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Page [unnumbered] University of California Publications in MATHEMATICS VOLUME I 191 2-1924 M. W. HASKELL D. N. LEHMER J. H. McDONALD EDITORS UNIVERSITY OF CALIFORNIA PRESS BERKELEY, CALIFORNIA

Page [unnumbered] THE UNIVERSITY OF CALIFORNIA PRESS BERKELEY, CALIFORNIA CAMBRIDGE iUNIVERSITY PRESS LONDON, ENGLAND

Page [unnumbered] CONTENTS PAGE 1. On Numbers which Contain No Factors of the Form p(kp + 1), by H enry W. Steger 1................................................... 1 2. Constructive Theory of the Unicursal Plane Quartic by Synthetic Methods, by Annie Dale Biddle.......................... 27 3. A Discussion by Synthetic Methods of Two Projective Pencils of Conies, by Baldwin Munger Woods.................. --- ——... --- —-. 55 4. A Complete Set of Postulates for the Logic of Classes Expressed in Terms of the Operation "Exception," and a Proof of the Independence of a Set of Postulates due to Del Re, by B. A. Bernstein.......... 87 5. On a Tabulation of Reduced Binary Quadratic Forms of a Negative Determinant, by Iarry N. W right...... ---—...... --- —---------------- 97 6.- The Abelian Equations of the Tenth Degree Irreducible in a Given Domain of Rationality, by Charles G. P. Kuschke............. -—.. 115 7. Abridged Tables of Hyperbolic Functions, by F. E. Pernot -... —.....-. 163 8. A List of Oughtred's Mathematical Symbols, with Historical Notes, by Florian Cajori............-...................-......... 171 9. On the History of Gunter's Scale and the Slide Rule during the Seventeenth Century, by Florian Cajori..................... ------- 187 10. On a Birational Transformation Connected with a Pencil of Cubics, by Arthur Robinson Williams........... --- —---—. -------------------- --- 211 11. Classification of Involutory Cubic Space Transformations, by Frank Ray Morris..............-..........- 223 12. A Set of Five Postulates for Boolean Algebras in Terms of the Operation "Exception," by J. S. Taylor.................................. - 241 13. Flow of Electricity in a Magnetic Field. Four lectures by Vito Volterra, Professor of Mathematical Physics at the University of Rome, with the cooperation of Elena Freda, Libera Docente of Mathematical Physics at the University of Rome... —..................... --- 249 14. The Homogeneous Vector Function and Determinants of the P-th Class, by John D. Barter............................ 321 15. Involutory Quartic Transformations in Space of Four Dimensions, by Nina Alderton................- - —.- ---------------------- 345 16. On the Indeterminate Cubic Equation x3 + Dy3 + D2z3 -3 Dxyz -1, by Clyde Wolfe......................................- 359 17. A Study and Classification of Ruled Quartic Surfaces by Means of a Point-to-Line Transformation, by Bing Chin Wong............. --- — 371 18. A Special Quartic Curve, by Elsie Jeannette McFarland... —......... ----- 389 19. A Study of Cubic Surfaces by Means of Involutory Cubic Space Transformations, by John Frederick Pobanz............. —... ----... -.. —.. 401 20. The Hyperspace Generalization of the Lines on the Cubic Surface, by Daniel Victor Steed......... --- —----------------------------------------- - ------------ 425

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 1, pp. 1-26 June 13, 1912 ON NUMBERS WHICH CONTAIN NO FACTORS OF THE FORM p(kp +l) BY HENRY W. STAGER "If pa is the highest power of a prime p which divides the order of a group G, the subgroups of G of order pa form a single conjugate set and their number is congruent to unity, mod p." The above well-known theorem, which was first established by Sylow,t suggests the segregation of numbers into two distinct classes: the one class to contain all numbers which have factors of the form p(kp + 1) where p may be any prime except unity and k may have any integral value greater than zero; the other class to contain all remaining numbers. Numbers of the second class will be found to have certain properties analogous to the properties of primes and will be denoted by P, numbers of the first class will be found to have certain properties analogous to the properties of composite numbers and will be denoted by C. If a C should be of the form p(kp + 1), and not merely a multiple of a number of that form, we will call it a "fundamental" C. Certain C's are fundamental for certain p's and not for others; e.g., 12 —2{2(1.2+ 1)} -3(1.3 + 1), and 12 is therefore fundamental for p = 3, but not for p - 2; whether a given C is to be considered a fundamental C or not will be made clear by the context. Two fundamental C's will be called "different" if the values of the p's and k's respectively are not both the same in the two C's. The present investigation is intended to develop some of the more fundamental properties of the P's and C's and to establish certain formulae for the number of P's within a given limit. COROLLARIES FROM THE DEFINITIONS It follows directly from the definitions above, that: 1. Any number which is a positive integral power, including the first power, of a prime p is a P. * Dissertation in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of California. t Sylow, Theor&emes sur les groupes de substitutions. Math. Ann. (1872). For the particular statement of the theorem used above, compare Burnside, The Theory of Groups of Finite Order, ~ 78 et seq.

Page 2 2 University of California Publications in Mathematics [VOL. 1 2. Every even number not a positive integral power of 2 is a C. Hence the only even P's are the positive integral powers of 2. 3. Every number whose last two digits are 05 or 55 is a C. Such numbers are evidently of the form 5(5n -- 1). THEOREM Every number which is divisible by any power of 3 and contains at least two other factors not multiples of 3 is a C. Let N- 3"paq......n n l, a — P +...... 2. Every odd prime is of the form, 3m -- 1, or 3m1 + 2. Hence, N is a C, if any prime is 2, or of the form 3m + 1. Also, since (3m1 + 2) (3m 2) 3m + 1, the theorem follows in any case. It is evident that in the expression a +1+...... 2 all the terms, except one, may be zero, provided that term is not less than 2. THEOREM The product of two consecutive odd primes cannot be a C. Suppose p1p, -lp(kp+ 1), where p, and p2 are consecutive odd primes and p2 > p1 > 2. Then, since we have only two factors, p1 and p,, and P2 > p1, 1 1, and P — Pl. Hence p1p2 - 1(kp1 + 1), and p2 kPl- + 1. But, since p, and p. are both odd, k must be even. Therefore A =- 2npl + 1, or p2, 2p, + 1. Now, by Bertrand's Postulate,* between x and 2x - 2, (x > 7/), there is at least one prime, whence p2 is not consecutive to pl, as was assumed. If pi 3, P2- 5, we have pIP2 15, which is not a C, and the theorem holds for all odd primes. *This theorem was first conjectured by Bertrand (Journal de l'Ecole Polyt., cah. 30) and afterwards proved by Tch6bicheff in 1850 in his Memoire sur les nombres premiers presented to the Academy of St. Petersburg and reprinted in Liouville, vol. XVII (1852).

Page 3 1912] Stager: Numbers with No Factors of the Form p(kp + 1) 3 COROLLARY It follows directly from the proof of the theorem that the product of two odd primes, p1 and p2, so related that p. < 2p1, cannot be a C. For, by the theorem p2 - 2p, + 1, or p. > 2p1, which is contrary to hypothesis. Further, the necessary and suflicient condition that the product of two odd primes, p1 and P2, pI > p1, should be a C is that p, 21np1 - 1. THEOREM No rational integral algebraic polynomial can represent only numbers which are P's. For, suppose f(x) - aoX'~ + aix-1 +...a.. -4-i + an, should be such a polynomial, then for some value of x, say t, the formula would represent a P. Whence f(:) _ ao"++-t + -...... +a-i + anP. But f(t + p2) - ao( + p2) )"+ al( + {p2)-.. * * * *.. + an-l(( + p2) + an, MP2 + P- P(MP + 1). The right member is a C whether P be a prime or a composite number. THEOREM If x and y are both odd or both even numbers, provided they are not both equal to 1 or 2K, then xn 4- y' is a C, for n > 1. Since any odd number is of the form (2k -t 1), it is only necessary to show that the sum contains at least one odd and one even factor to prove the theorem. 1~ Consider S = x' + y", where x and y are odd and x X y = 1. (a). Let n= 2v+ 1. Then = X2v-^ y2 + I 1, (X + _y)(X2V _X 2-ly+........ + y2). Now, the first factor is the sum of two odd numbers and therefore is even, and the second factor is odd since it has an odd number of odd terms. Therefore the sum S is a C.

Page 4 4 University of California Publications in Mathematics [VOL. 1 (b). Let n 2v. Then S- x2v + y2, - (2x' + 1)2v + (2y' + 1)2v. If we expand the parentheses, there will be (2v + 1) terms in each expansion, of which the last term is unity and all the others contain the factor 22. Therefore, we may write S 22X+ 1- 22Y+- 1, -2a2(X+ Y) + 1t. Hence S is a C. It is necessary to exclude the case where x and y are both unity, for in that case s = 1+1' — 2 and S is a P. 2~ Next consider S- x" + y", where x and y are both even and x z y = 2K (a). Let x = 2axf and y - 29y', where x' and y' are both odd and / > a. Then S 2an(x' + 2(9 - a)"y). The first factor is even and the second is odd. Therefore S is a C. (b). Let x - 2ax and y- 2ayf, where x' and y' are both odd. Then = -2an (xf" + y'/). But the second factor has already been proved a C by Case 1, therefore S is a C. The case excluded by the theorem, x - y 2K, gives S - Kn + 2Kn - 2Kn + 1 and S is a P in this case. In connection with this theorem, it is to be noted that if x is odd and y is even, then x"' + y" may be either a P or a C. Two illustrations will serve to prove this statement. If x=11, y= 4, thenx2+y2=-121 +16=137= P. x 17, y 20, then x2 + y2 = 289 + 400 689 - 13(4 13 + 1) C. THEOREM If p, q are odd primes, and a, P are positive integers, paqg is a C, provided r (q) a > -—, where 8 is the greatest common factor of ' (q) and indp with respect to q.

Page 5 1912] Stager: Numbers with No Factors of the Form p(kp. + 1) 5 Let N =paq, then must paq pa'q -' q(kq+ 1) i, where a' < a, whence pa-a' -kq + 1, and pa-a' -1 (mod q). But, by Fermat's theorem, p0(p) - 1 (mod ), pa-a' p*(q) (mod q), whence (a-a') indp - (mod q-1), or (a-a' ) —O (mod 1 ), where (q-1, ind p) =8. Hence, it follows that the least value of (a-a'), and consequently of a, is q-1, a quantity independent of the particular primitive root selected. Having discussed some of the fundamental properties of the P's and C's, we will now take up the problem of deducing certain formulae of enumeration. It is very evident from the nature of the C's that, if we attempt to carry out the analogy between P's and prime numbers, C's and composite numbers, a new interpretation must be placed upon many of the ordinary operations with numbers, in particular those operations which involve the processes of multiplication. This is due to the fact that the very important theorem, that a number may be resolved into prime factors in only one way, does not apply when we consider fundamental C's as our fundamental factors; thus 105- 5(4.5 + 1) 7(2.7 + 1). While these two divisors are entirely different C's in the sense of our definition, they are each equal to 105 and their product in the ordinary sense is equal to 1052. It seems most convenient for many purposes to define the product of two or more different C's as their least common multiple and to designate the operation by the usual symbols of multiplication; the powers of a C will have the ordinary interpretation. In case of division, in which divisor or dividend or both are products of C's, all multiplications must be performed first; thus, if C1 2(1.2 +1) and C2 3(1-3 + 1), C1C2 12, C1C2 = 12 1 2, and not, CC2 = C2 12. The operation of cancellation of C1 6 C1 C1C2 C's before multiplying will be indicated by the symbol-; thus, C2. C1 The divisors of a product of C's will consist, unless otherwise stated, of unity, the number itself, and any product of the C's therein contained.

Page 6 6 University of California Publications in Mathematics [VOL. 1 THEOREM The number of P's not greater than n is obtained by product ( C-)1 - )........ ((1- C1, (1 C~21,)....... C,1 ( C-,2... t (1 -- Ci) (1 - C,2')......... in which Ci,k-pi(kpi + 1), and then forming the sum part of each term with its proper sign. / expanding the double (1- ) C.Ki) (\ C2, K2) (1- C,) by taking the integral It is evident, in the first place, that for a given p and k, the number of integers not greater than n which contain the factor p,(jp1 + 1), where p, is a given prime, is given by Pl (jP pi+) 'r where [x] has its usual interpretation, the greatest integer in x. If we define 4p(n, j, pi) as the number of integers not greater than n, which do not contain the factor p, (jpl + 1), where p, and j have given values, then Ip(n, j, pl)-n -- pl(jpl 1) Now, let p remain fixed, but let k have a sequence of integral values, 1, 2, 3... Many numbers may contain two or more factors of the form p(kp + 1), as for instance, 48 4[3(1 3 + 1)1 3(5 3 + 1), and, moreover, the largest factor cannot exceed [n]. It will, therefore, be necessary to find the largest value of k for a given prime p and a given number n. It is evident that this maximum value of k will be obtained when, if possible, the factor is exactly equal to n, or n - pi(kpl+ 1), = kpl2 + pi, whence D -- 2 Pi Therefore, the maximum value of k for a given p and n will be given by L[ _,_

Page 7 1912] Stager: Numbers with No Factors of the Form p(kp - 1) 7 4p (n, p1) will now be defined as the number of integers not greater than n which contain no factors of the form p (kpi + 1), where p, is a given prime and k = 1 to n 2Pl, and therefore p (n, pl) will necessarily consist of the sum [n - Pi pl2 k — 1 plus some function which will provide for such cases as have two or more different factors of the given form. Let ~-i =-p1(k p, + 1)= l2p1(k2pl +1), where n1>l, k k2. Then 11 (klpI + 1)- 12(k2pI + 1). And since k1 _ k2, the two sets 11, k1p, + 1 and 12, k2p1 - 1 have factors common to each other. If 8 is the least common multiple of (k;1p + 1) and (k2p- + 1), any number of the form mp18 will contain factors of the two forms p,(kp1 + 1) and p1(k2plA- 1). A similar result holds for any number of factors of the form p(kp + 1), and also for the case in which both the p's and k's differ. We have already defined the product of two or more different C's as their least common multiple. If then we use the notation C1,k- pl (kp1 + 1), the sequence C],1, C1,2, *.* *. CI,K will correspond exactly with the sequence pI(p1 + 1), p,(2p, + 1),...... p (Kp + 1) where k and consequently the second subscript k of the C's has the values,,2. [n -pl It follows then from analogy with the manner of forming the function Oc(n; p, q, r) of ordinary primes that n n1 [L C1,1C1,2 J Product of different C's two at a time J _rn n — - [ C1,1C1,2C1,3 + [Product of different C's three at a time] -- etc.

Page 8 8 University of California Publications in Mathematics [VOL. 1 We may write the formula in the neater form p(n, pl) =- n] (- )(1 C )......(-C if we interpret this form to mean that we shall first expand the product and then take the sum of the integral parts of all the terms with their proper signs. Thus, rp(50, 2) = 50 - 27 + 9 - 2 - 30, which is easily verified as the number of integers not greater than 50 which contain no factors of the form 2(2k + 1). We will next define Op( a, X) as the number of integers not greater than n which contain no factors of the form pi (kp.i + 1), where pi is any one of the first first X primes, p -- 2, p.2 3,..... pi...... p, with a limit depending on n, and k runs independently for each prime, being limited as already shown. To find the limit of p for any n we assume k- 1, since this will give us the maximum value of p, such that p(kp + 1) _ [n]. Then p (p+1) n, p2 + p n, (p + )2<n + l, < /4n + -1, 2 whence the maximum value of p will be that of the largest prime not greater than /4n + - 1 -1. If, then, we consider the double sequence of C's, L 2 Ci, - pi(kpi + 1), where pi is any one of the first X primes, pi -2, p2 3,.... pi... p 1 < 4n + 1 —1 and where runs independently for each p from 1 to - p, Px= 2 wee 1 we can write down the desired formula at once, in that it must have the same general form as that for fp(nl, p,). Therefore, p(n, x)- -n { [c,] + A I[> [Product of different C's two at a timej [roduct of different C's three at a time] + etc.

Page 9 1912] Stager: Numbers with No Factors of the Form p(kp + 1) 9 Or, as before, we may write (i )()....... (i- ) ^1,1/ 1,2/ C1,Kj |(C-()1- C ) - C2,K2?(,1) (i2) I ' ' * (I)CiKi ( Cx l )I( CX. (1 C if we interpret this form to mean that we are to expand the double product and then form the sum by taking the integral part of each term with its proper sign. If p. has the maximum value of the limit already found, the formula will give the number of P's not greater than n. We will now define an important function / (N), analogous to Merten's function.* Let N- C1 C...... Cn be the product of any number of fundamental C's, not necessarily different, then we will define /, (N) as follows: c (N) 0, if two or more of the C's are equal. /u(N) = + 1, if all the C's are different and their number is even, also if N 1. /E(N) --- 1, if all the C's are different and their number is odd. LEMMA c (D) = 0, if the sum is extended over all divisors D (1 and N included) of a number, N, which is a product of fundamental C's. Let us assume N' - C1 C...... Cn, a product of n fundamental C's, then the total number of their products with an even number of factors (among this number we include the case of zero factors) is equal to the total number of their products with an odd number of factors. For the number of products with an even number of factors is n (n-l)) n (n- ) (n- 2) (n- 3) 1 2 1-2 3-4........... and with an odd number of factors is n n (n - 1) (n -2 ) 1 1.2.3 Their difference is zero, being equal to (1 - 1)". It follows, then, that in the expanded product (1 — C)(-......(1 — C,,) the number of positive terms is equal to the number of negative terms. * See Crelle, vol. 77, p. 289.

Page 10 10 University of California Publications in Mathematics [VOL. 1 Whence /Ic(D) =0. when this sum is extended over all the divisors D, a product of C's, of N'. Finally, let N C1IaC2a...... Cn Then all the divisors of N' are divisors of N, while all the remaining divisors of N contain at least the second power of a C, for which tc(D) - 0. It follows, therefore, that C Ic(D) =0, if the sum is extended over all divisors D (1 and N included) of N. THEOREM The number of P's not greater than x is given by the formula PHc(D)[ D where D is a divisor of II, the product of all possible fundamental C's not greater than x. Let f(n) and F(n) be two "zahlentheoretische Functionen"* which are only different from zero, when n is a divisor of a given number I, the product of any number of different fundamental C's. Thus, assume f(D)= F(KD) for every divisor D of II, the sum being formed for all divisors K of, where D all numbers are subject to our definitions of products, divisors, etc., of C's. Then we may form the sum Z /c(D)f(D)= - li(D)F(KD). D K-D The left member is to be summed for The right member is a double sum all divisors D of II. which is to be extended over all divisors D of I and also all divisors K of D Now KD will become equal to every divisor of I and if the terms of the double sum are collected in which KD is equal to the same divisor A of H, then the entire double sum becomes E GAF(A), where G. =aZ/ n(D), and is summed for all I A divisors of a. But, by the lemma just proved, GA 0 for all values of A, except ^- 1, when Ga - 1. It follows, then, that 2 c c(D)f(D) E G aF(A) =F(1). D A * Compare Bachmann, Zahlentheorie, Part II, p. 308, for the definition of this term.

Page 11 1912] Stager: Numbers with No Factors of the Form p(kp + 1) 11 Now, let x be any positive quantity and let II C1C...... C, be the product of n different fundamental C's. We may arrange the integers 1, 2, 3...... [x] in groups such that each group contains all the integers which have the same greatest common factor* with II, and no integer will appear in more than one group. Defining t (x, II, K) as the number of integers not greater than x which have with rI the greatest common factor, K, as already defined, we shall have [x] = ' (x, n, K), K the summation being taken over all divisors K of II. Let D be a given divisor II of II and K a divisor of -, then KD is also a divisor of I. But all the numbers D ' not greater than x which have the greatest common factor KDf with I will be found among the following rllnmbers: 1.D, 2.D, 3.D...... - D; and there will be just as many such numbers as there are integers not greater x TKDI H than - which have the greatest common factor, with. It follows that D D D x n KD (x, n, KD) ( D ' D ' D But r v 11 KD \ [D K D' D' D H where K is a divisor of D Whence [Dr ]x,- InH, KD), K where K is a divisor of-. * In determining the greatest common factor of two numbers, either or both of which are C's or products of C's, the greatest common factor may be any rational function of the C's, provided that in numerical calculations it is integral and a divisor in the ordinary sense of the quantities involved. In the particular case of any number n and II, the greatest common factor may be only a divisor of n as already defined, and, moreover, since two or more divisors of H, while different as products of C's, may be numerically equal, the greatest common factor may be only some one of the numerically distinct divisors of I. To illustrate this paragraph, let II = C1C2C3C4 6. 10. 12.18. Then the greatest common factor of any number n and II may be only some one of the numbers 1, 6, 10, 12, 18, 30, 36, 60, 90, 180; thus, the greatest common factor of 50 and I is 10. Again of the numbers 1, 2, 3, 6, only 3 has C1C3C4 6'12'18 36 -.3. _. = 3 as greatest common factor with C2C4 = 10 18 = 90. C1C3 6'12 12 t We only consider the distinct divisors KD. does not necessarily equal K,

Page 12 12 University of California Publications in Mathematics [VOL. 1 We have here an equation between two "zahlentheoretische Functionen" of the type already defined and so we can apply the results obtained. Hence it follows that OXI, I, 1) E /c(D) [ D, D D where the sum is extended over all divisors D of II, a product of C's, i.e., the number of integers not greater than x which contain no one of the C's which form the product II is given by E He(D)[ ]. D D If, now, we take II as the product of all C's, IICik -1 j piMpi + l)t i,k pi,k where pi is any one of of the first X primes, p, 2, P2= 3.... pi..... p^, and k-1 to x — JPi running independently for each prime, then L pi J ]p(XX)= e (D) [, D D where D is a divisor of II. Finally, if px is the largest prime not greater than '4 — 212 and the k's are determined as before, then pp(x,A) will denote the number of P's not greater than x. Therefore, the number of P's not greater than x is given by the formula (D) D)[]i, where D is a divisor of H, the product of all C's not greater than x. As an illustration of the method, let us find (x, ^n, 1), where x-50 and IH = 2(2 -t 1) 2(2.2 + 1)} )3(3 + 1) {2(4-2 + 1)/ - 6-10-12-18. ~(50, II, 1) = Z Lc(D)D- ], where D is a divisor of II=6 10.12 18, F 50 50 F 550] 50 r (All the other terms deL1I L6 1o L '10o 30tJ stroyed each other.) =50 - 8- 5+ 1- 38. Whence 38 of the first 50 integers are not divisible by 6, 10, 12, or 18.

Page 13 1912] Stager: Numbers nwith No Factors of the Form p(kp + 1) 13 COMPUTATION FORMULAE FOR fp(V, K, p1) AND sbp(n, A) While the formulae already obtained furnish the solution of the problem in question, the actual computations necessary to obtain numerical results when n increases soon becomes too extended and complicated for practical use. Therefore in the succeeding paragraphs we will obtain formulae which lend themselves more readily to numerical computation. In accordance with our previous definitions, we will define 4p(nK, p1) as the number of integers not greater than n which contain no factor of the form p, (kpl + 1) where p1 is a given prime and k — 1, 2, 3..... K. If we form the sequence of natural numbers in order from 1 to [n] and then exclude all those numbers which are divisible by factors of the form pl(kp, -+ 1) where k - 1, 2, 3...... (K- ), the number of integers remaining will be equal to Op((. K — 1, p,). It is still necessary to exclude the numbers which are divisible by p1 (Kp- + 1), which are the following: pl(Kpl + 1), 2pl( Kpl + 1 ).pl ( K pl p l 1+) However, we have already excluded all the numbers of this sequence which are divisible by other factors of the given form, so that for our purpose it is only necessary to consider those numbers of this sequence which are divisible by no other factors of the given form than p1(Kp, + 1). We may obtain this number by determining how many integers of the sequence (Ki+ ),2(Kp+l ), 3(Kpi+ 1) 3(Kp 1). l ](K p+ 1) are divisible by any factor of the form (kpl + 1), where k = 1, 2...... (K - 1), a result which is readily obtainable by a sieve process. For convenience we will define (P{A(Ki (+l ), (K- l),p 1 as the number of terms of the arithmetical sequence, whose common difference is (kpi +1) and the number of whose terms is v=L -(- 1) which have no pi(Kpi + 1) ' factors of the form (Kpi + 1), where k=1, 2, 3....... (K- 1). Then we shall have KPp(n, p) =, K-, p) - p(, K-, pP)- pIA(Kpi + 1 ), (K p),. 1 If we put ~~K-then- P then (p(n, K, pl)=-p(n, pi) =- p(n, K - 1, pi) -4MpA(Kpl + 1), (K-1), pi}. 1

Page 14 14 U-niversity of California Publications in Mathematics As an illustration, we will apply the method to determine Op(100, 3) = 81, Now PA 100, 3, 10, 9.'. Op(100, 3) =P(100, 10, 3). [VOL. 1 'p(100, (Ap( 100, 4P( 100, 4p(100, 4]p(100, <Ap(loo,,p(100, 1 10, 3) =p(100, 9, 3) -S, A(31), 9, 3} =82-1-81. 1 9, 3) (=p(100, 8, 3) —~pieA(28), 8, 3 }-82-0-82. 1 8, 3) =-p(100, 7, 3) —'p{A(25), 7, 3(25), 7, 3 83-1 82. 1 7,) = 1, 3) - (22100, 6, 3)p (22), 6, 3 84 —183. 1 6, 3)-,p(100, 5, 3) —, pIA(19), 5, 3/ =85 -1=84. 1 6 5, 3)=-, (100, 4, 3) — p}AA(16), 4, 3/ —85-0-85. 1 2 4, 3) = (100, 3, 3) - p1A(13), 3, 3-=87-2=85. 1 3 3, 3) =-p(100, 2, 3) — p{A(10), 2, 3 -89 —2=87. 1 4 2, 3)=<P(100, 1, 3) —bp!A (7), 1, 3 =92-3=89. 1 Op(100, 1, 3) =92. The numerical calculations must he made in reverse order, beginning with the last line instead of with the first. Interpreting fp(l, A) as already defined, we will proceed to find a formula for it in exactly the same manner as employed in the previous case. If we exclude from the sequence of integers, 1, 2..... [n] all those integers which are divisible by factors of the form p (kp --- 1), where p is any one of the first (A -1) primes and k is determined in the usual way, there will be left just 4p(n, - 1) numbers. The integers divisible by px(ikp -- 1) are the following: Px(P+- 1), 2Px(Px +1).. [.... P)]{ P + } Px(P-(P x+1 ), pX(Kp + 1), 2p,(KPX+ 1).. LP(Kp+. 1)] Px(KPx+) }. From this double sequence of numbers, by a sieve process, we can exclude all

Page 15 1912] Stager: Numbers with No Factors of the Form p(kp + 1) 15 those numbers which contain factors of the form p(kp + 1), p being any one of the first (X - 1) primes. We will then define v-PVk k =K A Px(kPX + 1), X-1 v —1 k-1 J n^-px1 r n where px is the Xth prime, k -=, 2 *.2, and v=1, 2 [ ( 1 + ) ]~ determined independently for each value of k, as the number of integers of the double sequence which contain no factors of the given form, where p is any one of the first (A - 1) primes. We may then write our formula, which is analogous to Meissel's computation formula for primes,* b(n, X) -p(, X — 1)_ A Px(Px + 1), X- -1. I v=l = 1 - If Px is the largest prime not greater than '1/-4n-1,-1 then (p( n, X) will give the number of P's not greater than n. As an illustration, we will calculate the number of P's not greater than 100. In this case Px L1/400+1 — 1] 9, i.e., - 7andX=4. v=vl k=1 ) '. (100, 4)= —p(100, 3) — P I A 7(7 + 1), 3 -=49 -0 49. v=1 k —1 v= -V k=3 4p(100, 3) =p(100, 2) -4p A 5(kj +1), 2 =50-1-49. v= 1 k= 1 v=1vo k-=10 p(100, l 2)= p(100,=ll 1) _-Ppl Ak1 3(kj3 + 1), I -56 -6= 50. ' '^ k==1 } v = 1 k=1 Mp(100, 1) = 56. The numerical calculations were made in reverse order, beginning with the last line instead of with the first. It is to be noted that _ n+ 5 Flog"n (n, 1) Ln-2- + L log2]' since sp(n, 1) is the number of integers less than n which contain no factors of the form 2(2n + 1) and, therefore, consists of all the odd integers not greater than ni plus the powers of 2 which are not greater than n. Therefore, the number of P's not greater than 100 is 49, a result which is easily verified. While the enumeration of the P's within a given limit, either by computation or by an asymptotic formula, is a matter of much theoretical interest, in many * Meissel: Math. Anialen II and III.

