. Trans. R. Soc., vols. 189 (1897) and 191 (1898).

Page  204 204 A. E. H. LOVE supposed to act upon the water. The position of a point on the surface is specified, by co-latitude 0, measured from the North Pole, and longitude J, measured eastwards from a fixed meridian. Let ' denote the vertical elevation, or radial displacement, at any point at time t; u, v the southward and eastward components of velocity at the same point and the same time. Then the equations of motion are known* to be - 2cov cos 0- at 2 co s a o =...........................(1), av) - + 20 cos 0 =... (2), at+ 2 coa sin 0 a........................ at a sin 0 () while the equation of continuity takes the form h a av (u sin 0) +.(3). at a sin in ) +,....................... 3). These equations will determine an oscillatory motion if u, v, ' are all proportional to simple harmonic functions of the time; but, since the oscillation is about a state of steady motion, not a state of equilibrium, the phases of the different simple harmonic functions are not the same, and are not related to one another in a very simple way. With a view to the formulation of the problem as a problem in the Calculus of Variations we introduce the notation a~ aq iu sin 0 = sin at........................... (4). at oat Then cosec 0 and r sin 6 are, with sufficient approximation, expressions for the southward and eastward components of the displacement of the water. We introduce also the notation cos 0 =,....................................(5), which is usual in problems involving spherical harmonic analysis. It is evident that: must vanish when A = + 1. Then equation (3) becomes a ((6) a................................. 6. When u, v, s are eliminated froml equations (1) and (2) by means of equations (4)(6) there result the two equations sI2n a_ h aa.(7), sin 0 coat-at - a" a - a............... siln 0 at+ 2 cot 0 -a=- a2 sin 0 -.......f a at" at a sin a ao apl. (8). Now, if we suppose that: and q are proportional to simple harmonic functions of t of period 27r/a, and write =: cos at +:2 sin at ) = j cos at + sin......................... * H. Lamb, Hydrodynamics, 3rd edn. (Cambridge, 1906), p. 313.

Page  205 THE APPLICATION OF THE METHOD OF W. RITZ TO THE THEORY OF THE TIDES 205 equations (7) and (8) give rise to the equations _2 - h ah la~ t a9 -.~ 21 — + 2&)00aw', ']72-'4' - 1 -/~2: a2 - o \ o'j + - a a............... (10), (1 - 2) ' + 2oa _ h g (ha2 _- w and these equations are the principal equations of that Calculus of Variations problem in which the integral J, where J=Ild1/~l2+a o0-2 ( ) + o22-+( - 2) (l2 +92.2+4) + (4 — 5.M ) gh (a27l ra2 a 22 N } 2 a -i aJo ap i a ~ is made stationary. The variations must, of course, be subject to the conditions that t, and:2 vanish at A = + 1, while all the quantities l1,,, r7, 72, considered as functions of q, are periodic with period 27r. These conditions imply that the two variations 81:, 8~o vanish at /u = + 1, and the two variations 8lj, 8l2 have the same values at = 27r as at, > = O. We write now r 4W)2a2 '2~ = f ' gh - = /..............................(12) 2( f' gh =. ( 12), and J = 4co21................................... (13), then the integral I to be made stationary is given by the formula I = fd/ d f2da [2 I_ + 2(1- ) (22 + 2 + + 2f/ (12 - ~e,) I I i at _ al7,\ I (8Dt _ @0)2}].(14), la alu ao/ ap a and we have l - o - La d - ~ f i (ad- a K2 d, P1J o L[0 a 'q-i 0 J 1 a a - PJ -1 L ' a+, ]o J -{1L(e a+) vd Jo + terms which vanish by (10)...................................... (15), so that 8I vanishes in virtue of equations (10), the conditions above laid down in regard to the periodicity of the functions, and the special conditions which hold at the limits of integration. The only difference that would be made in the above work if it were applied to the problem of free oscillations of an ocean of uniform depth bounded by vertical cliffs would be that the limits of integration in respect to, and p would have to be assigned in a suitable way, and that the variation would have to be conducted in accordance with the condition that the horizontal component of displacement along the normal to any boundary vanishes at the boundary. In the application to an

Page  206 206 A. E. IL LOVE ocean covering the globe we have merely to bear in mind that 4: and f2 must vanish at / = + 1. For this application we suppose in general that t1 = o0 COS S, =- -o sin so ( I = )0o sin sO, 72 = o cOSs )..*................... where s may be zero or any positive integer, and to, qo are functions of / only. When s 0, that is to say when the displacement is independent of longitude, it is convenient to eliminate 77 by means of the equation (1 - u2)f/ o + = 0.+.................. (17), which follows at once from (10). Then the integral Ito be made stationary is given by the equation = d..................... (18). It may be noted here that, when g is used as dependent variable instead of o0, the differential equation satisfied by C is the principal equation of that Calculus of Variations problem in which the integral -| tf -— C Y' kd/x] - fi ' dl........................ (19) is made stationary; and the difficulty noted by Poincare, as mentioned above, in regard to the critical latitudes arises through the presence of f2 - /2 in the denominator. The critical latitudes are those at which / = +f, and they are real if o < 2co, or the period is longer than half a day. We have now, in accordance with the general method, to substitute for 0o in (18) n a series of the form E AKU,(/J), where the u, are functions in terms of which an K=l arbitrary function can be expanded, and every one of the set of finctions 9,K vanishes in a sufficiently high order rat,L = + 1. The functions of the type dP, (1- *.2) where P,, or PK (/), denotes Legendre's Kth coefficient, possess the required properties. In fact the expression vanishes at x = + 1 for all values of the coefficients AK, and it has been shown by R. A. Sampson* that an arbitrary function of,/ may be expanded, within the interval 1 > > - 1, in a series of the form 00 dPK E AK (1 - ) d K=l I - We assume then for:o = E A (1-D ) d...........................(20), and observe that (18) has the form f1{= o f2 1 ao2 t2 d..................(21) * R. A. Sampson, "On Stokes's current function," London, Phil. Tran.. R. Soc., vol. 182 (1891).

Page  207 THE APPLICATION OF THE METHOD OF W. RITZ TO THE THEORY OF THE TIDES 207 Now it follows from well-known formulae that d'I /d70\2 1n 92a2 (I — b 1)2! (d-) d= 2( ) 1A K.....................(22), K=I 21Kq — I J 1 1~ f-1 2 +l............. (2), and we have also -1 K2dI df L P = -1 K=1 c + I(P W-2 / ( K K ( 1) 2 2 2 ) or 2 AK ~ * i - K=1 (2 + 1)2" 2C - 2K + K(K + +l)( + 2)( + 3) 2 +2AKAK+2 (2(c + 1)(2c + 5) 2/c+3...(24)' in which A,+2 is to be replaced by zero. On substituting in (21) and putting [I/3aAK = 0 for all values of K from 1 to n we obtain a series of equations to determine the ratios of the coefficients AK, and, on elimination of these coefficients, there results an equation connecting f and 3 which is an approximation to the period equation. Now equations (6), (9), (16), (20) give, when s = 0, h - ( cos at AKK ( + ) P....................(25), (t K=O and thus, if we put ~= cos atx Lt E C, PK.......................(26), we shall have a series of equations of the type CKe-2 C+ K+2 + ~.+ -- 2. (2.K —3) (2Kc- 1) +(2c + 3) (2K + 5),8+ C +K f (K + 1) (2K- 1)(2K + 3) which agrees with the sequence equation for the C's obtained by Hough (Part I. p. 221) when that equation is simplified by neglecting the effect of the self-attraction. In the more general case where s + 0, we substitute from (16) in (14) and find I = f.1 o2 o (1 l_ )} o + 2f/ o -o _(da - _s)oj d1...(28). J -1 l [1 ~ P- /~s / It is now convenient to write dfo d - o So =................................., so that h = cos (sf + o-t).............................( (30). Then (28) becomes I = [f2 f /~-2 + 2o + 1 ^ ' + 2 M s ( - ( \ d_2) -2 )2 o(1-2) '- 2f.. + ( ) o - ( 31). We now assume for to, ~o series of the form t n2 = A~ P (), = C ) ()..................((32), 1=l K=1

Page  208 208 A. E. H. LOVE where P(") (/) = ( - g')d"P" (/) where P (/) = (1.................. (33). We transform the integrals -1( - d2) (k 2, l o dfo [.(1 -^)()dit and di." by integration by parts into -a utse knon f (I -/;'k) diL and p1 o2 dT, and use known formulae* in regard to the "associated functions" P(') (/m). Then we find after some reductions I s 2 ( ) (K, + ) As)/, f 2 (K + s)'! A(U s2 2K +1 -(K- S)! K s 2K + 1 (K-S.) tK f2 c()A () ( 2) (K + + +s) 2 (K + )! S K K+l 2K + 3 2K + 1 (K - s)! +2f" C.) ( -l1)( -s) 2 (/ + s)! S2 K K-1 2/c-l 2K +l (K- S)! 2f 'S ()(), K + l + s 2 (K + )! 2f C(s) -s 2 (K + s)! Kc K+I 2K + 3 2 +l( (K-)! s K K-1 2K -1 2K + 1 (K - S) _ 1 (K +- 1)2 _ S2 C- 2 s 2 2 ) 1 2 (K + S)! () _ -S2 (2K + 1) (2K + 3) (2- 1)(2K + 1) - 2K + (K -s)!I K. _f, c () (c -.s2) (K + s + 1) 2 ( K+S) S2 K K+2 (2K + 3)(2 + 5) 2K+1 (K-)!................ The equations of the type aI/aA() = 0 enable us to express any of the coefficients A(S) by the formula while the equations of the type I/DG(), = 0 are found to be S K+s+l (1-l - ) =(- ) +3 KA+1 ( 21 -21 (K- )( K-S- ) )- ( +S + 2)(c + S + ) C( (2Kc- 1)(22 -3) K-2 (2K + 3)(2K 5) K-2 +Jl (K + 1)2 - 2 K-s2 s2 C {+ 1-(2 + 1)(2 + 3) (2 -1)(2+1) f+ 2} =.(6). On elimination of the coefficients A () there results a sequence equation for the coefficients Cs) and I have verified that this equation becomes identical with the sequence equation obtained by Hough (Part II. p. 153) for the same coefficients when his equation is simplified by neglecting the effect of the self-attraction. It will thus be seen that in the problem under discussion the method of Ritz yields very promptly the results already known. It is further noteworthy that the series assumed in this method to represent the components of displacement of the water do formally satisfy the differential equations of the problem, although this is not necessary to the success of the method, and need not be expected to hold in other applications. * The necessary formulae are all given by Hough in the second of the memoirs cited above.

Page  209 ON THE LAPLACEAN ORBIT METHODS BY A. 0. LEUSCHNER. The object of this paper is to set forth briefly to what extent the principles proposed by Laplace il his Mecanique Ce'leste, tome I, livre 2, may be made available in practice for the derivation of preliminary orbits of comets, minor planets, and satellites. Owing to its theoretical elegance, Laplace's method has engaged the attention of investigators at various times, but with indifferent success. In attempting to apply his principles to the formulation of practical orbit methods, mathematicians and astronomers have encountered difficulties which have led to the abandonment of Laplace's in favour of the more customary methods. It is not necessary to recite here the familiar history of Laplace's method, especially in view of the full and accurate historical account included by Charlier in his recent remarkable papers in the Meddelande of the Lund Observatory, Nos. 45 -47. It is my purpose, however, to show that the difficulties referred to are either not real or that they may be overcome without difficulty, and that the Laplacean principles admit of being cast into orbit methods of greater practical efficiency than the methods ordinarily in use. If this statement appears to be in direct contradiction to the opinion held by many eminent theoretical astronomers, the contradiction does not lie so much in an actual difference of mathematical results-for no mathematical problem, if correctly solved, admits of contradictory results-but in the difference of interpretation of the requirements of an orbit method. These requirements may be formulated either from an essentially theoretical or from an essentially practical point of view. The theoretical point of view demands methods which are applicable to the solution of an orbit from the necessary and sufficient conditions under any and all circumstances, as for instance for intervals of any length and in any ratio between the dates of three complete observations. The practical point of view demands methods which admit of as rapid and as accurate a solution of an orbit as the observational material immediately available in the case of a newly discovered object permits. In either case an orbit method is the more efficient the less numerical work is required in producing a result which represents the observational data. But since, in the present state of astronomical science, the observational conditions necessary and sufficient for the derivation of a preliminary orbit are available within a short time after discovery, the theoretical point of view must be of more consequence hereafter to the pure mathematician than to the theoretical astronomer. In other M. C. II. 4

Page  210 210 A. 0. LEUSCHNER words, astronomical science no longer requires the solution of the purely theoretical problem. What astronomical science does demand is a direct orbit method which admits of the derivation of the best possible preliminary orbit within a short time after discovery and of a method which admits of the convenient correction of the preliminary orbit on the basis of such additional observational material as may subsequently become available. To meet this demand of astronomical science has been the aim of my development of Laplace's principles into direct methods which I have termed " Short Methods." For several years these methods have been in process of printing in volume vii. of the Publications of the Lick Observatory, but their published appearance has been delayed owing to unavoidable obstacles in the State Printing Office of California. In a very condensed form they have been included in the third edition of Klinkerfues, Theoretische Astronomie, by H. Buchholz. Before proceeding to a discussion of the difficulties referred to above as having been encountered by various investigators in attempting to formulate practical orbit methods on the basis of Laplace's principles, and before demonstrating the advantages of the methods which I have termed " Short Methods," it is necessary to state that in emphasizing the practical value of the Laplacean principles I do not intend to detract in the least from their great but astronomically less important theoretical value. I shall therefore first consider in brief the general theoretical value of the Laplacean principles by means of a summary of their essential features. Laplace starts with the three linear differential equations of motion of the second order for the two-body problem, under the Newtonian law of attraction as applied to the motion of a material point (the object) about the Sun. He then expresses the heliocentric rectangular co-ordinates of the object in terms of its geocentric co-ordinates and the heliocentric co-ordinates of the Earth. The rectangular co-ordinates are then replaced in terms of polar co-ordinates, and thereby three equations are derived which give the geocentric distance p, its velocity p', and its acceleration p" at the epoch in terms of the observed co-ordinates (for which we may choose a and 8), their velocities (a', 8'), and their accelerations (a", 8"), and also in terms of the unknown heliocentric distance r of the object and known quantities depending upon the motion of the Earth about the Sun. If, therefore, for the present, we assume the co-ordinates a, 8, their velocities a', 8', and their accelerations a", 8" to be known, we have three fundamental equations for the solution of the four unknowns p, p', p", and r. The fourth equation is derived from the triangle Sun-Earth-object, and involves p, r, and known quantities. By elimination the problem reduces to an equation of the seventh degree in p with not more than two positive real roots, which may be interpolated to six decimals from a table which I have prepared for this purpose, so that the solution may be accomplished without the hitherto necessary laborious numerical approximations. The direct solution which has just been outlined corresponds to the so-called first hypothesis of other methods. It is evident that the accuracy of Laplace's direct solution depends upon the accuracy of the fundamental observational data for which we have chosen a,;, '; a", S". If the epoch is chosen to coincide with the date

Page  211 ON THE LAPLACEAN ORBIT METHODS 211 of one of the observations, then a, 8 are fixed numbers, and the accuracy of the Laplacean solution depends upon the accuracy of the adopted values of their velocities and accelerations or, which is an equivalent statement, upon the accuracy of their first and second differential coefficients. In practically all other methods the accuracy of the solution depends upon the accuracy of the adopted values of the ratios of the triangles. Perhaps the first shot was fired at Laplace's method by Lagrange in a letter (Oeuvres, t. xiv., p. 106), in which he says that while analytically Laplace's method constitutes the simplest solution of the problem, in practice it does not afford corresponding advantages because the differential coefficients could not be determined with the necessary accuracy. This far-reaching statement Lagrange intended as a mere opinion, which he proposed to verify later by mathematical demonstration. It is a remarkable fact that Lagrange's opinion, although never verified by himself, has been the chief cause of retarding the further development of the Laplacean method until recent times. It is quoted by Bauschinger, again without demonstration, in Die Bahnbestimmung der Himmelskorper, Leipzig, 1906, p. 393, and apparently confirmed by Bauschinger's statement (loc. cit.) that every application of methods based on Laplace's furnishes the proof of the laboriousness of the calculation and of the slight accuracy of the result. Harzer, to whom I am indebted for directing my attention to the Laplacean method by his valuable paper in A. N., No. 3371, has abandoned the researches which he commenced with such signal success, and has sought other lines of attack of the orbit problem, as indicated by his recent memoirs in the Publications of the Kiel Observatory. Bruns, who has made a most important contribution to the subject in A. N., No. 2824, particularly with reference to the barycentric parallax, clearly states the theoretical degree of approximation of the Laplacean methods, but renders a final verdict to the effect that the Laplacean methods will remain impracticable, because at least two direct solutions would be necessary to get rid of the parallax. To cite further authorities in explanation of why astronomers have been loth to adopt the Laplacean methods would seem superfluous. In analyzing the contention that the Laplacean methods are inferior theoretically, it is necessary to draw a distinction between the approximation of the so-called first hypothesis, direct solution, and that of subsequent hypotheses. The degree of accuracy of the Laplacean methods has been discussed by Poincare in a masterly manner in Bulletin Astronomique, t. xxIII., and also by myself in Publ. of the L. O., vol. vii. part 7, pp. 267-278. In agreement with Bruns' statement, it is found that for three observations the direct solution by Laplace's method is superior to the first hypothesis of Gauss, inasmuch as for unequal intervals Laplace's error in the geocentric distance is of the second order with respect to the intervals as compared to Gauss's error of the first order, provided the epoch be chosen to coincide with the mean date of the observations. For equal intervals the accuracy is the same, the error being of the second order. For unequal intervals, if the epoch in the Laplacean methods be chosen to coincide with the date of one of the observations (generally the middle), the error is also the same, being of the first order. 14-2

Page  212 212 A. 0. LEUSCHNER It is only recently that Bauschinger himself has published a discussion of the Laplacean methods in comparison to that of Gauss (Festschrift Heinrich Weber, Verlag von B. G. Teubner, 1912). Contrary to his previous contention (in his Bahnbestimmung), Bauschinger agrees in his Festschrift with the foregoing conclusions and independently demonstrates the same. It would seem, therefore, that the theoretical accuracy of the direct solution is no longer doubted, and this question requires no further comment. Nevertheless Bauschinger still considers it to be a fact, repeatedly confirmed by experience, that the methods based on Laplace do not accomplish in practice what they promise theoretically and supports this contention by the statement that although he has attempted many orbit solutions by Laplace's method, he has rarely succeeded in obtaining a satisfactory result, and that he has learned of similar failures on the part of other computers. In view of the conceded theoretical value of the direct solution, these failures on the part of Bauschinger are the more remarkable, as my own " Short Methods" in their present form have not failed in a single instance upon which they were tried. In parts 8 and 10 of the forthcoming volume VII. of the Publications of the L. O. no less than fifteen complete solutions by the "Short Methods" are reproduced in detail by Professor Crawford. Many of these were selected from the available computational material as severe test cases. For an additional number the accuracy of the results is carefully compared with solutions by other methods. Contrary to the experience of Bauschinger, in practically every one of these cases experience has demonstrated the superior accuracy and greater rapidity of the " Short Methods." It is very much to be deplored that the author of the Festschrift has refrained from publishing the details of the computations which he has marked as failures, so that the " Short Methods " in their present form might be tested on those examples. Such tests by direct solution, although very desirable, are, however, perhaps unnecessary in view of the following admission by the author (Festschrift, p. 10): "If it can be foreseen that one will succeed in the first approximation (for minor planets this will be the case for an interval up to 30 days between the extreme observations; for comets, for an interval of 6 to 8 days on the average), then neither of the two methods (Laplace's and Gauss's) has an advantage over the other; in particular no cause exists to designate Laplace's method as especially short (Leuschner's "Short Method "), nor Gauss's method as especially accurate." It is clear, therefore, that the failures did not apply to the intervals referred to above. Furthermore my own experience with the "Short Method" in its present form and with Gauss's method leaves no doubt that the solution by the former would have been not only more accurate, but also shorter. Now since it is conceded that for the intervals referred to the difficulties apparently encountered with the Laplacean method are not real, so that the " Short Method" is not inferior to that of Gauss, we may proceed to the examination of the concluding statement of the Festschrift:

Page  213 ON THE LAPLACEAN ORBIT METHODS 213 "But if a computation by approximations is in prospect, then it is not economical to work by Laplace's method, whether one carries out the calculation of hypotheses or desists from the solution proper (by successive hypothesis) and undertakes an orbit correction; for then Gauss's method maintains the advantage theoretically as well as in practice." Can this contention, which, aside from Bruns's verdict regarding the parallax, represents the only remaining criticism of the Laplacean method, be substantiated? The answer is yes, if applied to Bauschinger's own definition or interpretation of what ought to constitute " the process of the second hypothesis" in the Laplacean methods. The answer is emphatically no, if applied to the "Short Methods" in their present form. Bauschinger (Festschrift, p. 2) recognizes as Laplace's method that indicated by Laplace and later further developed by Bruns (and Schwarzschild), but not the transformations which are due to Harzer (and by his example to myself). He then proceeds to a discussion of the Laplacean method as thus defined and disposes of Harzer's and my own methods by declaring that they do not represent real orbit solutions in the sense of Gauss. Although the "Short Methods" are thus distinguished from the Laplacean methods, and although their convergence is not further discussed, they are included in his final judgment of the Laplacean methods, as quoted above. It is not my purpose to dwell on this inconsistency, but rather to emphasize that Bauschinger's difficulty arises from his own interpretation of what constitutes an orbit solution. He adheres to the purely theoretical point of view referred to above. But there can be no doubt that in astronomical science that method must be considered as the most satisfactory which solves the problem with the greatest accuracy and the least numerical effort, whether such method be developed in the sense of Gauss or any other sense. The following free quotation from the Festschrift (p. 2) will make Bauschinger's position perfectly clear: "Laplace and his successors accomplish the approximations following the first in a logical manner by improving the differential coefficients of the observed co-ordinates before obtaining the definitive Elements, and offer therefore a real orbit method in the sense of Gauss. Harzer and Leuschner, on the contrary, recognizing the slight convergence of this procedure, have adopted another course, in that they compare the observations with the results of the first approximation and correct the elements by means of the residuals; that is, then, naturally not an orbit determination in the strict sense, but an orbit correction and the problem proper is not solved"! Before proceeding, might it not be well to ask whether a, real solution has not been accomplished and whether the problem proper is not solved, no matter how the orbit has been produced, so long as the result is satisfactory and conveniently obtained? Would it not be rather illogical to adhere to Laplace's original process, after its slow convergence has been universally recognized? On p. 9 of the Festschrift we read: " In Laplace's method the first.and second derivatives of the observed co-ordinates

Page  214 214 A. 0. LEUSCHNER are improved by calculating step by step the third, fourth, etc. derivatives...," and also: "I am not aware that this process has ever been applied in practice; suitable formulae for computation are not to be found even in the more elaborate reproductions of Laplace's method." Bauschinger's failures with Laplace's method in the second hypothesis cannot, therefore, refer to his own interpretation of Laplace's method, but must have been experienced with Harzer's or my own methods. The limitations of Harzer's method were fully recognized by myself (see Publ. of the L. O., vol. vii., part 1), and gave the impetus to my " Short Methods." If, then, finally any of Bauschinger's failures are due to the " Short Methods," the computations must have been performed by the " Short Method" in its original form (Publ. of the L. O., vol. vII., part 1) and not by the "Short Methods" in their present form, in which they have appeared only recently for the first time in Buchholz's third edition of Klinkerfues. It is not difficult to state why the original " Short Method" may have failed in certain cases in the second approximation. The differential correction is based on series which in certain cases do not converge satisfactorily, as for instance for fast moving comets near perihelion and for long arcs. But this difficulty is now fully overcome in the present form of the " Short Methods" by the introduction of closed expressions. I must therefore contend that Bauschinger's final conclusion, quoted above, that it is not economical to work in the second approximation by Laplace's methods, does not refer to the " Short Methods" in their present form, as these were not known to him when he wrote the Festschrift. I must also contend (and the proofs of this contention are contained in Publ. of the L. O., vol. viI., parts 7 to 10) that in their present form the " Short Methods" answer all present requirements of astronomical science, in that they produce results of the greatest accuracy, as far as the available data will permit by any method, and in the most convenient form. The author of the Festschrift contends (p. 2) that to raise Laplace's solution to the level of Gauss's it would be necessary to invent devices, such as Gauss gained in so incomparable a manner by the introduction of the ratio of sector to triangle, upon which ultimately the whole brevity and the whole success of his method depends. It is not necessary to review here the familiar details of preparing for successive hypothesis by the computation of the ratios of sector to triangle in the method of Gauss; let it be sufficient to recall that each ratio must be computed by successive approximation, that their object is to express accurately the ratios of the triangles on the basis of the previous hypothesis, and that each hypothesis is worked out with the ratios of the triangles belonging to the previous hypothesis. This process, upon which Bauschinger lays great stress, is wholly indirect and is entirely dispensed with in the direct differential correction introduced by Harzer and myself. In this connection I may quote fromn my Introduction to part 7, vol. vii., of the Publ. of the L. O., as follows:

Page  215 ON THE LAPLACEAN ORBIT METHODS 215 "Instead of forming a second hypothesis in the usual sense by correcting the geocentric velocities and accelerations on the basis of the first solution, it is here proposed, as in part 1, to determine at once from differential relations such corrections to the geocentric distance and to the velocities in the heliocentric rectangular coordinates-which, together with the geocentric co-ordinates at the middle date, practically, are the elements of the orbit-as will remove whatever residuals the first solution may leave in the first and third geocentric places. This mode of approximation is not comparable to the method of forming successive hypotheses, but obviously leads directly to the required result. Hitherto it has been applied to the solution of more or less definitive orbits from normal places, with the aid of the differential coefficients of the geocentric co-ordinates with respect to the geometrical elements. "The advantages of the method of differential correction, over the method of successive hypotheses in improving an orbit, are exactly the same as the advantages of the method of differential correction over the ordinary trials in the successive approximations for the distances in Olbers's and Gauss's methods, etc. In the course of the ordinary trials, the final values of the distances of one trial form the initial values in the next trial, etc. In the method of differential correction, such corrections to the initial values of one trial are derived differentially from the differences between the initial and final values in the same trial, as will produce an agreement of the initial and final values in the next trial. But the number of approximations required by the ordinary trials is, in general, far in excess of that required by the method of differential correction. "Another simplification involved in the direct solution and the differential correction proposed here is that the consideration of various topics essential to the older methods becomes unnecessary, as, for instance, the consideration of the ratio sector to triangle, etc. "If it had been deemed desirable or essential, formulae on the plan of successive hypotheses might have been set up without difficulty. For this purpose it is not necessary to determine the corrections to the initial values of the velocities and accelerations from the results of the first solution. Instead, an expression of a = p cos 8 may be derived in which account is taken of the third and higher derivatives expressed in terms of the co-ordinates and their lower derivatives. The actual derivation of these expressions of the higher derivatives is somewhat complicated, although theoretically quite simple, but the resulting formula for K admits of a second hypo1 d(r thesis as soon as r and r' = - become known in the first hypothesis." k dt Preparatory to the differential correction in the " Short Methods" the residuals are not computed, as Bauschinger implies, after the elements have been derived, but they are obtained in the course of the direct solution from the geocentric distance p and the heliocentric velocities x', y', z'. The elements are computed after the residuals are found satisfactory, which is generally the case in the first hypothesis (direct solution) for moderate intervals. Nor should any emphasis be laid upon the fact (Festschrift, p. 8) that Laplace's direct solution may lead to considerable errors for fast moving comets, even for short

Page  216 216 A. 0. LEUSCHNER intervals, and to wholly illusory results for long intervals. Comet 1910 a was the fastest moving comet of recent years at the time of discovery. The orbit solution was performed without the least difficulty from one-day intervals by the "Short Method." The question of a direct solution from very long intervals does not arise in practice, as previous approximations are invariably available in such cases. Long intervals are apt to.lead to inaccurate results in any method, if applied to a first approximation instead of being used as a basis of orbit improvement. In computing from a long arc, every computer, no matter what method he may favour, will derive a first approximation of his fundamental data, be it the ratios of triangles, with or without the aid of the ratio of sector to triangle, or be it my starting values p, x', y', z', from existing preliminary orbits. As repeatedly referred to above, the solution of a first approximation from a long arc is of purely theoretical interest, and does not fall within the requirements of astronomical science. Nor does there appear to be any reason why for a long arc the velocities and accelerations of the observed co-ordinates should not be based upon more than three observations if four may become necessary in Gauss's method. The verdict of Bruns regarding the parallax, referred to above, pertains to a difficulty which I have readily overcome by a process of eliminating the parallax in the direct solution. A striking example of the effect of parallax was furnished by minor planet 1911 MT, where the direct solution (by Messrs Haynes and Pitman, first approximation), with the elimination of parallax, yielded the desired result. In this connection I ought to state that the computers find that they unfortunately committed a numerical error in the representation of their direct solution and then removed residuals which did not exist, so that the differential correction published in L. O. Bulletin 210 was wholly superfluous. The foregoing discussion inevitably leads to the conclusion that the difficulties that appear to have been encountered in the application of Laplace's method are either not real or that they have finally been overcome in the present form of the "Short Methods." It is also evident that whatever difference of opinion may still exist regarding the value of my own " Short Methods " resolves itself into a difference of interpretation of the requirements of an orbit method. At the outset of this paper I stated it to be my object to set forth to what extent the principles proposed by Laplace may be made available for derivation of practical orbit methods. Having discussed and disposed of the arguments against its practical value, I desire to direct attention to the fact that there have existed other and far more serious causes of failure not referred to by any one in his published criticisms, but fortunately overcome in my forthcoming papers in volume vii. of the Publications of the Lick Observatory. In this connection I may mention that hitherto comet orbits were computed by the general method of Laplace, without hypothesis regarding the eccentricity, instead of by a direct parabolic method; that closed expressions were not available for the second approximation so that it could not be applied to any and all conditions,

Page  217 ON THE LAPLACEAN ORBIT METHODS 217 particularly to arcs of any length; that the solution of the distances was performed by laborious approximations; that the accuracy attainable in each case could not be ascertained in advance; that numerical criteria did not exist to distinguish the physical from the mathematical solutions in the case of three roots in the parabolic method; that, as already referred to, no method existed for completely eliminating the parallax, if necessary; that no criteria were available to distinguish between the feasibility of solutions with or without assumption regarding the eccentricity; that no provision had been made for passing from one class of orbit to another in the course of computation without repeating the solution; that no provision had been made for taking immediate account of the perturbations, and so forth. For the details of these and many other improvements I must refer to my forthcoming papers and to the examples discussed by Professor Crawford in volume vii. of the Publications of the Lick Observatory. I trust that I may not appear to have been over-enthusiastic in proclaiming the possibilities of the orbit method initiated by Laplace, and too severe in refuting the objections that have been raised against it. I hope that its latest formulations, whether they still are to be counted among the Laplacean methods or not, will be given frequent and exhaustive trial and that these trials may lead to further improvements of importance.

Page  218 THE BALANCING OF THE FOUR-CRANK ENGINE BY G. T. BENNETT. ~ 1. The problem of balancing the primary and secondary hammering and the primary tilting of a four-cylinder engine with equal cranks has been treated by several writers. Special reference may be made to Schubert ("Zur Theorie des Schlickschen Problems," Mathematische Gesellschaft, Hamburg, 1898), Lorenz (Dynamik der Kurbelgetriebe, Leipzig, 1901), and papers by Mallock, Macfarlane Gray, Dalby, Schlick, Inglis and others in the Transactions of the Institution of Naval Architects (1886-1911). The present paper is in the main a geometrical study, having the particular requirements of the practical mechanical problem as its basis. The method differs considerably from those hitherto used; and leads not only to some graphical constructions which may assist the draughtsman, but also to some fresh formulae which may facilitate calculations. The results as a whole will, it is hoped, be found to supplement and complete those which are at present available. The material may best be presented in a sequence determined rather by convenience of theoretical development. At the end of the paper the chief results are summarized and rearranged in a form which may appeal more directly to the engineer engaged in designing a balanced engine. ~ 2. The problem consists virtually in placing four masses M1, M12, M3, M4 at the ends of equal cranks N1VQ, N2Q2, N3Q3, N4Q4, set out perpendicular to a shaft N1 N2N3N14, so as to satisfy the six equations MX1, = 0...............(1) I Xcos 01 = 0.............(7) M1yl = 0...............(2) M1 sin 01 = 0............(8) M1 (;1_- yl2) _ 0......(3) SM, cos 20 = 0............(9) MMI1y1 = 0............(4) or M1 sin 201 = 0.........(10) Mlzl, = 0............(5) V MXz cos 01 = 0.........(11) Mlylz i = 0............(6) Mlzl sin = 0.........(12) where M1 is the mass at Q1 (xlylzl) on the circular cylinder + y2 = a2; with x =a cos 01, yj = a sin 01 for the equivalent forms of equation in cylindrical coordinates; and similarly for the other three masses.

Page  219 THE BALANCING OF THE FOUR-CRANK ENGINE 219 Of the first six equations, or their equivalents, (1) and (2) secure the balance of the primary forces, (3) and (4) secure the balance of the secondary forces, and (5) and (6) secure the balance of the primary couples. It is convenient to take the origin at the centroid and to associate with the above equations the additional three M,= M....................................(13), S0,zi = 0....................................(14), Mlz = M c2.................................(15). The set of six equations (7-12) involve only eight essentially independent quantities; namely the ratios of four masses, the differences of four angles, and the ratios of the differences of four lengths. Any two of the eight, as data, should lead to the values of the rest, and any three should be related. In particular the four crank-angles are necessarily related (20-31) and the four masses are related (95-97). The distances between the cranks are independent, but the distances of the cranks from the centroid are also four related quantities (62-63). ~ 3. We may first consider (1), (2), (3), (4) alone. Then if Pi, P2, P., P, are the projections of Q1, Q2, Q3, Q4 on the plane z = 0 the equations may be interpreted, two-dimensionally, as meaning that masses MI, M2, M3, M4 at P1,, P2, P, P4 are "dynamically circular." The centroid is at the centre of the circle K - 2 y2 - a2 = 0..............................(16) and the moment of inertia about any diameter is the same and equal to 5Mix 2= sM ly2 = a2........................ (17). The product of inertia about any two perpendicular diameters is zero: but the general case of any two lines whatever giving zero product of inertia may be considered: say x + my +1=0 and 'x + m'y + 1 = 0 give zero product of inertia. So 2M (lx + my + 1)(l'x + m'y + 1) = 0, and hence jMa2 (I1' + mm') + M = 0; that is I1' + mm' + 2/a2 = 0..............................(18). Hence the two lines are conjugate lines with regard to the imaginary polarizing circle 2+ y2+ a2/2 =...........................(19); each line, that is, passes through the pole of the other with regard to the circle I. Pole and polar are on opposite sides of the centre 0 at distances having a product equal to a2/2. (In particular any equilateral triangle inscribed in K is self-polar with regard to I.) Take now specially the pair of chords P1P2 and P3P4, for which the product of inertia is zero. It follows that these lines are conjugate. The pair of chords P2P3 and PIP4 are similarly conjugate, and also P3P1 and P2P4. If the points Pi, P2, Ps

Page  220 220 G. T. BENNETT are placed arbitrarily on the circle K, the point P4 may be found thus:-Construct the pole F of PI P, with regard to the circle I; then P3F meets the circle again in P4 (Fig. 1). Permutations of Pi, P2, PS give three similar constructions all leading to the P1 P2 Fig. 1. same point P4. This construction gives a graphical form to the relation among the four crank-angles 01, 02, 03, 04. It may be remarked that the pole of any chord of K is on the diameter perpendicular to the chord, at the point where it is cut by the side of an equilateral triangle inscribed in K, with either end of the chord as the opposite vertex. ~ 4. With any four crank-angles 01, 02, 03, 04 are associated masses M1, M2, M3, M4 with ratios determinable from any three of the equations (7), (8), (9), (10). These mass-ratios may, however, be found graphically by further use of the method of products of inertia. Lines through P3 and P4 conjugate with regard to I give a zero product of inertia for the four masses, and therefore also for 1M and M2 alone. Hence the products of inertia for M1 and M2 are equal and opposite. If one of the two lines is drawn parallel to P1 P2 the other then cuts PIP2 in G12, the centroid of M1 at Pi and 112 at P2. Hence the construction:-Draw through P3 a line parallel to Pi P and find L its pole with regard to I. Then P4L meets PIP2 in G2,, the centroid of M, and M, at Pi and P2 (Fig. 1). The line G0,O will meet P3P4 in G34, and the ratios of the four masses may all be got from the segments of the lines PG12P2, P3G34P4 and G120G34: or the construction may be repeated for each pair of masses separately. ~ 5. The relation connecting the differences 012 01 - 02, etc. of the crank-angles may be put in various forms, each being an equivalent of the determinant eliminant of equations (7), (8), (9), (10). They are necessary for use later in connection with the formulae involving the masses and the crank-intervals.

Page  221 THE BALANCING OF THE FOUR-CRANK ENGINE 221 The chords P1 P' and P3P4 of circle K are cos (0, + 02) + y sin (01 + 0,)- a cos 012 = 0, x cos (0 + 0) + y sin - (0 + 04) - a cos I034 =0, and these are conjugate with regard to circle I (19) if COs I (01 + 0 -- 03 - 04) + 2 cos01cos 1 34 = 0...........(20), two other equivalent equations having the same form*. The symmetric equations cos (01 + 02- 03- 04) = 0......................(21) and E cos 012. cos 1034 = 0..........................(22) are equivalents of (20). Multiplied by 2 sin o- and 2 cos o, where 2o- = S0, (21) gives s sin(01 +02)= 0 or x1y2 = 0.....................(23), and 2 cos (01 + 02) = 0 or 2x1X2 = y'y2..................(24). From (21) or (22), by expanding, 3 (1 + tan i0, tan 01 tan 103 tan ~04) + E tan 01 tan 02 = 0......(25), from which is obtainable, without ambiguity of value, the fourth angle associated with any three. This last is a function of the differences of the angles, though not so in appearance, and may be written in each of four forms such as 3 + tan 012 tan 013 + tan 201: tan 2 014 + tan -104 tan ~01 = 0......(26). This equation gives immediately any one of the angles, 01 excepted, from known values of the other three. The equation (20) multiplied by 2 sin 30], gives sin 2 (03, + 04) - sin 1 (023 024) + sin 1 (2012 + 034) + sin 1 (2012 - 04) = 0, and so, pairing inside and outside terms, sin 2 (01 + 03)cos 12 - sin I (021 + 023) cos 014 = 0. Hence six formulae of the type sin 1 (012 + 013) cos1 07 si 2 021 + 0 ) COS 24..................... sin }0,~+0.) cos ~0, (27). By multiplication of three of these last equations sin 2 (013 + 0,4) sin 2 (02 + 024) sin 2 (032 + 034) - sin - (03 + 024) sin ~ (031 + 034) sin 2 (013 + 014)............(28), of which type there are four formulae. From (20) COS (01 + 02 - 03 - 04) = 2 (1 + cos 012) (1 + cos 034) - 1 = (1 + 2 cos 01) (1 + 2 cos 034)- 2 cos 02 cos 034. So (1 + 2 cos 01)(1 + 2 cos 034) = cos (01 + 02- 03- 04).........(29), * In any formula which is unsymmetric in the suffixes, they may be permuted, and give rise to a group of equations all of the same type. Any equation quoted by its reference number will be understood to belong to the type so named, but not necessarily to be the particular specimen given.

Page  222 222 G. T. BENNETT and hence from the identity 1 + 2 cos 0 = sin 30/sin 0 sin -082. sin 804 = cet. sim........................(30). sin 012,. sin 034 From (29) 1 + 2 cos 014 1 + 2 cos 013 COS 014-COS 013 sin 1 (013 + 014) 1 + 2 cos 024 1 + 2cos cos O3 - cos 823- sin (823+ 024) Hence six formulae of the type in 13 ) n 0sin (0+ 0) sin 24 (3. 2 2 2 o.oooooo........... (3 1). sin (023 + 024) sin 4024 sin -014 Further, an equation in 0 may be put down of which 01, 02, 03, 04 are the four solutions. With 2- = 20O the equation cos (20- o)- Z cos ( + 01 - ) + E cos I (01 + 02- 03- 04)= 0...(32) is identically satisfied by 01, 02, 80, 04. Hence, by (21), they satisfy cos (20 - a) = E cos (0 + 01 - o).....................(33). Putting E sin 0, = X sin e..............................(34), E cos 80 = X cos e..............................(35), so that X2 = 4 + 22 cos 01..............................(36) ~ sin 01 and tan e = 1 or I sin (01 - e) = 0..................(37), and cos(0 (33) may be written cos (20- o) = X cos (0 + e - o).....................(38). And further it may be noticed that X2 sin 2e = sin 201 + 2Z sin (01 + 02) = sin 201 from (23)......(39), X2 cos 2e = cos 203 + 22 cos (01 + 02) = 2 cos 208 from (24)......(40), hence X4= 4 + 2 cos 20..............................(41) Zsin 20, and tan 2e= 2 or sin 2 (01- e)= 0............... (42). n cos 20, ~ 6. Some of the equations of ~ 5, besides the parent form (20), admit of a geometric interpretation. The formula (30) shows the angles 01, 02, 03, 04 to be such that a pencil of directions given by 0i, I02, 203, 104 is homographic with a pencil given by 301, 302, 303, 304. The same result appears also from (38), which may be rewritten as a homographic relation between tan 40 and tan 30. Formula (28) shows an involution pencil to be given by directions 01, 03, 03 paired with (0 +04)(0+04), (03 + 04): and this result may be obtained directly from the figure and construction of ~ 3. For a quadrilateral may be taken whose sides are the three sides of the triangle PIP2P3 and the polar of P4 with regard to I; the three pairs of vertices being PI, P2, P3 paired with the poles of P1P4, P2P4, P3P4. Projection of these vertices from 0 gives an involution pencil having the directions of its rays given by the angles in question.

Page  223 THE BALANCING OF THE FOUR-CRANK ENGINE 223 A companion involution may be associated with the last. If angles 02, 02, 03 paired with f), 02, (3 give an involution, then b2, 03 may be replaced by 01 + 0,- 0 -and 0 + b - 12 respectively and a new involution is obtained. On the present occasion the new involution is given by angles 02, 0., 03 paired with 1 (01 + 04), x (301 - 03), - (301- 02). This is also shown independently by equation (31). The last two directions are obtainable by taking the images in OP1 of the perpendiculars from 0 on P1P3 and P1P2. This affords a fresh construction, alternative to the polar method of ~ 3, deriving P4 from PI, P2, P3; namely:-Let the images in OPI of the perpendiculars from 0 upon P1P3 and PIP2 meet these chords PP3 and P P2 in U and V, and let UV meet P2P3 in W. Then PIP4 is perpendicular to OW. Another construction for this involution may be made independently of the circle as follows. Draw a triangle (Fig. 2) with sides al, a2, a3 representing the given directions 01, 02, 03. Take the line which bisects the angle 04 between a, and a2 (observing the proper senses of a, and a2) and reflect it in a,, giving a line /3g. Similarly, reflect the angle bisector of al and as in a, and so get /32. Draw /3 joining the point /33/3s to the vertex a2a. Then reflecting a, (with sense included) in /, gives the direction 04. A slightly condensed variant of this method a may be used. Take the point of intersection of / 3 the angle-bisectors for the vertices on ac (being the centre of one of the circles touching the sides --- of the triangle ala2a) and take its image in the side a,. Joining this to the opposite vertex gives Fig. 2. /3i, and the reflection of a, in /i gives the direction 04. Formulae (34), (35), (39), (40) may be given a graphical interpretation analogous to the mass-quadrilaterals derivable from the equations (7), (8), (9), (10). From (7) and (8) it follows that a closed quadrilateral may be drawn whose sides have the directions 0,, 02, 03, 04 and lengths proportional to MI, M2, M3, M4: and (9), (10) show that another quadrilateral may be drawn with the sides unaltered and the angles doubled. Similarly (34), (35) show that four successive unit vectors in directions 01, 02, 03, 04 give a resultant vector X in direction e: and (39), (40) show that four unit vectors in directions 208, 20,, 203, 204 give a resultant X2 in direction 2e. This affords a test for the proper relation among the directions of the cranks. It has the appearance of a twofold condition; but if, for the doubled angles, the resultant comes squared, then (in general) its angle is doubled, and conversely. A similar interpretation may be given to (23), (24): namely that a closed equilateral hexagon can be drawn with its sides, say 12, etc., having directions given by 0 + 02, etc. If the sides are drawn (Fig. 3) in the order 12, 34, 23, 14, 31, 24 the alternate vertices (beginning with the first) lie on a line whose direction is represented by o, and the zero sum of projections of the sides on this line interprets equation (21).

Page  224 224 G. T. BENNETT A variant of this last arises on taking the resultants of vectors 12, 23, 31 and 14, 24, 34 as equal and opposite. Rotating both through the angle - (0, + 02 +) it follows easily that equal vectors in direc- 3 tions 01, 02, 03 give a resultant whose direction is - - 04 + r/2: deriving 4 24 again from 03, 0,, 08. 14 ~ 7. The zero determinant obtained 12 by eliminating the ratios of the masses from equations (1), (2), (3), (4) may be, interpreted as meaning that a rectangular hyperbola (Fig. 4) passes through the points PI, P2, P;, P4 and 0. Or, as equivalent, taking any conic F (x, y)= 0 through \ the four points P,, P2, P3, P4, then 2334 AM1F(x,,y) = 0: and hence, using (1), (2), (3), (4), it follows that if the conic passes through 0 it is a rectangular / hyperbola, and conversely. The polar equations of the hyperbola and the circle K give by elimination of the radius vector Fig. 3. the equation (38) for the vectorial angles of the common points. The polar equation of the hyperbola is therefore r cos (20 - o) = Xa cos (0 + e - o)....................(43). Fig. 4.

Page  225 THE BALANCING OF THE FOUR-CRANK ENGINE 225 The directions of the asymptotes are given by 6- - (7r/2 + o) (mod.,r/2)........................(44), and hence the directions of the axes by 0 _ o/2 (m od. 7r/2)...........................(45), being the mean values of 0i, 0,, 03, 04 (mod. 2rr). This agrees with the equal inclination of any pair of sides of the quadrangle PjP2P3P4 to the axes of the hyperbola. The tangent at 0 is 0 = 7/2 + a —.............................(46), and the diameter OAB is 0 = e, with (Xa, e) for coordinates of B: the diameter and tangent being equally inclined to the asymptotes. It will be found useful later to be able to construct the line OAB without making use of the hyperbola itself. From the resultant of unit vectors in directions 01, 02, 03, 04 given in ~ 6 it follows here that the resultant of vectors OP1, OPt, OP3, OP4 is OB. This is in agreement with a theorem, known of any rectangular hyperbola cut by a circle, that the centre of mean position C of the four common points is midway between the centres, A and 0. As alternative constructions, or as checks on the drawing, it may be remarked that the centre A lies on the nine-point circles of the four triangles formed by the points Pi, P,, P3, P,. Further, also, each of four such pairs of lines as P1O and P1B are equally inclined to the known directions of the asymptotes. ~ 8. It may now be shown that any transversal drawn parallel to the line OCAB (Fig. 4) cuts the radii OPF, OP2, OP,, OP4 in points Z1, Z, Z,, Z, giving a range similar to that of the ends of the cranks N\, N2, VN, N4 on the axis of the shaft. For this purpose the line OCAB may conveniently be taken, by rotation e of the former axes, as the line y= 0. The equations of the circle and the hyperbola are then K - 2 + y2 - a = 0..........................................(47), H - 2 + y2 + 2z.y + Xa (x - /y) = 0...............(48), where, = tan (2e - a)..............................(49). From these equations are derivable xz = Xa, Sy, =0, Ez,1y = 0; confirming the mean centre C and the equation (42). Let the cubic curve J - (x - Iy) K + (x + y+ a/X)H = 0..................(50) now be taken, passing through the four points PI, P,, P3, P4. It reduces to the form J\/a - 2X (1 + L2) xy2a + (X2 - 1) X2 + 2,ixy + (1 - X2,2) y2 = 0...(51) (in which only the terms, and not the precise coefficients, are here essential). Taking MlJ,/y,= 0 and quoting (1), (2), (4) it follows that M, 2/yl =................................. (52), or, as equivalent, since xI2 + y 2 = a2, IV /yl = 0.................................(53). M. C. II. 15

Page  226 226 G. T. BENNETT The equation (48) then gives EMHl/yl = 0, and so M lx,/yl =.M.................................(54). Take now a transversal y = h cutting OPI, OP2, OP3, OP4 in points Z1, Z^, Z4; and let the tangent x - uy=0 cut y= h in G (ph, h). Let the lengths GZ1, GZ2, GZ6, GZ4 be, by temporary anticipation, zl, Z2, Z3, z4. So +uh + z = h1l/y1, that is 21 = h (x2/y1 -.)..............................(55). Then there follow as consequences.Mz = 0 from (54)...........................(56), SMxil zi = SM1h (xl1/y - ix1) = 0 from (52)............(57), and Mlylzl = SMh (x1- yl)= 0........................(58). Hence lengths z1, Z2, Z3, z4 above constructed satisfy (56), (57), (58), and so have ratios which satisfy the equations (5), (6) and (14). A transversal, therefore, drawn parallel to the line of centres OA is cut by the crank-radii in points Zi, Z2, Z3, Z4 giving the spacing of the cylinders; and is cut by the tangent at 0 in the point G which represents the centroid of the four masses M1,, M2, M3, M4 at Z2, Z2, Z3, Z4. ~ 9. In terms of three dimensions it now appears, from (55), that the points Q1, Q2, Q3, Q4 lie on the rectangular hyperbolic paraboloid yz = h (x- - y).................................(59) as well as upon the circular and hyperbolic cylinders K- x2 + y2 _ a2 = 0 and H - x2 + y2 + 2pxy + Xa (x - Py) = 0. The two cylinders have four common generators each cutting the paraboloid in one finite point, and so giving the four points Q1, Q2, Q3, Q4. Eliminating x and y from the equations of the three quadrics there results a quartic equation in z, with roots Zl, z2, Z3, z4, which may be written X2 22 2z ) (22 _ p2) 2)2....................... (60), where z0 = - ph; so that G is (- z, h) and p = OG. This quartic is of the form f(z> z -p Z + 2 + + = 0........................(61), where p = Sz, q = -iZZ2, s = Zl2z3z44; and shows 212Z2 3= 0................................. (62), or, as equivalent, /zi = 0....................................(63). This is the relation referred to at the outset (~ 2) as connecting the distances of the cranks from the centroid. If the four points Z, Z2, Z3, Z4 are taken as data, and z, zl2, z2, z4, measured from an arbitrary origin, fail to satisfy the equation (63), then to correct the values and determine G as new origin the values z, Z2, z3, z4 must all be reduced by a value z determined by 2l/(z - - z) = 0............................ (64),

Page  227 THE BALANCING OF THE FOUR-CRANK ENGINE 227 that is (65). that is,d [(Z - z=) (z - 2) (Z- -3) (Z - z)] = 0.....................(65). This cubic in z has three real roots separating the four real roots of the quartic. The root to be chosen (see later, ~ 14) is the central one of the three. Its value may be calculated by any of the usual methods, Homer's or otherwise. The trigonometric method will be found to apply somewhat favourably; giving z = -Sz/4 + k cos (..............................(66), where 12k2= E (zi - z2)..............................(67) and 8k3 cos 30 = (z - Z2) (z, - Z3) (z - Z)..................(68). Comparison of the equivalent quartics (60), (61) gives 2X2 z (X2 +~2)p 2 (64 X2- = --.....................(69), p q s and hence p4 -qp2~ 3s= 0.............................. (70), 2z_ 3p 2 a quadratic for p2, and - =................................ (71), p 2p- + q giving z0. The equations (70), (71) serve to determine the position of 0 relatively to four given positions of Z,, Z2, Z3, Z4, and so determine the crank-lines as rays of the pencil projecting Z Z, Z,, Z4 from O. ~ 10. Equation (60) gives, for z = z, XOZ = (12 - 2)/..............................(72), and if GY,=GZ, with Y, on OZ1, then 0Y1. OZI= p2-z2, and so OY1=XGY, (Fig. 5). ~~Y1~~~~~~~Y Z4 Z2 Z, G Z3 Z Fig. 5. The sides and angles of the triangle OGY, give a geometric interpretation to the formula (38). Further it follows that the four points IY, Y2, Y3, Y4 lie on a circle Y - XS [(x - (h)2 + (y- h)2] _ ()2 + y2)= 0...............(73) with 0 and G for inverse points. The centre D on OG is (- p/2, -p/2pL) by use of (69); also GD = s/p3, DO/DG = X2; and the square of the tangent from G to the circle Y is sip2. 15-2

Page  228 228 G. T. BENNETT It may be noticed that, along with the pencil of lines OY,, OY, O}3, OY making angles 01, 02, 03, 04 with the original initial line, may be associated the pencil GY,, GY), GY3, GYT making angles 20,-e, 2 - e, 203-e, 284-, and the pencil DYI, DY,, DY3, DY4 making with the same line angles 308 - o — 7r/2, etc. ~ 11. If the equations (1), (2), (5), (6) are considered alone they represent the equilibrium of four centrifugal force vectors due to revolving masses M1, M1, Ms, M4, namely M,. QNlVio2 on QIN1, etc., the lines of action being all normal to the axis of the shaft. Four further conditions of equilibrium are necessary, of which one is geometrical. Any line meeting three of the forces must, for zero moments, meet the other also: and hence the four lines QNiV1, etc. are generators of a rectangular hyperbolic paraboloid, as has appeared analytically in ~ 9 (59). The four generators QiN1, etc. give the same cross-ratio for the pencil 0 (PIP.2P3P), representing their directions, as for the range NN3N23N4 in which the axis z meets them. This is accountable for the possibility (~ 8) of cutting the pencil 0(PlP2P3P4) by a transversal Z1Z2Z3Z4 giving a range similar to N3NoN3N4. The three further conditions give the ratios necessary for the forces, and hence for the masses. Taking moments about the line through N3 parallel to Q4N4, so that two of the four forces have no moment, MAz13 sin 014 + M2z23 sin 024 = 0, where z13 = - z3, etc.; hence M1_ xZ3 sin 04 M_ z23 sin 024 (74), M 2 Z1 sin 014 * **.. **..***** *..............*, i *., i Ml Z24 sin S3 and similarly 2 - - z4 sin 0, 111 -2 z14 sin 0r z13 Z24 sin 03. sin 024 so........ (75), z14. z23 sin 014. sin 03................... showing again the equal cross-ratios already referred to. A geometrical equivalent of (74) is M1 _ Z2Z3. ZZ. OZl M, ZlZ.Z, Z.ZOZ,' and hence the ratios of the masses are symmetrically given by M1.Z1Z. Z Z3. ZZ4/OZ = cet. sim...................(76), where OZ3, as in (72), has its sign determined by comparison of its sense with that of OPI. From (72) and (76) MIzz,2z,-3,l/(z -2 -P = cet. sim.......................(77). Now from (61) may be derived the identity zf' () - 3f(Z) _ _4 - qz2-3s - (2 - p2) (z2 + p2 - q) + (p4 - qp2 - 3s'). Hence, with (70), zf' (l) = (Z2 - p2) (Z,2 + p2 q), and so Zl1Z23zZ4u/(z2 - p2) = Z12 + p2 q....................(78). Hence M1 (z12 + p2 - q) = cet. sim.........................(79), a formula deriving the ratios of the masses directly from the spacing of the cranks.

Page  229 THE BALANCING OF THE FOUR-CRANK ENGINE 229 ~ 12. The equality of the cross-ratios of the range N1N2N3N4 and the pencil of rays 0 (PP2P3P4P) permits not only a direction for a transversal cutting the rays of the pencil in a similar range (~ 11), but also, by exchange of two pairs of elements, allows a direction for a transversal cutting the rays, in order, in 2Z3Z4Z3: and similarly two other directions giving Z3Z4Z1Z and Z4ZZ2Z,. These three directions, known as readily derivable from a mass-quadrilateral by taking the direction of a diagonal, suffice for their purpose but lack the direct correspondence given by the transversal of ~ 8. In the figure and construction of ~ 4 the line G20OG,3 is the line of the resultant of the vectors M1. OP, and M2. OP2, and also of M,. OP3 and M4. OP4; hence it is parallel to a diagonal of a mass-quadrilateral for which 11 and M2 are adjacent sides. The figure of ~ 4 thus gives at once the three directions of transversals which cut the pencil 0 (P1P,P:,P,) in a range of points giving the crank-spacings, but with double exchange of two pairs of elements. The transversal of ~~ 7-8, drawn parallel to the resultant of four equal vectors parallel to the rays, gives the spacing also, with the advantage of direct correspondence of angle and position. It may be useful to add here a statement of the relation of the four vectors and the four transversal directions for the general case of the quadrilateral construction used for balancing primary forces and primary couples only. Let p, P2, P3, P4,, rays of a pencil, give the directions of the four vectors; let tl4, t.4, t34 give the directions of the resultants of pairs of the vectors (being the same lines as t2,, t31, t2,) and let t have the direction of a fourth transversal giving direct correspondence. Then the four pairs of directions p, and p2, p3 and p4, tl4 and t24, t34 and t are in involution. There are three such involutions. These two tetrads of concurrent rays if cut by any conic, conveniently a circle, passing through their common point, give rise to two quadrangles of points PP.2P3P4 and Tf4T,24T3T on the conic, and these two quadrangles have the same diagonal triangle. This (or the usual ruler construction for the sixth element of an involution) gives a means of finding the transversal giving direct correspondence. In the case of Fig. 4 of ~ 7 the hyperbola H is itself a conic through 0; and the points P1, P2, P3, P4 give the first quadrangle. The second quadrangle consists of the three points in which OG34, OG24, OG34 meet H again, together with the point B, the end of the diameter OAB. This quadrangle has the same diagonal triangle as P P2P3P4. ~ 13. The product of inertia method of ~ 3 may now be extended to three dimensions and applied to the system of equations (1-6), (13-15), (17), which have the effect of making the disposition of the four masses at Q1, Q2, Q3, Q4 "dynamically cylindrical." Consider any two planes lx + my + nz + 1 =0, l'x + m'y + n'z + 1 = 0, and suppose the product of inertia of the four masses in relation to them to be zero, so that 2Mi (1Ix + my, + nz, + 1) ('x, + m'y, + n'z, + 1) = 0,

Page  230 230 G. T. BENNETT whence, from the equations referred to a(11+ mm)+ +c2nn' + 1 =0...................(80), showing that the two planes are conjugate with regard to the imaginary spheroid of revolution S - 2 (x2 + y2)/a2 + z2/c2 + = 0.................. (81). The product of inertia is zero if, in particular, one of the planes contains three of the four points Q1, Q2, Qs, Q4 and the second plane passes through the remaining point. Hence each point is the pole, with regard to S, of the plane of the other three points, and the four points form a self-polar tetrahedron. The same consequence appears otherwise on taking the quadric whose envelope equation is SMi (1lx + myi + nzi + )2= 0.....................(82), for its form shows the tetrahedron Q Q2QQ4 to be self-polar, and the equation reduces to the form associated with (80). The conjugacy of Q1 and Qo with regard to S gives 2 (x1x2 + yly2)/a2 + ZlZ2/C2 + 1 = 0, or in cylindrical coordinates 2 cos (01 - 02) + ZZ2/C2 + 1 = 0. Hence - ZZ2/c2= 1 + 2 cos 012...........................(83), or, as equivalent, - ZlZ2/c2 = sin 8012/sin i 012....................(84). From (83) z123 = 4C2 sin 012. sin (013 + 0,3)................. (85), and hence from (20) 12z34 = 4C2 sin 01,. sin 03............................(86), verifying (75) again. From (83) and (31) is also derivable z1 sin (013 + 014) 1 an -n _.......8...........(7), Z2 sin (023 + 024)........ and from (85) z 2 sin 1 01 sin (041+ 042)... 13 sin 1 013 sin 2 (041 + 0)) The equation (74) with (27-28) hence gives the symmetrical formula M1 sin 1,. sin 024. sin 4 (023 + 024). - 2 2 2 '" i ~~~~~~~~~~~~~~~~~(89) M2 - sin 0 33. sin 1014. sin (013 + 014)...... for the mass-ratios in terms of the angles. ~ 14. Equations (7), (8), multiplied by cos 01 and sin 0i and added, give M1 + M2 cos 012 + M3 cos 013 + M4 cos 014 = 0, which is obtainable also directly by resolving the four vectors parallel to the first. Using (83) - 2MC2 + M, (zZ2 + c2) + M, (z'z3 + c2)+ M4 (zz4 + c2) = 0, and hence, from (13), (14), M, (z2 + 3c2) = Mc2...........................(90), and z c2-= (M - 3M1)/M...........................(91).

Page  231 THE BALANCING OF THE FOUR-CRANK ENGINE 231 Equation (90) confirms the form of (79) and shows further, with (70), that = (p2- q)/3 = s/p2...........................(92), so that c2 is the positive root of the quadratic 3c + qc2- s = 0............................ (93). The product zlzzz, (= s) is necessarily positive, from (92), and the choice of G (~ 9) is restricted so as to have two of the points Z1, Z2, Z3, Z4 on each side of it. Divided by z1 (90) gives Mlzli + 3c2M/z11i = Mc2E1/z1, and hence, from (14) and (63), M, /z, =.................................(94). From (14) and (91) it follows that [M, (M - 3M )] = 0........................(95), the signs of the radicals matching those of the corresponding z's. This is the relation among the masses referred to in ~ 2. Again (91) and (63) give E [M1 /(M - 3M,)]=- 0...........................(96). These two equations, (95) and (96), must necessarily be only different forms of the same relation. And it may, in fact, be shown that the rationalized form of (95) is a factor of that of (96): so that (95) may be used as involving (96) as well. The rationalized form of (95) is found to be 1, (&13 - 41,2 + 9-33)2 = 4F4 (7/13 - 27/1.2 + 54L3)........ (97),,A(= M), /2, 3, /t4 being the sums of the products of MI, M2, M3, M4 taken one, two, three and four at a time. [The algebraic theorem which occurs here-of a form somewhat peculiar-is that if x = (a + b + c + d)/3 gives a root of the equation /Va~ ( - a) + V/b (x - ) + cc(x - c) + 2d- (x - d) = 0, it is necessarily a double root.] ~ 15. One or two extra formulae may finally be noted as close associates of some of the foregoing: o t e o g/ng= l M. /z3...........................(98), En1/zl3= MC 1/z1. ~~~.... ~~~~~~(98), of the same type as (14), (15) and (94); Zl/h = cot (0 - e) + tan (a- - 2).....................(99), giving, with (37) and (42), z's in terms of 0's, an equivalent of (55); tan (a- - 64 ~j2)sin - 0 + sin 2 + sin3.(3) tan ( - +7r/2)= —COS 01 -- COS 0+ cos 03............(100); cos 01~ cos 02 + cos 03 an immediate and unsymmetrical equivalent of (21), confirming the last construction of ~ 6; and (p + Z,) (p + ) (p + z2) (p z,)/(p + + ) =( - )p-z2) (p - z,)(p - z4)/(p - z,)...............(101),

Page  232 232 G. T. BENNETT verifiable from (69), (70). This last presents itself if the ratios of the M's are taken from (77) and are substituted in (3) and (4) by way of verification. ~ 16. The unbalanced secondary couple remains to be noticed. The vector polygon representing it has sides Mlzi, etc. in directions 20k, etc., and the direction and magnitude of their resultant may be found. Equation (38) gives Mlz, cos (20 - a) = XMl1z, cos (0, + e - o), and hence, from (7) and (8), cos aT. SMlzl cos 201 + sin a. 2Mlz, sin 209 = 0............(102), showing the resultant to have direction a7 +7r/2. As the shaft revolves, the secondary couple takes alternately a zero or maximum value as an axis or an asymptote of the rectangular hyperbola H becomes parallel to the plane of the axes of the cylinders. The square of the resultant is (M, zM cos 201)2 + (EMlz1 sin 201)2 = SM12zi2 + 2sMlM,2zz cos 2012. This may be put in terms of M's and z's by use of (83) and may be then reduced by use of (91) and (92) to (Mp)2. The resultant side closing the secondary couple polygon is thus iMp in direction a + vr/2. ~ 17. Some modes of application of the various results to the practical problem of design may now, in conclusion, be presented. Four chief cases may be sufficient to consider, differing according as the data consist of crank-angles or cylinder-spacings, and differing according as the work is to be analytical or mainly graphical. For each case a brief itinerary is here sketched, with references to the fuller details of the earlier paragraphs, and with indications of alternative routes. I. Graphical method: the directions of three cranks as data. (i) To find the direction of the fourth crank. Use the polarizing construction of ~ 3. Radii OPI, OP2, OP3 of circle radius a represent given cranks: and P3P4 must pass through the pole of PIP., with regard to circle I, centre 0 and radiussquared equal to - a2/2. (Fig. 1.) Three alternative constructions are given in ~ 6. (ii) To find the mass-ratios. Join P4 to the pole of the line through P3 parallel to P1P2. This line cuts PiP2 in G12, the centroid of Ml at Pi and MA at P2. (Fig. 1, ~ 4.) (iii) To find the cylinder-spacing. Find the resultant OB of equal vectors OP1, OP2, OP3, OP4 and draw a parallel to OB cutting the vectors in Z1, Z2, Z3, Z4. (~ 8.) For other constructions for the line OB see ~ 7. Also alternatively take OG12 and draw a parallel cutting the vectors in Z2, Z1, Z4, Z3. (~ 12.) As a check on (i), if the angles of the equal vectors of (iii) are doubled the angle of the resultant should come doubled and its magnitude OB2/OPI. (~ 6.)