Page 16 16 University of California Publications in Mathematics [VOL. 1 cases, especially in applications to the theory of groups, to know whether a given number is a P or a C is most important. This information will necessarily be best supplied by a table showing for each integer all its factors of the form p(kp + 1). The tabulation of the P's could be made mechanically by a process similar to that of Eratosthenes's sieve, which consists in writing down the integers in their natural order and then cutting out the successive primes, 2, 3, 5......; but, as in the construction of factor-tables, various devices are available which permit of simpler and more ready computation.* Such a table may be constructed in the following manner. Arrange the numbers by tens, as in a table of common logarithms of numbers, on a sheet of paper sufficiently long to contain the entire list and ruled in rows and columns. In the proper rectangle write the prime factors of each number, the p's of the C's. It then remains only to write in the various values of k for its corresponding p and the table is complete. If the paper has been accurately ruled, the k's may be written in very readily by measurement, for a given C repeats itself every 5C or 10C lines in the same column, according as the C is an even or odd number. Then, in order, take each C of the double sequence of C's, Ci, 7.- pi(kpi 4- 1) where the p's and k's are limited as in the preceding paragraphs. Compute its first appearance in each column, and finally enter it by measurement in the remainder of the column, writing the value of k after its corresponding prime. Many devices for checking the results will suggest themselves. On page 17 we give a sample of such a table.t * For an account of these devices, see the introduction to Professor Lehmer's Factor Table for the First Ten Millions. t For further details, the reader is referred to the writer's A Sylow Factor Table of the First Twelve Thousand Numbers which will be published by the Carnegie Institution of Washington and is now in press.

Page 17 1912] Stager: Numbers with No Factors of the Form p(kp + 1) 17 TABLE OF NUMBERS FROM 1 TO 99, SHOWING FACTORS OF THE FORM p(kp + 1) 0 1 2 3 4 5 6 7 8 9 lI _ 22-2 2 1 23 32 2 2 2 2 3 3 24 2 4 5 3 1 7 5 32 2- 2 2 5 23 5" 2 6 33 22 5 7 11 3 1 13! 7 2 7 25 3 2 8 5 22 2 9 3 4 3 3 I11 17 7 32 1 19 13 5 1 23 2 10 2' 32 2 11 24 72 5 3 2 11 5 23 3 5 1 2 12 3 22 2 13 52 23 36 2 14 52 17 13 33 11 7 1 19 29 22 2 15 26 5 2 22 3 3 1, 3 31 32 2 13 3 7 17 23 5 1 7 11 2 17 23 2 18 38 22 7 2 19 5 32 1 37 52 19 11 3 4 7 ______ _ 13 24 34 2 20 22 5 2 21 3 23 5 3 41 3 9 1,2 17 43 29 11 2_ _ ___7 I _ 2 22 7 22 3 10 2 23 5 25 2 24 32 32 3 13 23 31 47 19 3 1, 5 72 11 5 1 For each number in the above table the following data are given: 1. In the first column of each rectangle, the representation of the number as a product of powers of primes, thus 90 -2 * 32 5. 2. In the second column the value of k greater than zero for each prime greater than unity such that N= p(kp + 1), thus 90- 2(22 * 2 + 1), but there are no values of k greater than zero, for which 90 = 3(3k + 1) or 90= 5(5k + 1). 3. In the remaining columns, the values of k which give divisors of N of the form p(pk + 1), k > 0, p > 2; thus 90 is divisible by 3(3 3 + 1), also by 5(1.5 +1).

Page 18 18 University of California Publications in Mathematics [VOL. 1 Since there is a close analogy between ordinary primes and P's, a comparison between the number of primes and the number of P's within a given limit naturally suggests itself. In the following table (page 19), the number of primes and the number of P's for each century and for each thousand are listed in adjacent columns. The arrangement of the table is so simple that no explanation is necessary. In the appendix a list of all the P's less than 12,229 is given, so arranged that the number of P's between any two limits less than 12,229 may be easily obtained. The cut, facing the table, shows the number of P's and the number of primes by centuries plotted side by side. The abscissas give the century and the ordinates give the number of primes, or the number of P's, for that century. These two curves are approximate curves of frequency for the P's and primes.

Page 19 1912] Stager: Numbers with No Factors of the Form p(kp + 1) NUMBER OF PRIMES AND P'S FOR EACH CENTURY AND FOR EACH THOUSAND 19 0-3999 Century Primes* P'st Century Primes P's Century Primes P's Century Primes P's It 26 49 11 16 32 21 14 33 31 12 34 2 21 40 12 12 35 22 10 31 32 10 33 3 16 37 13 15 35 23 15 34 33 11 31 4 16 34 14 11 36 24 15 34 34 15 33 5 17 37 15 17 32 25 10 34 35 11 30 6 114 38 16 12 35 26 11 35 36 14 34 7 16 34 17 15 35 27 15 30 37 13 33 8 14 36 18 12 33 28 14 33 38 12 33 9 15 36 19 12 33 29 12 37 39 11 32 10 14 33 20 13 36 30 11 31 40 11 33 169 374 135 342 127 332 120 326 4000-7999 Century Primes P's Century Primes P's Century Primes P's Century Primes P's 41 15 33 51 12 33 61 12 29 71 9 33 42 9 34 52 11 31 62 11 36 72 10 34 43 16 35 53 10 28 63 13 33 73 11 34 44 9 31 54 10 35 64 15 30 74 9 33 45 11 33 55 13 30 65 8 35 75 11 32 46 12 33 56 13 32 66 11 30 76 15 24 47 12 32 57 12 32 67 10 31 77 12 32 48 12 31 58 10 31 68 12 31 78 10 33 49 8 33 59 16 33 69 12 33 79 10 31 50 15 33 60 7 33 70 13 31 80 10 34 119 328 114 318 117 319 107 320 8000-11999 Century Primes P's Century Primes P's Century Primes P's Century Primes P's 81 11 35 91 11 31 101 11 31 111 10 36 82 10 31 92 12 33 102 12 28 112 11 27 83 14 30 93 11 32 103 10 28 113 10 31 84 9 34 94 11 30 104 12 33 114 10 35 85 8 32 95 15 31 105 10 31 115 11 33 86 12 35 96 7 31 106 8 31 116 9 32 87 13 31 97 13 29 107 12 35 117 8 31 88 11 32 98 11 32 108 11 31 118 9 32 89 13 32 99 12 32 109 10 28 119 12 30 90 9 32 100 9 35 110 10 31 120 13 30 110 324 112 316 106 307 103 317 * The number of primes was obtained from similar tables in the introduction of Glaisher's Factor Table of the Sixth Million. t The number of P's was obtained from the list of P's given in the appendix. T The first century, 0-99, includes 1 as a prime and a P.

Page 20 20 University of California Publications in Mathematics [VOL. 1 ADDENDA A careful study of the curves of approximate frequency and the tables from which they were obtained, together with a study, as n increases, of the increase of the following four different types of P's; namely, (1) primes; (2) powers of primes; (3) products of two consecutive odd primes; (4) all other P's; brings out very clearly what appears to be an important relation between the number of P's and the number of primes within a given limit. Various methods of empirically noting the increase of the last three types of P's within the limits of the list of P's given in the appendix, especially of the logarithms of the prime factors involved, have brought out very interesting data, but so far it has been impossible to correlate these data into definite theorems. However, the data obtained and a study of the curve suggest the following theorem, no proof of which has been obtained: The number of primes and the number of P's within a given limit differ asymptotically by a constant.

Page 21 1912] Stager: Numbers with No Factors of the Form p(kp + 1) 21 APPENDIX LIST OF P'So In the following pages is given a list of P's in their order from 1 to 12,229 and so arranged for the ready computation of the number of P's between any two limits less than 12,229. The columns are numbered consecutively from 1 to the last column of the list in the first line at the top of the pages, and the lines are numbered from 1 to 50 in the first column on the left-hand side of each page. The last three digits of the P's are given in the columns, the hundredth's digit only being given for the first P in the hundred, the remaining digits are to be found at the top of the column; i.e., in the second line at the top of the page. In case the entry for any P is in heavy type, preceded by an asterisk, the other digits for the rest of the P's in that column are to be found at the top of the next column to the right. Thus the P at the intersection of column 35 and row 43 is 5127. * The P's for this list were obtained from the writer's A Sylow Factor Table of the First Twelve Thousand Numbers, already referred to (p. 16). I am indebted to Professor D. N. Lehmer for the method of listing the P's, which is identical with the arrangement to be used by him in a List of Prime Numbers from One to Ten Millions, of which he kindly showed me the manuscript.

Page 22 22 University of California Publications in Mathematics [VOL. 1 LIST OF P's, 1-2257 I 2 3 4 5 6 7 8 9 10 I1 12 13 14 15 16 1 1 1 1 1 1 1 2 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 1 1 2 3 4 5 7 8 9 11 13 15 16 17 19 23 25 27 29 31 32 33 35 37 41 43 45 47 21 22 23 24 25 26 27 103 235 371 511 643 787 929 079 07 39 73 12 47 89 31 87 09 41 77 15 49 93 33 91 13 43 79 17 53 97 37 93 15 45 83 19 59 99 41 97 19 47 89 21 61 801 43 99 21 49 91 23 65 03 47 103 23 51 93 27 67 07 49 07 25 56 95 29 71 09 51 09 27 57 97 31 73 11 53 11 28 59 401 33 77 15 59 15 31 61 03 35 79 17 61 17 33 63 07 37 81 21 63 21 35 65 09 39 83 23 65 23 37 67 11 41 85 27 67 27 39 69 13 45 91 29 71 29 41 71 15 47 95 33 73 33 43 77 19 51 97 35 77 35 45 81 21 53 99 39 83 39 49 83 23 57 701 41 85 41 51 87 25 59 03 43 89 45 53 89 27 63 07 45 91 47 57 93 31 65 09 47 95 49 59 95 33 69 13 51 97 51 61 97 37 71 17 53 *001 53 63 99 39 73 19 57 03 57 67 303 43 75 21 59 07 59 69 07 45 77 25 63 09 63 73 11 47 81 27 65 13 65 75 13 49 83 29 69 17 67 77 17 51 87 31 71 19 69 79 19 57 89 33 75 21 71 81 21 59 91 39 77 24 75 85 23 61 93 43 79 31 77 87 25 63 99 45 81 33 79 91 29 67 601 47 83 37 81 93 31 69 07 49 87 39 87 97 35 73 11 51 91 41 89 99 37 75 13 53 93 43 93!07 39 77 17 57 95 49 95 09 41 79 19 61 99 51 99 11 43 81 21 63 901 57 201 13 47 85 23 67 07 59 03 15 49 87 25 69 09 61 07 17 53 91 29 71 11 63 11 21 59 93 31 73 13 67 13 23 61 99 35 79 17 69 17 27 65 501 37 81 19 73 19 29 67 03 39 83 23 75 23 33 69 09 41 85 25 77 25 229 367 517 31 69 19 33 73 23 35 77 27 37 81 29 41 83 31 43 85 35 47 87 37 49 91 41 53 93 43 57 97 47 59 99 49 61 401 53 67 03 57 69 09 59 71 11 61 73 15 63 77 17 65 79 23 67 83 27 71 85 29 73 89 31 77 91 33 79 93 37 83 95 39 85 97 41 89 301 45 91 03 47 93 07 51 97 09 53 601 13 57 03 15 59 07 19 65 09 21 69 11 25 71 13 27 73 15 29 75 19 663 813 957 67 17 59 69 19 61 71 23 63 75 25 69 79 29 73 81 31 75 85 35 77 87 37 79 89 41 81 91 43 85 93 47 87 97 49 91 99 51 93 707 53 97 09 59 99 15 61 *003 17 63 09 19 65 11 21 67 17 23 71 21 27 73 23 29 77 27 33 79 29 35 83 31 39 85 33 41 89 39 45 91 43 47 95 45 53 97 47 57 901 48 59 03 49 61 07 51 63 09 53 65 13 57 69 15 59 73 17 63 77 19 69 79 21 71 81 23 75 83 27 77 87 31 81 89 33 83 93 37 87 95 39 89 97 41 93 99 43 95 801 45 97 07 49 99 11 51 101 103 11 13 17 19 23 25 29 31 37 41 43 47 49 51 53 57 59 61 65 67 71 73 77 79 83 87 91 95 97 201 03 07 09 13 15 19 21 25 27 29 31 37 39 41 43 45 49 51 57 I 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 49 51 53 59 61 64 65 67 69 71 73 77 79 2 81 83 85 87 89 91 95 97 99 101 31 33 37 39 41 43 45 47 49 51 57 61 63 81 21 83 25 87 27 89 31 93 33.95 37 99 39 501 43 03 45 07 49 09 51 11 57 13 61 i

Page 23 1912] Stager: Numbers with No Factors of the Form p(kp + 1) 23 LIST OF P'S, 2259-4687 I — I 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 259 407 63 11 67 13 69 17 73 19 79 21 81 23 83 25 85 27 87 29 91 35 93 37 97 41 99 43 303 47 09 49 11 53 13 59 15 61 17 63 19 65 21 67 23 71 27 73 33 77 35 79 39 81 41 83 45 89 47 91 49 95 51 97 53 501 57 03 63 07 67 09 69 13 71 15 75 17 77 19 81 21 83 27 87 29 89 31 91 33 93 37 95 39 99 43 401 45 03 49 551 713 8 57 19 61 23 63 25 67 27 69 29 71 31 73 33 75 35 79 37 81 41 87 43 89 47 91 49 93 53 97 59 99 61 603 65 09 67 11 71 15 73 17 77 21 79 23 85 27 87 29 89 33 91 37 95 41 97 43 99 45 801 47 03 51 07 57 09 59 13 357 011 151 59 13 61 61 17 63 63 19 67 67 23 69 69 29 73 73 31 75 75 35 77 79 37 79 81 39 81 85 41 83 87 43 87 89 47 91 91 49 93 93 51 99 97 53 203 99 57 09 )03 59 11 09 61 15 11 65 17 13 67 21 17 71 23 21 73 27 23 77 29 27 79 31 29 83 33 31 85 35 33 89 39 35 91 47 39 93 51 41 95 53 49 97 57 51 101 59 53 03 63 57 07 65 59 09 69 63 13 71 65 15 73 69 19 77 71 21 81 77 23 83 81 25 87 83. 27 91 87 31 93 89 33 95 317 473 19 75 21 81 23 87 25 89 27 91 29 93 31 97 35 99 37 501 41 03 43 09 47 11 49 15 53 17 623 25 31 35 37 777 929 79 31 81 35 85 37 87 41 085 43 91 43 47 93 47 49 97 49 51 99 53 53 803 57 59 07 59 61 09 61 65 11 67 67 15 71 69 17 73 87 29 85 91 31 87 93 37 91 96 41 93 97 43 97 99 47 99 101 49 403 03 53 09 09 59 11 11 61 15 15 65 19 17 67 21 19 71 23 21 73 27 27 77 29 29 79 33 31 81 35 33 83 39 35 85 41 39 89 43 41 91 47 45 93 51 47 95 53 49 97 57 51 99 59 53 303 61 57 07 63 59 09 69 63 11 71 227 381 531 33 35 37 41 43 47 49 53 59 61 67 69 71 73 77 79 81 83 89 4 59 21 71 21 77 61 23 73 23 79 65 27 77 27 83 67 29 79 31 85 71 33 83 33 87 73 37 87 35 89 77 39 89 39 91 79 41 91 41 95 83 43 95 45 97 85 45 97 47 *001 89 47 99 49 03 91 51 701 51 07 95 53 03 53 09 97 57 07 59 13 101 59 09 63 19 07 61 13 65 21 09 63 15 67 23 13 69 19 69 27 15 71 21 71 31 19 77 25 77 33 21 79 27 79 37 25 81 33 81 39 27 83 37 87 41 31 87 39 89 43 33 89 43 93 45 91 95 97 601 03 07 19 21 25 27 I 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 61 63 69 71 75 77 81 83 87 89 93 99 701 07 11 15 19 21 23 25 27 31 33 37 39 39 93 43 95 47 99 49 601 53 07 57 09 61 11 63 13 67 17 69 19 41 93 37 99 43 95 39 301 45 99 45 07 51 *,001 47 09 53 07 49 13 45 99 49 901 51 03 57 07 61 11 63 17 67 19 69 21 71 23 73 25 49 51 57 61 67 73 75 79 81 83 69 13 71 15 75 17 77 21 81 25 83 27 87 31 89 33 93 37 95 39 99 43 201 49 03 53 07 57 11 61 13 63 17 69 19 73 23 75 25 79 75 31 77 33 79 37 81 39 83 43 87 45 89 49 93 51 97 57 99 59 501 61 07 63 09 67 11 71 13 73 17 77 19 79 23 81 27 85 29 87

Page 24 24 University of California Publications in Mathematics [VOL. 1 LIST OF' P's, 4689-7181 I I 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 7 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 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 50 689 841 997 91 43 99 93 47 *001 97 49 03 99 53 09 703 57 11 09 59 13 13 61 15 17 65 17 21 67 21 23 71 23 27 73 27 29 77 29 33 79 33 35 83 35 39 85 39 41 89 41 47 91 45 49 97 47 51 901 51 53 03 53 57 09 57 59 11 59 63 13 63 65 15. 65 69 19 67 71 21 69 77 25 71 79 27 77 81 31 81 83 33 87 87 37 89 89 39 91 93 43 95 95 45 99 99 49 101 801 51 07 03 57 11 11 61 13 13 63 19 17 67 21 19 69 23 21 73 27 25 75 29 29 79 31 153 321 471 57 23 73 61 27 77 63 29 79 65 33 83 67 37 85 71 39 89 73 41 91 77 45 97 79 47 501 83 51 03 89 53 07 91 57 09 95 59 15 97 61 19 99 63 21 203 65 27 07 69 31 09 71 33 13 75 39 15 77 41 21 81 43 27 83 45 31 87 49 33 89 51 37 91 53 39 93 57 43 95 61 45 99 63 49 401 67 51 07 69 57 11 73 61 13 75 63 17 79 67 19 81 69 23 85 73 29 87 79 31 89 81 33 91 83 35 97 87 37 99 91 41 603 93 43 09 97 47 11 303 49 13 09 53 15 11 59 17 15 61 21 17 65 23 19 69 27 629 783 939 31 85 41 33 91 47 39 93 51 41 801 53 45 03 57 47 07 59 51 09 63 53 13 65 57 15 69 59 21 71 63 23 75 67 25 77 69 27 81 75 31 83 77 33 87 81 37 89 83 39 91 87 43 93 89 45 99 93 47 *001 95 49 07 99 51 09 701 57 11 03 61 13 07 63 17 11 67 19 13 69 23 17 73 29 21 75 31 23 77 37 25 79 41 29 81 43 31 91 47 37 93 49 39 97 53 41 99 59 43 903 65 47 09 67 49 11 71 51 17 73 53 19 77 61 21 79 65 23 81 67 27 85 69 29 89 71 31 91 73 33 93 77 35 95 79 37 101 103 251 407 561 715 877 039 07 53 09 63 19 81 43 09 57 13 69 21 83 45 13 59 15 71 23 87 47 15 61 19 75 25 89 49 17 63 21 77 27 91 51 19 65 23 81 29 93 53 21 69 25 83 31 99 57 25 71 27 87 33 901 61 27 77 31 89 37 07 67 29 81 33 93 39 11 69 31 83 37 95 43 13 71 33 87 39 99 49 17 73 37 89 43 601 51 19 75 39 91 45 07 61 25 79 43 95 49 11 63 27 81 45 97 51 13 67 29 85 47 99 53 17 77 31 87 49 301 57 19 79 35 91 51 07 59 21 81 37 93 57 09 63 23 85 39 97 61 11 67 25 91 41 99 63 13 69 29 93 43 101 67 17 71 31 97 47 03 69 19 73 35 99 49 09 73 23 79 37 801 57 15 75 29 81 39 03 59 17 79 31 85 41 09 61 21 85 33 87 43 11 67 23 87 37 91 47 15 71 27 89 41 93 49 17 73 29 91 43 97 53 19 77 33 93 47 99 59 21 79 35 97 49 503 61 23 83 39 99 53 09 67 27 89 41 203 59 11 73 29 91 43 07 61 15 77 33 95 45 09 65 21 79 35 97 47 11 67 23 83 39 99 51 17 73 27 87 41 *001 53 21 79 29 89 45 03 57 27 83 33 91 47 07 59 29 85 35 95 49 09 63 33.87 39 97 51 13 65 35 89 41 701 57 17 69 39 91 47 03 59 19 71 41 95 51 07 63 27 73 43 97 53 09 65 31 77 45 401 57 11 69 33 79 47 03 59 13 71 37 81 31 33 35 37 39 81 37 85 41 87 43 91 47 93 49 I

Page 25 1912] Stagyer: Numbers with No Factors of the Form p(kp + 1) 25 L LIST OF P's, 7183-9697 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 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.; 183 331 '487 669 87 33 89 73 93 39 93 75 95 41 95 79 97 43 99 81 823 9 25 27 29 31 79 81 83 85 87 -4 99 45 501 85 35 89 201 49 07 87 37 91 07 51 17 91 41 93 09 61 19 93 49 99 11 63 23 97 53 *003 13 67 29 99 59 09 17 69 37 703 61 11 19 73 41 09 63 15 23 75 43 11 67 17 25 77 47 13 71 21 29 79 49 15 73 23 31 81 51 17 77 27 33 87 59 21 79 29 35 89 61 23 83 33 37 91 65 27 85 35 41 93 71 29 89 39 43 97 73 33 91 41 47 99 77 37 95 45 49 403 83 39 97 47 51 09 87 41 99 51 53 11 89 45 901 53 61 15 91 47 03 57 63 17 93 51 07 59 65 21 97 53 09 61 67 23 601 57 11 63 69 27 03 59 13 65 73 31 07 63 15 69 77 33 09 65 19 71 79 35 13 67 21 75 81 41 15 69 25 77 83 43 19 71 27 79 89 45 21 73 29 81 91 47 27 77 31 83 95 51 29 81 33 87 97 53 31 83 37 89 301 57 33 89 39 93 03 59 37 93 43 95 07 63 39 95 49 97 09 65 43 99 51 99 13 69 47 801 57 101 19 71 49 07 63 11 21 75 57 11 67 13 23 77 61 13 69 17 25 81 63, 17 71 19 27 83 67 19 73 23 129 287 437 31 91 41 33 93 43 35 97 47 37 99 53 43 303 57 47 09 59 49 11 61 53 15 65 59 17 67 61 21 69 67 27 71 71 29 73 73 31 77 77 33 79 79 35 83 81 39 85 83 41 89 85 45 91 87 47 95 89 51 97 91 53 501 92 57 03 97 59 07 99 61 09 201 63 11 03 67 13 07 69 19 09 71 21 13 75 23 19 77 27 21 81 29 23 83 31 25 87 37 27 89 39 31 93 43 33 95 45 37 97 49 43 99 51 45 401 57 49 03 59 57 07 61 59 11 63 61 13 67 63 17 69 67 19 73 69 23 75 73 25 77 79 29 79 85 31 81 583 743 899 061 87 47 903 65 91 49 07 67 93 53 09 69 97 59 11 71 99 61 15 73 603 65 17 77 09 67 21 79 11 71 23 83 15 73 27 85 17 77 29 89 21 79 33.91 23 81 35 95 27 83 39 97 29 89 41 101 33 91 45 03 37 93 47 07 39 95 51 09 41 97 53 13 47 803 59 15 51 07 63 17 53 09 69 19 57 13 71 23 59 17 75 25 63 19 77 27 65 21 81 31 67 25 83 33 69 31 87 37 71 33 89 43 75 37 93 49 77 39 95 51 81 43 97 53 89 47 99 57 91 49 *001 61 93 51 07 63 95 57 11 67 99 61 13 69 707 63 17 71 09 67 19 73 11 69 29 75 13 71 31 79 f7 73 33 81 19 79 35 87 25 81 37 91 27 85 41 93 31 87 43 97 35 89 47 99 37 91 49 203 39 93 53 09 41 97 59 11 17 77 21 79 23 83 27 85 29 89 33 91 35 95 39 97 41 401 45 03 47 07 49 09 51 11 53 13 215 373 533 35 39 47 49 51 53 57 59 63 65 69 71 73 75 57 19 77 59 21 83 63 25 87 67 27 89 69 31 93 71 33 99 77 37 601 79 39 07 81 41 09 83 45 13 87 51 17 89 57 19 93 61 23 99 63 27 301 67 29 07 69 31 11 73 37 13 75 43 19 79 47 23 81 49 25 87 59 29 91 61 35 93 63 37 97 65 41 99 67 43 501 71 47 03 73 49 09 77 53 11 79 57 17 83 59 21 85 65 23 89 67 27 93 69 29 95 71 31 97 I L I

Page 26 26 University of California Publications in Mathematics LVOL. 1 L LIST OF P'S, 9701-12229 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 9 9 10 10 10 10 10 10 10 11 11 11 11 11 11 12 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 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 50 701 865 007 03 69 09 07 71 13 13 73 15 15 75 19 19 77 21 21 81 27 25 83 31 27 87 33 31 89 37 33 93 39 37 95 41 39 97 43 43 99 49 45 901 51 49 07 57 51 11 61 53 13 63 57 17 67 61 19 69 63 23 73 67 25 77 69 27 79 71 29 81 73 31 85 81 35 87 85 37 91 87 41 93 91 43 97' 93 49 99 97 53 103 99 59 09 803 61 11 09 63 13 11 65 17 17 67 23 19 69 29 21 71 33 23 73 35 27 77 39 29 79 41 33 81 45 39 83 47 41 85 51 47 87 57 51 89 59 53 91 63 57 95 65 59 97 67 63 *001 69 171 345 77 47 81 49 83 51 87 57 89 59 93 61 95 63 201 67 07 69 11 75 13 79 17 81 19 83 21 87 23 91 25 93 29 97 31 99 35 401 37 03 39 09 43 11 47 15 49 23 53 27 59 29 61 33 65 35 67 39 71 41 73 45 77 47 79 51 89 53 91 57 301 59 03 63 07 67 09 69 13 71 15 73 19 75 21 77 27 81 31 83 33 87 37 89 41 95 43 99 501 657 813 981 03 61 07 63 11 67 13 69 17 71 19 73 23 79 29 81 31 83 37 85 41 87 43 91 47 93 49 97 53 99 59 703 61 09 65 11 67 15 71 17 73 21 77 23 79 27 81 29 83 33 85 35 89 37 95 39 97 41 99 43 601 45 07 47 09 51 11 53 13 57 17 63 19 65 21 71 25 77 27 79 29 81 31 83 33 89 37 93 39 95 43 99 45 801 49 07 51 11 17 85 19 87 23 91 25 93 27 97 31 *003 37 07 41 09 43 11 47 13 51 15 53 17 59 21 61 23 67 27 69 29 71 31 73 33 77 35 83 39 89 41 91 45 95 47 97 51 903 53 07 57 09 59 13 61 15 63 19 65 21 69 27 71 31 75 33 77 37 83 39 87 43 89 49 93 51 95 53 97 57 99 61 101 63 03 67 07 69 11 73 13 75 17 77 19 79 23 125 301 447 29 03 49 31 09 53 41 11 59 47 13 61 49 17 63 53 21 65 57 23 67 59 27 71 61 29 77 71 31 79 73 33 83 77 35 85 83 37 89 85 39 91 89 41 93 91 45 97 95 47 99 97 51 501 201 53 03 03 57 07 07 59 09 09 63 13 13 65 15 15 69 19 19 71 21 21 77 27 25 81 31 27 83 33 31 87 37 33 89 39 37 91 43 39 93 45 43 95 47 45 99 49 601 ' 761 929 079 03 63 33 83 09 69 39 85 11 71 41 89 17 73 45 91 21 77 47 23 79 51 29 83 53 33 85 57 35 87 59 37 89 61 39 91 65 41 97 67 43 801 69 47 07 71 51 09 75 53 13 81 57 15 83 59 19 87 63 21 89 97 101 03 07 09 13 15 19 21 23 25 27 31 33 37 39 43 45 47 49 51 57 61 63 67 65 71 75 77 81 83 87 89 93 95 99 701 07 09 17 27 93 31 95 33 99 37 '001 39 03 41 07 45 11 49 13 51 17 57 19 I 49 401 51 19 51 07 53 23 57 09 61 25 61 11 67 29 63 13 69 31 67 17 73 33 69 19 75 35 73 21 79 37 79 23 81 41 81 25 87 43 83 29 89 47 87 35 91 49 91 37 93 51 93 41 97 53 99 43 99 59 61 21 69 63 23 73 67 29 75 71 31 79 73 35 85 75 37 87 79 39 91 81 41 93 85 43 97 87 47 99 93 49 203 97 53 09 99 57 11 903 59 15 09 65 17 11 15 17 23 27 67 19 69 21 71 23 73 27 77 29 I =

Page [unnumbered] VITA I, Henry Walter Stager, was born in the year 1879 at Nutley, New Jersey, the son of John Willis and Bertha Stager. My elementary education was obtained in the public schools of Nutley and the neighboring town of Montclair. I was prepared for college at the Montclair High School and graduated from the Leland Stanford Junior University in 1902 with the degree of A.B. In 1905, I returned to Stanford for graduate study, pursuing work under Professors Allardice and G. A. Miller, and received the degree of A.M. in 1906. In September, 1907, I entered the University of California and continued graduate work under the direction of Professors Iaskell, Lehmer, MacDonald, and Schilling. I wish to express my appreciation for the kindly interest of all my instructors. My thanks are especially due to the members of my Committee, Professors Lehmer, Haskell, and Schilling; and more especially to Professor Lehmer, whose encouragement and sympathetic advice have been a constant source of help in preparing my dissertation.

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 2, pp. 27-54, 31 text-figures September 24, 1912 CONSTRUCTIVE THEORY OF THE UNICURSAL PLANE QUARTIC BY SYNTHETIC METHODS BY ANNIE DALE BIDDLE INTRODUCTION In the following discussion the unicursal quartic is regarded from two points of view. Chapter I treats of the curve in its correspondence to a conic section through a quadratic reciprocal transformation. This leads to an interesting classification of unicursal quartics and affords a convenient and ready method for determining the form of the curve. Incidentally, it brings to light a geometrical application of the well known "Group of Four." In Chapter II the curve is defined as the locus of intersection of corresponding rays of two projective pencils of the second order. This develops properties of the curve not readily obtained in the other treatment. The discussion shows that the two definitions are not independent, but that each is supplementary to the other. CHAPTER I THE UNICURSAL QUARTIC IN CORRESPONDENCE TO A CONIC SECTION 1. A Quadratic Reciprocal Transformation.-The following well known geometrical construction of the general quadratic reciprocal transformation is of fundamental importance for the study of the unieursal quartic. Consider in the plane any two conic sections a and a' and a point P. The polars p and p' of P with respect to a and a', respectively, will intersect in a point P'. The polars of P' must likewise intersect in P. To P corresponds P', and vice versa. We say that P and P' are conjugate points of the transformation. By correlating all such pairs of conjugate points an involutory transformation is established between the points of the plane. 2. Theorem.-As a point P describes a straight line, its conjugate point P' generates a conic section as the locus of the intersection of corresponding rays of two projective pencils of the first order. As P describes a straight line 1, the polars p and p' of P with respect to a and a' describe projective pencils of rays L and L' of the first order. 3. Theorem.-As a point P describes a conic its conjugate point P' generates a unicursal quartic, as the locus of the intersection of corresponding rays of two projective pencils of the second order.

Page 28 28 University of California Publications in Mathematics [VOL. 1 As P describes a conic y, its polars p and p' with respect to a and a', respectively, describe projective pencils of rays K and K' of the second order. To determine the degree of the locus of P', the conjugate point of P, that is, the locus of the point of intersection of corresponding rays of the two projective pencils K and K' of the second order, cut it with a straight line 1. To I corresponds a conic A. The points in which I intersects the locus of P' correspond to the points in which A intersects y, the locus of P. But A and y can intersect in at most four points. The locus of P' is, therefore, a quartic. In establishing a one-to-one correspondence between the points of the quartic and a conic section, we have shown that the quartic is unicursal. 4. Theoremn.-Any curve of degree n is transformed into a curve of degree 2n as the locus of the intersection of correspondings rays of two projective pencils each of order n. As a point P describes a curve of degree n, its polars p and p' with respect to a and a', respectively, generate projective pencils of rays each of order n. To determine the degree of the locus of the point of intersection of corresponding rays, that is, of the point P', conjugate to P, we proceed as before in ~ 3, cutting the locus of P' with a straight line 1. To I corresponds a conic X. The intersections of I and the locus of P' correspond to the intersections of A and the locus of P. But of these there can be at most 2n. 5. Theorem.-Any point on a side of the self-polar triangle of a and a' corresponds to the opposite vertex and vice versa. For if P lies on a side s1 of the triangle S 2 S, S, self-polar with respect to a and a', its polars p and p' must meet in Sr, the vertex opposite to s,. 6. Theorem.-A curve of degree n corresponds by the above transformation to a curve of degree 2n having the vertices of the self-polar triangle as n-fold points. For from ~ 5 it follows that to each intersection of the curve of degree nt described by P with a side of the self-polar triangle corresponds the opposite vertex as a point of the curve described by P'. In special cases the curve of P may pass through a vertex of the self-polar triangle. The curve of P' then degenerates, the opposite side of the triangle appearing as a part of the curve. In general, a curve of degree n which passes in all k times through the vertices of the self-polar triangle goes into a degenerate curve of degree 2n, which contains the sides of the triangle counted k times and a curve of degree 2n - k. In particular, a line 1 passing through a vertex of the triangle corresponds to a line 1' passing through the same vertex. 7. The Quartics Qa and Qa'.-To the conic a (or a') itself corresponds a quartic Qa (or Qa') which passes through the four intersections of a and a', and also through the four points of contact on a' (or a) of the tangents common to a and a'. These curves can be traced at once for any given conics a and a'. They are, therefore, of great assistance in the construction of any curve C', for to the intersections of C (the corresponding curve) and Qa (or Qa') correspond the intersections of C' and a (or a'). These points of C' on a (or a') are the points of contact on a (or a') of one of the tangents drawn to a (or a') from the intersections of C and Qa (or Qa').

Page 29 1912] Biddle: Constructive Theory of the Unicursal Plane Quartic 29 8. Theorem.;-Conjugate points P and P' of the transformation project to any vertex of the self-polar triangle in an involution of rays. This follows from ~ 6, where it was shown that to a line I passing through a vertex of the triangle corresponds a line 1' passing through the same vertex, and vice versa. From this we see that it is possible to regard P' as the point determined by the two rays which correspond to the rays in which a point P projects to two arbitrary involution centers, SX and S2. The line SX S5 will correspond to some line s2 at S, and to a line s1 at S2, sl s2 determining the point S3. S1 S2 S3 is then a singular triangle of which each vertex corresponds to the opposite side and vice versa. The quadratic involutory transformation can be completely worked out from this point of view.* 9. Theorem.-The double or focal rays of the involutions at the vertices of the self-polar triangle are in each case the two common chords of a and a' which intersect at that vertex. If this is true as P moves along c, a chord common to a and a', P' must also describe c. The four intersections of a and a' are the self-corresponding, or invariant, points of the transformation. c is then a line passing through a vertex of the self-polar triangle and through two invariant points, I1 and 12. It must then correspond to a line passing through the same vertex and through I, and 12, that is, to itself. If a and at intersect in four real points, the double or focal rays are all real, that is, the involutions at the three vertices of the self-polar triangle are all hyperbolic; if they intersect in four imaginary points, at two of the three real vertices the involution is elliptic, at the third it is hyperbolic; if they intersect in two real and two imaginary points, the involution at the one real vertex is hyperbolic. 2/ 4 / 6 Fig. 1 * See D. N. Lehmer, "On the Combination of Involutions,'" Am. Math. Mo., vol. 18, no. 3 (March, 1911).

Page 30 30 University of California Publications in Mathematics [VOL. 1 10. The Form of the Curve.-If we regard the plane as divided into seven regions by the self-polar or singular triangle, we can, knowing the character of the involutions at the vertices of the triangle, determine in what region a point P' must lie that is conjugate to any given point P. Number the regions 1, 2, 3, 4, 5, 6, 7, and denote the vertices A, B, C, as in figure 1. If the involutions are all hyperbolic, a ray projected from a point P in 1 corresponds at A to a ray passing through the regions 1, 7, and 4; at B to a ray passing through the regions 2, 4, and 1 or 5; at C to a ray passing through the regions 6, 4, and 1 or 3. Any two of these three rays are sufficient to determine P' as a point in 4 or 1. In the same manner we can locate P' for any other point P in the plane, whether the involutions are all hyperbolic, or two are elliptic. If the involution is hyperbolic at A, B, and C, we find that: P in 1 or 4 projects to a ray at A passing through 4, 7 and 1 at B passing through 4, 2 and 1 or 5 P' is, therefore, in 4 or 1 at C passing through 4, 6 and 1 or 3 P in 2 or 5 A-2, 6,3 or 5 1 B 2, 4,1 or 5 P' in 2 or 5 C -2, 7, and 5 J P in 3 or 6 A -6, 2, 5 or 3 B -6, 7, and3 P' in 6or3 C -6, 4,1 or 3 J P in 7 A-7,1,4 1 B 7,2,5 P' in 7 - 7,3, 6 In the same manner we locate P' for P in any region when the involution is hyperbolic at C while elliptic at A and B, or hyperbolic at A while elliptic at B and C, or hyperbolic at B while elliptic at C and A. The complete results may be tabulated as follows: Involution is hyperbolic at A B C C A 1 or 4 4 or 1 6 or 3 7 2 or 5 P lies in 2 or 7 P' lies in 2 or 5 7 6 or 3 4 or 1 3 or 6 6 or 3 4 or 1 2 or 5 7 7 7 2 or 5 4 or 1 6 or 3 This investigation shows that the regions 1 and 4 are interchangeable; likewise, 2 and 5, 3 and 6. Indeed, we see that one can pass from 1 to 4 without crossing a side of the triangle; similarly, from 2 to 5, and from 3 to 6. We may, therefore, regard 1 and 4 as one region, R1; 2 and 5 as one region, R2; and 3

Page 31 1912] Biddle: Constructive Theory of the Unicursal Plane Quartic 31 and 6 as one region, R3. 7 stands by itself as R4. The above table may then be written: I I I. I _ Involution is hyperbolic at R1 P lies in RB2 R3 R4 ABC C A B R, R, R, B2 R, R, B3 R,, BR, BR, R, B,, BR, R, The transformation of P indicated by the first column leaves R1, R2, R3, R4 all unaltered; that is, it is effected by the substitution (1) (2) (3) (4). The transformation of the second column is effected by the substitution (1, 3) (2, 4); that of the third column by the substitution (1, 4) (2, 3); and that of the fourth by the substitution (1, 2) (3, 4). These four substitutions (1) (2) (3) (4) (1, 3) (2, 4) (1, 4) (2, 3) (1, 2) (3, 4) constitute the well known "Group of Four." Any given curve C intersects the sides of the triangle in a definite order, thus establishing the order in which the corresponding curve C' must pass through the vertices. This enables us to determine for a given branch of the curve C how the corresponding branch of the curve C' must pass through the region in which it lies. We can, therefore, for a given curve C determine at once the form of the corresponding curve C'. Consider, for example, the quartic that will correspond to the conic C in figure 2, where the involution is hyperbolic at every vertex. Reading from right to left, C intersects the sides of the triangle in the order a b c a b c. Denote the portions of C which lie in 1, 2, 3, 4, 5, 6, by cl, c2, c3, c4, C5, and c6, respectively, and the corresponding portions of C', c'1, c'2, c'3, etc., Fig. 2

Page 32 32 University of California Publications in Mathematics [VOL. 1 respectively. ct lies in 1 and between b and c. c'l, then, must lie in 4 or 1 and between B and C. Moving continuously along C' we may pass from B to C directly so that c', lies entirely in 4, as in figure 1, or indirectly by way of infinity so that c'1 lies partly in 4 and partly in 1, as in figure 3. In the same way we determine c'2, c'3, c'4, c'5, and c'6, and, excluding the possibility of infinite branches of the kind shown in figure 3, we find that C' must for the conic C in figure 2 have the form given in figure 6. /y /Y Fig. 3 11. Classification of Unicursal Trinodal Quartics. —If the given curve C is a conic or a curve of degree greater than the second, the position of the curve C with reference to the triangle affords a basis for classification of the curves C', since the order in which the curve C intersects the sides of the triangle is the order in which C' must pass through the vertices. The unicursal quartic is of particular interest here. A conic section according to its relative position may intersect the sides of the triangle in five essentially different orders. These are: (1) (2) (3) (4) (5) aabbcc ab cabc ababcc aacbbc acabcb

Page 33 1912] Biddle: Constructive Theory of the Unicursal Plane Qlartic 33 We may describe the position of the conic by noting the successive regions through which it passes. Thus the position of the conic in figure 2 is given by noting that it passes successively through the regions 1, 2, 3, 4, 5, 6. The same order may be given by more than one position of the conic. Order (1) is given by four positions, namely, 4 7 6 7 2 7, 3 2 1 2 7 2, 7 4 5 4 3 4, 5 6 7 6 1 6; order (2) by one position, 6 5 4 3 2 1; order (3) by three positions, 476543, 6 5 4 7 6 1, 7 4 5 6 7 2; order (4) by four, 3 2 3 4 5 4, 6 5 6 1 2 1, 2 3 2 7 6 7, 7 4 7 2 1 2; and order (5) by one position, 2 3 4 7 6 1. Any particular order represents more than one class of unicursal quartics, since each of the four possible forms of the involution may give rise to a different kind of curve for a given position of the conic. An analysis of all cases shows that a change of the conic from one position to another representing the same order is equivalent to a possible change in the character of the involution. It follows that all the classes represented by one order may be obtained from any one position of the conic which produces that order by changing the character of the involution. It is not difficult by the method indicated above to ascertain for the conic in any given position and under any form of the involution the number of classes represented by each order and the form of the curve (excluding the possibility of infinite branches such as those shown in figure 3). The complete results may be conveniently arranged in the following table:

Page 34 34 University of California Publications in Mathematics [VOL. 1 No. of Involution Form of Order Classes Position Hyperbolic at Quartic a a b b c 2 476727 A, B and C Fig. 4 C, or A, or B Fig. 5 321272 C Fig. 4 A, or B, or A, B and C Fig. 5 745434 A Fig. 4 B, orC,orA, B and C Fig. 5 567616 B Fig. 4 C, or A, or A, B and C Fig. 5 abcabc 2 654321 A, B and C Fig. 6 A, orB, or C Fig. 7 a ab cc 3 745672 A, B and C Fig. 8 C Fig. 9 A, or B Fig. 10 476543 A Fig. 8 B Fig. 9 C, or A. B and C Fig. 10 654761 B Fig: 8 A Fig. 9 C, or A, B and C Fig. 10 a a c b b c 3 323454 A, B and C Fig. 11 C Fig. 12 A, or B Fig. 13 656121 A, B and C Fig. 11 C Fig. 12 A, or B Fig. 13 232767 A Fig. 11 B Fig. 12 C, or A, B and C Fig. 13 747212 B Fig. 11 A Fig. 12 C. or A, B and C Fia. 13 acabcb 2 234761 A, B andC,orC Fig.14 A, or B Fig. 15

Page 35 1912] Biddle: Constructive Theory of the Unicursal Plane Quartic 35 4/6 6 obcabc F/6.-4 abbccll\./7 // 13 67aacb/c /\\ I/\ i \ F/6/+ \ Z /F /7 //3 x 7<y/ h 1 c__ Figs. 4-15

Page 36 36 University of California Publications in Mathematics [VOL. 1 In all, twelve classes of unicursal trinodal quartics are obtained in this way. Further distinction may be made, according as the intersections of the conic with the sides of the triangle are real and distinct, real and coincident, or conjugateimaginary. Special forms of the self-polar triangle give rise, of course, to special kinds of curves. CHAPTER II THE UNICURSAL QUARTIC AS THE LOCUS OF THE INTERSECTION OF CORRESPONDING RAYS OF TWO PROJECTIVE PENCILS OF THE SECOND ORDER 1. The curve is defined as the locus of the points of intersection of corresponding rays of two projective pencils of the second order. 2. Theorem.-The locus described is a unicursal curve having, in general, three nodes, and intersecting any line in the plane in at most four points. See Chapter I, ~~ 3 and 6. 3. Notations.-One pencil of the second order (and also the conic enveloped by it) will be denoted by K; the other by K'. The rays of each pencil will be denoted by a, /3, y,.... and a', /3', y',.. respectively. The tangents to K and K' from the points of the quartic and other than a, /, 7,.... and a', /', y',.... will be denoted by a, b, c,.... and a', b', c',.... respectively. The quartic itself will be denoted by Q. 4. Theorem.-No point of Q lies within K or K' This is obvious from the definition of the curve given in ~ 1.* 5. Theorem.-The quartic Q touches each conic K and K' in at most four points. Take a point S on K for the center of a pencil of the first order perspective to K. This generates with K' a cubic with a double point or node at S.t The cubic can intersect K in at most four other points besides S. These are easily seen to be points of the locus Q. 6. Problem.-Given K and K' with three pairs of corresponding rays, to construct Q. (See figure 16.) Let the three pairs of rays be a, a'; /, /'; and y, 7'. The intersection of a and a' will be a point A on Q. Likewise /3 and /' determine B on Q and y and y', C on Q. Draw through A to K the other tangent a; also to K' the other tangent a'. Take a perspective to K and a' perspective to K'. a and a' are two pointrows in perspective position since they have a self-corresponding point, A. Their center of perspectivity may be found by joining any two pairs of corresponding points on a and a', such as (a, /) with (a', /') and (a, y) with (a', y'). Denote this center of perspectivity by SA. Then to find the point X of Q determined by any ray 4 of K, we first join the point (4, a) with sA. This line will intersect a' in the point (4, a'). The intersection of 4 and 4' is the desired point X. * Unless otherwise stated, the sections referred to are those of Chapter II. See Drasch, "Beitrag zur synthetisehen Theorie der ebenen Curven dritter Ordnung mit Doppelpunkt," Wiener Berichte, vol. 85 (1882), p. 534. Also see D. N. Lehmer, "Constructive Theory of the Unicursal Cubic by Synthetic Methods," Trans. Am. Math. Soc., vol. 3, p. 372.

Page 37 1912] Biddle: Constructive Theory of the Uniclursal Plasne Quartic 37 /76. /6. 1F/6 /71 Figs. 16 and 17

Page 38 38 University of California Publications in Mathematics [VOL. 1 7. Theorem.-Every point P of the quartic Q has its corresponding point Sp. The locus of all such points will be denoted by S. 8. Problem.-Given K and K' with one pair of corresponding rays and the corresponding point of X, to construct Q. Let the given pair of corresponding rays be a, a'. The intersection of a and a' will be the point A of Q. As in ~ 6, draw a and a', taking them perspective to K and K', respectively. Using SA, which is given, as the center of perspectivity, the ray of K' corresponding to any ray of K, or vice versa, may at once be found, as before. 9. Theorem.-The tangents from SA to K meet a' in points of Q. Likewise, the tangents from SA to K' meet a in points of Q. (See figure 17.) This is seen to be true by drawing the rays of K' which correspond to the tangents from SA to K; similarly, those of K which correspond to the tangents from SA to K'. 10. Problem.-To find the fourth point of Q upon a or a'. (It is the point X in figure 17.) To do this we consider a as a ray 4 of K. 4 meets a in the point of contact of a and K. Join this point with EA. The line so obtained intersects a' in the point (4', a'). The intersection of 4' with 4 (or a) is the desired point. Instead of the point-rows a and a' and their center of perspectivity, SA, we may make use of any other such set, as b, b' and SB. In this case a, considered as a ray 4 of K, appears as an ordinary ray and 4' is found in the usual manner indicated in ~ 6. The fourth point of Q upon a' is found in precisely the same manner as the fourth point of Q upon a. 11. We are now in a position to state the following: Theorem.-The tangents fronm A to K meet a' in points of Q and a in points of E. Likewise, the tangents from SA to K' meet a in points of Q and a' in points of S. The point of S on a (or a') determined by one tangent from SA to K (or K') corresponds to the point of Q on a' (or a) in which a' (or a) is met by the other tangent from SA to K (or K'). (See figure 17.) The first part of the theorem was proved in ~ 9. Denote one tangent to K from -A by 8 and the other one by e. Then it follows from ~ 9 that (8, a') is D and (e, a') is E. d' and e' coincide with a'. In finding SD, draw the line (a, d- a' d'). This is a itself. Therefore, SD lies on a. Similarly, we show that SE lies on a. Instead of SA start with ED, drawing the tangents from ED to K. One of these is a. The other must meet d', which is a', in a point of Q. Obviously, it cannot meet d' in A or D. Nor can it meet d' in the point X found in ~ 10, since for X, a' is 4' and not x'. It must then meet d' in E and, therefore, the second tangent from SD to K is e. But e passes through SA. That is, SA and SD lie on e. In like manner, starting with SE, it may be shown that SE and SA lie on 8. Thus one tangent 8 from SA to K meets a' in a point D of Q and a in the point SE, while the other tangent e from SA to K meets a' in the point E of Q and a in SD. In precisely the same manner the theorem is proved for the tangents from SA to K' instead of K.

Page 39 1912] Biddle: Constructive Theory of the Unicursal Plane Quartic 39 L i-g/ /9.] N Figs. 18 and 19

Page 40 40 University of California Publications in Mathematics [VOL. 1 12. Prollem.-Given K and K' with one pair of corresponding rays, and the corresponding point of X, to find the point of Q corresponding to any particular point of E. Let the given pair of rays be / /', and the corresponding point of S, SB. Since f3 and /3' are known, b and b' are known. Let the particular point of S be SA. To find A, draw the tangents from SA to K and K', calling them 8, e, 0', and i'. Using NB as a center of perspectivity of the point rows b and b', construct 8', e', 0, and I, thus obtaining D, E, H, and I. D and E determine a' and H and I determine a. (a, a') is A. 13. Theorem.-The locus of the points 8, the centers of perspectivity corresponding to the points of Q, is a conic section from which Q is generated by means of a quadratic reciprocal transformation. (See figures 18 and 19.) A quartic generated from a conic by means of the quadratic reciprocal transformation discussed in Chapter I had its nodes at the vertices of the singular triangle, the involution centers. Consider the four tangents from a node of Q, two to K and two to K'. (See figure 18.) They are two pairs of corresponding rays of the pencils K and K'. Call them Iu, a', and v, v'. One pair determines the node as M, considered as lying on one branch of the curve, the other determines it as N, considered as lying on the other branch. Using I, I', and v, v', find SA for A. This is done by marking the intersection of the line (a, u- a', Iu) with the line (a, v - a', v'). The points A,y A; (a, ), (a', V'); (a, v), (a', t~')' are three pairs of opposite vertices of a complete quadrilateral and therefore project to any point of the plane in an involution of rays. They project to the node MN in three pairs of rays, two of which are I, v', and ft', v. But these lines are fixed and do not depend upon A or EA. Now consider the involutions at any two of the three nodes of Q. The lines from any given point 5A of S to the nodes correspond in involution to rays which intersect in the point A of Q corresponding to SA. As SA moves on X, A describes the quartic Q. Thus S is exhibited as that curve from which the quartic is generated by the quadratic reciprocal transformation and is, therefore, a conic section. (Chapter I, ~~ 3 and 8.) 14. Theorem.-The points of intersection of Q and S lie on the tangents common to K and K'. (See figure 20.) Let a common tangent of K and K' be a ray / in the pencil K. It corresponds to a ray /3' of K', in general distinct from /. b' then coincides with /. The tangents from SB to K meet 3 in points of E and b' in points of Q. But / and b' coincide. 15. Theorem.-K (or K/) and S are two conics such that a triangle circutmscribed about K (or K') is inscribed in S. This follows from ~ 11, the tangents there denoted a, 8, and e forming such a triangle, of which the vertices are SE, SA, and SD. (See figures 17 and 21, triangles SA ED SE and SA SF >G.) For the invariant relation connecting two conics related as in this theorem see Salmon, Conic Sections, ~ 376.