Page  233 THE BALANCING OF THE FOUR-CRANK ENGINE 233 II. Analytical method: the angles 01, 02, 03 for three cranks being given. (i) For the fourth angle 04 use (25) or (26) or (100). (ii) For the mass-ratios use (89). (iii) For the spacings use (87) or (88): or as alternatives (37) and (99). III. Graphical method: collinear points Z., Z2, Z3, Z4 giving the spacing. (Fig. 5.) (i) Find G so that 1/zl = 0 (two terms positive and two negative), where Z= GZ1, etc. (63, ~ 9.) (ii) With p = zi, q = -zi2, s = z1z2z3z4, solve the quadratic p4- qpo -3s = 0. Draw circle centre G and radius p. (70.) (iii) Take Z so that GZo = pp2/(2p2 + q) and draw the line perpendicular to Z1Z2Z3Z4 through Z0. (iv) From 0, either common point of circle (ii) and line (iii), project Z1, Z2, Z,, Z4. These rays give the crank-directions. To give them the proper senses take those of GZ1, GZ2, GZ, GZ4 in the order of occurrence of the Z's, and alternately repeat and reverse them. (v) The mass-ratios are given by M. Z1Z2. ZZ. ZZ4/OZ, = — cet. sim. (76.) Or as alternative draw GX perpendicular to ZZ2ZsZ4 and make GX2= 3s/p (=3c2) and then (90) M1, Z. X Z2 = M2.. XZ32 = M,. XZ=2. The properties of the circle Y (~ 10) may be used as a check on the working of (ii) and (iii). Its centre is at D on GO where GD = s/p3: and the foot of the perpendicular from D on Z1ZZ3ZZ4 is distant - GZ1 from Z0. The tangents from G to the circle Y have length c, the inertia constant, and the angle between them is 2 sin-1 X. IV. Analytical method: the differences of z1, z2, z3, z4 as data. (i) Determine the zero so that 1l/z1 = 0. (66-68.) (ii) Find the positive root of the quadratic in c2 3c4 + qc2- = 0. (93.) (iii) To find the angles use (83). The angles, as determined by their cosines, are ambiguous only in sign; and are to be taken so that 012 + 023 + 31 = 0, etc. (iv) To find the mass-ratios use (90). The selections above made are framed as being the most apt, possibly, for handling the practical problem of construction: but, as may be seen, they are not entirely exhaustive of the copious supply of simple material which is available for the purpose.

Page  234 234 G. T. BENNETT ~18. It may be useful to collect here also the various vector polygons which occur: (i) Four vectors (1, 0,), etc. give resultant (X, e). The angle e gives the direction of the shaft-transversal. (ii) Four vectors (1, 20,), etc. give resultant (X2, 2e). (iii) Three vectors (1, 0,), (1, 02), (1, 0:) give resultant in direction a- 04 + r/2 giving 04. (iv) Six vectors (1, 0, + 02), etc. give a closed hexagon. (v) Four vectors (Ji, 0,), etc. give a closed quadrilateral for the primary forces. (vi) Four vectors (M1, 20,), etc. give a closed quadrilateral for the secondary forces. (vii) Four vectors (Mlz,, 0,), etc. give a closed quadrilateral for the primary couples. (viii) Four vectors (Mlz,, 20,), etc. give a resultant (Mp, o+7r/2) representing the unbalanced secondary couple. The illustrative figures, though differing in scale, have all been drawn for the case of the exact values tan 1 = 1, = 715, l = 81, tan 02 = 0, z2=-195, M2= 165, tan 3 = -, z= 165, M = 169, tan,0 = 3, Z4=-429, M4= 125, with _, - - - - -. —4 X,=a, tane=-J~-,tano=-. with z, =-132, /h = 231, p = 33 /6-5, c = 65 V/33, / = 7, X2 = 65 tan =, tan f = 4. NOTE. In discussion of the above paper Professor Morley remarked that the directions of the sides of any triangle and of its Euler line (joining the circumcentre to the centroid) are symmetrically related (Mathematische Annalen, 1899, pp. 411 -412. "Some polar constructions," F. Morley); and that the relation is of the type given by the equations of ~ 5. From this follows a further construction for the direction of the fourth crank: namely, if a triangle is drawn with sides given by angles 02, 202,,03, the direction of the Euler line gives the angle 04. The construction may be immediately associated with Fig. 1 and the unit polygon of ~ 6 and ~ 18 (iii). The Euler line of the triangle P1P2P3 joins 0 to the centroid and so has direction a- 04 + 7r/2: and if the sides of the triangle and its Euler line are reflected in a line having direction 1 (01 + 02 + 0: + 7r) the directions obtained are j0,, 102, 103, 104

Page  235 SOME THEOREMS RELATING TO THE RESISTANCE OF COMPOUND CONDUCTORS BY T. J. I'A. BROMWICH. In what follows some of the familiar theorems for a system of linear conductors are extended to include cases of solid conductors, and also cases when both solid and linear conductors are present in the system. The only paper dealing with this topic with which I am acquainted is one published last year by Dr G. F. C. Searle*, in which the method of attacking the problem is quite different. In modern practical work with low standard resistances, the use of four-terminal standards has proved important, and it seems worth while, therefore, to sketch two simple applications of the method to such standards. I. The variational equation for the system of currents flowing in a solid conductor, or a system of conductors. The currents being supposed steady, the electric force (X, Y, Z) at any point of the conductor is derived from a potential V; thus if (u, v, w) denote the actual components of current and (u', v', w') any geometrically possible components of current, we find Xu + Yv + Zw = (u - a ( _V ) - a (v) - ( ), ax ay az a ') a a and Xu' + Yv' + Zu' = - (t V) — (v V) - (wV), because both (u, v, w) and (u', v', w') satisfy the equation of continuity atu av aw -+-+ =0. ax ay az Hence on applying Green's lemma we find the equation (X' + Yv' + Zw') dr =-(lu + mv'+ nw') VdS............(1), with a similar equation containing u, v, w. In equation (1) the volume-integral extends throughout the interior of any continuous conductor, and the surface-integral to the boundary of the conductor, I, m, gn being the direction-cosines of the outward normal. * Electrician, March 31-April 21, 1911.

Page  236 236 T. J. I'A. BROMWICH But we can now extend (1) to cover the whole system of conductors; for at any interface between two conductors the potential V and the normal component of current (lu' + mv'+ nw') will be continuous. Thus the contributions to the righthand side of (1) from the interfaces of adjoining conductors will cancel out, and we are left with a suifatce-integral extending only to the places at which current is led into or out of the system. In most cases of practical interest the leads may be regarded as confined to a number of isolated points; and if these points are denoted by 1, 2,..., n, the corresponding currents (which are supplied) being called (',:',...,.' (in our hypothetical system), we can write - (lu' + m' + nw') dS, = ', etc., so that equation (1) takes the modified form f(Xu' + Yr' + Zw') d- = V + ' V2+ * + V...+...........(2), where 0 = 1' + ' +... + n'. Since (as we have already remarked) this equation remains true when the actual current system replaces the hypothetical, we see that f(XZ, Y8 Z+ Yv + Zw)dT= VS V+ Ve_(+...+.........(3), where 0 = +.82 +... + 6, and u= ut'-u, Sv =v'-v, 8w =w'-w. This equation (3) is analogous to the equation of virtual work in mechanical systems; here it is simply an expression of the fact that X, Y, Z is derived from a potential-function. If we suppose the conductor to be isotropic (but not necessarily uniform), the relation between X, Y, Z and u, v, w is X = pu, Y= pv, Z= pw, where p is the specific resistance at the point considered. Thus if u', v', w' represents another current system in the same conductor, we have Xu + Yv' + Z' = p (uu' + vv' + ww') = X'u + Y'v + Z'w.........(4). More generally, if the conductor is crystalline, all experimental evidence shews that the matrix of resistance coefficients is symmetrical, so that we can write X = au +hv+gw, Y= hu + bv+fw, Z = gu +fv + cw, and equation (4) remains true. Combining (2) and (4) we then see that f1 VI + 2 rt2 +... + n Vn = t1 Vi + t2 +... + En Vn, which at once yields the reciprocal relation: If unit current is led in at 1 and out at 2, the potential difference between 3 and 4 (Vs - V4) is the same as that between 1 and 2 ( V1'- VF') when unit current is led in at 3 and out at 4.

Page  237 SOME THEOREMS RELATING TO THE RESISTANCE OF COMPOUND CONDUCTORS 237 This theorem has long been known for the case of linear conductors; it was first proved in its general form in the paper by Dr Searle already quoted (~~ 3, 7). Returning now to the variational form of equation we have the results fp (tSt + v v + w Sw) dr = EVrSr, or {auSu + h (uSv + vSu) +...} dT = Vr8. the former referring to an isotropic system, the latter to a crystalline system. Thus, writing H=J p (il + v2 + w2)dT or (a l2 + 2huv +...)dT.........(5), as the case may be, we see that when the variations are small we have the variational equation 3H== Vr,.6, where O= 8.........................(6), which is correct to the first order. And when the exact form of 3H is needed we can write (for an isotropic system) SH = (Uu + V8V + wW) dT + p {(8I)2 + (8V)2 + (8W)2} dT = E VT SA + S82, say.........(7). The necessary modification in (7) for a crystalline system is obvious. It is easy to modify these results to include the case when some (or all) of the conductors are treated as linear wires, some of which may contain batteries. For instance suppose that 1, 2 are points joined by a wire of resistance R containing a battery E, which carries a current I: then, = - I, ^ = + I, and V1- V+ E = RI. Thus (6) takes the form 3H, = EI - RIUI + V383 +... + Vlr,8,, where HI refers only to the solids. Hence if H=H1+ 1RI2, we find H = E + V3 +... + V..,, V. It is easy to generalize the last equation to include any number of wires and then we obtain the result;H = E s8Ur + V,8........................(6 a), where H = H1 + RIs2, and 0 =S. Similarly in (7) it is only necessary to add the term,E,8Is to the right-hand side, in order to include the effect of batteries. II. Deductions from the variational equation. A. If the system is free from internal batteries, and if the currents supplied from external sources are prescribed, we have in (6 a) E,=O, 8r =0,

Page  238 238 T. J. I'A. BROMWICH and so to the first order 8H=0. Thus H satisfies the condition of being stationary: and using (7) we see that accurately 8H=,2H, and 82H is essentially positive. Consequently with prescribed currents at the points of supply, the value of H is a minimum in the actual current system. This is the analogue of Kelvin's theorem for a dynamical system; it is of course a familiar result for a system of linear conductors. B. When batteries are present, it is easy to extend (A) to shew that (H - EIE,) is now a minimum, by using the extended form of (7). C. It follows from (A), that when zero currents are supplied and no batteries are present, no currents will flow in the system; for the case of zero current makes H zero, and (since H is in general positive) this current system makes H a minimum, and is therefore the actual system. Suppose now that Vr denotes the potential when unit current is supplied at r and taken out at n; then the function U = V- (~1v, + f2v2 +... + _n-lVn-1) will be a potential-function which corresponds to the case in which zero current is supplied at each terminal. Consequently U corresponds to the case of zero current; or U is a constant. Hence in general we can write V= (Elv + 22... + n-iv_,) + const. when the currents ti, 2,..., n-n are supplied at 1, 2,..., (n - 1), and the whole of them are taken out at n; this of course is equivalent to supposing:, o,..., i supplied at 1, 2,..., n, with the condition R1 + 5+ + + + h4i = 0. D. It is clear on substituting the value of V found in (C) that H becomes a quadratic function of 1, 2,,..., I,; and so (6 a) leads to the result 8H= V18+ 6 + V2 +... + Vl,_-i8n-i- Vn (86 + 82 +..+ 8n-i), provided that 8I = 0. Consequently aH 3H V 8aH - V V,-, -- = V - V.,..., -- v,- V.........(8) are the general equations for our system. III. Special consideration of four-terminal systems. In modern accurate work on standards of low resistance it is usual to employ conducting strips with four terminals: two of these terminals may be used as current leads, and the other two will then be used for determination of potential-difference. The so-called resistance of the standard will then be the potential difference corresponding to unit current supplied.

Page  239 SOME THEOREMS RELATING TO THE RESISTANCE OF COMPOUND CONDUCTORS 239 In such cases we have four points 1, 2, 3, 4 and 1 + 2 + 3 +4 = 0. Thus H can be regarded as a quadratic function of Al, ~2, 3; it will therefore contain six resistance coefficients and so the resistance of the standard depends on the arrangement of the terminals. Of course in practical cases two definite terminals (and only these) are used as leads, and the other two for potential readings only: thus, for instance, let the potential readings be taken at 2, 3, the leads being at 1, 4. Then we can write H = - (a 12 + b 22 + c 3 + 2f /;, + 2, 31 + 2h/ 1)2)............(9) in general; and so from (8) and (9) we have V1- =V4 a6l + h, + 9g) V, — V4 =h h + + f +f.......................(10). V,3 - V4 = g91 +f -+ c3)J Thus when unit current enters at 1 and leaves at 4, so that =, 2. = 0, = t= 0, we find V2-V3 = h - g. Hence in this arrangement of leads, the resistance of the conductor is (h - g). An interchange of the leads with the potential-terminals will not alter the result: this is evident from the reciprocal relation above (p. 236), or directly from (10) by taking the case = 0, ~2 = 1, 3 = - 1, which gives V, - V = h - g, or the same potential-difference as in the first arrangement. In some of the earlier experimental work it was supposed that any four-terminal system could be effectively represented by a model of five linear resistances with four terminals as indicated roughly in the figure. P Q R 2 3 Fig. 1. It is evident that this system cannot be equivalent to the general conductor, which contains six coefficients (instead of five): in this special case the form for H is (JP 2 2t + s (2 + T2) + Q (Al + ) + R ( + + )2i. Hence in this model f= R = g; which is not usually the case. In fact if the points 1, 2 were used as leads and 3, 4 for potential readings, it is evident that the apparent resistance (being g- f in general) would be zero in the model; a fact which may be seen from the figure by inspection. To conclude we shall give the theory (from our present standpoint) of one of the laboratory methods for finding the resistance of a four-terminal standard.

Page  240 240 T. J. I'A. BROMWICH A bridge is constructed as in the rough sketch; a battery is placed between LM, and the resistances R, S are adjusted until a balance is obtained when a galvanometer joins 2 to K. Then the galvanometer is placed between 3 and K, and the resistances R, S are adjusted to R', S' so as to again produce a balance. In the first arrangement (which is that sketched) 3 = 0, and so equations (10) give V, - V,4 (t= + ha, V2- V4= hV 4 + b2. R I l 2\ 3 P K Q S I L M Thus when the balance has been obtained (so that f, = 0), V,-V,=(a-h) 1, V,,-V 4=A. Fi Hence VL- V = (R + a - h), V,- V = (S + h), and so as in the usual Wheatstone Bridge method we have H I ig. 2. (R + a-h)/P=(S + h)/Q........................(11), because Va= VK. In the second arrangement we get similarly V,- V,=(a-g) 1, V,-V, =gu1, leading to (R' + a - )/P = (S' + g)Q........................(12). Hence on subtracting (11) from (12) we eliminate a, and find {(R'-R) + (h - g)}/P= {(S'-S)-(h-g)}/Q, which gives the resistance in the form h - g = {P (S'- ) - Q (R' - R)}/(P + Q). In the special case when P = Q, the formula reduces to h-g g = (S'- S) - ( -R)}, which is very convenient for practical work. It is noteworthy that only the changes R'- R, S'- S in the resistances employed need to be observed (as well, of course, as P and Q). This experimental arrangement and the final result are given in ~ 26 of Dr Searle's paper: those interested will find other examples (derived from experimental arrangements used in various standard Laboratories) in the concluding sections of his paper, and to any of these examples the analysis described above can be easily applied.

Page  241 DISPERSION AND DOUBLE-REFRACTION OF ELECTRONS IN RECTANGULAR GROUPING (CRYSTALS) BY P. P. EWALD. There are two ways of reducing the aeolotropic properties of a crystal to the properties of the molecules: first you may suppose every molecule to be aeolotropic having the same symmetry as the large crystal. For the theory of double-refraction this would mean that there are, in each molecule, three different periods of free vibrations in three principal directions. These aeolotropic molecules have to face the same direction in the crystal-this being the only restriction as to their arrangement. That these assumptions really lead to a definite value for the double-refraction has recently been shown by Langevin. The other possible explanation of crystalline properties has been given in the theory of crystal-structure by Bravais, Sohnke and Schonfliess. In this theory the ultimate parts constituting the crystal are assumed isotropic and the aeolotropy is produced solely by the arrangement of these particles in a regular space-grouping. This theory accounts for all classes of symmetry possible in crystals, and all qualitative conclusions from this theory seem to show the truth of the fundamental idea. I may add, that recent experiments by M. Laue on the passage of Roentgenrays through crystal slabs furnish the most direct evidence of the reality of the space-grouping. It seems therefore to be of interest to know to what extent double-refraction may be considered to result from the arrangement of isotropic molecules, and this is the question I have studied. It so chances that the preciseness of the problem leads to a stricter treatment than has hitherto been given, of questions arising in the general theory of refraction in transparent bodies. These questions, concerning mainly the formation of the refracted wave in the bordering layers of a body of finite extent, will be mentioned at the end of this lecture. Consider the simplest grouping which has the symmetry of the rhombical system. The planes of the grouping intersect at right angles and have the distances (a, b, c) in the (x, y, z) directions. (Fig. 1.) In each of the points of this grouping let there be situated a molecule with an electron capable of being excited by a periodic electromagnetic field. Let there be further a quasi-elastic force between the molecule and the electron and let this be equal for displacements of the electron M. C. II. 16

Page  242 242 P. P. EWALD in all directions. It is easy to extend the theory to cases where the latter assumption is not true, but we will here only consider isotropic molecules. 0 0 0 0 0 0 0 0 0 0 a 0 0 0 o 0 0 0 0 0 o 0 o o o o o0 0 0 0 o 0 0 o 0 0 0 o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o 0 0 o o 0 0 0 Fig. 1. When an electron in the grouping is displaced from its position of equilibrium there is not only the quasi-elastic force acting on it, but also a force arising from the action of the neighbouring molecules. This force will not be the same for displacements in the different directions x and owing to the difference of intervals a and b. This is how the interaction of the neighbouring molecules gives rise to doublerefraction. We have now to put these ideas in a mathematical form. For this purpose we assume that the oscillations of the electrons take place as if they were induced by a plane wave sweeping over the system with a certain (unknown) velocity q. This means that all electrons in a certain plane passing through the origin of coordinates have the same phase and that the phase of any other electron is given by its distance S from this plane. So that we may describe the field of any of the electrons by the well-known Hertzian vector-potential ~ +s -in (t- - + -) p e c f P=a R where a is the maximum electric momentum of the electron, R the distance between cm electron and point of observation, n the frequency, c = 3'1010 -; and where the sec term S/q in the exponent produces the phase difference demanded. The determination of the velocity q will presently furnish the refractive index v = c/q. Before determining v we have to consider the field produced by the assumed oscillations. The potential of this field is 2P, the sum extended over all electrons of the grouping. This sum can be formed by means of complex integrals and is found to be the product of two terms; one of these represents a wave of the same character as the wave we used for describing the phase of the electrons. The other term is periodically reproduced round each molecule and only comes in by its mean value, as the distance of the molecules is small compared to the wave-length. So that we find the optical field to consist of a plane wave of the velocity q. It is startling at first sight to find by this result that we may not introduce into our system the incident wave at all. This wave of velocity c (and perhaps other direction) would not combine with the field of the electrons to form a plane wave in

Page  243 DISPERSION AND DOUBLE-REFRACTION OF ELECTRONS 243 the interior of the crystal. The reason why the incident wave has to be left away will appear later on considering the influence of bordering layers. We now proceed to find the velocity q. For this we demand that the electrons be in dynamical equilibrium with the periodic field. This means that each oscillation must be produced only by the "exciting field," i.e. the field radiated to the electron under consideration by all other electrons of the grouping. The potential of this "exciting field" is U'P, where the accent denotes the process of leaving out one of the terms of the sum. We now have the differential equation for the oscillations: my + gy + mn0o2y = e x exciting field, -in t+ 1 where y =. e ( ) is the vectorial elongation, e the charge of the electron and e no its period of free vibration. The exciting field is found from the potential by twice applying the operation of curl, and as both sides of the equation depend on time in the same periodic way we get the equation a n (n -- n - in = curl curl 'P e2 \ m) in (t RS\ e ~ q/ = curl curl 'a R This is a linear homogeneous vector-equation for a and it is equivalent to three simultaneous scalar-equations of the same kind for the components of the amplitude a. These are only consistent if the determinant of the coefficients of a,, ay, az vanishes. These coefficients are formed of the second derivatives of the sum Z' and depend therefore on the value of q. The only values of q and of the refractive index v = c/q which are in accordance with the condition of dynamical equilibrium of the system can be found from this determinant. As the results of this theory I may state: we find a dispersion-formula 2=1 + N2 (20 ) - n2) + Ne2 which resembles closely the formula found by Lorentz and Planck. The only difference is found in the term #; this is a number depending on the ratio a: b: c of the intervals of the grouping and on the direction of propagation and polarisation of the optical wave. The numerical value of Jr can be calculated with a:b: c given. In the case of a cubical arrangement, =- - and we obtain the exact formula of Planck-Lorentz. The invariant relation due to T. H. Havelock, 1 _ D y y 1 -- 1= - y, between the refractive indices in the x and y directions follows from the formula given. D is independent of the number and nature of the electrons and of the frequency of the wave; by it we obtain the rational measure of Double-refraction caused by structure. 16-2

Page  244 244 P. P. EWALD It is of interest to find out whether the influence of structure produces the same order of magnitude of double-refraction as has been found in crystals. By calculating Ax - ry for a grouping of which the intervals have the same ratios as the crystallographic axes of Anhydrite this is found to be the case. Even if in many crystals we shall not feel entitled to attribute the whole of double-refraction to the arrangement of isotropic particles, we see from this example that the structural effect plays a great part in the production of optical aeolotropy. It can further be shown that in the grouping the refractive index varies with the direction of the optical wave in the same way as in crystals, the surface of wave-normals being the same in both cases. Let me add some remarks of interest for the general conception of refraction. We have found that electrons oscillating with the assumed connection of phases give rise to a plane wave of velocity q and that therefore the incident wave may not be superposed on the field due to the electrons. We conclude that in a slab of a crystal of finite thickness the incident light wave is annihilated by the electrons on the border of the slab. In the interior of this crystal there exists, therefore, only the refracted wave. In letting the thickness of the slab grow indefinitely we remove the border; but its influence is felt in the fact that the incident wave is not detected in the interior. This action of the bordering layer is perhaps formulated here for the first time. It seems of highest theoretical and practical interest to follow it in a more precise manner, because it shows us the exact process by which an incident wave of one direction is transformed into the refracted wave of other direction and velocity. By considering the bordering layers of a body we study this fundamental change, whereas in the usual theory of dispersion we find the conditions under which the changed state, if inaugurated, may continue to exist. The dynamical treatment of a system with a border is more difficult than that of an unlimited system, but I hope soon to present some studies of this-it seems to me, important-point.

Page  245 THE GRAPHICAL RECORDING OF SOUND WAVES; EFFECT OF FREE PERIODS OF THE RECORDING APPARATUS BY DAYTON C. MILLER. I have devised an instrument which may be called the "Phonodeik" for graphically recording sound waves. In the phonodeik the sound is in effect concentrated by a conical horn upon a diaphragm and the movements of the latter are recorded photographically or are projected upon a screen. It is well known that when waves of various frequencies are impressed upon a diaphragm the response of the latter is not proportional to the amplitude of the wave; the diaphragm having its own natural period, its response to impressed waves of frequencies near its own is exaggerated in degrees depending upon the damping. The theory for simple cases is complete, but what actually happens in a given practical case is much complicated by indeterminate conditions and cannot be predicted. It is also well known that the resonating horn modifies the waves which may be said to enter it. Therefore it follows that the resultant motion of the diaphragm is quite different from that of the original wave in air. These same difficulties are present in the reproductions of the talking machine and the telephone, perhaps in a doubled degree since they may occur once in making the record and again in reproducing it. It is argued that the photographic records of the phonodeik are more truthful than the best talking machine records because the movements of the first diaphragm which is nearly free are directly recorded without further mechanical disturbance. The justification of the phonodeik for research upon complex sound waves requires that it be shown that the diaphragm responds to all the frequencies being investigated; that it responds properly to any combination of simple waves; that it does not introduce fictitious frequencies; it is necessary to determine the relation between the response to a wave of any frequency and the relative intensity of that wave; it is necessary to show that the recording attachment faithfully transmits the motions of the diaphragm. The actual response of the phonodeik has been investigated by reference to an arbitrary standard scale of organ pipes. Two sets of pipes, one open diapason set of metal and one stopped diapason set of wood, were especially voiced and made of uniform loudness according to the judgment of a skilled organ voicer. The uniformity of these pipes has been improved and verified by two experimental methods. Each set contains eighty pipes, giving each musical semitone from a frequency of 129 to

Page  246 246 DAYTON C. MILLER that of 12400. The sounds from the various pipes of the two sets have been photographed and analysed; the open diapason pipes have a strong octave accompanying the fundamental; the lowest octave of stopped pipes have a weak third partial tone accompanying the fundamental and no other partials, while all the other pipes of this set give practically simple tones. With these pipes the response of the phonodeik under varying conditions has been determined by sounding the pipes in succession and making a photographic record of the amplitude of vibration. A curve is then plotted which shows the response to the successive tones. Such a curve is shown by the irregular line in Fig. 1; in this diagram the abscissae are proportional to the logarithms of the frequencies, and the ordinates represent the amplitudes of vibration of the diaphragm corresponding to these frequencies. The smooth curve represents an ideal response, the ordinates of this curve are the undistorted amplitudes of vibration for the various frequencies on the supposition that all the tones of the scale are of the same intensity. Constant intensity as here implied is defined by the condition that the square of the 20 10 129 n 259 517 1035 2069 4138 Fig. 1. amplitude times the square of the frequency is constant; that is the energy of vibration is constant. The departure of the actual response from the ideal is very marked; five causes were suspected and each was investigated. These distortions might be produced by unequal loudness of the pipes, by the attached vibrating mirror, by the mounting and housing of the diaphragm, by the free periods of the diaphragm, and by the free periods of the resonating horn. The effects due to the first three causes were found to be practically negligible in the analysis of the actual photographic curves; the effects due to the diaphragm and the horn are important. The response of the diaphragm, with no horn and with no cover on either the front or back side, was determined with the sets of standard pipes. Fig. 2 shows the results plotted in the manner described for Fig. 1. The curve shows three distinct peaks, and indicates that the response is very small for low frequencies. In order to learn the condition of vibration of the diaphragm, Chladni's method of sand figures was used; the figures were drawn and photographed for the eighty

Page  247 THE GRAPHICAL RECORDING OF SOUND WAVES 247 tones of the standard scale for frequencies from 129 to 12400. The first peak of Fig. 2 corresponds to the real free period of the diaphragm, when it vibrates as a whole most vigorously. The diaphragm then forms various irregular nodal patterns giving small response as the frequency is increased, till it forms one concentric circular nodal line, corresponding to the first over-tone, whose frequency is 3'28 times that of the fundamental; this corresponds to the middle peak of Fig. 2. With increasing frequency, smaller irregular nodal figures occur passing into two concentric circles, giving the second over-tone represented by the third peak in Fig. 2. The frequency of this tone is 6'72 times that of the fundamental. Nodal patterns of three, four, and five perfect concentric circles were obtained, corresponding to frequencies beyond the limits of the diagrams shown here. The presence of the three natural periods of the diaphragm was proven, also, by making a direct photographic record in the phonodeik of the vibrations resulting from a single impulse produced by a hand-spat in front of the diaphragm. 30 O 20 -10 129 n 259 517 1035 2069 Fig. 2. An elaborate study was made of the effects of the concentrating horn, and it is shown that it acts as a resonator, as does the body of a violin or the sound-board of a piano. But the horn adds its own free period effects; it has all the tones of the bugle, shown by at least ten peaks in the curve of Fig. 1; this makes it a very annoying adjunct to the phonodeik, but as it increases the response of the diaphragm several thousand-fold, it has seemed desirable to retain it. Experiments have been made with horns of a great variety of materials and of varying sizes and shapes. The results of these investigations must be given elsewhere. A long slender cone seems to be the best shape of horn for the phonodeik. The effect of such a horn on the response curve is shown in Figs. 3 and 4; the dotted line, a, shows the response of the diaphragm alone, as already described; the difference between this and the actual response, b, is mainly due to the horn; line c

Page  248 248 DAYTON C. MILLER represents the ideal response. If the ordinates of the curve b are multiplied by such factors as would cause curve a to coincide with curve c, the curve d is obtained; the difference between curves d and c shows clearly the peaks due to the complete series of partial tones of the horn; ten or more of these partials may be identified. 20 -10 C a 129 n 259 517 103 2069 Fig. 3. The practical application of these results in the quantitative analysis of sound is as follows. Photographs are made of the waves from the unknown sound; at the same time a response curve is made with the set of standard pipes, as described. The wave is analysed, and each separate component is corrected by being multiplied by a factor depending upon the frequency of the component, and determined by reference to the response curve. The factor is such a number, greater or less than 30 20 Fig. 4. unity, that when multiplied into the ordinate of the response curve for the specified frequency will make the product equal to the corresponding ordinate of the ideal curve, as shown in Fig. 1. The corrected amplitudes of the various components are those which would have been obtained if the response of the phonodeik had been equal to the ideal specified, and had been free from the effects of all free period and other distorting effects.

Page  249 THE GRAPHICAL RECORDING OF SOUND WAVES 249 Practical justification of the method has been obtained by photographing the same tone from a musical instrument with widely differing conditions of the phonodeik, as with diaphragms and horns of different free periods. The photographs so obtained are quite unlike in appearance; but after the components of each have been properly corrected new curves synthesized from these corrected components are similar. Detailed descriptions of the instruments and methods employed in these experiments, together with results of the analytical study of various musical and vocal tones, are to be given in papers which will be published in various journals devoted to physics.

Page  250 SURI LE MOUVEMENT IYUN FIL PAR E. TERRADAS. 1.Je vais m' occuper du mouvement d'un fil dans le cas o0i1 tons ses points de'crivent la me~me trajectoire relativement 'a des axes anime's d'un monvement de rotation*. On fera usage des notations suivantes: 8 masse par unite' de longuenr; a- arc de trajectoire par rapport 'a un point fixe; s are du flu; T tension; v vitesse comne' tons les points, fonction seulement du temps t; r distance d'un 'I'ment 'a l'axe z de rotation; p rayon de premie're courbure; w vitesse angulaire constante des axes; X,,u, v cosinus directeurs de la binormale; 'O,.12, 1J3 projections sur la tangente, normale et binormale des forces exte'rieures par unite' de masse. Les equations du mouvement relatif donnent 8 - = ~~~ 8w dr r 8 [V2 _pJ2] = T~ - L2 2L —j-21 -2wvp]. 1 a-=fvct +8; d.2 = dx2+ dy2~+dZ2. An moyen de ces 4equations on doit pouvoir exprimer x, y, z en fonction de o-, v en fonction de t, et il fant encore T > 0. Dans le cas v = const. ce sont les equations d'equilibre d'un flu dont la tension serait T - v.Quand S =2.2= 3L3 = v = 0, cc sont les equations de la corde 'a sauter. 2. Supposons d'abord 33 fonction de v on de t; la dernie're equation montre que, on. v est constant, on, v e'tant variable et 33 fonction line'aire de v, la projection de r sur la binormale doit eftre constante, et l'angle de la binormale avec l'axe doit e~tre aussi constant. Dans le cas oii 3J3 = 0, v pent ne pas &ftre constante quand les angles precedents sont e'gaux 'a 90', c'est-a'-dire que, si v n' est pas constante, la courbe doit e~tre une he~lice on. une courbe plane. Si c'est nne he'ice, J12 e'tant convenable, 8 pent snivre une loi quelconque, mais, en general, il fandra que 8 suive une loi exponentielle on soit constante. Si 3J3 est fonction senlement de a-, le monvement est, en general, impossible, 'a momns de satisfaire 13 'a certaines conditions. Si 313 e6tant 0, Z et i12 sont fonctions seulement de a-, elles ne peuvent pas eftre quelconques, Si elles satisfont 'a la relation Z - p K, K e'tant constante, le oveentestdonne' par dv = K. *V. aussi Arnoult, ThIse, Nancy, 1911.

Page  251 SUR LE MOUVEMENT D'UN FIL25 251 3. Dans le cas d'une courbe plane, supposons, en premier lieu, T3 et.A2 fonctions de v et 8 = 1. Le'lirnination. de T des deux premieres equations (1) conduit 'a une equation de la forme T, - T2S, - S, = 0, Tf' et T, e~tant des fonctions de t, S1 et 53 de a- seulement. En suivant la me~thode qui a e4te- indique'e par M. Appell dans son memoire des Acta*, on en de'duit que, ou la vitesse est constante dans la trajectoire K2 -S2 = K, les K e4tant des constantes, ou la traj ectoire est un cerele dont le K1 - S centre est 'a lorigine, la loi du mouvernent e~tant -- = 0. Toutefois, si ~2fct dt tel que 2wv - i2=K2, K, (etant iine constante, le mouvement scion la loi -d = K, dt est ossbledans la trajectoire K2 S K3. Si T et.J1 sont fonctions seulement de a-, il en re'sulte que, ou v est constante, on si T et J~2 satisfont 'a certaines conditions, la trajectoire pent eftre une spirale logaritlimique on uni cerele parcouru avec vitesse variable. Si T et JIl sont la Somme d'nne fonction de v et d'une fonction de a-, on arrive a des resultats semblables; si T et JJ2 sont le prodnit d'nne fonction de v et d'nne fonction de a-, on pent conside'rer divers cas scion le nombre des liaisons line~aires parmi les fonctions de t on de a- de le'quation qn'on obtient en e~liminant la tension. S'il n'y a qn'une relation, on Si leur nombre est 4, la vitesse est constante. 4. Nons allons nons occuper d'nn cas particulie~rement inte'ressant oii I'on pent realiser facilement la courbe. II suffit de prendre une chaline sans fin, et de la faire tourner en la pendant de l'avant-bras. La chaine a une figure permanente par rapport 'a des axes entraine's snivant la rotation. 1i est admis qn'il n'y a pas de glissement entre la chaine et son support. La vitesse constante des e6lements, on vitesse du fil, est v = wr1, ri etant le rayon dn cylindre qui sert d'appni "a la chaine. On obtiendra les equations qni conviennent "a ce cas en posant T=J2=0; v=wr,. Eliminant T entre les deux premieres equations (1) on obtient apre's integration, et puisque 2- da-r (p distance de l'origine "a la tangente) r2 + 2pp ~ 4rip = hi, Ah 6tant constante. Mais commep = r dr on. aura, e~liminant p, inte'grant et en egard "a ce que pour r = r, p-r1, (p + 2rj) (h1 _ r.2) = ri (hi - r12).............(2). * V. Appell, Actanmathemnatica, tome x.

Page  252 252 E.TERRADAS On pent remarquer que w n'initervient pas; c'est-ah-dire, que la courbe trajectoire du flu est inde'pendante de la vitesse de rotation. En augmentant w on augmente T scion la relation suivante oii z = r T-=Ii +"2,-z On se rend compte facilement de ce que la forme de la courbe soit inde'pendante de w. En effet: les forces exte'rieures sont uniquement la force centrifuge ordinaire et celle de Coriolis; la premie're est proportionelle 'a W2, la deuxie'me aussi en vertu de v = wrl. Donc, la direction de la resultante est inde'pendante de w. De e'~quation (2) on en de'duit, 6 ktant l'angle polaire compt6 'a partir du minimum rmde r, + ri 2z ~IR 1dz, 1? = (Z _ z1) [z2 - (2h, + 3z,) z ~ (h1 + zJ)2] = (Z - zj) (Z - Za) (Z - z6). La coiidition R > 0 *jointe 'a T > 0 donne les conditions limi tatives de r. On pelt distinguer deux cas: (1) h1 > zj. Les trois racines z6, za, z1 sont r6elles, supposons Z6 > Za > zi. La courbe est comprise dans' la couronne r = rl, r = ra. Elle est forme'e d'une infinite' de branches sime'triques on congruentes 'a la branche qui va de r = r'ar = ra. Dans les points correspondants, elle touche les cercies limites de la couronne. II y a un point oii p s'annule, dont le rp est donne par 2rP2=h 1 Le rayon de courbure donne' par 2p=(hi- Z)2 est maximum pour r = ri, se re'duit 'a la quatrie'me partie pour r = rp et devient minimum pour r = ra. La courbe a la forme de la figure (1). On pent l'appeler courbe exte'rieure. Fig. 1.

Page  253 SUR LE MOUVEMENT D'UN FIL 253 (2) hl < z. (a) - z1 < h, < z,. Dans ce cas, z > z > Za. La courbe est entierement comprise dans la couronne r = ri, r= r,. Elle passe aussi de concave a convexe par rapport a l'origine, le rayon de courbure est aussi maximum pour r=r, et minimum pour r=ra. Comme dans le cas ant6rieur il ne s'annule jamais. La courbe a la forme de la figure 2. On peut l'appeler courbe interieure. Fig. 2. (b) hi =-z,. C'est aussi une courbe interieure dont l'equation est + =1 rl-r -r1 + r Elle tend asimptotiquement a la circonference r= ri. I1 n'y a pas d'autres cas possibles. Cependant il est a considerer a part le cas hi = z1, qui correspond a la circonference r = ri, ce qui se deduit de la valeur de p. g -2h, + 4 o2n p+ 4ss 5. Considerons la courbe exterieure. Avec z = x/4x + j, j =2, on passe a la forre de Weierstrass. En introduisant l'integrale = dx. 3 xR on trouve r /4p (I + ) +j....................(3). Si l'on fait -j = 4p (v), on a p' (v) =- ir (h, + z,), et 6= r-i() (v+ W - 2 (-(W2+, -+ ) - (2w,-v) 0= [/4r,-i~ (v)] I- - 1 2 a-(I + 2+v) a- (w2- v)' Si l'on veut profiter des tables, on pourra se servir des valeurs suivantes: e2 e__ p _______ fV'. r_2 _?12 K2 en2 =sen22 = S2 = s n 2 [I v/el - e3] 2= w = - el - e2 ' =g r" Ve - e vVe1s l-~-e3 rb-2 -r2 senoa' = 1 - 1 2, sen2b' = sn2 rb2 i rb2 —

Page  254 254 E.TERRADAS Avec lesquels on aura w1, v et diverses valeurs de I pour les valeurs correspondantes de r. La valeur de 0 pent e&tre mise dans la forme: =(~/_4 ri -7V)I + 1J...............(4), cosh. WV - 3q2 cosh. 2iw, v. avec 7r ~ (Wj — (3Wv senh7 senh*iw 2AB K' I A2- B2 ' e K -T V WT 2W?) 2W A=12q cosh..?Cos — I +2q'cosh. — cos-1I... 1W1 WI 1'W1 WI B=-2qsenh.- Ysen2-I +2q1 senh. v 2 1W1 W1 1w1 WI C'est avec ces formules que la courbe fig. (1) a e'e calcule'e. La valeur de 1'angle 0m, que font les rayons r = r, et r ra, resulte de =m (4r, - WVY) WI. La longueur est donne'e par 2o4 V_ z dz = (hi -j)1I- 2 ___ _ r) + ~4_ 4~'/ 4 ~VR2r La demi-longueur 1 d'une branche est: S'il y a it branches, la longueur totale de la courbe sera L = 2-n[l +Wrr, 6rn. 6. Ces courbes dependent de deux parame'tres, par exemple, z1 et h1. Toutes les courbes pour lesquelles Z, - X a la me'me valeur, sont homote'tiques. On obtient la totalite' des courbes exte'rieures, en faisant varier XV de 1 'a 0, et les courbes inte'rieures en faisant varier X de 1 'a - 1, et en y ajoutant toutes les homote'WT tiques. A mnesure que h1 augmente, r, ktant invariable, Om augmente, jusqu'ah - dans 2 les exte'rieures. Lorsque h1 diminue dans les inte'rieures, 0.. augmente sans limite. A tonte courbe exte'rieure definie par les valeurs de r, et ra", on pent faire correspondre une courbe inte'rieure, qnon ponrrait appeler conjugu'e et pour laquelle rl=ra et ra'= ri, les accents se rapportant 'a la courbe inte'rieure. Une fois construite l'une de ces conrbes, p. e. 1'exte'rieure, on obtient la conjugu'e par les formules Vffectivement: des formunles qni donnent za, et Zb on en de'duit en e'liminant h1 Zb Zi +Za+ ~ 2 2~ Ia

Page  255 SUR LE MOUVEMENT D'UN'FIL 255 Les trois racines de I? sont communes aux courbes conjugue'es. Par consequent, j, vl w1,, J sont les me'mes aussi. Les formules ante'rieures s'obtiennent immediatement en restant les valeurs de 6 et 1 correspondantes. La figure 2 est la courbe conjugue'e de la figure 1. On de'duit encore h,=Za-z+1+ Vz1ia; h11=z, Za VlZlZa. 7. On peut exprimer x + iy en fonction uniforme de I. On a en effet re =o-2(I~(V) [~ W2 - v) cc + iy = re~~~~~~~~0 - i 0 (V cc + jy est, par constequent, fonction doublement pe'riodique de seconde espe'ce a multiplicateurs non spe'ciaux. Avec cette valeur de xc + iy on pent analiser s'il n'y aurait pas des courbes alge'briques qui correspondraient "a des valeurs de v qui seraient des fractions de la pe'riode W2. On sait que si le coefficient de I dans l'exposant est v W2et v est f 2nw2I 6gal 'a i on pelt mettre (cc + iy)1Th en fonction alge'brique de V (I + w2), c'est-at-dire de ec + y2. Mais ii faut pour cela en employant les notations conrantes * 2n, De cette dernie're on en de'duit que, s'il existe une courbe alge'brique, elie doit e~tre ferme'e et telie que 209m soit les - de la circonference. En effet: la condition de fermeture est scion (4) 20,,, = 277 = 2w, Kr4i- + i (i~(, ce qui, avec la derniere condition, se reduit ai m 'in Toutefois, s'il est bien evident qu'i'l existe des courbes fermnees, ces courbes ne correspondent pas 'a = 2' w2, du momns en general, car les deux conditions (5) doivent e'tre satisfaites par une valeur de XI les n et in e'tant entiers. Elles sont donc transcendentes et ferrniees. Le cas n = 1, m = 4 (particulie'rement simple, car gd(V) 2 W2) est facile 'a caiculer) conduit 't Z, (Za + Zb) = ZZb, et d'autre part, le deuxi me (5) conduit pour la courbe exte'rieure "a I, = 3z,, deux conditions incompatibies. On arrive au meme re'sultat pour la courbe inte'rieure. * Haiphen, Fonctions elliptiques, tome ii, p. 180.

Page  256 DAS GRAYITATIONSFELD VON MAx ABRAHAM. Naclidem es Maxwell gelungen war, die Fernikrdfte aus der Elektrodynamik zu verbannen, entstand die Aufgabe, auch die Gravitation auf Nahewirkungen zurtickzufuhlren. Eine solehe Nahewirkungstheorie muss Differentialgleichungen des Schwerkraftfeldes angeben, durch deren Integration das Newtonsche Qesetz, wenigstens mnit der durch die astronomischen Erfahrungen gewiihrleisteten Annaherung, folgt. Sie muss ferner die vom Gravitationsfelde an die Materie abgegebene Energie und Bewegungsgrtisse aus dem Energiestrom und den fiktiven Spannungen, sowie aus Dichte der Energie und Bewegungsgr6sse des Feldes ableiten. Der erste Ansatz fuir eine Theorie des statisehen Gravitationsfeldes rtihfrt von Maxwell selbst her*. Er ist der Elektrostatik nachgebildet; entsprechend dem versehiedenen Vorzeichen der Kraft zwischen gleichnamigen Massen wird indessen die Energiediebte des Schwerkraftfeldes negativ, wenn man sie bei versehwindendem Felde gleich null setzt. An dieser Sehwierigkeit seheitert jeder Versuch, die Theorie des Sehwerkraftfeldes nach dem Vorbilde des elektromagnetisehen Feldes zu konstruieren, d. h. sie auf die Wechselwirkungen zweier Yektoren zurtickzuftihren, weiche dureli Differentialgleiehungen von der Art der Maxwellsehen miteinander verkntipft sind. Eine gitickliche Idee von A. Ejusteint bot die Aussicht dar, von einer anderen Seite aus dem Problem der Sch-werkraft, beizukommen. Er nahm an, dass eine Funktionsbeziehung zwischen dem Gravitationspotential und der Liehtgesehwindigkeit bestehe. Hieran ankniipfend, habe ich Ausdriieke fair die fiktiven Spannungen, die Energiedichte, den Energiestrom ()und die Impulsdichte (g) des Sehwerkraftfeldes angegeben, welehe diese durek die ersten Ableitungen des Gravitationspotentials nach den Koordinaten und nach der Zeit ausdrticken+i; diese Ausdrticke umfassen audi die Theorie des zeitlich veriinderlichen Feldes. Dabei wird zwisehen Impulsdichte und Energiestrom die in der Elektrodynamik giiltige Beziehung Ct-/C2)agnomn In der Sprache der vierdimensionalen Vektoranalysis kann man sagen: Die zehn Gr6ssen, niimlieh die sechs Spannungskomiponentein, die drei Komponenten des Vektors cq = i2/c, und die Energiedichte E bilden die Komponenten *J. Cl. Maxwell, Scientific papers I, S. 570. ~ A. Einstein, Ann. d. Phys. 35, 898, 1911. M. Abraham, Physik. Zeitschr. 13, 1, 1912.

Page  257 DAS GRAVITATIONSFELD25 257 eines vierdimensionalen Tensors. Dabei haingen die Komponenten des Gravitationstensors von denjenigen eines Vierervektors ab, namlich des Gradienten des Gravitationspotentials, wabrend diejenigen des elektromagnetischen Zehnertensors durch die Komponenten eines Sechservektors bestimmt sind. Jch habe mich soeben der Sprache der Relativitiitstheorie bedient. Doch wird sich zeigen, dass diese Theorie mit den bier vorgetragenen Ansichten fiber die Sehwerkraft nicht zu vereinbaren ist, schon darum nieht, weil das Axiom von der Konstanz der Lichtgesehwindigkeit aufgegeben. wird. Ich habe in mieinen friiheren Arbeiten tiber die Gravitation versueht, wenigstens im unendlich kleinen, die Invarianz gegentiber den Lorentz-Transformationen zu bewahren. Doch habe ich mich davon iiberzengt, dass ineine Bewegungsgleichungen des materiellen Punktes sich nicht mit den Prinzipien der analytischen Mechanik vereinbaren lassen. Andrerseits hat sich die von Einstein zugrun de gelegte " Aquivalenzhypothese " ebenfalls als nnhaltbar erwiesen. Ich m~iehte es daher bier vorziehen, die nene Gravitationstheorie zn entwickeln, ohne auf das Raum-Zeit-Problem einzugehen. Ich gehe aus von den geeignet veraligemeinerten Ansdrticken der zehn Komponenten des Gravitationstensors und von den Lagrangeschen Bewegungsgleichnngen, und suche mii iibrigen die Voranssetzningen mn~gliehst wenig einzuschr~inken. Es sei wv=wv(c).(1).................. eine zunachst beliebige, Funktion der Lichtgesehwindigkeit c, a eine universelle, auch von c unnabhiingige Konstante. Unsere Ansdruicke fuir die fiktiven Spannungen lauten I V 2 i/aw I7 a\2 1 a W2 ~ - ~ - ~ (\~ ). (h ), 2 ax 2aw 2!( Z 2 1 (aWt aW 2 I aW 2 ~ )2_2K~t2 2ax, 2c( ay aW c w 2 2~~~ 29 x,= c(2- aw _ Die Yektoren, nergiastrom un Imusd=t see bestimmt.........d..re.bb nDdie VkonEnergie srmudIpdichte Eeenbescitduc ac =(grad )2+19t.(d'). Es ist bemerkenswert, dass diiese anch im zeitlich ver~inderlichen Felde stets positiv ausfiullt. M, (1, I 1. 1 7

Page  258 258 258 ~~~~~MAX ABRAHAM Zur Berechnung der Schwerkraft f, die auf die Volurneinheit der Materie wirkt, bedienen wir uns des Imipulssatzes, indem. wir setzen fX=a-xX ax,~, +axz a~q ax a1 (az at 3y X = ay -.............(2). ax ay ai+ at aZX aZy a3Z. aq~ ax ay az at Entsprechend berechnet sich aus dem Energiesatze die vOin Gravitationsfelde an die Volumeinheit d( r Materie abgegebene Energi e at. Setzt man in (2) uind (2 a) die Ausdrtieke (1 a-d) emn, so erhalt man l aw a (law\ a Ilaw I aw1 oder, wenn man abkiirzuingshalber schreibt, +wa, +2 a2w 1 a JlaIV.......... aX2 ay z2 - at -.) aw a!ya1aw aIa und es ergibt sich ax Es folgt also aus demn Jmpulssatz (2) die Sehwerkraft pro Volunmeinheit El — W. grad w................(4), andererseits berechnet sich aus dein Energiesatz (2a) die pro Ranum- uind Zeiteinheit an die Materie abgegebene Energie* i nW a-W................(4 a). a ~at Noch haben wir keine Annahme tiber die Funktion w (c) gemacht. Wir wollen annehinen, es sei w gleich CX. Wir sagen dann, w ist " vom. Grade X." Es mag hinsielitlich der Liingenmessung die folgende Festsetzung getroffen werden: dieselbe hat so zn geschehen, dass die Abmessungen eines hinreichend kleinen K~irpers die * Obwohl die Impulsgleichungen (2) uind die Energiegleichung (2 a) nicht die geeignete Form haben, urn ailgemein aus einem Zehnertensor einen Vierervektor abzuleiten, so haben doch in diesem besonderen Falle infolge der Relation (3 b) die Ausdriicke (4), (4 a) fiir die Sehwerkraft die Form eines Vierervektors.

Page  259 i)AS GRAVITATIONSFELD25 259 gleichen bleiben, wenn er an versehiedene Orte des Sehwerkraftfeldes gebracht wird. IDann ersieht man auis (1Id), dass die IDichte der Energie, uind foiglich auch die Energie selbst, voirn CiAade 2X sind. JDasselbe wird avieh von der Energie der Materic gelten mniissen. Es sei nun E=f 4c..................(5) die Energie eines rnhenden materiellen Pnnktes im statisehen Sehwerkraftfelde; M sei eine, dem inateriellen Punkte individuelle, von c unabh~ingige Konstante (Massenkonstante), vofl zuniiehst unbekannter Dimension. I ann ist, wie E,, aueh PD, das Gravitationspotential, vom Grade 2K. Aus der Arbeit bei1 Versehi ebung (Iles Punktes im Felde - dE MdP fol~gt ffir (lie Seh werkraft der Wert Si' -M grad 1P................(5 a) oder naeh (5) E SLI- O ad(14?................ (5b). Wir wollen nun annehmen, dass dieser letztere Ausdruek ffur die Sehwerkraft auch im Falle der JBewegung gelte, sowohl fuir einen materiellen Punkt als auch fair emn beliebiges physikalisehes System von hinreichend geringer Ansdehnnng. Diese Annahme besagt: Es soil aligemein die Sehwere eines Systems proportional seinem Energieinhalt sein. Die Gesetze der Erhaltung der Energie uind der Erhaltiing der Sehwere xverden dann identisch. IDoch list natiirlich das Verhailnis von Energie uind schwerer Masse von c, d. hi. vomn Orte im Gravitationsfelde, abhdingig. Wir wenden nun, iin liber den Grad (2X) von 1D, und damit caueh jiber den Grad (X) von wv, Atifiehluss zu erhalten, die Gleichungen der analytisehen Meehanik an. Es sei die Lagrangesehe Funktion des materiellen Punktes L =L (v, q)........I.........(6). D ann si'nd lBetrag des Imp1ulses (G) und Energie U a~~1L Da man hat UL aL aq aL _ so ergeben die Lagrangesehen Gleichungen d (aL aL die Vektorgleichung -dt + ga(d P.() wobei. das erste Glied die Triigheitskcraft, das zweite die Schwerkraft (larstellt. I 7 -2

Page  260 260 MAX ABRAHAM Durch Yergleichung mit (5 b) folgt E=-.D(7Za). Hieraius und aus (6 a) erhalten wir L = + av 4q).(8)............. Es muss also, nach einem Satze von Euler, die Lagrangesche Funktion eine homogene Funktion ersten Grades von v und c1 sein. Wir k~unnen diese, ohne EinschrAnkung (der Aligemeinheit, schreiben Wo f(0)=. Da das \Torzeiehen von v nicht von Einfluss sein kann, wird die Reihenentwicklung von f nach Poteuzen des Quadrats (V/qP)2 fortschreiten 1)ie Koeffizienten dieser Reihenentwieklung mtissen universelle Konstanten sein; denn die Massenkonstante ist die einzige dern materiellen Punkte individnelle Konstante. Seinun d1 = C2X1...................(10); alsdann ist dei Koeffizient a im allgemeinen keine dimensionslose Griisse, sondern die (4X - 2)te Potenz einer Geschwindigkeit. Man wird also dazu gen~itigt, eine universelle Koitstante c, von der Dimnension einer Gesehwind~igkeit einziifiihren iind( Zn setzeia = ~. C0~-,-2................. (10 a) (~eine reine Zahi). Diese neue Konstante c0, geht in den Ausdruek der trAgen Masse des materiellen Punktes emn, deren Grenzwert ftir kleine Geschwindigkeit (Ruhmasse) definiert ist duirch W0= limn (G/v) =lIimii'L aus (9), (9 a), (10) und (10 a) folgt niimlieh in-M 2X 2.............(lO b). Wenn man die Einfiihrung einer universellen Geschwindigkeit von zweifelhafter physikalischer IBedeutung in den Ausdruek der Ruhinasse vermeiden will und diese als lediglich durch die Massenkonstante Ml und durch die Lichtgesehwindigkeit in dem betreffendon Punkte bestimmnt ansieht, so bleibt nichts anderes iibrig, als zu setzen 2X= 1, 2=4 Daunnwird PD= c.....................(11) und Ufl(1 w~~~~~~~~~q = \%/C...................(11I a).

Page  261 DAS GRAVITATIONSFELD 261 Dainit ist die Funktion w (c) bestirnmnt, von der die Koinponenten (1 at-d) des Gravitationstenisors und diej enigen der Schwerkraft (4, 4a) abhAngen*. Die Ruhimasse ist, niaeh (10 b), D Ie t r dge R uh la ss c i s alIs o veinm G ra de - I In 1C, entspreehend ineiner friiheren B3ehauptungt. Kehren wir nun zuriiek zu den Ausdrtiiken (4), (4 a), welehe die pro Rauin- und Zeiteinheit voni Schwerkraftfelde an die Materie abgegebenen Betrage von Iinpuls und Energie bestimnienl. Fiir den von Materie und Energie leeren Raum folgt aus ihnen w=0...................(12), mit (w = ~~~~~c) als Differentialgleiehung des Gravitationsfeldes (veraligemeinerte Laplacesehe Gleichung). We aber Materie mit der Energiedielite ij den Raum erfitilt, miiissen wir setzen w ciw =2aij.................(12 a) (verailgemeinerte Poissonsehe Gleichung) +. Es folgt niiinlieh dann ans (4) fair die Sehwerkraft pro Volumeinheit f- 2,qgrad w= 77grad c............(18), w c und durch Integration tiber das Volumen ergibt sieh die Gesainitkr-aft k= ---grad c...............(1 3a), in Ubereinstimmung mit (05 b) und (11). Entsprechend folgt aus (4 a) =, aiw q 3c w at c at, und durch Voluimintegration K E ac(1b als Wert der in der Sekunde voin veriinderliehen Sehwerefelde abgegebenen E iierg ie. *Die Ausdriicke (1 (L-d) mit w =,jc gehen aus denen meiner ersten Note hervor, indlem man setzt,j =-C2 =i-jV4, wy=aCc2-ov6, dabei bedeutet ~y die Gravitationskonstante im. gewi5hnlichen Sinne, die somit vom Grade 3 in c ist. t Mv. Abraham, Physik. Zeitschr. 13, 314, 1912. Allerdings sehe ich die dort angegebene Ableitung nicht mehr als stichhaltig an. In (12 a) ist die linke Seite invariant gegenfiber Lorentz-Transformationen, die rechte Seite aber nicht, weil die Energiedichte keine Invariante im Sinne der vierdimensionalen Vektoranalysis ist. Es ist demnach gerade die physikali-sch so plausibele Hypothese der Proportionalitlit von Energie und Schwere, weiche dazu zwingt, der Diff erentialgleichung des Schwerefeldes eine der Relativitiitstheorie nicht entsprechende Gestalt zu geben.

Page  262 262 MAX ABRAHAM Es lauten somit lInpuls- und Energiesatz ftir einen rmateriellen Punkt (und ftir emn System, das einern soichen aquivalent ist) d ~ -Egrad c................(14), dE- Eac.- - 1II1'(14 a). Es bedingt also emn Geffille voin c emn An-wachsen des Jmpulsvektors, eine lokaice zeitliche Zunahine von c emn Anwachsen der Energie des Punktes; im statisehein Felde bleibt die Energie des bewegten materiellen Punktes konstant, w'enn ausser der Schwerkraft keine andere Kraft wirkt. 1)ie Feidgieichungen (12), (12 at) gehen fiir den speziellen Fall des statischenl Feldes in die voii A. Einstein angegebenen* iiber. Doch erseheinen dessen Eintwicklungen als einigermassen willktirlich und nicht widerspruchsfrei, insofern, als sie auf der Aqnivalenzhypothese beruhen, die sich gerade imit diesen Feldgleichungen nicht vereinbaren Iiisst. Auf die Blewegungsgleichungen (14), (14 a) kommen wir ani Schiusse zurtiek. Wir betrachten zwei ruhende materielle, Punkte, P nnd P', von den Massenkonstanton M uiid M'. Auf P wirkt, nach (5 at) und (11), die Kraft S = -1Mfgrad c= -2Mw gradwv............(15), in dcii von P' crregten Felde; dieses Feld bestimnint sich goindiss (12 a) durch Intcgration voni a~w a2w a2w 2orn + ~ = X2+ ay2 a2 w fairi dcii Grenzfall cincr in PI konzentrierten Energie E' = MVc. Es f'olgt a M'wV' wo IV, den Wert von Vc thir r- = cc anigibt. Aus (1 5) erhalt iman ffur den ]3etrag der Kraft, mit der PI den Punkt P anzieht, aw a 3/1w. M4'w' K= 2Mw.............. (15 b). IDa nun aber, nach (15 a), w selbst von r abhdngt, so ist die Anziehungskraft nicht strong proportional zu r~2 Man hat vielmehr ja M M ' x - a ''.........( ) Nach der hicr cntwickclten Theoric gilt also das Newtonsche Gcsetz nicht strong; es ist durch emn Gesetz von der Form A B =....................(16) zu ersetzen. * A. Einstein, Amt. d. lPhys. 38, 443", 1912. f Streng genolinmen hiingt auch w' von r- ab; jedoch in dern Falle, wo JIl klein gegen itt' ist (z. B. wenn P den Planeten, P' die Sonne bedeutet), ist dieser Umstand ohne Einfluss.

Page  263 DAS GRAVITATlONSFELD23 263 Ftir die nuinerisehe Berechnung des Quotienten B/A gentigt es, Zn setzen: a = 7wy/c 3 (y Gravitationskonstante), M' =m'c, wo mn' die Masse des anziehenden K~irpers bedeutet, = 1.,, Man erhdIlt dann ffir die Sonne, wenn r in astronoinisehen Einheiten gemnessen wird, B/A = ym'/2c02 = 1 08. Die in (16) eingeftiihrte Korrektion des Newtonschen Gesetzes dtirfte mithin zu klein sein, urn die Planetenbewegung merklich zu beeinflussen. Ich m~ichte betonen, dass die hier vorgetragene Theorie von besonderen Annalimen tiber die Lagrangesehe Fnnktion nnabhiingig ist. Letztere ist, wie wir gesehen haben, eine homogene Fnnktion von* v und 1D = c, die wir schreiben k~nnen (vgl. 9, 9 a) L-Mcf (j3), /3=V(1) mit f (3) =I-1~ )2........?....... (17a). Setzt man speziell*, indem manl ftir konstantes c, d. h. ausserhaib des Schwerefeldes, die Relativtheorie als gliltig ansieht, f() = ~2+(18),2 so wird L=-M.Vc v.............. (18 a), und die IBewegungsgleichung (14) niminmt dann die von A. Einstein angegebene Form an. Da hier ~= 1 ist, so folgt aus (11 b) fti~r die Ruhmasse der Wert rn't =M- = Eo (E. Ruhenergie)............(18 b). C C Doch kiznnte schr wohl ~= 1 sein, ohne dass die Koeffizienten der h~5heren Glieder in der Reihenentwieklung der Funktion fQ(3) mit deii von der Relativtheorie geforderten tibereinstimmnen. In meiner Dynaimik des Elektrons istt man hat also ~,und daher o4 M _ 4E, I ) r03 c 3 C2.(9) Von den sehwebenden Fragen der Dynamik des Elektrons ist die oben entwickelte Theorie des Gravitationsfeldes ganz unabliangig. Ihre Voraussetzungen sind, ausser der Grundannahme, dass das Sehwerkraftfeld dureh die Lichtgeschwindigkeit bestimmt sei: Die Gtiltigkeit der Ausdrticke (1 a-d) fair den Gravitationstensor, die Prinzipien der Meehanik (d. h. die Impuls- und Energiesdtze und die Lagrangesehen Gleichungen) und endlich die Hypothese der Proportionalitait von Sehwere und Energie. *M. Planck, Verh. d. D. Phys. Ges. 8, 140, 1906. t M. Abraham, Theorie der Elektrizitlt ii, 2. Aufi. S. 167, 1908.

Page  264 AETHER, MATTER AND GRAVITY BY S. B. MCLAREN. I denote the coordinates of a point in space of three ldimensions by x, y, z, in four by x, y, z, s. The four-dimensional space is "Euclidean," the distance from the origin being (x + y + + y2 + 2). The vector notation is reserved for three dimensions. r or t is the time. The velocity of a point is denoted by dr u dt' or in four dimensions it has the components dr ds dr with ds dr dT dv and dvds are elements of volume. I write the suffix a to denote reference to a particle of aether. d d dt or d, denotes the time rate of change at a moving point. ~ 1. Time and Space. For mathematical physics time is merely the independent variable in a scheme of differential equations where the measurable variables are positions. So long as the form of these equations is unaltered so are all the results of measurement. With Minkowski the physical universe is a steady state in four dimensions. The time variable is the distance of any of a series of parallel sections from one of them. For him all the physical characteristics of any section of the universe are fixed by the value of this independent variable. For us too the time t, as it appears in physics, is still the independently variable distance s of the sections. If we write s=ct.......................................(1), that merely indicates some scale of measurement of velocities. But though I suppose the state of the universe to be steady, it is a state of steady motion. Aether and matter are both incompressible fluids moving in unchanging streams. To fix the coordinates of any particle of aether or matter the absolute time T is required. In any stream line therefore x, y, z, s are functions of T. Thus I have one more disposable variable than Minkowski, but otherwise I have fewer physically

Page  265 AETHER, MATTER AND GRAVITY 265 ultimate conceptions. Electric charge is explained as a flux of matter, electric current also. The time of physics is still s, for this is still the independent variable whose value gives the "present" physical state, though it does not give the actual coordinates of any particle of matter or aether. The equation of continuity is Div ( dr\ d { ds'\ Div I dr\+ d In ds = 0........................(2). dT) ds( dr) Along any stream line r is a function of s. Let ds Let m d = pa.................................... (3), where a is a constant. (2) becomes D iv d + =0..............................(4). By using (1) the equation of continuity reappears in the ordinary form. Thus the density in three dimensions is proportional to the flux along the fourth dimension. p I identify with the density of electric fluid; m is proportional to the absolute weight per unit volume. ~ 2. The equations of Motion. Maxwell's electromagnetic scheme is extended from three to four dimensions, and all the formulae are deduced from the principle of least action. To begin with three dimensions, L dvdt =.................................(5) =dF ) \2 L = (87r)- t + V) - (87r)-l (Curl F)2 + pc-l (uF-co) + ep2 (u-c2)...(6). The last term assigns intrinsic energy and momentum to electricity and makes internal equilibrium of the electron possible. (5) requires (pdv)= 0. E and H the electric and magnetic intensities are given by dF E =- cdt V, H =CurlF........................(7). (6) is invariant under the Lorentz-Einstein substitution ct t = 3 (et - ax), 2 ( - a) = 1; I = 3 (x -act), a<l 8 r..................(8).:l= Z, } (5) and (6) give Maxwell's electromagnetic equations together with d- (Ep2u dv) + V ( Ep2 2- e2p'U2) dv = E'pdv...............(9), E '= E + c- [uH ]..............................(10).

Page  266 266 S. B. MCLAREN The electric fluid has per unit volume the momentum ep2u and the energy ep (c2 + u2) with the apparent pressure Ep (C2 u2).............................. (11). The integral expression (epu + c-1F)dr round any closed curve moving with the fluid is constant. Permanent "irrotational" motion is possible such that ep U +c X^, 6p + C-1^ - ax..................(12). epu + c- F VX, p+C -l cdt(12) With (12) Maxwell's equations give IdF 1 )2 47r\ (n J \ -( —Cdt2 —e 1 -PU |(5) and (6) can be extended by analogy to four dimensions 8 j( Ldvdsd, = 0................,. 5> 1, -T (87invos a r e to a)-(87r)-l (Curl F + ol - F,+(d 01-aJ,) Transflation as arm by the substitution U7 =,1 (a71 - O f1), 92 (as-,1) = 1, S-= ('1 - 1r,71 d> 1, = r=rl. This involves a reference to axes moving with a velocity aa, greater than a. It is referred to these axes that I suppose the state a steady one. There is translation as a whole with a velocity greater than that quantity a which corresponds to the velocity c of light in (6). Then (2), becomes transformed into (2) and I now postulate that (5), is to hold under the restriction that, is a constant, or that the liquid is incompressible; this condition introduces a liquid pressure Pm in matter.

Page  267 AETHER, MATTER AND GRAVITY 267 (5)1 therefore becomes for steady states sf If L dvdsdr..............................(14), L = - (s8)-' (VJ)' + (87r)-1 (dJ2 + rmJ + (8_r)-1 d+ V - (8tr)- (Curl F)2 dr _dds,~ I k +, F - mas ^ + 1 k - -r d — ds: * _......(15), (dr dT ( ) { [dT) (d() } 2 ) with 8 (dvds)= 0.................................(16). Variation of F, ~ and J gives Maxwell's electromagnetic equations together with (-c2 -{) — J + 47rT = 0.......(.....17). Here (1) and (3) are used. Variation of the distribution of matter gives four equations of motion. These involve the pressure of matter p,, whose amount can be found by an integration as in Hydrodynamics = - a + ep2 (C2 u2) +.....................(18), where /c'a2 = eC2 and a is a constant of integration. The condition at the free surface of matter is pi, = 0 or ep2 (c2 - u2) = a - m J...........................(19). Three equations of motion are dt' (ep2udv) + Vp,,,dv = E'pdv + Vd...............(20). By (18) these are reduced to (9). Regarding the left of (19) as the pressure, there appear to be the externally applied pressure a, and tension mJ at the external surface of matter. ~ 3. Aether a polarizable Jflid; matter an aether sirk. If in (6) all terms were omitted except the last, then (12) would become dx epu=V, epc=-cdt It is now obvious how the terms involving differential coefficients of J in (15) are to be interpreted. Write anl - = Ua (4r) J ds ) ds j(21). d, a-d =..= -.(47.r) J dr ds)

Page  268 268 S. B. MCLAREN In (21) I understand mia to be the absolute density of aether, ac the density in three dimensions. (17) is now Div ( u ) + d - (47r) m..........................(22) or Div 1(ma )dT + d- (ma d (47r m........... (23). Thus by (22) the quantity (47r)2 m of aether disappears per second and per unit volume of space. Or by (23) the absolute volume dvdsa shrinks at a rate given by d (dId): - (4-} mdvadsa Ma ~dT- (dVadss) =-(47r)2 mdds a..................... (24). The aether thus absorbed is conceived of as annihilated, its momentum being given to matter. It may be shown that it is possible for the aether to move "irrotationally" even when matter is present. The equations of motion for aether can be deduced by putting for its Lagrangian function La, /Idr \2 ds \2 La =- n 2 a(t + ma2 a + ( )-) + (8.(EJ-H2).........(25), and introducing what appears as the external friction on unit quantity of the aether (47r)2 mua. This force is really the rate of transfer of momentum due to the destruction of aether by matter. The pressure pa is found to be a = b + o-2 (c2 - Ua2)- (87r)-l (E2 -H2)...............(26). On restoring the Newtonian Potential J the principles of least action and conservation of energy are saved by a mathematical fiction which attributes energy to matter of density -mJ and gives (14) and (15) once more. ~ 4. Optical Phenomena not affected by the motion of the aether. The gravitational energy is wholly kinetic, it is per unit volume i 2 a(dr 2(ds) The electromagnetic energy is wholly potential (87r)- (d- + V) + (Sr)- (Curl F)2. These two forms of energy may be put down as due to the motion of translation of the atoms of a fluid and their polarization. If the polarizing apparatus be regarded as centred about each atom on a universal joint, then there is a statical balance of the electromagnetic forces in spite of the aether's motion. They produce also a resultant force on aether of intensity - (87)-l V (E2- H2) in unit volume.

Page  269 AETHER, MATTER AND GRAVITY 269 I would also urge for those whom this does not satisfy that the absolute velocity of the aether can be made as small as we please by increasing n,. The force of gravity is the only phenomenon imposing any restriction. It is dr,, _ ma or (47r) 2VJ, (XT and we therefore know only the product of mass and velocity. ~ 5. Cavities in the Aether, Electron and Magneton, Positive and Negative charge. It may be that cavities exist in the aether. The surface of any such cavity must also be a boundary for matter. Conditions sufficient to satisfy the principles of conservation both of energy and momentum are Et+c- [unH ]=O, Hn=0........................(27), dJ dJ P a+ P = O....................................(29). The suffix n denotes a normal and t a tangential component, u, the velocity of the surface normal to itself. By (27) the surface is a perfect reflector; by (28) its normal velocity is equal to the normal velocity of the aether itself and by (29) the total pressure of aether and matter on the surface of the cavity vanishes. Such a surface may carry what is effectively an electric surface charge. For by Maxwell's equations the total normal electric induction is invariant. If it is multiply connected it may serve as a permanent magnet, for the total magnetic flux across any aperture is invariant. Also there is here a possible distinction between two kinds of electricity. The one a volume distribution, the other an integral induction over the surface of the cavity. (14) and (15) lead to the same expressions for the energy and momentum of the Gravitational field as have already been given by Max Abraham. It is therefore now unnecessary for me to do more than refer the reader to his paper in this volume for these results which I had myself obtained independently.

Page  270 SOPRA UN CRITERIO DI CLASSIFICAZIONE DEI MASSIMI E DEI MINIMI DELLE FUNZIONI DI PJU VARIABILI Di C. SOMIGLIANA. Se si osserva che in una, regione montuosa le linee, che dividonro i diversi bacini idrografici, ossia le creste, vanno a cong~iungersi nelle vette della, regione, nasce spontanea, lidea di classificare le vette mediante il numero delle creste che in esse concorrano. Anche le linee di valle, cosi chiamando le linee percorse dalle correnti d'acqua, hanno in generale origine dalle vette e sono per ogni vetta in numero uguale a quello dalle creste. Si hanno cosi vette di 1', 2', 30,... ordine secondo che da esse partono iuua linea di cresta, ed una di valle, oppure dute linee di. cresta e due di valle, oppure tre linee di cresta, e tre di valle e cosi via. Portando questo criterio nel campo delle funzioni di dule variabili per ottenere una, classificazione dei massimi e dei ruinimii occorre anizitutto definire in mnodo preciso analiticainente le linee di cresta e le linee di valle. A CR)' si arriva considerndo-opraogni linea di livello-i massimi ed imiinimi deli' angolo che la verticale fa colla normale alla superficie. I luoghi di questi inlassimni e di questi minirni costituiscono rispettivamente sulla superficie le linee di cresta e le linee, di valle. L'equazione complessiva di tutte queste linee sia di cresta che di valle e' la seginente ax aiy ay ax.() dove p (x, y) e' la funzione che Si considera, e La determiinazione di queste linee non richiede percio' alduna integrazione. Nelle vicinanze di un massimo o di un minimo la equazione (1) Si pub porre in generale sotto la forma: n1, (x, y) VI (x, y) u2" (x, y) v2 (cc, y).,. t,, (c, y) v,, (x, Y) = 0. Iinuinero n rappresenta, allora l'ordine del nmassimo o del minimo considerato. Non e' difficile estendere la definizione di linee di cresta e di valle alle funzi.oni di un numero qualunque di variabill.

Page  271 CLASSIFICAZIONE DIM MASSIMI E DEI MINIMI DELLE FUNZIONI DI PIU VARIABILI 271 Quneste linee hanno iriteresse anche sotto un altro punto di vista. Esse possono essere considerate come le linee secondo le quali, a partire da un massimo, o da un mimimo, la funzione decresce, o cresce, c olla mnaggiore o colla minore rapidita'. Esse portanro quindi in certo modo ad irna estensione del concetto di massim-o e di minimo per le funzioni che dipendono da p~i di una variabile, in quanrto rappresentano degli enti, non p~iu a dimensione nulla, come sono i massimi e minirni ordinari, ma a dimensione finlita (uguale ad uno) caratterizzati da una certa prevalenza, o deficienza, in essi dei valori della funzione. Sotto questo punto di vista la considerazione delle linee di cresta e di valle pub tornare assai utile nello studiare l'andamento di una fauzione di due o p~i variabili. nel campo in ciii essa e' definita.

Page  272 ON A LAW OF CONNEXION BETWEEN TWO PHENOMENA WHICH INFLUENCE EACH OTHER BY W. ESSON. 1. It is proposed to measure the influence of one phenomenon upon another by the ratio of the percentage increase of one to the percentage increase of the other. 2. One advantage of this method is that the measure is a pure number independent of the units in terms of which the phenomena are expressed. Another advantage is that the mode in which one phenomenon influences the other need not be known. 3. If A and B are the number of units of the two phenomena present at any time and a small increase dA determines a small increase dB, the measure here proposed is the ratio A-ldA/B-'dB = m. 4. In the case in which the changes of A and B do not influence either the nature of the phenomena or the mode in which they react upon each other it is reasonable to suppose that m is constant. In this case the relation between the phenomena is expressed by A/Ao= (B/Bo). 5. In other cases in which the conditions affecting the phenomena are not so simple mi will be a function of A/Ao and B/Bo. 6. This method of computing the mutual influence of two phenomena occurred to the author forty years ago during the course of a research conducted by Harcourt and himself upon the connexion between the conditions of a chemical change and its amount. The results of this research were published in the Bakerian Lecture of the Royal Society in 1895. 7. In the early part of this year Harcourt made a further research on the variation with temperature of the rate of a chemical change and requested the author to discuss his experiments as well as the experiments of other chemists on the same subject. This discussion has been published in the Transactions of the Royal Society, Series A, Vol. ccxII. pp. 187-204. 8. In the Bakerian Lecture it was shown that the rate of chemical action k is connected with the temperature t by the relation k/ko = {(c + t)/(c + t0)}m and further that the value of c is 272'6, so that the relation may be written k/ko = (T/T0)m, T being the absolute temperature. Thus it was shown that the zero of chemical energy is the same as the zero of the molecular energy of temperature.

Page  273 ON A LAW OF CONNEXION BETWEEN TWO PHENOMENA 273 m is a constant depending on the kind of medium in which the chemical action takes place and is the measure k-ldk/T-ldT of the effect of thermal energy upon chemical energy. If u' is the mean velocity of the atoms or electrons the energy of which causes the chemical action and v is the mean velocity of the molecules, the measure of the effect may be expressed as m = v-ldv/u-ldu. This relation is satisfied by the results of experiments by other chemists in which due precaution has been taken to keep constant the nature of the chemical action and of the medium. It is hoped that the relation may be of use in the measurement of the mutual influence of other than chemical phenomena. M. C. II. 18

Page  274 SELF-CONTAINED ELECTROMAGNETIC VIBRATIONS OF A SPHERE AS A POSSIBLE MODEL OF THE ATOMIC STORE OF LATENT ENERGY BY L. SILBERSTEIN. This comnunication will be published in the Philosophical Magazine in the course of 1913.

Page  275 MULTIPLY-CHARGED ATOMS BY SIR J. J. THOMSON. In the photographs of the positive rays (see, for example, those given in the Phil. Mag. Aug. 1912) the mercury line is remarkable for the exceptionally small displacement of the head of its parabola. Even when the electric and magnetic fields are strong enough to produce deflexions of several millimetres in the heads of the parabolas corresponding to the other elements, the head of the mercury parabola is so little deflected that at first sight it seems to coincide with the origin. When, however, the electric field used to deflect the particles is made very large, in our experiments from 5000 to 10,000 volts per centimetre, the head of this parabola is distinctly displaced, and on measuring the electrostatic displacement it is found to be 1/8 of the normal displacement of the heads of the parabolas corresponding to the other elements. The displacement due to the electric field is inversely proportional to the kinetic energy of the particle displaced, so that the atoms which produce the head of the mercury parabola must have eight times the maximum amount of energy possessed by the normal atoms. This could be accounted for if some of the mercury atoms in the discharge-tube had lost 8 corpuscles, for then the energy communicated to the atom by the electric field would be eight times the energy communicated to an atom with the normal charge. Eight is a very large number of corpuscles to lose, much larger than the number lost by the other elements which get multiple charges. I had previously to these experiments never found a case in which this number exceeded three; so that in my paper in the Phil. Mag. for August 1912, I suggested another explanation of the behaviour of the mercury line. A study of the plates taken with large electrostatic deflexions has revealed the existence of 7 parabolas due to mercury, corresponding to the mercury atom with 1, 2, 3, 4, 5, 6, 7 charges respectively, the parabola corresponding to 8 charges has not been detected, but as the parabolas get in general fainter for each additional charge on the atom, it is probably there but too faint to be detected. Fig. 1. taken from a photograph when the gas in the tube was the residual gas left after exhaustion by the Gaede pump, shows these lines very well. The measurements of r/e for these parabolas gave the following values: mle 200 102 200/2 66-3 200/3 50-4 200/4 44 this not a mercury line but is due to CO2. 39-8 200/5 33-7 200/6 28-6 200/7 18-2

Page  276 276 SIR J. J. THOMSON It will be noticed that the heads of the parabolas corresponding to 1, 2, 3 charges respectively lie on a straight line passing through the origin, indicating that the velocities of the particles producing the heads of these parabolas are all equal, and therefore, since each particle is an atom of mercury, that the kinetic energy of the particles at the heads of the parabolas is constant. This is what we should have expected, for the heads of all the parabolas are due to particles which had originally lost 8 corpuscles, the particle at the head of the parabola corresponding to one charge has regained 7 of these after passing through the cathode, that at the head of the parabola corresponding to two charges, six, and so on; these particles when in the discharge-tube were all in the same condition, and so acquired the same amount of kinetic energy. The question now arises as to how the mercury atom acquires these very various charges. When a mercury atom is ionized, can it lose any number of corpuscles from one to eight? Taking for example a mercury atom with 5 positive charges, has it got into this condition by losing 5 corpuscles when it was ionized, or did it originally lose the maximum number 8 and regain 3 subsequently? The photographs prove, I think, that the second supposition is the correct one, and that in the discharge-tube there are two, and only two, kinds of ionization; in one of these kinds the mercury atom loses 1 corpuscle, while in the other kind it loses 8, and that there are no indications of ionization of such a character as to deprive the mercury atom of 7, 6, 5, 4, 3, or 2 corpuscles. The evidence for this is as follows: let us suppose for a moment that atoms with any charge from 1 to 8 were produced by the ionization of the mercury atoms in the discharge-tube. Consider now the parabola due to the mercury atom which, when it passed through the electric and magnetic fields, had one positive charge. This parabola would result from atoms of the following kinds: 1. Atoms which had lost 8 corpuscles on ionization and regained 7; 2. Atoms which had lost 7 and regained 6; 3. Atoms which had lost 6 and regained 5, and so on, the eighth and last members of the series being atoms which had lost one corpuscle on ionization and had not regained it. The parabola on the photographic plate would be due to the superposition of the 8 parabolas due to these types of atoms. If 8d is the horizontal distance from the vertical axis of the head of the parabola due to the atom which lost one corpuscle on ionization, the horizontal distances of the heads of the parabolas due to atoms of the 1st, 2nd, 3rd types will be respectively 8d/8, 8d/ 8d 8d/6, 8d/5, 8d/4, 8d/3, 8d/2, 8d. Thus up to the horizontal distance 8d/7 there would be only one parabola, at 8d/7 another parabola would be added, this would produce an abrupt increase in the intensity of the photograph, there would be another abrupt increase at 8d/6, another at 8d/5, and so on. Thus the intensity of the parabola on the photograph would not be continuous, there would be places where the intensity was suddenly increased giving a beaded appearance to the photograph. The abrupt increase at 8d is very marked on this parabola, the others are not visible; but as the intensity of this parabola is very great it might be thought that they escaped detection owing to the breadth of the parabola. Let us

Page  277 MULTIPLY-CHARGED ATOMS 277 therefore consider one of the finer parabolas, say, that due to the atom with four charges, to which this objection does not apply. This parabola might arise from atoms which had lost 8 charges on ionization and regained 4, from those which had lost 7 and regained 3, and so on, the last being atoms which had lost 4 and not regained any. Then if d has the same meaning as before, the horizontal distances from the vertical axis of the heads of the parabolas would be 4 x 8d/8, 4 x 8d/7, 4 x 8d/6, 4 x 8d/5, 4 x 8d/4; there would, therefore, be abrupt increases in intensity at 32d/7, 32d/6, 32d/5, 32d/4. The photographs show, however, that the intensity is perfectly continuous, and not one of these abrupt increases is to be seen. We conclude, therefore, that there are no atoms which begin with 7, 6, 5, 4, 3, 2 charges, and that in this case there are two and only two types of ionization, in the one type an atom loses a single corpuscle, in the other it loses 8. This result suggests that ionization takes place in the discharge-tube in two ways. In the first method the ionizing agents are the rapidly moving corpuscles which constitute the cathode rays, these very small particles penetrate into the atom and come into collision with the corpuscles inside it individually, the collision in favourable cases causing the corpuscle struck to escape from the atom; this type of ionization results in the atom losing a single charge. In the other type of ionization we suppose that the mercury atom is struck by a rapidly moving atom and not by a corpuscle; after the collision the mercury atom starts off with a very considerable velocity, which at first is not shared by the corpuscles inside it. The tendency of the corpuscles to leave the atom depends only upon the relative velocity of the atom and the corpuscles inside it, so that the ionizing effect produced by the collision is the same as if the atom were at rest, and all the corpuscles were moving with the velocity acquired by the atom in the collision. Thus if there were eight corpuscles in the mercury atom connected with about the same firmness to the atom, the result of the atom acquiring a high velocity in a collision might be the detachment of the set of eight leaving the atom with a charge of 8 units of positive electricity. We see in this way how the cathode particles might produce one type of ionization resulting in singly charged atoms, while the atoms forming the positive rays might produce another type of ionization resulting in multiply-charged atoms. All the elements I have examined give multiply-charged positive atoms with the exception of hydrogen, on which I have never observed more than one charge; in no other case, however, have I observed charges approaching that possessed by mercury. The majority of the elements seem to acquire only two charges; this is the number acquired by helium, and this case is interesting since in the vacuum-tube the helium atom occurs with both single and double charges, whilst as an a particle it always seems to have two charges, suggesting that the process by which the a particle acquires its charge is analogous to the process by which multiply-charged atoms are produced in the discharge-tube. I have observed nitrogen atoms with three charges, but the parabola due to the triply-charged atom is exceedingly faint. Argon shows triply-charged atoms very distinctly as can be seen from fig. 2, where the parabolas I, II, III due to Arg+, Arg++, Arg+++ are all very distinct; the Arg++ parabola has probably a parabola due to neon superposed on it. This plate shows the helium line, and thus incidentally gives us an estimate of the sensitiveness of this method of detecting small quantities of a gas. The volume

Page  278 278 SIR J. J. THOMSON of the discharge-tube was about two litres, the pressure 1/300 of a mm. of mercury: there was thus in the discharge-tube about 1/100 c.c. of argon at atmospheric pressure. This is about the amount in 1 c.c. of air at this pressure, and as the helium line was visible along with the argon, we see that this method can detect the amount of helium in 1 c.c. of air, which, according to Sir William Ramsay, is about 4 x 10-6 c.c., even though this is mixed with an enormous excess of argon. The tube used for this photograph was not at all well adapted for detecting small quantities of an impurity, as the cathode had been in use for several weeks and the tube through it was almost silted up by the sand-blast action of the positive rays, and was just about to be replaced by a new tube. An interesting result, which is now being investigated, is that when very pure nitrogen is in the discharge-tube the mercury line corresponding to the atom with five charges becomes abnormally bright, brighter than those for the atom with four or even three charges, though in other gases the greater the charge on the atom the fainter the line. I have much pleasure in thanking Mr F. W. Aston, B.A., Trinity College, for the great assistance he has given me in these investigations. Fig. 1. Fig. 2.

Page  279 ON WAVE-TRAINS DUE TO A SINGLE IMPULSE BY HORACE LAMB. The object of this note is to call attention to some graphical methods by which the general character of the wave-train generated by a single impulse in a dispersive medium of any given constitution can be ascertained. For simplicity, cases of propagation in one dimension are alone considered. The properties of the medium are sufficiently determined, for the present purpose, by the relation between the constants a- and k in the formula u = cos os at.................................(1), which represents an indefinitely extended system of standing waves of simpleharmonic type. If r denote the period, and X the wave-length, we have of course 7= 27r/a, X = 27r/..............................(2). The velocity of propagation of a progressive train of simple-harmonic waves = cos k (Vt + x).................................(3) is V = X/7 = al/.................................(4), and the corresponding value of the group-velocity is U= d=V- X..............................(5). dc dX.(5) By superposition of wave-systems of the type (1) we can obtain the result corresponding to an initial impulse (at time t = 0) whose space-distribution is given by an arbitrary relation = (x ).................................... (6), viz. we have u = r cos at cos kxf(k) dk + - cos at sin kx F (k) dk.........(7), where f(k) = J (a) cos kada, F(k) = f (a) sin kada.........(8). If the initial distribution be symmetrical with respect to the origin 0, we have F (k) = 0; and if we further assume it to be concentrated in the immediate neighbourhood of 0, in such a way that ( a)da= l................................. (9), -e

Page  280 280 HORACE LAMB where e is infinitesimal, we have finally =- cos at cos kx dk...........................(10), vrJo which is the expression to be elucidated. It may be noted that the integral will not converge at the upper limit unless the wave-velocity (a-/k) is infinite for infinitely short waves (,= oo ). The difficulty, when it arises, may sometimes be met by assuming, in ((j), 1 13, — 1 _.... (1]), d)( 2) = 4 2............................. which makes f(k)= e-, F(k)...........................(12), and examiining the form which (7) assumes as 13 is diminished. In some cases, e.g. that of sound-waves, we cannot expect a determinate result from an absolutely concentrated impulse. The integral in (10), even when convergent, cannot be evaluated except in a few special cases, the most important of which is that of deep-water waves*; but Lord Kelvin has shewn-t how to obtain an approximation, under certain conditions, in the general case. The disturbance is, in the first place, analysed into two progressive systems of simple-harmonic waves, travelling to the right and left, respectively; thus 1 f3 1C31 U==, cos (t- x) dk + | cos (at + kx)dk.........(13). In strictness both these integrals have to be taken into account in calculating the disturbance at any given place and time; but usually the first term is alone important at points lying to the right of the origin. It may be remarked, however, that if the group-velocity is negative, as it may be in certain imaginable cases, it is the second integral which is predominant at such points. Taking, however, the more usual case, and ignoring the second term, we have, for x > 0, 2 = 2 cos (ot- kx) dk.....................(14). The disturbance thus represented is made up of an aggregate of simple-harmonic trains, of wave-lengths varying from oo to 0, each travelling with its proper velocity. Owing to differences of phase between the various trains, we have "interference," in the optical sense, and the main effect, at time t and place x, will be due to those trains whose wave-length is such that the phase is stationary, or nearly so. The equation dk tdk-x=0, or x=Ut........................(15), therefore determines the wave-lengths which are predominant at the time and place considered. Lord Kelvin proceeds to shew, by a process which need not be * Discussed by Poisson and Cauchy; see Proc. Lond. Math. Soc. (2) vol. 2, p. 371. t Proc. Roy. Soc. vol. 42, p. 80 (1887); Math. and Phys. Papers, vol. 4, p. 303. + See Proc. Lond. Math. Soc. (2) vol. 1, p. 473 (1904).

Page  281 ON WAVE-TRAINS DUE TO A SINGLE IMPULSE 281 reproduced*, that corresponding to every value of k which satisfies (15) we have a term 1 V{2wt d/dF }* cos (o-t-lcx - *r)................ (16) ~~~~42tin the value of a. in the value of i. The process and the result may be illustrated by The simplest, in some respects, is based on a slight modification of a diagram already employed by the writert. We construct a curve Vt with X as abscissa and Vt as ordinate, where t denotes of course the time that has elapsed from the beginning of the disturbance. To ascertain the nature of the wave-system in the Q neighbourhood of any point x, we measure off a length OQ equal to x, along the axis of ordinates. If PAr be the ordinate corresponding to any given abscissa X, the phase of the disturbance at the point x, due to the elementary 0 wave-train whose wave-length is X, will be given by the gradient of the line QP; for if we draw QR parallel to ON we have various graphical constructions. P N Fig. 1. PR PN- OQ Vt -x at- kx.................... ( 17). QR~ ON x 27r.) Hence the phase will be stationary if QP be a tangent to the curve; and the predominant wave-lengths at the point x are accordingly given by the abscissae of the points of contact of the several tangents which can be drawn from Q. These are characterised by the property that the group-velocity has a given value x/t. If we imagine the point Q to travel along the straight line on which it lies, we get an indication of the distribution of wave-lengths in the medium, at the instant t for which the diagram has been constructed. If we wish to follow the changes which take place in time at a given point x, we may either imagine the ordinates to be altered in the ratio of the respective times, or we may imagine the point Q to approach 0 in such a way that OQ varies inversely as t. We may consider a few particular cases. In the case of deep-water waves, where 2 = gk, V2t2 gt2. X/2r........................(18), the curve is a parabola (Fig. 2), and one tangent can be drawn from any position of Q. It appears at once that in the procession of waves at any instant the wavelength diminishes continually from front to rear; and that the waves which pass any assigned point will have their wave-lengths continually diminishing. It will be found that these statements are reversed in the case of waves due to capillarity only where C2 = T:T, VF t2 = 27r t2/........................(19). * See also Rayleigh, Phil. Mag., vol. 21, p. 181 (1911). + Manchal. Mem)oirs, vol. 44 (1900); Hydrodynamics, 3rd ed., pp. 362, 439,

Page  282 282 HORACE LAMB When gravity and capillarity are both operative, we have a2= gk + Tk3, V2 = g/2r + 2rT/\..................(20). Vt Vt Ut 0 N Fi. 2. Fig. 3. Fig. 2. Fig. 3. The curve is shewn in Fig. 3. It appears that two tangents can be drawn from a point Q on the axis of ordinates provided O Q < Uot....................................(21), where U0 is the minimum group-velocity, corresponding to the point of inflexion on the curve*. We have, then, two predominant wave-lengths at any point of the train, which is moreover limited in the rear. Finally, we may notice the case where there is a limiting value to the wavevelocity, as in the case of gravity waves on water of uniform depth. The relation is now oa = gk tanhkh..............................(22); and the curve is shewn in Fig. 4. The condition of stationary phase is now fulfilled approximately for all very long waves, and we infer that the maint disturbance begins with a solitary wave about the time x/(gh)2 followed by a train of oscillatory waves whose general character towards the rear approximates to that which is characteristic of deep-water waves. The construction above explained has the defect that it gives no indication 0 X of the relative amplitudes in different Fig. 4. parts of the wave-train. For this purpose * Manch. Memoirs, I.c. See also Rayleigh, I.c. t This is preceded by a minor inequality, whose initial stages have been investigated by Rayleigh; see Phil. Mag., vol. 18 (1909), and Sc. Papers, vol. 5, p. 514.

Page  283 ON WAVE-TRAINS DUE TO A SINGLE IMPULSE 283 we may construct the curve which gives the relation between at as ordinate and k as abscissa. If we draw a line through the origin whose gradient is x, the phase due to a particular wave-train, viz. at- kx, will be represented by the difference of the ordinates of the curve and the straight line. This difference will be stationary when the tangent to the curve is parallel to the straight line, i.e. when da t =x..................................(23), o-t o- t = kx o0 k Fig. 5. as already found by the previous method. It is further evident that the phasedifference, for elementary trains of slightly different wave-lengths, will vary ultimately as the square of the increment of k. Also that the range of values of k for which the phase is sensibly the same will be greater, and consequently the resulting disturbance will be more intense, the greater the vertical chord of curvature of the curve. This explains the occurrence of the quantity td2a/dk2 in the denominator of Kelvin's formula (16). The annexed Fig. (3, which corresponds to the case of water waves under gravity at\ / Fig. 6.

Page  284 284 HORACE LAMB and capillarity combined, indicates that, of the two systems of waves in the more advanced portion of the procession, one consists of short and high waves, the other of long and low ones*. It may naturally be asked whether the above methods are capable of throwing any light on the theory of earthquake waves, or on the interpretation of seismometric records. All that can be said, at present, is that they do afford a means of testing, to some extent, particular theories which attribute the observed effects to dispersion. In particular it may be noted that the period of the waves which pass any particular point will become shorter or longer as time goes on, according as the curve which shews the relation between V and X is convex or concave upwards. The same conclusion may be obtained analytically as follows. If we regard X, and therefore k, as a function of x and t, we have, by differentiation of (15), do- d2- ak + t 0............................(24). de dk2 dat(24) Hen~ce~ ~o do- a- (doi 2 d2o at d1c at \dl dk.(25). But from (5) we find dk = X dd2................................(26) Hence ao-/at is positive, i.e. the frequency at the point x increases, if d2V/dX2 is negative. It may be noted that if an analytical representation of the wave-system, in any case of dispersion, could be obtained, the relation between a- and k could be deduced as follows. We should have -t- kx = f'(x,t).............................(27), a known function of x and t, with the condition dotd =.................................. (28). Hence d (ta t).(29). ~Hence CT ~- - k = /f t, t...........................(29). Since the right hand side must be a function of k only, the function f(x, t) must be homogeneous in respect to x and t, of degree one. Hence, putting f (Xt) = tF()(30) /(x. t)= f..............................30 we have = k dk + F ()..........(31), which is of Clairaut's form. This is illustrated in the case of gravity waves, or capillary waves. A formula which includes several interesting cases is 1 F = {1-..(32)./\ * It indicates also that the shorter waves have the greater amplitude, hut this circumstance is greatly modified (by interference) in the case of an initial disturbance which is diffused over a finite range of x instead of being concentrated at a point.

Page  285 ON WAVE-TRAINS DUE TO A SINGLE IMPULSE 285 This leads to _ is_- 1- (33). ~~~~~This leads to - c-.............................(33). tm,-l + (kc)n-1L-1 icc Thus if n = 2, we have (34); Is+ ieb................................. if n =,2 2= C2 + m 2.......................... (35), which is the case of waves propagated vertically in an isothermal atmosphere; if n = which is the case of a tense chain destitute of stiffness, but possessing rotatory inertia. 1 + b................................. which is the case of a tense chain destitute of stiffness, but possessing rotatory inertia.

Page  286 HOW ATWOOD'S MACHINE SHOWS THE ROTATION OF THE EARTH EVEN QUANTITATIVELY BY JOHN G. HAGEN. The Treatise of G. Atwood in which the machine, known by his name, is described and mathematically explained, was published in Cambridge as early as 1784, or six years before Guglielmini's experiments (1790-92) in Bologna. It is a curious fact that Benzenberg in his book, Die Versuche iiber die Umdrehung der Erde (1804), mentions Atwood's machine, illustrates it by a figure (1, Plate VII) and (p. 94) praises its advantage of reducing the velocity of falling bodies to any desired minor quantity. And yet he, who had a double experience in Hamburg (1802) and Schlebusch (1804), with velocities from heights of 76 and 85 metres, never thought of profiting from this advantage. On the contrary, he proposed both the Pantheon and St Peter's for a repetition of his experiments. Perhaps contrary to expectation his double wish was fulfilled a century later, at Paris in 1903 and at Rome in 1912, but not with the heights that he had in view. In the Pantheon only 68 m. were utilized, and in Rome only 23 m., in a building close to St Peter's Basilica. Even that height was unnecessarily large. The free fall would have given us an easterly deviation of 1'8 mm.; this was reduced to one half, or 0'9 mm., by the machine. The experiments proved that less than one half of this quantity can be observed quite exactly with the new method. I. Description of the Apparatus and the Experinments. 1. The pulley was mounted in an upper room of what is called the Nicchione. The floor of the room consists of a stone vault which covers a triangular staircase, 23 m. high. Within the staircase a shaft was built of wood and cloth to protect the descending weight and thread from air-currents. The shaft communicates with the room above by a circular opening in the floor. 2. The pulley consists of a thin, solid disc of aluminium, with steel-points in the axis, and was made new for the purpose. The rest of the apparatus is part of an old seismograph: a brass tube, which holds the pulley excentrically so that the weight hangs inside the tube, and is thus protected from lateral currents. A bell-jar, put over the whole apparatus, excludes vertical air-currents. The tube has three legs and is screwed to a wooden structure which is firmly imbedded in wall and floor. The old weight was an empty cylinder, with a hole in the centre of the cover for the

Page  287 ROTATION OF THE EARTH 287 thread. It was filled with shot until it weighed about 50 gr. The counterweight was made three-fourths as heavy; it is, at its upper part, provided with an arrow-shaped spring, and at its lower end with a hook. The spring just mentioned is part of the damping-apparatus. If the counterweight were arrested directly and suddenly, the thread would be apt to break or, in consequence of its expansion and following contraction, jump off the pulley. The damper consists of a brass plate which is lifted by the counterweight to a height of about 70 cm., as soon as it enters the upper room. The thread passes through a central hole in the plate and pulls the arrow-shaped spring through the same, but not the weight. The plate is thus firmly caught between the spring and the weight. When the plate is lifted, it glides between two guiding bars, which also prevent it from falling back. For this purpose another pair of springs is needed. In our case the springs were attached to the guiding bars, under the supposition that the plate would always reach the same height. This is not the case on account of its varying friction against the bars. The springs should be attached to the plate and the bars provided with rack-incisions. 3. The shaft reaches to the ground-floor, where it is connected, air-tight, with a wooden box. Inside, at the bottom, are the binding posts of an electric current, which hold the fusible wire on which the counterweight is hooked. Higher up, the box is provided with two windows, in the shape of biplanar glassplates. They are on a level with a theodolite which is mounted outside against the wall and independent of the floor. Through the theodolite the thread of the descending weight is observed from two directions. Viewed from the north, the easterly deviations are seen, and viewed from the east, the deviations within the meridian. The narrow locality necessitated the use of a mirror, which was mounted on the wall, north or east of the thread, opposite either of the windows in the shaft. The mirror can be turned on vertical hinges, like a door, by means of a rod from the eye end of the theodolite, until a certain mark, inside the shaft, throws its image on a certain line of the micrometer scale. In the shaft there are two marks, in the shape of black threads suspended against the white wall: one in the meridian and the other in the prime vertical of the descending thread. The micrometer scale is divided on glass and visible, with the descending thread, through the strong illumination inside the shaft. 4. The experiments proceeded in the following way. The observer closed the circuit of the fusible wire by a switch close to the theodolite. About ten seconds later he saw the thread appear in the field, but never the weight. During the first trials the entire attention was directed to the oscillations of the pendulum after the counterweight was arrested. One fact revealed by these trials was that, viewed from the north, the first oscillation always went from east to west, and never once from west to east. This fact alone constituted a qualitative proof of the terrestrial rotation as it had never been reached by observing the free fall of bodies.

Page  288 288 JOHN G. HAGEN It was found out, however, that the oscillations of the pendulum did not represent the easterly deviation quantitatively. The amplitudes grew alternately smaller and larger; and the same thing was observed from the east side. The weight, after its descent, evidently described spherical ellipses with revolving axes. The cause of this was the expansion of the thread, to the extent of one-third of a metre, and the bouncing back of the weight. At the same time the thread began to spin and to revolve the weight about an axis which may not have coincided with its geometrical axis. The thread would sometimes show vibrations like a musical chord. It became evident that oscillations after the arresting of the counterweight would not yield any quantitative measures, except the direction of the vertical. Five elongations on either side were deemed sufficient to fix the position of the plumb-line on the micrometer scale. Fortunately, after some practice, it was found that the fall of the thread could be observed with great exactness, before the counterweight struck the damping apparatus. 5. For an instant the thread would appear on the micrometer scale as a perfectly steady fine line, whose position could be estimated to tenths of a scale division, and even to half tenths. The value of one division was determined by placing a millimetre scale in contact with the thread and perpendicular to the line of sight. The number of millimetres covered by the micrometer gave the following results: View from E., distance 6'24 m.; one division = 1-697 mm.,,,, N.,,, 4-30 m.;,,,, = 1170 mm. 6. The crucial test for all the methods of measuring the easterly deviation of falling bodies is the southern deviation. Both Gauss and Laplace have proved, from theory, that the southerly deviation of falling bodies is inappreciable. Beginning was therefore made with this, and special care was taken to avoid windy days, and also hours of the day when the Museum, close by, was frequented by visitors. After the method had stood the test of 22 experiments, with no deviation to the south, a larger number of experiments was made for the easterly deviation, and less care was taken in the selection of quiet days and hours. In computing the final results no experiment, which had succeeded mechanically, was discarded. They are as follows: Deviation S.; 22 experiments: + 0-010 + 0-027 (1 exp. + 0-128),, E.; 66,,: +0899 +0027 (1,, +0-217). All measures are expressed in millimetres. The stricter choice of the circumstances is evidenced in the smaller P.E. of one experiment for southerly deviations. II. The Theory of the Experiments. As the theory of these experiments will be published in detail in a forthcoming volume of the Vatican Observatory, it need only be outlined here. The coordinate system is connected with the place of the experiments: the origin is the centre of

Page  289 ROTATION OF THE EARTH 289 gravity of the descending weight in its original position, and the axes have the directions: + x=south, +y=east, +z=nadir. 1. The following notations are used: o = angular velocity of earth.... 0'000073, q = latitude of the place of experiments = 41~ 54', g = acceleration of free fall....= 980 m., Y=,,,, retarded fall...= a + 1z. The fundamental equations for the relative motion of a body suspended from a string are, as deduced from Gauss and Binet: $ = - (g - y). x/z = + 2 cos. - (g - y). ylz.....................(1). f = + C The equations are only approximate: firstly, all terms with powers w2 are omitted; secondly, in the fractions x/z and y/z the denominator z should strictly be replaced by z + d, where d is the distance of the weight from the point where the string leaves the pulley. In the experiment the distance d should be kept as small as possible. A third approximation will be the assumption of a constant value for y. It was well known to Atwood that y increases constantly on account of the thread adding continually to the descending weight*. In his Treatise (1784), 110, he gives 7y the linear form: y = a + /z... (a and / positive constants)............... (2). There is no difficulty in integrating the third equation (1), as was known to Atwood from the writings of Euler. Yet the substitution of the exponential functions in the first equations of (1) would make the latter unmanageable. The assumption of some equivalent constant value y0 for z yields the following integrals: x=0 \ Y - 2 cos. t3.........................(3). Z — Zot2 z = 70+t Evidently, when yo approaches g, the equations will represent the free fall of bodies. 2. The hypothetical constant value y,, which will approximately satisfy the system (1), has to be determined from experiments by means of the last equation (3). A priori it would be impossible to say, for what value of z the experiment should be made. The safest way seemed to be, to deduce y, for a number of equidistant heights * In the forthcoming volume it will be shown how y can be kept constant during the fall. 19 M. C. II.

Page  290 290 JOHN G. h-AGEN and to take their mean. Accordingly the height of 22'96 m. was divided into 3 nearly equal parts, and the duration t of each fall observed. These are the results: 7-47 m. 19-94 22-96 t s. 6-52 8'71 10-48 Mean = To 0-352 rn. 394 418 0-388 im. Adopting y, = 0388 as the nearest approximation possible to an equivalent constant value of the acceleration, we can now compute the easterly deviation y from the second equation (3), as follows: 72 ~ = 0-014229 m., 0w cos = 0-00005426, 27o +. t = 10-48, t = 1151-0184, y = + 0-889 mnm. Incidentally, it may be observed that the three determinations of y7 serve also to compute the constants a and /. After the exact equation z = a + 3z is integrated, the three pairs (z, t) give three conditions, which yield, as the most probable solution: y = 0-315 + 0-036z. The variable y reaches the value y0 = 0'388 m. after a distance of only 2 metres. The main results may be tabulated in the following manner: Deviation By theory,, experiment South, 22 Exp. East, 66 Exp. x=0 x= + 0-010 ~ 0-027 mml. y= + 0889 mm. y = + 0899 + 0-027 rnm. Exp.-Theory + 0-010 mm. + 0-010 mm. Although the close agreement between experiment and theory seems to be partly accidental, yet it certainly is greater than it ever was in experiments with freely falling bodies. It is quite likely, therefore, that, after Atwood's machine has proved its adaptability to the experiment proposed by Newton, trials with freely falling bodies will not be resumed again.

Page  291 REZIPROKE KRAFTEPLZXNE ZU DEN SPANNUNGEN IN ETNER BIEGSAMEN HAUT VON WILHELM BLASCHKE. Nach einigen alteren Versuchen, die his auf Johann Bernouilli zurtickgehen, ist vor nunmelir etwa dreissig Jahren von Lecornu und Beltrami, die Statik der biegsamen und undehnbaren Haute entwickelt worden*. Ich m~ichte hier zeigen' dass man dieser nicht bloss ffir den Geometer sondern vielleiclit auch fdr den Techniker interessanten Theorie eine besonders durchsichtige Fassung geben kann, wenn man beMerkt, dass die in die graphische Statik von J. C. Maxwell eingefiihrten reziproken Krafteplane sich auch zur Bestimmung de Spnugn In eie iegsamen Haut anwenden lassen. Dabei komint man auch von einer neuen Seite aus auf die von L. Bianchi zum Studium der Flachendeformation benutzten " assozi'ierten " Fliichent. Denk~en wir uns emn StUck eines Polyeders etwa von lauter Dreiecksfldchen begrenzt. Sind seine Kanten aus einem starren Material ausgef-dhrt und in ihren Endpunkten gelenkig verbunden, so haben wir emn besonderes rdumliches Faehwerk vor uns, das A. F6ppl als emn Flechtwerlc bezeichnet. Wirken auf dieses Flechtwerk nur in den Eckpunkten seines Randes dussere Krdfte emn, so kann man ffir den Fall des Gleichgewichts zu den Spannungen in semnen Kanten einen reziproken Krdfteplan entwerfen, in dem die Spannungen durcb Strecken dargestelit sind, die zu. den entsprechenden Kauten. parallel laufen. Das Polyeder unterwerfen wir nun einem Grenztibergang, bei dem es in eine krumme Fldche tibergeht. Sehen wir das Polyeder als Fachwerk an, so miissen wir in der Grenze der Fldche notwendig folgende mechanische Eigenschaften zuschreiben: die v6illige Biegsamkeit und die Undehnbarkeit. Ftihrt man diesen Grenzprozess in geeigneter Weise durch, so geht auch das Polyeder des zugehiirigen Kriifteplans in emn Fliichensttick tiber, das man als Krafteplan zu den Spannungen in der ersten Fldche ansehen kann. Die Verhdltnisse liegen ndmlich, wie man das nach diesen Grenzbetrachtnngen schori vorhersieht, so. * L. Lecornu, "1Sur 1l6quilibre des surfaces flexibles et inextensibles," Journal de VE'cole Polytechnique, cahier 48 (1880), S. 1-109. E. Beltrami, "Sull' equilibrio delle superficie fiessibili ed inestendibili," Opere matematiche, 3. Bd, S. 420-464 oder IReidiconti del Istituto Lombardo, Serie 2, Bd 15 (1882), S. 217-265. Man vergieiche dazu such L. Bianchi, Lezioni di Geometria differenziale (Pisa 1903), Bd 2, S. 31 U. if. t Man siehe etwa Bianchi (Lukat), JVorlesungen fiber Differentialgeonietrie (Leipzig 1910), Kapitel 11. 19-2

Page  292 292 292 ~~~~~WILHELM BLASCHKE Es sei eine, biegsame und undehnbare Haut, nennen wir sie H, unter dem Einfluss ausserer Krafte, die nur an ihrem Rande angreifen, im Gleichgewieht. Dann kann man zur Flache H eine zweite Fliiche K mit folgenden Eigenschaften anifinden. Wir bilden H derart punktweise auf K ab, dass in entsprechenden Punkten die FlIhchennormalen parallel ausfallen. Will man nun zu. einem Linienelement ds der Haut H die hier wirksame Spannung angeben, so braucht man nur das entspreehende Linienelement d~ von K aufzusnehen: die Richtung von d2~ gibt bei zweckmiissiger Vorzeiehenwahl die -Richtung und das VerhhIltnis d~: ds gibt die Gr~sse der auf die Lingeneinheit bezogenen Spannung. So kbnnen wir also mit Recht K als reziproken Kriifteplan der Spannungen in H bezeichnen. " Reziprok " deshalb, weil die Beziehung zwischen den beiden Flichen H und K weebselseitig ist, man kann namlich in derselben Weise auch H als Kriifteplan zn K ansehen. Nimmt man H als gegeben, K als gesuelit an, so ergibt sich ftir K eine partielle Differentialgleichung zweiter Ordnung, die auch in Hilberts Untersuchungen iiber Integralgleichungen eine Rolle spielt. Die auf den Rand von H wirkenden ausseren Krdafte liefern (Ilie zugeh6rigen Randbedingungen. Ans dieser partiellen Differentialgleichung ersieht man aber, dass K niehts anderes ist als die assoziierte Flache einer geeigneten " infinitesimalen Verbiegung " von H. Es sei darauf hingewiesen, wie man diese assoziierten Flachen Bianehis, fair die wir eben eine statische Dentung geftinden haben, audi in einfacher Weise kinematisch erkiaren kann. Denken wir uns die Fliehe H stetig ohne Dehnunug und ohne Knick verbogen und greifen wir einen Atigenblick dieses Biegungsvorgangs heraus. Ist mit einem Flachenelement von H emn starrer K6rper verbunden, so ftihfrt er in dem Augenblick eine Sehraubung aus (die " tangierende " Schraubung). Die mit dieser Schraubnng verbundene Drehung stellen wir in iiblieher Weise durch einen Vektor dar, der zur Drehaehse parallel lauft. Zn. jedem Pnnkte von H geh6rt in dieser Weise ein Vektor. Alle diese Vektoren tragen wir von einem festen Punkt aiis an, dann erfiillen die Endpunkte im Allgemeinen eine Flache und die ist die assoziierte Flache K der Biegung von H. 1st H eine geschlossene konvexe Flache, so reduziert sich, wie man nnschwer nachweisen kann, die zugehiirige Flache K notwendig anf einen Punkt. Dentet man diese Beziehung (zum Beispiel) kinematisch, so findet man den Satz von Jellett und Liebmann fiber die Lfnverbiegbarleeit einer geschlossenen kconvexen Flache. Einige Formeln werden die vorgetragenen geometrischen und mechanisehen Tjberlegungen deutlicher machen. Nehmen wir an, unser reelles Flachensttick H hainge einfach zusammten und werde durch parallele Normalen eineindeutig auf die Kugel abgebildet. Nennen wir die Richtnngskosinus der Norinalen X, Y, Z und die Entfernung der Tangentialebene vomn Koordinatenanfang p, so k~innen wir die FlAche durch eine soiche Gleichung darstellen p =H (X, Y, Z). Dabei bedeutet es keine Einschrdnkung, wenn wir von der Funktion H annehmen, es sei H (ex, c Y, CZ) =CH (X, Y, Z)

Page  293 REZIPROKE KRA~FTEPLA~NE ZU DEN SPANNUNGEN IN EINER BlEGSAMEN HAUT 293 ffir alle positiven c. Emn Punkt von H hat dann die Koordinaten x = H, y==H, z-=Hz wenn die Mai-ken partielle Differentiationen andeuten. Wir setzen das Fliichensttick H als biegsam voraus-die Undehnbarkeit ist unwesentlich-und nehinen an, II sei im Gleichgewicht unter der Einwirkung iiusserer Krafte, die tangentiell am Rande von H angreifen. Es sei nun S die Spannung mit den Komponenten d.~ds, d-/ds, d~/ds, die auf das Linienelemient ds von H wirkt. Soil Gleicligewiclit herrschen, so niifssen Ilangs jeder gesehiossenen Kurve auf H die drei Integrale Null ergeben. Erstreckt man daher diese Integrale von einem festen Punkt P0 zu einern veranderlichen Punkt P der Flache H, so sind die Integrale unabhangigIvom Weg und nur abliangig von der Lage des Punktes P. Beschreibt P die Flache H, so durchlauft P {~Z, ~, ~} eiue Fldche 1if Da die Ricbtung {di?9 d d} der Spannung S in der Tangentialebene an H liegt-gerade dadurch drtiekt sich die Biegsamkeit der Fldche aus-so sind die Flachen H und H so auf einander bezogen, dass in entsprechenden Punkten P und -P die Tangentialebenen parallel sind. Deshalb kann man H analog wie H so darstellen P=H (X, Y, Z). Die partielle Differentialgleichung ffir die Flache H, die nichts anderes ist als der Krafteplan K zur Haut H, gewinnt man durch folgende Betraclitung. Langs jeder gesehlossenen Kurve auf H mtissen die Integrale tiber die Momente der Spannung um die Koordinatenaehsen x* =fd~i-zd~, i*= fd& - x dzi, z* fd~ -y d., verseliwinden oder, was dasselbe bedeutet, diese Integrale sind erstreckt zwischen den Punkten P0, und P von H unabhaingig vom Integrationsweg. x*, y*, z* sind also Funktionen der Richtungskosinus X, Y, Z der Flachennormalen in P. Die drei Integranden oder die zweireihigen Determinanten der Matrix Hx Hxd xd xd Hy Hyd yd yd Hz HzxdX + HzydY + Hz~dZ mtiissen vollstaindige Differentiale sein. Aus der Homogeneitat von H und H folgen die identischen Beziehungen I~yz 4zy + HzzHyq Hxyzx Hzx y - HxxHyz -HyzHxx xx Yz Hz~x- 2Hxz lxHzz~ Hvzx 4- Hxy~ - Hy~ - Hxy yy zx - xx H,,, - 2Hxy Hxy ~Hyy xx zx Hz 41 Hz - Hzz Hx - Hx R, zz x Y

Page  294 294 ~~~~~WILHELM BLASCHKE Ftihrt- man naeh Hilbert ftir diese sechs gleichwertigen Ausdrttcke die Abkttrzung (H, H) emn, so reduzieren sich die Integrabilwittsbedingungen auf die einzige Gleichung (H, H) =0o. Deiselben partiellen Differenbialgleichung fuir H gentigen aber, wie ich gezeigt habe*, die den IBiegungen der Flache H assoziierten Flachen. Zum Schluss sei noch erwiuhnt, dass man auch die voin Punkte P* jx* y*, z~j beschriebene Flache, die auf H1 unter Orthogonalitat entsprechender Linienelemente bezogen ist, in gewissem Sinne als Kriifteplan ftir die Spannungen der Haut H verwenden kann. Derartige KrAfteplane sind fuir Fachwerke ktirzlich von L. Henneberg betrachtet wordent. *Emn Beweis fuir die Unverbiegbarkeit geschiossener konvexer Flkchen, G~Jttinger Nachtrich~ten 1912. U iber das Glcichgewicht an Seilnetzena..., H. J~eber-Festschr'ift (Leipzig 1912), S. 111-129.

Page  295 EXTENSION DE LA NOTION DES VALEURS CRITIQUES AUX EQUATIONS A QUATRE VARIABLES D'ORDRE NOMOGRAPHIQUE SUPERIE UR PAR FARID BOULAD. Dans notre me'moire intitule: "Application de la notion des valeurs critiques a la disjonction des variables dans les 6quations d'ordre nomographique supe'rieur" (Bull. de la Soc. Math. de France t. xxxix. 1911, p. 105), nous avons expose une m'thode permettant, an moyen de cette notion, d'effectuer la disjonction des trois variables de 1'e'quation F223 = 0 * en vue de la construction d'un. nomogramme 'a points aligne's, dans le cas oii cette equation se pre'sente sous la forme pratique ge'neralet F1. F23 + GI. G2-3 + [II11123 =0. Nous nous proposons ici d'e'tendre cette notion tre's f~conde, due 'a M. le Professeur d'Ocagne, 'a la resolution du proble'me de la disjonction des 4 variables z1, z2, Z3, z4 de 1'e4quation F2234 = 0 en viie de sa representation par un nomogramme a double alignement, dans le cas fr4equent dans la pratique, oii cette equation est mise sons les denx formes suivantes nomographiquement rationnellesli. F1234 = F, F234+ GI. G234 + HI H 1234 = 0) F1234=F4 F123+ G4. G123 ~H4.HN3 = Of.(1)..... D'apre~s le principe du double alignement e'tabli par M. d'Ocagne~, pour que le proble~me ci-dessus soit re'solu, il suffit de determiner deux syste~mes de trois fonctions (F2., G2, H2) et (F3, G3, 113) des deux variables Z2 et z3 et un syste~me de trois fonctions (F0,, G0, H.) d'nne senle variable anxiliaire z0, tels que e'6quation propose'e (1) puisse e'tre obtenue par 1le'lirnination de cette variable zo entre les (Ieux determninants F, GI H, F,, G4 114 F2 G2 H2 =0....(2), F., G, 13 =0......(3). F0 G0 Ho F0 Go Ho *Nous employons ici la notation de MV. d10cagne qui consiste, lorsqu'on affecte un indice 'a chacune des variables zj, z2, Z3,..., a placer 'a la suite do chaque signe fonctionnel les indices de toutes los variables sur lesquelles porte ce signe.I t Cette me'thode a fait pr6c6demment l'objet d'une communication par nous 'a l'Acad. dos Sc. do Paris (Comptes rendus du 14 Fevrier 1910, p. 379). + 13Si, toutefois, 1'6quation F 24Oso pre'sente seulement sous l'une dos deux formes ci-dessus (1), la notion des valeurs critiques peut S'tre utilis6e aussi d'une fa~on plus ge'n6rale pour effectuer la diojonction des variables do cette equation, ainsi quo nous 1'avoliz, indique' dans notre recent m6moire "ISur los 6quations 'a 4 variables d'ordro nomographique sup6rieur" (Bull. de la Soc. Math. de France t. XL. 1912). ~ Trait6 de Nornographie par M. d'Ocagne, p. 213, et son Cours de Calcul Graphique et Nornographique, p. 305.

Page  296 296 FARID BOULAD Or, remarqnons que si flous conside'rons les deux formes (1) comme e'tant les deux de'veloppements respectifs des deux determinants (2) and (3) par rapport aux elemnents de leurs premieres lignes, nous, pourrons admettre que chacun des elements Fo, Go0, Ho figurant dans le determinant (2) est une fonction des deux variables z, et z4 et que chacun de ces i-nemes elements figurant dans le deterininant (3) est une fonction des deux variables z3 et2 z; avec la condition que, si l'on e'crit, d'apre's cette remarque, les deux determinants ci-dessus comme suit: F2 GI HI < F4 Cr4 14~ F1234= F2 Cr2 H2 =0....(2'), F2234= F3 G3 H3 O....(3') F34 G34 HL34 F'12 G12 1112 ii fant que l'on ait l'nne ou lantre des conditions sufivantes: 10ou~o bien F34 G34 H_ 4 1134 F32C12 G 12H2.() 20-ou bien, en rapportant 'a un me'me syste'me d'axes (Ox, Qy) les deux nomogrammes a simple alignemient repre'sentatifs des deux equations (2) et (3), il faut que les equations des supports des deux e'chelles (zo) de ces deux nomogrammes soit identiques afin (4ue lon puisse superposer ces deux nomogrammes partiels en faisant conucider les supports de ces deux e'chelles sans se preoccuper de leur, graduation. De cette fa~on on obtiendra un nomogramme 'a double alignement repre'sentatif de I equation (1) ayant comme charnie're le support commun des deux e6chelles (zo). L'e6quation du support de e'6chelle (zo) du nomogramme (Z1 z2 z11) s'obtient en e'liminant (23) et (z4) entre les equations suivantes: X= F34, y = Cr34, z =134.............(5), de'flnissant cette e'chei~le en coordonne'es cart6siennes et homoge'nes. De me'me l'equation du support de le'chelle (zo) correspondante an second nomogramme (zoz3z4) s'obtient en e6liminant z1, z, entre les equations snivantes de cette dernie're e6chelle, X = F32, y=-G12, =112.........I.....(6). 1i est inte'ressant de remarquer que chacune des deux conditions (4) n'est autre chose que e'6quation (1) misc sons la forme CPI2 = (P34. Cela pose', les deux syste'mes des fonctions inconnues (,F-, Cr2, 112) et (F,,, G34, H14) s'obtiennent en resolvant les deux identite's snivantes: F2. F234 + G2. G2334 +112. 11H234 0 (quels que soient z. et z4)....(7), F34. F234 + Gr34. Cr234 +H134. 11234 0 (quel que soit 23).........(8), lesquelles avec le'quation (1) donnent par elimination le premier determinant cherche' (2'). De me'me, les deux syste'mes des fonctions inconnues (F3, Crs, 113) et (F2,~ G1~2, H12) sont fournies par les deux identite's suivantes: F3. F123 + G3 CrG123 +H13. 11223 0 (quels que soicnt zi et 22)......(9), F12. F123 +Cr2. CrG123 +H1122 H133 0 (quel que soit 23).........(10),

Page  297 EXTENSION DE LA NOTION DES VALEURS CRITIQUES29 297 lesquelles avec le'quation (1) donnent e6galement par elimination le second d6terminant (3'). La resolution des identite's fonctionnelles ci-dessus s'effectue ais'ment par la m'thode de la determination des valeurs critiqe exose dans nos deux nm'moire precite's. Remarque importante. Ii arrive dans la pratique que les 6quations des supports des deux 6chelles, (zo) des nomogrammes (z1 z2 Z) et (zo zz 2) rapport's 'a tin meme syste'me d'axes sont re'elles et homoge'nes en xy z, mais non identiques. Dans ce cas, si ces eqjuations repre'sentent des droites ou des coniques, ces deux nomogrammnes pourront etre toujiours superpose's d'une infinite' de manie'res en ayatit recours an principe de l'homographie. Pour cela, il suflira de faire subir respectivement aux deux determinants g'ne'rateurs (2') et (3') les deux transformations homographiques les plus generales obtenues par la multiplication de ces deux determinants respectivement par les deux determinants transformateurs suivants: ~~~a/3'" =1=0, A2-lX& =10. En identifiant les deux equations obtenues pour les, deux 64chelles transforme'es (z,,on aura les valeurs des parame'tres c, /3, v. ", / qui re'alisent la coincidence de ces deux 6chelles et par suite la representation par double alignement de l'equation donne'e. Application. Equation a' 4 variables d'ordre nomograpitique total 60 Ia plus ge'nerale. Cette equation s'e'crit sons la forme suivante: f, ( f4A23 + g4B23 + h4Cl23) +gI (f4A23' +g4B23'+ h40C23')~ h/if4A23/' +g4B231'+h4C2311)=0 (11) d'ordre 2 par rapport 'a chacune des deux variables z, et 24 et d'ordre 1 par rapport a chacune des deux autres variables 22 et 23, et dans laquelle on a d' une manie're g~ne'rale A 23 = alf2f3 + a2 f2 + a3f3 + a4, B23 = bifif3 + b2f2~+ b3f3 +b4, 023 = C1 f2f3 + C2f2 + C3f3 + C4, C2 I= c111 f~fl + C2"fi + C3"'fi ~ C4", f2 et f3 de'signant des fonctions quelconques des deux variables 22 et 23 et a1, a2,... C3"1, 04" des constant es quelconques. Cette Meme equation pent s'e'crire aussi sous la 20 forme, f4 (f1A23 + giA23' ~ hIA23"') + g4 ( f B23 ~glB23'-+ hlB23"')+h4 (fI 23 -igl 03' ~h1 023")= 0 (12). Cherchons par la m6thode ci-dessus "a mettre les deux formes (11) et (12) de cette equation respectivement sons les deux formes des deux determinants (2') et (3'). Pour cela prenons les elements: PF, =/ji, G, gi, HI~ =hi F4f4, G4=g4, 114= h4.