Page 41 1912] Biddle: Constructive Theory of the Unicursal Plane Quartic 41 I F/G. z A 1'11~ \ QY 6F/G. 2/. Figs. 20 and 21

Page 42 42 University of California Publications in Mathematics [VOL. 1 16. Theorem.-WThen SA lies on K, a' is tangent to Q and a is tangent to S; similarly, when SA lies on K', a is tangent to Q and a' is tangent to S. (See figure 22.) In this case the two tangents from iA to K, 8 and e, coincide. Consequently, a' meets Q in two coincident points, D and E, and a meets S in two coincident points SE and ED. Similarly, for SA on K'. Differently stated, the latter part of this theorem tells us: The tangents common to K (or K') and Y are obtained by drawing the tangents to K (or K') from those points of S where the tangents to K (or K') at the intersections of K (or K') and - again intersect E. Thus K (or K') and S have as many common tangents as intersections. 17. Theorem.-If K (or K') and S do not intersect K (or K') lies wholly within Y. (See figures 23 and 24.) If K (or K') and S do not intersect, either > lies wholly within K (or K') or K (or K') lies wholly within E. Otherwise common tangents could be drawn. But. cannot lie wholly within K (or K') since a triangle circumscribed about K (or K') is inscribed in E. Therefore, if K (or K') and S do not intersect, K (or K') lies wholly within S. 18.-Theorem.-a and a', a' and a determine an involution of rays at A in which A sA corresponds to the tangent to Q at A. (See figures 24 and 25.) Let D and E be the points of Q on a determined by the tangents from SA to K. K' and F and G the points of Q on a' determined by the tangents from SA to K. Consider, then, the three pairs of points A and SA, E and SE, F and SF. They are obviously not the three pairs of opposite vertices of a complete quadrilateral, since A, SA, E, and SE determine a quadrilateral of which D and SD are the third pair of opposite vertices; and A, SA, F and SF determine a quadrilateral of which G and SG are the third pair of opposite vertices. The locus of points from which the three given pairs of points are seen in involution is a general plane cubic C, passing through all the six given points and through D, SD, G and:SG. From ~ 13 we know that the nodes of Q also satisfy the condition for points of C. The cubic C is tangent to Q at A and to S at SA. In the quadratic reciprocal transformation a point A of Q goes into SA of S. The tangent line to Q at A goes into a conic through the three involution centers, the nodes of Q (see Chapter I, ~ 6), and tangent to S at SA. A point of the cubic C passing through the three involution centers goes into the conjugate point of C.t A and SA are conjugate points of the cubic C. The tangent line to C at A corresponds to a conic through the three involution centers and tangent to C at EA. If the tangent lines to C and to Q at A coincide, the corresponding conies must coincide; that is, if C is tangent to Q at A it must be tangent to E at EA. The cubic C is tangent to E at EA. Construct the tangent to S at SA by means of Pascal's theorem, using the points SA, D,, EF, G, G, and numbering them 16, 2, 3, 4, 5, respectively. (See figure 25.) * See Schroeter, Ebene Curven Dritter Ordnlung, Leipzig, 1888; ~ 1, 6; ~ 2, 1-4. t See D. N. Lehmer, "On the Combination of Involutions," Am. Math. Mo., vol. 18, no. 3 (March, 1911).

Page 43 1912] Biddle: Constructive Theory of the Unicursal Plane Quarlic 43 F/C LEE _' _sl _, _ W KF/6 LE3 Figs. 22 and 23

Page 44 44 University of Californtia PublicatioJns i}& Mathematics [voL. 1 The tangent line to the cubic C at SA is that ray which corresponds to the ray SA A in the involution determined at SA by two pairs of conjugate rays, as: (>A E) and (>A SE); (A F) and (>A F).To obtain this tangent line to C at:A, construct a complete quadrilateral, two of whose pairs of opposite vertices will project to:A in (SA E) and (>A E); (SA F) and (SA SF); and of the third pair, one vertex in (SA A). (See figure 25.) The other vertex must lie on the ray (SA SA), the tangent to C at EA. Such a complete quadrilateral is determined by the four lines: a, a', (SE:F), and the Pascal line found in constructing the tangent to E at SA. But the vertex which projects to SA giving the tangent to C at:A is the intersection of the Pascal line and the line (3, 4) in the construction of the tangent to S at YA; that is to say, the point which with:A gave (6, 1), the tangent to S at SA. The tangent to E at SA and the tangent to C at SA are, therefore, one and the same line; that is, the cubic C is tangent to S at:A. It is, therefore, tangent to Q at A. Hence, to construct the tangent to Q at A, we need only to construct the tangent to C at A. This is done, as in the case of the tangent to C at SA, by constructing the ray which corresponds to the ray A SA in the involution determined at A by two pairs of conjugate rays, as: (A E) = a, and (A:E) -- a'; (A F) - a' and (A SF) = a. This may be done in the following manner. (See figures 24 and 25.) On the line A, A select any point other than A. Through it draw two lines I and I'. Mark the intersections of I with a and a' and those of 1' with a and a'. The intersection of the lines (la -l'a) and (la' - 'a') must lie on the tangent to Q at A. 19. The Cubic C.-For every point of Q there is a cubic C such as that described in ~ 18. It may be denoted by CA, CB, etc., the subscript indicating the point of Q at which the cubic is tangent. We observe that we have at once all the intersections of CA, both with the quartic Q and the conic S. The three nodes of Q account for six intersections of Q and CA, the tangency at A for two more, and the remaining four are at the points denoted by D, E, F, and G in ~ 18. CA is tangent to E at MA and intersects it in the four remaining points SD, >E, SF, and cG. Incidentally, the existence of the cubics C gives a method of determining the nodes of Q since through them all the cubics C must pass. 20. Theorem.-The class of Q is 6. A tangent line to S from a node of Q, that is, from an involution center Sx, corresponds to a tangent to Q from S,. Since from any point in the plane only two tangents may be drawn to a conic section, only two tangents may be drawn from S1 to S and, hence, only two from S, to Q. There are two tangents to Q at S,, each of which counts as two tangents to Q from a point of the plane not on Q. In all, then, at most four tangents may be drawn to Q from a node S,, or at most six from an ordinary point of the plane. Consistent with this is the number of tangents common to K (or K').and Q. * See Schroeter, op. cit.; ~ 2, 6.

Page 45 1912] Biddle: Constructive Theory of the Unicursal Plane Quartic 45.0, 0.010000 "'~O Figs. 24 and 25

Page 46 46 University of California Publications in Mathematics [VOL. 1 From ~ 16 we know that every intersection of K' (or K) and S yields a tangent common to K (or K') and Q. Since E and K' (or K) can intersect in at most four points, this gives rise to only four tangents common to K (or K') and Q. The remaining eight lie at the four points where Q may touch K (or K'). (See ~ 5.) 21. Theorem.-The six tangents drawn from the three nodes to the quartic are tangent to one and the same conic. (See figure 26, conic I.) The lines in question correspond by involution to the six tangents from the three nodes to the Y conic. Denote the nodes by S, 82, and S3; the tangents from them to E by 1, 2; 3, 4; and 5, 6, respectively; and the lines corresponding to 1, 2, 3, 4, 5, 6, by 1', 2', 3', 4', 5', 6', respectively. Since the lines 1, 2, 3, 4, 5, 6 are tangent to a conic, the lines (1, 2 -4, 5), (2, 3 -5, 6), (3, 4 -6, 1) are concurrent. (Brianchon.) But these lines are (81-4, 5), (2 -5, 6), (S3-6, 1). To them correspond (8- 4', 5'), (S,-5', 6'), (S- 6', 1'), respectively. Since the first three are concurrent the second three must be. But the second three are (1', 2'-4', 5'), (2', 3' -5', 6'), (3', 4' 6', 1'), and therefore 1', 2', 3', 4', 5', 6' are tangent to one and the same conic section. (Brianchon.) 22. Theorem.-The tangents to Q at a node correspond in involution to the lines joining the node with those points of, which lie on the line joining the other two nodes. For from Chapter I, ~ 5, it follows that EM and EN, corresponding to the node M N, lie on the side of the singular triangle opposite to M N, that is, on the line joining the other two nodes. M YM corresponds to M M, the tangent to Q at M, and N EN corresponds to N N, the tangent to Q at N. 23. Theorem.-The six tangent lines to Q at the three nodes are tangent to a conic section. (See figure 26, conic II.) As before in ~ 21, denote the nodes of Q by 8, 82, and 83. From Brianchon's theorem we know that if the six lines in question circumscribe a conic section the three lines joining the opposite vertices of the hexagon they from all pass through one and the same point. Number the two tangents to Q at S8, 1 and 2; those at S2, 3 and 4; and those at S3, 5 and 6. The lines (1, 2 -4, 5), (2, 3-5, 6), (3, 4-6,1) join opposite vertices of the hexagon. In the involutions at S, 82, and S3 there correspond to 1, 2, 3, 4, 5, 6, lines which we shall call 1', 2', 3', 4', 5', and 6', respectively. If the lines 1, 2, 3, 4, 5, 6 are tangent to a conic section, the lines 1', 2', 3', 4', 5', 6' are also tangent to a conic section, and vice versa. For the point (4, 5) goes into the point (4', 5') and the line

Page 47 1912] Biddle: Constructive Theory of the Unicursal Plane Quartic 47 Figs. 26-28

Page 48 48 University of California Publications in Mathematics [VOL. 1 (S8 -4,5) corresponds to (S -4', 5'). Similarly, (S2 6, 1) corresponds to (S 6', 1') and (83-2, 3) to (3- 2', 3'). The intersection of any two of the three, (s- 4, 5), (2 -6, 1), (83-2, 3), as (S -4, 5) and (2- 6, 1), corresponds to the intersection of the corresponding two, as (S1 4', 5') and (82-6', '). It follows that if the three lines (8S-4',5'), (S2-6',1'), (S3- 2',3'), that is, the lines (1', 2' —4', 5'), (3', 4'-6', '), (5', 6'- 2', 3'), are concurrent, the three corresponding lines ( - 4, 5), ( - 6, 1), ( - 2, 3), that is, the lines (1, 2-4, 5), (3, 4-6, 1), (5, 6-2, 3) are also concurrent, and vice versa. If the lines (1', 2'- 4', 5'), (3', 4'- 6', 1'), (5', 6'-2', 3') are concurrent, the lines 1', 2', 3', 4', 5', 6' are tangent to a conic section. In that case, the lines 1, 2, 3, 4, 5, 6 are also tangent to a conic section. 1', 2', 3', 4', 5', 6' are lines joining the vertices of a triangle, each with the points in which the opposite side intersects a given conic section. (See ~ 22.) Any six such lines circumbscribe a conic section. (See figure 27.) Mark the six intersections of the three lines (1', 2',- 4', 5'), (2', 3'-5', 6'), (3', 4'- 6', 1') with the given conic, numbering them in any order (1), (2), (3), (4), (5), (6); say, so that (1) and (4) lie on the line (1', 2'-4', 5'),.(2) and (5) on the line (3', 4'-6', 1'), and (3) and (6) on the line (2', 3'- 5', 6'). These six points form an inscribed hexagon whose opposite sides must, therefore, intersect in three collinear points. (Pascal.) The three collinear points are: {(1) (2) - (4) (5)}, {(2) (3) - (5) (6)}, {(3) (4) - (6) (1)}. Now make use of the following notation:* (1) A { (2) - {(2)(3) (6)(1)} - C (4) -A', (5)=B'1 {(5)(6) - (3)(4)} =C', (3) -A2 (4) -B2 {(2)(3) - (4) (5)}- C (6) A'2 (1) - B'2 { (5) (6) -(1) (2) } C'2 (5) - A, (6) =B {(4) (5) - (6) (1)} = C (2) -A'3 (3) = B'3 {(1)(2)-(3)(4)} C'3 * Care must be taken to have corresponding vertices lie on the lines (1', 2' - 4', 5'), (2', 3' -5', 6'), and (3', 4'- 6', 1').

Page 49 1912] Biddle: Constructive Theory of the Unicursal Plane Quartic 49 A1 B, C7 and A'1 B'1 C'1 are two triangles so situated that their corresponding sides intersect in three collinear points. Therefore, the lines joining corresponding vertices, that is, the lines A, A' =- (1', 2' - 4', 5'), B B' -(3', 4' - 6', 1') and C, C'\ are concurrent. (Desargues.) For a similar reason the lines A2 A'2= (2', 3'- 5', 6'), B '2 - (1', 2'- 4', 5'), and C2 C'2 are concurrent; likewise the lines A3 A'33 (3', 4'- 6', 1'), B B'3 (2', 3'- 5', 6'), and C3 C'3. But the triangles C1 C. C3 and C'1 C'2 C' also fulfill the condition for Desargues' theorem, and therefore the lines C1 C'1, C2 C'2, and C3 C'3 are concurrent. Accordingly, the lines (1', 2' —4', 5'), (2', 3'-5', 6'), and (3', 4'- 6', 1') are concurrent and the lines 1', 2', 3', 4', 5', 6' circumscribe a conic section. But this is the condition that the lines 1, 2, 3, 4, 5, 6 should also circumscribe a conic section. 24. Theorem.-If Q is a tricuspidal quartic the three cuspidal tangents meet in one and the same point. (See figure 28.) To a tricuspidal quartic corresponds a conic tangent to each of the three sides of the triangle S1 S2 S3. If the three points of contact on the sides sl, S2, s3 be denoted by L, M, and N, respectively, then to the lines S, L, S2 M, and S3 N correspond the three cuspidal tangents. But Si L, 82 M, and S3 N are three concurrent lines (Brianchon), and therefore the three corresponding lines, the three cuspidal tangents, are concurrent. 25.-Theorem.-If Q has three bitangents, their points of intersection may be joined with the three nodes, one with each node, in such a way that the three joining lines pass through one and the same point. (See figure 29.) If Q has four bitangents their points of intersection in sets of three may be joined with the nodes in such a way that the joining lines form a complete quadrangle of which each pair of opposite sides intersects in a node. (See figure 30.) Each bitangent of Q corresponds by the quadratic involutory transformation to a conic passing through the three nodes, S1, S2, S3, and having double contact with the S conic. Denote the four bitangents to Q by 1, 2, 3, 4; their points of contact on Q by A, B; C, D; E, F; and G, H, respectively; and their corresponding conics through Sl, S9, S3, by I, II, II, and Iv, respectively. Denote the fourth intersection of i and ii by U; of ii and In by V; of in and i by W; of I and iv by X; of ii and iv by Y; and of im and iv by Z. Consider first the case of three bitangents, 1, 2, 3. Then the lines SA SB? SC SD2 and one side of the triangle S1 S2 S3, say 82 S3, and S1 U all pass through one and the same point and form a harmonic pencil.* Similarly, for the lines * See Salmon, Conic Sections, ~ 263.

Page 50 50 University of California Publications in Mathematics [VOL. 1 / J6 30 1 F/6.Zg. Figs. 29-31

Page 51 1912] Biddle: Constructive Theory of the Unicursal Plane Quartic 51 Sc SD, SE SF, S3 S1, and S2 V; likewise, for the lines SE SF, SA SB, S1 S2, and S3 W. From this it follows that the lines S1 U, S2 V, and S3 W are concurrent. For on S, W the lines SA YB, SC SD, S2 S3, and S1 U determine four harmonic points. Likewise SE SF,, C 1D, S S3, and S2 V determine four harmonic points on S3 W. But on S3 W, Zc E D in each case determines the same point; S, S3 and S S3 both determine S3; YA SB and E SF determine the same point since the lines YE SF, SA SB, S1 S2, and S3 W are concurrent. Therefore, S, U and S2 V determine the same point on S3 W; that is,,S U, S2 V, and S3 I are concurrent. But if these three lines are concurrent, the lines to which they correspond in involution must be concurrent. That is, the lines (S- 1,2), (5 —2, 3), (3 -3, 1) pass through one and the same point. Consider now the case of four bitangents to Q, 1, 2, 3, 4. Proceeding as before, we find the following sets of harmonic lines: For I, II, and IIII and II II and II IIn and I S2-S3 S3 S, S1 S2 S1 U S2 V S, W SA SB Sc ED:E.F XC SD SE SF SA SB For i, IT, and ivI and II II and iv Iv and I 2S S3 S, S2 S, S3 S, U S3 Y S2 X SA SB E:c SD:G:H EC ED SG SH SA SB For i, III, and IvI and III iII and Iv I and iv S, S2 S2 S3 S, S, S3 W S, Z S2 X SE SF GE EF SG sH SA SB SG SH SA SB For II, III, and-IvII and III iII and Iv iv and II S3 S, S2 S3 S1 S2 S, V S, Z S3 Y Sc ESD >E SF Sc SD >E SF SG SH 1G SH

Page 52 52 University of California Publications in Mathematics [VOL. 1 From this we obtain the following four sets of three concurrent lines: s1 U s1 U sz sZ 1 S2 V S2X S2 X S2 Vj S3 W S3 Y S3 W S Y The configuration determined by these six lines is a complete quadrangle of which each pair of opposite sides intersects in a node of Q. Each set of three concurrent lines, one of which passes through each involution center, corresponds in involution to another set of three concurrent lines, one of which passes through each involution center. Therefore, the figure corresponding in involution to a complete quadrangle of which each pair of opposite sides intersects in an involution center is another complete quadrangle of the same character; and the lines S1 U, S2 V, S3 W, S1 Z, S2 X, 83 Y correspond in involution to lines joining the points of intersection of the bitangents of Q to the nodes. 26. Degenerate Cases.-In general, the pencils K and K' have no self-corresponding rays. That is to say, a tangent common to K and K', considered as a ray of K, corresponds in general to some other ray of K', and considered as a ray of K', to some other ray of K. But it may happen that of the three pairs of corresponding rays of K and K given to construct Q (~ 6), one, two, or all three may be self-corresponding. Also, in general,. the conies K and K' are distinct. But the two pencils of the second order may envelope the same base conic; and in that case, too, there may or may not be self-corresponding rays. In all these special cases both Q and S assume degenerate forms. A. If K and K' have one self-corresponding ray, a a', Q consists of a cubic C and the ray a a'; 2 consists of a line a and the ray a a'; points of Q on a a' correspond to points of Y on a and points of Q on C to points of E on a a'. (See figure 31.) Since a and a' coincide, the four tangents from >A to K and K' meet a a' in four points of S, showing at once that all the points of a a' are points of X, since more than two of the points of a a' are points of S. That is, E degenerates into two straight lines, one of which is a a'. If from points of S as D, YEE, F, etc., on a a' we draw tangents to K and K', such tangents must meet the lines d', d, e', e, f', f, etc., in points of Q. But since always a a' itself will be a tangent from such points both to K and K', it follows that half of the points so obtained will lie on a a'. a a' is then a part of Q. The remaining part is, of course, a cubic C. The construction shows at once the correspondence between the different parts of Q and E. B. If K and K' have two self-corresponding rays, a a' and t /', Q consists of the two lines a a'. and f /P' and a conic A; S consists of the two lines a a' and /, /'; points of Q on a a' correspond to points of S on / / ' and points of Q on,3 /' to points of E on a a', while the points of Q on the conic X all have their corresponding points of E at the intersection of a a' and f/ /'. C. If K and K' have three self-corresponding rays, a a', t /3', and y y', Q consists of the four tangents common to K and K'; S consists of the three vertices of the triangle formed by a a', t /3', and y y'. Points of Q on one of the three lines

Page 53 1912] Biddle: Constructive Theory of the Unicursal Plane Quartic 53 a a', /f /P', or y y' correspond to the opposite vertex of the triangle formed by those lines. Points of S corresponding to points of Q on the fourth tangent common to K and K are indeterminate. To prove that the fourth tangent common to K and K' is a part of Q, find the point P on a for which a is a ray 7r of K. p and p' must coincide. Suppose they do not. p, considered as a ray p of K, determines a point R of Q, and p', considered as a ray e' of K', determines a point X of Q. None of the points P, R, and X can lie on any of the rays a a', /3 /3', y y', since every point of Q on a selfcorresponding ray is determined by that ray. Since Q is of the fourth degree, P, R, and X must lie in one and the same straight line. This can happen only if p and p' coincide. D. The two projective pencils of the second order, K' and K", envelope the same base conic K. (1) If there are no self-corresponding rays, S degenerates into two straight lines a( and a2 tangent to K, and Q into these same two lines and a conic section X. The construction shows that points of Q on a, correspond to points of S on (2, and points of Q on a2 to points of E on al, while points of Q on A all have their corresponding points of S at the intersection of ao and a2. The conic A is tangent to K at the points of tangency of a1 and 2-. Here a coincides with a', a' with a, b with /', b' with f/, etc. It follows that the line (a, / - a', /3') coincides with (b', a' - b, a); (a, y - a', y') with (c', a'C, a); and (b, y - ', y') with (c', /' - c, /). These are three lines joining the opposite vertices of a hexagon circumscribed to a conic section and are therefore concurrent. (Brianchon.) Therefore, EA, YB, Sc all coincide. The tangents from SA, ~B, Sc, etc., to K determine points of Y on a, a', f3, /?', y, y', etc., and points of Q on a, a', b, b', c, c', etc. Because of the coincidences mentioned all these points lie on two lines, ao and a2, the tangents from SABC, etc-, to K. (2) If K' and K" have one self-corresponding ray a a', a a' coincides with aU or 02, and SA is indeterminate. In other respects Q and Y are as described in (1). (3) If K' and K" have two self-corresponding rays a a' and /3 /', a a' and / i' are the lines ao and a2, and SA and SB are indeterminate. In other respects Q and S are as described in (1). (4) If K' and K" have three self-corresponding rays, a a', 3 /3', and y y', everything is indeterminate. 27. The Unicursal Curve of the Fourth Class.-It is of interest to apply the principle of duality to the results of this chapter, considering then the unicursal curve Q' defined as the envelope of lines joining corresponding points in two projective point rows of the second order, K and K'. It has, in general, three bitangents and four nodes and is of the sixth order and fourth class. (Cf. ~~1, 2, 20, 25.) Given three pairs of corresponding points in two projective point rows, we can construct the curve (cf. ~ 6) and establish properties entirely analogous to all those of the curve of the fourth order. The more interesting and important ones may be noted.

Page 54 54 University of California Publications in Mathematics [VOL. 1 Every ray p of the rays enveloping Q' has its corresponding ray of perspectivity S'p. The ensemble of rays:'p envelope a curve:' of the second class. (Cf. ~~ 7 and 13.) Any ray a joining corresponding points A, and A'1 has a second point A2 in common with K and a second point A'2 in common with K'. The lines joining the points of intersection of the ray S' a and K with A'2 are rays enveloping Q'; with Al, are rays enveloping Y'. Likewise, the lines joining the points of intersection of the ray S' a and K' with A2 are rays enveloping Q'; with A"', are rays enveloping S'. The ray tangent to N' which joins A1 (or A'1) with one point of intersection of:' a and K (or K') corresponds to the ray tangent to Q' which joins A'2 (or A2) with the other point of intersection of Y' a and K (or K'). (Cf. ~ 11.) The common rays of the sets enveloping Q' and S' pass through the points of intersection of K and Ki, two through each point. (Cf. ~ 14.) The points A2 and A'", A'2 and A, determine an involution of points on a in which the point (a, S' a) corresponds to the point of tangency of a on Q'. (Cf. ~ 19.) The six points of Q' other than the points of tangency which lie on the three bitangents to Q' all lie on one and the same conic section. (Cf. ~ 21.) The six points of tangency on the three bitangents to Q' all lie on one and the same conic section. (Cf. ~ 23.) If the two points of tangency coincide on each bitangent the three points of tangency all lie on one and the same straight line. (Cf. ~ 24.) If Q' has three nodes the lines joining them intersect the three bitangents, one each bitagent, in such a way that the three points of intersection lie on one and the same straight line. If Q' has four nodes the lines joining them in sets of three intersect the bitangents in such a way that the points of intersection form a complete quadrilateral of which each pair of opposite vertices lies on a bitangent. (Cf. ~ 25.) Transmitted October 3, 1911.

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 3, pp. 55-85 February 28, 1913 A DISCUSSION BY SYNTHETIC METHODS OF 11TWO PROJECTIVE PENCILS OF CONICS* BY BALDL\\,IN ilUN(GERI \()ODS CONTENTS PAGE Introductory Outline................................................. 56 I. The cubic as a locus of intersections of a pencil of rays and a pencil of conics projective to it.................. --- —--------—. --- —------------—................... 56 Introduction. —....................................-........... —....... --- —--—. — 56 1. Locus of intersections of pencil of rays and pencil of conics......................-..... 57 1I. The quartic Q as the locus of intersections of two lprojective lpencils of collics........ 58 2. Reference to analytical discussion........ —........-...........5........8.....- 5S 3. Point-rows through a fixed point of a pencil of conics.......................................... 58 4. Double l)oints of quartic.-.................................... —....... ----- 59 5. Synthetic discussion of order of Q................-).........-...... --- —--------- --------- 59 6. P'roof of order of L. —... —.....().............. --- —. --- —- ---------... --- —------ --------. 11I. Discussion of the quartic L witl two double points................................-............. 61 7. Deficiency of L.................-........... --- —--------------------------------------—. 61 8. Bearing of quadratic transformation theory............................-....... 62 9. Degeneracy of L with double rays corresponding............................... 62 10. Single-branched cubic....................................... --- —--------------------------- 63 11. Conditions of tangency of a line to L............-.......... ---. --- —--------—. 63 12. One-to-two involutory correspondence of points on a line from this viewpoint 64 13. Four-to-four transformation of the plane.......-.................... ---------- 65 14. Two-to-two semi-involutory correspondence of two pencils of rays...........-.. —... 65 IV. Locus problem suggested by the discussion of the quartic L. —........................... 66 15. Locus problem synthetically...................-...-................. --- -.. — 66 16. Locus problem analytically.-...............................- 68 17. Quadratic transformation of plane resulting therefrom............-............ —...... 69 * A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in the College of Natural Sciences of the University of California.

Page 56 56 University of California 'Publicationls in Mathiematics [Vol,. 1 INTRODUCTORY OUTLINE THE PROBLEMS TREATED IN THE DISCUSSION The following study of pencils of conics falls naturally into four parts, concerned each with the discussion of a particular problem. The first part-with the exception of its introduction, which deals with the geometry of a one-to-one correspondence between pencils of conics-is concerned with the locus of intersections of corresponding elements of a pencil of rays of the first class and a pencil of conics. This locus is shown to be a cubic curve; and, from the construction, a discussion of the one-to-two involutory correspondence of points on a line is obtained. In this problem, Pascal's Theorem is found to be a useful tool; and, indeed, in the succeeding discussions, the method of attack is often by means of this theorem. The second part is concerned with the locus of intersections of corresponding elements of two projective pencils of conics. This locus is shown to be the general quartic curve, and conditions are obtained for the double points. The solution is found to depend on the locus of intersections of two involutions of rays where there is a one-to-one correspondence between pairs of rays of the two. This locus is studied in the third part and is shown to be a quartic curve with two double points. When the involutions are in half-perspective position, the locus is the single-branched cubic. From the discussion of the problem of part three, a locus problem is suggested which is solvable with the aid of Pascal's Theorem, and which gives rise to a certain quadratic transformation of the plane which is studied by analytic methods. This is the essence of the fourth part. I!THE CUBIC AS A LOCUS OF INTERSECTIONS OF A PENCIL OF RAYS AND A PENCIL1 OF CONICS PROJECTIVE TO IT Introduction In the following discussion, the term "pencil of conics" will be used to designate the totality of conics that can be constructed passing through four arbitrary fixed points in a plane. When two such pencils of conics are so related that there is a one-to-one correspondence between the conics of one pencil and those of the other, the two pencils will be said to be projective to each other. The geometrical construction used to determine this one-to-one correspondence will be set up as follows: If an arbitrary line be drawn through one of the fixed points of a pencil of conics, it is obvious that every conic of the pencil will meet this line in one point besides the fixed point, which is common to all. Conversely, every point

Page 57 1913] Woods: Synthetic Discussion of T1'o Projective Pencils of Conics 57 of the line will determine a single conic of the pencil passing through it. In particular, the fixed point of the pencil of conics considered as a point of the arbitrary line will be assumed to determine the conic of the pencil which is tangent to the line at the fixed point of the pencil. It will be shown later that two such point-rows are projective in the ordinary sense. If this line, considered as a point-row, be projectively related to a pencil of rays of the first class, there will be a one-to-one correspondence between the rays of the pencil of rays and the conics of the pencil of conics. In this case the pencil of rays and the pencil of conics are said to be projective to each other. If, in a second pencil of conics, an arbitrary line be similarly drawn through one of the fixed points, the conics of the two pencils of conics can be put in one-to-one correspondence by merely considering the two arbitrary lines as projective point-rows. This construction, as outlined, is employed throughout the following discussion. 1. Locus of Intersections of Pencil of Rays and Pencil of Conics The first problem to be discussed is the following: Required, the locus of intersections of corresponding elements of a pencil of rays of the first class projectively related to a pencil of conics. Consider a pencil of conics through the points A1, A^, An, and A4 (see fig. 1), and a pencil of rays through S, given projectively related to it. The ray by which the projectivity is set up is denoted by A1Q. Hence, the pencil of rays at S is projectively related to the ray A1Q, considered as a point-row. Consider an arbitrary cutting ray I in the plane. It may be taken as a point-row perspective to the pencil of rays S, and will therefore be projective to the point-row A1Q. But points on 1 where a ray of S meets its corresponding conic are points of the required locus. Now, if a ray revolves about S, its corresponding point P moves along 1, cutting out a point-row projective to the point-row Q cut out on A1Q by the corresponding conic. If P falls on the conic of the pencil determined by Q, I' is a point of the locus. In other words, whenever the six points P, Q, A A2, A::, and A, lie on a conic, P is a point of the locus, and the six points named will satisfy Pascal's Theorem regarding a hexagon inscribed in a conic. Number the points as indicated in the figure. Call the point of intersection of 12 and 45, L; of 23 and 56, Ml; of 34 and 61, N. As P moves along 1, 34 revolves about the fixed point 3, cutting out a pointrow N on the fixed ray 61, perspective to the point-row P. Similarly L cuts out on a, a point-row perspective to P. Likewise, since the point-row Q is projective to P, M cuts out a point-row on the fixed ray 56 perspective to Q, and hence projective to P. Hence, the point-rows L, MI, and N are projectively related to one another, and there will, consequently, be at most three rays which pass through corresponding points of the three point-rows. For these three cases, P lies on the conic of the pencil determined by Q, and is a point of the locus. There are at most three such points on an arbitrary ray 1. Hence, the

Page 58 58 University of Californlia Publications in Mathematics [VOL. 1 Theorem': The locus of intersections of corresponding elements of a pencil of rays of the first class and a pencil of conics projectively related to it, is a point-row of the third order. That this locus is the general plane cubic is easily demonstrated both analytically* and synthetically.'It should be noted that the four fixed points of the pencil of conies are on the locus; since one of the intersections of the line SA1, for example, with its corresponding conic must be at A1. Similarly, S is on the locus, since the conic of the pencil which passes through S must meet its corresponding ray there. By the conditions of the problem, we observe that to any point A of I (see fig. 2) considered as a point of intersection with a ray of S, there correspond two points B1 and B2, considered as the points of intersection with I of the conic corresponding to that ray; and that, conversely, to this same pair of points B,;and B. corresponds back again the starting-point A. Hence the following 'Theo)remV: In an inlolutory one-to-two correspondence of points on. a line, ll/tre (aCre at nmost three points where corresponlldiing points are coincident. II TIlE (QUARTIC Q AS TIlE LOCUS OF INTERSECTIONS OF1 TWO I'RO.JECT1VE PENCILS OF CONICS 2. Reference to Analytical Discussioin The next problem to be studied is that of the locus of intersections of corresponding elements of two projective pencils of conics. This locus is easily shown analytically to be the general quartic curve. ' Let us denote it by Q. That it is a point-row of the fourth order will presently be demonstrated synthetically. 3. Point-rows through a fixed point of a 'Pecil of Conics Before proceeding to this, however, let us examine a few properties of the figure, and enunciate a theorem that is of use later on. Theorem: If, in a pencil of conics through fou f fixed poits, rays are passed through the several fixed points, the point-rows described by the other intersections of the conics with the rays are projective to one another. In fig. 3, represent the fixed points of the pencil of conies by A,, A.,, A:, and.A, and two of the fixed rays by a, and a::. Consider any conic of the pencil, and call the points in which it intersects al and a1, 2 and 5 respectively. The six points 2, 5, A,, A2, Aa, and A4 must satisfy Pascal's Theorem. Hence, numbering the points as indicated in the figure, we have 12 and 45 intersecting at L, 23 and 56 at M, and 34 and 61 at N. Both L and N are fixed points, therefore this Pascal line of the pencil of conics is fixed. As 2 moves along the ray a, * See Emch, Introduction to Projective Geometry and its Application, p. 182. t Schroter, Theorie der Ebenen Crvcu dr'itter Ordung,, 1). 58. * See Emch, loc. cit., p. 181.

Page 59 19131 Woods: SniJthetic Discussion of Tw'o Projecctice Pencils of (Conlics 59 determining the various conies of the pencil, it projects to A, in a pencil of rays, giving a point-row M on the Pascal line perspective to the point-row 2. The point-row M projects to A, in a pencil of rays giving a point-row 5 on a;. perspective to the point-row M and, hence, projective to the point-row 2. Therefore, the pencil of conies cuts out on a, and a:, point-rows that are projectively related to each other. This may be extended to include rays through the other fixed points, or several rays through the same fixed point. 4. DoZtble P'oints of Quartic Consider further the two pencils of conies determined by the points A,, A,,.4;, and A, anld B B1, B, B (fig. 4), projecttively related to each otlhe byy means of point-rows a, and b) through A, and B, respectively. Theorem': 7ihe ei(h!t fi.red points of] /e t oo projectclce pen ils of coniCs a (c on Ile loccas Q of intersectcsc ionSs )of co')rc.spoltdinJ conris. This is evident since the conic of the first set passing through B,, for examlllle, must meet its corresponding conic of the second set there. Similarly, for the others. Tlheorem: If the two projecti:le pencils of conics have a fixed point in common, this is a double point of the locals Q. Call the common point (AB1) (see fig. 5). The projectivity of the two pencils of conies may now be referred to two point-rows through (AB,), say a, and bi, which are projectively related to each other-since these are projective to any other point-rows through any of the fixed points of either pencil. Since the projectivity between the pencils of conics is arbitrary, the p)oittrows al and b1 are not, in general, perspective to each other, and the point (A1B1) is not, in general, self-corresponding. Hence (A1B,) considered as a point of a, corresponds to another point of b1, say R, giving (A1B,) as a point on the locus Q, determined by the conic of the first pencil tangent to a,, and the conic of the second pencil through R. Considered as a point of b,, (A,B,) corresponds to another point of a1, say P, and hence gives (AB1) as a point on the locus Q determined by an entirely different pair of conics, the conic of the first pencil through P, and the conic of the second pencil tangent to b,. Hence. (AiB1) occurs twice on the locus or, is a double point. There may be in this way as many as three double points. When there are three, the curve is evidently unicursal, since there is but one movable point of intersections of corresponding conies, and the correspondence is continuous. If there are four common fixed points, the locus degenerates,.in general, either into these fixed points, or into two conics through them. 5. Sy/nthetic Disc ssion of Order of Q With ihis introduction, we proeedl to the synthetic discussion of thel order of the locus. In figure 6, denote by A,,A::A A and B1B_,B:B4 the fixed points of the two pencils of conics, by a. and b. the point-rows determining the projectivity of the two pencils of conics, and by 1 an arbitrary cutting ray of the plane. Let a

Page 60 60 University of California Publications in Mathemtatics L Vol. 1 pair of corresponding conics, determined by P of a.. and R of b,, cut 1 in A' and A", and B' and B" respectively. As P moves along a,, A' and A" will describe an involution of points on I; B will move along b,, and B' and B" will describe another involution of points on 1. There is a one-to-one correspondence between pairs of points on I set up in this way, such that to any pair of points on I determined by a conic of one pencil corresponds a pair of points determined by the corresponding conic of the other pencil. Since the pencils of conics are projectively related, the correspondence of point-pairs on I is involutory. Hence, our problem is reduced to that of discovering how many coincidences of corresponding points of an involutory two-to-two correspondence of points on a line can occur. For, if a point of 1 be at once a point of both involutions, it is an intersection point of corresponding conics and, therefore, a point of the locus. Join the points A' and A" to an arbitrary point, say A1, and B' and B" to B1. Now, as P moves along a., and R moves along b,, projective to P, we shall have an involution of rays at A1 and also one at B1, with a one-to-one correspondence between pairs of rays of the two involutions. The locus of intersections of corresponding pairs of rays of these two involutions-call it L-will meet 1 in points where corresponding conies intersect; in other words, in points of the original locus Q of intersections of corresponding conies. Hence, the order of L is the same as that of the original locus, and we shall confine our attention for the moment to L. 6. Proof of Order of L Theoremn: The locus of intersections of correspondiing pairs of rays of tzwo involutions of rays, where there is a one-to-one correspondewce between the pairs of rays of the two involutionss, is a point-row of the fourtlh order wtith t'wo doublepoints; or, a qulartic curve of deficiency on1e. In figure 7, denote by I, and 1. the centers of the involutions of rays and let X be an arbitrary cutting ray of the plane. Construct through I, and., any conic-call it r-which is tangent to A. There will be a double infinity of such conies. Now, the rays of I, for example, will cut out an involution of points on A. Draw from each of these points the remaining tangent to r. Now, to a given pair of rays of I1, there will be a pair of points on A, say G1 and G., and hence a pair of tangents to r, say g, and g.. If four points G, of X be taken, the four points of tangency of the four tangents g, will project to any point of r in four rays with the same anharmonic ratio as the points G1, since the points G, are chosen on a tangent to r. Hence, the points of tangency of the various pairs of tangents of.g and g2 will constitute an involution of points on r. Now, the rays joining corresponding points of an involution of points on a conic pass through a point,* say S,. Hence, the pencil of rays S, will determine our involution of points on r, and, consequently, our involution of points on A, determined by the rays of I,. Of course, S, varies with the different conies r that may be taken. Similarly, there will be a pencil of rays S,, determined by * See Reye, Geometrie der Lage, p. 147.

Page 61 1913 I IV'oods: Silthctic Discussioii of 'lo lrojlectile l'cPecils of ('onics 6il tangents drawn to r front the involution of points H, and H,, on A, determined by the involution of rays 1,. Since there is a one-to-one correspondence between the line-pairs of the two involutions, there will be a one-to-one correspondence between the rays of 81 and S,. The point of intersection of corresponding rays of S, and S., will therefore describe a conic, say X, which will intersect r in, at most, four points. These points are coincident points of the two involutions of points on r and, consequently, since tangents g and h coincide here, the tangents to r at these points will meet A in points where G and II coincide; that is, in points of L. There can be at most four such points on an arbitrary A. Hence, the first part of the theorem above is established, and the two following theorems may be added as direct consequences of the discussion,Theorem: Inl a two-to-two involutory correspondence of points on a line there are at most four points 'where corresponding points coincide. Theorem: The locus of intersections of corresponding conics of two projective pencils of conics is a point-row of the fourth order. The conditions for double points have been established above. Ience, the locus is the general quartic curve. III DISCUSSION OF THE QUARTIC L WITH TWO DOUBLE POINTS 7. Deficiency of L In article 6 of the preceding part, we have shown that L is a point-row of the fourth order. Let us proceed to a discussion of this quartic beginning with the determination of its deficiency, which may be established as follows. Consider the ray 11I, as a ray of the involution 11. Considered as a ray I,G,, it meets its corresponding ray 12H1 at 12; likewise, considered as a ray IkG,, it meets its corresponding ray I.H, at 12. 1.111 and 1.H., are in general different rays. IIence I, occurs twice on L or, is a double point. Similarly witlh ii. A third double point does not in general exist; but conditions for its existence are easily determined. A pair of rays of 1i has its corresponding pair of rays of 12. The four intersection points-say P), Q, R, S-are points of L. If the two rays of 1. should fall together as a double ray of the involution at 1, the points P and Q, and R and S would fall together in pairs as indicated, and the rays of I, corresponding to a double ray of 1, would be tangents to L. There are not more than two double rays of 12. Hence, 'Theorem: Fronm either double point of L at most four tanglents may be drawln to L. These are tihe rays of the ilnvolttion at one double point corresponding to tie double 'rays of the ilol1ition; at thle other double point. If a double ray of I, corresponds to a double ray of 12, they are each tangent to L at their point of intersection, and their point of intersection is, consequently, a (loulble point of L. Hence,

Page 62 62 Unidvcersity of California Publications in Mathecmatics I VOL. 1I 7'ltoro)n: 1f' a dloiUb)lr?aJ 1 Iof 1 J coeJlsp'l)Ons to (1 dotblo ra'if of /l., L has Ihlr'c double points, v'iz: A, I., andt the point of intersectioJn of the corrcs)ponding double rays. 8. Bearing of Quadratic Transformation Theory Before discussing other cases of L, let us consider the bearing of the quadratic transformation theory of the plane on this discussion. Ordinarily, if two involutions of rays, 11 and L,, be given, and the point of intersection P of a ray of 1I and one of 1I move on a point-row of order n, the intersection point R of the corresponding rays of I1 and 1. will move on a point-row of order 2n. 11 and., will be vertices of a fundamental triangle of the transformation, of which the third vertex, say 13, will be the center of an involution of rays that may be used in place of either 1, or 1, in the construction. Now, each time the point P of the point-row of order n passes through one of the points I,,.,, or /., the point R of the point-row of order 2n describes a straight line as part of its locus and the order of the remainder is reduced by one. In the locus L, under discussion, the points P' and R traverse the same point-row, a point-row of order four with two double points. Comparing this with the theory mentioned above, we find it to be consistent; for, if P move along L, Q will move on a point-row of order eight. However, P passes through 11 twice and I1. twice. Hence, the locus of Q degenerates into a point-row of the fourth order and doubly into the lines in the quadratic transformation which correspond to the points I, and Io of the fundamental triangle. 9. Degencracy of L with Double Rays Corresponding Certain degenerate cases of L are interesting and present themselves naturally. Suppose, first, that each double ray of I, corresponds to a double ray of I,. Let the first double ray of 1l meet its corresponding double ray of 1, at A, and the second double ray of 1I its corresponding double ray at B (see fig. 8). IBy previous reasoning 1., _,, A and B are double points of L, hence-with four double points-degeneracy is to be expected. The ray AB has the same involution of points described on it by the rays of I, and 1., since the double points A and B of the involutions are the same. Hence, the corresponding rays of Il and 1.. will meet on AB and it is a part of the locus. Let the points of intersections of corresponding pairs of rays of 11 and I., be P, Q, R, and S. Two of these, say P and R, are obviously on AB. Now, as P moves along AB, it is always an intersection point of corresponding rays of I1 and 1.. As it crosses II,, the point of intersection of the corresponding rays of 1~ and 1. is indeterminate, and 1l1., is thus a part of the locus. The remainder is a conic through I7, 1,, A, and B, and there are not only four but five double points to the locus in this case.

Page 63 1913] Woods: Synthetic Discussion of Twlo Projcctive Pencils of Conlics 63 10. Single-branched Cubic Another and more important case of degeneracy is that in which the ray Il, is self-corresponding, but is not a double ray of either involution. In this case, the locus L degenerates into the ray II., and a point-row of the third order. This last is a single-branched cubic and has been discussed by Schriter from this point of view. This position of two involutions of rays is ternmed by him ''half-perspective position.' ' In figure 9, a case of this type is shown. From T', a Ipoint on a singlebranched cubic, tangents t, and t., are drawn to the curve. (There are always points 7T from which this is possible.) Calling the points of tangency A, and A,, we know that the rayA A., meets the cubic in another point, say I1, which is conjugate to T1. If, now, any point P of the cubic be joined to At and A.,, the lines so drawn will each meet the cubic in one other point. Call these B, and B,, and join them to T, and T.'. As P moves along the curve, the rays T'B, and T1B. will give an involution of rays at T,, and T.2BI and T:B2, an involution of rays at T,. Since T1T2 is self-corresponding, it is part of the locus. The ray AA, is a double ray of T2; hence, the corresponding rays of T1 should be tangents to the curve as, indeed, they are. 11. Conditions of Tangc(,cy of a Line to L Let us turn again to a discussion of the properties of L as revealed il figure 7. If the conic 1, described by the intersection point of corresponding rays of St and S.: is tangent to the conic I', two of the points of intersection of the conics will be coincident, and two of the points of intersection of X with L will be coincident. Hence A will either be a tangent to L or a secant through a double point. That this last state of affairs might occur is readily shown. In figure 10, let the double rays of 11 meet A in A and B, and the double rays rays of I., in C and D. Then, since the tangents to r from these points determine the double points of the two involutions of points on r, it is obvious, for instance, that the tangent to r from A joining, as it does, two corresponding points on r, contains 81. Similarly, the tangent from B contains S,. S8 is therefore determined as the intersection point of the tangents to r from the points where the double rays of,1 intersect A. S., may be determined in like manner. Now, if one double ray of 1I correspond to a double ray of I,, their point of intersection A is a double point of L. Consider any cutting ray A through this point, and construct the conic r as before. A tangent to r from this double point contains both S1 and S.,. Now, since the point A is a self-corresponding double point of the involutions of points on A, the common ray SIS., of the pencils of rays S, and S., is self-corresponding, and the conic S degenerates into two lines, one of which is SS., the tangent to r from A. This is obviously true for any ray A through A and, hence, E and r are tangent if A be tangent to L, or pass through one of its double points. *See Schr6ter, Theorie der Ebenen Curven dritter Ordnung, p. 148.

Page 64 64 University of California Publications in Mathenmatics [VOL. 1 The same remarks apply to the general quartic determined by two projective pencils of conics, since the intersections of A with the general quartic are the same as its intersections with the L determined for that particular cutting line. If the cutting line should pass through one of the double points of L, a single involution only of points would be determined upon it, and our reasoning regarding the coincidences of intersection points of Q and L with A breaks down. Since the selection of double points for L is arbitrary, this difficulty is obviated by moving them. As a consequence of the identity of the intersections of L with A and of the general quartic with A, we may revolve A about a fixed point, and enunciate the following Theorem: The geeral quartic curve may be described as the locus of inltersections of a pencil of rays acnd of a pencil of quartics of deficiency one with arbitrary, fixed double points. 12. One-to-Two Involutory Correspondence of Points on a Line from this Viewpoint Before leaving the discussion of L, let us apply this machinery to the discussion of the one-to-two involutory correspondence of points on a line, previously discussed with the aid of Pascal's Theorem. In figure 11, let I1 be the center of a pencil of rays of the first class, and I, the center of an involution of rays. Select any cutting ray A and construct a conic r, containing 1, and 12, and tangent to A. The points of tangency of tangents to r from the points H1 and H., where pairs of rays of 1, cut A. will give an involution of points on r. The rays joining corresponding points of this involution will pass through a poinlt, say S.-. The points o( tangency of the tangents from the points G of A where rays of JI meet A will p)roject to 1, in a new pencil of rays projective to the original one. By tile conditions of the problem, the new pencil of rays at I, and the pencil of rays at 82 are projective to each other. They determine a conic I which cuts r in at most four points, one of which is I,. Tangents to r at the other three points of intersection of E and r meet A in points where corresponding points of the one-to-two involutory correspondence are coincident. The tangent at I, does not in general cut X in such a point, for the tangent (from a point of A) whose point of tangency is I,, does not uniquely determine a ray of the new pencil at 11, since any ray through 11 answers the requirements. Ience the following Theorem: The locus of intersections of correspondinig elements of a pencil of rays of the first class and an involution of rays, where there is a one-to-one correspondence between the rays of the first and the pairs of rays of the second, is a point-row of the third order with one double-point; or, a unicursal cubic curve. 12 is the double point, since the ray II., considered as a ray of Il, meets two rays of 1, at I1. Hence, 1. occurs twice on the locus, or, is a double point. It is obviously also on the locus, since the ray 1,21 of I, meets its corresponding ray of I at 1l. The last theorem may be stated inversely in the form in which it is given when approached from the other side.

Page 65 1913] Woods: Synthetic Discussion of Tico Projective Pencils of Conics 65 Theorecm: The poilts of intersection. with the curve of rays through any point of a unicursal cubic project to the double point in an involution of rays, the double rays of which are determined by the tangents from the arbitrary point to the curve. The rays of the involution, corresponding to the ray joining the arbitrary point to the double point are the tangents of the curve at the double point.* 13. Four-to-Four Transformation of the Plane The constructions of figures 7 and 10 by which L was discussed, contain and suggest several problems and loci. The points S, and S2 are determined by constructing a conic through two points and tangent to a given line. Let us inquire whether it is possible to choose SA arbitrarily and, if so, how many points S2 will there be corresponding to a given S\. We note that with a given Sl, r is determined as a conic which shall be tangent to S$A, SXB, and A, and shall, in addition, contain the points 1, and I.. There are, in general, four such conics.t For a given conic, it is obvious that S, is uniquely determined. Hence, to a given SX, arbitrarily chosen, there are four S.,'s, and conversely. This gives a four-to-four transformation of the plane, which may be called "onequarter involutory," since any one of the points S, determined by an arbitrary S1, gives back this same S, and three others. 14. Two-to-Two Semi-involitory Correspondence of Two Pencils of Rays Suppose the point of tangency of r to A be fixed and be denoted by K. Then the figure furnishes an example of what may be termed a "semi-involutory" two-to-two correspondence of rays, as follows,-suppose an arbitrary ray of A be chosen. On this ray there are in general two points S,, since two conies of the pencil are in general tangent to the ray of A chosen. The conics r now constitute a pencil, since they all pass through 11 and I1 and are tangent to X at K. Also, each conic determines a ray of B, tangent to it, and hence an S,. To either of the rays of B so determined, two conies of the pencil are tangent, one of which is the conic tangent to the original ray of A. Hence, to each ray of A, there are two rays of B; and, to each of these rays of B, there are two rays of A, one of which is the ray of A with which we started. An easy geometrical example of this is obtained by using either two points and two rays which do not contain them, or two points and a non-degenerate conic which does not contain them (see figures 12 and 13). To the ray a, of A, there are two rays b1 and bo of B, and to the ray b, of B there are two rays a, and- a of A-one of which is a ray of A which determined the ray b1 of B. In figure 12, the rays joining A and B to the point of intersection R of the rays p and q of the construction are corresponding double rays. In figure 13, the tangents from A to the conic have double rays of B corresponding to them, *Proved by D. N. Lehmer, "Constructive Theory of Unicursal Cubic by Synthetic Methods," Transactions of American Mathematical Society, 1902. I See Salmon, Conic Sections (ed. 10), p. 389.

Page 66 66 University of California Publications in Mathematics [VolI. 1 and, similarly, the tangents from B to the conic have double rays of A corresponding to them. This correspondence can be studied further from this point of view. IV LOCUS PROBLEM SUGGESTED BY THE DISCUSSION OF THE QUARTIC L. 15. Locus Problem Synthetically By altering slightly the construction and interpretation of S,, we obtain a problem whose solution is interesting as furnishing another example of the method of using Pascal's Theorem in problems involving pencils of conies. The power of this method is obvious from the several uses of it already made in this discussion. Suppose that S1 is a point determined as the point of intersection of tangents to r at the points where it meets the double rays of I1. If the point of tangency K of r be fixed and r be determined as a conic through 1I an(d., and tangent to A at K, what is the locus of S as the various conies r of the pencil are taken? This problem is not directly connected with the study of L previously undertaken, but comes in naturally as a problem connected with this particular pencil of conies. In figure 14, let r cut AI, in R, and BI1 in R'. Call the tangents at R and I', a and f respectively. Their points of intersection is S,. The points I,, L,, K and R must satisfy Pascal's Theorem. Number them, and call the intersection of 12 and 45, L; of 23 and 56, M; of 34 and 61, N, as indicated in the figure. The points L, M, N are on the Pascal line, and N is a fixed point of this line for the given pencil of conics. Similarly, construct the Pascal line L'M'N', replacing R in the construction with R'. As R moves along A1, it cuts out a point-row determining the conies of the pencil. This point-row I)rojects to 1. in a pencil of rays which determines on KI (a fixed line) a pointrow L perspective to the point-row R. The point-row L of K11 projects to.N in a pencil of rays which describes a point-row M on AB, perspective to the point-row L of KI,, and hence projective to the point-row R on Al,. Therefore, the ray RM envelopes a conic, as R moves on AI,, determining the various conies of the pencil. Similarly, R'M' envelopes another conic. Since there is a oneto-one correspondence between the tangents RM and R'M', S8 is determined as the locus of intersections of two projective pencils of rays of the second class. This curve is the unicursal quartic, and has been discussed from this point of view.* If a common ray of the two pencils is self-corresponding, it is, obviously, part of the locus, so that, if all four common rays be self-corresponding, -they constitute the locus. Indeed, if three are self-corresponding, the locus consists of them and of one additional line. We shall show in this problem that the locus is degenerate and consists of the lines AB, KI,2-counted twice-and another line. * See Annie Dale Biddle, "Constructive Theory of the Unicursal Plane Quartic by Synthetic Methods,' Univ. Calif. Publ. Math., vol. 1, no. 2, 1912.