Page  298 298 298 ~~~~~FARID BOULAD Pour avoir les elements (F3, G2, H2) du d4eterminant (2'), substituons ces lettres respectivement 'a fi, g,, h, dans la forme (1 1). Ensuite ordonnons nomographiquemnent par rapport aux 6 quantite's on groupe de quantite's f~4 f~g4, f3, f4, g4) h4. En exprimant que l'identite' obtenue a lieu quels que soient Z2 et Z3 nous aurons un syste'me de 6 equations line'aires et homoge'nes en F2, 02, H, que nouls appellerons (:~). Cela pose', pour determiner aussi les e'leinents F34, G314, H34 substituons ces derniers respectivement 'a fi, g,, h, dans le'quation (1 1). Puis rendons inde'termin~ee la variable z, dans l'identite' obtenue, nous aurons un syste'me des deux equations line'aires et homoge'nes en F3, G34, H34 que nous appellerons (4)). A present, posons d'une manie're generale Ai= aix + ai'y + ais", Bi = bix~+ bi'y + bill, Ci = cix + Cl~y + ci". Je dis que, si les rapports B1 C, _ A2 _ B2 _ 0 sortt identiquernent eigaux, quels que soient x et y, les e'quations du syste'me (1) sont compatibles et les e'chelles (z2) et (z.) du nomogramme (z~ ~zz0) correspondant a' la forme (11) sont situe'es sur une me~me conique deitnie par 1'e'quation Al B, 1) A3 AI~(1) En effet, substituons, dans le syste'me d'e6quations (Z), x, y, 1 respectivement a FG, 0,12. Puis e'liminons f, entre la preminire equation de ce syste'me et chacune des 5 autres, nous aurons e'~galite' des 6 rapports A1=B3=Ci A2 B2 _ 2 A3 BA 07A4 B4 04 Or, en remarquant que chacune des cinq relations qu')on obtient en 6galant l'un quelconque de ces 6 rapports, le premier par exemple avec les 5 autres, doit repr&' senter le'quation du support de 1'4chelle (z2), il re'sulte que pour rendre compatible le systeme d'e'quations (Y.), il faut et il suffit que l'on rende identiques ces cinq relations. En effectuant ainsi cette identification, on obtient pre'cise'ment les conditions (1 3). D'autre part, substituons w, Y) 1, respectivement, "a F24, G34, H.,, dans le syste'me d equations (4)). Puis ordonnons par rapport aux 6 quantite's ci-dessus f. 4 f3g4, f3, f4, g4, h4.En ayant e6gard aux conditions (13) on trouve imme'diatement pour le support de I echelle (zn) la me~me equation (14) que pour le support de e'~chelle (z2). Done ces deux e4chelles (z,) et (z2) sont situ6es sur une me'me conique. A present, si l'on cherche les elements F3, G3, H3 et F12, 032, H12 du determinant (3') correspondant 'a la forme (12) et si l'on pose d'une manie're generale, Ai= ai x + biy +0c,> Ai= ai'x + bi'y + ci,f Ai,= aij'x + billy + ci',/

Page  299 EXTENSION DE LA NOTION DES VALEURS CRITIQUES29 299 on de'montre, par une inarche analogue 'a la precedente, que si les conditions Al A11 L\.3 A3' ZA3" __.~~~..............(15) sont remplies, e'~quation (1 2) pelt eftre misc sous la forme du determinant (3') et que les deux 6cehelles (z.) et (23) du nomogramme (ZI 23 Z4) correspondant 'a ce determinant sont situe'es sur une meme conique de'finie par le'quation, Comme les deux e'chelles (zo) correspondantes des deux nornogrammes partiels (ZI Z2 ZO) et (zII Z3 Z4) ont leurs supports conicjues il re'sulte, en vertu de la Remarque ci-dessus, que ces deux nomogrammes pourront e~tre superpose's par l'homographie de fa~on 'a obtenir un. nomogramme 'a double alignement representatif dle e'~quation (11). De la', le th6ore'me suivant: Si les conditions (13) et (15) sont satisfaites, l'e'quation (11) a' 4 variables d'ordre total 60 la plus ge'nerale est repre'sen table pa'r un nornogramme a' double alignement sur lequel la charnie're est une conique servant de support commun aux deux e'chelles (22) et (23). Exemple pratique: Comine cas particulier de ce nomogramme signalons celni construit par M. Wolff pour la representation de e'6quation suivante: V2 bd +d2 4O2 s3 b + V2d (Gours de Galcul Graphique et Nomographique, par M. d'Ocagne, p. 3.32), qui donne la vitesse d'e6coulement V d'un. canal 'a section trape'zoidale avec talus 'a 1/i en fonction de la largeur au plafond b, de la hauteur d du plan d'eau et de la pente longitudinale s.

Page  300 INTEGRALS IN DYNAMICS AND THE PROBLEM OF THREE BODIES BY SELIG BRODETSKY. ~ 1. The general dynamical problem is defined in the following manner. We have n "coordinates" qj, q2,..., qn; n "momenta"p pP, p,..., p; and the independent variable t. Between these variables we have the Hamiltonian system of equations: dqr_ H. dpr_ - H. dt ~= Apr; d~tt ~; (r= 1, 2,..., n)......... (1); dt apr dt aqr H is a given function of the 2n +1 variables. The problem is to find a function dW W(ql, q2,... qn, pI, p2..., Pn, t) such that d = 0 in virtue of the equations (1). Thus if W=const. is an integral we have 9W W fa dqr a+W dpr at r=l \ardt d r dt = At r=l aqr a pr apr afr i.e. 3t + (W, H ) )= 0..................................... (2), using the Poisson bracket expression. dH aH H9 This gives dt - t (H, H) a. dH Thus if H does not involve t explicitly, d- =0, and H= const. is an integral, the dt energy integral. ~ 2. H can involve the momenta in any manner we choose. One case is however of great importance in natural problems, namely when H is a quadratic function of the p's, involving only squares; i.e. H = 1Pp12 + P2p22 +... + Ppn2p + P............. (3), where PI, P1,..., Pn, P are functions of qj, q2,..., qn, t.

Page  301 INTEGRALS IN DYNAMICS AND THE PROBLEM OF THREE BODIES 301 When H has the form given by equation (3), we can prove an elementary but fundamental property of the integrals. Let W = const. be an integral not involving dW one of the momenta, say pi. dt = 0 gives aw a3W3 a H WH H\ at +aq apql rp - \aqr apr a ar aw aUW a W i1 a, IaP P a laT\ W i.e. — t +PpAl +.. P -- 4 q+ -+ apn2 + -- - 0. 8at a1, 1=2 8 8, L2 a'" 2 8r ar_] a-pl Now pi occurs to the first power only in the term Plpl a. Otherwise it occurs either in the form pi2 or not at all. Thus we must have P1 -=0; Wa i.e. - = 0, and W does not contain ql. In other words if an integral of a system (3) can be found which does not contain a certain one of the momenta, then it does not contain the corresponding coordinate. The converse is of course not necessarily true. ~ 3. This result is true of the problem of three bodies reduced by means of the integrals of linear momentum. We here have n = 6; and H=T- U, where T = (2 + p2 + p) + (p 342 + p52 + 62); U = mlm2/r2 + nnr3 + m /rl + mm/r23; and rl2 = (ql + q2 + q2) r,= q2 + q2 +q62)+ (qlq4 + q2q5 + q3q6) + -m --- (q2 + q22 4-q r3 = (q42 + q52+ q6 2)- + (ql q4 + q2) ( + q3q6) + (q + q+ 2)}; r:(+ + )-+ +3)+M + M. M+ +D n + m m = ml mIn/(im + nt2; fu = m3 (ml + m2)/(mn + mn + mn3); ml, m2, m3 are the masses of the bodies. H is of the form given by (3). The result of ~ 2 holds, and if W = const. is an integral not containing pi, say, then it does not involve q,. The differential equation for W is now aw p2a3W paW p4aW +palW p.3W at / aq2 /Z a8q3 A '4 a, 8aq.5 aq6 a_ W mm2q2 Mil ( m '12 ( 2 ) q- ap2 Lr12 r133 iml + 2 I + m2 + m q2 q5 r2 n M, + in2 (,n M m YLr~~a (itm+? Z'- m, P~

Page  302 302 SELIG BRODETSKY aw r1n, yq3 mi f/ m2 \2, M2 W fre me r + r13 + m ( 2 q3 + q6 r2m,3 r3m, m, ) Ml+m, ) r233 (\l+ q3 ml + m2 aw I rns n3 (m \ m ml _ a XL ~r a {(m-+ m ql + q4 + r 3- {(m mql '+ -q4 ap4 [ r133 KmI+2 r23+3 { +)m2 awr MOln3 f( m, m2m3 ml ~ I[ *r r:,.3 " q2 + 16 + 0 + -,,Li.,-Tm(,, q: - + " +... a-P i r1' {(ml3? + m2) } r233 (m + M21 }]........(4). qi is present only in the square brackets and in terms of 1. ql. 1. ql. 1 r123 r13 13 r 3 ' r23 23 r12, r.2, r3l cannot make one another disappear in any manner; they are also all irrational to qj. Thus the five forms in which q1 appears in (4) must separately aw aw vanish. Hence a = 0, which also gives - = 0. Further we get ap4T ---— 4 aw aw q2 - + q3 = 0_...............................(5). m. \/ Ml W aw aw aDW 3W\. aW "a TV and ( ml \ )( w aw aw aw aw aw and - ) q5 a- + '2 4 p + q3=-6- + 5 + q6 - = 0. nt\W ina + aq ~ + 5 3 q6 P 05.ps These two equations are equivalent to aW aW q5ap- + q6.. = (60.)............................(6); 9W 9W 9W 9W and ~aw aw aw a.() and ap, + ap5 +ap +qp 0.....................(7). Finally we have the equation aW p,2W pWpW paw (8,aw) a_ W -- -p2aW + PlaW- aW paW 0................... at J taq2, Zaq3 /L aq.5 PL aqi Equations (5) and (6) give us W = W' (q23 - 3p23, q5,p6- q6p5, q.2, q3,, q, t). Substitute in (7). We get aW' W' -=0; a (q2p3 -q3p2) a (q5p6 - q6P5) hence W = W" (q2p3- q12 + qp6 - q6p5, q2, q3, q5, q6, t) = W" (L, q,, q5, q3, t),

Page  303 INTEGRALS IN DYNAMICS AND THE PROBLEM OF THREE BODIES 303 where L= const. is that integral of angular momentum which does not contain p,. Substituting in (8), we get aW" pa W" pW"\ p(W" p W I' 9t + '/M (-8 + ps3-AL) * i,, ( 9q3 - p2 3L ) i.e. d~ p/ + + pa- + 0 at I aq2 P aq, PI aq5 d aq6 in which we look upon L as a constant. This equation is possible only if aW" aw" aW " sw" aW" - ~ = - '- -" _ "-= 0. at aq2 aq3 aq5 aq6 Hence W = W" (L). Thus an integral of the problem of three bodies in the reduced form given above, from which p, is absent, is merely a function of that integral of angular momentum from which pl is absent. The same is true for any one of the momenta. It is seen that the momenta are associated in couples. When p, is absent, then p4, as well as q1, q,, is absent, and the integral is a function of L. When p2 is absent, then ps, as well as q,, q5, is absent, and the integral is a function of M, where M q3p - q1p3 + q4 - q4p6-. When p3 is absent, then p6, as well as q3, q6, is absent, and the integral is a function of N, where N qP2 - q2P + q4p5 - q5p4. There are no integrals in which momenta from more than one set are absent, e.g. p, and p2, or p3 and p5, etc. It follows that integrals independent of the integrals of angular momentum must involve all the momenta. The argument here given applies equally well if we are given that W= const. is an integral from which one of the coordinates, say qj, is absent. We get q, present in terms of the form ql. 1 ql. q. 1 12 123 13 r13 r23 r23 To make the term in gt- go out, we must make a =0, and the same results follow. r1.23 api Thus if q1 or q4 is absent from an integral, then the integral is a function of L; etc. It follows that integrals independent of the integrals of angular momentum must involve all the coordinates. Hence we have the important result: In the problem of three bodies, integrals independent of the integrals of angular momentum must involve all the coordinates and all the momenta. This is true of n bodies, n > 3; also for all laws of force which give U in odd powers of r12, r 23, 723; i.e. for which the law of attraction is according to a positive or negative even power of the mutual distance.

Page  304 304 SELIG BRODETSKY ~ 4. The energy integral is an integral independent of L, M, N. It is seen that it involves all the coordinates and all the momenta. Do other such integrals exist? Bruns has shewn that when the coordinates, the momenta and the time are all supposed to occur algebraically, then no integrals exist independent of the energy integral and L, M, N. Painleve has extended this to include integrals in which only the momenta are supposed to occur algebraically, the coordinates and the time occurring anyhow. We shall here consider the more general type of integrals in which only one of the momenta is supposed to occur algebraically, the other momenta and the coordinates and the time occurring anyhow; we shall however restrict the occurrence of the single momentum thus considered to even powers only. Moreover we shall consider not only the problem of three bodies and the allied problems mentioned at the end of the last article, but also all problems defined by H = Pp2 + P2p22 +... + P...............(9), where P1, P2,..., P are constants, and P does not involve the time. There exists then the energy integral H= const. As in Bruns' theorem we need only consider integrals rational and algebraic in pi, the momentum supposed to occur algebraically. We write then W - W,/W2= const. is an integral, where W1, WV are rational and algebraic in p,2 containing only zero and positive powers. We say p,2 because we suppose pi to occur only in even powers. ~ 5. We suppose then that W - W/ W2 = const. is an integral of a system given by (9). We can suppose that W contains no factor H in the numerator or the denominator, for if H does occur as a factor we may put the energy constant for it and the result equated to a constant must still be an integral. Now we can write W1 - HW1'+ W1 WW2 - HW' + W2; where W1' W2' are all of the same form as W1, W2, but containing pi two degrees lower respectively, and where W14", W2" do not contain pi at all. We have dW -0; dt d i.e. dt [(H W1' + W,")/(HW,' + W,")] = 0; d d i i.e. (H W' + W2") -t(HW' + W,") - (HW,'+ WT,') (HW2,' W") = 0; i.e. W21 _ t " d = H dt dt-H

Page  305 INTEGRALS IN DYNAMICS AND THE PROBLEM OF THREE BODIES 305 multiplied by a certain function, since dH 0 dt aH aP aH aP Now, in (9), - aq' aq2 - etc. - dw a3w ( a q w aw aaP Hence d- = -a — + E Prpr d-t = ) dt at q,8 ap 8/-q dw so that if w does not contain pi, dt is linear in pi. Further WT", W2" do not dt dW/ ' d WI I ' dW' contain pi. Hence dt WI dt is at most linear in pi. It cannot then be a multiple of H unless it vanishes. Thus we must have d dt( W," /w") = o. Hence W'"/ W" = const. is an integral. But it does not contain pi. Thus Wi"/ W" is a function of those independent integrals which do not contain pi. Call this function L1. We get W = (HW,' + L1 W2")/(H W' + W2") = L, + H(W1' - Li W2')/(HW' + W2). W = const., L = const. are integrals. Hence H (WT' - Li W2')/(HW' + W")= const. is an integral, i.e. (W' - LI W2)/(HW' + W") = const. is an integral. Call it W = const. Suppose W1 is of order 2n in pi, W2 of order 2m. We may suppose 2n >2m, since if this is not so we can use the inverted form of W for our integral. W' now has pi2n-2 at most in the numerator, and p,2m in the denominator. If pi still occurs to a higher degree in the numerator than in the denominator, we repeat the above process for W'. We get, say, W' L2 + H W", where L2 is a function of integrals independent of pi, and W" = const. is an integral with pi211-4 at most in the numerator, and p12m in the denominator. We repeat the process till we have reduced W to the form W= L + H(L+ H (L3+ H (... + H (L+ + HW(A))...))), where Wt() = constant is an integral with p12m in the denominator and lower powers of pi, say up to p'f21' in the numerator. We invert W(X), and we reduce it by the same process to 1/WA" = Lk+i + H(L,+2 +... + H(L, + HW ))...), where W() = const. is an integral with p?21' in the denominator, and lower powers of pi, say up to p2'"T in the numerator. We invert W() and repeat the process. Proceeding in this manner we must ultimately get W in the form TW=Li+H L2+... +H L A(,+ H...H. VLV LA+H Lk-(LA + H+WL,+ I + ).. + H W(V) M. C. II. 20

Page  306 306 SELIG BRODETSKY where W( = const. is an integral not containing pi at all, i.e. W(v is a function of the independent integrals independent of p,, say Lv+l. We get then W in terms of H and the functions L1, L2, A, +..., L,, L+, L,+l,..., L~, L,,+1; i.e. W is a function of the energy integral and those independent integrals which do not contain pi. This is true for any momentum. If two momenta occur in the manner indicated, then W is a function of the energy integral and the independent integrals independent of these two momenta. Similarly for any number of momenta occurring in this manner. If all the momenta are involved algebraically and only in even powers, W is a function of H only, because integrals independent of all the momenta reduce to mere constants. In these results if a momentum does not occur at all, it is equivalent to its occurring rationally algebraically in even powers. Thus in problems defined by (9), after we have found the independent integrals in which one or more of the momenta is absent, then there are no new integrals of the type considered except the energy integral H=const. ~ 6. We apply this to the problem of three or more bodies with the proper laws of force, the Newtonian being one. Here the only integrals in which one or more of the momenta is absent are L = const., M = const., N = const. Thus if p, occurs in the manner indicated in ~ 5, W is a function of H and L; the same holds for p4, and in fact the property is reciprocal-if pi occurs in the prescribed form then p4 occurs in this form, and vice versa. Similarly for the couples P2, p5 to which corresponds M, and for P3, P6 to which corresponds N. If two momenta out of different sets, say pi and pa, occur in the prescribed manner, then the integral is merely a function of H, and then all the momenta occur in this manner. The conclusions here arrived at are of great generality, and should throw light upon the unknown integrals of the problem of three or more bodies. ~ 7. We consider once more the problems given by (9). Suppose that the power series in pi W _ Asp1i + Apl~n- +... + Ap.. + A0 = const..........(10) is an integral. We have dW dA, dAn dA_ dAo dt — pl d+p1 -+.. dt- + d-t P)-P aP ap - nPin-iAn- - (n Il - pi a "2A - A - aq, a(n ap dA, dA, d d, A Now An, An_1,..., AD, Ao contain no pl. Hence dA, d are, dtr' dt. d

Page  307 INTEGRALS IN DYNAMICS AND THE PROBLEM OF THREE BODIES 307 using (9), linear in pi. We can write dA. a' An dAn aA}; etc. dt =Pl q + {dt - q; etc. We get dW aL n a _dA aAn ____ dA,1 _i __ It -pin Plplq + - -Pipi Ij ] + pi LPnP1 p A + dA PlPl dPl + ( a- -PPl a- ) aj +' "+ PPa- + J.- iP ai - [nP;-I A +(n-j)P/-~A + [ +AAl]pP -. The expressions in the curly brackets contain no pi. We now equate dW to zero. dt Equating coefficients of powers of pi we get the results: [an.t2]- Pp D-A = 0; \ dA, P aAnan-1 -0 * [an+,] = Y - 1j i - + PIp aql =O; dAn- 1p 8 i p aAn2 a [am] -{ d ]- pl-1 - -l +4- Plpi -- - n A -= 0; etc.; a] - dt 'paq, aq a [an] dAm-1 _ aAM-1J f l~p aAmA 2 aP [am] {d -Pp l +P aA, e - =0; etc. dt liAq1 }+Pi 0 [ai] dt - 2AA =0; [al dAo aA-l = - ldt aT- aqq We get [al] as a case of [am] by using the value A_ =0. In the same way [a,+1] and [an+2] are cases of [am] in which A,,+ and An+2 are taken to be zero. [a~+2] =0 gives us An independent of ql. [a,+] =0 gives us by a partial integration, An-_ in terms of An. [an] = 0 gives us A_2 in terns of An and Ani, i.e. in terms of An. Thus working backwards till [a2] = 0 we get Ani, An_,..., Al, Ao all in terms of An. The last equation gives another condition which An must satisfy besides being independent of qj. We can use the equations (11) upwards. [al] = 0 gives us A id^o_ p MO 'a.o -Al~ Plpl t ' ~ _ / r3) SIaql' [a] = 0 gives A, = |_t- PI l + 1 PP 2 J / q 1; i.e. we have A2 in terms of differentials of A0. Similarly we get As, A4,..., An in terms of differentials of Ao, given by the equations [a3] = [a4] =... =[an] = 0. The two equations [a,,0,] =[a,,+2]=0 give us two conditions which A0 must satisfy in order that W=const. may be an integral. The conditions [a,,+,] = [a~+2] = 0 are really equivalent to getting An+, = An+2 = 0 from the general equation A [, = dAml, P Am-l} ai ' m (12). AM = dt PIP1aql I + pi aql aqi......1 20-2

Page  308 308 SELIG BRODETSKY Thus if A0 is chosen such that on obtaining A,, A2, etc. from equations of the type (12) we get two consecutive A's, say An+, An+2, zero, then A0 will "generate" a finite power series which is an integral of the system. We may call A0 a "finite generating function." If Ao is such that we never get two consecutive A's vanishing then we do not get a finite power series in p, as an integral. But assuming convergency, we can obtain an infinite power series which equated to zero will give us an integral. Any value of A0 will lead to such an infinite power series, and may be called an "infinite generating function." But A0 must not itself be an integral. For if A0= const. is an integral, then as it contains no pi, it also contains no qj. Thus dAo= - =0 We get dt aq- - AI = A2 = A, = etc. all zero. Thus we do not get a power series at all. In particular A0 must not be a mere constant. Thus we cannot have Ap, + A2p2 +... = const. as an integral, whether as a finite or as an infinite power series. It is obvious that the algebraic sum of any number of generating functions will be the generating function of the algebraic sum of their integrals, whether finite or infinite. Further, assuming convergency, the product or quotient of two generating functions will be the generating function of the product or quotient of their integrals. This can be extended to any number of operations. Thus we get the general result that any algebraic (or other well-behaved) function of generating functions is the generating function of this function of their integrals. We assume the necessary convergency. ~ 8. It seems that this result applied to the problem of three or more bodies with the appropriate laws of force, should enable us to extend Bruns' and Painleve's results mentioned above (~ 4) to shew that integrals independent of the classical integrals must contain all the coordinates and all the momenta, not one of the latter occurring algebraically. I have not yet, however, been able to construct a perfectly rigorous demonstration. ~ 9. We now consider the general theory of integrals in dynamics. If the system is defined by dqraH. dpr aH dt ~.pr dtd qr then W = const. is an integral if aW /(awaH awa9 0 at r=1 aq pr pr apr /qr i.e. if + ( W, H) = 0, using Poisson's bracket expression. Now we may discuss a dynamical problem from two points of view. One way is the consideration of integrals already exemplified. This gives us general solutions independent of the conditions special to the problem, such as initial positions and

Page  309 INTEGRALS IN DYNAMICS AND THE PROBLEM OF THREE BODIES 309 motions. But in any given problem we may also take into account given conditions aw and solutions already known. Thus it is not necessary to have a + (W, H)= 0. 3W If -W + (W, H)= -WT, where W, is known to vanish for our problem, then W= const. Wat is still an integral. When a- +(W, H) 0, we may call W = const. a primary at integral. 3aW If a + ( W, H) _ M W, there are two cases (1) M — L: then d (LW)= 0, and L W = const. is a primary integral. L dt dt We may call W a sub-integral (Forsyth). (2) If we cannot get M _ L dL,then W = 0 may be an integral if W vanishes dW for a given moment of time. For then d will vanish at this moment. Also ddlW d 4/ W d ll. dd W d (MW)= M + W dM=0 dt tt dt(M- M dt dt at this moment. And so on we may shew that all the derivatives of W with respect to the time (totally) vanish, so that W= O is true of the problem. Painlevd calls such an integral particularised; we may add the epithet primary, since it is possible to have alW a+ (W, H) M + W1, where W1 vanishes in virtue of the conditions or known solutions. In this case too we get sub and particularised integrals. But a distinction has to be made. This distinction we obtain as follows. 9W If a- + (W, H)= WT, where W, vanishes because of conditions or integrals, but W1 is independent of W, then W= const. is an integral of our problem, which we aw shall call secondary (Forsyth calls it composite). If then - +(W,H) MW+ W, then WV= const. may be a secondary particularised integral, or it may be possible I dL to get a function L such that M - dL, so that LW = const. is a secondary integral. In this case W is a secondary sub-integral. These definitions may be extended. ~ 10. Cases of primary integrals are of course very well known. The energy integral where it exists is one. Secondary integrals are e.g. Liouville's integrals for the form: T=1M(QQq2+...); M(V-C)=Fl(q) +F(q2,...). In fact if W,= C, dW, aM is the Liouville integral, we get dt 1 aq, (T + V -); hence if T+ V - = 0 is the energy integral, Liouville's integral exists, but only in virtue of the energy integral. It is therefore secondary.

Page  310 310 SELIG BRODETSKY Again, if T is homogeneous in p's of order 2, and in q's of order m; and V is homogeneous in q's of order - m - 2; then Jacobi has shewn that aT aT qa4 +..+.+qn a(m +2)Ct- A =0 is true of the problem. This also depends on the energy integral and is in general secondary. Examples of particularised and other integrals will also readily occur to one. ~ 11. For the further discussion we suppose the constant of integration if it 9W exists to be absorbed in W. Now let -a +(W, H)- W. We use the operator 3W (H) W - + (W, H). Then (H) W= WT. If now W= 0 is true of the problem, at so is W1=0. Again if (H)2 W-(H) W, - W, then W2=0 must be true. Proceeding in this way, it follows that if (H)n W_ (H)n-1 W... =- (H) Wnl - Wn, then Wn = 0 is also true of the problem. This argument also works backwards. Thus if W, = 0 is true, then (H) W = 0, so that W=0 is true (with constant of integration absorbed). If (H)-1 W_ W_, then W_= 0 is true. In general if (H)-n W_ Wn, then W_- = 0 is true of the problem. Now positive powers of the operators give us nothing new. In fact the relations (H) W, (H)2 W,..., (H)n W,... =0 are merely conditions that W = 0 may be true. But negative powers of (H) can give us new results, as is seen in the case of Liouville's and Jacobi's integrals already exemplified. Thus in a dynamical problem, if W==0 is an integral, we can take FW, where F is any function and find (H)-1FW, (H)-2FW, etc. These functions equated to zero will give us new integrals of our problem. In general we have the energy integral which may be used for this purpose. But when no energy integral exists, we can use another integral for the purpose. It is easy to shew that integrals given by ignorable coordinates are of no use in this connection. ~ 12. When no energy integral exists and no other integral is known, we may construct an artificial energy integral by increasing the order of the system by one. Thus Whittaker has shewn that when H (qD, q2.'',n qnp, p. * *, pn) + h = 0 is the energy integral of a problem, then if we solve for pi in the form pi + K (qq, q2,.. qn, Pl, p2,..., p n, h)= O, the problem can be represented by a Hamiltonian system with variables q2,...,q, p2,...,pn and independent variable q1, in the form dqr _ K. dPr_ K dq1- a ' d~ - (r= 2, 3, r,..., n). dq, -p' dqI - vq '

Page  311 INTEGRALS IN DYNAMICS AND THE PROBLEM OF THREE BODIES 311 If now t occurs in H in a dynamical problem, we reverse Whittaker's process. Let us then have dq_ aH. dpr. aH.. dt ap- 3 dt- 3q'r 1 H involves t explicitly. Take a new independent variable r, and a new momentum T. Let H' + h'- (T+ H)F, where F is any function of the q's, p's, T and t. We get a new system dq, aH'. dp, aH' d- = ap- d-r = —; (r= 1 2,..., ) ) dt aH' dT aH' dr~ 3T; dr~: 3t d aT T' do at H' does not involve r. We have then an energy integral H'+ h'= O. We can now use H' + h' for deriving secondary integrals. When obtained these may be reduced back to the problem as originally stated with independent variable t and variables qi, q2,..., n,) P, pp2,.., pn. The method of obtaining secondary integrals here sketched is useful in obtaining particular integrals. Thus if (H) W W1 and W1T can be made to vanish by means of some particular condition, we have an integral of a particular case. ~ 13. The notion of secondary integrals can be used to obtain integrals involving the energy where an energy integral does not really exist. Let H dH aH involve t explicitly. Then as seen in ~, dt - If now the conditions of the dvt ot aH problem make a- zero, H=h is an integral in spite of t being present in H. at We may call such an integral a "quasi energy integral." If H can be split up into a kinetic energy, which is (as is usually the case) independent of the time explicitly, and a potential energy V, then the condition is that at =0 should be true of the problem. We get a particular solution. Thus let a particle move in a plane under a central force due to a potential which involves the time. Using polar coordinates, H = 1 (+2 + r2O) + V (r,t). There is no energy integral. But if a[ = 0 is true of the problem, then at H=h is a quasi energy integral, if 3= 0 and H= h are compatible. r20 = const. is true at of the system. Let the constant be K. Put V' = V + K2/2r2.

Page  312 312 SELIG BRODETSKY The conditions governing V' are found to be 2 (h- V')= VT '2/TVrt 2; vnd /CJTI CJtl3 = VCJ~T~ t\............(14); and Vr Vrrt' -= Vrrt' V t'2 -2 Vrtt' Vtt' Vrt' + Vttt Vrt 2 where suffixes denote partial differentiations. It is easy to find V' from these differential equations. The result is of no particular interest except as an illustration of the existence of a quasi energy integral. ~ 14. Again we may have a "quasi ignorable coordinate." Thus if W-p, - C = O OH aH is to be an integral, then we must have -- = O. If then - contains as a factor an expression which is known to vanish for the problem, we get the quasi ignorable coordinate q1, and p - C, = 0 is an integral. A simple illustration is the following. A particle moves in a plane under a central force varying inversely as the cube of the distance. With polar coordinates H- (12 + r262) +. We convert into the Hamiltonian form, by putting r = qg, 0 = q2, i, r=p, r2 =P2, H = (p +q +ql q2 is an ignorable coordinate. Let p2- C = 0 be the integral; then d a _ p2 + 2L C22 + 2L dt ( la- q13 - 3 for our problem. Thus if 2L=-C22, we get the quasi ignorable coordinate ql, and p - C, = 0 is true of our problem, as a particular solution. We get f = C,; r20 = C,. Hence r = Ct + C,'; 0 = C2/(Ct + C,')2; so that 0= - C/ (1Clt + Cl) + c2=2/ Cr - + 02. Thus we get the reciprocal spiral Or = constant by a proper choice of the zero for 0. Several problems can be solved in this manner, and integrals can be obtained replacing the energy integral when it does not exist, as exemplified in ~ 13. But we shall conclude with an extension of the well known Liouville's integrals. ~ 15. Let H = Pipl2 + P2p22 +...+ Pnn2 + P, where Pi, P,..., P,, P are functions of (ql, q2,..., q). H= h is an integral. We shall examine when we can have an integral of the form W- L + ~Lpi2 + }L2p22 +... + ~Lnpn2= const. depending on the energy integral; L,, L2,..., L, L are also functions of (q,, q2,.. qn).

Page  313 INTEGRALS IN DYNAMICS AND THE PROBLEM OF THREE BODIES 313 dW We put dt =(M+Mp +... +Mnpn)(H-h), where M,, M,..., Mn, M are functions of (q,, q2,..., qn). We get ppn iaL IL1 9lnn L P I aP? 1 aPn o\ pq( + 208p2+ '' +2 a-n 2 -2EttL2(aq _y 2 2'.... (M + Mpl +..+ Mpn) (P - h + 1 pl2 +. + pn2). Equating coefficients of powers and products of the p's, we get at once M1= 0. We also get a _ p a=M_(P-h); (r=,, 2,.,n); ayL, arP and P I =1P- 7M; and t Pr _ — Lr Jr = Mr q; (r, s= 1, 2,..., ). aqr a4r These give us - {4 (P - h) =a ( 1, 2,.,n). We put p(P -h)= L +Fri (qq..., qr-1,qr. *qn); (r = 12,. n). r Fr does not contain q,. We then obtain after a little reduction ar (F.- Fs)} = (r,, = 1,2,..,). aqp (P-h We put pS^(Fr-Fs)=Fr(ql,q2,... qr-_l,qr+l, qn); (r=1,2, s...,-1,s+1,...,n). F. is again independent of r. Thus if we can find quantities Fr, Fs,, r = 1, 2,..., n; s = 1, 2,..., n; r = s, such that Fr, Frs do not contain r, and PsP -h)= F (Fr- F)........................(15), then we have an integral W - L + ~Lp2 +.. + +Lnp 2 = const., where Lr = (L + Fr) Pr/(P - h)........................(16). Equation (15) gives us (n-1) values for P/(P -h). There must then be (n - 2) relations among the quantities F., F2,..., F.,, FF+,,..,F.,, Fs F -, sF, Fs+,, s.s., * *Fns. The integral is n L + 2 2 (L + Fr) P/(P - h). p2= const., r=l Using the energy integral H - h = 0, we get n PrFrpr= 2K(P-h). r=l

Page  314 314 SELIG BRODETSKY Equation (15) defines a generalised form of Liouville's system. For n = 1, we get the energy integral. For n = 2, we get a Liouville's system as usually defined. For n = 3, we get the first extension of a Liouville's system. Equation (15) gives PJ/(P-h) = F2/(F. - F,)= F3,,(F3- F1) for s= 1. Hence F2,/(F2- F,) = F3/(F - F1). Similarly we get FI2(F1 - F2)= F32/(F- F2); F13/(F - F3) = F23/(F2 - F3). Thus we get F12. F23. F31 +F. F3 F3 =..................... (17). It is to be remembered that each F is independent of that coordinate which has for suffix the first number in the suffix of F. If there is only one number in the suffix of F, then we take this number as the suffix of the absent coordinate.

Page  315 CONTRIBUTION A LA THEORIE DE LA CHUTE DES CORPS, EN. AYANT ItGARD XA LA ROTATION DE LA TERRE PAR ALFRED DENIZOT. C'est Newton qui a de'couvert la deviation vers l'est d'un corps tombant Sur la terre. Dans la the'orie du mouvement relatif on rame'ne ce phenomene 'a leffet d'une force fictive, 'a laquelle on a donne' le nom de la force centrifuge compose'e. Le calcul donne encore une deviation du corps vers le sud qui se pre'sente comme l'effet de la meme force fictive, mais qui est trop petite pour e~tre confirme'e par 1'expe'rience. A d'autres occasions, specialement dans une seance de la " Berliner mathematische Gesellschaft f," j ai montre' qu'on a ne'glige6 dans les equations generales quelques termes contenants le carre' de la vitesse angulaire de la terre. Ces termes sont l'expression d'une autre force fictive que j 'ai nomme'e la force centrifuge instantanee. Surtout jai appele' lattention 'a cc que la force centrifuge compose'e ne peut pas expliquer la rotation apparente du plan d'oscillation d'un pendule, tandis que la force centrifuge instantane'e peut servir pour donner une telle explication. Aussi dans le proble'me de la chute libre d'un corps les termes, neglig&s totalement j usqu'a' present, conduisent 'a des re'sultats re'marquables. En conservant ces termes dans les equations diff6rentielles du mouvement, leur inte'gration s'Ieffectue d'une manie're tre's simple. Prenons d'abord les 6q uations qui se rapportent au systeme suivant des coordine'es rectangulaires: L'origine 0 soit un point de la verticale du lieu sous la latitude 0, l'axe O.'7 perpendiculaire 'a l'axe de rotation de la terre, pris positif dans la direction oppose'e an centre du parall~el, l'axe OZ parallel 'a l'axe de la terre dans un sens convenablement choisi. Les deux. axes sont alors situe's dans le plan meridien; l'axe OHf sera dirige6 vers l'est, dans la direction normale "a ce plan. En signifiant par g la pesanteur et par w la vitesse angulaire de la terre, on regoit par rapport au syste'me mentionne' les equations suivantes g O A2 w jj ~ ~ ~,q-2(t.(1)........... =-g sin ~ *Sitzungsberichte, 1906, p. 78 (aussi v. Bulletin de l'Acad. d. Sciences de Cracovie, 1904, p. 449; Annalen der Physik, 1905, p. 299; Physikalische Zeitschrift, vi. 1905, p. 342, vii. 1906, p. 507).

Page  316 316 A. DENIZOT Les integrales gen6rales de ces equations differentielles donnent les expressions = (At + C) sin wt - (Bt + D) cos cot + -9cos cp) =(At + C)coscot + (Bt + D) sint w(2). =1 gt2 sin + Et + F) L'etat du corps au commencement du mouvement determine les constantes A, B, C, D, E, F. Dans notre cas, an moment t = 0, le point mat6riel doit coi'ncider avec ]'origine 0 des coordinees, c'est-a-dire ~o = flo = ~o =0, et les vitesses relatives doivent e'tre to = ii = o= 0. Ces six conditions donnent les constantes. On trouve =-cos (1 - Coscot - cot sinact), - cos (sin cot - cot cos cot) _ _ - t2 sin g Si nous posons g - -- cos = 4:, nous aurons au lieu de 4 4 21= -- cos ~ (cos cot + ct sin ct), et nous voyons que la trajectoire du point dans le plan =OH est la developpante du cercie, dont le rayon est egal ' - cos b. En se servant des series pour sin ct et cos ct, on trouve ensuite 4:= _I gt2cosp jgco2t4 CoS} n g= 5g3 Cos......................(4). j= - gt2 sin 0 Ordinairement on rapporte les equations diff6rentiefles a un systeme des coordonn6es rectangulaires 0 (X, Y, Z), dont deux axes sont situ6s dans le plan horizontal et le troisie'me axe coi ncide avec la verticale du lieu. Nous choisissons le systeme suivant: L'origine des coordonnees coincide avec l'origine precedente, de meme l'axe 0 Y avec l'axe OH. Les axes OX et 0Z se trouvent done dans le plan m riidien: l'axe des x est la trace du plan horizontal men6 par l'origine 0 et dirige vers le nord, l'axe des z coincide avec la verticale. Les nouveaux axes sont lies aux precedents par les relations suivantes 4 = - x sin z - z cos r7= Y.(5). 5= wxcosob-zsino J A laide de ces relations nous transformons les equations (1) et nous obtenons c2 (x sin z cos )sin - 2co sin p Yj- c ~2y + 2(w;.gsinp + icosq4. (6). g + oz (x sin j+ z Cos os 2cos o

Page  317 CONTRIBUTION A LA THI~ORIE DE LA CHUTE DES CORPS37 317 Si nous voulons determiner les integrales de ces equations, ii nous faut 'a laide de (5) transformer les valeurs (4) de ~, q, ~' en x, y, z. De cette manie're lions aurons x =-IgW2t4 Sin~bo sbCO z _ ~t - 0g2 t4csq Comme nous voyons il y a une de'viation (y) vers 1'est et une autre (x) vers le sud. La valeur pour la deviation vers l'est est la me'me que donne la the'orie adopt~ee jusqu'a' present; mais cette dernie're donne pour la deviation vers le sud la valeur J gW2t4 sin 0 cos f, done la valeur trouve'e auparavant est pius petite de 1got4 Sin co COS Cette diff~rence, i est vrai, ne jolle aucun role quant 'a 1'expe'rience, parce que les erreurs des observations la surpassent, ne'anmoins elle presente quleique int're't quant 'a la question, dans quel de'gre' les deux forces fictives contribuent an phenomene. 'Supposons d'abord que le temps de 1'observation. soit si court que l'influence de ces forces fictives reste insensible, nous aurons les equations du mouvernent I j=O0,..g dont les integrates seront x=O, y =O z3 Z t2................(a), Si la chute du corps dure plus longtemps, nous aurons des influences des deux forces fictives, mais nous voulons d'abord supposer que ce ne soit que la force centrifuge composee qui se de'veloppe et que la force centrifuge instantanke reste encore sans effet. En ne'gligeant alors cette dernie're, nous nous servons d'un proce'de qu'on a employe' jusqu'a' present. Done nous aurons les equations x -2co~ sinc = 2co (b sin sb+i cos) z g + 2wo cos. On revoit par 1'inte'gration. les expressions x= g ((2t4 sin 0cos~ y = ijg&t3 COS.............(b). = 1 gt2 - gW2t4 COS2fJ C'est le me'me re'sultat, auquel Gauss est parvenu le premier. Pour la deviation vers l'est nous avons obtenu la valeur deja' indique'e sons (7); done on peut conside'rer la deviation vers lest comme l'effet produit seulernent par la force centrifuge COMPOSee. Enfin nous voulons admettre que dans les equations generales (6) on puisse supprimer la force centrifuge compose'e et ne conserver que des termes representants la force centrifuge instantane'e. Dans ce cas on a les equations suivantes = + cW2 (x sin + z cos ~)cos

Page  318 :318 A. DENIZOT Nous trouvons x =Lgwt2 t4 sin 0 cos z= ~g2+ -LgW2t4 Cos~f Maintenant nous avons re~u une deviation du corps vers le nord, mais aucune vers 1'est. En faisant alors la somme des x, y, sous (a), (b), (c), nous obtenons les expressions dej'a' indique'es soils (7), re~ues par nous comme re'sultat de l'inte'gration des equations generates (6). En etudiant le phenomene de Newton nous sommnes parvenus an re~sultat suivant: La de'viation vers lVest est l'effet de la force centrifuge compose'e, la deviation vers le sud est une superposition des effets de la force centrifuge cornpose-'e et de la force centrifuge instantane'e.

Page  319 UBER ASYMPTOTISOHE INTEGRATION VON DIFFERENTIALGLEICHUNGEN MIT ANWENDUNG A{JF DIE BERECHNUNG VON SPANNUNGEN IN KTJGELSCHALEN 'VON OTTO BLUMENTHAL. Das mechanische Problem, das ich behandein will, ist die Bestimmung des elastischen Spannungs'zustandes einer Kugelkalotte unter der Navierschen Annahme, dass bei der elastischen Verzerrung Normalen zur neutralen Easer geradlinig uind normal bleiben. Der Nachweis der Zulassigkeit dieser Annahmne bei klienen Wandstarken und der Ansatz des Problems ist in den ailgemeinen Formein enthalten, die Love fflr die elastisehen Deformationen diinner Schalen gegeben hat*. Reissner hat kiirzlieh gezeigtt, dass die volistaindige Ldsung dieses Problems des Qleichgewiehtes du-nner Kugelsehalen im Falle rotationssymmetrischer Belastung auf die Integration einer gewdhnlichen linearen Differentialgleichung 4. Ordnung ftihrt. Ich werde mich hier mit der Behandlung dieser Gleichung 4. Ordnung besehAftigen, denn sie ist nach den gewdhnlichen Jntegrationsmethoden dureli Reihenentwicklung praktisch nicht durchfifihrbar; man muss vielmehr andere Methoden heranziehen, die ich als asymptotiscite Integration bezeichnet habe. 1. Es handelt sich inn folgende Aufgabe und folgende zu berechnende Grdssen. Eine Kuigelkalotte von dem Radius a und der Dicke 2h mit dem. Offnungswinkel 0" erfahre keine ausseren Krafte, nur an dem. freien Rande 00 miigen tiber den ganzen Rand gleichmiissig verteilte Spannungen und Biegungsmomente angreifen, unter deren Einfluss die, Schale im. Gleichgewicht ist. Es soil der Spannungs- und Dehnungszustand in (ler' ganzen Kalotte berechnet werden. Die in Betraeht kommenden Komponenten der Spannungen und Verritekungen sind folgende: Normalspannungen T~, T.,; eine Schubspannung N; Biegungsrn-omente G1, G,; Versehiebungen u, w. (Siehe Figur auf der folgenden Seite.) Die Ltisung des Problems wird zuriiekgeftihrt auf die Integration einer Differentialgleiehung fiur N: sins ON"" ~ 2 sin' 0 cos O N"' - sin 0(- sing- 0) N" ~sin 0cos 0(3~+2si )N~sin in 3)-0 (A). =4 (1-o2) (i + h)' a- = Poissonsehes Verhailtnis * Love, Treatise on Elasticity, 2nd ed., p. 488-510. t Festschrift Milller-Breslau (Leipzig, Krdner, 1912), S. 181-193.

Page  320 320 OTTO BLUMENTHAL Aus N berechnen sich dann die iibrigen Grossen wie folgt: T1 C aa NT2 T,= Ncot, T2=N', GI = - 3 + cot 0 N" - - -ot2 0 + N +( - + 1 +r --- sin2 cot 0 N,| 2 3a (l -a) [S 1 + + t B N + (r - 1 COt2 +- o1 — N + (o 1+ " + cot ON h l+21 sin2 cot0, G2=lar +cot N"+ cot2 1- ) N = 2hE sin0(B+3), =- la2h,[rotN- N'+( +cr)cos6(B+3)],.........(A') B = willkiirliche Konstante, E = Elastizitatsmodul, 3 = 0 ^d(Ncot -N')d. Die Abz/ihlung der Integrationskonstanten ergiebt in einfacher Weise die geometrischen und dynamischen Randbedingungen, unter denen Gleichgewicht moglich ist. Durch die Festsetzung, dass Spannungen und Biegungsmomente im Punkte 0 =0 der Kugel endlich bleiben sollen, werden 2 Integrationskonstanten absorbiert, wie die Reihenentwicklung der Integrale der Gleichung (A) in der Umgebung von 0=0 zeigt. Es sind dann noch 3 Konstanten iibrig, von denen eine nur in die Verschiebungen, nicht in die Krafte eingeht. Dies

Page  321 UBER ASYMPTOTISCIIE INTEGRATION VON DIFFERENTIALGLEICHUNGEN 31 09.1 O" en-tsprieht der Tatsaehe, dass der ganzen Schale noch eine Translation erteilt werden kann. Abgesehen von dtieser Willktirliehkeit sind noch zwei Bedingungen am Rande vorgebbar, (Ilie sieh entweder at-if die anrefenden Krafte und Momente, oder auf die geomettrisehe Gestalt des Randes beziehen kdnnen. Emn einfaches Bedingungssystem ist Z. B., dass der Rand etwas auseinander gedehnt ist, und 'kein Moment an ihmi herr-schen soil. Es soil aber hier ant spezie'lle Probleime nicht Cingegangen w erden. 2.. 1)ie Sehwiverigkeit besteht Puin (larin., (lass (Ile Gleichuoig (A) ftir N einen nach Voraussetzung sehr gr-ossen. Paramleter. b4 enthijit. Da ni-imlich bei der ganzen Theorie die Kugelsehale als sehr dtinn vorausgesetzt werden muss, ist -notwendig eine grosse Zahi und dem-gernass b4 gross. Dies hat zur Folge, dass die fibliehen Reihenentwiekiungen nach Potenzen von 0 (oder sin 0) nur in der naehsten Unmgebung des Nullpitnktes mit praktiseh brauehbarcr Sehnielligkeit konvergieren werden, wihrend in cinigermassen grosserer Entfernungy z. B. 0=4)infolge des grossen Koeffizienten b4 die Glieder der Reihe zunaehst sogar stark anwachsen werden, umi erst naeh einer mnit waehsendem b gegen Unendlich gehenden Zahl von Gliedern abzunehmen. Diese Reihen sind also zur Ubersieht tiber den Verlauf der Erseheinuing giinzleh ungeeignet. Hier' greift die asymptotisolte Integration emn. Die asymptotisehe Integration setzt sich zur Aufgabe, gerade die grossen Werte des Parameters zu bertieksiehtigen, indemn sie die Integrale naeh absteigenden Potenzeu des Parameters entwiekelt. Das Verfahren wird wohi am tibersiehtliehsten und fair die Reehnung am bequemsten so dargestelit *. Dnrch die Substitution N ---- wird zuerst das Glie-d mit dem 3. Differentialquotienten weggesehatft nnd fair M1 folgende Gleichung erhalten: M"" + a, M" + a1, J' + (b4 + a0o) ill 0, 8 1 5 CosO0 63 1 9 1 9'... (B). "2 ~ +- a, =3 (to +- - ~ 2 sin 0 2' sinO L 16 sin0 8 sin2 16) Whehst nun bei festemn 6 > 0 der Parameter b beliebig an, so werden diel Funktionen ato, a1, a12ime mehr gegen b4 versehwinden. Man kann daraus schliessen, dass die In-tegrale Ml n.aherungsweise itbereinstirumen mit den Ldsungen der vereinfachteii Gleichung M""/ + b4 M = 0)..................(B'). Die Theorie der asymptotiseheni Integration hat die Aufgabe, diesen Sehiuss genau zu begrtinden und den Fehier der Niihermiig abzusehiitzen. 3. Ich will (lie Fehlerabsehatzung ausfiihrlieh geben, weil ieh sic in hoheim Masse xverde benutzen mtissent. Seien *Siehe auch meine Darstellung des Verfahrens in Arch. Math. Phys. (3) 19 (1912), S. 137 f. [Ich habe diese Methode der Restab~schdtznngo in A rch. Math. Phyis. (3) 19 (1912), S. 141 ff., verdientlicht und damals flir nou gohalton. Seitdem bin ich durch Herrn Bkhchr in freundlicher Weise darauf aufmerksam gemacht wvordeii, dass die:-elbe Mothodo bereits 1908 von Birkhoff, Trans. Am. Mjl h. Soc. 9, angegeben worden ist. M. C. IL 21

Page  322 322 1OTTO BILUM ENTH AL die 4 Wurzeln der Gleichung X4 + 4 = 0, und iV1 das Integral von (B), das durch eAx, asymptotisch dargestellt wird, schliesslich?1. = M l- e i................................ ( ). Das Fehlerglied ql gentigt der Beziehung q'"//' + b4r1 = - ((12 M1// + al 2]1' + ao ll,), und ein dicse Gleichung befriedigender Ausdruck ist 711 = i x jie- e ((111 ( M1 0J d..........(2), wo mit 01 die untere Grenze des Integrationsintervalls bezeichnet sei, das sich von dort nach 00 erstreckt. Mit Einsatz dieses Wertes in (1) findet sich aL,2 / + t M1' + aoMi- = (Xa, + X1a,2 + ao) ex'0 + 4b (Xi(j2 + ai + Xiao) exi e-^i~ (aM1" + alMl' + aoM) dO.. (2'). Aus dieser Gleichung finden wir nach einem auf Liouville zuriickgehenden Verfahren zunachst eine obere Grenze fiir a2M1" + aM' + caoM, Ie-x". Dazu gebrauchen wir eine Abschhtzung der Integrale der rechten Seite, die ich sofort in einer ftir die praktischen Zwecke besonders geeigneten allgemeinen Form geben will. Sei ( (0) eine im Intervall (0,, 6,) nirgends verschwindende, positive, stetige Funktion. Dann ist e-xio (a. /M" + a, JM, + (ao ) d0 = e e(A-A 0 (0) do, J~~~~~,(a +M +1 J"M e- ( \) d ' und daher, da e(Al-A)0 nirgends in dem Intervall grdsseren absoluten Betrag hat als am Punkte 0, 01 | ei I (.a/211 + a, ll +- o,,l) d0E1 < e( - ai) max a.l.l + (tll, _- ao.l e- (0) dO. 0(e0,0) f(0) Daraus folgt aber, bei Einsatz in (2'), Xla2 + Xi, a + a, I11(1 t 21'11 ~ ('A/h + Xi, e <(0,,0) 4b (0) maxll - X ____.;- + _+) W e-__ <_ _+__t (01. 0) I (0) I X 1 l 2 Xi (X21q, + X Otl + o) 0 ) 4b. Max 4 (0) do 46~b4 (0)0) g o h, 2 22 i max - — m <(0,, 0,) o (0) b9 b V2b3 (0,, 0,) (0) ( 0 und dieselbe Ungleichung gilt auch fur M.2, M3, Ma. Setzen wir die gefundene

Page  323 UjBER ASYMPTOTISCHE INTEGRATION VON DIFFERENTIALGLEICHUNGEN 32 323 obere Grenze nochmals in (2) ein, so erhalten wir fair die Restey folgende Abschiitung: 2b, b (I ~ (t 2 ~ + () I~~~0 max eAO --- ~~~~~(01,00) p 3 f4(0) dO b2 1 max 2 0 /2 h (0) dO,\/2b3 (o.(3). Lu Falle der Gleichung (2') erhal~t mhan eine glinstige Absehimitzng von m, indem man (0) = 1set zt. Es ergiebt sich so it' nesentliehen Yernachhissigungen: '3b2 13b,3~ 1 ~~~~~4 \V2' '16(3Sin2 0, Nco012 ~ ~3b2 13 b, 6 3 1. (d 1 V2 b 4 V2 C~ 116 si Xe1 1)ie Ungleichung (3) besagt, dass q von der Ordnung -is.Aus der Ungleichung (3') schliesst mnan spezieller, dass der Rest mit waehsendem b gegen das Hauptglied versehwindet, xvenn man innerhaib soleher Intervalle (0,, 0,) bleibt, fuir die b sin 0 mit b untendlich wird. Dasselbe gilt ftir die Differentialquotienten aller Ordnnngen. 4. Es lassen sieh also mnit relativ versehwindendem Fehier 4 Integrale der Gleichung (A) in folgender reeller Form ansetzen: 1 b -0 b 1 - b I V~e e/2os - 0, N2- e 2 sin- 0) VS111 0 \2 2Vsin V2' 0~ co A ba. Vsin 0 si<2snnV~U Jetzt aber kommt der schwierigste Punkt. Es handelt sieh darum, wie sieh das den Endliehkeitsbedingtingen fhir 0 0 nnd Randbedingungen ftir 0 = 00, gentigende Integral von (A) ans diesen partikuldren Integralen zusamimensetzt. Die Randbedingungen machen keine Sehwierigkeit, da ja fdir 0 = 00 die asymptotisehe Darstellung giiltig ist. Dagegen ist es unmnglieh, ant direk tern Wege zu entseheiden, welehe der Integrale (4) die Endlichkeitsbedingumgen am Punkte 0 = 0 befriedigen: denn an diesem Pnnkte versagt die asymptotische Darstellung augenseheinlich, denn der Rest (3) wird dort grdsser als das Hauptglied. Man kann aber die Entseheidnng znniachst auf unstrengein Wege auis der physikalisehen Ansehauung ableiten, indem man als Tatsaehe annimmnt, dass die Spannungen N von dem Rande naeh dem Pole 0 = 0 abklingen. Die einzigern Fnnktionen (4), die diese Eigenschaft haben*, sind die Fnnktionen N1 und N2, w~ihrend N3 uind N4, mit waehsende r Entfernung von doei * Wir betrachten dibei nur soiche Intervalle (6, 0o), die in genfigender Entfernung von 0-0 1)leibefl sodass das langsame Abnehnmen des Nenners \'idnicht in Betracht konimt. 21-2

Page  324 324 824 ~~~~~OTTO B3LUMENTH AL freien Rande iinlner grdsser werden. Unser Integral der Gleichung (A) wird also asymptotisch ats T=AI,+A 2.................(5 anzusetzen sein, und die Konstanten A, Ufld A2 werden dann zur Erftillung zweier IBedingungen amn freien Rande gerade ausr-eichen. Die Kriifte, und ebenso, nach den Gleichungen (A'), die JBiegungstnomente und Verrtiekungen, stellen sich daher als sehr sehnelle Sehwingungen. dar, die, von demn frelen Rande naeh dem Pol exponentiell sehr raseh abklingen. Das logarithunisehe 1) Dekrement auf die Bogeneinheit ist narnlieh ~. Dieses tibersiehtliehe Resultat hiitte init Htilfe der gewdhnliehen Integrationsmnlethoden dureh Potenzreihen nieht gewonnen werden kdnnen. 5. Es bleibt tiibrig, streng naehzuweiseni, dass der Ansatz (5) riehtig ist, d-h. dass die beiden Integrale N:3. und N4 am Punkte 0 = 0 die Endliehkeitsbedingungen nieht befriedigen. Eine Aufgabe dieser Art findet sieh wohl in den meisten prakti'seh wiehtigen Probleinen. asymptotiseher Integration. Sic lasst sich ailgememn so formnulieren: ein exaktes Integral einer Difflerentialgleiehung ist festgelegt dureh Bedingungen can einem Punkte, anl dern die asymptotisehe Darstellung versagt; trotzdem sollen atif cinem Umweg die Integrationskonstanten der asymptotisehen Darstellung bestimmt werden. Aligemnein zn reden,ist das Problem hdehst sehwierig, und ieh sehe keine aligerneine Methode, es zu behandein. Emn in mnanehen praktisehen Fallen anwendbares Verfahren habe ich an demn Beispiet der Legendresehen Polynoine in meiner eitierten Arbeit durehgeftiihrt *. Dieses V~erfahren ist aber bei unseremn jetzigen Beispiele nieht anwendbar. Wohi aber kain ieh die Fr~age, aneh hier exakt entseheiden, wenn aCueh auf einem nieht eleganten Wege. Ich gehe dazu ans von der lDarstellimng (8') des Restes. Man sieht leiclit, dass ffir O,?- der Rest kiciner als die HaIfte des Hauptgledes ist. Andererseits lasseli sieh die arm Puinkte 0 = 0 den Endliehkeitsbedingungen gentigenden Integrale ftir 12 solehe Werte von 0, die etwa kiciner sind als ohne, Sehwierigkeiten nueIse 7;bereehnen. Ic/h werde zeiqen, dlass 11an darch namerische Berechnung der Werte eines den Entdli'chkeitsbed'ingaritgeni genfigenden -Integrals jfib- 4 Patk-te des Bereicites -< 0< - 1 b b be weisen kaim, d/ass die deni Entdlichk-ei'tsbed'ingantigeni geniigeniden Integrale iiotwendig dlie Formi (5) habe~n rnitssen?. In der Tat haben aite Integrate (5) and nur diese folgende eharakteristisehe i-ligenschaft. Ersetzen Nvir' die aCsymptotisehe Darstellung (5) dureh Zuftigung der Reste, dureh eine exakte, so lautet diese, wvie leiehte Rechnung zeigt, fair die Funktion 31 = V-sin6O -N M- eO K,2 Ai eosj bo~+A,, Sinl jb0) + (a, (0) eos ~ b04 a, (0) sin j O *Lie. S. 150-163.

Page  325 UBER ASYMPTOTISCHE INTEGRATIOTN VON DIFFERENTIALGLEICHUNGEN 32 325 obI, mit Einftihrung der in (3) definierten Restgro~sse -a —=(0 a, (0)2 + a, (0)2 = T (0)2 (Al22 A22) ist. Wir betrachten die. zwei Werte Oj =7 lind 02" = _ Dfann ist b ~ 2 b Al (021 )2 + e 731M(0/1 )2 = [(AI + a, (01/))2 ~ (A2 + a2 (02/) )21 e,, bO2' a,(021/)2 ~ a (02")> = (Al2+A2 ic\/ A2t A22 )eb' = (A 12 + A2 2) (1 - V2T, )2 e,/ 2 b 0' wo e erne Zahl vom absohtiten iBetrage <1I bezeichnet und -r eine Grtisse, deren absoluter Betrag den griisseren (eId beiden Werte Tr (02'), 7 (01") nicht iibersteigt. Machen wir dieselbe Rechinung fur 0,' = 9w 0. "nddiidire, V2g b 2~ b<nddvdees og M (0.'2+ W ()2 2 ~ i'7~ (7r 23N M(02'~ M M(01'")2 (-V7)2 \ 2) () wo T eine obere Grenze fuir die beiden Zahien -r nind Tr2 bezeichnet. Alle 4 Punkte 02' 01" 02' 021" liegen ii dem Intervall <bi) b Deimiach ist aber 2 2 n ae der betrachtete (Quot'ien yro~sser als 1. Dies Ihisst sich rechnerisch fuir die beiden den Endlichkeitsbedingungen geniigenden Integrale _N verifizieren, und damit beweisen, dass diese in der Formi- (5) darstelibar sind. Die genauere Ausftihrung ist folgende: Vor allem. ist zu bemerken, dass der Nachweis, dass die den Endliehkeitsbedingungen geniigenden Integrale die Ungleichung (6) befriedigen und daher nur aus NV, und NV2 zusammengesetzt sind, aber N3 und N4, nicht enthalten, nar filr geni~gend grosse iWerte vont b geftthrt werden mfuss. Aus den Entwicklungen des ~ 3 meiner citierten Ar-beit folgt nam-lich, dass dlie Koeffizienten der I arstellunig dieser lntegrale durch die asymptotischen Fundamentalintegrale analytische Funktiovemi von b sind. Sind daher die Koeffizienten von N, nnd N, fuir genii gend grosse Werte von b Null, so sind sie es fuir alle Werte. Nun schreiten dlie einfaehsten Entwicklungen fuir die den Endlichkeitsbedinguingen geniigenden Integrale N-aus denen sich ja die entsprechenden M sofort ergeben-nach uingeraden Poteuzen von sin 0 fort. Die Rekursion fuir die Koeffizienten sehreibt sich am einfachsten in folgender Form. Ich setze zuerst N - 5 ~c,, sinn0, dann c, sin210 =/V dlsodass N=! I, wird, schliesslich sin 0= a/b. IDann findet sich (n -2) (it 3) a2 - FQ - 2) (i- 4) (n - 5)a 4 a4 _ 1d (a ~ 1) (it ~ 1) b2 A-2 2 L ( + 1Q )2 b4 + n+ 1) ( jI1 - 3)j Nuin brauehen wir die Werte der Integrale N nur mit sehr geringer Genanigkeit (vielleicht 10 0/0 Fehier). iDaher kdnnen wir zuiniiehst bei geniigend grossemi b den sin 0 durch 0 ersetzen und haben also N fuir die vier Werte a = wV\2, 3v/V,,2, 2wV2, 5-w/V2 zu berechnen, die aile zwischen 3 und 12 liegen. Aus den Rekursiorisformein lasseni sich dann miit Eeiehtigk~eit fuir geniigenid grosses b Nhheruingswerte fuir (lie

Page  326 326 OTTO BLUMENTHAL ersten dn ableiten. Die rechte Seite dieser Formeln reduziert sich namlich praktisch auf das zweite Glied der Klammer. Man findet, dass die dn im allgemeinen infolge von a >3 zuerst zunehmen, dann aber rasch abnehmen. Die Summe N berechnet sich leicht mit geniigender Genauigkeit aus wenigen (hichstens 7) Gliedern. 6. Die asymptotische Darstellung ist nicht, wie wir bisher getan haben, auf das erste Glied beschrankt, sondern lasst sich zu beliebiger Gliederzahl fortsetzen. Wir kommen allgemein fiir die Integrale M der Gleichung (B) zu folgender asymptotisehen Naiherung: ^=e (i~f1(B+- )~ bzf(0)~ -...~ () +............(7). Die Funktionen f werden gefunden, indem man den Ausdruck (7) in (B) einsetzt und die Koeffizienten aller Potenzen von b bis einschliesslich g__ Null setzt. Dies ergiebt die folgenden n Gleichungen, wobei zur Abkiirzung ei = Xi/b eingefiihrt ist: 4~efi, K+3 + e,? (6f,', K-+ + ac2, K+) +, (4/f'i, K + 2a2/f'i, K+ + a.f., K+1) + (f/;, K + a2f. K + af'i, K+ afi, ) = 0 ( = 1, 2,..., n - 3)......(8). Aus diesen berechnen sich die Funktionen f der Reihe nach durch Quadraturen Fiir unsere Gleichung ergiebt sich z. B. - 4e 2 cot +20 ) ' und auch alle weiteren Funktionen f^,, lassen sich in geschlossener Form anschreiben. Die Reste qrji der Ausdriicke (7) befriedigen die inhomogene I)ifferential1 -gleichung 1I7 il + (t',7 in + qal/i n + (b4 + aO) 'i/in - bn-2 Fin, Fi,, = {(f "",,-i- + (-Cf"i,,'2-2 + lf i', n-2 + Ca,,f, n-2) + E6 (4f"', 1-1 + 2at2/f', _ -1 + (af, i-,_) + Ie, (6f", + a2f;,,)} -I- 7 {(f/"j,..-I + ( -ii ~o + 2,+,^,1 + ^oA,) +e (4f ', 4 + 2a2fi,,, +.,/;:, n)} + {.f i"", ~- +.v f, ~(-1 + "(ofi, 1- + f}(, "}' Nach der Methode der Variation der Konstanten folgt hieraus z. B. fuir ll7, die Darstellung I -- M i ie F n........................(9), b'0-2 1 Jo1, wo /Li bekannte Determinantenquotienten sind, in die die Mi nebst ihren drei ersten Ableitungen eingehen. Die Gleichung (3) oder die,quivalente M i= i (1 + b 1)..........................(.... zeigt, dass die /li von der Form _ scn mssen. Zu ihrer wirklichen zeigt, dass di 41)4e I + b sein ii I I~U \ D

Page  327 UBER ASYMPTOTISCHE TNTEGRATION VON DIFFERENTIALGLETCHUNGEN 327 Berechnung-mit genauer Abschiitzung der Fehlerglieder a-aber sind die Determninantenauscririeke wenig geeiget Man geht dazu vielmelir besser davon aus, dass die 1-ti Integrale der adjungierten Differentialgleichung d02(a2tk) dO (a1l ) + (b4 + (1.o) ALk = 0 sind ~.Diese ist von derselben Form wie (B), und man erhj~it daher asyniptotische Darstellung und Rest ihrer Integrale nach Formel (3). In unserem Falle ergiebt sich das besonders githstige Resultat, dass (B) mit ihrer adjungierten zusammenfaillt. IDaher 1st his auf konstante uind fuir unsere Zwecke belanglose Faktoren der Form 1+ einfach 1= - 4IJV.-. Darnit aber wird augenscheinlich, nach den in Nr..3 b 4b4 gebraucbten Methoden der ]Integralabschatznng, N </2 b' + 1 b3,jI, O.1) wo) 0, maX die in (3') ange gebene obere Grenze der I i bedeutet. Urn mtiglichst gute asymptotische Nitherung zn erhalten, hat man fuir jeden Wert von b und 0, dasjenige n zn bestimlmen, das die kleinsten oberen Grenzen fuir die Reste qi~,, liefert. * Schlesinger, Ilandlbuch dler DifJerentiallihugnhS.5-6

Page  328 THE METHOD OF PERMANENT AND TRANSITORY MODES OF EQUILIBRIUM IN THE THEORY OF THIN ELASTIC BODIES BY J. DOUGALL. 1. The Mathematical Theory of Elastic Solids contains no chapters more fascinating or with greater claims to the attention of the physicist than those which deal with the approximate laws of strain in thin bodies subjected to given forces. The results are simple and should, one cannot but think, be susceptible of correspondingly simple rigorous proof, that is to say, formal deduction from the fundamental differential equations. But as a matter of history the processes by which they have been obtained are to a marked degree tentative and incomplete, and it has only been by the gradual elimination of errors and inconsistencies that they have come to take the forms in which they stand at present. Even now it may not be too much to say that one's belief in the broad correctness of these Approximate Theories rests not on any confidence in the reasoning by which they are attached to the general equations, but rather on the evidence of their agreement with direct experiment and with various exact solutions that have been given for special cases. With a view to the improvement and development of the existing evidence of the latter sort I have worked out exact solutions of the general problems of equilibrium for certain bodies in analytic foforms which make it possible to apply them immediately to the derivation of approximate results, and I propose to give here a brief statement of the nature of the method, illustrated by an example of its application to a simple problem. The bodies for which I have worked out, or am at present engaged in working out, the solutions are the plate, circular cylinder, spherical shell and circular cone. In the present paper I refer only to the isotropic plate and circular cylinder*. For the necessary leisure to prosecute the investigations, which are more tedious than might be expected, I have to express my great indebtedness to the Carnegie Research Schelme in connection with the Universities of Scotland. 2. The essential feature of the method is the determination of solutions for any point source of strain (i.e. for a force applied at a single point either in the interior or on the surface of a body), in terms of fundamental solutions, each of which is a solution of the body and surface equations of equilibrium under no forces. * The following paper has already been published: "An Analytical Theory of the Equilibrium of an Isotropic Elastic Plate," Edinburlh Roy/l,Sc. TrM.s. Vol. XT,, Part I (1904), pp. 129-228.