Page 67 1913] lWoods: Synthetic Discussion of Two Projectirle Pencils of Conlics 67 Denote by S the conic enveloped by RM, or a, and by,', the conic enveloped by R'M', or /3. AB and Al1 are tangents to S, and AB and BI1 are tangents to S'. Hence, AB is a common tangent of Y and S'. Now, in the conic 2, the points corresponding to A, considered first as a point of AB and then as a point of AI,, are the points of tangency to E of AlI and AB, respectively. If R moves to A, RI2 becomes AI2 and cuts KI in a point denoted by Q. NQ determines M, as the corresponding position of M. This construction will be referred to later. Hence, M1 is the point df tangency of AB to E. Moreover, if R moves to A, the conic r degenerates into AB and 11.,, and R' moves to B. lHence, the common tangent AB of Y and 2' is self-corresponding and is therefore part of the locus. When R' moves to B, R'1. becomes BL2, and intersects KIZ in Q'. N'Q' determines M1' as the point of tangency of AB to S'. If R moves to N, L moves to K, M moves to K, and r degenerates into KIl and KI,. Since R is at N and M at K, KN is a tangent of X, being a ray RM. Since, when R is at N, r degenerates into KI, and KIZ, R' is either at I, or N', as it is always on the conic r. If R' is at IZ, M' is at B, and R'M' is not a tangent to r. As this is contrary to hypothesis, R' is at N'. Consequently, L' is at K, and M' is at K. Hence KN is a tangent of 2', that is, KN is a selfcorresponding common tangent of S and S', and is therefore a part of the locus of S,. We shall apply Brianchon's Theorem in the following way to determine its points of tangency to Y and S'. Let AB, BC, and CA of figure 15 be three tangents to a conic and let S and T be the points of tangency of CA and AB, respectively. Then R, the point of tangency of BC, is found by drawing through A, a ray passing through the point of intersection of BS and CT. In figure 14, the point of tangency of AB to E has been found to be M1. The point of tangency of AlZ is found by moving M to A. If this is done, L moves to I, and R moves to I,. Hence AI1 is tangent to: at I,. To discover the point of tangency of KN to S, apply Brianchon's Theorem as follows, remembering that R is at N. Join M1 to N and K to l,, these lines meeting in Q. AQ cuts KN in the required point of tangency, which is seen to be I,, from the way in which Q was previously determined. To discover the point of tangency of KN to -', it is necessary to find the point of BI to V'. If M' moves to B, L' moves to 11, and R' moves to I,. Hence, BI, is tangent to S' at I1. The point of tangency of AB to S' has already been discovered to be M'. Now, to discover the point of tangency of KN to V', apply Brianchon's Theorem, remembering that R' is at N'. Join Mi' to N' and K to Il, giving Q'. Q'B cuts KN in the required point, which is seen to be I^. Hence, S and 2' are tangent to each other at I., and KI., is a double common tangent which is selfcorresponding. Hence, finally, the locus of S. is composed of the lines AB, KI, (counted twice) and another. That is to say, S, moves in general on a straight line.

Page 68 68 University of California Publications in Mathematics [VOL. 1 Similarly, if an S2 be determined by tangents to r at the points where the double rays of 1, cut r, S, will move on a line. As different points K of A are taken as points of tangency for r, S, and S. will move on lines which will correspond in pairs. Of this, more will be said later. 16. Locus Problem Analytically The analytic discussion of this problem reveals a further property of these rays. This is inserted temporarily, as a complete synthetic discussion has not yet been obtained. In figure 16, denote by I, and I., the fixed points, by AI1 and BI the fixed lines through I1 and by K the fixed point of tangency of conies through 1, and I to an arbitrary line 1. The tangents a and P at M and N will determine S,. Choose 12, I4, and K as the points (1:0:0), (0:1:0), and (0:0: 1) respectively. The equation of r may be put in the form yz + zx + xy — 0 AB is the tangent at (0: 0: 1). Hence, its equation is AB x + y-0. Choose for coordinates of A and B, (1: -1:/ ) and (1:-1: v) respectively. The equations of AI, and BI, are: AI1 z - Zx =- 0 BI1 z - vx = 0. Hence, the coordinates of Mi are (/ + h):-ui: i(i + XA). The coordinates of N, similarly, are (v + A):-V: v (v + A). This gives as the equation of a,[U(,it + A)- Aj,] x + (/, + A) y +,Az 0 or Lju2x + ( A) 2 y + Az = 0. Similarly, the equation of / is v2x + (v + A)2y +Az=0. The coordinates of S1, the intersection point of a and 8/, reduce to-,U + v +2A:- (u + A):X A + Av + 21uv. Let the absolute coordinates of S, be (x1, y1, z); then x, = + +v+2. (1) g yl = — (M + A) (2) z= =A( + v) + 2uv (3) whence (a + v) 2 x + (- V) 2 y - 2(, + v)z — 0 results as the equation of the locus of S,. This represents a line which cuts x+y=0O in the point (1:-1: 2- ). A + v

Page 69 1913] ~Woods: Syntthetic Discussion of Two Projective Pencils of Conics 69 Now the line I,12 cuts xi + t ( 0 in the point (1:- 1:0). The harmonic conjugate of (1: -1:0) with respect to A(1:-1: t) and B(1:-1:v) is (1:-1: 2 ). tA + V Hence the following Theorem: As the point of tangency K of r moves along AB, the line which is the locus of S, revolves about the harmonic conjugate, with respect to A and B, of the intersection of I112 and AB. If a similar construction for S2 be made, the fixed points on the tangent AB being C and D, the line which is the locus of So will revolve about the harmonic conjugate, with respect to C and D, of the intersection of I1I2 and AB. 17. Quadratic Transformation of Plane Resulting therefrom If, now, the point of tangency of r to AB be allowed to move and Sh be chosen at random, we inquire as to the number of points S2 to a given 81, and, also, regarding the locus of S, when S, moves, for instance, on an arbitrary line. In figure 17, let I1, I2, and A be the vertices of the fundamental triangle; the points (1:0:0), (0:1:0), and (0:0:1) respectively. Let the fixed rays of IT meet, in A and B, the fixed line AB which is tangent to r; and the fixed rays of I meet this line in C and D. Call the tangents at the several points of intersection, a, /3, 7, 8, as indicated in the figure: a and /3 determine S, and y and 8 determine Se. The equation of any conic through I1 and I may be written az2 + byz + czx + dxy- 0. The equation of the fixed line 1 through A is y - x - 0 where A is a constant. That this line may be tangent to the conic r is represented by (b + c)2 —4adX (1), since the equation obtained by solving az2 +- byz + czx + dxy 0 and y - x - 0 simultaneously, viz: az2- + bAzx 4- czx + d.x'2 0 must be a perfect square. The equation of a tangent to r at a point (.r: ': y:z') on the curve is (dy' + cz')x + (bz' + dx')y + (cx' + by' + 2az')z = 0.

Page 70 70 University of California Publications in Mathematics [VOL. 1 The ray Al, whose equation is y — 0, meets r at (a:o: -c). Hence, the equation of a, the tangent at this point, is c2x + (b - ad) y + acz - 0. Denote B by (1:A: X), since it is an arbitrary point on 1. The line BI has for its equation y- - z= -0. Its point of intersection with r (besides I) is [-L- (ar + bX): (cp + dx): -(cu + dx)] Hence, the equation of f/, the tangent to r at this point, is (CdA + dX)2x- + (b - ad)p2y + (acx2 + bdX2 + 2adXp)z- 0. The intersection point of a and / is S1. Denote its coordinates by (x1: y1: z1). By absorbing whatever multiplier there may be (say 7) into the constants of r, we may write x ---(bA+ 2alz) (2) Yx- cA (3) z = 2c + dA (4) and the first condition: (bX + c)2= 4adX (1) The equations (1), (2), (3), and (4) determine the parameters a, b, c, d of the conic r. Since only one of these is quadratic in a, b, c, and d, the others being linear, there are two conies to a given S1, and hence two points S2 to a given point S1, as there is but one S2 for a given conic. Conversely, to a given S2, there are two points S1. This construction gives, then, a semi-involutory two-to-two transformation of the plane. Suppose the point S1 to move on an arbitrary line of the plane, let us discover the nature of the locus of S,. Let the points where the fixed rays of 1.2 meet the line 1 be C(l:X:- ) and D(1:A:p), where q and p are constants. The equations of CI, and DI2 follow: CI2 z - x - DI., z - px 0. CI., meets r in the two points (0:1:0 b4 + d: — (ap +c): (b4 + d) DI, meets r in the two points j0:1:0 l bp + d:-p(ap + c):p(bp + d) The equation of y is -2(bc - d)x + (bp + d)2 y + (abha2 + 2ad p + cd)z 0.

Page 71 191.31 Woods: Synthetic Discussion of Two Projective Pencils of Conics 71 The equation of 8 is p(bc —ad)x + (bp + d)' y + (abp2 + 2adp + cd)z 0. Denote S2 by (x2: y: Z ), whence, by reduction.2,: 2:., - 2d +- b(p + ): 2app + c(p + )): 2bpr + b(p + o) Let S, move on the line L.rx + My, + -Nz = 0. Its coordinates will satisfy this equation, and the several equations determining the locus of S. are -L(b + 2aL) -+McX +N(2ct — d) -0 (1) (b + c)2- 4adX (2) rx2 -2d + b(p+ q) (3) Ty2 -2ap + c(p + ) (4) TZ2 2bpH + d(p + ) (5) The parameters to be eliminated are a, b, c, d, and r. As all the equations )ut one are linear in these, and the exceptional one is quadratic, the locus of S., is a conic. lence, we may write the Theorem: The locus of S,, as S, moves on a line, is a conic; or, in general, the locus of S,, as S, moves on a point-row of order n, is a point-row of order 2n. From equation (2), we note that d-0 gives the equation of a tangent to the S2 conic. If d-0, we have, by elimination from (3) and (5) Xi-=PH-+ + or 2px - (p + )z -0. Z 2po This particular line is obtained regardless of which line S, moves on and is therefore tangent to all the S, conies. It is the line through 12, which passes through the harmonic conjugate, with respect to C and D, of the intersection of 11I, and AB. Similarly, there is a line through I,, which is tangent to all the 81 conies which correspond to lines described by,8. Hence, the following Theorem: By means of the conics tangent to an arbitrary line and passing through two arbitrary points, through each of twhlich two arbitrary rays arc chosen, a quadratic transformation of the plane may be established. To every point S,, there are two points S,, and, conversely, the correspondence being semiinvolutory. If 81 move on a point-row of order n, S, moves on a point-row of order 12. All the S8 loci are tangent to a certain invariant ray of the secondl of the two fixed points. Similarly, all the S1 loci corresponding to arbitrary paths of S, are tangent to a certain invariant ray of the first of the two fixed points. In particular, however, if 8, describe a ray passing through a certain point of the fixed tangent line, S, describes a ray' passing through anothe, certain point of the tangent line. Thus, in this quadratic transformation, there is a particular pencil of rays which goes into a pencil of rays.

Page 72 72 University of California Publications in Mathematics [VOL. 1 VITA I, BALDWIN MUNGER WOODS, was born in Lampasas, Texas, on September 22, 1887. I studied in the public schools of Fort Worth, Texas, until 1904, when I entered the University of Texas, from which I received the degree of Electrical Engineer in 1908. During the year 1907-08 I held the position of Assistant in Applied Mathematics in that institution. In 1909, I was appointed John W. Mackay, Junior, Fellow in Electrical Engineering at the University of California. I held this position until January, 1910, when I was appointed Assistant in Mathematics. From July, 1910, to the present I have been Instructor in Mathematics in the University of California. In 1909, I received the degree of M.S. in Electrical Engineering from the University of California. In the University of Texas, I studied under Professors Porter and Benedict and Mr. Rice in mathematics, and Professors Scott and Taylor in engineering. In the University of California, I have studied under Professors Stringham. Haskell, Lehmer, and Putnam in mathematics; under Professors Cory and LeConte in engineering; and under Professor Raymond in physics. To all these I wish to express my thanks,-especially to Professors Lehmer and Haskell, who, in their supervision of the present work, have been a constant source of inspiration. On April 29, 1912, I passed the public final examination for the degree of Ph.D.

Page 73 L - -- / / / I-_. / \ -*1 --- -_ M \ \ / --- - _A4 -\A3/ AS. 2 Fig. I IBi - - 'B2 i I A r 1g. II Figs. 1 and 2 I 73 ]

Page 75 Fig. III A-lo ~0o z oA4 Fig. TV 0o4 Fig. V Figs. 3-5 [75]

Page 77 -l L- l 3. -l 0 0 s l C l w \Ic 0 P

Page 79 tJ Fig. IX Fig. X Figs. 9 and 10 [791]

Page 81 Fig YLI Fig. XIII Figs. 11-13 L 81 ]

Page 83 L -1 X I Ict C-.,h Q 0) -J_

Page 85 o:0o) Fig. XVII Figs. 16 and 17 [85]

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 4, pp. 87-96 May 15, 1914 A COMPLETE SET OF POSTULATES FOR THE LOGIC OF CLASSES EXPRESSED IN TERMS OF THE OPERATION "EXCEPTION", AND A PROOF OF THE INDEPENDENCE OF A SET OF POSTULATES DUE TO DEL RE* BY B. A. BERNSTEIN INTRODUCTION The first complete set of independent postulates for the logic of classes-the logic invented by Boole and perfected by Schr6der-is due to E. V. Huntington. In his "Sets of independent postulates for the algebra of logic" (Transactions of the American Mathematical Society, July, 1904, p. 288) he presents, and proves the independence of, three sets of postulates for the logic of Boole: one is a modification of a set due to Whitehead, in which two operations, E and 0, are assumed; another is founded on Schroder's axioms, postulates, and definitions, which are all expressed in terms of the relation @; and a third is Huntington's own set, in which only the operation D of logical addition is assumed. Quite recently another set of independent postulates for Boolean logic was proposed. In a paper, "A set of independent postulates for the Boolean algebra,"' H. M. Sheffer presented a set of postulates all expressed in terms of the operation "per", denoted by the vertical line 1, a b being ultimately identified with ab"neither a nor b." Of course, by virtue of Peirce's duality principle, the substitution of 0 for E, together with the interchange of V and A (the "whole" and "zero" respectively), will transform Huntington's third set of postulates into a set in which the single operation assumed is that of logical multiplication. Likewise, the * Read before the American Mathematical Society (San Francisco section) October 25, 1913. 1 Read before the American Mathematical Society December 31, 1912. An abstract of the paper is given in the Bulletin of the American Mathematical Society, March, 1913. Dr. Sheffer was kind enough to let me see his postulates. The complete paper appeared in the October (1913) number of the Transactions of the American Mathematical Society.

Page 88 88 University of California Publications in Mathematics [VOL. 1 substitution of the operation 1+ (say) for 1, a 1+ b reading "either not-a or not-b," will change Sheffer's set into one in which the single operation involved is the operation which forms from the logical elements a, b the element usually denoted by a + b. It is thus seen that of the six essentially different operations which form a new element from two given logical elements by means of logical addition alone or of logical multiplication alone, that is, of the operations which form from a, b the elements a + b, a - b, a + b, ab, ab, ab, Professor Huntington has chosen one and Dr. Sheffer another as basis for a set of postulates. Four of the six operations have thus been utilized for postulate building, the operations O, 1, carrying with them their respective duals 0, 1+. It therefore suggests itself to one to complete this group of postulate sets by constructing the pair based on the two remaining operations. Accordingly, in part I of this paper I present a set of postulates involving the operation "exception", -, a- b reading "a except b," or "a which is not-b"; which, also, carries with it the dual set involving the operation —, which I call (having in mind Euler's diagrams) " adjunction", a. b reading "either a or not-b."2 In my set, as in Dr. Sheffer's, the existence of "zero" and of the " negative" is proved; the "whole", however, is postulated. (Professor Huntington postulates the existence of all three of these elements.) In proving the independence of my postulates I employ the usual methodthat of exhibiting for each postulate a: "system" which contradicts this postulate but which satisfies all the rest.3 To show that my set is "sufficient" for the algebra of classes I deduce from it the set of postulates4 from which Professor Del Re develops his Algebra della Logica (Naples, 1907). In part II of this paper I substantiate Professor Del Re's (mere) assertion5 of the independence of his postulates by exhibiting the necessary proof systems, hoping thereby to add to the completeness of his admirable Algebra. 2 We have the identities a - b a, aa - b a + b. So that the operations exception and adjunction are not logical subtraction and logical - division, the inverses of logical addition and of logical multiplication respectively. a - b and a - b, as results of the operations exception and adjunction, are identical with the logical difference and the logical quotient only when, respectively, b < a and a < b. 3 Among the writers who used this method are to be mentioned especially (only the earliest papers known to me being cited): PEANO, "Sul concetto di numero," R. d. M. (Rivista di Matematica), vol. 1 (1891), p. 93; BURALI-FORTI, "Sulla teoria delle grandezze,'" R. d. M., vol. 3 (1893), p. 78; PADOA, R. D. M., vol. 5 (1895), p. 185, note (in paper of Vailati, "Sulle proprieta caratteristiche delle varieta a una dimensione"); PIERI, "I principii della geometria di posizione," Mem. Accad. Sci. Torino, vol. 48 (1898), p. 60; HILBERT, Grundlagen der Geometrie, Leipzig, 1899; HUNTINGTON, "A complete set of postulates for the theory of absolute continuous magnitude," Trans. Am. Math. Soc., vol. 3 (1902), p. 278; VEBLEN, "A system of axioms for geometry," Trans. Am. Math. Soc., vol. 5 (1904), p. 347. 4 See below. The postulates are Huntington's first set in which the distributive law x + yz= (x + y) (x + z) is replaced by the two laws: x + x = x, x + (y + z) = (x + y) +z. 5 Algebra della Logica, p. 8.

Page 89 1914] Bernstein: Postulates for the Logic of Classes 89 I A SET OF POSTULATES IN TERMS OF THE OPERATION EXCEPTION Let us take as undefined ideas a class K of elements a, b, c,.... and an operation "-", a - b reading "a except b." Let us understand by a -b that a, b can be interchanged at pleasure; by a == b that this interchange cannot take place. Then the logic of classes may be defined as a system (K, -) which satisfies the following six postulates: Postulates. 1. a- b is an element of K whenever a, b are elements of K. 2. If postulate 1 hold, there exists an element 1 in K such that a- (b-1) -— a, whenever a, b are elements of K. 3. If postulate 1 hold, (a — a) - c —(b b) - d, whenever a, b, c, d are elements of K. 4. From a - b b - a follows a- b, whenever a, b, a - b, b - a are elements of K. Def. 1. a' - 1 - a. 5. If postulates 1, 2 hold, and if the element 1 is unique, then (a -b) - (c-d) - {[(a-b) -d']'- [(c'-a') -b]}', whenever a, b, c, d are elements of K. 6. If postulate 2 hold, there is an element a in K such that a + 1. Consistency of the Postulates. As a system (K, -) satisfying all the postulates 1-6, thus proving their consistency, we have: K the class of two elements e,, e1, with e, -ej defined by the "exception" table: e0 el eo eo eo e1 e1 e0 The element 1 required by postulate 2 is el, which is unique. Postulate 5 is seen to hold by noting that the substitution of eo or of e1 for a, b, c, or d will reduce 5 to an identity. Theorems. The following propositions, deducible from postulates 1-6, will throw into our hands Professor Del' Re's eleven postulates as theorems, and will thus prove the sufficiency of our postulates for the algebra of classes. 6 In Peano's Formulaire de Mathematiques, -a is used for a', the "negative " of a. Def. 1 makes clear the distinction between the sign "-" considered as belonging to an element and the operation "-" involving two elements. -a is merely an abbreviation for 1- a.

Page 90 90 University of California Publications in Mathematics [VOL. 1 7. a-a= b-b. For, (a-a) - (a- ) =a- a, and (b-b) - (a- ) b -b (by 2). Also, (a-a) - (a-1) (b-b)- (a-1) (by 3). Therefore, a a- b -b. Hence, we may define 0, "zero": 8. Def.2. a-a-0. 9. a —1 0. For, a - - (a-1) - (a -) =0 (by 2, 8). 10. a-0-a. For, a- O a -(a-1) —a (by 9, 2). 11. O-a 0. For, 0- a (0-0)-a- (0-0) -0 0-0 0 -(by 8, 3). 12. The element I of postulate 2 is unique. For, suppose two such elements: 1, and 12. Then 11- 12 -12 - 1l = (by 9). Therefore, 11 - 12 (by 4). Cor. 1. a' - 1 - a is unique. Cor. 2. From a - b follows a' b'. 13. 1' -0; 0' - 1. For, 1'=1- 1 -0 (by Def. 1, 8); and 0'= 1-0=1 (by Def. 1, 10). 14. (a')'-a. For, in 5, let b - 0, c 1, d = 1; then the left member will reduce to a and the right to (a')' (by 2, 10, 13, 11). Def. 3. a"- (a')'.

Page 91 1914] Bernstein: Postulates for the Logic of Classes 91 15. (a —b) -c- (c' — a') -b. Proved by letting d - 0 in 5 (by 10, 13, 9, Def. 1, 14). 16. a-b - b'-a'. Proved by letting b -0, c b in 15 (by 10). 17. a - (b - c)-[(a - b)' —(a - c')]'. Proved by writing a, 0, b, c respectively for a, b, c, d in 5 (by 10, 16, 14). 18. a-a' — a. Proved by writing b - 0, c - a, d - a in 5 (by 10, 8, 14). 19. (a-b) -c (a-c) -b (by 15, 16, 14). 20. 0==1. For if 0 1, then, whatever a is, a a - a -1 -0 - 1 (by 10, 9); contrary to 6. 21. a= a'. For if a- a', then a a - a' - a = 0 (by 18, 8); hence a' =0' - 1 (by 12 Cor. 2, 13). Therefore, 0 - a - a' - 1; contrary to 20. 22. Def. 4. ab a —b'. Def. 5. a + b-(a'- b)'. Hence (by 22, 14), 23. (a + b)'a'b'; (ab)' — a' + b'; the well-known De Morgan's formulae. Del Re's Postulates. The propositions I-VIII following, taken by Professor Del Re as basis for his Algebra della Logica, may now be derived as theorems. I. xy is an element of K, if x and y are elements of K (by 22, 1). I'. x + y is an element of K, if x and y are elements of K (by 22, 1).

Page 92 92 University of California Publications in Mathematics [VOL. 1 II. There is an element T in K such that xT x for every element x. (The element T is 1, by 22, 13, 10.) II'. There is an element N in K such that x +N —x for every element x. (The element N is 0, by 22, 10, 14.) III. xy yx, if x, y, xy, yx are elements of K (by 22, 16, 14). III'. x +- y —y + x, if x, y, x + y, y + x are elements of K (by 22, 16, 14). IV. x + (y+ z) -(x+y) +z, if x, y, z, and their combinations are elements of K. (For, x + (y + z) -- [x' - (y'-z)']' - [(y'-z) -x]' [(y'- x) -z]'- [(x' - ) -z]' [(x' - y)" ]'- (x + y) + z, by 22, 16, 14, 19.) V. x (y + z) =xy + xz, if x, y, z, and their combinations are elements of K. (For, x (y + z)-=x - (y' —z)"= x - (y' -z) [(x - ')' - (x -z') ]' x y + xz, by 22, 14, 17.) VI. x + x= x, if x, x -+ x are elements of K (by 22, 18, 14.) VII. If T, N of postulates II, II' exist and are unique, then for every element x in K there is an element x such that xx - N, and x - x T. (The elements 1, 0, x' will serve for T, N, x respectively, by 22, 8, 13.) VIII. K has two distinct elements (by 6). The whole algebra of classes, having been derived from I-VIII, may thus also be derived from postulates 1-6. It can easily be verified that propositions 1-6 can be derived from I-VIII, so that the two sets of postulates are, in fact, equivalent. Independence of Postulates 1-6. The systems (K, -) following prove the independence of postulates 1-6, since there is for each postulate a system that is false for that postulate and not false for the rest. Denoting the elements of K by e>, the following tables define these systems (eq + ej if i + j).

Page 93 Bernstein: Postulates for the Logic of Classes 93 1. - I eo el eO eo eO e1 ~ 1 a where a is not in K. All the postulates, except 1, are satisfied by this system. 2. 1 el e2 e, el I el e el e1 e1 e1 e2 e3 el e2 e3 | e3 e, el Postulate 2 does not hold, since no element e, can serve as the element 1. In fact, e2- (ei - ei) - e3, whatever ei is. Postulate 5 is not contradicted, since no element 1 exists. 3. - e1 e2 I, eI e2 e61 el e e2 e2 e2 Postulate 3 is not satisfied; for, (e1 - el) - el == (e2 - e2) - el. Postulate 5 is not contradicted, as the element 1 is not unique. 4 1 eo el e2 e, eo eo eo eo eo el el eo el eo e62 e2 eo eo eo e3 e3 eo eo eo Here e2 - e3 = e - e2, but e, + e3,; so that 4 is false. Postulate 2 holds, either e1 or e3 serving as the element 1. 5. - eo el e2 e eo eo eo eo eo e, el eo el eo e, e2 eo eo el e.3 e3 eo eo eo Here 1-=el, unique. Postulate 5 is false, since (e2 - e) -(e- e) == {[( -e2 eO) - e'l - [ (e' - e2)' - e })'. 6. The class of one element e,, with eo - eo = e. Adjunction. The Duals of Postulates 1-6. The operation which forms the element a + b' from the elements a, b will be called adjunction and will be denoted by. We have: Def. 6. a- b - a + b'. Since a + b' (a' - b')', and since a - b ab' (a' + b)' (a'. b')', we get (a b)'=a'- b'; (a —b)'=a'- -b'; formulae that play exactly the same role in connecting the operations exception and adjunction as De Morgan's formulae do in the case of logical addition and logical multiplication; indeed, they are De Morgan's formulae expressed in different language. From this follows (with the help of 12 Cor. 2):

Page 94 94 University of California Publications in Mathematics [VOL. 1 The Law of Duality. For any proposition in the logic of classes expressed in terms of - and - there is a "dual" proposition which is obtained from the former by interchanging the signs —,, and the elements 1, 0. We thus have the following: Duals of 1-6. 1'. a - b is an element of K whenever a, b are elements of K. 2'. If 1' hold, there exists an element in 0 in K such that a- (b- 0) - a, whenever a, b are elements of K. 3'. If 1' hold, (a -:-a) - c - (b -b) — d, whenever a, b, c, d are elements of K. 4'. From a-b-b — a follows a = b, whenever a, b, a. b, b -a are elements of K. Def. 1'. a'- O - a. 5'. If 1', 2' hold, and if the element 0 is unique, then (a-b) -- (c d {[(a-b) ( d)- d' ]'. [ (c' a') - b]}, whenever a, b, c, d are elements of K. 6'. If 2' hold, there is an element a in K such that a = 0. It is evident that propositions 1'-6' might have been taken as postulates and propositions 1-6 derived from them. II PROOF OF THE INDEPENDENCE OF DEL RE ' POSTULATES The following systems (K, 0, ) prove the independence of Del Re's postulates I-VIII above. (The signs X, + without circles around them will denote arithmetical multiplication and addition.) I have aimed to make all the systems concrete. These "near algebras of logic" may be of interest in themselves. The systems are: I. K =the class of the two numbers 1, 2; x 0 y =xy; x 0 y-G. C. D. of x, y. Here N - 2, T - 1. Postulate I is false, because 2 X 2 is not an element of K. I'. The same as for I, with the interpretations of 0 and 0 interchanged. II. K the class of concentric circles enclosed by a circle W of center C, including the point-circle C but excluding circle W; x o( y the largest circle contained in x, y; x E y the smallest circle containing x, y. Postulate II does not hold, since no element T exists. II' holds, the element N being the point-circle C. Postulate VII is not false, for T does not exist.