Page  329 PERMANENT AND TRANSITORY MODES OF EQUILIBRIUM 329 The existence of such solutions for the physical problem before us-they are common enough in other departments of physical mathematics, the theory of vibrations for example-seems at first sight to conflict with the well-known result that the only solutions of the elastic equations of equilibrium under no forces are the rigid body displacements. The exemption of the fundamental solutions spoken of here from this theorem has its origin in the fact that they possess singularities, at infinity in the case of the cylinder, at infinity or at some right line in the case of the plate. In each case the fundamental solutions fall into two classes, consisting of what may be called permanent and transitory modes of equilibrium respectively. From one point of view the former are degenerate forms of the latter, but they are distinguished from these by the possession of certain important properties. In a circular cylinder, for example, the transitory modes are defined in terms of harmonic functions of the form eaZJm, (ap) cos m (c- o0), where the fundamental numbers a are real or complex, being the roots of a certain transcendental equation whose form depends on the value of the integer m. The permanent modes correspond in a certain way to a = 0, and involve only rational integral functions of the rectangular coordinates of a point. They are in fact the six modes of equilibrium discovered by Saint Venant, along with the six rigid body displacements. A solution for any given source can be found which is expressed in two distinct analytical forms on the two sides of the source, so that only those transitory modes occur on either side which vanish at infinity on that side. Unless some condition at infinity is laid down, the coefficients of the permanent modes are to a certain extent arbitrary, but the diference of the coefficients with which each of the twelve permanent modes occurs on the two sides of the source is determinate. In a plate, the trasitory modes are defined in terms of harmonic functions of the forms (e z, e-z)f(x, y), where (- +a + )f and the fundamental numbers K are roots of the equations sinh 2Kch + 2Kh = 0, and sinh 2c, = 0, where 2h is the thickness of the plate. The permanent modes are rational integral functions of z, and involve functions of x, y which are solutions of the equation a2 all)e (ax% + F=o. For a point source in the plate, the functionf of the transitory modes is o, (KR),

Page  330 330 J. DOUGALL where G, (a) is that solution of Bessel's equation which tends to zero when the imaginary part of a tends to + x i, and the function F of the permanent modes is IR2 logR- 1R2, where R2 = (x - x')2 + (y - y)2. 3. It is usually easy to determine an infinite number of modes of equilibrium under no forces, and when this is so, if we assume that any deformation within a part of the body (bounded by two transverse sections) where no force is applied is expressible in terms of these modes, the well-known reciprocal property known as Betti's Theorem leads to a simple method of defining the coefficients under which they occur in terms of the given forces applied at other parts of the body. It might be possible, say by an application of the Theory of Integral Equations, to complete this process by justifying the assumption referred to, and this would be useful, for the permanent modes can easily be found for many forms of cylindrical boundary for which it is not at all likely that the transitory modes can be determined in manageable forms, and an approximate theory can be rigorously deduced from the knowledge of the permanent modes and of the expressibility of any source solution in terms of these and transitory modes. The method I am to describe, however, does not depend on the above or any similar assumption. 4. The first step is to find a solution for given surface tractions, and especially for those tractions which arise from the known solutions for a single force at a point of an infinite solid, in terms of particular solutions having no singularity anywhere in the solid. These particular solutions of which this, the classical form of solution, is composed, involve functions of the type eiKz, cos mn (co - co,), in the circular cylinder, Pit (cos 0), cos m,( - 0o), in the spherical shell, where K and n are real. Solutions of this class have been given by previous writers, notably for the spherical shell by Lame and by Lord Kelvin. The solutions for the plate by Lame and Clapeyron, and for the circular cylinder by Pochhammer* belong to this class, but they are not perfectly general. From this classical form of solution the solution in terms of permanent and transitory modes is deduced by an application of Cauchy's Theory of Residues. 5. The permanent and transitory modes are distinguished from each other most strikingly by their respective characters at a distance from the source, for the disturbance due to a transitory mode diminishes with great rapidity as the distance from the source increases, in virtue of the presence of a factor which, either exactly or approximately, is an exponential function of a negative multiple of the ratio of this distance to the thickness of the rod or plate. But even close to the source, although the transitory modes have an important local effect, yet in general their * "Beitrag zur Theorie der Biegung des Kreiscylinders," J. f. Math. (Crelle), Bd 81 (1876).

Page  331 PERMANENT AND TRANSITORY MODES OF EQUILIBRIUM 331 contributions to the strain are of a higher order in the thickness than those from the permanent modes due to the distant sources. As a consequence of this it can be shown that even if we neglect the transitory modes altogether and in the solution for a continuous distribution of applied force, which is obtained by integration from the source solutions, retain the permanent terms alone, the values of the displacements, and of the most important stress components are found correctly to a first approximation. 6. The solutions for an infinite cylinder or plate give of course particdlar solutions for the body force and the traction at the surface of a finite cylinder, or at the faces of a finite plate. If we retain the terms derived by integration both from the permanent and the transitory modes of the source solutions, the particular solution is exact, but by means of integration by parts in the case of the cylinder, and an analogous transformation in the case of the plate, this exact solution can be transformed into a solution in the form of series of terms of ascending order in the thickness, with a remainder. These series may or may not be convergent according to the character of the distribution of force, but they can be used to determine the displacement and stress to any assigned degree of approximation. At a cross-section at which the applied force or any of its space derivatives becomes discontinuous, certain terms of the transitory type are introduced in the course of the integration by parts referred to a few lines above. These transitory terms, however, do not interfere with the first approximation to the displacement and stress. 7. Lastly, it can be shown by an application of Betti's Theorem similar to a well-known application of Green's Theorem in the Theory of the Potential that the deformation in a finite cylinder or plate due to tractions at the ends or edges alone is expressible in terms of the permanent and transitory modes found for an infinite cylinder or plate. And the conditions at the ends or edges which define the permanent modes (which alone are of importance except close to the ends or edges), can be deduced, either exactly as in the case of the cylinder, or with sufficient approximation in the case of the plate. Thus, for example, by the use of the Green's function method, I have found* to a second approximation the edge conditions for the permanent flexural mode in a plate, originally given by Kirchhoff to a first approximation in correction of Poisson. I have prepared and hope to have published soon a somewhat elaborate account of the method as applied to the circular cylinder. The illustrative example given below is only meant to give a general idea of the nature of the analysis. 8. Infinite circular cylinder. Solutions in terms of harmonic functions. If (x, y, z) or (p, w, z) be the rectangular or cylindrical coordinates of a point the equations of equilibrium under no body forces are pV2 (UX, it,z) + (x + /) ( aX ay, ) =...............(1). * Edinburgh Roy. Soc. Trans. I.c. Article 46.

Page  332 332 J. DOUGALL The following three general solutions are easily verified: 02- au 2 a20 a2 _ 2 ( + 2/L) -qx= a22) X - it =Y-" — = - ' a+- o aZx Y 2 uz... X (2)? a9 a9 a9 itx = it,== ao Itz - ao x =, z az......................................(3); x= ay' U -,=.......................................(4); where q, 0,, are harmonic functions. We shall use the symbol v for -(4 + ) =(1-)4 ((5). In (2) au + + z = (2 - v) ax ay aZ a-; in (3) and (4) A = 0...................................... (6). The displacements in cylindrical coordinates can be written down at a glance from (2), (3) and (4). They are a2 ao 1 aq, p P a2 ap p aw I ao a* U,, = - --..................... (7). p ao ap a2 q a a o % Va p. uz = - p a-p - i' + - oaapaz %z az Since 2p =XA+ 2/A -p a, p aWit= =+/-,p, p ( PP =AA +PpL ap p a ] -a ~&p )(8), " a + 3pC E"\ p p p w) ' \dz op we obtain from (6) and (7) PP(2)a2P0 a"p ~2 a2 2 aB2 2 a* PP=( - 2) + 2p +2 +2 2 a3z2 *a, apap ap p apao P2 a p5 _ a' 2 a20 2 a_ a2, i af 1 a a2 L awaZ2 p pa~w p2 ( ap2 ap p2 )2 =- a3, aa 1- a20 1 a2f P = 2p V- ++ +2 20 + 1- 2+ /A pawza3 appz p aW2az apaz p awaz 9. Solution for normal traction N (z) cos (o - w'). We shall suppose that the function N (z) vanishes unless z lies between z, and z2, where Z2 > z. Then by Fourier's Theorem N(z) = 1 dc N () cos K (z - z') dz'..............(10). 7f J d.f 2, In view of this theorem, we begin with the simplified problem with surface conditions pp = cos (z - z') cos ( - ), =, 0.p = 0............(11). Assume ( ) = (A ) J, (ip) cosp K (Z (- ') i)............(12).

Page  333 PERMANENT AND TRANSITORY MODES OF EQUILIBRIUM 333 These values of q, 0, r give at p = a, from (9), pp -a = A (2 - vKc2aJ-2iKt3a3J') + B (- 22a2J") + C(-2) (J- iKaJ')} PP A cos KC (z - Z) Cos (O) - (A) pwo - = {A KtacJ + B. 2(J- iKaJ') + C (Ka2J" + i/aJ'-J)} (13) cos K (z - z') sin (o - o') pz a = {A (2KaJ + iKcaJ'+ J) + B (- 2iKaJ') + C(- J)} sin K (z- z') cos ((o - ') where the argument of the J functions and derivatives is itca. In (13) we shall write the coefficients of A, B, C for brevity in the forms Aal + Bbl + Cc, A a + Bb2 + Cc,..............................(14). Aa3 + Bb3 + Cc3 A solution of the problem with surface conditions (11) is therefore (12), with a2 A, aCB B, a2 C, A..... B = C-._(15), AD= ' = - C D.................(1), a,, bl, c1 where D = a2, b2, c2...........................(16), a3, b3, c3 and A,, B1, C1 are the first minors multiplying a,, b,, cl in D. In order to obtain a solution of the problem of normal traction N (z) cos (w - o') it is now natural to try the result of multiplying the values in (12) by N (z')/r and integrating as to z' from zl to z2, then as to K from 0 to oo. It will be found, however, that unless the form of the function N(z) is considerably restricted, the K integrations cannot be carried out, on account of the form of the functions p, 0, s defined by (12) and (15) near K = 0*. But the difficulty is easily surmounted. For the terms of negative degree in the ), 0, r of (12) and (15) can contribute nothing to the surface tractions at p = a, simply because the values of pp, pW and pz in (11) contain no terms of negative degree in their K power expansions. Moreover these terms of negative degree in q, 0, f of (12) and (15) are of the form H4/K4 + HJ2K~, so that all we have to (o is to subtract them. Write the terms to be subtracted as (i,, 00, 0o)-. We thus obtain for normal traction N (z) cos (w - o') the tentative solution (,0 a 2 [x 2 ~ 1 Al^ii-i\ Bo )o~ T ) (~ 0) ~= a d/K I){(ADC Bi) J ((i(Kp)cosC( (z - Z) sin)N()d -- 7/?'/ 0 zi - \ 9o sin......(17). It is not difficult to show, by calculating p p, p5, for any internal point from (17), and proving that these pass continuously into N (z) cos (co - '), 0, 0, respectively as p approaches a, that (17) is actually a solution of the problem proposed. * In Pochhammer's solution I.c. K is an integer, and he avoids a similar difficulty to that which arises here by restricting the form of N (z).

Page  334 334 J. DOUGALL The order of integration in (17) can be changed, and we can put (t0) -| ( 0) N(z') dz'........................ (18), where JI (i) COS KD\[ 0, } ( c (- Z') COS (ft -^ (1)dc...(19). (4' aI)~, =C, a(I{ l) J ipcsin *o Equations (18) and (19) define a solution of the problem proposed at the head of this article, of the type referred to in Art. 4. 10. Expression of the definite integrals as complex integrals. In each of the integrals of (19) the integrand is a uniform even function of K of the form E (K) cos c (z - z') - Eo (c), where E0 (K) stands for the terms of negative degree in K in the power expansion of E (K) cos c (z - '). Now f E (K) cos (z - z') - Eo (K)} de = - f E (K) cos K (z - z')- E, (K)} dK...(20). The latter integral can be regarded as a complex integral, and its path of integration can be deformed. We define a first and a second path of integration as passing from - oc to cc along the real axis, except near the origin, where they follow a small semicircle above and below 0 respectively; also a third and a fourth path as passing along the imaginary axis from - cc i to cc i except near the origin where they follow a small semicircle to the left and right of 0 respectively. Then the right-hand member of (20) = ~f {E (K) cos K (z - z') - E (/)} dK (first path) = -fEKc cos K (z- z') d. (first path)......................... (21) (for evidently K or iC over this path vanishes). \ --- -~ —` —J J fC2 J /C4 This again = fE (K) e-iK(z-z' dKe (first path) + 4 fE (K) eiK (z-') dK (first path)..... (22) = lfE (Kc) e-iK(z-z' dic (first path) + fE () iK () de- K -z d (second path).. (23), as we see by changing K into - K in the second integral of (22). The result may be written f0x K {E (K) cos K (Z - z') - E0 (K)} d = 4 fE (c) e-iK(-') dK (first and second paths)...(24). In the integrals on the right of (24) change the variable of integration from Kc to /3 where /3 = i a....................................(25).

Page  335 PERMANENT AND TRANSITORY MODES OF EQUILIBRIUM 335 If E (ic) thus becomes Es, we obtain 1- i f(z-z ') {E (K) cos K ( - z') - Eo (K)} dK = 4 e-a e a d/3 (third and fourth paths) (26). 11. Transformation of the complex integrals into series, Applying the transformation (26) to the integrals of (19), we find __ 0A ^ ^o _ c/A1B f (3p -')cos ((,, a JI A Bipe a i (W - o') d1 (third and fourth paths) (27), \ ^if 47(rip D G a sin where the functions D, Al, B,, Cl defined at (16) are to be expressed in terms of /3 by means of (25). The integrand in (27) vanishes at infinity in such a way that, by Cauchy's Theorem, each integral can be replaced by (- 27ri) (sum of the residues of the integrand at the poles of D to the right or left of the path according as z - z' is positive or negative). Hence, if z > z', 01 a EAl B) J qP COS e a A B o \ V~ ~dDldf C, a sin ) -- coefficient of lin D B, J~ 1 p e a o ( - a) (2/) ) (A B P e a sin) where E denotes summation over the zeroes of D whose real part is positive; it 13 can be proved that these are all simple. If z < z', the corresponding results are found most simply by interchanging z and z' in (28), retaining the meaning of S unchanged. 12. Permanent terms of the solution (28). The principal terms in the expansions of A,, B1, 0C and 1/D near P = 0 are ' 4 7 A6 A 1184_ 1 '86, A 186 + 3 '8} Al= 4-T 6, A2=-.34- 6,4, A3-= 7. 8 + 1 6 ( 64- 7v _...(29). D = (8 )(8 - v~ ) 8-/- + 2 4 With the help of these results, the partial values of 0,, 01, *l in the second line of (28) are easily calculated, and lead by (7) to the displacements (z > z'), 'tp= { (- _ z') + 34 (z -') p2 cos (o - o) ) with the rigid body rotation nZp = (z - z') cos (O- &7) /L /( *^ v 2 v-20v+92{ 1 u, = - (z - z ) sin (w - W ) 7r ( -) }..... (31) uzpc05(O c04 7,rr(8-v) 3wtj (8 - v) a tz = - p2 COS (co - o ) For z < z' the signs of all these displacements have to be changed.

Page  336 336 J. IDOUGAL, The functions of v which occur here are easily expressed in terms of the familiar Poisson's ratio a and Young's modulus E by the relations 4 —v 4 p (8- v) = 2E v2- 20v+92 2u + 2 (8- v)2 E E2........................(32). )............ 13. Transitory terms of the solution (28). The function D possesses an infinite number both of real and of complex zeroes. For the large real roots = N'n + 83=N7r+74r_,1, /19\ } and for the large complex roots.(33), 13 = Nr +- log 4N- + i (lo I4N7r + approximately, where N is a positive integer. The asymptotic forms of the high transitory modes in the first line of (28) are different in the two cases of real and complex values of 3. Real roots of D. 11= 4(.+2) 2(.+2)2 14(v2)N AI1 --- — W 2B CN (N 4 (v +2) d......(34),. = 8 V27r- * cos N~- (N)TT d,8P ap - u, = - N7r =Q- \_p \(p p al/at 1 2) si* (Nrp UZ= -- P Fj p asa P1 N (2p_1 tjb \a2 p, os (,, Cos ( c, os (,, sin (,, sin ( - -a) cos (0) -- ') 4 a,, ) sin(,, ),, )cos(,, ) (- ))cos(,, ),, )sin(,, ),, )cos,, ) v+2 1 2/e, N4qr4 - (e+V7r) Z - e \ 4 a...... (35) where z > z'. Complex roots of D. A1= 4N37r2, dD d/3 B1 = - 4iN4w3, Ci = - 4NV3r2,/2 7r- cos N e 4 (N') ( 36i)r l6V-w3c.....e..N.)..(36), 16 4,27r t- ~ cos A7r e- 4 (N7,r).~......

Page  337 PERMANENT AND TRANSITORY MODES OF EQUILIBRIUM 337 ('l,, UZ) — ' % Y- ) (p -1 ip (z' ) cos 1 1 - / It a o Cos N( rip...... (37), P ) a I a \ 7r\p J /...... (38), where z > z'. 14. Solution for distribution of normal traction. The solution for normal traction N (z) cos (o - o') from z = z2 to z = Z2 is given by (18), in which we can take for (,, 0i, r1 their values as in (27) or in (28). Let any displacement u be calculated from (27) and (7), and let the result be I U f-(z-Z') u=- 4 —jD e a d,/ (third and fourth paths).........(39). Here U is a uniform function of 3, even for Up and u,, odd for uz. For definiteness suppose now that u is up. The transformation by Cauchy's Theorem gives, for z > z', L U. 1(z-z') U P(z-z') uz,= coeff. of in - e a + -e.........(40), 1 U ( ) U (-z') for z < z, p = coeff. of -in e a + dD e a............(41). 2 D -dD/d(4 The value of u, corresponding to (18) is now (when zI < z < Z2), u, - N (z') {u, of (40)} dz' + ~ N(z') u, of (41)} dz'......(42). From the asymptotic forms of u, in Art. 13 it will be found that the integrations as to z' here can be performed term by term. Thus, for zi < z < Z2, ~z ~ 1 U a T -) z,( 1 U (z-z') Uo ^= ~ coeff.of-in - e a AN(z')dz' + coeff.of-in e a N(')dz' Jt 1z - z ') ) J z (z') LD+ d l {:e a N(z') dz' + e a N(')d'}j...... (43), for z > Z2, coeff. 1 N U i oeff.of in e a (') dz' + e a ()...(44) and, for z < zj, r2 1 77 — Zh 7 Z, _( 1 ) uo =I { coeff. of in D e a N (z') dz' + E d N (z' ) dz......(45). The values of up in (40) and (41) are simply what we should get by calculating 1p from (28); and similarly with us and u,. M. C. II. 22

Page  338 338 J. DOUGALL The nature of the solution for z > z, or z < z, is obvious. It is compounded of permanent and transitory modes, being obtained by the integration of such with respect to a variable and between limits which do not depend on the current coordinates. For z1 < z < z2 the case is different. 15. Nature of the solution within the region where theforce is applied. Consider the second line of (43), say Ur r jz _( -z) rz j ( -Z') - dI=E adD \ly e a N (')dz' + e a N(z) dz'...(46). Repeated integration by parts gives IU= E{d2iD-{-a + N (z)+ N... to the term in a'-l or a'l = LWIH / +3 dZ E (-i) e ( +. (aZ) a2 d a-n d n-' N( 13 LdD/d, (/3 /2da d1 / I dzn1t-i _ U _ e 2-^ )a N 2 d7 an d n-i \(,~) dz2 N) d~-' N (Z2)}] dDIdJ )32 dzN (xN2 Rn df l2 —l rdU Da - z-z'-) dn afn e-( )z' ) d+E /1 (-1 Ja ' da N (z') + dz' + d - N () d'J......(47). Consider separately the four series in the four lines on the right of (47). In the first line, the sum of the series can be found in finite terms, for it is I N2a 2a3 d2 [Residue at / ofD - N (z) + 3 N(z) +... to the term in an-l or a 4 (48). The function of which the residue is taken here is an odd uniform function of / which vanishes at infinity in such a way that the sum of all its residues vanishes. But the residues at + /3 are equal, so that E [ ] in (48)=- residue at 0 of the function of f in [ ] in (48) = - coeff. of in u (z) +2 d N (z) +... to term in an-' or all 2 - COeff1of$i (z) +......(49). Now the ascending power expansion of U/D has the form U = U(- -4 U(-)3-2 U(O) 4o( + U (2) 2 +........(50). Hence the first line of I in (47) a=-aU() N(z)-a3U(2) d 2 N(z) -.. to the term in a'-l or an...(51). This will be considered in next article in combination with the first line of (43). In the second line of (47) the general term is proportional to the displacement 3(z -,) Up= Ue a belonging to one of the transitory modes.

Page  339 PERMANENT AND TRANSITORY MODES OF EQUILIBRIUM 339 When a is small the corresponding deformation is insensible except very close to the section z = z,. Similarly the third line of (47) represents a local perturbation at the end z = z,. Lastly the fourth line of (47) contains a series of the same general character as the series (46); but this series is at once of a higher order in a on account of the factor a," and more highly convergent on account of the factor /3- in the general term. 16. The asymptotic particular solution within the region of applied force. We consider now the first line of (43) along with (51); let the part of Up arising from the sum of these be denoted by up). By (50) U = U(-4)3-4 + U (-2/-2 + U(~ )3~ + U(2) +.... Also e a = 1 (- ) + 3... a 2a 6a3:-;~~~~~~~~~~~~~......(523). + { — a-3 U(-4) (Z - Zy)3 + a1 U- (-2) (z -z)} N (z') dz' It is convenient to introduce a function F, defined as follows: when z < z,, F = f (z - z') N ( z) dz' whenz < Z < < z, F= z - Z') N (z') dz' - 12 (z z) N (z') dz'....(54). Z1 functions of z, and that d4F dz4= (z). We shall write, as equivalent to the definition (54), F = D -4 (Z).................................( 5) Also we shall write Dz-2 NV (z) for dz2 Thus (53) becomes - a-4U(-4)D-4N (z) - a-a U(-2)Dz-2 N (z)................(56). Hence the sum of the first line of (43) along with (51) is dz,p(~o) = -a-3 U(-4) Dz4-N (z) - aU(- Dz-2 N (z)-aU(~)N (z) - a3U(2) N(z) -... to the term in an-1 or an......(57). The functions U(-4), U (-', U(~), etc., are all of order 0 in a. 22-2

Page  340 340 J. DOUGALL The value of up(~) in (57) along with the fourth line of (47) gives the value of up in an exact particular solution. In (57) the terms of negative order in a, and involving DZ-4 and D-N21 (z), are just those which come from the permanent terms in the source solution, i.e. the terms in (30) and (31). Hence if these permanent terms only are used, the displacements neglected are of an order in a higher by two units than any of those retained. It is not difficult to show that this must be so, independently, by consideration solely of the dimensions of the terms in the unit of length. If N (z) is a rational integral function of z the series (57) continued indefinitely terminates, and we obtain a particular solution in finite terms. In the general case when N(z) has an unlimited number of successive derivatives it may happen that the remainder (in the fourth line of (47)) tends to zero with increasing n. The condition for this can be shown to be that the power series in ' E a dn N (z) fn n dZ should have a radius of convergence greater than 1/ 3o I, where,0 is the zero of D with smallest modulus. In general, however, the series (57) is only asymptotic.

Page  341 COMMUNICATIONS SECTION III (b) (ECONOMICS, ACTUARIAL SCIENCE, STATISTICS)

Page  342

Page  343 EQUILIBRIUM AND DISTURBANCE IN THE DISTRIBUTION OF WEALTH, BY R. A. LEHFELDT. It has been said that the element of time is the centre of difficulty in economic problems. The following is an attempt at a method of expressing these time relations in mathematical form, so that they may be apprehended with the clarity that always attaches to mathematical language. The paper arose from Prof. Edgeworth's recent articles in the Economic Journal (Sept. and Dec. 1911). I. In a single business, the entrepreneur must be regarded as a constant factor. If by association with quantities x, y,... of other factors he generates an amount of product f(x, y,...); if p be the price of the product, and I, 7,... the prices of the factors, then the profit is v= Qbf - F - vy -... First assume that all the prices are constant; then to make v a maximum, we must have -df = 0, -df _-0, etc., the solution of which is, say, X =, y=y0, etc. The business should therefore grow to and stay at this size, and the quantity Vmax. will be the return to the entrepreneur. The question has been raised whether this quantity tends to a normal level: this is considered below, but while dealing with a single business, it is best to put the entrepreneur in a different category from the other factors. The assumption of constant prices depends on the insignificance of the business, compared with the industry or the nation of which it forms part. There are, of course, limits: a very great increase in a single manufacturer's product might lower its price, or a very great increase in his demand for skilled labour result in higher wages. Usually the assumption is practically true in those respects, but fails sooner in respect of capital. A very great increase in the manufacturer's demand for capital would probably make the lenders (the banks) hesitate, and raise their price for it. Otherwise vmax. might conceivably become indefinitely great.

Page  344 344 R. A. LEHFELDT If all the conditions are constant, there is no problem. The real question is, What are the consequences of something new happening? There may be a gradual or a sudden change in conditions: we take the latter first. Let x be altered abruptly to x1. In general, its price will alter also, say to ~'. Then the equation of profit becomes v'= Of — '/ -y... and there is a new state of equilibrium characterised by dV'0=o=df-' dv- = -, etc., dx dx dy dy whose solution is = x', y = y', etc. If xi < x, then:' ': and x0' will usually lie between x0 and x,. (I omit complications, such as functions with more than one maximum.) yo' may be >= or < y0 according to the nature of the function. The business will then move towards the new maximum. Consider particular cases, in order to bring out the meaning of these equations. The abrupt change may occur within or without the business. Changes within the business do not, however, usually alter prices, and hence do not cause a change in the function v; this is nearly always true as concerns labour; changes of capital require more consideration. (a) Some of the capital may be abruptly destroyed, as by a fire (not covered by insurance). Then the entrepreneur will probably have no difficulty, if the business was making normal profits, in persuading his banker that the damaged buildings or machinery should be renewed, and the banker will lend the money at normal interest. (b) A sudden influx of capital, by floating a debenture issue. If the business were really in equilibrium before, this money could only be used to fund floating liabilities, and any balance over would have to be lent outside. No appreciable change would occur in the price of capital used, and the profits function would tend towards the same maximum as before. Next take changes outside the business. Capital might become suddenly scarce by destruction on a large scale, say by a war or an earthquake: or plentiful by a boom on the Stock Exchange turning investors' attention to the industry. (Similar, but smaller changes are constantly happening and showing themselves in fluctuations of the bank rate.) Then the price of capital will be changed and the business will tend towards the new maximum. Labour, also, may change in price, owing to emigration, immigration, sudden demand for some other industry, famine, epidemic and so on: or temporarily through a strike. Another cause leading to disturbance and a new position of equilibrium is invention. The adoption of a new process may be expressed conveniently by introducing a new term in the equation, say - z (z having previously been zero); dv' and this is usually characterised by the fact that d-v is exceptionally large, so that duz

Page  345 EQUILIBRIUM AND DISTURBANCE IN THE DISTRIBUTION OF WEALTH 345 as soon as that direction is rendered available by the invention, there is a strong tendency for the business to move along it (with the corresponding readjustments in x,y,...)* The business then will, after the disturbance, tend to move towards the position of equilibrium (the new one if a price has been altered, the original position if not). To take the simple case of two variable factors, the tendency will be to move in the direction which makes with the axis of x an angle whose tangent is (dv' dv' dy dx (x=x,, Y=Yo0) The tendency, or driving force is in this direction, but the resistance to changes along x and along y will not in general be the same (these resistances include economic friction of various sorts). Hence the movement of the business will not be in the direction above indicated. Call the driving force dv = X. Then we may write as a first approximation dx X=h d dt as an equation of motion (t = time, h = constant)t. X depends for its amount on the departure from equilibrium x - x'. If we took it to be r (y)(x - x') the solution would be in the well-known exponential form, but as near states of equilibrium, the surface is curved about a maximum, it would be better to take X = r (y) (x - x)2, which yields the solution - ~(X~ - X =q say. -(x - 0') = (y) t t Here q represents the slowness (or inertia) of recovery from a disturbance of the given type. The constant q in the case of disturbances within the business, would express the delay in reconstructing machinery after a fire, or in taking on new hands after a strike, and so on. In the case of outside disturbances, the much longer delay in saving new capital, or in growing and educating new skilled workmen. By means of it, greater precision can be given to the ideas associated with such terms as "normal price," "in the long run," etc. In any case the direction in which a business will move after a disturbance depends on the q's as well as on dv'/dx, dv'/dy. If, as is often the case, one q is much greater than another the business will move, first, approximately in the direction of the smallest q, then in that of the next larger, and so on. Gradual changes in the conditions will differ from abrupt changes, in that x- x the departure from equilibrium will never be large. The business continually adjusts * -I may be called the "profit gradient " in the direction of x. ~~~dX~~~~~~~~dx t x as well as y, f, etc. are really fluxes or velocities; hence d corresponds to acceleration in dynamics, and 1t is in some aspects a measure of inertia rather than of friction.

Page  346 346 R. A. LEHFELDT itself towards the equilibrium state, i.e. x tends towards X0, but as the latter is changing continuously, x never reaches it. The departure from (or lag behind) equilibrium, in any direction, will depend on the value of q in that direction. II. From the point of view of an entire industry or of the community at large, there being a number of entrepreneurs, it is convenient to regard their services as paid for at normal rates, according to ability, and introduce a term vu where u represents the quantity and v the price of this factor of production, giving v = Of - vu - x - rqy -.... Although the entrepreneur gets paid in quite a different way from the capitalist or the workman, there can, I think, be no doubt that in the long run his receipts usually tend to an average depending on his productivity, so that the symbol v has a definite meaning. If this is so, then on the average and in the long run v in the last equation will be zero (provided the number of separate businesses is enough to justify statistical treatment). At any time when v is not zero, it constitutes a surplus or deficiency in profits belonging to all the entrepreneurs. More generally, if a change in conditions alters: to ~', then the whole surplus accruing, as a consequence, to a unit of x is ((- ) dt, J which could be found if the relation of r to x (i.e. the elasticity of demand) and q be known.

Page  347 I CARATTERI MATEMATICI DELLA SCIENZA ECONOMICA Di LUIGI AMOROSO. (ABSTRACT.) Proposing to apply to economics methods which have succeeded in mathematical physics the author obtains a partial differential equation relating to the distribution of incomes, from the following postulates: I. There exists a continuous function z = F(x, t) with continuous derivatives such that F (x, t) dxdt represents for every value of x and t the number of individuals who during the interval of time between t and t + dt have an income comprised between x and x + dx. II. If x0 and x, are two very closely neighbouring values of x, the number of persons who at the moment t pass from income x0 to income x, is, other things being equal, proportioned. to the difference between the number of persons who at the moment under consideration have the income x, and the number of persons who at the same moment have the income x0. From these postulates the author deduces partial differential equations, such as dz d'2z dt =P d2 (p a constant), for which it is claimed that like the partial differential equations in the physical sciences-that of heat for example-they may lead to important practical conclusions, such as the distribution of incomes at any moment t if three data are given: 1. The initial distribution at moment to. 2. The increment of population in the same interval from to to t. 3. The constant p. It is supposed that in the interval to, t no radical perturbation of incomes happens for any reason whatever. These perturbations would render the integral not regular.

Page  348 REDUCTION OF ERRORS BY MEANS OF NEGLIGIBLE DIFFERENCES BY W. F. SHEPPARD. Abstract. The so-called "graduation" practised by actuaries consists in replacing each term of a sequence by the weighted mean of a number of terms, so as to give an opportunity for neighbouring errors of opposite signs to balance one another. This process is not legitimate unless the difference between the original and the substituted term is a difference which would be negligible if the original terms did not contain errors. The underlying assumption therefore is that differences of the true values, beyond those of a certain order, are negligible; and the substitution is only legitimate if the mistake thus introduced is expressible in terms of these negligible differences. In the same way the existence of alternative formulae for quadrature, etc., implies that the difference between two formulae represents negligible differences of the terms of the true sequence. Conversely, any formula expressing a quantity by a linear function of a set of terms of a sequence may, within certain limits, be modified by the addition of terms representing the negligible differences. The added portion will involve arbitrary constants, and we can choose these so as to give the best result for the new formula. For comparison of goodness, some criterion is required. We take the mean square of error as the criterion; the problem therefore is to find the values of the coefficients when the mean square of error of the total expression is a minimum. It is not possible to give more than a formal solution for the general case; but, for the standard class of cases in which the errors of the terms of the sequence are independent and are all equally variable, the solution is obtained explicitly in terms of central differences. The method differs in essentials from the method of least squares and the method of moments, but it is found to give, for certain problems, the same results as would be given by these methods. The formulae obtained may therefore be regarded as giving (for the standard class of cases) a general solution, in terms of central differences, of the equations which arise in these methods. The method is extended to particular classes of cases where the errors are not independent or are not all equally variable.

Page  349 REDUCTION OF ERRORS BY MEANS OF NEGLIGIBLE DIFFERENCES Contents. 349 PAGE I. INTRODUCTORY. 1. Material to which the method applies 2. "Graduation" as an example of the method 3. Conditions to be satisfied.. 4. Criterion for best formula..... 5. Modification for terminal values 6. Statement of general problem. 7. Alternative initial formulae 8. Notation....... * *... 350 351. 352 352 352 353. 353 354 II. FORMAL SOLUTION OF GENERAL PROBLEM. 9. Statement of problem, for odd number of given values. 10. Method of solution... 11. General conclusions..... 12. Value of mean square of error. 13. Method for even number of given values 356 356 357 358 358 III. CASE OF ERRORS INDEPENDENT AND EQUALLY VARIABLE. (A.) General Solution. 14. Separation of portions representing differences of even and of odd order, for odd number of given values.... 15. Solution as regards differences of even order.... 16. Solution as regards differences of odd order.. 17. Solution for even number of given values... 358 359 362 363 (B.) Applications. 18. Graduation. 19. Interpolation 20. Quadrature 21. Other applications 365. 366 367 368 IV. RELATION TO METHOD OF LEAST SQUARES AND METHOD OF MOMENTS. (A.) Standard system (errors independent and equally variable). 22. Comparison of methods.. 23. New formulae as a solution of the moment-equations 24. Finality of the moments..... 368 370 372 (B.) Other systems. 25. Modifications of the method of least squares and of the method of moments V. TREATMENT OF NON-STANDARD SYSTEMS. 26. Special cases...... 27. Frequency-distributions..... VI. GENERAL OBSERVATIONS. 28. Relation between central and terminal conditions... 29. Cases of failure....... TABLES. Table I. Formulae for vo etc. (standard system).... Table II. Values of reduction-ratio for central term (standard system) 373 375 376 377 377 378 384

Page  350 350 W. F. SHEPPARD I. INTRODUCTORY. 1. The preliminary property, on which the methods considered in this paper are based, may be seen by differencing an ordinary mathematical table. The following is an example, the position of the decimal point being, as usual, ignored in writing down the differences. 1st 2nd 3rd 4th 5th 6th 7th 8th x | 0logo0 X diff. diff. diff. iff. diff. diff. diff. diff. 5-4 '7324 -1 -2 + 8 - 32 +80 - 1 +4 -16 5-5 '7404 -2 2 -8 +33 +78 +1 -4 +17 5-6 '7482 -1 -2 + 9 - 38 +77 -1 +5 -21 5'7 '7559 -2 +3 -12 +46 +75 +2 -7 +25 5-8 -7634 0 -4 +13 -44 Here the successive differences at first are regular and decrease (numerically), and then become irregular and increase. The reason of this is that each term in the column log0, x consists of two portions, viz. the true value and an " error" which has to be added (algebraically) in order to obtain the corrected final figure. The effect of this error may be seen by writing down the true value to 7 places of decimals, and the error, for each term. x 5-4 5-5 5-6 5-7 5-8 107 loglo 7323938+062 7403627+373 7481880+120 7558749+ 251 7634280 - 280 1st diff. +79689+311 +78253 —253 +76869+ 131 +75531 - 531 2nd diff. - 1490+0490 -1436- 0564 - 1384+0384 - 1338 - 0662 - 1291+1291 3rd diff. +54- 1054 +52+0948 +46-1046 +47+1953 It will be seen that the differences of the true values decrease numerically (the apparent irregularities in the 3rd differences being due to our having only taken the true values to 7 places), while the differences of the errors increase. This property, which is the preliminary condition for the application of the methods of the paper, may be stated as follows. (i) We are dealing with a sequence ~... ' —i, U U, u 2.... (ii) The terms of this sequence correspond to those of another sequence... U_1, U0, U1, U2,..., but differ from them by errors... e_1, eo, el, e2,..., so that, if Ur is any term of the former sequence, Utr = Ur + er.

Page  351 REDUCTION OF ERRORS BY MEANS OF NEGLIGIBLE DIFFERENCES 351 (iii) The terms of the second sequence, i.e., the true values of the u's, are so related that their differences beyond those of a certain order, which we shall throughout denote by j, are negligible. We can define more precisely later on (see ~ 3) the meaning of the word " negligible." The sequences which possess this property are of many different kinds. We have taken a mathematical table as an example; but this is rather because our knowledge of the U's enables us to illustrate and test our methods than because of any actual utility of the methods in reference to such tables. The cases to which our methods have to be applied cover, on the one hand, cases in which the U's are actual magnitudes, such as lengths, pressures, velocities, etc. measured more or less imperfectly at a series of points of space or of time, and, on the other hand, cases in which the t's are statistical data which are taken as varying from hypothetical U's by errors due to the paucity of the observations. 2. The problem which we have to consider is a generalisation of, and includes, the problem of what is-not very happily-described by actuaries as "graduation "; and we can consider the nature of this particular case before proceeding to the general problem. The conditions are as follows. The U's are certain statistical ratios, whose values, as found from the limited data available, are the u's. The U's, if we had them, would proceed regularly, in the sense that condition (iii) of ~ 1 would be satisfied. But the u's do not proceed regularly, since their errors produce an irregularity, analogous to the irregularity discussed in ~ 1. " Graduation" consists in replacing the sequence... u_-, Uo, u,... by another sequence v_1, Vo, vI,..., such that each v is a linear compound* of the corresponding u and n others on each side of it, i.e., that VO = PnUn +Pn-Un-_ + p... + P -nU-n, where Pn, pn-i,... P-, are numerical coefficients; but the process may be regarded in two different ways. Usually the object seems to be to obtain a sequence which is more regular than that of the u's; i.e., the process is merely one of "smoothing." But this is not a scientific method, unless some criterion is adopted for deciding which of the possible new sequences is the best; and it is difficult to apply such a criterion except to the sequence as a whole, in which case the process becomes one of " fitting." The better view, which enables us to treat each term separately, is that we wish to apply a correction to the term in order to make its error as small as possible. If we can do this, the new sequence will be smoother than the old one, not merely because the new errors will be smaller, but also because the errors of adjoining terms will be more closely correlated. But this smoothness should be regarded rather as an incident than as an object. In any case, certain conditions must be satisfied in order that it may be legitimate to replace u0 by vo. We have therefore to see (1) what these conditions * I use this term in preference to the ordinary term "linear function," since we are not concerned, for the most part, with functionality; what we have usually to consider is the proportion in which the various u's, or the differences of various orders, have to be compounded in order to produce the required result.

Page  352 352 W. F. SHEPPARD are, and (2) what criterion we are to adopt for determining whether one set of p's which satisfies these conditions is better than another. 3. Since Ur-= U+ er................(...............(1), we shall have Vo = Vo +fo.................................. (2), where Vo P- p + Un +...p U _- +..................(3), fo pn en + Pn-ien-i +. +Pne-n.....................(4), so that in substituting Vo for u, we really substitute Vo for UO and f for eo. Assuming (as a condition of our method) that the mean value of each e is zero, it follows that the substitution of Vo for u, is not legitimate unless it would also be legitimate to substitute V0 for Uo, i.e., unless the difference between Vo and Uo is negligible. This difference can be expressed in terms of Uo and its central differences up to that of order 2n inclusive. But, by hypothesis, differences of orders 1 to j are not negligible, while those of higher orders are negligible. Hence, for practical purposes, the necessary condition for replacing uo by v0 is that VF - U0 should only involve differences of Uo of orders j + 1 and upwards; i.e., that o, - uo should only involve differences of u, of orders j + 1 and upwards. We may therefore say that vo is formed from u, by adding terms involving these differences. Whether, even then, the substitution is legitimate in the particular case, will depend on the actual magnitude of these differences of U0 and on the magnitude of their coefficients in the formula. We must therefore, in describing the differences as negligible, remember that this description is a relative one, and that the differences may cease to be negligible if the numerical coefficients by which they are multiplied become too great. 4. The condition stated in ~ 3 implies j + 1 relations between the coefficients p-1n, p-n+i,..-. p in the expression for vo; and, subject to these relations, we want to choose the p's so as to make fo, the error of v0, as small as possible. This we cannot do. But it is necessary to adopt some criterion to determine whether one formula for v0 is better or worse than another. For reasons familiar in the theory of error, we take the mean square of error as a criterion, and say that one formula is better or worse than another according as it gives a smaller or a greater mean square of error of v0. Hence the problem of "graduation," in the more correct sense explained in ~ 2, is to find the p's so as to make the mean square of fo = pn en + pn-i en-i +... + P-n, e_, as small as possible, subject to the conditions stated above. (The adoption of the above criterion is, of course, only a delimitation of the cases we have to consider. We must adopt some criterion, in order to apply mathematical methods; if in any particular case the criterion is unsuitable, this only means that our results do not apply to that particular case.) 5. This problem leads to its own extension, if we consider what we are to do at the extremities of the range. Suppose, for instance, that j = 3, m = 9; i.e., that differences beyond the 3rd are negligible, and that we are ordinarily using 9 consecutive u's to improve the value of the middle one. Iet the values at the beginning of'the sequence be denoted by u1, u2, u,.... Then the above statement of

Page  353 REDUCTION OF ERRORS BY MEANS OF NEGLIGIBLE DIFFERENCES 353 the problem applies to the improvement of u5, u6, u,...; but what are we to do with U4, ut, u2, and u,? (i) We can leave them as they are. This means that the improvement of the sequence is incomplete. (ii) We can improve u4 and us by using 7 and 5 u's respectively. But this does not enable us to do anything to Ate (since the first 3 terms only give 1st differences and a 2nd difference) or to u,. (iii) The difficulty is met by extending our method so as to enable us to use the 9 terms ul, u2,... u, for the improvement not only of u5 but also of u,, u3, U2, and ua. Taking any term, such as t2, we compound it with differences of orders 4 and upwards; and we then choose the coefficients in the resulting expression so as to make its mean square of error a minimum. 6. This leads us to the general problem, which may be stated as follows. Our data are m consecutive u's, corresponding to m consecutive U's. There is a quantity W which either (a) is one of the U's, or (b) is a linear compound of two or more of them, or (c) can be approximately (i.e., within the limits of accuracy within which we are working) represented by a linear compound of this kind. Let w be the first approximation to W, obtained by replacing the U's in W by the corresponding u's. Then the problem is to determine the form of a quantity z such that (i) z is a linear compound of the u's, (ii) z differs from w by differences of u of order j + 1 and upwards, and (iii) the mean square of error of z is as small as possible. For any particular set of data, the particular problems included in the above fall into two classes. The first class comprises those in which the total number of u's is greater than rn, so that we apply one process to a number of different sets of u's, dropping one u at each stage and taking up another. This class includes the problems of (1) "graduation," (2) calculation of ordinates when the data represent areas, or of areas of consecutive strips when the data represent ordinates, and (3) determination of direction of tangents, for the purpose, e.g., of determining modal values (values corresponding to maximum ordinates). The second class comprises the cases in which we have only the m i's and are dealing with them all together. Here the specially important problems are those of determining the best formulae for quadrature and for calculation of moments. Interpolation is a matter which requires to be dealt with specially. 7. In the cases coming under (c) of ~ 6 there will usually be several alternative formulae for W, i.e., there will be several linear compounds of the U's, any one of which would, if the U's were known, give W with sufficient accuracy. But this will not affect our results, since the differences between the alternative expressions will themselves only involve differences of U of orders exceeding j. The following cases will illustrate this; j being taken = 2. (i) Suppose that the U's correspond to values... x_-, x0, i,... of a known quantity x, these values differing by a constant increment h, so that Xr = x, + rh; and suppose that there are intermediate values of U, which can be represented by a M. c. II. 23

Page  354 354 W. F. SHEPPARD polynomial in x of degree not exceeding m-1. Then, if Uo is the value of U corresponding to x = xo + Oh, where 0 < 0 < 1, we have as possible formulae for Uo Uo= Uo+ 0a Uo + 0 ( - 1) 2 Uo, Uo= Uo + OA Uo + 0(0-1)A2 C. But these differ by 0 (0- 1) A3 U_. (ii) On the same suppositions, let m = 5, and let the required quantity be the area of the figure bounded by the ordinates U1 and U,. Then the values of U' (d U/dx), and of U,' (dU/dx), will be given by hT1'=(U ( U-)- ( U3-2 U, + U1)=- (U3-4 +3 U), hU5' = ((U - U4) + - (5 - 2U4 + U) = 1 (3U - 4U4 + U3); and therefore, by well-known formulae, we have as alternative expressions for the area W=.h(U + 4U2 2U+44 + U,), W -=. h(Uu + 2 U22U3+ 2 U4+ U5)-^ -. h (3U1- 4U2 + U3) -2 4. h (U3- 4U4 + 3 U5) =. h (9 U + 28 U + 22 U+ 28 T4 + 9 9 U). It will be found that these differ by 4 (U, - 4 U4 + 6 U, - 4 U2 + U,) = 4. A4 U1. 8. Notation. (i) Our formulae will be expressed in terms of central differences; and therefore, supposing the data are m u's, the formulae will be of different kinds according as m is odd or even. (a) Let m = 2n + 1. Then we take the u's to be un, un+1,... un, so that the central u is Uo. The central differences of u0 are /S, 82Uo, U,83Ua0,..., where J8O 2 {(?t1 - UO) + ( U_0 - U-)}, 82UO 1- 2o0 + '-1, tL82r+1 Uo _,8. 82X uo, 82r+2 t0-o 82 82r o. We can then express z- pnun + pn, u^n_ +... + p-n-,, in the form z = qo Uo + q1 1/to + q2 82Uo +... + q2n 82' Uo. (b) Let in = 2n. Then we take the u's to be uz-+1, u-I_+2,... ut, so that the two u's in the middle are Uo and i1. The central-difference formulae then involve iu.L and its central differences 8ul, g82U, 83u1,..., where GUA — 2 ('it + u.o), 8 U, - it(, 82r~ U 11,. 82rU, 82r+1U_ --. 82rU1; and z pNun + n- Uni +... + P-_n+i u-n+ can be expressed in the form z = qo/, + U +... + q + n- _822"1-t. (ii) We can deal with the powers of a and 8 as operators obeying the laws of arithmetic. If, e.g., we take... v_i - 8~ui, vo = 82o, v1- ~u1,..., and then take:1- g82V_, we shall have ): =/84U,. We can therefore detach the operators, and can, e.g., write u0 -+ 82u + 2~40 in the form (1 + 2 8) 0o. (iii) For stating the relations between the p's and the q's in (a) or (b) of (i), and for our purposes generally, it is convenient to express the binomial coefficients, and

Page  355 REDUCTION OF ERRORS BY MEANS OF NEGLIGIBLE DIFFERENCES 355 certain coefficients derived from them, in a notation suitable to central differences. If we write* (n, r) = n (n - 1)... (i - r + 1)/r!, [,, r ] _ n (n + 1)... ( + - 1)/!, we can also write (n, 2s + 1] _ (n- s)(n - s + 1)... (n + s)/(2s + 1)! = n (n2 2) (n2 - 22)... (n2 - s2)/(2 + 1)!, (n + -, 2s] = ( - - + 1) (n - s + 2)... (n + s)/(2s)!; this latter being equivalent to (n, 2s] = {n2 - ()2} tc2 - (3)2}... n2 - (s - -)2}/(2s)!. These definitions hold whether n or i + 1 is an integer or not. If in (n, r] we give n a series of values with constant increment 1, we obtain a sequence whose 1st differences are the values of (in, r-1] for intermediate values of n; i.e., (n + 1, r] - (n, r] = (n +, r - 1]. If now we construct central differences of these coefficients in the usual way, we obtain a new set of coefficients which may be written [n, 2s) -1 {(n + ], 2S] + (n,- ], 2s]} = n2 (n2 - 12) (n2 - 22)... n2 - (s - 1)2}/(2s), [n + -, 2s + 1) {(n 1, 2s +1] +(n, 2s +1]} =( + ). (n- s + 1) (n- s 2)... ( + s)/(2s + 1)!; this latter being equivalent to [n, 2s + 1) = n2 n- (1\1 2 - (3)2}.. Jn - (s - )2}/( + )!. (iv) We then have the following relations: (a) For (a) of (i), r being a positive integer, 2ru = Ur- (2r, 1) U,_ + (2r, 2) u,_2... + (-)r (2r, r) uo +.. -r. _..(5), 2r. *2'r-1u0 = r,. - (r - 1) (2r, 1) Ur- + (r - 2) (2r, 2) u,._... - r_,. (6), U. = u, + (r-, 1] o8u + [r, 2) 82Io + (r, 3],3uo +... + [r, 2r)) 82rU...(7), -r = 'o - (r, 1] LtO + [r, 2) 82U - (r, 3],3uO +... + [r, 2r) 82r0...(8). Formulae (5) and (6) give the p's in terms of the q's, and (7) and (8) give qo = (P + (P' + P-1) + ( +P-2 p) +... + (, +-p-n) (9), q = [0, 2s)po + [1, 2s) (pI + P-1) + [2, 2s) (2 +P-2) +... + [n, 2s) (pn + p1-n)) q2s-1= (1, 2s - 1] (pi-p-1) + (2, 2s - 1] (p2-p-2) +... + (n, 2s - 1] (pn -P-n) (10). (b) For (b) of (i) 82r- lU= U,. - (2r - 1, 1) Ur-1 + (2'- - 1, 2) r-,2 -.. - +1......(11), (2r4 + 1),'U. =(r + ~) r+ - (r- ~) (2r + 1, 1) r + (r - -) (2r + 1, 2) ur_1-... -(r 4- -) 1_. (12), ur+1 = U1 + [r +.2, 1) ut61 + (r + 1, 2] l82u +... + [rI +, 2r + 1) 8"2u1 (13), lL/a~-[r+4z 1>3u2 2 2 u-_r = - [r + i, 1) ul + (r +4-, 2],S2'1-...-[r + 1, 2r + 1l) 82r+luL (14). *The alternative notation for (n, r), [n, r], (t, r], [i, r), is n ), [], (7, l' [). But the former are more convenient for printing. 23 —2

Page  356 356 W. F. SHEPPARD Formulae (11) and (12) give the p's in terms of the q's, and (13) and (14) give qO = (Pi +PO) + (P2 +P-) + (p +p-2) +... + (pi +P-n+l)).15 q2= (, 2s] (pi + Po) + (2, 2s](p2 +p_-1) +... + (n - 2, 2s] (pn +p-_+, )J q2s-l= [o,,2s -1) (pi — P) + [ 2s.-1) (P2 P-) +..-. -+ [n-, 2s-1) (p - -n+)...(16). II. FORMAL SOLUTION OF GENERAL PROBLEM. 9. We take first the case in which there are m - 2n + 1 u's, which we denote by u_,, u_-,+,... un. Since, by hypothesis, W is a linear compound of the U's, we can write it in the form W = CUo + cP8Uo+ c,2 U0 +........................(17). Then our first approximation to W is W = Co o + Cl,/Ao + C 82Uo +...........................(18). Now suppose we take z - pnUn + - +... P+ _2-nu_........................(19) - 0uo0 + qlSuo + q208'2t +... + q2n82918...............(19 A). Then, in order that w and z may differ by differences of the u's of orders exceeding j, we must have 0= C0, l = Cl, q2= c,... qj = j..................(20). The error of z is f= pen + n-len-i +... + p-e_.....................(21). Let the mean square of error of ir be denoted by 7r,,.a2, and the mean product of errors of u,. and u. by 7rr,a2, where a is a quantity of the same kind as the e's, so that the 7r's are numerical. Then, if the mean square of error of z is R2a2, we have R2 = TT,prps..............................(22), r s where S and S denote summations for all integral values of r and of s from -n to n, r s so that the term involving 7rr,, is 7Trrpr2, and the term involving 7r, is 2rr,sPrP.s The problem is then to determine the values of the p's which will make R" a minimum, subject to the conditions (20). 10. If in (20) we express the q's in terms of the p's, in accordance with (19) and (19 A), we shall have j + I conditions Pn + Pn-i +... + P* + Po + P-1 +... + P- = Co (n, ]pn + (n -, 1] p,1_- +... + (1, 1]pi - (1, l]p- -... - (n, l]p-n = c, [n, 2) p,, +[n - 1, 22 ) pp-I +. +[1,2) p +[1, 2)p_ +... + [n, 2)p_, = c2 (23), (i, 3] p + (n - 1, 3] p,_ +... + (1, 3] pi -(1, 3] —... - (n, 3] p-n = 3 etc. the last being (',.J] p + (in - 1, j] p +... + (- 1, j] pn = Cj or [I, ) + [1 -,j) p- +.. +- n, j) P-n j,

Page  357 REDUCTION OF ERRORS BY MEANS OF NEGLIGIBLE DIFFERENCES 3577 according as j is odd or even. We shall, for shortness, assume j to be odd; the general results will be independent of this assumption. To make B2 a minimum, subject to the above conditions, we proceed in the usual way, by taking differentials and equating them to zero. From (22) we obtain IV 7Tr,s]Js} dpr = 0; S and hence, by (23), we find that we must have, if j is odd, 77n,itPn + 7T7n, n-i Pn-i +... + 77-n, -n]-n = A + (n, 1] X1 + [n, 2) X2 +. ~ (n1, j] Xj 7rit-1,unP + 7rn-i,n-1Pn-i +... + 7Tn-i, nP-n = Xo + (n - 1, 1] XI -+ [n - 1, 2) X. +... + (n -i, Xj]~ 7T7-n,4J1in + 7T-nn-iPn-i+.+ 7r-Wn, -nP-n = Xo + ( VI, 1] X, + [- n, 2) X2 + ~ (- n,.]] Xj...... (24), where the X's are unknown multipliers. These are 2n + 1 equations by means of which we can formally (using determinants) express pm, Pn-1,... p- in terms of xo, X,,... Xj. Substituting in (23), we have j + I equations to determine X0, X,,... Xj in terms of c0, cl,... c1; and these give Pit, p,1_,... Pn_, so that the solution is theoretically complete. II. The result of these operations will be to give the p's, and therefore tile q's, as linear compounds of the c's. This leads to the following conclusions. (i) Expressing the q's in terms of the c's, let z, as given by (19A), become z = cOtt + c1tt + ct2 +... + c tj....................(25). Then the forms of the t's are independent of the values of the c's, and therefore, if g is any one of the numbers 0, 1, 2,... j, we should obtain z = t, by putting cg = 1 and each of the other c's = 0. Hence to, tI, t2,... tj are the values of z obtained by making B2 a minimum, subject to the respective sets of conditions (1o, q1=0, q20, O,... qj= 0 (,I(= 0, qI 2, -= 0, q3=0,... 0 (Io = 0, qI =OY, (b=1... q..=0,.... = (26). (f)=-=O, l1=01......... (I: =0, qj=l (ii) Let W be (J,, the true value of one of the given it's. Then, by (7), w = Ur. = Uo + (r, 1],LL&u0 + [r, 2) 82U0 + (r, 3] U83Uo..........(27). Hence, if we denote the new value of u, by v,, we shall have v," = to+ (r, 1] t1 + [r', 2) t2+ (r, 3] t3 +...................(28), the series being continued up to the term in tj. But this will be a polynomial in r of degree not exceeding j. The differences of the v's of order exceeding j will therefore be zero, so that we shall have Vr = V0 ~ (r, 1] Iuivo + [r, 2) tS2vo 4.....................(29), the series being continued up to the difference of order j. But, since this is true for all the v's, we see by comparing (29) with (28) that we must have to = Vo, t1 = /18VO, t2 8'V,.

Page  358 358 W. F. SHEPPARD (iii) Hence we get the following result. Let us, as in (19A), write z - qoUo + q + q1 0 U q o82 +... + q282nUo. Taking each set of conditions in (26) separately, find the values of q which, for each set, make R2 a minimum. Let the resulting values of z be denoted by V0, /Lvo, sV82,..., up to 82k0 or,"82k+lv0 according as j = 2k or = 2k + 1. Regarding /v$o 82vo,... as the central differences of Vo derived from a sequence v_n, v_,n+,... vn, whose differences of order j + 1 are zero, construct this sequence, or suppose it to be constructed. Then, whatever compound W may be of the U's or of U,, /A Uo, 2U,,..., z will be the same compound of the v's or of Vo, _AkVo, Vo,.... There may, as mentioned in. 7, be alternative formulae for W, but this will not affect our results, provided that the alternative formulae differ by differences of the U's of order exceeding j; for, since the corresponding differences of the v's will be zero, the formulae will all give the same z. 12. To find the mean square of error of z, we have, by (22), (23), and (24), RJ2= S ({ T,gsps} pr r s = 2 {X0 + (r, 1] XI + [r, 2) \X +... (r, j] Xj} p, r = X, p,. + Xi (r, 1]pr+. +.+ Xj(r,j ]pr ma r rX = X0CO + XlCl + X2C2 + \XsC +.. + XjCj. Let the value of Xf obtained from the (g + 1)th set of conditions in (26) be Xfg. Then the total value of Xf is Xf,oCo + Xf,lC1 +... + Xf,jCj, and therefore R 2 = Cf gXf,.............................(30)..f g 13. We shall obtain similar results for the case in which there are qn 2n u's. We denote these by Unn+, un+;,... Un. Suppose that W = Co/tGL1 + + C cUU2a + C2U1 +..., 2 2 2 and that we replace this by z - qoZ2 + qu + q q2u U + q2 +... + q2n82n-1U1. Then the mean square of error of z will be found to be a minimum when Z = CO/LV1 + C18V1 + C2/82v1 +..., the last term being c2kLS2kvl or c2k+8i2k+l according as j = 2k or =2k+ 1; the 22 quantities /uvl, 8vl, /L2vl,... being the values of z obtained by making its mean 2 2 2 square of error a minimum, subject to the sets of conditions (26). The equations involved will be (23) and (24), with the omission of the terms involving p-n and the last equation in (24). III. CASE OF ERRORS INDEPENDENT AND EQUALLY VARIABLE. (A.) General Solution. 14. The simplest case is that in which the errors of the u's are independent and have all the same mean square, which will be denoted by a2. We shall first suppose that m = 2n + 1.

Page  359 REDUCTION OF ERRORS BY MEANS OF NEGLIGIBLE DIFFERENCES 359 We can write z in the form Z = PnUn + U + p -... + P... + p-n = Po{o( + 1(p ]) l + p-) (11 + t-) + (p2 ( + p-U) +...(- + UP) + + Pn+-n) (Un + U-,) + 2 (Pi P- ) (I - U-1) + 2P ( -P-2) (UM - U-2) +.. + (pn -Pn) (Un - u_,).........(31). Also, if the mean square of error of z is R2a2, R2 = f P1 -- + pn... + - +,= + I (p + p-)2 + t (p +P-2)2 +... + I(p, + p-,)2 + '(p p-)2+ (P2 -p-2)... + n + p-)2.........(32). From (31) we see that z may be regarded as based on two separate sequences, in one of which, if r has any of the values 0, 1, 2,... n, the coefficient of u., and of u_- is }(Pr + pr), while in the other the coefficients of u,. and of ur are respectively l (Pr -p-r) and I(P-r-pr). And from (32) we see that the total value of R2 is the sum of the values for these two sequences. The first sequence gives rise to the portion qoUo + q28u0 +... of z, and the second to the portion qluSuo + q3,usUo +.... We can therefore treat these portions independently, and make their mean squares of error a minimum separately, the respective conditions being q0=C0, q2= C, q=C4,..., and q = c, q; = c, q= c,... The effect is the same as if we first took W = Co Uo + C2 2Uo +..., z = q0uo + q2 82uo +... + qUn, 8211o, and then took W = c1, Uo + cS8 Uo -+..., Z = ql2Ut0o + q3s1u83 +... + q2,_i/82n-s2lu and then added the results obtained by making the mean square of error of each z a minimum, subject to the above conditions. 15. Taking first W= cU + cU +...........................(33), let us write z = qoUo + q2 82Uo +... + q2n '....................................... (34) - Po U, + pi (tl + u_-,) + P2 (U2 + 't-2) +... + Pn (Un + -a).. (34 A), where the new p,. is the old I (p,. + p-r). (i) The mean square of error of z is R= po2 + 2p2 + 2P22 +... + 2p,2.......... (35......(35), and the conditions are, if j = 2k or 2k + 1, po + 2pl + 2P2 +... + 2p, = Co [1, 2)p, + [2, 2)p )+... +[n, 2)p,=?z c1, [1, 2k)pl + [2, 2k)p., +... + [n, 2k) n = 1 ck

Page  360 360 W. F. SHEPPARD Making B2 a minimum, subject to these conditions, we find that Po = Xpi = XO + [1, 2) X2 + [1, 4) X4 +...+ [1, 2k) X2k p2 =Xo + [2, 2) X2, + [2,4)X4~..+[2, 2k-) X2k........ (37). pn= o+[n, 2)X2+[n, 4)X4+.. +[n, 2lC)X2k For any given value of Ic, we could find the p's from these equations in the manner explained in ~ 10, and thence we could find the q's.- But this requires each formula to be worked out separately. The following is the general solution. (ii) The value of z is, as in ~ 11, Z = COV0 + C S2Vo +... + C~k 82V...................(38), where vo, 82V0,... 32kV0 are the values of z obtained by making its mean square of error a minimum, subject to the conditions qo =, q2=O,q4 = 0,...q2k-2 = O, q2k = 0, q0 = 0, q2=l, q4= O,...q~2 = 0, q2k = 0, qo = O, q2=O,q4 = 0,...q2k-2 = 0, q2k = 1. (iii) The conditions for S2tvo are qo = q2 =... = q2t-2 = 0, q2t= 1, q2t+2=: =q2k = O............(39). Making B2 a minimum, subject to these conditions, we obtain equations of the form (37). These equations show that p, is a polynomial in r2 of degree k; and it may therefore be written in the form p2.= a0 + a2 (r, 2]+ C(r, 4]... + C2k (r, 2k]..............(40). Hence, since Z = P0oUo + Pi (U1 + it-,) + P2 (a2 + U-2) +... + Pn ('un + U0), it follows that z is of the form Z = a0:ou,- + ct2Y (r, 2] u,. + a4c' (r, 4] u. +... + (r, 2k] u... (41), the summation being in each case from r = - n to r, = n. But it may be shown that 8=n2 ~ 3,2g] 1, (4 + 2) (it + -1, 2g 4. 1] Y, (n + 8, 2s]1(2s +~ 2g~ + 1). 82OUc,... (42); S==O and therefore, by substitution in (41), we see that 8v= fl X ki (s)/{(2s~+ 1) (2s + 3)... (2s + 2k + 1)}.(n ~ Z, 2s] 2SUO, 8=0 where 4 (s) is some polynomial in s of degree k. In other words q28 = (s)/{(2s + 1) (2s + 3)... (2s + 2k + I)}.(n + 1, 2s]......(43) for all values of s from 0 to n. Now introduce the conditions (39). Then these, omitting q2t =1, show that s (s) contains s, s - 1)... s- t + 1, s- t - 1,... s - k as

Page  361 REDJ3CTION OF ERRORS BY MEANS OF NEGLIGIBLE DIFFERENCES 361 factors; and the condition q2t =1 determines the numerical coefficient. Thns q,, is completely determined, and we find that 82tVO = (-)k-t t+1 k ( (n + 1, 2s] 81suo..((44). t 1 (ke - t) 1 ('11 + g,2t] 8=0 (s - 0) [S +, + 11 (iv) Hence we find that, in the complete expression for z, s having any value from 0 to n, s (s — 1. (s - le~) (n + ~ 2s] L-lk [t + 12, e + c Ct q24S = '!~ (_)k-t (k, t) (4 ) q= k~! [s + k +1] t=o (n+, 2t] s- t We might have obtained this without using the method of ~ 11. We should find, as in (iii) that q28 is of the form given by (43); then the conditions q0 = CO, q2 = C2,... ~ q2k - C2k would give the valucs of 4 (0), b (1), b (2),... 0 (k); and thence, by Lagrange's interpolation-formula, we should obtain the value of 0 (s), which would give (45). (v) We see from (44) that the process of calculating the values of v0, 82V0..., for any value or set of values of n, is very simple. We can start with the formula for v0 for the case of k = 0 (j = 0 or 1), viz.:S=111 = (v + ~, 2s]/(2s + 1). 82,1?.....................(46) 8=0 s=n 1 1(n~ + Jj [n1 + k, 2s+1)820Uo.(47), s=O which is, of course, equivalent to VO = (Un + Un-1 +... ~ u + 1)/(2n + 1). To obtain the formula for 82tV0, for any other value of k, we divide the series in (47) by the coefficient of 62tfuO, and then multiply the terms by a set of multipliers, which depend on k and t but are the same for all values of n. These miultipliers, written for each set as a series, are as follows, for values of j from 0 to 5. (a) For v,. 3 2 9 12 18 21 8 27 (j=2 or3)[-3(s- 1)/(2s+3)= 1+ - 1 --- 7 3 11 13 17 19 7 23 ' 5 45 6 (j=4 or 5)[15(s-i) (s-2)/{2 (2s+3) (2s+5)=]1+0-0+ + 143 + 13 10 225 15 - 20 108 +j4+323 19 23 115 (b) For 3v2V 10 5 20 25 35 40 15 50 + + + $-2+~i- + __ + + (j = 2 or 3) [5s/(2s~ 3)=0l + + + +p +2+17 J9+ 7+ 2 + 35 280 35 56 (j=4 or 5)[-35s(s- 2)/{(2s~3)(2s~5)}=]0+ 1+0- 33 143 13 17 1225 80 105 112 323 19 23 23

Page  362 362 W. F. SHEPPARD (c) For 84v. 21 378 42 (.=4or 5)[63s(s-1)/2 (2s +3)(2s + 5)}=] 0+0+ +1 + + 1 63 1323 84 108 567 +17+ 323 +19 23 +115+ The coefficients in the resulting formulae, for values of n up to 9 (m = 19), are given in Table I. A (i). The expressions so found are connected by certain linear relations. If P2k = —( + -, 2k]/(2 ). 2k.....................(48), so that P0== VO as given by (47), and P2, P4,... are found by multiplying the coefficients in Po by the values of 5s/(2s + 3), 63s (s - 1)/{2 (2s + 3) (2s + 5)},..., as given above, then the values of vo, v82V, 4vV... for k = 1, 2,... are given by i.; = 2 o~ 3) (. + 1, / Pt. } (=2 or3)i vo = Po -P~...........................(49), ~( ro =Po-P2 +P4 ) (j= 4 or 5) (t + t, 2]/3.8Vo= =P P,2.*- -(50), t( + -, 4]/5. 8,= P= etc., so that it would be sufficient to tabulate Po, P2, P4,.... But it is simpler to calculate the terms of Vo, 82Vo, 84Vo,... for each value of le by multiplications by the appropriate factors, either independently or successively. 16. Next taking W = cl3Uo + ct83 Uo +...........................(51), we find in the same way, using the formula S [r, 2g + 1) i, = 2 (n +, 2g + 1] E 2s (n +, 2s]/(2s + 2g + 1). 2,1-uo...(52), s=l that z = ClfSv0 + C3Lp83Vo +... + C2k-lp-2kVI o..................(53) = q1,ih0O + q3 83uo +... + q,_8'-tl- u...............(54), where =O-. -\- [t +, k] n S (s - 1)... (s -k) (,2s].( t (e - t) I(n + 1, 2t] =1 (s - t) [s + - I,] 2] and that the coefficient of /82-1'u0 in the complete expression for z is s (- 1)... (s - ) (n +, 2s] t- [t + k] t q-l = [s~2 ()k-t (k ) 2.(56). [ — +, kn ] - ) ( +, 2t] s-t - The formula for,Avo for the case of j = 1 or 2 is therefore found from the formula for Vo in (47) by omitting the 1st term, altering 32uo, 64o,... to avuoc, 8'tUo,..., dividing this new series by the coefficient of /L8uo, and then multiplying the terms by 1, 2, 3,.... The formula for 182t-1Vo, for any other value ofj, is obtained by dividing this series by the coefficient of /2t-1Uo and then multiplying the terms by a set of

Page  363 REDUCTION OF ERRORS BY MEANS OF NEGLIGIBLE DIFFERENCES 363 multipliers independent of n. These multipliers, for values of j up to 6, are as follows. (a) For uSvo. 5 10 15 4 25 30 5 40 (j =3 or 4)[-5 (s-2)/(2s +3)=]1+0- -- 1 -1-3 - 3 17 -193 3 9 11 13 3 17 19 3 23 35 7 (j = 5 or 6) [35 (s-2) (s-3)/{2 (2s + 3) (2s +5)}=]1 +0- 0+ 4 + 3 14 350 25 35 196 + 17 + 323 19 + 23 115 (b) For p,83v. 14 21 28 7 42 49 8 63 (j=3 or 4)[7 (s- 1)/(2s + 3)=]0 + l +- -- + + ~ - + -3 + + 189 168 (j = 5 or 6) [-63 (s - ) (s -3)/{(2s + 3) (2s + 5)} = ] 0 + 0 13 65 63 1512 105 144 3969 17 323 19 23 575. (c) For pjt8vo. 297 198 (j=5 or 6)[99(s-1)(s-2)/{2(2s+3)(2s+ 5)}=]0+0+1+ 1 + 143+ 66 1485 99 132 3564 17 323 19 23 575 + The coefficients in the resulting formulae, for values of n up to 9, are given in Table I. A (ii). 17. For m= 2n, suppose that W = co, U, + cl U + c.2,82 U1 +.....................(57). 2 2 Then it will be found (cf. ~ 13) that Z = CopIVil + C28I8V + c.8v +.............................(58) 2 2 2 = qolUi + ql u1 + q2.juIt +... + q2l + 82n-l...........(59), where, forj = 2k or 2k + 1, s(s-l) -... - k) (n, 2s +, c1] c2t q2S = --- --- [S ^-+T ] 3^ ( t(k,? 2^+1] ^-^ * *...(60), u= /! [s+2VkL] t=O )1w, 2t + 1] s - t and, for j = 2k + 1 or 2k + 2, s(s -...(s-k) (n, 2s +i] _ [t + k + 1] 2t+ uq2Stl- —:+,:0 +I)(it, 2t+ 1] s-tso that vi, 8, Bvl, u82,... are obtained by a method similar to that of ~~ 15 and 16. (i) For FAvi, f8J2vl,... the standard series is 2 -2 s=nl-1 (j=0 or 1) v= 1l/n.. (n, 2s + 1] SU,............(62); 2 ^ =s=O 2 and other expressions are obtained by a process similar to that of ~15(v), the multipliers (after dividing by a coefficient) being as follows.

Page  364 364 W. F. SHEPPARD (a) For /vj. 3 2 9 12 18 21 8 (j= 2or 3)[- 3(s -1)/(2s +3)= ]l +0 - 1 - 1 -1 - 7 3 11 13 17 19 7 5 45 (j=4 or 5)[15(s- l)(s-2)/{2(2s + 3)(2s + 5)}=] +0-0+ 33 + 143 6 10 225 15 20 + 13 + 17 + 323 + 19 + 23 +... (b) For /8i. 10 5 20 25 35 40 15 (j= 2 or3)[5s/(2s+3)=]0+1+ 1+ +3+ -11 + 2 + 1+ + 7... + 35 280 35 (j = 4 or 5) [- 35s (s - 2)/{(2s + 3) (2s + 5)} = ] 0 + 1 + 0- - - 33 143 13 56 1225 80 105 17 323 19 23 (c) For A84qv. 21 378 42 (j= 4 or 5)[63s(s-1)/{2(2s +3)(2s +5)}=]0+0+ + 1 + 3+ 13 63 1323 84 108 +17i 323 +19 23 +.... (ii) For yvl, 'vl,... the standard series is s=n-l (j = 1 or 2) 8v) = 1/n. E 3 (n, 2s + 1]/(2s + 3). u....... (63), 2 s=0 2 the coefficients in which are obtained from those in the standard series of (i) by multiplying by the values of 3/(2s +3), i.e. by 1, 5,, 3, 1-1' "; and the multipliers for other expressions (after dividing by a coefficient) are as follows. (a) For 8v,. 2 5 10 15 4 25 30 5 40.i = 3 or 4) [- 5 (s - 1)/(2s + 5)= 1 + 0 - 5 - 1 -_ 15 - _ - 25 - 30 - - - 40 - '; (j=3or4)[-5(s-1)/(2s+5)=]1+0 —9 11 13 3 17 19 3 23' 35 7 (j=5 or 6)[35(s- 1)(s-2)/{2(2s + 5)(2s +7)=] + 0 -0+ 14+ 3 14 350 25 35 196 +- ~+ + -6 + 2~ + - +.... +17+ 3+19+23 1+l15 (b) For 8'vI. 14 21 28 7 42 49 8 63 (j= 3 or 4)[7s/(2s+5)=]0+ 1 + + 1 + 3 + + 1+ 23 + 189 168 63 (j=5 or 6)[-63s(s- 2)/{(2s+ 5)(2s+ 7)}=]0+ 1+0- 18 65 1 1512 105 144 3969 323 19 23 575

Page  365 REDUCTION OF ERRORS BY MEANS OF NEGLIGIBLE DIFFERENCES 365 (c) For 85vl. 27 198 66 (j=5 or 6)[99s(s-1)/{2(2s+5)(2s+ 7)}=]0+0+1+ + 5 +17 1485 99 132 3564 _ _ + + i~ +323 +19+23 575+ The coefficients in the resulting formulae, for values of n up to 10 (m= 20), are given in Table I. B (i) and (ii). (B.) Applications. 18. "Graduation." Here W = U0, so that z is vo as given in ~ 15; i.e. by (44), 1.3.5...(2k + 1)8 (s-1)(-s32)...(s-k) n s ] 1.2...k =k (s + 1)(2s + 3)...(2s + 2/ + 1)......(64). (i) The ratio of the new mean square of error to the old is (cf. ~ 12), by (37), R2 = p02 + 2p12 + 2p22 +... + 2pn2 =poXo + 2pi {Xo + [1, 2) \2 +... + [1, 2k) X2k} +...... + 2pn \Xo + [n, 2)X2 +... + n, 2k) X2k =X0 {po + 2p + 2p2+... + 2pn + 2X, {[1, 2)P1 + [2, 2)p2 +... + [n, 2)pn} +... + 2Xk {[1, 2k) p + [2, 2k) p +... + [n, 2k)pn} = =po.............................................................(65), by (36) and (37), since o= 1, C2= C, =... = 2k = 0. But po is the coefficient of uo when 82k+2Uo, 82k+4utt, 8... 82uo are expressed in terms of the u's. Hence we find that 2 S=n ( ).3.5...(2k1) (n — s1+l)(n-s+2)... (n+s) s=k+l 1.2... s!(s-k-1)! s (2s + 1)(2s + 3)...(2s + 2+1).........(66). Denoting this by Rk2, we shall have, in the usual notation of hypergeometric series, 1.3.5... (2k- 1) Rk2- Rkl (4k + 1) 1 * k3 5 (2-1) () (1 - s + 1)(n - s + 2)...(1+s) s=k s!(s- k)! (2s + 1) (2s + 3)... (2.s +2k +1) 1.3.5...(2k-l) (n-k+l)(n-k,+2)... (n+) 1.2.3... k k! (2k + 1) (2 + 3)... (4k+ 1) xF{-n,+k, n+k+ l, k+; 1, k+1, 2k+,; 1}...(67). The value of F I } is given by the formula* F{-n,20 +n- 1, - - 0; 1, 0, 0; 1} = [20 - b, n]/[O, n]..... (68), * See Proceedings of the London Mathematical Society, series 2, vol. 10, p. 475.

Page  366 366 W. F. SHEPPARD so that Rn;2 _- Rk 2 =.l_. 35...(2n- I)2 (-k + 1) (n - k + 2)... (n+k) Bk2 B- k- = (4k I) 1 -.23-... (2n- 2 + 1) (2n - 2k + 3).. (2n + 2 + 1) = (4 + 1) 1. 3 -.5...(2k- 1)L 2 (n2.- 12)(m,2-32)... tm2-(2k- 1)2} 2.4.6... (2/) M (Wn2- 22) (m2 - 42)... {m2 -_ (2k)2}.........(69), and thence M,,1,2 in (M2 - 22) \2. 4 / (m2 - 2') (m2 - 42) 1 (1<~ "2-12 2(1. 2 (_ -12)(n22-32) ~ +(4k ) J1. 3.5...(2k1-l)(2 (n —) (m2 - 32)... {- (2k- 1)2} l 2.4. 6...(2^) (m2 - 2-2)(2- 42)... 2 - (2/C) The resulting values of R, for m = 3, 5, 7,... 31, are given in Table II*. (ii) For practical purposes it is more convenient to express vo in terms of the u's, and also to replace fractional by decimal coefficients. I propose to deal elsewhere with this, and also with the comparison of the resulting formulae with some of the best known "graduation "-formulae. 19. Interpolation. We take U0, as in ~ 7, to be the value of U corresponding to x = x, + Oh, where 0 < 0 < 1. (i) The ordinary (Lagrangian) formula for us is obtained by taking a polynomial in x of degree j and equating it to j + 1 consecutive u's. If, therefore, m u's are available, we have a choice of m -j ordinary formulae. It will be found that the most accurate formula, i.e. the one which gives the least mean square of error for u,, is one of the j formulae which use both u0 and U1, but is different for different portions of the range (0 to 1) of values of 0. If we denote the roots of the equation in t g=j (j,g - 1) (, g) (t - 1, j)/(t- g) = g=l by tl, t2,... tj_-, and if we write 06 - t - 1, 02 t2- 2,... 0j_i - t,- (j - 1), then 01, 02,... Oj_ all lie between 0 and 1, and are in ascending order of magnitude; and they divide the interval (0 to 1) into j portions such that, if 0 lies in the gth portion, the most accurate formula is the one which uses ug+1, ug,+,... Ujg+1. (ii) The central-difference formula is therefore not the most accurate formula, except near the middle of the interval. It is, nevertheless, the best for ordinary purposes, since (1) its form is convenient for calculation, (2) it is the only formula (of this type) which throughout the whole interval makes the mean square of error of u0 less than that of the given u's, and (3) the improvement effected by using any other formula is not much greater except at the extremities of the interval. Moreover, if * By putting m=2k + 1, we obtain Rk2=1, i.e. 1 1 +4 k(k +1) + 9 (1.3 (k-1)k(k+l)(k+2) 2k + 1 \1 (2k - 1) (2k + 1) (2k + 3) + 9. 2 (2k - 3) (2k - 1)... (2k + 5) "'+(4k+1) 1.3.5-...(2k —1)[2 1. 2.3..(2h:).*+(^i 1.2.3...k 1.3.5...(4k++ )' This series contains k + 1 terms. Inspection of Table II suggests that when k =21+ the sum of the first 1 +1 terms is ~, and the sum of the second 1 + 1 terms 2; but I have not been able to prove this.

Page  367 REDUCTION OF ERRORS BY MEANS OF NEGLIGIBLE DIFFERENCES 367 we are subtabulating (i.e. constructing a table for which the interval is a submultiple of h), we obtain a smoother table by using the central-difference formula (or, if m is even, the mean of the two central-difference formulae) than by always using the most accurate formula. It should, however, be mentioned that, so long as we keep the original u's unaltered, smoothness and accuracy are antagonistic. The effect of using the most accurate formula for each interpolated value is to produce the maximum irregularity in passing from one interval to the next; and, conversely, the effect of using a special smoothing process, such as that of " osculatory interpolation," is to increase the inaccuracy. (iii) Our new method uses,i consecutive u's; and, if these include the 2j from itj+l to uj, the resulting formula will be more accurate than any of those considered in (i). We therefore take as our basis the central-difference formula (m = 2n + 1) U 0 = UO + o8U + 02/2!. 2U0 +..., (m = 2n) Us =, U + (9 - E) 8 U + 0 ( - 1)/2!. 2 U. +..., and obtain the new formula by omitting terms after the difference of order j, and then replacing U by v throughout. (iv) This formula is only suitable for improving isolated values. If we try to apply it to subtabulation, we shall introduce discontinuities, on account of pairs of values being found for quantities such as U0 (m = 2n + 1) or U1 (m = 2n). For subtabulation, therefore, it is better to improve the data by the "graduation" method and then to apply the ordinary central-difference formula to each interval separately. 20. Quadrature. The ordinary formulae for quadrature are of two types, examples of which have been given in ~ 7. Both may be regarded as based on the formula for the " chordal" or the " tangential " area, which would be a correct formula if j were = 1; for other values of j the formulae of the one type are obtained by altering the coefficients in a certain cyclical manner, while those of the other are obtained by introducing terminal corrections. Each type has certain advantages. It is sometimes stated that the "best " formula is the one obtained by treating u as a polynomial in x of the highest degree allowed by the data; but this does not rest on any clear definition of what is best. The most accurate formula, in our sense of the term, is obtained by the methods considered above. There are four cases, according as the data are the bounding ordinates of the strips or the mid-ordinates, and according as their number is odd or even. In each case the formula is obtained, as before, by altering U's to v's. (A.) Bounding ordinates given: m strips. (i) in = 2n. Area = 2nh U, + (nh)2 Uo"/3! + (nh)4 U,,v/5!+...} = 2nh {U, + 1./6. 62 U, + (:324 - 5n)/360. 84 U, + (31t6 - 21n14 + 28n2)/15120. 86Uo +....

Page  368 368 W. F. SHEPPARD (ii) mr=2n+1. Area = mh { Ul + (lmh)2 U,"/3! + (l-nth)4 UIrv/5 +...} = mh U,1a + (mn2 - 3)/24. 82U1 + (3m4 - 50m2 + 135)/5760. G84U + (3m6 - 147m4 + 1813mn2 - 4725)/967680. u6U1 +...}. (B.) Mid-ordinates given: 'm strips. (i) m = 2n. Area = 2nh { U, + (nh)2 Uo"/3! + (nh)4 U0oV/5! +... = mh {t Uo + (im2 - 3)/24. a2 Uo + (3m4 - 50m2 + 135)/5760. 84 Uo +.... (ii) m= 2n + 1. Area = mh i U, + (nmh)2 U1"/3! + (mh)4 Uv/5 +... = mnh U, + m2/24. 82U, + (3m4 - 20m')/5760 84., + (3m6 - 84m4 + 44812)/967680. 86 U +...}. The resulting formulae can all be adapted for numerical calculation in the manner explained in ~ 18 (ii). 21. Other Applications. The following are other applications; the method in all cases being the same, viz. to obtain the true formula in terms of central differences of the U's, and then to turn the U's into v's. (i) Differential Coefficients. If U is the area measured up to the ordinate, U' is the ordinate. The ordinates enable us to trace the curve, and their differential coefficients show the position of the maximum ordinate, corresponding to the " modal" value of x. The formulae are well known. (ii) Moments. The formulae for moments of areas can be improved in the same way. The U's may be either ordinates or areas, and the formulae will be different according as the moments are of odd or of even order. There are altogether 12 general formulae. (As to certain particular cases, see ~ 24.) IV. RELATION TO METHOD OF LEAST SQUARES AND METHOD OF MOMENTS. (A.) Standard system (errors independent and equally variable). 22. The results in ~~ 18-21 may be regarded as obtained by treating U as a polynomial in x of degree j, and then, after performing any processes of integration, differentiation, etc., replacing the U's by the v's. But we should get the same result if we altered the order of these operations. It follows that, for any set of m consecutive u's, and for any given value of j, there is a polynomial v - Ao + Ax + A,2 +... + AjxJ, such that by taking U =v throughout we obtain a less mean square of error for the resulting value of W than we should obtain by taking any other formula for U; and that this polynomial is the same whatever W may be, provided that W can be expressed as a linear compound of the m U's.

Page  369 REDUCTION OF ERRORS BY MEANS OF NEGLIGIBLE DIFFERENCES 369 It may be shown that this polynomial is the same as is given by the method of least squares or the method of moments; it being understood that in the lastmentioned method the moments are not the moments of the area of which the u's are taken to be the ordinates, but the moments of the u's themselves. Hence these latter methods give the best results not only for the fit of the required polynomial to the given sequence as a whole, but also for the separate terms and for any quantity which can be expressed as a linear compound of them. Although the new method and the old method (in either of its two forms) lead to the same result, they are based on quite different principles. (i) The old method is a method of " fitting "; it considers the data as a whole, and finds the polynomial which fits them best. The new method deals separately with any particular u or with a quantity derived from some or all of the u's, and finds the best expression which can be substituted for it. (ii) In the old method we consider that we are finding the real U, and that (e.g.) Ur is actually (taking j = 2) of the form Ao' + Al'r + A2'r2. In the new method we do not assume that v,. is identical with Ur; indeed, we say definitely that it is not, and we have in the coefficients of the "negligible" differences a means of measuring the extent of the mistake we make in substituting v, for ur. (iii) In the new method the preliminary assumption (in the above case) is that Vr differs from ur by a linear compound of differences of the 3rd order and upwards, and the fact that it is of the form Ao + Ar + A2r2 is a deduction from this assumption. In the old methods this latter is the preliminary assumption, so that the process is reversed. To show that the latter assumption leads to the former, let wr = Ao + Air + A2'r2...........................(71) be the assumed value of U,.. Then, in the method of least squares, we make X (W,- Ur)2 a minimum. But r,. = o r. + r. + r/2!. 82tto + b (r), where b (r) is a definite linear compound of differences of u of order exceeding 2. Hence Wr -U= (Ao'- u) + r (Al'- 8ui) + r (A2 - 2,) - (r). Keeping wr - Ur in this form, let us make E (w,. - Ur)2 a minimum. Then we shall have a set of linear equations to determine Ao' - o, Al' - p8uo, and A2' - u,,; and these equations will give these quantities as linear compounds of the 4's. In other words, we shall have Ao'= u, + o, Al =,ou, + #1, A2' = 820o + 2.........(72), where o, i'l, 2 are linear compounds of the higher differences mentioned above. Hence, finally, w, will be of the form W.= A,' + A,'r + A2'r2 = (u, + ruou, + r2 r'O) + (o + r#1 + r 2)............(73), =u,.+ (r) where (r) is a linear compound of the higher differences, with coefficients involving r. M. C. II. 24

Page  370 370 W. F. SHEPPARD 23. Since the new method gives the same result as the old methods, our formulae for v0, 8Svo,... or for Lvi, 3vi,... give an explicit solution of the equations which occur in the application of the old methods; and, conversely, these formulae may be expressed in terms of the moments, if on account of m being very great it is more convenient to use the latter. To illustrate the use of the new formulae, let us take the example given in M. Merriman's Method of Least Squares (8th edition, page 131) of the velocities at successive tenths of the total depth of a river. If u is the velocity in feet per second, multiplied by 10,000, and if the unit of measurement of x is 0'1 of the depth, the differenced table is as follows: x u all &2u 3,11 au 65 lu 7tu 81 69t 0 31950 +349 1 32299 -116 + 233 - 38 2 32532 -154 + 18 + 79 - 20 + 37 3 32611 -174 + 55 -229 - 95 +35 -192 + 835 4 32516 -139 -137 +606 -2372 -234 -102 +414 — 1537 +5630 5 32282 - 241 +277 - 931 +3258 - 475 +175 -517 +1721 6 31807 - 66 -240 +790 -541 - 65 +273 7 31266 -131 + 33 -672 - 32 8 30594 163 - 835 9 29759 __ -____ _ _ ___ ____ __ -____ - _ ___ ___ _ - ____I. _ ____ __. The most laborious method is to take x = 0 as our zero; assuming U to be of the form Ao' + A,'x + A,'x2, this leads to the equations 10A' + 45A,'+ 285A2'= Mo= 317616) 45Ao'+ 285A,' + 2025A' = M1= 1408957i, 285Ao' + 2025A' + 15333A' = M, = 8828813 the solution of which is A,' = (136MO - 57M, + 5M1)/220 = 31951-327, A' = (- 684Mo + 437M, - 45M,)/2640 = 442-530, A,' = (12Mo - 9M, + M2)/528 = - 76530. A less troublesome method would be to take moments with regard to the middle of the range, i.e. to take the values of x to be - 4, - 3-,... + 41. To apply the new method, we take the values of x to be -4, - 3,... + 5, and we find the values of 2/ULv, Svi, 2/ut8v, from the central differences of the interval (0, 1). The coefficients are given in Table I. B (ii), under m = 10, j = 2; and thus we obtain 2,v = (64798) - (140) - (- 325) - 5 (886) = 64747-68,

Page  371 REDUCTION OF ERRORS BY MEANS OF NEGLIGIBLE DIFFERENCES 371 vil = (- 234) + -2 (- 102) + 5- (414) + -f (- 1537) + - (5630) = - 246-24, 2/av, = (- 380) + (140) + (- 325) + IT (886) = - 30612. Since 2/,vi = vl + vo, 8vi = v,- v, this gives us the two middle v's; and thence we construct the table x v v 8v62v -4 '31951-32 + 366-00 -3 32317-32 -153'06 +212-94 -2 32530-26 - 153-06 + 59-88 -1 32590-14 -153-06 - 93-18 0 32496-96 - 153-06 -246-24 1 32250-72 - 153-06 - 399-;30 2 31851-42 - 153-06 -552-36 3 31299-06 - 153-06 -705-42 4 30593-64 - 153-06 - 858-48 5 29735-16 - 153-06 - 1011-54 6 28723-62 These values of v have can then be reduced. to be divided by 104; and the number of significant figures In this example, I have followed Merriman in taking U to be of the 2nd degree in x; but it is doubtful whether this is justified by the data. The proper method of finding when differences of U become negligible is to take out, and difference, values of u at regular intervals, so as to increase the successive differences in increasing ratios. If, in our example, we take even values and odd values of x separately, we get the following results: 31950 32299 + 582 -+ 312 32532 -598 32611 - 641 -16 - 95 - 329 -46 32516 — 693 32282 - 687 - 709 +189 -1016 +196 31807 - 504 31266 -491 -1213 -1507 30594 29759 ____________________ __ __________ ___ The parallelism of the 3rd differences makes it very doubtful whether we are justified in assuming that 3rd differences of U are zero. If U can be represented by a 24 —2

Page  372 372 W. F. SHEPPARD polynomial in x, it seems more likely that this ought to be taken as being of the 4th degree in x. The calculation of 2pvx, 8v,... 2,8z4vl would then be as follows: 64798- 380+140- 325 - 886 ~2/zv 1+ 0- 0+ 8/33+ 9/143 = +64774-974 2/I82v 0+ 1+ 0-14/33-14/143 =- 328-862 2,a34v 0- 0+- 1+ 8/11+18/143 =+ 15'161 -234-102+414- 1537+ 5630 av, 1+ 0- 1-16/33- 9/143 = - 257-124 83, 0-0+ 1+ 7/6+14/33+ 7/143 =+4 4-534 L 2 This gives the following table: x I vav 8 2v 63v 64v -4 31945-93 4-+361-06 -3 32306-99 - 134-81 +4226-25 - 18-21 -2 32533-24 -153-02 +7-,58 + 73-23 -10-63 -1 32606'47 - 16365 + 7'58 - 90-42 - 3-05 0 32516-05 -166-70 + 7-58 -257-12 + 4-53 1 32258-93 -162-17 + 7 58 -419-29 +12-11 2 31839-64 - 150'06 +7-58 -569-35 +19-69 3 31270-29 - 130-37 + 7-58 -699-'72 +27-27 4 30570-57 - 103'10 +7-58 -802'82 +34-85 5 29767-75 - 68-25 -871'07 6 28896-68 l l 24. Since the results obtained by the new method are the same as would be given by equating moments, it follows that the method cannot be applied to the improvement of the moments themselves; in other words, if M=g rg^u, is the gth moment, its mean square of error is already a minimum, and cannot be reduced by adding to Mg terms which involve differences of orders exceeding j. But it should be observed that (1) this only applies if g j, and (2) it only applies to the moment as defined above (cf. ~ 22). If the U's are ordinates of a figure, the moment of the area

Page  373 REDUCTION OF ERRORS BY MEANS OF NEGLIGIBLE DIFFERENCES 373 of the figure is different from Mg, and our methods apply to the determination of its best value. (B.) Other systems. 25. It has been assumed, throughout ~ 22-24, that the data belong to the standard system, under which the errors of the u's are independent and have all the same mean square; and it is only for this system that the method of least squares or the method of moments, as ordinarily used, will make the mean square of error of z a minimum. For any other system of errors, these methods will be correct, in the sense that they would give the true result if the data contained no error; but, in order that they may give the most accurate result, certain modifications are required. (i) Denote the given u's by uf, uf+l,... Uf+m_-; and let us find the best value for U. by the new method. We take s or t to be one of the numbers f,f+ 1,... f+m-1, and g or h to be one of the numbers 0, 1, 2,... j. If the best value for U, is Vr = I pss....................................(74), 8 then (~ 9) the p's are found by making R2- 7rs, tpspt s t a minimum, subject to certain conditions which may be written EPs =1 8 Y.sPs =r es2 pp = r2v"......................... s * I. Zsj pS = rj This (~ 10) gives m relations of the form S7r,,t Pt = Xs..............................(76). t g Treating these as equations to give the p's, we find that Pt = Fg (t)}................................. (77), where, if P r/,f 7Trf,/f+ 7f,f+2 *.....................(78), 7+f+l,f 7rf+l,f+l qrf+l,f+2 2 7'rf+2,f 7rf+2, f+l 7rf+2, f+2. Ps, t cofactor of 7r,t in P........................(79), then F, (t) = s P, tP.............................(80). Substituting for Pt from (77) in the (h + 1)th equation of (75) in the form.thpt = rh, t we have rh= {g, h} Xg................................. (81), g

Page  374 374 W. F. SHEPPARD where {g, hJ E_~thFg (t) t EStPt...........(82). s t Substituting from (81) in (75), for h = 0, 1, 2,... j, we have j + 1 equations to determine the X's; and, if we write Q i I0, O~ 11, {2...(83), {o, 11 11, 11 12, 1... {0, 21 {1. 2j {2, 2} h, h= cofactor of {g, h} in Q.........(84..........(8), these equations give QXg = Qg,o + rQg, +... + rjQgj......................(85). Hence, substituting in (77), Qpt = Y-Qg,,hr g (t)........................... (86); g h and finally, by (74), Qvr= Qyptut= SQCqg, hhi jq(t) qtt.... (87) t tgh = Q Q 7-h Sg Ps,t ut/P........................ (88). s t g h (ii) To adapt the method of least squares, we use the known property that the probability of occurrence of errors ef, ef+1,... ef+m,,1 in the u's (each error being taken to a definite number of decimal places, so that we can omit the differentials def, defl,1...) is proportional to exp (-,VPs,teset/P)...........................(89), st where P and P,,t are defined as in (i). This suggests that, if U, is assumed to be of the form U, = w, - A, + A,r + A,r2 +... + Aj7ri............... (90), we should find the A's by making I Ps, t (w8 - U.) (Wt - ut)........................ (91) s t a minimum. Differentiating (91) with regard to each of the A's, we have j + 1 equations of the form Y. Fg (t) (wt - ut) = 0......................... (92), t for g=0, 1,2,. j. Hence, if we write Y., Fg (t) u t Mg..............................(93), t we obtain I {g, hI Ah = f...(94). h Solving these equations in the A's, we find that QAhL = t Qg,hMg......................... (95), U

Page  375 REDUCTION OF ERRORS BY MEANS OF NEGLIGIBLE DIFFERENCES 375 and thence Qwr = sh I Qg, h Mg = s Qg, hr Fg (t) Ut...........................(96). t q h Comparing with (87), we see that this method gives the same result. (iii) Also from (92) and (93) we see that the method of moments gives the same result, provided that in finding the gth moment we multiply Ut and wt by Fg (t), i.e. by ZPs,ts9 (with any constant factor), instead of by tg. s V. TREATMENT OF NON-STANDARD SYSTEMS. 26. The formulae obtained in ~ 25 are so complicated that their practical application, as general formulae, would seem to be out of the question. We conclude that, except possibly in a few simple cases, the only practicable method of obtaining the best results is to convert our data to the standard system, in which the errors are independent and have all the same mean square. A specially important case is that of data which represent frequency-distributions. Before considering this case, the following simpler cases may be noted. (i) Suppose that the errors are independent but have not all the same mean square. Let the mean square of error of Ur be a,.2 (a) If ar2 is known, we can tabulate a/ar. Ur, where a is a quantity which is the same for all the u's; and our formulae will then apply. But it will here, as in all other cases of this kind, be necessary that the fundamental condition as to existence of negligible differences (~ 1 (iii)) should continue to be satisfied. (b) If a.2 is a definite function of Ur, let us aim at obtaining Y, a function of U. A small error er in the value of Ur will produce an error (dY/dU)rer in Yr, and the mean square of this error will be {(dY/dU)r}2ar2. We therefore choose Y to be such a function of U as to make dY/dU= 1/a, for all values of a; if yr is the corresponding function of Ur, the mean square of error of y, will be 1, and our method will apply to the sequence of values of y. If, for instance, U is a physical measurement, the " probable error " of u might be proportional to U or to / U. In the former case we should have d Y/d U proportional to 1/U, so that we should tabulate log10 u in place of u, and then apply our formulae. In the latter case we should tabulate Vu. (ii) Suppose that the correlation of the errors is due to the fact that each error U,. consists of two portions er and Or, such that the e's either are all equal or else form a sequence with regular differences, while the ~'s are independent. In such a case we cannot make any reduction in the e's, and we must regard U + e as the quantity whose value we are seeking, and ' as its error. The case therefore comes under the category of independent errors. Under this head would come a sequence of astronomical observations, all containing an error due to the "personal equation "; or a sequence of nautical observations based on dead-reckoning subject to an unknown current.

Page  376 376 W. F. SHEPPARD (iii) Suppose that 2nd differences of the U's are negligible, and that the mean product of errors of u,. and a, is of the form A + B (r + s)- C r-s, the mean square of error of u,. being A + Br. We have then a choice of two methods. (a) The mean square of error of p 'tp22+... +pm`Um is a minimum when P =P3 P=-... =, = 0; the values of p1 and p, being then determined by the conditions of the case. If, for instance, m = 2n + 1, and we are improving the u's, the improved value of un+i is I (u, + U2n+i). (b) The errors of the 1st differences are independent and have all the same mean square, so that we can (usually) treat these as the given quantities. 27. Next take the case of a frequency-distribution. Let N be the total number of measurements or observations, and let Nur be the number for which x is less* than x., so that u ranges in value from 0 to 1. The data will then ordinarily be arranged in the form of a table, the numbers in the compartments of the table being proportional to the 1st differences of the u's. (i) The fundamental assumption is that the data are such as might be produced by random picking of N individuals out of an indefinitely great number for which the proportion classed as below Xr would be Ur. (ii) The mean square of error of u, is therefore U (1 - Ur)/N, and the mean product of errors of t,. and u, is Ur (1 - Us)/N or Us (1 - Ur)/N according as r < s or s < r. Also, if y and y' are two of the 1st differences of the u's (i.e. if Ny and Ny' are the numbers in any two of the compartments of the table), and if Y and Y' are the corresponding 1st differences of the U's, the mean squares of error of y and of y' are Y(1 - Y)/N7 and Y' (1 - Y')/N, and their mean product of errors is - YY'/N. (iii) Hence, if the extreme values of x are x0 and Xm, so that the observed u's are u- O0, ul, U2,... um-, u -- 1, the probability of occurrence of errors el, e2,... e,_is proportional to exp (- E2/N ).................................(97), where 2_ e (e2- e)2 (e3 e 2)22 em_ - em_,)2 e2m-. (9 8 E ~ + _ + J u- +'" + ~ gi_ +l- _ '(98); U i U2- Ul U3 U2 Um-l - Um-2 +1 Um-1 or, denoting the errors of the y's by e'l, e',... e'm-, s=m E2 = E e'2 S- /Y _..............................(99). s=1 The form of this is the same as if the errors e'A, e,... e'_ were independent; but they are not really independent, since their sum is zero. (iv) It follows from (98) that (disregarding a constant multiplier) Fg (s) = {(s + 1)Y - s}/( U - Us) - {so - (s - 1)9}/( Us - Ul)...(100). Since, however, this involves the unknown U's, it is not of practical use, unless we proceed by successive approximations. * In practice it is usually more convenient to make u range from 1 to 0, substituting "greater" here for "less." This involves some changes of sign.

Page  377 REDUCTION OF ERRORS BY MEANS OF NEGLIGIBLE DIFFERENCES 377 (v) The correlated errors can be analysed into independent errors. If, for instance, we tabulate log (1 - u) - 0, log (1 - ui), log (1 - u),... log (1 -,n-,), the errors of their 1st differences will be independent, provided that N is so large that, for each value of r, the error of u,. is small in comparison with 1- U,. This is useful for construction of control examples, but, for actual cases, the difficulty remains that the mean squares of error involve the U's. (vi) If the number of compartments is large, so that YY' is always small in comparison with Y(1 - Y) and Y' (1 - Y'), we see from (ii) that the correlation of the errors of the y's may be ignored, so that we can treat these errors as independent, the mean square of error of each y being Y (1 - Y)/N. Hence (~ 26 (i) (b)) we can convert the system to the standard system by replacing the values of y by those of 2/rr. arc sin \/y, or, sufficiently well for practical purposes, by those of Vy. (vii) This gives us a criterion for deciding what intervals should be used for the purpose of tabulation. It has been suggested* that it will generally be sufficient if the interval is so chosen that the whole number of classes lies between 15 and 25. But this will not usually be sufficient for our purposes, at any rate if N is large. We take the square roots of the 1st differences, and then apply our formulae; and the mistake so introduced into the quantity we are calculating ought to be small (say about one-fifth) in proportion to the probable error of the quantity. Roughly, in the ordinary case of a " double-tailed" distribution, with observations numbering (say) from 1000 to 4000, it will be sufficient for most purposes if the interval is such that no compartment contains more than about 7 per cent. of the total number. VI. GENERAL OBSERVATIONS. 28. In deciding what order of differences we are to treat as negligible, we must remember that the terms which we introduce involve differences at the centre of the range, and that the conditions may be different from the conditions at the ends of the range. Suppose, for instance, that we require the gth moment of a double-tailed figure whose ordinates are the u's. We can take Sxgu as an approximate formula. But the reason of this is that the true value differs from xgU by an expression involving differences or differential coefficients of the extreme U's. These are very small, whatever their order may be. But at the centre the differential coefficients of low order will not necessarily be small. 29. The whole method depends on the fundamental assumption (~ 1 (iii)) that differences beyond those of a certain order are negligible. If this condition is not satisfied, the method fails, not because it is wrong, but because the interval of tabulation is not small enough, or because the independent variable is not suitably chosen. In such cases the only method is to substitute a new sequence, by a process analogous to that mentioned in ~ 26 (i) (b). This is a familiar process, of which the treatment of the tail of a distribution by taking logarithms is an example. We can then apply our method to the substituted data, though there may be some difficulty in finding the best formulae. * G. U. Yule, Introduction to the Theory of Statistics (1911), p. 79.

Page  378 -'S 7 8 378 ~~~~~~W. F. SHIEPPARD TABLE I. Formulae for v0, etc. (standard system). This table relates to the standard system, in which the errors of the u's are independent and have all the same mean square. The highest order of differences of U that is not negligible is denoted by j. The total number of u's involved in the formula being denoted by m, the cases of m odd and m even are dealt with in parts A and B of the table. Part A shows (for j = 1, 2, 3, or 4) (i) the coefficients of a10, 82U0, 84... in the expressions for v0 (m = 3, 5, 7,...), 82 V (mn = 5, 7, 9,...), and 84V (m = 7, 9, 11,.),and (ii) the coefficients of b~Uo ~...in the expressions for,u~v, (m = 5, 7, 9,...) andfSV ( = 7, 9, 1,.);the it's being u-n, U-n1,... m- Part B shows (for j = 1, 2, 3, 4, or 5) (i) the coefficients of 2u1, p~2u4,,... in the expressions for ILv1 (m = 4, 6, 8,...) 2 2 ~~2 2.L82V, (m = 6, 8, 10,... ), and /~14V (m = 8, 10, 12,...), and (ii) the coefficients of &t1, 83u1, 85u1,... in the expressions for 8v, (m =4, 6, 8,...),3V (n (= 65 8, 10,...) and 85V, (m = 8, 10, 12,...); the u's being u-,+, U-,,2,... 'un For an example of the use of the table, see 23. A. m j Co. u0, 12110o 541O. 3 0,1~~~ 5 0,1~~~~ +03 7 0, 1 1 +2~+1 + 7 3 2 2,3 ~~~7 21 01+- +'r' 7 42 4,5 1+0-0+ — 4, ~~~ 231 o + 1 + 0 663 11 3 9 2,3 ~~9 2 -1 7 3 11 + 7 2 33 5 5 4, 5 1+0-0+-+ 33 143 0+1+0- - 29 00 11 143 (Continued on page 379) in = 2n + 1. M1 J Co. g371o, Ao33uo, M,L5150.. 1 ~~~5 7 1 2 01 + 1 14 8)4 1+0- -5 -7 4 O? 9 992 (Continued on page 380)

Page  379 REDUCTION OF ERRORS BY MEANS OF NEGLIGIBLE DIFFERENCES 379 A (i) continued. mz j Co. U, 82U, 4U,.... 11 0, 1+5+7+4+1+11 2,3 1+0 8 9 12 2, 3 1+0-3 -3 11 143 4 4 5 0+1+2++ - + 143 -3 +-1 +143 20 45 6 4, 5 1+0-0+3 + +143 33 143 143 28 56 7 0+1 14+0 11 143 143 13 13 O, 1 +7+14+12+5+1+1- - 45 12 1 2, 3 1+0-6-8 — 12 -11 13 13 20 20 100 25 2 7 +7 + 77 91 91 20 225 6 10 4, 5 1 +0 - 0+ 11 + 143 + 1221 20 200 5 8 11 143 13 221 18 135 3 9 0+0+1-+ + — 11 143 13 442 28 126 55 1 15 o, 1+ 8 + 126 +30+ +6+1 + -1 54 72 6 2 1+0 --- 20-15- -1 — 235 13 85 27 75 25 225 3 1 o+i+ 3+ + + ++ +^ 0+ 1 + 17 + 14 + 7182 +14+ 68 50 75 36 10 15 4, 5 1+0-0+ + + +3 11 13 13+17+323 75 50 45 6 35 0+1+ 22 13 26 17 1292 25 275 10 5 7 0+0+1+ +- + - + 6 +4- 6 ++ 11 143+ 13+34+646 17 0, 1 1+12+42+66+55+26+7+1+ 17 18 21 2,3 1+0-18-44 - 45-24-7 - — 3 55 25 25 7 35 10 0 + +- +3 6 6 204 969 4,5 1+0-0+10+225 70 225 15 13 17 323 323 35 350 35 98 1225 20 0+1+0 6 39- 6 51 3876 969 45 21 63 2 0+0+1+3 + 4+2+ + +63- 3 Con13 34 o 646 323 (Continued on page 380)

Page  380 380 W. F. SHEPPARD A (i) continued. m j Co. u0, 2u0, 4U0,.... 19 0, 1 1+15+66+132+143+91+35+8+1+-9 19 7 + 8-117-84-35 17 -19 133 44 44 52 35 14 56 8 1 28 56 49 392 1960 16 7 3 3 3 51 - 969 - 57 4- 37 42 63 49 735 1764 14 18 I 1 1 + 11 374 3553 209 4807 A (ii) continued. m1 j Co. l.lUo, 3113U, SU... 14 12 4 1 11,2 1+ +- + + 5 5 5 11 3,4 4 8 15,4 1+0 --- - 3 11 143 4 6 10 0+1+4+ -- + 3 11 143 36 20 5 6 13 1, 2 +4+362+5+ 3 4 1o- 720 200 75 8.3,4 1+.7 77 91 91 15 5 1 o+1+2+5 + +2 11 13 26 15 1 1+27 135 55 45 9 1 5 14 7+14+ 14+20 75 50 6756 5 3, 1+0- 14 7 182 7 68 25 25 50 5 7 0+1 + _ + _ + _ + 9 9 39 + 1-839+ 8306 1 733 55 65 7 7 + 2 17 1,2 1+7+2+3+ 6+2+12+51 55 50 25 14 175 20,3 2 3 204 323 11 10 7 7 14 3 3 6 34 969 9 1 2 1 44 132 572 91 14+ 56 + 8 + 3 19 1- +- + ++14++ 44 104 56 280 16 1 3, 4 1+0- 0 35 - 3 3 3 51 19 19 +1+14+ 91+245 245 196 98 2 3 11 33 66 187 627 209

Page  381 REDUCTION OF ERRORS BY MEANS OF NEGLIGIBLE DIFFERENCES B. m = 2n. 381 (i) in j Co. /tLi, %2/11, I4Ui,.... 4 0, 1 1+ 21 6 0, 1 1+ 4+,3 3 2,3 1+0 — 3 1+0 0 + I + 5 14 8 0, 1 + + 3+9 1 03 I+0+1+ 6 1 1 + I+ 6+1 4,5 1+o-0+- 1 — 7 0+ t+0-4 -66 16 12 40 8 0+0+1+ 7 10 0,1 1i+4+ 21+ + 9 16 9 1+0+1-+ -+ 0+ I +0 143 1 8 18 11 143 2 0 1 1~ +3+ +. +6+ +6 15 2 2,3 10-4-4- - - 11 13 16 12 40 5 o+1+-+ 7 +7 7 77 91 ~45 510 75 1 5 1+0-0+- + + 11 143 13 12 80 I 11 143 13 27 135 3 o++ 1+ + + 22 286 52 (Continued on page 382) (ii) m I-iMn j Co. 6U,:3 u 5, 1..... 4 1, 1+ 1-0 4 ~ 4 11 6 1, 1+.;, 4 1+0- 6-5 2 7 5 3,463 O + I + 2i5,6 1+0-0+13 35 5, 6 17 0-+4- 1 71 --- 5 5 15 55 16 9 0+1+ —33143 41 42 420 0+1+- ++143 715 0, 1+0-+1+ —+6 1086 12 9 8 33 12 1, 2 1 +7+4+2 + 25 + 5 -- 3, 1+o-20 20 75 2 33 14339 14 1 0+1+-+1+ +I9 11 143 39 5670 35 7 5, 6 1+0-0+ — + 7 01+143 20 108 48 9 143 7143 22 0+0+1+ +-+ 26 26 884 9(Continued on page 383)39 (Continued onz pa~ge ~383)

Page  382 382 W. F. SHEPPARD B (i) continued. om j Co. LUa, 6L2Ui, A41,.... 120 55 12 1 14 0, 1 1+8+18+ + + 7+ 54 80 45 144 1 2, 3 1+0 -7 7 91 7 45 25 25 75 1 0+1+ —+-+ —+ +14 7 14 182 28 200 225 72 10 4,5 1+0-0+7 + 91 + +1l 11 13 26 17 20 15 4 1 o -t 0+o+ +- + +-3 + ~ q21 63 165 55 39 7 1 16 0,1 1 + -+ — + 2+ ++ 2 2 4 2 4 4 8 1+1 + +7 +42 21+ 14 3 204 25 225 9 35 225 4,5 1 +00 + 262 + 34 +2584 25 200 5 28 175 0+1+0- 39 - 2 -51 - 3876 5 30 7 21 0+ 0+1- 3- + +7 2 +0++ 3 + 34 1292 40 154 715 364 35 16 1 18 0, + + - +88 + -- + - + + 2,3 1+0-22- -65- -— 2 -3 3 3 3 17 57 176 112 35 32490 7 2, 0+1+0- 22-3 - 65 3 3 17 969 57 1 1 65 35 14 1 0+1+ +11+ -++-+-+ +2 6 + 3 4 51001 57 + 1 40 56 350 400 5 4, 5 1+0-0+2 +5+ +51. +32 + + 35 49 49 490 2 3 6 17 969 57 36 45 28 315 504 2 1 + -1+ 22 + + 4 + 37 5 + 4 209 33 396 858 1001 273 68 9 1 20 o, 1 1 + — ++- - +- +-5 + — + 2 + 1188 572 819 72 189 4 2, 3 1+0- -- 126 - 56 — 35 5 5 5 95 35 48 52 728 175 224 56 48 1 0++-7+-3 + -- - + +-~- + ~+0-9 + 7 560 180 27 2 1 +0-0+26+63+63+ 17+ 9 +19-9 364 784 245 6272 1960 96 7 33 - 11 - 56 -627 - 209253 91 147 245 490 147 21 3 2+ 2 + 44 + 187 +- 209 +209+ 506

Page  383 REDUCTION OF ERRORS BY MEANS -OF NEGLIGIBLE DIFFERENCES B (ii) continued. m j Co. 5i.-, 3,11i, 51,...., +24 54 40 15 36 1 14 1, l+ -+ + + + +-1. 7 7 7 77 91 35 30 400 225 48 5 3, 4 l+0 4 77 91 91 119 5 25 25 5 1 (}+1 + 200 15 72 10 0+ 1+2+11 +26+26 +68 5, 6 1+o-+-+43 ~13 221 + 323 225 15 135 9 143 13 442 323 20 11 44 11 0+0+-+-^+ + 13 13 221 646 3 27 55 15 9 7 3 16 1 1 + l +- + 4 2 4 + 20+ 136 a, 15 25 225 35 45 2 2 26 68 1292 10 25 +100 5 7 35 0+++:- + + -~ + — 3 6 39 6 51 3876 175 105 63 245 75 5, 6 1 +0 —0+ 2 + 6+ + 64- + 2584 -75 40 45 84 25 26 13 34 323 1292 55 22 11 77 11 + 26 13 +17 46 1292 18 1,288 65 28 16 1 18 1, 2 1+8+22+ — +-+-+- + ++5 - 110 80 112 175 160 5 3, ]t 1 + () 25 - 3, 1 9 3 9 351 323 171 77 35 49 49 98 1 o+1 + 8 +7 +- + + 68+- + +17 280 35 392 2450 400 35 5, 6 1 + +o-+ + + + + + 39 3 51 969 969 1311 63 147 441 70 6 13 34 323 323 437 36 28 315 24 2 0+0 + 1 + - 3+ 8 + - + 6- 9 + 4 3 -7 9920, 1188 286 273 63 56 12 27 1 20 1- + + - +9- + 7 10 35 5 5 2 5 5 95 70 132 280 72 9 4, 4 1+0- — 52-63-42- 7 17 19 19 161 16 364 392 245 1568 392 16 1 3 33 33 33 561 627 209 253 147 441 3920 60 189 14 5,6 1~+0 —0+ 14 + + - + -3 — 33+ + 4+ 5 + 5 + 17 + 323 ~19 437 575 84 784 2205 18816 280 864 63 0+1+0. 11 55 187 3553 209 4807 6325 7 49 245 490 7 21 3 2 10 68 +323 19+437 1150 383

Page  384 384 W. F. SHEPPARD TABLE II. Values of reduction-ratio for central term (standard system). [The table gives the value of R, where a2 is the original mean square of error, and R2a2 is the reduced mean square of error, for different values of m and k; m _ 2n + 1 being the number of terms used in the formula, and 2k or 2k + 1 being the highest order of differences that cannot be neglected.] l | number Highest order of differences that cannot be neglected m = number of values of u in -- --.. the formula 2n+1 1st 2nd or 3rd 4th or 5th 6th or 7th 8th or 9th (k=0) Ik=1) (k=2) (k=3) (k=4) 1 3 -577 1000 1'000 1'000 1'000 2 5 -447 697 1'000 1'000 1'000 3 7 -378 '577 '753 1 000 1 000 4 9 -333 -505 -646 '787 1'000 11 ' 302 '455 '577 '690 -810 I 6 13 '277 '418 -528 '625 -721 7 1 -5 '258 -389 '489 '577 '660 8 1 17 ' -243 '365 '459 '539 '614 9 19 '229 ' 345 '433 '508 '577 10 21 '218 '328 '411 -482 -547 11 23 '209 '313 -393 '460 '521 12 25 '200 '300 '376 '441 '498 13 27 '192 -289 -362 -424 -479 14 29 '186 -279 '349 '408 '461 15 31 '180 -270 '338 '395 '446 1 __, l l _____________________.

Page  385 SUIR UNE METHUDE D'INTERPOLATION EXPUSEE PAIR HENRIt POINCARE~, ET SURt UNE APPLICIATION POSSIBLE AUX FONCTIONS DE SURVIE D'ORtDRE it PAR ALBERT QUIQUET. Dans son (Jalcul des Probabilite's*, Henim Poincare' a expose' la inu'thodc suivaiito d'interpolation. Je crois devoir la rappeler h la section actuarielle du Ve Congres international des Mathe'maticiens, car clle mn'a paru susceptible d'une application, que je ferai ensuite connai'tre, 'a la recherche des " fonctions de survie ": ce nom' pent etre donne' aux fonctions qui repre'sentent analytiquement le noinbre 1 (x) des vivants a iage x, pour un nombre donne' de naissances, ou pour un noinbre donne' d'individus ayant accomrpli en MeMe teMpS un 'age determine'. Voici succincternent 1'expose' de Poincare't. Ii s'agit d'appliquor- la me'thodc des inoindros carre's 'a la rocherche d'une fonction. inconnue f (x). On a mesur' fin valours do cotto fonction: f (a1) =Al, f (a2) =A2,) f arn) = AM~. On aurait ainsi un certain nombre de points de la courbe y =f x). On pourrait toujouirs par ces in points faire passer uric courbe, inais cotte solution ne scrait pas la meilleure; on fait passer une courbc pre~s de cos points, aussi continue que possible. On peut vouloir quo cette courbe soit de degre' h aussi petit que possible, f(X)=Co+CiX~ - ~GhX h *La deuxi~me 6dition (1912) de ce Calcul des Probabilitds a Wt peut-e'tre lune des derni~res ceuvres du regrett6 mailtre. II avait professe' ce cours At la Sorbonne en 1893-1894, et, Ai cette 6poque, ii avait bien voulu me laisser ihbonneur de r~diger les leqons re'unies dans la premie're 6dition. Pr~s de vingt ans plus tard, avec une conscience dont je lui garde une respectueuse reconnaissance, il avait maintenu mon nom sur la couverture du volume, cependant revu et augmente' par lui. + Chapitre xv. Th~orie de l'interpolation. 25 M. C. IL

Page  386 386 ALBERT QUIQUET hest plus petit que rn - 1, car si h 'tait 'gal ' ft - 1 on aurait une fonctio satisfaisant exactement aux conditions. Quelle valeur attribuer 'ah Cette valeur est arbitraire..On la choisit d'abord assez petite, puis, si elle est insuffisante, on introduit un terme de plus dans le second mnembre, et ainsi de suite. Supposons choisie une premie're valeur de h. Nous de'terminerons les coefficients du polynorne de telle fa~on que Ef [(a~) - A] soit minimum. f (x) est line'aire, par rapport aux coefficients C. Poincare' rattache cette question au de'veloppement eni fraction continue dii rapport F' (x) oui F (x) = (x - a,) (x - a,,)... (x - a,.). Si, dans cc de'veloppement, F' (x)1 F (x) Q __ Q2 +1 Q3 + QQM on appelle Di ie, de'nominateur de la iC re'duite, Di est un polyno'me de degre' i. Ce sont ces polyno~mes Di qui doivent retenir notre attention. Poincare' s'en sert pour e'crire f (x) sons une forme spe'ciale. Si f (x) est un polyno'me d'ordre h, ii pent toujours, dit-il, etre mis sous la forme f (x) =GCo+0CDj(x) + C2D2 (X) +... +GChDh (X). Poincare' etablit ensuite que l'on rendra minimum si les coefficients Ci ont l'expression suivante: ci = A Di (a1) + A2Di (a2) +... + A mDi (aln) Pour terininer l'expose de la me'thode, empruntons encore an Galcu'l des Probabilite's * l'explication de l'avantage des polynomes D. "Je suppose que l'on ait - essay6' d'abord de representer les observations par un polynome de degre4 h: on a trouv6" alors 00, C1, *.., Ch On constate ensuite que la Somme des carre~s des erreurs commises est inadmissible: on se refsigne alors 'a poursuivre avec un polynome de degre6 h ~ 1. Tout serait 'a recommencer, Si l'on *Page 290.