Page 95 1914] Bernstein: Postulates for the Logic of Classes 95 II'. K -the class of positive rationals; a C a c a c a c a c - -. -; -0 = —, the mediant of-,b d b d b d b+d b d Postulate II' is false, since no element' N exists. 1 The element T is the rational-. 1 III. K - the class of point-groups in a plane area A, including the "null" group of no points; x O y group x; x @ y - the smallest group containing the points in x and in y. Postulate III does not hold. The element T is not unique, and so VII is not contradicted. III'. K the same as for III; x ( y = the largest group of points contained in x and in y; x @ y - group x. Here N is not unique. IV. K - the class of letters ei, the subscripts i being 0, 1, 2, 3, 4 and the combinations of 1, 2, 3, 4 taken two, three, and four at a time; ei o ej ep, where p is composed of the integers common to i and j; ei 0 ej = es, where s is composed of the different integers found in i and j with the exception that e12 Q e, = e: ( e12 - e3. Postulate IV is not satisfied. For, (e, 0 en) 0 e3 e3, while el ( (e2 0 e3) e123. Te 1234; N eo. V. K the class of linear spaces7 passing through a point P in a space of n dimensions, say S3 for concreteness, including the space S3 and the space of zero dimensions P; x y = the space of greatest dimensions contained in x and y; x 0) y the space of least dimensions containing x and y. Postulate V is not satisfied. For, let x, y, z be any three distinct co-planar straight lines; then x 0 (y 0 z) =x, while (x 0 y) 0 (x 0 z) = P. T= S3; N P; P= S3; S3=P; for x P, S3, the element x is any space not containing x which determines with x the space S3. VI. K=the class of point-groups in a plane area A, including A and the "null" group N; x o y = the largest group common to x, y; x 0 y the smallest group of points in x and in y, excluding those points common to x and y. Postulate VI does not hold, for x 0 x N for all groups x. 7 A linear space is one which contains the straight line joining any two of its points.

Page 96 96 University of California Publicationls in Mathematics [VOL. 1 VII. (K, o, ) =-the system of circles given for II, with the exception that 1W is not excluded. No element x exists for every x, such as postulate VII demands. N- C; T= -W. VIII. K -the class containing the single number 0; x O y xy; x y- x + y. Transmitted December 20, 1913.

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 5, pp. 97-114 July 6, 1914 ON A TABULATION OF REDUCED BINARY QUADRATIC FORMS OF A NEGATIVE DETERMINANT. BY HARRY N. WRIGHT. The purpose of this paper is to study certain relationships among positive reduced binary quadratic forms of a negative determinant and to derive enumeration formulae for the determination of the class number of such forms. The discussion is arranged in four parts. Part I contains a description of a useful tabulation of the forms studied and a determination of the number of solutions of the congruence x2 = D (mod a), in which the usual restriction that [D, a] = 1 is removed. In Part II is derived, from the results of Part I, an expression for the number of reduced forms for given values of a and A. Then this is summed for all values of a _ [4A] and a method of computing the number of reduced forms for a > u4 is described. Part III is a development of formulae giving the number of single forms on any value of A. In Part IV theorems are developed showing the distribution of reduced forms of the first and second sorts, and the relationship between the number of single forms of the first sort and the number of those of the second sort is determined for values of A containing no square factor. PART I. The well-known conditions necessary for ax2 + 2bxy + cy2 to be reduced when b2 - ac < are that: 1 2b1 < a, a < c and if 12b = a or a = c then b > 0. Such forms may be arranged in a table with a and A as arguments, as shown in the accompanying tabulations, and in which it is necessary to enter only the values of b and c. The forms in a given column, i. e. having a given value of a, arrange themselves in periods, each period covering a values of A and containing a forms. This is seen from the following consideration: if the form (ai, bi, cl) occurs on A1 then (al, b1, cl + 1) occurs on A2 = A1 + al; for from the equation ac - b2 = A we see that a change of one unit in c causes a corresponding change of a units in A. Therefore the forms of a given column with a given value of b occur at intervals of a places. In a column there are just a possible values of b, viz: O,:1, =:2, -3,, +2, or a-1 0, 2) 2y 3y...... a or a: 2 2 2

Page 98 98 University of California Publications in Mathematics [VOL. 1 the last named value depending upon whether a is even or odd. Now a form with each of these values of b will occur once, and only once, within a section of the column covering a consecutive values of A, where A > a2. This follows from the fact just noted that the forms of the column having a given b occur at intervals of a places. Therefore this section contains a forms which are reproduced periodically as A varies. For the purposes of our discussion we define a period as beginning with the place occupied by the form (a, 0, c) and ending with the place preceding that occupied by (a, 0, c + 1). The foregoing remarks apply to what we will call the complete periods of the table, viz: all periods for A > a2. But in those periods where A < a2 the forms are missing, first in which c < a and second in which c = a and b < 0. From the equation ac - b2 = A it is clear that these conditions may occur on any value of A < a2 but not on A > a2. Therefore we call those periods incomplete which extend from the least value of A for the column to A = a2. To determine this minimum value of A for a column we put the minimum value of c and the maximum value of b in ac - b2 = A and get, for a even: 2 3a2 a2 = = 4 4 and for a odd: 2 a - 1 2 3a2 + 2a - 1 a -t42 4=A. From these we may write the inequalities for a even: 3a2 A > and a < for a odd: > 3a2 + 2a - 1 -1 + 21 + 3A iA > --- and a -— 3 <3 4 3 The inequalities for a even are the ones usually given for a both odd and even, but they do not furnish exact limits for a odd as do the ones given here. It is interesting to note that in the tabulation all forms, (a, 0, a), occur on the parabola A = a2, which therefore is the boundary between the region of the complete periods and that of the incomplete periods. And, since in each complete period there are a forms, the number of forms in the region of such periods and for all values of A < A1 is approximately the area included by the curve, its axis and the ordinate A1. One may devise various methods for rapid construction of the table. One of these is to construct a complete period for each value of a within the limits of the table, then repeat these as often as necessary. In writing the forms in the incomplete periods one needs to remember the inequalities defining reduced forms. Another method, which is of service especially in building a table covering a comparatively short range of large values in A, is to begin for each a on a A which is a multiple of a with the form (al, 0, c1) and from this locate the other forms (al, 0, c) at intervals of a places. Then the form (al, b, cl) is b2 places above (al, 0, cl) and from it all forms (a1, b, c) may be located. Or one may find it advantageous to write all forms (a1, b, c1) first, which may be done rapidly by using the fact that

Page 99 Wright: Tabulation of Reduced Binary Quadratic Forms 99 beginning at zero successive squares may be located by the addition of successive odd numbers. Combinations of these and other properties of the table will suggest themselves as being useful to one in tabulating the forms. From the equation b2 - ac = - A = D we may write the congruence x2 = D (mod a). Assuming that the solutions of this congruence are expressed in such form that the absolute value of each is less than or equal to a/2, and that we refer only to forms in the complete periods, it follows that the solutions are 'the values of b for the forms of the given a and D and therefore that the number of forms with this a and D is the number of solutions of the congruence. Thus the problem of finding the number of reduced forms for given values of a and A resolves itself into the determination of the number of solutions of the quadratic congruence, with the understanding that the usual restriction that [D, a] = 1 does not apply. We now proceed to the solution of this problem. Lemma.-If the congruence x2 = D (mod a) has any solutions and if a = a' * pn D = D' * pK, where a' and D' are each relatively prime to p, then K cannot be odd and less than n. For let x1 be a solution of the congruence and assume K to be odd and < n, then xi2 must contain p as a factor at least K + 1 times. Then from the equation x2 - a * c = D it follows that p divides D at least K + 1 times which contradicts the hypothesis. Theorem I.-If a = pn (where p is an odd prime) and [a, D] = p", then the number of solutions, if any, of x2 - D (mod a) 1) is 2pM when K = 2/ < n; 2) is pA when K = 2/ or 2/ + 1 = n. We note by the above lemma that K cannot be odd and less than n. 1) When K = 2i < n. Put D = D' - pK. To enumerate the solutions of (1) x2 - D (mod pn) consider those of (2) x2 - D' (mod pn-2). Let xl + 1 * pn-2^ be a solution of (2): Then it follows that pA(xi + I. pn-2") is a solution of (1). Let 1 take various values, and as many of the resulting expressions are independent solutions of (1) as are incongruent modulus pn. Let r and s be two values of 1, such that p'L(x1 + rpn-2i) - pA (X + S. pn"-2A) (mod pn); then (r - s)p"-' 0 (mod pn) and r - s 0 (mod pa). Therefore, if s = 0, r = pA is the smallest value of I yielding a congruent solution modulus pn. i. e., for I = 0, 1, 2,... p - 1 we get solutions of (1) incongruent modulus pn. Therefore each solution of (2) yields pA solutions of (1). Moreover, all solutions of (1) are obtained in this way, for any one of them must be divisible by pA; and when divided by pA the quotient is a solution of (2). But by well-known results (2) has two solutions, if any..'. (1) has 2 ~ pT solutions, if any. 2) When K = 2/ or 2n + 1 = n.

Page 100 100 University of California Publications in Mathematics. [VOL. 1 (i) When K = 2t. Here the argument is identical with that in case 1), excepting that pn-2l = p0 = 1, with the result that congruence (2) has but one solution, and therefore (1) has pi solutions. (ii) When K = 2At + 1. Here the argument is the same as that of case 1) modified for subcase (i) with the additional change: Instead of multiplying the solution of (2) by p* we must multiply it by p(c+)/2 = p(+l)/2. Then we get r - s 0 (mod p(n-l)/2). But p(n-l)2= p'. Therefore (1) has pl solutions. Note that by the above theorem when K = n the number of solutions is one half of the number when K < n. Theorem II.-If a = 2n and [a, D] = 2", then the number of solutions, if any, of x2 = D (mod a) is 27+*, where pt = [K/2] and a = 0, 1 or 2, according as n - K <, = or > 2. Recall that by the lemma K cannot be odd and less than n. The argument of Theorem I, case 1) applies here with the change: Congruence (2) becomes x2 = D' (mod 2n-c), which we know has 20 solutions where a = 0, 1 or 2, according as n - K <, = or > 2. Therefore the number of solutions of x2 = D (mod 2") is 2. 2" = 20'+. Theorem III.-If n a = Hi pi (where po = 2), i=0 the number of solutions, if any, of x2- D (mod a) is 2"-~+a II pis where -=o n [a, D] - IIpii; ui = [K/2]; 0 v = the number of times Ki = ai for p an odd prime; ao = 0, 1 or 2, according as ao - 2o <, = or > 2. We get a solution of (1) x2 = D (mod a) for each set of solutions of x2 - D (mod 2ao) (2) x2 _ D (mod plal) x2 D (mod pn") By Theorems I and II the first member of (2) has 2a+~O solutions, and each succeeding congruence has 2pig' solutions, except in the v cases where Ki = ai, when this number becomes Pie', then there are as many sets of solutions of (2) as the product of these results, which is * 2y+o *. 2pl1 *. 2p2A2 * * 2pn. * 2- which may be written in the form n 2nv+or II pa'. 0 Note that in case [a, D] = 1, v =..o = A0 = * = n = 0

Page 101 Wright: Tabulation of Reduced Binary Quadratic Forms 101 n and 2f-v+ fII pi- becomes 2,"+f which is the well-known formula for the number 0 of solutions in this case. PART II. We have shown that the number of reduced forms, if any, for given values of n A and a in the complete periods is 2n-V+a II pi'. Until stated otherwise, we will 0 assume D = - A to contain no square factor. Then LlO =.l1 = *. = n = 0, and the possible number of forms is 2"-Y+r. This may be written in the form i=l Pi \iP)] where (D/pi"?) is an extension of the Jacobian symbol for the quadratic character of D with respect to pi, and is equal to 1, 0, or - 1 according as D is a residue, a multiple or a non-residue of pi'. The difference between the Jacobian symbol and the one used here arises when D = 0 (mod p). In this case the Jacobian symbol, (D/pii), becomes 0, but as we shall use it, (D/pai) = 0 only if D -= 0 (mod pai) and is equal to 1 or - 1 otherwise, depending upon whether D is a residue or non-residue in the extended sense (i. e. D not necessarily prime to pi) of pi?". In particular, since we have assumed that D contains no square factor, (D/pi) = - 1 when D = 0 (mod p) and D * 0 (mod pa); for if we could solve x2= D (mod pa) when D = 0 (mod p) and a > 1, D would contain p2 as a factor. Also we will later give meanings to (D/poa), i. e., for pa = a power of 2. The sum 1 + (D/piai) equals 2 for each time that D is a residue of pia and is not divisible by it, which happens n - v times, unless x2 D (mod pi) has no solutions in which case (D/pi) = - 1 and 1 + (D/pic') = 0. The factor 20, which is the number of solutions, if any, of the partial congruence in which 2a0 is the modulus, may be expressed as a binomial similar to the other terms of the formula by giving (D/poao) values as follows: for ao < 2 let (D/po~a) = 0, for ao = 2 let (D/poa~) = 1 for D = 4n + 1, and - 1 for D = 4n + 1. for ao > 2 let (D/poao) = 3 for D = 8n + 1, and - 1 for D = 8n + 1. Then 1 + (D/poao) is the number of solutions of x2 D (mod poao); for, in order that there should be any solutions, it is both necessary and sufficient that D = 4n+ 1 or 8n + 1, according as ao = 2 or > 2; and if ao = 1 there is always one and only one solution. By substituting the various values of (D/poao), it is seen that if there are solutions, 1 + (D/poo~) = 20 and, if there are no solutions, 1+ (D/po"a) = 0. n Therefore we may write 2n"-v+ in the form II [1 + (D/poao)], which not only 0 gives the number of reduced forms for given values of a and A when there are any, but is equal to zero when there are none.

Page 102 102 University of California Publications in Mathematics [VOL. 1 n To sum II [1 + (D/poao)] for the various values of a, it is of advantage to have 0 it in terms of the quadratic symbols (D/pi) rather than (D/pai), where at > 1, for all values of p except p = 2. We will now make this change. Comparing values of (D/pi) and (D/pi), we see that if (D/p) = 0, (D/pia') = - 1, for since ai > 1, and D does not contain a square factor (D/piai) = 0, and since D 0 (mod p) (D/pia) = 1. If (D/pi) = 1, (D/pia) = - 1. And conversely, if (D/p'ai) = 1, (D/pi) = 1; and if (D/pai) = - 1, (D/pi) = 0 or - 1 according as D is or is not divisible by p. If (D/paiL) = 0, ai = 1. Now let r ffn U A = ki and a = fIpai = hs i kiksi, 0 0 0 1 where h and k are primes, ho = po = 2, Si > 1 and the k"s are those odd k's which occur among the p's with exponents greater than one. Let 9[1 D)] ISa [ (D + piai ) Then.:[ ( h( )] [1+( hoii )]II [1+( ' )] By the above comparison of values of (D/hi?,) and (D/hi) we get h 0-[ —)].[l-()]h[1]( D (1) Sa [1+( h+o~) ]*I1 + (i]r [ )I hogi 0I ki'~i and if there are no values of k' this becomes (2) Sa [= + o I +( If there are one or more values of k', Sa = 0 for (D/k'i8i) = - 1; while if we replace (D/kTc'i) by (D/kc') we get sa=[X+hol ( D i ( k i = [i (_0)] i+( )] for (f)=o. Therefore, making this substitution in (1), we must subtract the latter result as a corrective term, getting (3) D (-[l - ( h-o) ] [ (1 + (i) ]. For greater simplicity in formulae we now define (D/ho) to mean (D/ho$o), and (2) and (3) become respectively (4)[ 1 + )] [+ ) o o Pi

Page 103 Wright: Tabulation of Reduced Binary Quadratic Forms 103 and a-(5) Sa= [ 1 + (h)] [ 1 + -(k- ) ]- [ 1 +(-) ] = [1 (i )]- [ (i)] -0 for =0. k') =0 If there are no values of k', the expression for Sa in (4) is the number of reduced forms for a given a and A in the complete periods. It may be put in better form for computing as follows: n 7, D D /D ID, II [1 (p)] 0(P ) j -(p i)(p') (pO)(p) (p P)' Let Pi * p * pk = P. Then D D D D D (p )(D). (.. k) ) = p__ pk) ( ). \iPi Pi ' Pi'P...Pk P If P is odd, (D/P) = 1 or - 1, when [D, P] = 1, and equals zero when [D, P] 4 1. If P is even and equals poao P', D ) (D ) D \ V P / \P oaO P in which (D/P') is evaluated as is (D/P) for P odd, and (D/poa") has the values given in its definition. Then, if we define 1 = (D/1), we may write n D D 0 Pi / E P where P runs through unity, the odd factors of a containing no square and such of its even factors as are divisible by pOaO and contain no odd square. To put this result in general form so that it will apply whether or not there are values of k', we need only to multiply it by the symbol (D/k'), which is zero when we have a value of k' and does not exist when there are no values of k'. Therefore, in either case, (6) Sa ) = E )1 represents the number of reduced forms on a and A in the complete periods. We proceed as follows to get a formula which sums this result for all values of a belonging to a given value of A and lying within the complete periods. For the present disregard the factor (D/k') in (6), and write Later correction will be made for the error thus introduced. Put A= [/A],

Page 104 104 University of California Publications in Mathematics [VOL. 1 the largest value of a within the complete periods, and let A S' = Sa. a=l Then, recalling the range of values of P, we see that (D/1) occurs once in each sa or A times in S'; (D/3) occurs [A/3] times and in general (D/P), when P is odd, occurs [A/P] times in S'. When P is even (D/P) occurs [A/P] - [A/2P] times in S', for, although P is a factor of the values of a which are multiples of 2P, it is not divisible by the highest power of 2 contained in such a's. Then, denoting the even values of P by Pe and the odd ones by Po we have A A D A A D <7) St L Sa I = +' (7) SZ a = (=) ( ) + i [( Pe) 2Pe)] (Pe To simplify this we now prove that A A A+x) [( ) 2x)] 2x (i) When [A/x] is even: A= q+r then =q+, and A ) A x x 2X 2x 2x A+x A 1 r 1 r 1 A + 2x =2+ q+2+2, but therefore..2x) = q 2x 2x 2 +2x +2' 2x <2 2x (ii) When [A/x] is odd: A r A q-1 r+x x A\ A\ q+1 = q + where q isodd. A= +2x, and - (A = 2 x x 2 2 + 2x (/ 2x 2 A+x A 1_q-1 r+x 1 q+ 1 r 2x 2x 2 2 2x 2 2 2x' Therefore (A+x q+1 \ 2x 2 * Using this in (7), we get A D A+Pe D (8) S'- ( Po ) ( LO J P O 2Pe )( Pe) In making this summation getting S' we assumed that none of the a's contain factors k'2, thus omitting the symbol (D/k') in (6). With this symbol present sa = 0 and without it sa ^=piy therefore when it is omitted and a contains a factor k'2 our result for Sa is too great by (D/P). Then S' should be diminished by this amount for each value of a containing a k'2. The a's which are multiples of ki'2 are ki,, 2k,2, 3kOi, * *, aki,2 * *, A ) k \ Aik'

Page 105 1914] Wright: Tabulation of Reduced Binary Quadratic Forms 105 By expanding E (D/P) into its separate terms we see that its value for a = a'ki'2 P is equal to its value for a = a', independently of whether or not a' is itself a multiple of the squares of one or more values of k'. Therefore the excess introduced into S' from these multiples of ki'2 is the summation of E (D/P) for P a 1,2,3,..., ( k2 which is the above formula for S' with A replaced by [A/ki'2]. Denote this by ci(1) _= ( A D + A+ ki2Pe D Then if, as we sum ci() with respect to i, X values of ki'2 are factors of the same a our result includes the correction from this a X times when it should contain it but once. This may be remedied by applying the principle of cross-classification. Let ci(v) be the sum of the corrections for all a's which are multiples of the product of the squares of v values of k', the subscript i denoting the particular combination of k's just as when v = 1 it denotes the particular k'. Then E Ci(1) - Ci(2) + E Ci(3) -. + (- 1)+1. E Ci(A) i i i i represents the correction for all a's which are multiples of no more than X factors of the type k'2, and letting X be the largest number of such factors in any a this becomes the total correction to be applied to S'. Then, denoting the number of reduced forms in the complete periods for a given value of A by S we have (9) S = S - - Ci(1) + E Ci(2) - ci3) + '' + (- 1)^Ci() * i i i But S' = ci() = c(O) i and we may write (10) S= {(- 1)V ci()}. v=O i Let i(y) = the ith combination of v factors k'2, then (11) C liP (P O T 2 *(V) Pe) (P ci(,,) =o P 2 e )' Put (11) in (10) and get (12) S t( —l ~ IV^AA D I y/A + l)Pe D (12) S = = ^= [ pjp+ (2.(-) Po -. (e ] ~ ' V=O i PO 0 2 ( )(A +i(v) Pe Pe(D] In finding S by this formula first compute the terms for v = 0. In most cases this completes the computation. If there are values of k', they are each odd factors of A and must be less than or equal to A1 for k'2 is a factor of one or more values of a and a _ At. There will not be a large number of such factors and the number of products of the squares of two k's which divide one or more a's is

Page 106 106 University of California Publications in Mathematics [VOL. 1 much less. e. g. the smallest possible k' is 3 for which A > 81; if k' = 5, A > 625; the smallest possible value of li(2 is 32 - 52 = 225 for which A _ 50625. Moreover when v > 0 the summations with respect to Po and Pe contain very few terms because of the rapid decrease in size of the greatest integer coefficients. The value of (D/Po) must be obtained where Po runs through the odd numbers which contain no square and are less than or equal to A. (D/Pe) may then be evaluated from (D/Po) for Pe ( 2ao) (Po) where Po' runs through the smaller values of Po. All terms containing (D/Pe) where Pe = 2 (mod 4) may be omitted for (D/2) = 0, and similarily all terms containing (D/Po) may be omitted where [D, Po] 4 1. We have derived (12) with the understanding that S is the number of reduced forms for D on the values of a from 1 to A = [4A] inclusive, but A may be given any value less than [VA] and the resulting value of S is the number of reduced forms over the corresponding range in a. We now consider the number of reduced forms on those values of a lying in the incomplete periods. Here aruns from [4] +1 to [i4A/3] or to ( 1 +- 2 1 3A) inclusive, according as the greatest value is even or odd. In computing use [ 44A/3] first; if this is odd use ( + 2 +1 - 3 ). The number of a's in the incomplete periods is approximately 3/20 1A. The number of solutions of the congruence b2 - D (mod a) is not in general the number of reduced forms for a and A in these periods for here A < a2 which, as seen from the equation b2 = ac - A, makes it possible for c to be less than or equal to a. Therefore the summation made over the complete periods cannot be extended over the incomplete periods. Theoretically all we can say is that the number of forms for a given a and A, where a > A, may be determined from the number of solutions of x2 - D (mod a) whose squares are greater than or equal to a2 - A; for if c is greater than or equal to a, b2 or ac - A is greater than or equal to a2 - A. Practically the computing of the number of forms for A in these periods is comparatively easy. Assume that we have first computed the number for the complete periods. Of the values of a in the incomplete periods those which have one or more factors (including a itself) which are values of k'2 or of which D is a nonresidue will have no forms. Such are readily found, principally by the use of the previously evaluated quadratic symbols. The remaining values of a will have forms belonging to them unless ruled out by the condition that c > a. For each of these a's we must find the number of values of b such that b2 = ac - A, where c > a and b2 < a2/4. To do this form the numbers a * a - A, a(a + 1) - A, a(a + 2) - A, or a2 -A, a - 2-+a, a2 a2- + 2a,... a - + i a, so long as a2 - A + i. a < a2/4.

Page 107 1914] Wright: Tabulation of Reduced Binary Quadratic Forms 107 (Note that i < 1/a(A - 3a2/4) which shows i to decrease rapidly as a increases.) The perfect squares among these numbers are the values of b2, each of which yields two forms excepting b2 = a2 - A or a4/4 in which cases there are no forms for the negative values of b. Example I.-Using (12) to find the number of reduced forms and from this the class number, h for - D = A = 2010 = 2 * 3 * 5 * 67 A = 44. (1) terms for v = 0: =4-6+-3-+ —1=++i+1 + A D D A ) D D D D D 5 ( 2Pe ) (,PJ ( 4 ) + ( 8 ) + (l6) + (28) + (32) + (44) o/\ D\ D+ D D =6( (,4; +3 (8;) + (16) + (4;) (7 ) ( 32) +DWD\ but for A even and ao > 1 (D/2ao) = - 1.'. this becomes = -6- 3-1 + 1 - 1 - 1 = - 11. (2) terms for v = 1; li(l) = 3 or 5: when li^() = 3, put [A/9] = 4 for A and repeating computation as in (1) get for Po -4()=-4 forP.e- (4) 1. when li(l) = 5, put [A/25] = 1 for A, and get for Po: - (D/1) = - 1. There are no further terms for v = 1 and none for v > 1..'. collecting the above results we have S = 47 - 11 - 4 +1 - 1 = 32. To find the number of reduced forms in the incomplete periods: here a = 45, 46, 47, 48, 49, 50, 51. By (6) and the values of (D/P) already found we readily see that none of these a's, except possibly 51, can have any reduced forms. For a = 51 form the numbers a2 - A, a2 - + a, etc., which are 591, 642. neither of which are squares, therefore there are no reduced forms on a = 51 and hence none in the incomplete periods,..h = S = 32.