Page  387 SUR UNE METHODE D'INTERPOLATLGN EXPOSEE PAR HENRI POINCAR9 387 avait eu recours aun proce'de' quelconque; ici, au contraire, on n'a qu'a' aj outer un terine Ch+, Dh+I (X): les precilents coefficients ne chaiigent pas, comine on le voit sur I'expression de cii." Les applications des mathe'matiques conduisent fre'quemme-nt 'a de semblables th'tonnements. La technique des assurances est peut-6~tre une de celles qui ont le plus 'a y recourir: on con~oit sans peine l'inte'ret que peuvent porter les actuaires 'a tout proce'de' capable d'abre'ger leurs essais, notamment quand ce proce'de n'annule pas les efforts ante'rieurs. C'est pre'cise'ment le cas des polynornies D, et c'est pourquoi in ' a sembl' indifferent de montrer un exemnpie professionnel de La base des operations viageres, c'est-a'-dire le nombre 1 (x) des vivants, est rarernent susceptible d'e'tre figure'e par un polyno'me de degr6' h. Ce degre6 devrait en general e~tre assez e'leve', s'il s'agissait de repre'senter une table e'tendue de survie; et le nombre de terines qu'il entralinerait rendrait la formule pen maniable. L'on recourt plto a certaines fonctions transcendantes, qui ont d'abord l'avantage de la concision, puis qui posse'dent des proprie'tes appr'cie'es, que l'analyse leur de'couvre, et dont les polyno~mes semblent momns bien pourvus. Je n'ai pas besoin de citer ici le prix, pour ainsi dire journalier, qui s'attache, dans les operations courantes d'assurances, aux ce'lebres lois de deux actuaires anglais, Gompertz et Makeham. Si les polynornies n'offrent qu'un inte'ret secondaire pour l'interpolation de 1 (x),J uls ne sont pas cependant 'a 6arter de tous les problemes relatifs 'a la mortalite' humaine. Le nombre des vivants n'est pas la seule fonction de l'~ge x dont on fasse usage. Le taux de mortalite', qui derive de 1(x), pent e~tre cite' comme une autre de ces fonctions. Pour nous borner "a un seul cas d'application des polyno'mes, realise' effectivement, de 0 'a 25 ans le taux de mnortalit4' des Assure's fran~ais (table AF) a e'te interpole' par une formule alge'brique du sixie'me degre'. Peut'-etre unesi des polyn~mes D efit-il indique' si cc degr e' povait pas ^tre abaiss'. Je me permettrai de citer encore une autre fonction de la~ge x, le logarithine de 1 (x). Dans la the'se que j'ai soutenue pour devenir membre agre'ge de I'Jnstitut des Actuaires fran~ais *, j'ai pu 6tablir le parti qui est 'a tirer d'une certaine fonction zx, la diff6rence seconde de cc logarithme. Lorsque zx obe'it 'a une relation de recurrence line'aire et d'ordre n., Bozx + Blzx+1 +... + Bl~zx+n = 0, j'ai donne 'a 1(x) le nom de " fonction de survie d'ordre nt." Ces fonctions fortment une classe inte'ressante au point de vue de linterpolation des tables de survie: elles coruprennent notamment les lois de Gompertz, de Makeharn, de Lazarus, de Janse, etc., et ge'neralisent les proprie'tes de ces lois, en faisant ressortir les liens qui les unissent. Or un cas, en quelque sorte limite, est 'a remarquer an point de vue de l'application des polyn6mes D. Ce cas est celui oui le6chelle de recurrence des *Albert Quiquet, Representation alg~briquse des Tables de Sur'vie; g~njiralisation des lois de Gonspertz, de Alakehcam, etc. Paris, 1893. 25-2

Page  388 388 ALBERT QUIQUET fonctions zx n'admet qu'une racine multiple d'ordre n et 6'gale h 1unit6 z' est alors un simple polyno'nme de degre' i - 1, et par suite le logarithme do 1 (x) est un polyno'me de degre' n + 1. Cette forme particulie're laisse cependant 'a 1 (x) le caracte're d'une " foniction. de survie d'ordre a,." On peut done avoir inte'ret 'a ladapter 'a des observations brutes; pour savoir si cette adaptation est possible, on commencera naturellement par essayer los plus petites valeurs de n. Si elles ne conviennent pas, on passera ai la suivante. C'est ainsi que s'introduiront les polyno'mes D: grace ai eux, cette recherche progressive s'ope'rera avec regularite6 et sans cesser d'utiliser, an fuir et 'a in~esure, los calculs successifs quo l'on a laborieusement obtenus depuis l'origine. Ces considerations sembleront assure'ment bien modestes dans un Congre~s destine' surtout aux plus hautes speculations dos mathe'matiques pures. Mais notre section envisage los applications les plus inmmediates des theories que nous transmettent nos maitres. A l'heure oii la science universollo deplore la perte re'cente de l'un d'eux, l'on voudra bien m'excuser, je l'espe're, d'avoir apporte6 ici le souvenir des le~ons qu'il nous a laisse'es, et d'avoir montre' leur utilite' possible, me'me dans des ope'rations aussi concre~tes quo los assurances sur la vie.

Page  389 ON THE FITTING OF MAKEHAM'S CURVE TO MORTALITY OBSERVATIONS BY J. F. STEFFENSEN. Amongst all formulae which have been proposed by actuaries for graduating mortality tables, none has a more established reputation than Makeham's Curve y = a + / c....................................(1). Its authority rests less on the philosophical considerations on which it was originally founded than on the practical test to which it has been put, and on its well-known advantages for calculating complicated benefits. In this paper I shall not deal with the question, what kind of mortality experience the formula may be presumed to fit; nor shall I discuss which are the best equations for determining the constants. I shall assume that the method of Least Squares has been chosen, and my main object is to find a practical and simple way of solving the resulting equations numerically. Let o- be an observed value of yx, and let 1F be proportional to the weight of ox. We have to determine a, / and c so, that the expression (yx - )2P r...............................(2) is a minimum, that is =o; n=o; o 0. Replacing yz by its valule (1), these equations become acrF +,3c2xx - Scrxx = 0. (3). aScXrX + 8..C2XrI X -C rXC ~ = *................... (3). aSXCXrX + 1xcC2x F1 - Zoz-xcXrX = 0 This system is linear with respect to a and A, but algebraic of a high degree with respect to c. Previously to the method proposed by me two other methods have chiefly been employed for dealing with these equations. The first method is, in principle, the classical Gaussian one and requires that the equations be rendered linear by introducing approximate values of the constants. It is well known, that a strict application of this principle to the equations (3) involves considerable numerical work. This was, for the first time, carried through with consistency in a thorough paper by Joh. Karup*. * " Die Ausgleichung der Sterblichkeitserfahrungen der Gothaer Bank nach der Gompertz-Makeham'schen Sterblichkeitsformel." Rundschau der Versicherulngen, xxxiv. (1884). See also Journal of the Institute of Actuaries, xxIv. (1884), p. 70.

Page  390 390 J. F. STEFFENSEN The second method seems to have been proposed simultaneously by Rosmanith* and Draminskyt. The guiding idea is to adopt a succession of trial values of c, and to determine the corresponding sets of a and 3 from the two first equations (3). For each set of constants thus obtained the value of the expression (2) is calculated, and it is evidently possible by continuing this process to get indefinitely close to the absolute minimum of (2). In order to obtain an actual solution of the three equations (3) it is, however, necessary (as pointed out by Rosmanith) to employ some sort of criterion for the approximation obtained to the absolute minimum, which complicates the calculation. The method I have personally employed+ seems more direct and less laborious than either of the two previous ones. I eliminate a and / from (3) and obtain a single equation in c Er, Ccxrx 2,Xrx D - ScXr, c2$rF So,-cxr =...............(4). ixc.FX S.XC2XrF Eo-fXCxrX To solve this equation numerically with any desired degree of accuracy presents no difficulty, remembering that log c = -04 is generally a good first approximation. Three trial values of c will as a rule produce about five correct figures of this constant, by interpolation with Newton's divided differences; if it is further known that an increase of c means a decrease of D (as is the case in the experience dealt with below and may be presumed to hold good for similar experiences), we may often by means of only two trial values and linear interpolation between these be able to obtain c to about five figures. Having thus determined c, the values of a and 38 follow fiom any two of the equations (3). For actual calculation we transform (4) as follows. We remember that f(x) = 2f (x).................................(5), if the summation is commenced from below. It is here assumed, that the observations begin with x = 1; otherwise E2 must be interpreted as + (t- 1).................................(6), where t stands for the first argument at which the observations begin. We may now instead of (4) write I Eg CcxrF azxeP D - cxFPx Sc2X r Eaoxcxr 0 =0...............(7).:2&c]X Y 2C2Xfr Y2OxcXFX Each new trial value of c requires the writing down of the following ten columns, which are readily formed, four-figured logarithms being as a rule sufficient for this part of the arithmetical work: * Mitteilungen des Oesterreichisch- 1Ugarischen Verbandes der Privatversicherungs-Anstalten, vol. 2 (1906), p. 54; Berichte des V internationalen Kongresses filr Versicherungs-Wissenschaft, vol. 2, p. 329. See also a recent paper by John S. Thompson in the Transactions of the Actuarial Society of America, xiI. (1911), p. 225. t D;delighed efter Forsikringsart og Forsikringstid, Copenhagen, 1906, p. 99. + Dansk Forzsikriigs-Aarbog, Copenhagen, 1909, p. 45.

Page  391 ON THE FITTING OF MAKEHAM'S CURVE TO MORTALITY OBSERVATIONS 391 log cx log C xr crx SCx rx 2 Cx rX log c2xFX C2X r YC2X Fr groups of, say, five years of age; I have, however, avoided this in the example below, possible without much increase of labour. The further details of the method will best be understood from the example on which I have employed it. The table in question is the largest and most recent Danish experience, called DM(5) (healthy male insured lives, excluding the first five =-log(l- )..........................(9). As the observations would give 289 = 1o, the observation for age 95 has been left out. For the weights wn we have used the well-known approximate formula 1 E x.............................. (0), and rF was taken as ox F2-PxcrX groups of, say, five years of age; I have, however, avoided this in the example below, For reasons which we need not discuss here it is inexpedient to use the ungraduated values eof po and q in the formule (10) and (11). A preliminary graduation should therefe urher de; fr this purpose eit is uslly sufficient to determine a, l and c from four iencegr, alues of log The preliminary formla used lv e relimin this yW - ol p = = - +/2 C.............................. ( S), conse was For the Colpw s = w0007e6 + 10h039e54+ te 39ellno n aroimate formula 1 case was col = 00075 E6.................. (12). * Dodlighetstabeller enligt nitton skandinaviska och finska Liffrsilkringsanstalters Erfarenheter, Stockholm, 1906, p. 33.

Page  392 392 J. F. STEFFENSEN The values of log F, according to this formula are stated in column (5) of the adjoined table, while column (6) gives the mean error m, or log e m-/IP X............................ As the first trial value for log c it was natural to take the result of the preliminary graduation or logc= 039654; the value of D for this and two other trial values, namely log c = '039 and log c = 040, was calculated. Putting for convenience 10000 (log c -'039) = x.4) 10-2D=y\ *****3.********** *(14), 10-22 D = y the following table with divided differences was formed x y ay 2y _______________I _______,________ ___ _____ 0 00 652 - 96-33 6-54 22 -9-85 - 194'79 10-00 -652 I The value of x for which y vanishes is found by Newton's formula* y = yo + x $yo + x (x - x) 82y0 = 0, where x, = 6'54. We find x = 6-68 and finally log c = -039668. For a and /3 we find thereafter by (3) a 13 From the 1st and 2nd equation -00090332 -000046237 1st,, 3rd,, 90330 46237,,,2nd,, 3rd,, 90313 46238 The agreement between these values shows the accuracy of the process. As final values were adopted a= 0009033 log / = 5-66499..............................(15) logc -039668 whence col px = *0009033 + 10 039668x+5 66499.(16). In order to test the success of this graduation I have first examined the column headed (yx - o-x):m, with respect to the distribution of the signs + and -. If such a distribution is quite casual, the average number of isolated sequences of the rth * As pointed out by Dr W. F. Sheppard, the solution of an equation for x may be avoided by putting y = 0 in the inverse formula: x- xo) + (y -?,,) I.r, +- ( y - y?) ) a (Y t,.

Page  393 ON THE FITTING OF MAKEHAM'S CURVE TO MORTALITY OBSERVATIONS 393 order is n: 2r+, where n denotes the number of elements*. The average number of isolated sequences of all orders is n. In the present case we have n= 75 and find Number of Sequences Order -_ calculated observed Difference 1 i 18-8 19 -0-2 2 | 94 9 +04 3 i 4-7 6 - 1:3 4! 2'3 1 + 1'3.; 1-2 3 i - 18 6 06 0-6 +0-6 all 37-5 38 -0'5 The agreement is as close as can be wished. I have secondly compared the magnitude of the deviations with the typical Law of Errors. In the small table below is indicated how many of the deviations y,- ax are found, according to observation and to theory, within k times the mean error. The table is easily verified by means of column (10) in the larger table, containing (yx - ~x) 'm. It is seen that the distribution is nearly typical, and would be still more so, if the observation for age 26 were left out. This observation is suspect a pr'iori, as it has a deviation of more than six times the mean error. The occurrence Number Percentage observed calculated observed calculated 0 2 7 11 9 093 -159 0-4 22 23-3 -293 - '311 0'6 32 33-8 427 -451 0-8 40 43-2 533 -576 1-0 48 51-2 640 -683 1-5 63 650 840 -866 2'0 67 71-6 893 ' 954 2-5 72 74-1 -960 -988 3 0 74 74-8 987 -997 of this sort of abnormality in a mortality table cannot entirely be guarded against; it may be due to local connexions between observations, to clerical errors or to many other causes. On the whole I think the graduation may be considered successful, especially considering that the formula contains only three constants, to be determined by 75 observations. Commutation columns at 31 ~/o will be found in my Danish paper on the subject. * Czuber: JWahrscheinlichkeitsreclinung, 2 Aufl. vol. I, p. 146.

Page  394 (1) x 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 I 94 (2) 47 75 111 179 260 337 506 699 1027 1456 2245 3112 4091 5071 6068 7034 7873 8600 9157 9547 9913 10108 10252 10236 10186 10040 9816 9602 9241 8941 8442 8077 7695 7332 6932 6557 6198 5802 5460 5113 4707 4403 4099 3863 3584 3300 3057 2777 2521 2291 2075 1849 1640 1447 1252 1090 935 774 637 535 443 358 295 223 160 122 95 52 36 23 15 10 6 5 (3) (4) ox o 0*00000 1 582 0 000 1 244 3 504 0 0.00000 9 780 2 124 5 212 6 179 10 0-00194 12 168 18 192 22 189 34 244 31 0-00192 36 199 51 258 58 275 38 173 42 0-00185 55 237 76 323 69 294 68 291 83 000361 89 396 99 450 84 397 104 508 106 0-00549 112 607 118 671 123 735 104 656 117 0-00782 148 1050 110 0831 153 1234 120 1031 123 0-01130 141 1413 128 1378 152 1743 119 1466 136 0-01827 133 1932 165 2661 138 2445 121 2357 126 0-02721 126 3065 132 3644 128 4022 116 4222 109 0-04576 109 5383 86 5115 71 5132 57 4893 59 006207 47 6112 56 9142 46 10033 32 09691 19 0-07352 33 18533 9 08254 11 15837 8 18564 5 017609 3 15490 1 07918 3 39794 0 00000 o Graduation of the Dm(5) Table. (5) (6) (7) (8) log 1x - Yx - a'x 4-2845 0-00313 0-001190. + ~001190 4757 251 1218 - 4602 6333 210 1248 + 1248 8275 168 1281 - 1159 9756 141 1317 - 3723 5-0735 0-00126 0-001357 + 001357 2341 105 1400 - 6400 3578 091 1448 + 0208 5072 077 1500 - 0620 6404 066 1557 - 0233 5-8090 0-00054 0-001619 - 000321 9305 47 1688 + 0008 6-0281 42 1763 - 0157 0992 39 1845 - 0045 1542 36 1935 -- 0505 6-1944 0-00035 0-002034 +'000114 2186 34 2142 + 0152 2313 33 2260 - 0320 2322 33 2390 - 0360 2230 34 2532 + 0802 6-2114 0-00034 0-002688 +'000838 1911.35 2859 + 0489 1678 36 - 3046 - 0184 1369 37 3251 +1 0311 1041 39 3476 + 0566 6-0664 0-00040 0-003722 -- '000112 0245 42 3991 + 0031 5-9823 44 4287 - 0213 9326 47 4610 + 0640 8846 50 4965 - 0115 5-8256 0-00053 0-005353 - 000137 7718 56 5779 - 0291 7156 60 6245 - 0465 6592 64 6756 - 0594 5992 69 7316 + 0756 5~5388 0-00074 0-007929 + 000109 4779 79 8601 - 1899 4125 85 9338 + 1028 3489 92 0-010144 - 2196 2830 99 11028 + 0718 5-2094 0-00108 0-011997 - '000497 1426 117 13058 - 1072 0733 126 14220 + 0440 0092 136 15494 - 1936 4-9380 148 16889 + 2229 4,8633 0-00161 0-018418 1+ 000148 7911 175 20093 + 0773 7101 192 21928 - 4682 6286 211 23939 - 0511 5473 231 26143 + 2573 4,4645 0-00255 0-028557 ~ 001347 3743 282 31201 +- 0551 2818 314 34099 - 2341 1870 350 37274 - 2946 0833 395 40752 - 1468 3-9821 0-00443 0-044564 - 001196 8742 502 48739 - 5091 7506 579 53314 + 2164 6241 670, 58327 + 7007 5063 767 63819 + 014889 3-3818 0-00885 0-069837 ~ 007767 2465 1034 76429 - '015309 1191 1198 83653 - 007767 2-9540 1448 91567 - 8763 7657 1799 0-100238 + 3328 2-6034 0-02168 109738 ~'036218 4496 2588 120147 - 65183 1422 3687 131552 + 49012 1P9361 4676 144047 - 14323 6944 6175 157738 - 27902 1-4609 0-08079 172737 - 003353 2360 10466 189172 + 034272 09644 I 14309 207178 + 127998 8345 1 16619 226906 l - 171034 0835 39455 i 248521 ' + 248521 I (9) 1 (10) nix + 22-9 +0038 -- 137-6 -1-83 + 53-6 +0-60 - 77'9 - -069 - 351 9 -2-64 + 160-7 ~1107 - 1097-2 - 610 + 47 -4 ~0023 - 199-4 -0.81 - 101.8 -0-35 - 206-7 -0,59 + 6-8 + 0-02 - 167,5 -0-37 - 56'5 -0-12 - 720-2 -1-14 + 178-3 +0033 + 251P4 ~0045 - 545-0 -0-96 - 614,5 -1.08 ~ 1340 3 +2-39 ~ 1363-3 +2-46 + 759,3 + 140 -- 270-8 -0.51 + 4263 +-0-84 + 719.3 ~1147 + 130-5 ~0028 + 32-8 + 0-73 - 204-5 - 0-48 + 548-0 ~P136 - 88-2 -0-23 - 91-7 -0 26 - 17241 -0-52 - 241P6 -0-77 - 271-0 -0-92 + 300-4 ~1.10 + 37.7 +0*15 - 570-9' -2-40 + 265-8 ~ 1-20 - 490-4 - 2 -39 + 137-7 +0-72 ~ 80 5 +0-46 - 148-9 -0-92 + 52 1 ~0035 - 197 7 -1-42 + 193 2 + 1*51 + 10-8 ~ 0 09 ~ 47-8 ~ 0 44 - 240-2 -2-44 - 21-7 - 0-24 + 90,7 ~1.11 + 393 ~ 0-53 + 13-1 ~0020 - 44-8 -0-75 - 45 *3 -0-84 - 17 '8 -0-37 -- 11.5 -0-27 - 381 -1*01 ~ 12-2 + 0 37 ~ 29 -5 ~1 05 ~ 47 -8 ~194 -F 187 +0088 27 0 ~1 48 - 102 -0-65 7 *9 -0-61 ~ 19 ~019 14 5 ~1-67 - 18-4 - 252 ~ 6-8 ~133 - 12 -0-31 - 14 -0-45 - 01 -0-04 ~ 0-6 ~0033 ~ 1 2 ~0089 1.2 - -103 ~ 0-3 ~ 0-63

Page  395 ON THE APPLICATION OF THE CALCULUS OF PROBABILITIES IN CALCULATING THE AMOUNT OF SECURITIES AND IN UTILISING STATISTICAL DATA FOR THE SAKE OF CLASSIFICATION OF LEGAL ACCIDENT-INSURANCE RISKS IN THE PRACTICE OF THE DUTCH STATE INSURANCE OFFICE* BY J. H. PEEK. The Dutch Act regulating the Insurance of labourers in case of accidents, allows, by provision of the articles 54 and following, private institutions and employers to undertake under certain restrictions the risk of insurance according to the provisions of that act. One of the conditions to be satisfied before the undertaking or transferring of such risk is permissible, is that the " Rijksverzekeringsbank" (State Insurance Office) shall hold from the future insurer a certain sum as a security for the obligations resulting from the law being observed. It has been thought desirable to fix these securities at a fraction of the average net value of the probable charge resulting from accidents to be expected per annum among the labourers at his risk, augmented by such a multiple of the mean deviation from that charge, that greater deviations have a smaller than an assigned probability. Two problems arise from this: (1) the determination of the mean deviation from the above said net value, (2) that of the law relating to the distribution of deviations. As these problems have proved to be of like importance in other cases daily occurring in the practice of legal accident-insurance, I think them of sufficient importance for communication. The compensations provided by the Dutch Insurance Act consist chiefly of medical attendance, temporary weekly payments, which consist of a constant fraction of the weekly wages for the first six weeks after an accident, and afterwards of an amount depending on the degree of incapacity, and pensions in cases where permanent disablement results from the injury by accident (dependent upon the degree of incapacity) and, if death results, a pension to each of the dependents. If, in a group of insured persons of a constant number exposed to a constant danger, on the average n, + n, + n3+... + nI accidents occur per unit of time (year) of which the cost is * As far as no more recent results are concerned, this paper is an extract from an ample report, offered in 1902 to the Directors of the State Insurance Office by the author as Mathematical Adviser attached to the Institution.

Page  396 396 J. H. PEEK respectively mn, m2, m3,... ml, per accident, then the yearly premium due for the insurance of this group will be: G = n1mnl + n-m2, + nm. +.. + nml. Taking into consideration an indefinitely short period dt, the numbers ndt, ndt, n3dt,...n1dt, may be taken for the probabilities that during this period an accident may occur of which the value of the cost (pecuniary importance, in what follows briefly called "importance") will be mn, min, m3,... ml. The probabilities that two accidents may occur in so short a period being of second order, it seems permitted to neglect them. If during this period an accident occurs of importance mi, the difference from the average value will be vi, - Gdt; if no accident occurs, the difference is: -Gdt. The probability of the first of these deviations is ndt, the probability of the second 1 - dtEn,. Consequently the square of the mean deviation: dtV (m, - Gdt)2n, + (Gdt)2 (1 - dt n) or, neglecting powers of dt: dt. mM2ns. By integrating, with respect to t, from 0 to 1 we find the square of the mean deviation per annum to be M= = m s'2ns. In order to facilitate as much as possible the calculation of this quantity we will write now M =, m2ns, = a pSn2; where a denotes the number of persons belonging to the constant group (the number of Vollarbeiter; travailleurs-types; years of labour) and p, the number of accidents per Vollarbeiter of importance Im,. Division by a gives the mean error per Vollarbeiter, denoted by Let the cases be divided now into three groups: Group I, cases of temporary incapacity, characterised by index 1,, II, fatal accidents,,,,,, 2,, III, accidents resulting in permanent incapacity characterised by index 3 at the left at the head of the symbol concerned; then S pm,82 = E Ips,82 + 2p.,m i2 + E.3pi m,2, 2p.s being understood as the product of two quantities, namely 1p: the average number of accidents per annum of kind n per Vollarbeiter and ^q,: the probability, that an accident of kind n will necessitate the expense n,, then Epm, 2 = 1p IqS.,2 + 2 q. nm + p3pE. mq'- +'p m2. Here 'lq.. in,2 is the average value of the square importance of a case of temporary incapacity, to be denoted by: lM2; -2q.,mS2 the average value of the square importance of a fatal accident to be denoted by 2M2; 23q, mn2 the average value of the square importance of a case of permanent incapacity to be denoted by 'M2; so that S p.,1,2 = lpf2Il + 2)-M1 12+ 311p:2.

Page  397 ON THE APPLICATION OF THE CALCULUS OF PROBABILITIES 397 The values of 1M', 2M2 and 3M2 depend on the amount and distribution of wages, the nature of accidents, the distribution of ages, the proportion of married and unmarried workmen and the formation of families. Consequently they might be quite different for different employers and that might be an impediment for application to practice. If we restrict application to larger groups of labourers we may expect less different values, especially if dividing by the square of the average annual wages L. Let the quotient of M2 and L be denoted by rgJI2, then pmnz,2 = L2 (lp lS2 + 2pY 2S 2 + 3p3t ). Our first problem is now solved, if we know for every group of insured workmen the value of the average annual wages L, the numbers 1p, 2p and 3p (being the numbers of accidents per annum and per Vollarbeiter), and the values 1Ji", 2V2 and Vg2 (being the average square of the importance of an accident of the different groups per unit of average annual wages). We have supposed (with respect to a previous calculation, executed before any experience had been acquired as to the effect of the Accident Insurance Act) that the values 1^iJ2 and 29J"2 are invariable. Moreover the term apo39N2 was transformed into 3pt.t2 3sJ2 in which 31t represents the number of accidents per Vollarbeiter, where a permanent incapacity of the fraction t of the total resulted, while 39J12 represents the average square of the importance of a case of total incapacity per unit of average annual wages. This transposition supposes that the average square of the expense of a case of permanent partial disablement of the fraction t may be represented by t2.3%,l2. The quantity 39t12 too was supposed to be invariable. The calculation of the quantities y2e can be effected most easily and most correctly from the expense of each of the individual accidents. As this information was not at hand when the previous calculation was made, we had to recur to the official Austrian statistics on the subject. These statistics relate to great numbers of cases and contain no data relating to the cost of each case individually. It was therefore impossible to obtain the quantities S9)2 in a direct way, but to obtain them we could make use of the property, that the square of the mean deviation of an observed quantity from its average is equal to the average square minus the square of the average of the quantity. It may be well to recall the proof of this well-known property. Let mn denote the observed quantity, )n2 its average square, /2 the square of the mean deviation, and mn the average of the individual values nm; ps the probability that the quantity m, will be observed, then 2 = (mn - m1)2ps = EmM2p - 2mnmsp,+ wnl2ps, and 2mp> = m, 1p, = 1, therefore u2 = I2 - mn and 9)3J2 = i+ m2. Now the square of the mean deviation may be approximated to by means of the property that this quantity taken over a cases is equal to a times the square of the deviation per case. Consequently we had to calculate the value of the square of the mean deviation for the different territorial Austrian accidents insurance institutions from the abovementioned statistical publications, containing the number and the charge of accumulations of accidents belonging to each group, at that time (1901) disposable over

Page  398 398 J. H. PEEK five successive years. From the value obtained we could calculate the average square, the average value m being easily obtained. We will call this the indirect method. If we dispose of individual data, then we do better in following the direct method by which we may obtain more accurate results by calculation of the desired average from the square of the importance of each accident. So, as soon as a sufficient experience was at hand, according to this direct method, a calculation of 2J2 and ~39Nl2 was made for several branches of industry, in order to verify whether these quantities for an approximate calculation could really be considered as invariable. For different branches of industry 5 values were calculated for 29JN, the extremes of which were found to be 5'9 and 6'8; their arithmetical mean being 64. Though the statistical data were rather incomplete, the results were not so very different. For 3J9, 12 values have been calculated for different branches of industry. The extreme values were 10'3 and 12*6, the arithmetical mean being 11'2. Again it appears that the above supposition was not altogether wrong. A similar calculation of 19NJ has not been made. The importance varies here between 0 and thousands of guilders. The high results are comparatively scarce but have an important influence on the final result. Therefore the value of '1N which will appear to be of little importance in the subsequent calculations has been obtained by the above explained indirect method from more ample statistical data than could have been taken into consideration, when calculating it in the direct way. It will be understood that the divergence of the different values acquired in this way, was far greater than that of the values 2)Jt and s9D, so much the more as we know the amount of the compensations of Group I to be in a high degree subject to personal estimation. So far we have found the mean deviation to be: VL2 (1p.1't2 + 2p.2St2 + 3p.3V2), in which formula 25J and "3$ may practically be considered invariable, at least as far as cognate branches of industry are concerned. If we divide by L we shall find that in those branches the mean deviation only varies when 1iJt (being of small importance) varies or when lp, 2p and 3p vary. Even when considering the same branch of industry, these values may differ a good deal, as will appear in due time. Greater differences will disappear if we divide by the net-premium due for the insurance per 100 of annual wages. This premium n = 100 (1p1m 2 + m + 3m); in which formula 'm, 2nt and 3%n stand respectively for the average importance of cases of temporary incapacity, of fatal accidents, and of cases of permanent incapacity per unit of annual wages. The result of the division is:,|2_ = 1p193l2 + 'p 2J2' ~+ 3p''. 100 ('plm + + m + m)' If we consider 1'JI, 29r, )J39 to be invariable for the same branch of industry as well as the quantities lm, 2m and 3m, then the value of M2 appears to depend solely on the mutual proportions of 1p, 2p and 3p. If these proportions were invariable too, in the same branch of industry, then M2 might be considered invariable under the same circumstances. It has been proved already that even in much different branches

Page  399 ON THE PPLLICATION OF THE CALCULUS OF PROBABILITIES 399 of industry o2J and 3S).? vary very little. On investigation the same appeared to be the case with regard to the value of m. A priori it is not improbable that the mutual proportions of 'p, 2p and 3p should be nearly invariable. If the frequency of accidents should vary within the same branch of industry and in the same territory a difference in these proportions may arise from intensity of trade or economical depression, from different carefulness with regard to protection against accidents, etc. As long however as the technical character of a trade remains unaltered, these influences may have their effect on the frequency, but not on the issue of accidents. This, doubtless, holds true with respect to the proportion of the numbers of cases of permanent incapacity and of fatal accidents which are the most important. Moreover if we pay special attention to the quantities in the formula for M', it is evident that on the whole 3p3s2 and 3p.3m are predominant because of the slightness of 2p and of 1l')2 and lm. So the differences in the mutual proportions of the values of p would have to be enormous to cause any important differences in the values of M2. The following table may prove that as a rule the mutual proportions of the heterogeneous cases never differ much, even when the numbers of accidents show great differences. Assuming the number of accidents causing permanent incapacity or death to Number of be 100, we find for Branch of industry accidents per one 1000 (name of the trade) the number (name of the trade) Vollarbeiter" the number the number (years of labour) the number of cases of of cases of of faccidentsal permanent temporary incapacity incapacity Masons in the large cities 165 18 82 2497,,,, other places 98 15 85 2782 Dock labourers in Rotterdam 538 14 86 2554,, 1, elsewhere 261 14 86 2584 Ship-builders in the north 142 13 87 4362,,,,,,,, south 213 12 88 3542 Painters in the large cities 47'3 31 69 1540,,,, other places 39-1 28 72 2881 Railway Company A 54-7 39 61 3105 B 61-0 48 52 3177 The building-trade in the large cities 148 15 85 2878,,,,,,,, other places 142 12 88 3363 Inland navigation northern provinces 35-1 47 53 1100,,,, southern provinces 53'5 51 49 877 When calculating the values of the quantity M for the railway companies A and B (where the proportions of the number of accidents show the greatest mutual differences), it appears how slight the influence of differences like these is on the value of M. For company A, M amounts to 0,23, for B to 0'25. Consequently we may apply the values of M calculated from statistical data for the whole of a branch of industry to larger groups belonging to the same branch. We may even expect to find only slightly deviating values for different branches of industry.

Page  400 400 J. H. PEEK Up to the present, in the development of V m52ns, no allowance has been made for the so-called "multiple accidents " (those, by which more than one workman took injury). The necessary information to make this allowance in calculation is wanting; an approximate calculation has proved that in the railway-trade (that is a trade where relatively perhaps more "multiple accidents" occur than in any other) these accidents add to the value of M much less than 3 ~/o of the total value. So we are justified in calculating without these. So far we have been operating exclusively according to what is called the combinatorial principle, though what may be called the physical principle has already been applied in calculating the quantities 35 according to the indirect method. After in 1901 previous calculations had been made by application of the above explained combinatorial principle to Austrian statistics, it was felt to be necessary to compare the result to approximation of the same quantity. For this purpose we made use of the ' Revidierte Unfall-Statistik " of 1890-1896, in which for several trades the total charge (B)* of all accidents, the total amount of wages (1), and the total number of " Vollarbeiter" (a) for each of the seven years (1890-1896) are communicated for each of the seven official territorial insurance-establishments. By totalizing the a and B for a certain establishment over all the seven years, we SB find the probable value of the charge per Vollarbeiter exposed to risk. The YB deviation of B from the average charge is then B- a a-. If a were invariable it would be easy to calculate from these the mean deviation for a workmen, from which the mean deviation per workman would be obtained by division by V/a. Approximate equality of the number a could often be attained by adding the data for different establishments in opposite order of time. Perfect equality not being attainable, the square of the mean deviation per workman has been calculated by the formula E (B - E a n-i in which n denotes the resultant number of independent groupings. By dividing the square of the result by (- ) and by 100- 1 we finally obtain the value of M2, comparable to the value obtained according to the combinatorial principle by means of the quantities 19N2, 2Jn2 and 3S)I2. The average of the values obtained in this way for the same branch of industry, but for different institutions, was not considered to be the most probable, before the utmost care had been taken that these values should depend on statistical experience of the same extent, that deviations not depending on the number of observations (symptomatical deviations) were eliminated by inverse summations, and that the individual values should be as nearly as possible of equal weight. * Charge is understood in this paper to be the value of all expenses resulting from accidents which occurred in one year of insurance, calculated at a rate of interest of 3t per cent. per annum.

Page  401 ON THE APPLICATION OF THE CALCULUS OF PROBABILITIES 401 The following table of the values of M2 proves that, if not always, still in the principal cases, where ample statistical information was at hand, the expected conformity as to order of greatness was indeed found to exist*. Value of M112 calculated after Branch of Industry — E the first method the second method Ironworks 0074 0-091 Mechanical Brick-works 0-088 0-081 Glass-works 0'086 0-063 Mechanical forges 0063 0-026 Engine-works 0-072 0-064 Gas-works 0-063 0'072 Wool-manufactories 0-082 0032 Spinning-mills (cottoll) 0-082 0'081 Paper-mills 0-088 0'067 Saw-mills 0-085 0-046 Sugar-factories 0'083 0'056 Builders 0'083 0'102 Mechanical printers 0-065 0-043 Average 0-078 0-063 Moreover we may conclude from the first column that, according to expectation, the value of M obtained by the first method will not differ very much for different kinds of trades. If we know the value of M for a certain trade we may find the square of the average deviation for a certain group of persons in that trade by multiplying M2 by the net-premium taken from that group per 100 guilders of wages, by the square of the average annual wages calculated for that group and by the number of labourers; that is according to the formula aL2nM2, if a denotes the number of labourers, L = the average annual wages, and n = the net annual premium for 100 guilders of wages. As soon as sufficient statistical material concerning the legal insurance against accidents in the Netherlands was at hand, calculations have been repeated. This time calculations have been made exclusively according to the combinatorial principle, the quantities 1t) having been obtained according to the above called direct method. When applying the physical method the increase of the frequency of accidents which has been noted in our country as well as in Austria and Germany has a particularly disturbing influence. These symptomatic variations in the annual net-charge, though eliminated as is above said as much as possible, continue to be of so much importance that, at least in ample statistical material, the accidental deviations are quite overruled by them. This makes the results of * Hitherto approximate conformity of combinatorial and physical results had only been obtained by Lexis as to the sex-ratio at birth (Jahrbuch fiur National-Oekonomie, 27 and 32), by the author as to the number of yearly deceased persons among the Dutch population (Das Problem vom Risiko in der Lebensversicherung, Baumgartner's Zeitschrift, 1899). M. c. II. 26

Page  402 402 J. H. PEEK the physical method, where of course the statistical material should not be too extensive, unreliable in a great measure. Throughout this series of calculations we used the same average value for 1$92 and 2S)M2, which, as far as the value of 2s"2 is concerned, seemed to be quite admissible. Besides the material was too small to derive a sufficient number of reliable values of 25))2, while moreover fatal accidents have a comparatively small influence. That these values approximately might be considered invariable, does not hold good for 1I n2 as it does for 2)j 2, but we noticed already that: (1) it is impossible to find reliable results for one single branch of industry, and that: (2) the term in 1S 2 does not influence the result very much. As far as s3J) was concerned it has proved desirable to have at our disposal values that reflect the individual differences which, according to the statistical reports for different trades, appear to exist. It is true that the different values found for 3S)12 were only slightly different, but the introduction of this quantity rested upon a supposition which may only approximately be called correct, viz. the supposition that the average square of the importance of a case of permanent incapacity of the fraction t could be denoted by t23Ms)2. A proportion as assumed here is certainly incorrect for the expenses of the first years after accident; for neither the expenses for medical treatment nor the payments during the period of sickness and recovery are at all connected with the percentages of resulting permanent incapacity. Moreover it has been proved that the older labourers are, the higher is the degree of incapacity, so that the average-age of the permanent invalids with a high percentage of incapacity is considerably higher than that of invalids with but slightly reduced capacity. Moreover, these circumstances being of great importance as to the result, it is desirable that not the average distribution of wages over the whole industry be contained in them, but the distribution of wages only over the branch of industry concerned, which certainly shows a closer dispersion. Therefore the value of 3\)t2 is to be calculated directly from the material for every trade in particular. The values found for 12 in this way are given in the following table as far as they could be calculated from sufficiently extensive statistical material. Industries Ml Mechanical tile-works and brick-works.. 023 Mechanical printers..... 0'22 Painters....... 027 Whitewashers and plasterers... 025 Engineering works...... 0'27 Masons........ 025 Carpenters...... 0-23 Builders....... 024 Chemical industries..... 0'23 Saw-works...... 0-22 Forges........ 023 Engine-works....... 024 Non-mechanical ship-builders.... 0'26

Page  403 ON THE APPLICATION OF THE CALCULUS OF PROBABILITIES 403 Industries M Mechanical ship-builders.... 025 Paper-mills.... 0-27 Weaving-mills (cotton)... 0'25 Distillers...... 0'23 Dock-labourers.. 0*23 Railways.... 0-26 Carriers and draymen... 0'23 The extreme values are 0'22 and 0'27, which satisfactorily agrees with the supposition. The law regarding the distribution of deviations of the real charges from the most probable or average value is I+ 7r z 1 2 ) -62 64 -6 1 cos (0a) cos 0 oss 0.. e 4! d* where a denotes the value of the deviation and the letters y, z, u, v, etc. consecutively the quantities msns8; msns; m 2n ns;; SmnM4's, etc. In applying this general law we do not suppose, as we should do in applying a priori Gauss's law, that accidents of every importance are represented in large number. However, to apply it would give much trouble. It is evident that where very ample charges are concerned, combinatorial deviations will follow Gauss's law. Where only comparatively small charges are in question, we had at first to put up with what could be deduced about the probability of deviations from the Theorem of Tchebychef. However, from this theorem we obtain no information about the probability of deviations itself, but we obtain a deviation for which the probability of being surpassed remains below a given fraction. It soon appeared that this theorem was by much too rough for practical purposes and that indeed the dispersion of deviations was by so much more restricted that research for another expedient was required. Mr K. Lindner, officer in chief of the Mathematical and Statistical department of the State Insurance Office, who has applied my theory developed above to the statistical results and further to the insurance-practice of that institution, has tried therefore to find an empirical frequency-curve, from the experience of the middle-sized charges commonly appearing in practice. For this purpose he calculated the values of M briefly communicated above and applied them to the data contained in the statistical reports of the financial results of the insurance of the years 1903-1907. In those statistical reports the total insured wages I, the total net charge B, and the number of "years of labour" a, are collected for a great number of trades and for each of the years 1903-1907. By means of these data could be calculated the average net charge 100,B 1 per hundred n = sl-, the average annual wages: - and the mean deviation from I /Ta the average charge of each year:/ a nM2. * This formula is obtained according to the method explained in Traite du calcul des probabilites by H. Laurent, p. 247 sq., by considering accident-insurance as a succession of constant contracts, each of infinitesimal duration dt. 26-2

Page  404 404 J. H. PEEK The deviation in the annual charge for each year could be calculated to be B- Thnl and this deviation could be expressed as a multiple or a fraction of the corresponding mean deviation. Out of 405 deviations, 190 were positive, 215 negative. The distribution as to the value was as follows: Observed deviations Deviations per cent. of _____ the mean deviation positive negative 0- 20 31 30 21- 40 33 41 41- 60 23 33 61- 80 22 26 81-100 25 36 101-150 36 36 151-200 12 12 201-400 7 more than 400 1 1 Total amount 190 215 The asymmetry as to positive and negative deviations disappears for the greater part if we exclude observations dependent on smaller probable charges. The following table relates to the deviations from probable charges of 20,000 guilders at least. Observed deviations Deviations Distribution per cent. of the - -- - -- according to mean deviation Gauss's law positive negative 0 20 19 17 18 21- 40 20 26 18 41- 60 15 18 16 61- 80 17 15 14 81-100 13 19 13 101 —150 24 15 21 151 —200 5 6 10 201-400 3 5 more than 400 1 - Total amount 113 117 115 For the purpose of a comparison of the dispersion with that according to Gauss's law, the latter is put in the third column next to the observed distribution. There appears to be sufficient concordance so that it seems to be admissible to assume this law as available in the case of not too small charges. As rather satisfactory concordance with Gauss's law was found even for a charge of but 20,000 guilders, it was judged superfluous to draw up an empirical law. Moreover we may see in the obtained result another proof of the usefulness for practical

Page  405 ON THE AP'PLICATION OF THE CALCULUS OF PROBABILITIES 405 purposes of the chosen approximate calculations of mean deviations. This is important for another question where the mathematical department of the State Insurance Office originally met with no small difficulties and for which an apparently sufficient solution has since been found by Mr Lindner. The State Office has to charge premiums from the branches of industry fairly; that is, counterbalancing as nearly as possible the charges of the insurance. The following xules prevail here. By Royal Resolution the different branches of industry are arranged in a great number of classes. Every class contains usually five (only in the lower classes three) degrees of risk, which are respectively about 10 and 20 ~/ above or below the middle degree. The tariff contains for every degree of risk a proportional premium: With exception of a few special cases (to be mentioned hereafter) an establishment has to be arranged in the class in which the branch of industry practised is arranged. It depends on the individual character of the establishment as to which degree of the class premium shall be charged. Only when special reasons exist, an establishment can be arranged in another than the prescribed class as soon as special Royal Authorization has been obtained. At first it did not seem an improbable supposition that the risk in establishments belonging to the same branch of industry should vary between rather near limits, and from that idea the prescription for classification arose which was in a large measure borrowed from the Austrian Insurance Act. It soon appeared in giving effect to the Dutch Insurance Act that this plausible supposition was by no means supported by experience and that in the same trade the risks were often very much different. It is not exclusively the State Office that is an authorised insurer, as is the case with the Austrian official Accidents insurance institutions, and the existence of private insurance institutions prevents the State Office from expecting that evil risks will be compensated by good. Therefore it was felt necessary to profit as soon as possible by the statistical experience concerning the insurance of the different establishments individually. Consequently we were soon seriously in want of a conclusive answer to the question when and in what circumstances the premium to be charged from an establishment may be deduced from statistical experience. In Austria the officials have been occupied in answering this question while attempting the general classification of industries. According to the official statistical report of the empire, entitled "Ergebnisse der zum Zwecke der Gefahrenclasseneinteilung iiberprtiften Unfallstatistik der Jahre 1890-1896" (Seite Vl), they go by the following criterion: "die aus den Unfalls- oder Belastungsziffern abgeleiteten Verhaltniszahlen fur die Unfallsgefahrlichkeit der einzelnen Betriebsgattungen konnen nur dann als sichere, von Zufalligkeiten nicht mehr beeinflusste Ergebnisse angesehen werden, wenn die Beobachtung sich auf mehr als 10,000 Vollarbeiter erstreckt. Auch die aus Beobachtungen von mehr als 5000 Arbeitern abgeleiteten Verhaltniszahlen konnen immerhin noch als massgebende Ergebnisse der Erfuhrung angesehen werden." Though we do not know on what considerations this criterion depends, it appears that for non-dangerous industries it demands insufficient security; for very dangerous industries an exorbitant one. If the insurance of an average establishment causes great loss one year after the other, it is not right to go on charging the same premium from it until statistical experience finally runs over 5000 Vollarbeiter (years of labour), nor is it fair to

Page  406 406 J. H. PEEK classify an establishment belonging to a non-dangerous kind of industry satisfying the Austrian criterion, lower than normally, because of statistical reports though running over 2-5 millions of wages over perhaps no more than 5000 guilders of charge; in such a case of small charge the casual favourable issue of one accident may already have caused the deviation from the average value. Therefore we have tried to meet the difficulty in a different way. Let b denote the statistical charge and I the corresponding total sum of wages, 100C then the percentage of the premium: l-, may be derived from these data with an accuracy determined by the proportion of the mean deviation of the probable charge from the charge C. The first thing that is to be done now is to make a reasonable choice of the degree of accuracy required here. In our case the choice was determined by certain legal prescriptions. Our purpose is to find the class in which the risk has to be arranged and, with reference to what has been communicated above, the question has to be answered, under what circumstances it is to be expected with sufficient probability, that the net premium necessary to meet the probable charge will be less than 20~/o higher or lower than the percentage of charges 10, deduced from statistical data. We assume a probability greater than I to be sufficient, because then already there will be more chance for the establishment to be arranged in the right class than in a wrong. The mean deviation in the probable charge C may be approximated by the term /aL2nM2 if a denotes the number of labourers, L the average wages and n the charged net-premium aL. If aL = I be the amount of wages to which the charge C relates, then aLn = In is the hundred-fold of the probable charge C; so that we may write VL.M /100 C for VaL2niM2. Then the probable deviation may be expressed by 0-675 VL. M. V/00 C, consequently greater deviations than these have a probability less than 1, and such a deviation will coincide with the limits of the normal class, if it be equal to 20~/o of C. From this follows the condition that a statistical experience will begin to furnish sufficient indication of the class, if the expected charge satisfies the equation 0-675 /L.M100 C = 0, from which C= 1139 LM2*. If we know the average annual wages L, and the value of M, classification on exclusively statistical grounds will be allowed as soon as the expected net charge amounts to 1139 LM2, in which condition, instead of the expected charge, the most probable value of it, the real net charge, must be substituted in case of application to practice. Now the question still remains in what way allowances have to be made for statistical experience of smaller extent. If no other data were at hand the premium * It must be observed that 1139LM2 is always large enough for an approximate application of Gauss's law, as follows from the preceding investigation, so that the factor 0-675, inferable from that law, was justly applied.

Page  407 ON THE APPLICATION OF THE CALCULUS OF PROBABILITIES 407 100 0 would have to be fixed at 1. In most other cases however we have other data at our disposal. If the classification of a separate establishment is concerned, an accessory datum may be found, for instance in the classification of the whole branch of industry that is in the average experience of all similar establishments together; if a branch of industry is concerned in most cases another branch may be found to which it may be compared and which may be classified on ampler statistical grounds. The experience of the State Insurance Office shows that this proceeding leads to results of less probability; as it has been proved that establishments appertaining to the same branch of industry may show the most different risks. Nevertheless, in case of scanty statistical experience we also have to operate after this method. Then 100 C a premium must be chosen between l and for instance the normal tariff-premium. It follows from the nature of this problem that whichever solution may be chosen here, it will remain comparatively arbitrary, as the probability of the applicability of normal classification in the case of a separate establishment is unknown; and even cannot be expressed numerically. For a practically serviceable solution the following consideration may be available. Though d priori the statistical charge is the most probable value of the risk 100 C, we are safe in assuming that this value shows a deviation from the real risk of sign contrary to that of the difference between the real risk and the net tariff premium and of such an extent that greater deviations will have only a probability 2, if by this supposition the risk should indeed be estimated right. This supposition takes for granted an influence of statistical data on classification increasing with experience; and only a few obvious restrictions are necessary always to obtain acceptable results. It is evident, however, that any method will show equally serviceable results, if only this method admits of the charges C< 1139 LM2 having an influence on the net premium, dependent on the value of C. Our object was only to show that the theory developed in the preceding pages has in practice proved to be tenable and serviceable.

Page  408 NOTES UPON THE CURVES OF CERTAIN FUNCTIONS INVOLVING COMPOUND INTEREST AND MORTALITY BY ROBERT RAYNAL BRODIE. It is well known that the rate of mortality is very high during the first few years of life and that it diminishes continuously till about age 10 or 12, thereafter increasing steadily right up to the end of life. This characteristic is common to all tables of mortality though the actual rates of mortality at different ages vary slightly according to different tables. If the rate of mortality were the same at all ages there would be no limit to the duration of life; and it would not be necessary for Life Assurance Offices to accumulate large reserves-the contributions of any year being simply applied to meet the claims of that year. Since, however, the rate of mortality does increase from year to year (omitting the first few years after birth) it is evident that a level periodic contribution for a sum payable at death must in the early years be greater and in the later years less than the amount required for the actual risk from year to year; in this way the effect of an increasing rate of mortality is to create a Fund out of the excess contributions of the early years so as to help to pay the excess claims due to high mortality in the later years of life. The amount of the Fund at any moment therefore must represent the difference between the contributions (with interest) received and the benefits paid and must be accumulated so as to help future contributions to pay future claims. Just as past contributions exceed past claims so future claims will sooner or later exceed fiture contributions. It is this feature of mortality that necessitates the accumulation of large Reserves by Life Assurance Societies; these Funds do not represent surplus; they must be on hand to meet the liabilities of the future. In the case of benefits depending on mortality there are two forces operating on the reserve value that accrues from year to year, viz. Compound Interest tending to increase the Fund, and Mortality tending to reduce the Fund, and the progress of the Fund will depend upon the relative strength of these two forces. The object of the present note is to shew the resultant effect of these two forces upon the curve traced by such a Fund. The following is the notation used in this paper: Vt the reserve value at duration t arising out of past contributions in respect of each unit of benefit. 7r the amount of yearly contribution payable by continuous instalments for each unit of benefit.

Page  409 CURVES OF CERTAIN FUNCTIONS INVOLVING COMPOUND INTEREST 409 aut the intensity of mortality (per year) at moment t. 8 a year's interest (payable momently) per unit. When a benefit becomes payable at duration t there will be Vt on hand to meet the claim in that particular case so that the general strain on the Funds of the Society will only be 1 - Vt. The probability of death occurring within a year at moment t is /t. The expression 1/t (1 - Vt) is termed the "death strain" and represents the "loss" due to mortality. On the other hand the "gain" for the year due to premium and interest on reserve is 7r + Vt. We therefore have the relation vt t = 7 + 8 Vt - Lt (1 - Vt)..................... (1). This relation is quite general and applies to all classes of benefit. It indicates separately the operation of the two forces of interest and mortality. Under any contract payable on death interest and mortality act in opposite directions. If, however, the benefit were payable only on survivance (such a benefit being known as a pure endowment) formula (1) would become dt Vt = 7r + 8 Vt - t (0 - V), that is to say, the death strain in such a case becomes negative, and mortality instead of opposing the operation of interest acts in harmony with it. If under a pure endowment the accumulated contributions were repayable on death the element of mortality would be entirely eliminated and the death strain would be zero throughout. Suppose that somewhere on the curve of V d Vt passes through the value zero then at that point there is either a maximum or a minimum value, and there we have the relation + Vt = t ( - Vt).............................(2). At that point the forces of increment and decrement exactly counterbalance each d other, and there the reserve curve has a turning value. If d Vt pass from negative to positive the point will of course be a minimum, or if from positive to negative then it will be a maximum. If a benefit be payable in event of death only during a certain time the reserve value is zero at the expiration of the term, and since it must commence at zero, assuming a periodic contribution to be payable, it is evident that in such a case a maximum value will arise somewhere on the curve. If contributions are paid in one sum we shall have in place of formula (1),writing At instead of V, d J At = A t - /t ((1 - At)........................... (3) and the condition for a maximum or minimum value will be SAt = ^t (1 - At). In the case of an assurance payable at death effected at a very early age the curve of the reserve value will pass through a minimum point, due to the rapid

Page  410 410 ROBERT RAYNAL BRODIE decrease in mortality. It is possible, however, by giving sufficient weight to the element of interest to eliminate this minimum value. Consider, for example, two single-payment assurances for ~100 each, opened at age 3, one payable at death and the other payable at death or on attainment of age 21. The minimum value occurring about age 5 in the former case disappears under the endowment assurance, -the disappearance being caused in the latter case by the larger accumulative element. The following table shews the operation of the two forces of mortality and interest round about age 5, d A in the former case passing from negative to dx positive and in the latter case being positive throughout. The figures are based on the Institute of Actuaries' Text Book 3 ~/ Table. TABLE I. Assurance payable at death Assurance payable at death or age 21 Age ___ _____ __ _ Age attained attained ( A) — x x ((1 -A — x 8 4 (1-Ax) - Ad | A -X 1 Al-A:2_-) dAx:21-l 3 -782 1-314 -'532 1-808 -694 +1-114 3 4 '771 1-019 - -248 1-846 -518 +1-328 4 5 '766 -846 - *080 1-888 413 + 1475 5 6 -'767 -685 + 082 1 934 -320 +1-614 6 7 -771 -553 + -218 1-984 -246 +1-738 7 8 -779 I 447 + 332 2-036 -189 +1 847 8! _______ ___ Differentiating both sides of the expression marked (1) we get dt2 - dt - dt Lot Vt) The condition for a point of inflexion on the curve of V is d Vt dF -t.1. Rdt dt -.....................(4 ). If at that point t2 pass from positive to negative the curve thereabouts will be of the form form or if it pass from negative to positive the curve will be of the If the assurance is by single payment At will take the place of V, in formula (4). Most of the calculations made by Life Assurance Offices are based upon what are known as " Select Tables." These tables trace the mortality among a body

Page  411 CURVES OF CERTAIN FUNCTIONS INVOLVING COMPOUND INTEREST 411 of lives " selected" by medical examination. During the first few years succeeding the date of medical examination the mortality among such lives is much less than among lives of corresponding age in the general population, and the proportional increase in mortality due to the wearing-off of selection is correspondingly large. This rapid disappearance of the traces of selection tends to introduce a point of inflexion on the curve of V. As an illustration shewing the difference between the curve of V according to a table where selection is present and a table where selection has worn off we may trace from ages 30 to 35 two single payment assurances for ~1000 each payable at death on lives which entered at 20 and at 30. In the former case the effect of selection has worn off during the 10 years since date of entry. On the O[M] 3 ~/ table the curve of the former will be of the form and of the latter as will be seen from the following figures:TABLE II. Age 20 at entry Age 30 at entry Age ____ _______ _ Age attained attained (x) d- d - d2A A d - d2Ax (x) -Ax [x (1 - Ax)] dx x )] dx2 dx I 30 -210 032 + 178 '315 2-604 -2-289 30 31 '215 i 033 + '182 -267 -764 - '497 31 32 ' 220 I 037 + 183 -256 *345 - ~089 32 33 -226 -044 + 182 '255 -235 + -020 33 34 '231 i 051 +-180 *256 -221 + -035 34 35 -236 -058 +'178 -257 '212 + -045 35 An increase in the rate of interest has always an accelerative effect on the ascent of the curve of V, and it is evident therefore that a point of inflexion will be hastened or deferred by an increase in the rate of interest according as the curve thereabouts is of the form or We have seen that under a select table there are two opposing forces operating in the early years, —selection tending to give the curve the form / and interest giving the curve the form / and the resultant shape of the curve will depend upon the relative strength of these two forces. If the assurance is payable only at death it may not be possible to counteract the convex-upward shape of the curve in

Page  412 412 ROBERT RAYNAL BRODIE the early years resulting from selection, but under an endowment assurance payable at death or a certain age we can by making the term of the assurance sufficiently short give the accumulative element sufficient weight to smooth away the influence of selection upon the shape of the curve. This point may be made clearer by tracing for the first few years the reserves under three endowment assurances by single payment for ~1000 each according to the O[M] 3 /o table, opened at age 30 and maturing at ages 60, 45 and 40 respectively. TABLE III. d --- -- d2A Duration - d[ (1 - )] A dt A dtdt2 maturing at 60 0 - 365 1-176 — 811 1 i -346 -872 - -526 2 -338 -469 - -131 3 -337 -243 + 094,maturing at 45 o0 540 -732 - 192 ]1 -539 -427 1 +'112 2 -~546 204 + '342 3 i -558 '043 + 51]5 maturing at 40 0 ~ -630 -504 + 126 1 *638 '250 +-388 2 -652 -067 + 585 3 -669 - 058 +-727 Here it will be seen that the effect of changing the maturity age from 60 to 45 is to hasten by about two years the point of inflexion resulting from selection; and by reducing the maturity age to 40 the point of inflexion is made to disappear altogether and all trace of selection upon the curve is eliminated. It was already mentioned that under a temporary assurance with premiums payable throughout the term a maximum value arises. If such an assurance is effected by single payment it can easily be shewn that the reserve curves may assume any one of the following general shapes depending on the length of the term of the assurance and the nature of the mortality: (1) no point of inflexion, no maximum value, (2) a maximum value not preceded by any point of inflexion, (3) a point of inflexion followed by a maximum value, (4) two points of inflexion followed by a maximum value. As an example of another type of curve we may take the case of an assurance for ~1000 effected by single payment on a life aged 1 and expiring at age 25. No minimum value will arise under such a policy,-the expiry age is too low to admit of that,-and the reserve curve will descend throughout. In the early years dtA dt

Page  413 CURVES OF CERTAIN FUNCTIONS INVOLVING COMPOUND INTEREST 413 is positive; the curve then is at first convex-downwards and as the curves under such temporary assurances are invariably concave-downwards eventually a point of inflexion must arise. The position of the point of inflexion will be seen from the following figures and the curve will be of the type. The figures are based upon the Institute of Actuaries' Text Book 3 ~/o table. TABLE IV. Age I -, attained A [ (-A1 )] /^ ' d) (l 2o-x: d: x! (i x: 525- xl 12 - 0568 - 2115 + 1547 13 - 0539 - 0794 + 0255 14 -*0541 + 0538 - 1079 15 - 0060 + '1744 - 2350 In the case of assurances payable certain sooner or later (such as whole life or endowment assurances, as distinct from temporary assurances) it will be seen from the relation Vt= At- A that any peculiarities that may occur on the curve of the reserve-value where a single payment has been made will be reproduced at the reserve-value where a single payment has been made will be reproduced at the identical points when a continuous premium is payable; that is to say, if a point of inflexion or a turning value has arisen in the former case it will occur also in the latter case at the same spot although the curves in the two cases will naturally ascend or descend at quite different rates. As regards temporary assurances it is only when the period of the assurance approaches the complement of life that the curves of benefits by single and continuous premium tend to have the same peculiarities. On comparing the initial curve of the reserve arising under a pure endowment by single payment effected at an early age with the curve under a single payment assurance by a select table effected later in life it will be found that they are more or less of the same form, viz. convex-upwards. In the former case the shape of the curve is the result of rapidly decreasing mortality and a negative death strain; in the latter case it is the result of rapidly increasing mortality and a positive death strain.

Page  414 CALCULATION OF MOMENTS OF AN ABRUPT FREQUENCY-DISTRIBUTION BY W. F. SHEPPARD. 1. The formulae adopted for calculation of the moments of a frequencydistribution-or, more generally, of a trapezette in respect of which our data are not ordinates but areas of strips-are not always correct. The formula for the case in which the figure is "double-tailed," i.e. has close contact at both extremities with the base, is fairly well known; the cases to be considered are those in which the figure, at one or both ends, is "abrupt," i.e. either ends with an ordinate which is not zero, or forms an angle between its upper boundary and the base. 2. The following notation will be used. (i) y is the ordinate, corresponding to abscissa x, of the figure of frequency. (ii) Observations are taken at intervals h in x. (iii) A is the area of the strip of the figure which is bounded by ordinates corresponding to abscissae x- Ih and x + -h, and u is the average ordinate of this strip; i.e. Ax+h yd A -hu =f ydx. J x-.L (iv) S is the total area from the ordinate y up to some definite ordinate; i.e. S = ydx. (v) xo denotes x0 + Oh; and ye, Ao, lu, and So are the corresponding values of y, A, u, and S. (vi) The range of values of x is from x_-, o- nh to = x, + mnh, where im +n (but not necessarily m) is an integer; so that there are m + n strips whose mid-ordinates correspond to the values of Xr for r = - n + -, - n +-,... -.- The data are the areas of these strips, i.e. are the values of Ar for these values of r. (vii) Roman superscripts and D denote differentiation with regard to x, and 8 and aj denote the central difference and the central mean for increment h in x; i.e. yI = Dy =_ dy/dx,;f (x) =f h - + (x) + ) -f( - h f) f( + h) +f ( - h)}, so that, at any rate if we are dealing with polynomials in x, 8= 2 sinh ~hD, = cosh ~hD. 2'""> r" 2~~~

Page  415 CALCULATION OF MOMENTS OF AN ABRUPT FREQUENCY-DISTRIBUTION 415 (viii) E denotes summation for the values of r mentioned in (vi) above. (ix) - denotes approximate equality. 3. The area and the successive moments of the figure are fydx (= EAr,), fyxdx, fyx2dx,..., the limits of integration being x = x_, and x = xn. But in actuarial work, and in statistical work generally, we have to deal with more general expressions of the form JyP (x)dx, where + (x) is a function of x which possibly is not given explicitly but is tabulated for a series of values of x. If, for instance, we are valuing an annuity fund, b (x) may be such an expression as (1 + i)x-t lt/lx, ydx being the total value at age t of the annuities conditionally payable to persons who are now between ages x - ~dx and x + ~dx. The expression whose value we wish to find will therefore be denoted by f y (x) dx. 4. If the figure is double-tailed, the proper formula is fy (x) dx I A A, 0 (xr), where 1 h7 31, (x) < ()-2 h2(TT () + 5760 hpV ()- 967680 (). = iu () - /tI (x) - () + 3- h4LV (x- ) -+..(. =1- ~+6 - 360 15120} x 1 2 a 3_ 5 86 24 + 640 6 -7168 '" 1 = 1 4 6 ~= i,1-16S6 + 30 140 -} the third form of 4 (x) being used if (x) is tabulated for the mid-values x-_+t, x_,,1+,... m_, and the fourth form if it is tabulated for the bounding values x_n, X-n+, *... X.* * For moments, the first or the second form of 1 (x) is the more convenient according as we are taking moments with regard to a mid-ordinate or a bounding ordinate. Let Vp be the true pth moment f yxpdx; and let pp be the ordinary "raw" pth moment A,.xp, obtained by massing the area of each strip along its mid-ordinate, and pp' the alternative raw moment YE, Ar (xP_ + xPr+ ), obtained by placing half the area of each strip along each of its bounding ordinates. Then the formulae are 1 7 31 Vp " pp -1 () h ph2 + 240 () h pp4 - 1344p 6 h6p-6 +..., l 7 31 Vp p - p) h2 pp-2 + 15 p(4) h4p p-4 - 21 p (6) h p p_ +..., where p (tdenotes p!/{t!(p - t)!}. The terms after the first, in either expression, may be called the "corrections for massing." The two formulae give identical results.

Page  416 416 W. F. SHEPPARD 5. The formula in ~ 4 is obtained by the following steps. (A) If A - F(x), then y = F (x), and therefore 1 1 1 6yVi m = Alh = 8/(hD). y = y + -- hyJIJ + h4'#' + h2y"'6 +... 24 1920 322560 1 7 31 6V+ hD8. = u- h2u + 5 4uIV h + 24 5760 967680 (B) Sfy (z) dx =f(hD8S. iu) (x) dx. (C) f (hJD/. u) 0 (x) dx | tft hD/8. (x)} d;. (D) fu hD/I. 4 (x)} dx =fuf,(x) dx, where q0 (x) is defined as in ~ 4. (E) _6,1 (x) dx - hEaZrt, (Xr). Of these steps, (A) and (B) are true absolutely, at any rate in the cases with which we are concerned; certain conditions as to continuity etc. being assumed with regard to y as a function of x. And (D) is true under similar conditions with regard to + (x), which are of course satisfied if + (x) is a power of x or a polynomial in x. But (C) and (E) only hold if the figure is double-tailed; (E) being the ordinary formula for quadrature by means of the "tangential area," and (C) being based on the fact that {hD/8. 1t} (x) - t {thD/8. 0 (x)} consists of terms of the form (omitting powers of h and numerical coefficients) tit ^> (x) - t" 1( (x), tIV (x) - u0v (x'), which are the derivatives of U' (X) - U (X), ut'LT (x) -_ "II (l ) + LIII ( ) - U(JII (),.... The formula cannot therefore be employed unless the figure is double-tailed. 6. But it does not follow that, in the cases in which the figure is not doubletailed-which are the cases to be considered in the remainder of this paper-, we can, as is sometimes suggested, content ourselves with treating fyo (x) dx as being equal to fut (x) dx, and then using the tangential formula for this latter; for this involves two separate mistakes, in that we first omit f(Y - u), (x) dx = f {hD/8 - 1} u. (x) dx = (- h24 + 5-760 h4 _...)(x dx, and then use an incorrect formula for calculating fuit (x) dx. These two mistakes are of different kinds, and do not balance one another, except in some very special cases. 7. To express the complete formula, we use D1 and D, to denote differentiation of u and of ( (x) respectively, so that, for differentiation of a product of u or any of its derivatives by q (x) or any of its derivatives, D=D, +D,.

Page  417 CALCULATION OF MOMENTS OF AN ABRUPT FREQUENCY-DISTRIBUTION 417 The formula is then* f f (x) dx ilAriA {hD/8 (x,' )}+ (P (xr) - C)D (X,) 1 7 31 15~~ A r 0 ( Xr) h ) h Mov' (Xr) +68 'F~r Xr) -24 h (X)+5760h4i 9x)-67 6h80 ( )+ + (P (Xmn) - ( (XA), wheret (1 lhD1 I __'D2 jhD sinh2-hD1 1 sinhA2 A4 (x) (x) 2sinh~~ ~~~ 2 D2hD 1hD2 sinh h~D1D2 coth IhD - pD)-(coth 1hD2 -hD)} AO (x). Performing the differentiations, we find that I -011 + h1 ( -. (P (X) = h4' (x) - M ~ k h5ov (x) +-1A J12 240 6048 240h (w)~ 3024 hc/ (x) + ~ 61 13- -JO4V } h2All 1+440 120960 362880 41 1 (X) - 5 h401v (X) +...} h3A"' + h1 b-96-0 '-2576-0 31 1051 1547 + 8 -h~'(x) - -IX~ -38218 1~20 2903040038210 103 201 61 4 + 5806080 h'q" (x)+ 47900160 + etc. With regard to this formula, the following points should be noted. (i) If p (x) is tabulated at intervals h, and its derivatives would have to be calculated from the differences of the tabulated values, it is best to express the coefficients of A, hA',... in terms of these differences. The coefficients of the (central) differences in the successive portions of (P (x) are given in Table I. (ii) The values of A, hA', h2Aii,... for x-,,, and for x,, have to be calculated from the given values of A for x-,_t,, x-~n+,,... and for rn-X, W M 3. Usually we can get fairly good values by using an auxiliary function, as in the example given in E 9 below; when this is not possible, we have to use the actual differences of the A's, and these will not generally give such good results. In these latter cases, the alternative method considered in ~ 11 may be regarded as simpler. 8. For moments, the formulae of ~ 7 give V1 Pi + (DJ (Xm) - (DJ (X-,) pi, + (D, (xm~) - qPi (x_,,), Where the symbol - is used, in the remainder of this paper, the expressions which it connects are absolutely equal if y and q5 (x) are polynomials in x. t This differs from the expression originally given by me in Proc. Lond. Math. Soc. xxix. 358-9, in being expressed in terms of the A's instead of in terms of the y's. M. c. II. 27

Page  418 418 W. F. SHEPPARD TABLE I. Co(ffici or I 4 Apaq) (.r) 0 1 2 ~24 (X) o I 0 " q0 (-) +_ () ents of products of A, hAt, h A1l,... by Ih (x), /~ (x), 83a (x),..., * by 3 8(xi), I2 4 (x), /3 (x),..., in ) (x) (~ 7). tA-l1 11 2A'l" /1,a1l' h4 4AV 0 1 240 I 29 24192 0 I 0 41 +60480 0 7 1.440 19 26880 -- - 1009 8294400 1440 0 5,: +40:320 I I 0 2351 - 7257600 0 41 +120960 0 31 161280 0 2567 58060800 i 0 103 5806080 0 47 10817 414720 +1532805120 2901.13 0) +' - i I:30656102400~ 31 0 ~ + 0;~ 161280 41 103 i+ 0 120960 5806080 01981 0 0 290(:30400 1.03 1 3709) 1451520 0 7066402560 1-4941 0 + 70 766402560 _~.___~~. __ _. __ _ __ __ __ - _- - -: I I i3+ (.;,) 1 I _, _^:/ a S ( ) -!!!84 0(.XV) I, 8'I F,(h(X) i1 13 720 - I I 0 241! 60480 1 a P2 - 2 h p' + 4' (Xn) - (2 (X-n) 1 7 ^ p. -; ^h2p + (Po (Xi )-l,2 ($-n)n Y; P3 - /t f* + (3 (43) - ()P (3X-n) 2 p, h2p' + (P (X.,) - (3 (-X-), r4 P4 - h2 p + 240 h4po + (4 ("n) - ) (-4 () P' - //2p2' + P ' + 4 ' + (xI,) -(- (P)4 (X-.),

Page  419 CALCULATION OF MOMENTS OF AN ABRUPT FREQUENCY-DISTRIBUTION 419 5 7 *' P5 6 h - + 4 hpl + D5( -) - ) ( -,) P p hp2p - / + 7 h4p1' + q) (xin) - q5 (x-n), etc., where ( (x), D, (e),... are functions whose values are given by Table II. The terms due to the )'s may be called the "corrections for abruptness"; the terms which precede them (after the first) being still called the "corrections for massing," though really the two sets of corrections together form the complete correction for massing. If the figure is single-tailed, it will usually be simplest to take the abrupt end as the zero for x. The given A's will then be A1, Al,...; and from these we determine A,, hAoI, hAo,....The coefficients of A0, hAoI, h2Aj,... in the quantities to be added to p,/h, p/h - 1/12.p,,.. in order to obtain v,/h, v2/j2,... are given in Table III. TABLE II. Coefficients of A, hA', 2A ll,... in, (x)/h, p, (x)/h,.,. (~ 8). x/h.) CO. 1 co. A - Co. h'1A c. h4 111 c o. hA" co.;1v c 1 >(')// + _ 0 0 |i 12 1440 161280 <F i'L +;( ]2( - 0(, -+ 60+8 --- - __ __ +1 14 I-7 41 31 103 +' 20 1008 40320 483840 4 i480 41 53760 103 (x)/61+0 20160 967680 '12 + 41051 40 20160 - 4838400 1 1 41 31 103 I o 20 ~ 360 ~100804 +40320 483840 1 61 31 1051 161 10o 126 3040 30240 1209600 1995840 0 __64 + 2256 4 +- 4 8 4 0 - -- 1^I1 1 61 1051 6048 1 290304! iJ 4 4 i 5 I+ 2016. 31 483840.1 -t- ~ ~ ~ ~ ~7+ + t+ |;., ^^+ 6048^ +3991684 252 | 3024 3193344 27-2

Page  420 420 w. F. SHEPPARD TABLE III. Coefficients of A,, hA,', h A,11,... in - CP, (x,)/h, - I,(x,)lh,... for xo =0 (~ 8). co., Aco. hA,' co. h2A II -41(O)/k -08333 3333 0 + 00486 1111 - )2 (0)/h2 0 + 00833 3333 0 - cR(0)/h' + 02500 0000 0 - 00302 5794 4)4 (0)/ 4 0 -00793 6508 0 <)5(0)/h' -.01984 1270 0 +-00429 8942 co. h3A,"' Co. h4A,1 IV --- ----- --- I 0 — 00019 2212 - 00067 7910 i 0 0 i +-00021 7221 co. hl5A, v 0 +'00003 5480 0 + '00102 5132 0 0 — 00048 4445 - 00008 0668 0 9. As a simple example, take the half of the ordinary figure of error; h being half the standard deviation. Taking h as the unit, the equation to the curve, if the total area is 10,000, is y = 10' V{1/(22w)1. exp. (- x'/8). This gives the values of A shown below on the left, and the values of 2FtA that may be used for calculating the raw moments (pl', p,',...) with regard to x= 0. x A X 2/LA 2 3829 0 3829 11 2998 1 6827 2' 1837 2 4835 3T 881 3 2718 4 -L '331 4 1212 5' 97 5 428 6f 22 6 119 7 f 4 7 26 81 1 8 5 9 1 10000 20000 To find the values of A, hA', h2A",... for x = 0, we take logarithms: loglo A Ist diff. 2nd diff. 1 2 1~ -f 305831 3-4768 3-2641 2-9450 -1063 -2127 -3191 -1064 -1064 -1061 We have therefore, approximately, log,,A = 3-5964 - *0532x',

Page  421 CALCULATION OF' MOMENTS OF AN ABRUPT FREQUENCY-DISTRIBUTION 41 421 whence, by differentiation, A' = - 245OAx, A"' - -2450A'x - -2450A, A"l' = - -2450A'Tx - -4900AI, AJv = 2450AhuIx - 735OAuJ, v=- 2450AJ~x - -9800A"I'. Hence, for x = 0, we have A = 3948, Al'=0, AI'-967, All'-=0, Alv = 711, Av= 0. The following is a comparison of the raw moments as ordinarily used (pi, p2,,*. with the corrected values after allowing for massing and for abruptness, and with the true values; these latter being found by calculating the true moment of each strip of the original figure, and then altering this moment in the ratio of the integral number taken as the area of the strip to the true area of the strip. 1st 2nd Raw (p)........... 162.92'00 40840-00 Corrected for massing.... 16292-00 40006-67 Corrected for massing and for 1586 4067 abruptness..... True............... 15958-13 40006-43 Moments.3rd;131717 00!127644,00:127745'78 127741-31 4th 5thl 501019-00 2158528'25 480890-67 2051140-00 480890-67 2051057-17 480830-04 2050388-68 Hence we deduce the following values regard to the mean. of the mean and of the moments with Moments with regard to mean Me an 2nd 3rd From raw moments (p's) 1-629200 14297 18595 4th I 81695 74839 5th 266434 260938 For raw moments corrected for massing..... 1P629200 13464 18595 m-assing, and for abruptness... From true moments...I 1,595816 14540 17495 82189 255326 1155813 14540 P59 ~~~~17492 82154 255038

Page  422 422 W. F. SHEPPARD 10. With regard to the above example, the following points should be noted: (i) The " corrections for massing," taken alone, do not affect the third moment with regard to the mean. This is always the case; for this moment as deduced from the raw values (p's) is 3- 3 (pl/pO) p2 + 2 (pl/po)3 po, while, as deduced from the raw values corrected for massing, it is (P-3 -,hp,)-3 p)( P2- 12h,,)) +2 (p, and these are equal. (ii) The corrections for abruptness are specially important for the purpose of determining the mean; and, for the moments generally, the most important part of these corrections is that which is due to A0. We can, in fact, usually get fairly good results by using A0 alone, which can be determined with tolerable accuracy, and ignoring hA 01, "2AoI.... (iii) For the second and fourth moments with regard to the mean, we certainly get better results by omitting both sets of corrections than by the "corrections for massing" alone. This may seem to justify the ordinary practice; but it is due to the nature of the example. The peculiarity would be still more marked if the curve descended steadily from the initial ordinate. The more common case is that in which the bounding ordinate of the figure is appreciably smaller than the maximum ordinate; and then the correction for massing becomes important. Taking only the leading corrections, the fully corrected second moment with regard to the mean is P2 - 2 'Po - PI - j /hA Po, so that the correction is approximately 12 h2po + 6 hA PI The two portions of this are of opposite signs, so that they tend to counteract one another; but, if A0 is so small that p1Ao is less than hpo2, it would even be better to make the " correction for massing" alone than to make no correction at all. The corrections for massing are, of course, more important if the raw moments are pi, P2,... than if they are pi, p2,.... 11. If the values of A, hAI, h2AIl,... would have to be determined directly from the differences of the given A's, no suitable auxiliary function being available, it may be simpler, for calculation of a few moments, to use an alternative method. Let S be the area defined in ~ 2 (iv), so that S' =- y. Then we know the values of S for x =x_,, x-_,+1,... x,,. The general formula is

Page  423 CALCULATION OF MOMENTS OF AN ABRUPT FREQUENCY-DISTRIBUTION 423 mm X-n '= I Sb' (x) dx- {Sin (xn) - Sb (x-n)} h I SA ~S h + rI ' (X. 1 - o (X,) + H) (X), I 1+ v j1-r-(X) where ~ (x) S- (x) + 2 hD - 0 + 340h5D...} S (x 12 720 30240 For moments, let Ck be the "chordal" (trapezoidal) area of the figure whose ordinate is Sxk, i.e. C(- - h 2I{S,_. _S. + S.+1 x.+2} Then, for k > 0 the kth moment with regard to the ordinate at the lower extremity is VkkCk,- C.- )k (x,,) + H)k (-n), I l 1 =-'X,+$ k + 60 - 7 -...} Shxk-. For a single-tailed figure, we should usually take the abrupt end as the zero for x, and the other end as the zero for S; and vk would then be formed by adding to kCk_G the value of 7 l 71 1 + 191 F I. h k': 'i-2 ~ - 720S G0'+6(480o a-.. Si at the abrupt end. In the example considered in ~ 9, for instance, we have the for the calculation of the second moment:x S Sx 1st diff. 2nd diff. 3rd diff. 0 10000 0 + 171 1 6171 6171 -5996 + 175 +3483 2 3173 6346 -2513 -2338 +2663 3 1336 4008 + 150 -2188 + 838 4 455 1820 + 988 -1.200 - 246 5 124 620 + 742 - 458 - 411 6 27 162 + 331 - 127 ' - 231 7 5 35 8 1 8 9 0 0 following data 4th diff. - 820 - 1825 -1084 - 165 + 180

Page  424 424 W. F. SHEPPARD Taking - 820 to be the constant 4th difference at the commencement of the table, we obtain x Sx 1st diff. 2nd diff. 3rd diff. 4th diff. +16470 +5123 0 0 -10299 -820 + 6171 +4303 1 6171 - 5996 -820 + 175 +3483 2 6346 - 2513 -820 and thence r2 = 2. 19170 + 1/12.22641 - 11/720. 9426 = 40083. This is not so good as the value obtained in ~ 9, but it is better than the ordinary raw moment. The objection to the method, when several moments have to be calculated, is that the values of Shxa- have to be differenced separately for each value of k. 12. By performing the differentiations in 0 (x) we obtain a formula analogous to the formula of ~ 7, but involving the derivatives of S instead of those of A. It will be found that, in the general formula of ~ 11, { 201 1 (X) _ 1 h4 IV (X) + O + h (x ) - h / - () ) + } 7 -S + h() - ( 7) 0 I ( + 38)0 3v h- } h(S)+ -72 h24- ( x) + 3024 (X) -.. h hs ~+ ~ 164 2h2- ~2 (S tu) + ~x2 ()-~_n), +1'-V3 3(72-p 3 (x r) r 1 h"()+ 1 ) hsx-n h) w1 he () + h4. ( vn t30240 57600 380160 + 17280 h20" (x) + IV (X) -... hS 1 i -728-9 0-0- 570240 + etc. For the moments, this gives PI:_ Co - OI (X.) + o01 (.-0), V 20, - 0, (Xi) + 0, (x_-), 3__ 3C2 03 (X1.) + 0, (x-n), P4.r 4C3 4 04 (XX.) + 04 (X-.), 5.r 5C0 - 0s (Xm) + 05 (X-_L), ete, where 0, (x), (), (x),... are functions whose values are given by Table IV.

Page  425 CALCULATION OF MOMENTS OF AN ABRUPT FREQUENCY-DISTRIBUTION 425 TABLE IV. Coefficients of S, hSI, h2SI,... in ~1 (x)/h, 2 (x)/h2,... (~ 12). ( x/h.) CO. h2 1 CO. l1S IlI e (x)/L co. S + + V + I 1 +::<+4 co. 1hS co. h4SIV 1 12 0 1 720 0 1 30240 02 (,.*)//(2 e (.r)//j2 e:, (' I/e co. h5Sv co. O6SVI 4 (x)//4 + 4+ 62 ]. 30 5 + 4 +f~+ 1 4 1 40 1~ I - $2 4 5 252 1 120 1 _1 40 1 126 5 + 5 126 1 360 1 240 1 504 1 - I /3 180 1 + $ 1 4 144 5 +, $2 1 1 288 1 +3024 +-I 1008 + $2 504 1 1440 _ — -- 1 I + 15120-$ 10080 9600 I 1 +_. 'iGO^" 2400 6048 f 1, 960 $ 1+ 3168 0 1 86400 28800 14400 14400 1 + — L 23760 I 86440 7 1 1 4-7 4752 + 5 s 1512 - Oe, (,)//h 1 6f 1 288 The combination of kCk_, with the terms involving S gives the same result as the raw moment with the "corrections for massing"; and the remaining terms are the "corrections for abruptness." In fact, if we omit the first line of ( (x), the remainder, with sign changed, is identical with < (x) of ~ 7, A being equal to - S. 13. For the purpose of the formulae in this paper, it has been assumed that the extreme values of A_n+i and A m_ actually represent the areas of the corresponding strips. Sometimes this is not the case; the value of A may be measured from or to an ordinate lying inside the strip, or the curve of frequency may cut the base of the strip, so that the true (algebraical) area of the strip is less than the area represented by the given value of A. In such cases special methods have to be employed. 14*. A special application of the method of ~ 8 is to calculation of the first moment, all deviations being taken to be positive. Let K be the true first moment with regard to y, and Lt the raw first moment with regard to yt, both moments being calculated in this way; i.e. (replacing x, y by I, 7), let * This section has been added as a result of discussion at the Congress.

Page  426 426 W. F. SHEPPARD K-f m d I -x = - (x- +)d + (-) d4, X_n X-n X r=t- r=l-m- ^ Lt=-Ar Xr-Xt-= 2 Ar(Xt-X,)+ E A,(X,-Xt) r = - n - r=t Then it will be found that Kt Lt + ~ Di (xin) - 2 P1 (Xt) + 1i (x-n), where (1D (x) has the value stated in ~ 8. In the case of a double-tailed figure, if the differences near At are fairly regular, Kt will be approximately equal to 1 11 191 /4 Lt - hAt + — 0 hAt 3 0 4 hA +.... For calculating K for intermediate values of x, we have KI= St- t, K = 2t, Kt Yt = 2y..., where St' is the total area from yn to yt, and St is the total area from yt to y,. If we use an auxiliary function, we have 7 iv hKtI = 2At- - h2At11 + - h4A tV -..., hcJitl = 2hAt' -12 1hAtUI + 2887 lh5A t - etc. Or, if we use central differences, hKI — = 2/u, t - + 84A, -..., I 1 h2KtII = 28At- -83At, +45 8slAt-. h:1KtV = 28 - A, / + " -,..., h14K = 283At - 1 85A +..., etc. For the median ordinate, K is a minimum. Let the corresponding value of x be Xt+o. Then the minimum value of K is Kt + h (St' - St) + 2 {h2yt 2/2 + hyt1 03/3!+... }, 0 being given by 0 = (St' - St) + hyt + hy02/2 + hytI 03/3!+.... The successive approximations to the value of K, supposing hytI to be small in comparison with yt, are Kt, K-4 (S'-St),/yt, Kt - (Syty. t- (S,' - St)2/yt - (St,' - Sc)P YcYt3. 4 24~~2~

Page  427 A METHOD OF REPRESENTING STATISTICS BY ANALYTICAL GEOMETRY BY F. Y. EDGEWORTH. I. Introduction. The statistics which it is here attempted to represent are of a kind that commonly occurs in biology, sociology and other statistical sciences. This general type is characterised by simple curves of frequency, such that if the abscissa denote the size of an object or attribute, and the ordinate the frequency of each size, the ordinate has only one maximum, and becomes zero at each extremity. (Compare Pearson, Mathematical Contributions to the Theory of Evolution, No. XIV, pp. 4, 5; where an additional property is attributed to the general type.) The proposed method is a development of that which has been presented by the present writer under the title of "Method of Translation" (see Journal of the Royal Statistical Society for Dec. 1898, March, June and September, 1899, and March, 1900). The method consists in the transformation of a normal (Gaussian) curve of error by substituting for each of the magnitudes grouped according to such a law some assigned function of that magnitude. If, as will usually be the case, the origin from which the magnitudes to be operated on are measured is well outside the sensible area of the normal curve which is to be "translated"-say at a distance from the centre of the normal curve of more than twice its modulus, or thrice its standard deviation-then any points on the abscissa of the transformed curve, P, Q, R... divide its area into the same proportions (form the same percentiles) as the corresponding points p, q, r... on the abscissa of the generating normal curve. Accordingly, the median of the transformed curve may be taken as coincident with the centre of the generating normal curve, say 0; and to every abscissa of the normal curve, say I, measured from 0, there will correspond, measured from the same point, an abscissa of the transformed curve, say X, being a certain function of 4. If X = ad where a is a constant, the transformed curve is still normal, the scale only or size of the modulus being altered. Since the forms with which we have to deal do not for the most part deviate widely from the normal type, there is a propriety in assuming for the assigned function a Taylorian expansion of the form a + k12+ 4 3+..., where k and I are small with respect to a. Or, if for the small quantities k/a, 1/a we put respectively K and X, we may write the operator a (5 + C2 + X)3).

Page  428 428 F. Y. EDGEWORTH The modulus of the generating normal curve may be taken to be unity without loss of generality. Thus each element or small rectangular strip of the generating curve at the point e with base A:, and of height - exp.- _2 is translated to the distance a (~ + K2 + X3 +...); its base being changed from Ad to dX Ax Xd= =a (l +2 +3X2 +... ), and its height altered in the inverse ratio. It will not in general be necessary to proceed beyond the third power of 4 in the expansion of X, The transformation may be also conceived as the reverse of the following transformation. Consider the curve which represents the integral of the represented curve; and transform this integral by simply writing for its abscissa X (measured from the central point, 0) ~ multiplied by the factor (a + k + - l2). Whence (by reversion of series) the abscissa ~ (relating to the integral of the normal curve) may be equated to an expression of the form A + BX + CX'2 +.... The rationale of this method consists not merely, and not principally, in its exactly representing those cases il which each member of the frequency group under consideration is a definite function of some member of a group distributed according to the normal law of error. For instance if the velocities of winds, or of any other objects, are distributed normally, then the corresponding energies will be distributed according to a frequency curve which is an exact translation of a normal curve. So if the diameters of oranges or other spherical bodies vary normally, the solid contents, proportional to the cubes of the diameters, will be distributed according to a law which is given by putting X in the above written formula 3b2c( + Kc2+ Ec2~3); where b is the mean diameter (the average of the normal group of diameters), c is the modulus of that normal group; and K = c/b, a small fraction in the case of many natural objects (as pointed out by the present writer, Philosophical Magazine, 1892, Vol. xxxIv, p. 435). The cases in which translation is the formula are doubtless not uncommon (compare Journal of the Statistical Society, Vol. LXI, 1898, p. 678). But the reason of the method lies deeper. It consists in the affinity of the formula to that universal law which is ever and everywhere approximately fulfilled throughout the whole vast realm of Statistics-Statistics proper as distinguished from mere arithmetic by sporadic or fortuitous dispersion. This "generalised law of error " may be written (in the simple case of a single variable, x) 1 d3 1 d4 Y = y0- 3! kl3 Yo+ 4!k 2d4 Yo+... where yo is the normal error function -- exp. - x2/c2; k, and k2 are closely related to the important coefficients designated by Professor Pearson as /,l (" Contributions to the Mathematical Theory of Evolution," No. II, Transactions of the Royal Society, 1895, et passim), and r (Biometrika, Vol. Iv, p. 177). Provided that the essential attribute of Statistics, plurality of independent agencies, is present in some degree, it may be expected that this law will be approximately fulfilled for a portion of any frequency group, a central body as distinguished from the extremities

Page  429 A METHOD OF REPRESENTING STATISTICS BY ANALYTICAL GEOMETRY 429 which obey no general law. (See the present writer's exposition of the " Generalised Law of Error" in the Journal of the Royal Statistical Society for 1906, referring to his paper on the "Law of Error" in the Cambridge Philosophical Transactions for 1905.) For example, suppose a group of magnitudes to be formed by batches of twenty-five digits taken at random from mathematical tables, or the developments of a constant such as 7r. The sum of the twenty-five (generally n) digits may conveniently be divided by 5 (generally /n). Here, the group being symmetrical,,1 (= i/323) = 0, (=,2/l 2 - 3) =-.04896. Accordingly the generalised law of error applied to the data gives the equation 1 x2 i '04896 x2 2 x4 Y^ v vK ele K4e 165 (- 5 +3(165yj ) = v /1' e1665 4 2-2 -65 3 (16 ' )2 16 neglecting higher terms of expansion affected with the coefficients k4, k,, etc.; where x is the number of digits in excess of the mean number (in a batch of 25/5), viz. 22'5. (If instead of 25 any other number n of digits is taken and the sum divided by /n, the equation remains unaltered, except as to the factor outside the square brackets, which is inversely proportionate to n.) According to the method of translation, a normal curve with unit modulus being placed with its centre at the point (on the abscissa) 22-5, it is transformed by the operator 4'08 (~ - '004:3); the constants being determined by one of the methods presently to be described (below, p. 430). There are, then, to be distinguished three loci: (1) the particular real form, in the present case a polynomial of unmanageable complexity, in general unknowable; (2) the generalised law of error, which is approximately true up to a certain extent of the abscissa, and ultimately becomes unmeaning, if not false; (3) the particular representative curve, which is approximately true up to a certain extent of the abscissa, and ultimately becomes false, if not unmeaning. Thus the above written expressions both for the generalised law of error and for the particular representative break down long before the extremity of the real locus is reached, the point 45 (measured from zero), corresponding to the probability of all the twenty-five digits proving to be 9's! The contrast between the generalised law and the particular representative is more significant when r is positive and large. The use of such a representative particular, true only as to the main body of the frequency group, but with fictitious extremities-as it were handles of inferior material-seems to be a requirement of mathematical statistics. As a good way of meeting that requirement the Method of Translation is recommended. II. Determination of the Constants by Moments. For the determination of the constants in the formula for translation there is available the use of moments; which Professor Pearson has shown both by theoretical considerations and practical success to be not only serviceable for the fitting of curves in general, but also particularly appropriate to frequency groups of the kind here considered (see his " Observations on the Fitting of Curves" in Biometrika, Vol. i, Part III, Vol. II, Part I, and his Contributions to the Mathematical Theory of Evolution, Nos. I, II, XIV, et passim). There seems indeed a peculiar propriety in the use of moments in the case of a method which claims affinity with the (generalised) law of error. For the simplest

Page  430 430 F. Y. EDGEWORTH and perhaps most findamental proof of the validity of that law is afforded by the coincidence between the moments of the real and those of the approximate locus. (See the present writer's "Law of Error," Cambridge Philosophical Transactions, 1905; and compare Poincare, Calcul des Probabilites Lecon xv, the priority of which-in respect of the 'normal law of error-not known to the writer in 1905, has been acknowledged by him in the Bulletin of the International Institute of Statistics, 1909). This proof virtually assumes that a locus which has the same moments as another is the same locus. The observed moments are connected with the sought constants a, K, X, by the following simple propositions. If Y=f(X) is the required frequency-curve, then by construction Y=f(:) d; where f({) is the normal error-function with unit constant, and X = a (I + s2 + X3). Therefore X'f (X) dX (the symbol c being used to denote an extreme limit not necessarily at an infinite distance from the origin), the 'nth moment of the curve Y=f (j) about its median = Ja( + t2 + X ~3) )f(~) da. -ox The moments about the median, say M1, M1,, M3 etc., can thus be deduced in terms of the required constants from the well-known values of the moments of the normal error-curve about its centre; and the moments of the translated curve about its centre of gravity, say /kD,, /2, 3, can be deduced from the moments about the median by a process which Professor Pearson has made familiar. We have thus M3 = a K, 1I2 = a32 (K + — f3 K+ 32 X + X2),X2)...................................... Whence /1 = Ml = /K, /, =M2- M1 2 a2 (1 + ic2 - 3X + - A-X2), 3 = M3 - 3M13- 2 = a3 ( + 9 + + 9X + X2),............................................................ We have accordingly for the constant /3, hereinafter written simply /, the equation 82_ (; + It + 9X + -jX2)2 3(= 13L2/3)= 8 ( +_9X I s ( = / (l ~K 2 + 3X + 3+5+X2)3 an expression which may be simplified by putting = ='c2- By parity we may deduce the expression for r (= //l/,2 - 3), and thus obtain the fundamental equations connecting the observed moments with (two of) the required constants:(1) 8X ( + X + 9 1X2)2 + -) - (1 + X + 3X + -X\2)3 = 0, (II) 4 (6X + 3X + 3x2 + 54XX + 27X2 + 135XX2 + 44~X3 + 1 —T5X4) - (1 X+ + X + 3x + X)2= 0. These equations are not so formidable as they appear at first sight; owing mainly to the incident that the required values of X and X are usually small fractions. Accordingly the higher powers of the variable may often be neglected for a first approximation. For example, some statistics of barometric heights which seem to

Page  431 A METHOD OF REPRESENTING STATISTICS BY ANALYTICAL GEOMETRY 431 be fairly typical of a moderately skew frequency-curve (the statistics discussed in the papers already referred to in the Journal of the Royal Statistical Society, 1898-9) present for x and X respectively '13 and '34. These values being substituted in the fundamental equations we obtain from the abridged equations (I) 18x - 13 (1 + 3X + 9X)= 0, (II) 24X + 12- 34 (1 + 2 + 6X)= 0, neglecting all but the tirst powers of the variables, the approximate solution X = '0084, X= 0146. And these values remain correct to the second place of decimals, when the omitted terms are replaced in the equations. Commonly it is sufficient, for a first approximation at least, to take account only of the second (in addition to the first) powers of the variables. Abridged methods of solution will no doubt suggest themselves to the practical statistician. Difficulty from the occurrence of imaginary roots is not to be apprehended, as there is reason to believe that real values of X and X may be found, if not for every conceivable pair of values for /3 and rj that is consistent with the general description of the frequency-curves with which we have to do, at least for all probable and usual data. For instance approximate roots to the equations are readily found in the case of all the examples of the Pearsonian Types given by Mr Palin Elderton in his Frequency-curves, and in the case of the examples given by Professor Pearson in his second Contribution (except Nos. X-XII, which are to be treated according to one of the methods prescribed below for exceptional cases, and Nos. IX and XV, for which the moments up to the fourth are not explicitly given). The whereabouts of the required roots having been ascertained, it is not very troublesome to proceed to nearer approximation by means of the expressions d/3 die d7 d7q for dK and dX' dX In fine, if the method of translation were generally d/c d\ dX' d&' adopted, mathematicians with more leisure and industry than the present writer would soon construct a Table showing for every assigned pair of values of /3 and rv (within the limits of ordinary practice) the corresponding values of X (= c2) and v. It is hardly necessary to add that when the values of /c and q have been obtained, the value of a is given by the equation 2 = 221/(1 + K2 + 3X + -L5X2). III. Determination of the Constants by Percentiles. A start on the quest for the values of K and X may be obtained from a summary use of percentiles. Thus, suppose the position of the median to have been observed; and that the point on the axis of X below the median, between which and the median are intercepted '3414 of the total area (the point corresponding to the standard deviation on the generating normal curve), is at the distance S from the median. Likewise let the quartile above the median be at the distance Q from the median, and at the distance R let there occur the decile above which there is just a tenth of the area (corresponding to 1-28155 times the standard deviation-or '906 the modulus-of the generating curve). We thus obtain the three linear equations for the three quaesita: a ('707 - '7072K + '7073k) = S, a ('477 + '4772K + 44773X) = Q, a (906 + -9062 + '9063) = R.

Page  432 432 F. Y. EDGEWORTH The solution thus easily obtained is inadequate, both because it does not utilise all the data, and because it does not take into account the difference in weight and the correlation of the errors incident to the several equations. For a more perfect solution the required constants must be connected with the correlated errors incident to the whole system of percentiles (as to which correlation see W. F. Sheppard, "On the Application of the Theory of Error," Transactions of the Royal Society, 1898, Vol. 192 A, p. 114 et seq.). Beginning with the outermost percentile, say at the upper extremity of the group, cutting off to the right the fraction v of the total area, we know that its error is due to the incident that the area to the right of the true, the ideal percentile, is not, as a priori most probably, v, but v plus (or minus) a deviation therefrom, say ev (on the principle underlying Laplace's "Method of Situation," as pointed out by the present writer, Philosophical Magazine, 1886, Vol. xxII, p. 374. Compare Sheppard, loc. cit.). Likewise the error of the percentile next in order depends on the deviation of the area to the right of the corresponding true percentile from the true area. Thus the errors of portions of area corresponding to the percentiles form a system analogous to the errors of the numbers or proportions which are presented when from an indefinitely large me'lange of differently coloured balls there is extracted at random a large sample numbering N. To 'fix the ideas, suppose that there are mixed up balls of seven colours, say those of the prism, violet, indigo, blue, green, yellow, orange, red; in the proportions v, i, b, g, y, o, r; Nv being the number of violet balls in the supposed indefinitely large bag or jar, Ni the number of indigo balls, and so on. The probability that the number of violet balls in the sample numbering N will exceed (the most probable number) Nv by Nev is approximately proportionate to exp. - Nev2/2v (1 -v). The probability that when Nl(v + e,.) violet balls are drawn, there will be a certain excess Nei of indigo balls above that number of indigo balls which is now most probable, viz. Nr(1 - v - e,) is proportional to exp. - N (1 - v - e,) ei2/2i (1 - v - i). This probability, multiplied by the above written probability of ev, gives the probability of the double event e, and ei. Now put ei = ei + evil(l - v), so that the deviation of the number of indigo balls from the originally most probable number thereof, viz. Ni, becomes Nei; and we find for the probability of the two errors ev and ei: Const. x exp.- N v (1 --- v ) + i + - -t))). v (1-) i(1 - v - i) neglecting in the numerator of the latter part of the expression terms which are of the third order of magnitude. (Compare as to the orders of magnitude " Law of Error," Cambridge Philosophical Transactions, 1905, p. 58.) Multiplying out, we have for the probability of the double event ev and ei Const. x exp. I (e~ (I)+ 2e e + ei(1v - -). e +1 v/ Proceeding similarly with the remaining colours up to the sixth inclusive-the number of the seventh, red, is given when the errors of the other six numbers are assigned-we find for the system of six errors ev, ei..., eo the probability Const. x exp. -N (e (r + v) e( + 2ei) + 4 )/

Page  433 A METHOD OF REPRESENTING STATISTICS BY ANALYTICAL GEOMETRY 433 This expression will be found to agree with that obtained by Professor Pearson in connexion with his Criterion of Random Sampling (Phil. Mag. July, 1900). If green is so to speak the median colour, viz. v+i+b< I, v+ i+b+g>4, we might write for g, u,, and for the proportions y, o above the median colour, or class uI, Iu2, etc. and for the proportions below u_i, _-2, etc. mutatis mutandis; and for the corresponding errors eo; el, e2; e_1, e_2. We have now to connect these errors of area with errors of the percentiles involving the sought constants. The error pertaining to any compartment, of which the content is ideally ul, is made up of two errors, one at its upper and one at its lower limit, each of which may be regarded as a little rectangle of which the base is the deviation of the corresponding percentile from its true position, and the height is the ordinate of the translated curve at the point. Thus the rectangle error at that boundary of the tth compartment which is nearer the median has for base the difference between the observed distance from the median, sav 2it, and the ideal distance act + k 4t2 + lt3, if at is the abscissa of the normal error-curve corresponding to the position of the lower limit of ut (the point above which the area of the normal curve = ut + 1 +t+ + ut+2 +...). Also the ordinate of the transformed curve at the point under consideration is f (t) id) where f(at) is the ordinate of the normal curve with unit modulus for abscissa at, say 't. Therefore the transformed ordinate = t/(a + 2kt + 31t2), and the error of the tth compartment, viz. et, (a+t + ct + l- n,) t a + 2ktc- t + 31lt' plus a similar product which is formed by substituting t + 1 for t in this expression (and changing the sign of the expression). The expression thus formed is to be substituted for et in the above written expression for the probability of the system of errors; and similarly formed expressions are to be substituted for all the other e's. Let the logarithm of the resulting function of a, k, I be called - P. Then the simultaneous equations, dP dP dP - =, - 0, 0, da dk dl give the values of a, k and I which make the probability of the observed complex event a maximum, provided that the second term in the expansion of P fulfils the condition of a minimum. The equations become simplified when, as very commonly, k and 1 are small with respect to a, and accordingly it is allowable as above to put, for X, a (f + Kg2 + X)3), where K and X are small fractions. The exposition may be completed by considering one or two simplified examples. First suppose that Kc and X are each zero. The problem then reduces to the simple and familiar one (treated in papers above referred to): to determine the modulus (or standard deviation) from observed percentiles. In this simple case the ordinate of the transformed curve reduces to pt/a, where a is the sought modulus, and the equivalent of et becomes _ t (at - ) t) a a 28 M. CJ. II.

Page  434 434 F. Y. EDGEWORTH In the expression for P made up of squares and products of e's of this type, it is proper to equate to zero the differential coefficient of P with respect to a; the complete differential, the a in the denominator not being treated as constant (in accordance with the principles of Inverse Probability as stated by the present writer-after Laplace and Pearson-in the Journal of the Royal Statistical Society, 1908, Vol. LXXI, p. 388, and context, p. 499, etc.). The resulting value for a, and the expression for the error to which it is liable, will be found to be in substantial agreement with Mr Sheppard's determination of the weights which make the mean square of error of the standard deviation a minimum. An introduction to the general method, and estimate of the trouble it is likely to require, is afforded by the case in which only one of the constants, viz. X, is supposed to be zero. For further simplicity it may be supposed that there are only three percentiles, the median and the two quartiles. It will be found convenient in this case, and generally, to treat the observed median provisionally as the true one, to measure the ~'s in absolute quantity on either side of the median, and to form the expressions for El, E_,, in general Et, E_t, where Et is the error incident to the whole area cut off to the right by the tth percentile, and E_t is the error of the area cut off to the left by the percentile designated as - t. The E's can readily be obtained from the e's above defined. It will be seen that the error introduced by treating the observed median as the true one tends to disappear by compensation. When the values which have been determined for k and I (or mutatis mutandis K and X) have been substituted in P (above defined) the probability of any system of errors, say Aa, Ak, Al, in those values is given by the normal (hyper-) surface z = Const. x exp. - d2 Aa2 + 2 d2 AaAk +... ). rPkd\da2 dadk + ) This surface may be compared in respect of tenuity with that which pertains to the determination of the moments (as to which see Pearson, "Contributions," No. IV, Transactions of the Royal Society, Vol. 191 A, p. 266 et seq.), or to the determination of K and X as functions of the moments. It is not to be hastily assumed that in this comparison the moments will have the advantage, because this is so in the case of the constant pertaining to the normal curve, as shown by Mr Sheppard. For in the case of the normal curve it is given by Inverse Probability that the most probable value of the modulus (or standard deviation) is a function of (the first and second) moments (cp. Journal of the Statistical Society, 1908, loc. cit. p. 388). As the (normal) curve of error for that determination rises to a higher apex, so it spreads less than the curve pertaining to other determinations. But the most probable values of the constants entering into the formula for translation are not simple functions of low moments. The method of moments which has been exhibited is a good method of determining the constant, but not necessarily the very best in respect of accuracy. In the practical choice of the method for determining the constants, such a difference of probable error as may exist in the abstract between the method of moments and that of percentiles will not appear decisive to those who hold with a recent writer that " there are very few arguments made from probable error which would be seriously affected if the probable error were altered by 25 per cent." (Rhind, Biometrika, Vol. VII, pp. 127 and 386). Much more serious differences

Page  435 A METHOD OF REPRESENTING STATISTICS BY ANALYTICAL GEOMETRY 435 arise from the imperfections occurring in concrete statistics. Suppose the data are not finely graduated, say only four or five degrees or percentiles given; or suppose that at the extremities there occur heterogeneous measurements, such as dwarfs and giants may be in the case of anthropometry. In such cases the method of percentiles can utilise all the sound data, and omit those that are suspicious, without manipulation; but the method of moments has to be eked out by hypotheses. Percentiles, too, admit of abridged methods which may often be convenient and sufficient even when the data admit of greater precision. In the light of the theory practical statisticians could no doubt devise a procedure intermediate in respect of facility and accuracy between the very summary method mentioned at the beginning of this section and the very elaborate method afterwards set forth. For instance, to constitute each of the three equations employed in the summary method two percentiles instead of one might be employed. Thus in the barometric statistics already referred to, in the neighbourhood of the point on the upper arm of the curve which has above it a tenth of the total area, there occur two percentiles with the corresponding proportions of the modulus unity obtained from the ordinary tables, =.8958, ~ = 1-0796. Whence we obtain the following two equations: a '8958 + k *89582 + 1 '89583 = 4-5, (1-0796 + kl-07962 + 11-07963 = 5-5. By adding these two equations we obtain a compound equation presumably better than either of the components; which is to be combined with two other equations each similarly compounded. But caution must be exercised lest by taking in additional observations unweighted we should make the result worse (as pointed out by the present writer, Philosophical Magazine, 1893; where, as noticed by Mr Sheppard, loc. cit., p. 135, there was committed the error of treating the percentiles as uncorrelated. The matter had been rightly stated by the writer in an earlier paper, Philosophical Magazine, 1886, Vol. xxI1, p. 375). IV. Exceptional Cases. Percentiles are not so available as moments for the treatment of exceptional cases in which the generating normal curve is not only extended (or contracted), but also in part folded over in the process of translation. For instance, suppose the coefficients a and I to become both zero, the operator taking the form X = k'2. Then the lower (left) half of the normal curve will be swung round the axis of Y at the centre of the (original) curve so as to be coincident with the upper half; while both are distorted in such wise that the initial ordinate now becomes infinite. The infinity of a single ordinate, it may be remarked, is not fatal to the representative character of a curve. To determine / we have now the following equations (X being as before measured from the centre of the generating normal curve, as origin, and f/() denoting the normal error-function with unit modulus): = 2 XF (X) dX = 2 kf() d = Iei; l =2= X2F (X) dX _= 2 k2f () d4= 3 -2o J i3= 2 X3F(X) )dX = 2 k 76j'()dt-= Ujtk3. 28 —2

Page  436 436 F. Y. EDGEWORTH Whence 2M = M- M2 = ak; 43= M- (M1 + 3~M12) = 3; / = 3/ = 1/ =8. The same result may be deduced from the fundamental equations (above, p. 430). For now, X vanishing, k becomes infinite. Accordingly, the fraction which is equated to 8/ becomes in the limit equal to 8. The corresponding value of q obtained in either way is 12. Now let I receive a positive value, a being still zero. And first let the value of I be small relatively to k. Then the elements of the normal on the left of the central axis will continue to be swung round to the right up to the point at which dX becomes zero, that is when 2k = 3X^. Thus practically the whole of the area d4 on the left is swung round to the right when ki/l > 2, say, or 2-12 (corresponding to 3 times the standard deviation), or l/k < ~. The transformed curve will, however, have lost the typical character of uni-modality (cp. above, p. 427) at an earlier point, dY the point at which the expression fordX changes from negative to positive (becoming infinite at the point 2 = 1k/l). By varying the values of k and I within the limits indicated the formula may be adapted to a considerable variety of data of the sort instanced by Professor Pearson in the tenth, eleventh and twelfth examples of his second Contribution (Transactions of the Royal Society, 1895, Vol. 196A). Further latitude is obtained by supposing a to have a small value (relatively to k or 1). Thus, suppose that a/k is a small fraction, say, while I is zero, then the whole portion of the area on the left of the point ~ defined by the condition a - 2kc = 0 is swung to the right round the axis of Y at the point distant from the origin on the negative side. The resulting curve will have a mode other than that formed by the infinite ordinate at the extreme left, which, if a/k be sufficiently small, may well fall within the limits of the first of the given compartments-often comprising a considerable percentage of the whole area-and so escape notice. By assigning different relations to the constants a great variety of cases, perhaps " magis mathematicae quam naturales," can be manufactured. It must suffice here to specify one more anomaly. Suppose I is negative, while k is zero: the operator dY is then a ( -X3). And the expression for dX becomes 61;~ + (6X - 2) (1 - 3X2)3 t being measured positively, or in absolute magnitude, to the left (as well as to right) of the origin. If X >, this expression is positive in the immediate neighbourhood of the origin, and continues positive up to the point at which ~2 = 1/3X. For instance, if X=, dX is positive on either side of the origin up to the point = 816, where Y becomes infinite. We have thus a symmetrical U-shaped curve. If, now, k becomes sensible, we have an unsymmetrical inverted curve, provided that the values d Y of k and X are such and so related that the cubic equation (corresponding to dX = 0) 6X3- 4k2 + t(6X - 2) - 2k = 0, has only one real root, corresponding to a single inverse mode.

Page  437 A METHOD OF REPRESENTING STATISTICS BY ANALYTICAL GEOMETRY 437 To him who grounds the use of a particular representative curve on its affinity to the generalised law of error these freaks of analysis are of secondary interest. They serve, however, to dispel the suspicion that the method is deficient in the power of adaptation. V. Frequency-surfaces. The analogue of the preceding method in cases where the given statistics relate to two (or more) characters, say X and Y, utilises, in addition to the mean powers of those variables, mean products, such as the sum-integrals of X2Y and XY2. Constants corresponding to these additional data are presented by adding to the expression for X hitherto employed terms involving rv (as well as ~) the coordinate of a normal surface with unit moduli and coefficient of correlation r. Thus for powers and products of the third dimension we may write X = a (4 + K1t2 + Kc1'-), Y= b (r7 + K21 + K2't). For the fourth dimension, as there are three additional data, it would seem proper to introduce three new " X's," X' the coefficient of 3r/, X\' the coefficient of r34, and X" the coefficient of 27'" in the expressions for both X and H. Substituting these expressions in the values of the mean powers and products given by observation, we obtain as many equations as there are variables; of which the solution may be practicable, if the coefficients of the type K and X are small, in which case the coefficient of correlation, r, differs by a small quantity, say p, from what it would have been if K,,... were all zero, say r' (= 11n/V/,2/v2). Thus we may substitute r' +p for r wherever r occurs in the expressions equated to our data-for instance in the integral of 02X2. We thus obtain equations comparable in respect of the smallness of the required roots with equations (I) and (II) of Section II, which usually prove manageable. The solution may be simplified in cases where we know that mean products of the type X2Y, XY2, X3Y may be ignored (perhaps even we only do not know the contrary), for then the distribution will depend only on the mean powers. Let there be obtained from the third and fourth (together with the first and second) mean powers (moments) the coefficients of translation for the two curves which represent the frequency of X irrespectively of Y (with equation Z = F (X, Y) A Y) and the -00 corresponding frequency of Y irrespectively of X. Say the coefficients are respectively Kc, XI and K2, X2, then the given surface may be regarded as generated by translating a normal surface, say z=f(, q), with unit moduli and coefficient of correlation, r, as follows. For every element (small rectangular column) of the generating surface standing on base A4Aq1 there is substituted an element of equal content at the point X = a ( + K1i2 + Xf3), Y= b (r + K2q2 + X273); and with area AXAY correspondingly modified. The coefficient r is thus connected with the observed product-sum YYXY (and the other data). By construction SEXY, say p12 = i a (+ +K1,2+ X13) b ( + K2q+2 + X2v) dd77 J= f + - +dd00 = r |rab (~ + KK2,,2 + x,q3 + x2 + X2X3 +:3) 3) dtd, J - -X. -J

Page  438 438 F. Y. EDGEWORTH omitting integrals which reduce to zero. Assigning to those which are retained their respective values (for a normal surface with unit moduli) we have: /12= ab r +KK + 2r) (1 + Xr (9r + + ) + There is thus obtained for r a cubic equation, with of course a real root. But if this root is greater than unity, the solution is not admissible. VI. Comparison with the received method. The method of translation labours under a heavy disadvantage in that it has not been accepted by the highest living authority on mathematical statistics. Professor Pearson's objections, it must be admitted, had force when directed against the method as first presented. But they are not equally applicable to the method as now developed. It cannot be now objected-if it ever was against the present writer, as distinct from one with whom he was bracketed-that the curve proposed is not a "graduation formula," or that it is not "determined by constants of which the probable errors are easily deducible." (See K. Pearson's "Rejoinder," Biometrika, Vol. IV, pp. 201, 202, 211.) Nor can there now be urged the objection: "What is x of which the observed character X is a function?...What is the character which obeys the normal law?... The function x has no real existence as a biological entity" (loc. cit. p. 199). This objection might be applicable if the proposed form was advocated as a particular real curve (above, p. 429), related to some real attribute distributed normally, say, as energy is to velocity. But the objection is not equally applicable to the position now taken. The best defence of this position is that it is the same as Professor Karl Pearson's. For his Types, as here interpreted, are but particular representative curves (above, p. 429), formed by a judicious divergence from the normal law of error (a divergence well indicated by himself, loc. cit., pp. 179 and 210 (viI)). In the Pearsonian system the normal curve is transformed by substituting, for the abscissa, say in our terminology I, of the differential coefficient of the logarithm of a normal curve, an abscissa X divided by a quadratic expression, say be + bX + b2X2; in the system here proposed, the normal curve is transformed by substituting for the abscissa 4 of the integral of the normal curve an abscissa X multiplied by a quadratic expression, say a, + a, X + a2X2 (above, p. 428). Is there anything to choose in respect of rationale between the two transformations? Are they not both equally true or equally false (as maintained in a former paper, Journal of the Statistical Society, 1906, Vol. LXIX, p. 512 and context)? The question for the practical statistician is whether they are equally useful. A use special to the method of translation is to confirm some of the results obtained by the received method; not by a servile repetition, but as an independent witness giving a similar but not an identical version. This would be a strange mode of verifying ordinary mathematical formulae; but in dealing with Probabilities, it may be appropriate. Thus Professor Pearson has pointed out, with special reference to his Type III which does not utilise the fourth moment, that commonly the distance of the mean from the median is about a third of what it is from the mode (Contributions, No. II, p. 375. Compare Yule, Introduction to the Theory of Statistics, p. 121: "there is an approximate relation between mean, median and mode that

Page  439 A METHOD OF REPRESENTING STATISTICS BY ANALYTICAL GEOMETRY 439 appears to hold good with surprising closeness for moderately asymmetrical distributions...expressed by the equation Mode = Mean - 3 (Mean - Median)"). Now the case in which the fourth moment is not required on our system is when X = 0. In that case, the mode is given by the equation I 1=0 2+ I _ +-I= o. This equation gives for the distance of the mode, in the negative direction, from the median = - / nearly, X = a (~ + ct2) = - a =- k (nearly), a small fraction. But the distance of the mean from the median in a positive direction is found to be I k. Therefore the distance of the median from the mean is about a third of the distance of the mode from the mean. The Pearsonian Type III is situate so to speak on the dividing line between two types which utilise the fourth moment with range on the one side finite, on the other side unlimited. The transition is marked by the sign of the "criterion," 6 + 3/, - 2,82 (where 8/ is our /8 and /32 = /42//,L2 =our v + 3). The transition corresponds in our system to the passage of X through zero. But the criterion of this transition is not so simple as the above-written one. It is the relation between / and r which is given by putting X = 0 in the fundamental equations I and II (above, p. 430) and eliminating X. In the common case, however, when X (=,C2) is small, this eliminant is much simplified. We have then, neglecting powers of x above the first: (I) (18 - 3) = a, (II) X (24 - 2r]) =. Whence 24/3 - 18 + rq3 = 0. Neglecting the product,/3 as of the second order of magnitude and writing V = 2 - 3, and dividing by a constant, we have the Pearsonian criterion 6 + 3/3 - 2/a2. As for the range, there is a distinction on our system, too, between limited and unlimited; and the dividing line is exactly at the vanishing point of X. When X is zero, the representative curve may be considered to stop or break down at the point where dY dX changes sign a second time as we move in a negative direction from the origin, the median; the point defined by giving to the radical in the root of the above-written equation for the distance of the mode from the origin, the negative sign, that is 2 nearly = a/2K/-a point at which the curve may be regarded as no longer sensible when 1/2k > 2-1 (above, p. 436), say K < j. But when X is not zero, the position of the mode is given as a root of the cubic equation 6X~3 + 4kf2 + ~ (2 + 6X) + 2/ = 0 (the above-stated quadratic plus terms involving X). The criterion distinguishing limited from unlimited range is the condition that this cubic equation should have two impossible roots; and it will be found that there are or are not impossible roots according as X is negative or positive. The distinction, then, between X positive or negative, marks a real difference in the data. But of course by those who consider that our general knowledge of approximate forms terminates with the generalised

Page  440 440 F. Y. EDGEWORTH law of error-refers to the body and not the tail of a frequency-curve-exact propositions about range must be taken cum grano. Other features of the Pearsonian types may be traced in the variant representation now proposed. When they cannot be so traced, it is suggested that they are of secondary importance. With regard to the general purposes of the statistician, the advantage seems by no means to be all on the side of the received method. It is true that in that method the connexion between the observed moments and the required constants is particularly simple. By a stroke of singular power and felicity, Professor Pearson has expressed that relation by a series of simple equations which contrast favourably with our many-dimensioned system (above, equations I and ii). On the other hand, the relation between the constants and the form, or (integral) equation, of the representative frequency-curve is not so simple in the Pearsonian system. A. slight variation of quantity in the constants is attended with convulsive changes in the form of the functions. Thus in dealing with statistics which seem to be in pari materia, frequency groups nearly coincident with the normal error-curve (such as the anthropometric measurements dealt with by Miss Fawcett in the first volume of Biometrika), a single step may plunge the practitioner from one unfamiliar form to another, from Type I to the stupendous Type IV. But on the plan here proposed the corresponding changes are quite gentle: presumably from a small positive value of the constant X to a small negative value thereof, with corresponding small changes in the constants a and K. Data which exercise the mathematician on one system present no significant peculiarity on the other. Thus the example of Types V and VI given by Professor Pearson in his contribution No. X (Transactions of the Royal Society, 1901, Vol. cxcvII) differs in our system from ordinary cases principally in having peculiarly large values for K and X. No doubt the distinctions marked by the Pearsonian criterion K, do correspond to real differences in the statistical material as well as the distinctions marked by his criterion Kc. And for some purposes it may be useful to take account of those differences; but for other purposes it may be useful to ignore them. Not only is the analytic form employed in the method of translation uniform, simple and familiar, namely the error-function (together with a rational algebraic expression of the third degree), but it is particularly useful as connecting the percentiles of the given group with the constants (when ascertained by moments) of the representative curve, by simple relations, with the aid of ordinary tables. The method of translation has also an advantage in being capable of application (as pointed out already, p. 435), when the data are imperfect. Even when the data are not imperfect it may often be convenient to roughly characterise a given frequencycurve without employing the laborious apparatus of moments and criteria. One deficiency indeed, in the method of translation, is felt and acknowledged; one thing it yet lacks-authority.

Page  441 CONTRIBUTION TO LAPLACE'S THEORY OF THE GENERATING FUNCTION BY D. ARANY. I. The expression Q(x)=l/{-x!(1-x)!t.pxq1- (p+q=l).....................(1) is only defined for integral values of x, and is arbitrary for other values. In the Calculus of Probabilities it is convenient to replace Q (x) by an expression that shall be equal to it if x is an integer and shall have a definite meaning whatever the value of x may be. The ordinary method is to use Stirling's theorem i! = n'1e-Tl V(2rrn) exp tB,/(1. 2). 2 - - B,/(3. 4). n-~ +...,......... (2) where B1, B9,... are Bernoulli's numbers. Substituting in (1), we obtain Q (x) = ll/{1 - xx )l. / ( 1 (1 - x) —1. p' (2)- - ( )-} l-x exp B,/(1. 2). (I1- -] -(1- )-...}....(3) Writing x = 1p + t, 1 - x = lq - t, this becomes 1 t2/l 1 ) x= - - - approximately...................................(. ) / t2 exp 1~ 2- ' 2 - p- approxim ately...............................................(7) 27Trlpq II. It can be shown that both (6) and (7) can be obtained by extending a method due to Laplace, called the method of Generating Functions. As the original method is applicable to every function of x which is defined for integral values of x, we will give the extension of the method in the general case, and then check the result by applying it to Q(x). Let f(x) be a function of x whose values are given for x = 0, 1, 2,... 1. Then we define f(x) for other values by the equation ru=a+27ri ( t=l t 27rif (x) = e- J f (t) e du,..................... (8) t a — - t=U

Page  442 442 1). ARANY which can be proved to hold for any integral value of x and for any value of a. We then write t=l q (t) - S f(t) etu, t=o and call this the generating function of f(t). For a, which is quite arbitrary, we choose a value which makes ( (a) = (a + 2Sri)= 0. We then, for any particular value of x, define a value s of it which makes the first derivative of e-xu"((u) zero. This gives ' (s)- X (s) = 0, x = ' (s)/ ()....................(l) We also define a new variable y related to u by the equation (for this value of x) e-x"t Qb (a) = e-xs. (s) e-.................................. (12) We then have 27rif (x) = e-x8 (s)f e-Y2 (d) dy......................(13) It then remains to develop (d ) in a Maclaurin-series in powers of y. This is done by introducing certain functions (i) and O (), explained in the paper*, such that (du/dy)y=o =i I(s)} 2 (dudy) = i2 [d (u)} -du]............. (d38/dy3)Y=O = i3 [d2 { (u)} 2]u= etc. Substituting in (13), we obtain e-xS = (s) { - ( ( 1- { L "(u)} - */7_2 i / 2!2L du2 U=S 1 1.3 d4 ) 2 c 4!2.2 L dit4 J - }-(22) We can prove that -() 0!dx 2!ds 1!d2x ( ).................................(26) 3! dex 2!d3x S" (s) ds etc. * (1L)=log A (.) = (log (u) 2 ( +!) = 3 (S)(2 + ( ( -)) + - + ete. where k (S) = dk"k t =, s

Page  443 CONTRIBUTION TO LAPLACE'S THEORY OF THE GENERATING FUNCTION 443 and therefore (22) becomes approximately e-Xs qS(s) f X) = f-.................................... /27r dc/d. ds We can transform this into e-f" dx( (x ) = - B. -...............................(29).(t'( ) = Q (.), For the value of s is t \l \t l )'I l _ 1 s / \]., 1) (I 2-F,p2 (I2 Also B = 1, and therefore as in (7). t2 e 21pq A%/2,7rlp/9

Page  444 RiEPER{TOTIRE BIBLTOGiRAPHIQUE DES SCIENCES MAT1ITIW~ATIQUES PAR A.. GE"RARDIN. A la niernoire de notre reyrett4' Pre'sident Henri Poincctre'. Le RC'pertoire Bibliographique des Sciences Mathe'matiques est destine' 'a 4arger ux ravillursde longues et p'nibles recherches. II contient les titres des Me'moires relatifs aux Mathe'matiques pures et applique'es de 1800 'a 1900 inclusivemient, ainsi que des travaux relatifs 'a l'histoire des Mathe'matiques depiuis 1600 jusqua~ 1889 inclus. Le Repertoire parait par series de 100 fiches 8,5 x 13,5 sur papier fort, chaque fiche renfermant en moyenne neuif titres complets. Actuellement, vingt series sont parues, comprenant 2000 fiches et l'indication de 18523 articles. La se'rie 21, 'a la composition, est re'serve'e 'a la The'orie des Nombres, ch la bibliographic complete dii dernier the'ore'me de Fermat, et 'a la Ge'ome'trie. Apre~s avoir rendu un juiste hommage an labeur de mes pre'decesseurs, MM. Hlumbert, C. N. Laisant, Raffy, je vais donner ici par matie're, mais sains entrer dans le detail, les chiffres des articles classes. CLASSE A. (1355 articles.) Alge'bre e'lementaire, the'orie des equations alge'briqnes et transcendantes; groupes de Galois;1 fractions rationnelles; interpolation. CLASSE B. (752 articles.) Determinants; substitutions line'aires; elimination; the'orie alge'brique des formes; invariants et covariants; quatertilons; e'quipollences et quantite's complexes. CLASSE C. (779 articles.) Principes dii calcul diff~rentiel et integral; applications analytiques; quadratures; integrales multiples; determinants fonctionnels; formes diff~rentielles; operateurs diff4~rentiels. CLASSE I). (1279 articles.) Theorie generale des fonctions et son application aux fonctions alge'briques et circulaires; series et de'veloppements infinis, comprenant en particulier les produits infinis et les fractions continues conside'rees an point de viue alge'brique; nombres de -Bernouflli;fornetiotis sphe'riques. et analogues.

Page  445 RE1PERTOIRE ]31BLIOGRAPHIQUE DES SCIENCES MATH9MATIQUES 45 445 CLASSE E. (220 articles.) Intitgrales de'flnics, et en particulier inte'grales eule'riennes. CLASSE F. (207 articles.) Functions elliptiques avec leurs applications. CLASSE G. (191 articles.) Functions hyperelliptiques, abe'liennes, fuchsiennes. CLASSE H. (1297 articles.) Equations diff~rentielles et aux diff~rences partielles; equations fonctionnelles; equations aux diff~rences finies; suites re'currentes. CLASSE I. (1163 articles.) Arithme'tique et The'orie des Nornbres; analyse inde'termin~e;e the'orie arithin tqedsfre t e rcin otinues; division du cerele; nombres complexes, ide'aux, transcendants. CLASSE J. (650 articles.) Analyse combinatoire; calcul des probabilite's; calcul des variations; the'orie generale des groupes de transformations [en laissant de cote les groupes de Galois (A), les groupes de substitutions line'aires (B) et les groupes de transformations geometriques (P)]; the'orie des ensembles de M. Cantor. CLASSE K. (1040 articles.) Ge'ome'trie et Trigonome~trie e'lementaires (e'tude des figures fbrme~es de droites, plans, cereles et sphe'res); Ge~ome~trie du point, de la droite, du plan, du cerele et de la sphe're; Ge'ometrie descriptive; Perspective. CLASSE L. (526 articles.) Coniques et surfaces du second degre". CLASSE M. (617 articles.) Courbes et surfaces alge~briques; courbes et surfaces transcendantes sp~eiales. CLASSE N. (117 articles.) Complexes et congruences; connexes; syste'mes de courbes et de surfaces: g~mtre 'num frat've CLASSE 0. (912 articles.) Ge'ome'trie infinite'simale et ge~omeftrie cinefinatique; applications ge~omnetriques dii calcul diff~rentiel et du calcul intefgral 'a la the~orie des courbes et des surfaces; quadrature et rectification; couirbure; lignes asymptotiques, g~od~siques, lignes de courbure; aires; volumes; surfaces minimna;- syste'mes orthogonaux.

Page  446 446 446 ~~~~~~A. GERARD)IN CLASSE P. (213 articles.) Trallsforinatio us g~oni~triques; homnographie, homnologie et affinite'; correlatiori et polaires re'iproqtues; inversion; transformations birationnelles et autres. CLASSE Q. (246 articles.) Ge'ome'trie, (livers;, ge'ometrie h -i dimensions; geonietric non euclidienne; analy-sis situs;- geonuetrie de situation. CLASSE R. (1319 articles.) Me'anique gemerale; Cine'inatique; S-tatique comnpren'ant les centres de gravite' et les moments d'inertie; dynamique; mecanique des solides; frottement; attraction des ellipso~des. CLASSE S. (953 articles.) Me'canique des fluides; hydrostatique;- hydrodynainique; thermnodynarnique. CLASSE T. (2444 articles.) Physique Mathe~natique; eflasticite; resistance des mnate'riaux; capillarite; lumi~re; chaleur: electricity. CLASSE U. (1065 articles.) Astronomic; Me'canique ce"Ieste et ge'ode'sie. CLASSE V. (851 articles.) Philosophic et histoire des sciences mathe'natiques; Biographic. CLASSE X. (327 articles.) Proce~des de calcul; tables; Nomographie; calcul graphique; planinie'tres; instruments divers. En resume'; Analyse Mathe'matique 7893 articles; Ge'onmetrie, 3671, et 6959 pour les Mathe~natiques applique'es. La classification detaillee et aussi complkte que possible de tontes les questions dui domaine inathe'niatique a kt' e'tablie inde'endaniment des mn'thodes.

Page  447 COMMUNICATIONS SECTIONS IV (a) AND IV (b). JOINT MEETING (PHILOSOPHY, HISTORY, DIDACTICS)

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