Page 108 108 University of California Publications in Mathematics [VOL. 1 The computation of this result by the known formula-see ~106 of DirichletDedekind, Zahlentheorie3 h = 2 (a ) = 2 (a/A) where the limits of a are defined by A/8 < a < 3A/8, would be much more difficult as one may see from the following considerations. The range of a is from 252 to 753 inclusive, i. e. over 502 values. Then excluding all a's which are not relatively prime to A we have left about 152 values of a such that (a/A) = 1 or - 1. Also A = 2010, a comparatively large number, which complicates the evaluating of (a/A). Example II.-To find h when A = 827, a prime. A = [1827] = 28 terms for v = 0: A /D D D +5 D D D D PO PO P 28 1 )+9(3 5 +4(7 )+(211 +2( 13 + (15)+ (17)+ DD ) = 28 + 9 - 5 - 4 + 2 - 2 - 1 - 1 + 1 - 1 + 1 = 27 6 (A + a) (D ) = 4 ( 2) + ) + )+ (16 ) ( ) ( ~2Pe)(D 4 12 24 - 28 4-2 + 1 -1 -1 - 1 1 = -1. Since A is a prime there are no values of k' and therefore no terms for v > 0.'. S = 27 - 1 = 26. In the incomplete periods: [I]= 33 which is odd therefore we find [ - 1+21 + 3 2 and the values of a in these periods are 29, 30, 31 and 32. We see, by formula (2), that there are no forms for a = 29, 30, or 32 for ()=-1, ( = —1 and — ( 29 ) ' ( 5 ) ( 32 But (D/31) = 1. Then we form the numbers a2 - A + i * a which are: 134, 165, 196, 227, of which 196 is a perfect square giving b = - 14.

Page 109 Wright: Tabulation of Reduced Binary Quadratic Forms 109.'. we get 2 reduced forms in the incomplete periods. Adding this to S we get 28 as the entire number of reduced forms for A = 827. But 827 = 8n - 5.'. h = 3/4 * 28 = 21. To get this result by the known formula we must compute 4 h = (aA) 0 where a runs from 0 to 413 which makes 413 symbols, (a/827), to evaluate. PART III. We define single reduced forms as reduced forms which have no opposites. Such forms belong to the three types (1) (a, b, a), (2) (a, a/2, c), (3) (a, 0, c). It is our purpose now to determine the number of reduced forms of this kind occurring on a given A. Theorem I.-The number of single reduced forms of type (1) on A is the number of times A can be expressed as the product of two factors, the same in parity, the smaller of which is greater than or equal to one third of the larger. The form is (a, b, a). A = a2 - b2= (a - b)(a + b) let a-=b = and a + b = 3 from which a -y + and b= 2 2 2 But b _ a/2 then 2- <7~ and 7>. 2 = 4 = 3' And conversely given values of y and 3, the same in parity, and satisfying these conditions, an a and b may be found giving a form (a, b, a). y and 3 must be alike even or odd to give integers for 7+3 3-7 a= - and b= Theorem II.-The number of single reduced forms of type (2) on A is the number of times A can be expressed as the product of two factors, the same in parity, the smaller of which is less than or equal to one third of the larger. The form is (a, a/2, c) where a = 2a' a2 A = ac - = 2a'c - a' == a'(2c - a') c. > a.'. 2c - a' > a'. Let a' = 7 and 2c - a' = 3 then 7+3 c = - and a = 27. But c _ a then > + 2y or y <

Page 110 110 University of California Publications in Mathematics [VOL. 1 Conversely, given factors 7 and 6, the same in parity, and satisfying these conditions we can get a form (a, a/2, c). Theorem III.-The number of single reduced forms of type (3) on A is the number of times A can be expressed as the product of two factors. The form is (a, 0, c) and A = a * c from which the theorem follows. We need to note the possibility of one form belonging to two types. This can happen only in cases of (1) and (2), and of (1) and (3). If a form belongs to both (1) and (2) y = 6/3 and A = 3r2; if it belongs to both (1) and (3), y = 6 = a and A = y2 = a2. Thus we have one of these special cases only if A is a square or 3 times a square. Theorem IV.-If n A = fi piti i=l n and is odd, the number of single reduced forms is II (1 + ai). 1 n The number of factors of A is 1 (1 + ai). Then the number of times A can be n expressed as the product of two factors is 2 *I (1 + ai), except when A = a2. 1 n By theorems I and II we get 3 * I (1 + ai) forms each belonging to one of the 1 n first two types. By theorem III we get 3 * II (1 + ai) forms of type (3). Then n if no form of (1) or (2) belongs to (3) we have in all f (1 + ai) single forms. 1 As already noted the only time a form of (1) or (2) belongs to (3) is in case of (a, 0, a). Here A = a2 and therefore the number of times it may be expressed as the product of two factors, counting the case of A = a * a, is (1 + a) + 1. 2 [1 We then have - [l (1 + a) + 1] forms of types (1) and (2) and the same number n of type (3), making - (1 + ai) + 1 in all. But since (a, 0, a) belongs to both (1) n and (3) we have counted it twice. Therefore subtracting one we get II (1 + ai) single forms as before. Theorem V.-If A = fiPiai 0 (where po = 2), the number of single reduced forms is ao I: (1 + ai). 1 n First assume A =:= a2. The number of factors of A is (1 + ao) I1 (1 + ao) and 1

Page 111 1914] Wright: Tabulation of Reduced Binary Quadratic Forms 111 therefore the number of forms of type (3) is 1 The number of times we can write A = - * 6 where y and a are the same in parity is ao- 1 "2 -I( l+ a,). 1 For y and a must each be even and not divisible by 2ao, and the number of factors of A of this kind is n (ao -1) + (1 + -t). 1 Then there are - (1 + ai) 2 1 forms of the two types (1) and (2). Therefore the entire number of single forms is (ao + ao ) 1 ( 2 + 2 ) i+ o(1 + a) =. + i) 2 2 If A = a2 it may be shown, as in the proof of theorem IV, that the result is unchanged. In particular when A contains no square factor the expression for the number of single forms becomes 2" where n is the number of odd primes in A. PART IV. All reduced forms on a value of A which contains no square factor are primitive, i. e. [a, b, c] = 1. Primitive forms are said to be of the first or second sort according as [a, 2b, c] = 1 or 2. We will now determine the distribution of these two classes of forms in the tabulation. We know that primitive forms of the second sort can occur only on values of A where A = 8n - 1 or 8n - 5. Therefore we will confine our discussion to values of A of these types. Theorem 1.-When A = 8n - 5 and contains no square factor, and a = 2 (mod 4) all forms are of the second sort and for a * 2 (mod 4) all forms are of the first sort. If (a, b, c) is of the second sort a and c are even and b is odd. a * c =A +b2 but b2 = 8m+ 1 then a * c = 8n - 5 + 8m + 1 = 8(n + m) - 4 and ac _ 4 (mod 8) which may be only when: (1) a_ 4 (mod 8) and cis odd

Page 112 112 University of California Publications in Mathematics [VOL. 1 or (2) a 2 (mod 4) and c 2 (mod 4) or (3) aisodd and c 4 (mod8). But we can have forms of the second sort only under (2), since a and c must both be even. Conversely whenever a 2 (mod 4) we have forms of the second sort. For if a - 2 (mod 4) and ac - 4 (mod 8) c must be even. And since A is odd b must be odd. Theorem II.-When A = 8n - 1 and contains no square factor, one half the forms on a 0 (mod 8) and all forms on other even a's are of the second sort. a * c = A + b2 = 8n - 1 + 8m + 1 = 8(n + m) which can be only when: (1) a - 0 (mod 8) and c odd or even or (2) a 4 (mod 8) and c O (mod 2) or (3) a 2 (mod 4) and c O (mod 4). It is evident that all forms on values of a under (2) and (3) are of the second sort. We investigate forms on values of a 0 (mod 8) as follows: If (a, b1, cl) and (a, b2, c2) occur on the same A, then ac - b12 = ac2 - b22 and b22 - bl2 _ 0 (mod a); and conversely if b22 - b12 - 0 (mod a) and (a, bl, cl) occurs on A, (a, b2, c2) occurs on the same A. Let (a, b1, cl) be on a given A and let a be a factor of a. Then if a/a - b = b2 the form (a, b2, c2) is on the same A when a( _ b )-b0O (mod a) 7 - 2a ---i =0 (moda) a a a b --- 0 O (mod a) which is true when a -- 2b- 0 (mod a). a

Page 113 Wright: Tabulation of Reduced Binary Quadratic Forms 113 Now all values of b on a= 0 (mod 8) are odd. Put b1 = 4hl 1 and a = 8k. Let a = 2 then a a - = -0 (mod 4) a 2 and -2bl O (mod 2). Therefore b1 and a/2 - b1 = b2 occur on the same values of A and a. b2= a/2 - bl = 4k - (4hl: 1) = 4(k - hi) = 1 = 4h2 = 1 in which h2 = k - hi or hi + h2 = k. Now b22 - ac2 = bl2 - aci b22 - bl2 = a(c2 - ci) = 8k(c2 - cl). Also b22 - bl2 = (4h2 - 1)2 - (4h =F 1)2 = 16(h22 - h12) 8(h2 + hi) = 8(h2 + h1)[2(h2 - hi) - 1]. Therefore C2- c = 2(h2- h1) - 1 and c2 and c1 are different modulus 2. Then one of the forms (a, bl, c1) and (a, b2, c2) is of the first sort and the other is of the second sort, i. e., for each form of the first sort on a given A = 8n - 1 and a, where a _ 0 (mod 8), there is a form of the second sort on the same A and a and conversely. It is known* (1) that for A = 8n - 1 and for A = 3 the number of forms of the first sort equals the number of those of the second sort; and (2) that for A = 8n - 5 (excepting A = 3) the number of forms of the second sort is one-third of the number of those of the first sort. Attempts to prove these relationships by the methods of this paper have been unsuccessful, but the following theorem may be proved and is of interest in this connection. Theorem III.-If A = 8n - 1 or 8n - 5 and contains no square factor the number of single forms of the first sort equals the number of single forms of the second sort. From Part III recall the three types of single forms, viz: (1) when a = c, (2) when b = a/2, (3) when b = 0. Now a form of type (1), (a, b, a) is second sort, for if a and b were both odd A would be even and if a were odd and b even we would have A = a2- b2 = 8m+ 1-4k = 8n - 3 or 8n - 7. Therefore a is even and b is odd. * Dirichlet-Dedekind, Zahlentheorie, ~ 97.

Page 114 114 University of California Publications in Mathematics [VOL. 1 A form of type (2) is of the second sort, for: (a, a/2, c) may be written (2b, b, c). Then A + b2 C= 2b in which b is odd and b2 = 8m + 1. Then 8n- 1 + 8m + 1 c= 4o2b = 4n+m or b 8n - 5 + 8m + 1 or -- 2b 2(n + m) - 1 b Therefore c is even. A form of type (3), (a, 0, c) is first sort for otherwise A would contain a square factor. Therefore all forms of types (1) and (2) are of the second sort and those of type (3) are of the first sort. But in Part III we saw that the number of forms of types (1) and (2) equals the number of those of type (3), from which the theorem follows.

Page [unnumbered] \a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 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 1. 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 10 0 11 0 12 0 13 0 14 0 15 0 16 0 17 0 18 0 19 0 20 0 21 0 22 0 23 0 24 0 25 0 26 0 27 0 28 0 29 0 30 0 31 0 32 0 33 0 34 0 35 0 36 0 37 0 38 0 39 0 40 0 41 0 42 0 43 0 44 0 45 0 46 0 47 0 48 0 49 0 50 r I 2. 1 2 ) 2 1 3 0 3 1 4 0 4 1 5 0 5 1 6 0 6 1 7 0 7 1 8 0 8 1 9 0 9 1 10 0 10 1 11 0 11 1 12 0 12 1 13 0 13 1 14 0 14 1 15 0 15 1 16 0 16 1 17 0 17 1 18 0 18 1 19 0 19 1 20 0 20 1 21 o 21 1 22 0 22 1 23 0 23 1 24 0 24 1 25 0 25 3. 1 3 0 3 ~1 4 0 4 ~1 5 0 5 16 0 6 A=1 7 0 7 81 8 0 8 ~1 9 0 9 -1 10 0 10 d=l 11 0 11 -1 12 0 12 =-1 13 0 13 -1 14 0 14 ~1 15 0 15 =I l 16 0 16 i 1 17 4. 5. 14 0 4 =1 5 0 5 -1 6 0 6 -1 7 0 7 ~18 0 8 =t1 9 0 9 =1 10 0 10 -1 11 0 11 41 12 0 12 2 4 2 6 2 7 2 8 2 9 2 10 2 11 2 12 2 13 25 1 5 05 -2 6 -1 6 0 6 -2 7 -Z1 7 07 ~2 8 -1 8 08 42 9 1 9 0 9 -2 10 ==1 1C 0 1C 6. 7. 8. 3 6......... 2 6........ 3 7........ 1 6......... 0 6..........2 7........ 3 8...............3 7....1 7.......... 0 7............2 8........ 39 27......1 8........ 0 8 1 74 8...........2.9........ I 1 1 a 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 94 95 96 97 98 99 100 1. 2. 0 51 1 26 0 52 0 26 0 53 1 27 0 54 0 27 0 55 1 28 056 0 28 0 57 1 29 0 58 0 29 0 59 1 30 0 60 0 30 0 61 1 31 0 62 0 31 0 63 1 32 0 64 0 32 0 65 1 33 0 66 0 33 0 67 1 34 0 68 0 34 0 69 1 35 0 70 0 35 0 71 1 36 0 72 0 36 0 73 1 37 0 74 0 37 0 75 1 38 0 76 0 38 0 77 1 39 0 78 0 39 0 79 1 40 0 80 0 40 0 81 1 41 0 82 0 41 0 83 1 42 0 84 0 42 0 85 1 43 086 0 43 0 87 1 44 0 88 0 44 0 89 1 45 0 90 0 45 ft91 1 46 0 92 0 46 0 93 1 47 0 94 0 47 0 95 1 48 0 96 0 48 0 97 1 49 098 0 49 0 99 1 50 0 100 0 50 3. 4. 5. 6. 7. I 11 0 17 +1 18 0 18 =1 19 0 19 -1 20 0 20 -1 21 0 21 1 22 0 22 -1 23 0 23 -1 13 0 13 -1 14 0 14...... ~1 15 0 15 -1 16 0 16 -1 17 0 17 2 14 2 15 2 16 2 17 2 18 = 2 11 4=- I1 11 0 11 - 2 12 =i=l 12 0 12 -2 1.3 -1 13 0 13 -2 14 -1 14 3 10 -1 9 0 9 L2 10 3 11 -1 10 0 10 -2 11 3 12 ~1 11 0 11 ~2 12 3 13 I II I =i=2 8 -3 9 -1 8 0 81 2 9 - 3 10 =i=1 9 0 9 -2 10 -3 11 =-l 10 I I. I.I I, I. I, I 8. 9. 3 8............ 4 9............ 2 8.................. 1 8 =- 3 9............ 0 8 4 10........................ 4 9...... =t2 9.................. 10................................................................................................................... 11. I I I I I I I I I I I................ 0 14...... 0 10.................. 1 24 1- 18.... 42 15 1- 12...... -1 9 -+3 10...... 0 241 0 1812 19 0 12 1> -1 25 0 25 -1+ 26 0 26 r-1 27 0 27 -1 28 0 28 -1 29 0 29 =-=1 30 0 30 i1 31 0 31 -1 32 0 32 -1 33 0 33 -1 19 0 19 =..l 20 0 20 = 1 21 0 21 =i=l 22 2 20 2 21 2 22 1=1 15 0 15 -2 16 -1 16 0 16 = 2 17 -1 17 0 17 i 2 18 -2 13 3 14 -1 13 0 13 -2 14 3 15 -1 14 0 14 -2 15 3 16 I.I 0.I I I: I I 5 i..... I =1=2 11 -+-3 12 -1 1.1 0 11 -2 12 -3 13 -l 12 0 12 -2 13 -3 14 =l 13 0 13 -2 14 =i=3 15 -1 14 0 14 ____ II II i I I 0 91 4 11.......1 =-c2 10 =itl 10 0 10 -2 11 =t3 11 4 12 ~4 10 2 9 1 9 0 9 -4 11 )I I 3 9................................................ -3 10........................ 5 10 4 10 5 11 \a 101 102 103 104 1.05 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 0 108 0 54 0 109 1 55 0 110 0 55 0 111 1 56 0 112 0 56 0 113 1 57 0 114 0 57 0 115 1 58 0 116 0 58 0 117 1 59 n nlson 51 0 36 -1 37 0 37 ==1 38 0 38 =-1 39 0 39 r r 51 k( 0 119 1 60 ~1 40 0 120 0 60 0 40 0 121 1 61...... 3 122 0 61 -1 41 0 123 1 62 0 41 3 124 0 62...... 0 1251 63 11 42 0 126 0 63 0 42 0 127 1 64...... 0 128 0 64 -1 43 0 129 1 65 0 43 0 130 0 65...... 0 131 1 66 -1 44 0 132 0 66 0 44 0 133 1 67...... 0 134 0 67 =1= 45 0 135 1 68 0 45 0 136 0 68...... 0 137 1 69 =+l 46 0 138 0 69 0 46 0 139 1 70...... 0 140 0 70 1=1 47 0 141 1 71 0 47 0 142 0 71...... 0 143 1 72 -1 48 0 144 0 72 0 48 0 145 1 73...... 0 146 0 73 1-i 49 0 147 1 74 0 49 0 148 0 74...... 0 149 1 75 -1 50 0 150 0 75 0 50 1. 0 101 0 102 0 27 =-l 28 0 28' ~1 29 0 29' ~1 30 0 30 ~1 31 0 31 =-l 32 0 32 -1 33 0 33 -1 34 0 34 -1 35 0 35 -1 36 0 36 =141 37 0 37 2. 1 51 0 51 3. =-l 34 0 34 i. 0 10311 52...... 0 104 0 105 0 106 4. 0 52 1 53 0 53 =-1 35 0 35 0 26 -l 27 ~l1 261. 0 10711 54 =i-l 36 5. 2 27 2 28 2 29 2 30 2 31 2 32 2 33 2 34 2 35 2 36 2 37 2 38 ~2 21 -1 21 0 21 -2 22 -1 22 0 22 -+2 23 -1 23 0 23 ~2 24 4i= 24 0 24 =4=2 25 -1 25 0 25 -2 26 -2 18 3 19 ~ 3 16 6. 1=t 17 0 17...... I~11 8 0 18 -2 19 3 20 -1 19 0 19 -2 20 3 21 — 1 20 0 20 =1=2 21 3 22...... -1 21 -0 21 -= 1 15 0 15 -2 16 ~3 17 -1 16 0 16 -2 17 -3 18 =-l 17 0 17...... -2 18 -3 19 =t1 18 0 18 i I I I II I I -1............ ==2 10...... ~1=j 11 -3 12............ I 7. -2 15 i. I 0 22 2 23 -1 23.... 0 23 2 24 -1 24.... 0 24 2 25 -1 25.... 0 25 2 26 -~4 13 I.I 8................... I 9. I -1 18 =1=1 15 0 18 0 15 =2 19............ -2 16...... 3 17 ~1 19...... 0 19 -1 16 =1=2 20 0 16....... 2 17 i-1 20 3 18 0 20...... 10. ~-3 11 I i I I -1 131=3 141...... 0 11 =12 12...... -1 12 0 12 -4 —2 lo 4 13............ = l 10...... 0 10...... ==4 12 ~,-3 13 — 2 11 4 14...............l.11... 11.... 11 I I -3 11 =t3 12 3 10.....4.11.... 5 12.... 2 10 5 11 1 10.... 0 10|.... I...... - 2 22...... -1 26 3 23 42 19 0 261............ =-2 27 =-l 22 -3 20...... 0 22 -1 19............ 0 19 ~1 27 -2 23...... 0 27 3 24...... 42 28...... =2 20...... - 1 23............ 0 23 -3 21 L — 28...... = 1 20 0 28 i 2 24 0 20 =^2 29 3 25............ -1 24 ~2 21 =11 29 0 24...... 0 29...... -3 22 ~2 30 -+2 25 -1 21...... 3 26 0 21.................. -1 30 -1 25...... 0 30 0 25 -2 22 0 13 =2 14 ==l 14 0 14 -2 15 -1 15, 0 15 =t2 16 -=l 16 0 16 -2 17 -1 17 0 17 -i=2 18 -1 18 0 18 -2 19 4 15 ~3 15 4 16 ~3 16 4 17 -3 17 4 18 -3 18 4 19 ~3 19 4 20 ~2 12 -1 12 0 12 ~4 14 -2 13 -l 13 0 13 ~4 15 -2 14 -=l 14 0 14 -4 16 ~2 15 ~1 15 0 15 -4 17 42 16 -1 16 0 16 ~f4 18 ~2 17 -3 13 -3 14 -3 15 ~3 16 -3 17 I I I ~4 12 5 13 -2 11 =t1 11 0 11 ~3 12 ~4 13 5 14 ~2 12 -1 12 0 12 ~3 13 ~4 14 5 15 ~2 13 -1 13 0 13 -3 14 -4 15 5 16 -2 14 -~:l 14 0 14 -3 15 ~4 16 5 17 ~2 15 =-=1 15 0 15 11. V. 4 11...... t5 12 1...... 3 11...... ~.4 12...... 2 11.......5 13...... 1 11...... 0 1. 1...... -3 12.......4. 13........2 12...... =15 14...... =1=1. 12...... 3 12........3 13............ 3. 12 -4 14...... ~2 13...... -5 15 2 12 ~1 13...... 0 13 1.. 1. 2...... 0. 12 -3 14..............3.13 -4 15...... - 2 14...... 6 12 5 12 6 13 4 12 ~5 13 6 14 -4 13 -5 14 6 15 I 2. 13. 14................................................................................................................................................................................................................................................................................................................................. 6 13 -. - -.................................................................................................... 5 13.............. =1= 6 14.......... 7 14.............................. I I I.1 I - i I I I

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 6, pp. 115-162 July 6, 1914 THE ABELIAN EQUATIONS OF THE TENTH DEGREE, IRREDUCIBLE IN A GIVEN DOMAIN. OF RATIONALITY. BY CHARLES GUSTAVE PAUL KUSCHKE. CONTENTS. PAGE Introduction...................................................................... 115 The Irreducible Abelian Equations of the Tenth Degree................................ 118 A. <p = 0 (i = 1 to 5)........................................................ 119 I. r = 0............................................................... 121 1. S = 0........................................................... 121 a. j = 0 (j = 1, 3, 7) (type I).................................... 122 t3. b7 7 0; 2 $ 0 (type II)........................................ 123 2. S 0 (no type)..................................................... 126 II. r 0................................................................ 128 1. p2 = p3 = 0 (type III).............................................. 129 2. p2 = 0; p3 $ 0 (type IV)............................................ 132 3. p2 $ 0; p3 $ 0 (no type)............................................ 135 B. pi are all distinct (i = 1 to 5)............................................... 137 I. R 2 = 0............................................................... 140 1. S 2 = 0........................................................... 140 a. 2 = 0; p4 $ 0 (type V)........................................ 140 f. p2 $ 0; p4 0 (type VI)........................................ 144 2. S2 $ 0 (no type)................................................... 149 II. R 2 $ 0.......................................................... 150 1. P8 = P2 = 0 (type VII)............................................. 150 2. p8 = 0; P2 $ 0 (type VIII).......................................... 156 3. p8 $ 0; p2 $ O (no type)............................................ 162 C. C onclusion................................................................ 162 INTRODUCTION. In the following pages is found the number of types of Abelian equations of the tenth degree irreducible in a given domain of rationality. The various types are discussed with respect to their construction and the character of their roots. The rational domain does not need to be the natural domain; in fact it will appear that there is no irreducible Abelian equation in the natural domain of rationality R(1). If f(x) = 0 is an irreducible equation of the tenth degree, then all the ten roots

Page 116 116 University of California Publications in Mathematics [VOL. 1 xi = 1 to 10) must be distinct, for otherwise f(x) = 0 and d f(x)= would have a common root, which is impossible, as - f(x) is only of the ninth degree in x.* From the irreducibility of f(x) follows further that its Galois group is transitivet and as 10 = 2.5 this group must be imprimitive.: Our first question therefore is: What are all the imprimitive groups of the tenth degree? These groups are found by Cole~ and divided into three types. 1. The groups which contain only 5 systems of intransitivity. 2. The groups which contain only 2 systems of intransitivity. 3. The groups which contain both 5 and 2 systems of intransitivity. To type 1 belong thirteen groups, the largest of which is of order 3840 containing all the other twelve as subgroups To type 2 belong fifteen groups. The largest group is of order 28 800 and contains all the other fourteen groups as subgroups. Type 3 includes eight groups, of which the largest is of order 240. Nettoll showed that the group of an Abelian equation is an Abelian group and vice versa. Our first problem is, therefore, to search for all Abelian groups among the above thirty-six imprimitive groups. Expressing the group of order 240 in type 3 as a substitution group, we see the following relation between the eight groups of type 3, which we may call G240; G120; G'120; G40; G20; G20; G10; G'10 where the subscripts indicate the order: G240 contains all the remaining seven groups as subgroups. G'20 is a subgroup of G'120 whereas G20 is a subgroup of G120 as well as of G40 and contains both G'1o and Gio as subgroups. Writing this relation in a better form we have the following scheme IG120 G FGio G240 G40J [G o l G'12o G'20 As G'lo is the dihedral group of order 10 and G'2o contains five subgroups of order 4 which are conjugate by Sylow's theorem, it follows that the cyclical group Glo is the only Abelian group of type 3. We shall use as G1o the ten powers of S = (X1X7X3X9X5X 6X2X8X4X1) * Weber, Algebra, vol. I, p. 454. t Weber, Algebra, vol. I, p. 482. t Netto, Substitutionentheorie, ~237. ~ Cole, Quart. Journ. Math., vol. 27, 1895, p. 39; but also Miller, Bull. Amer. Math. Soc., vol. 1, pp. 67-72. [ Loc. cit., ~180, and C. Jordan, Traite d'Analyse, ~402.

Page 117 1914] Kuschke: Irreducible Abelian Equations of the Tenth Degree 117 and in dealing with substitutions we shall write only the subscripts of the x's, replacing the subscript ten by zero. Whenever we can show that a given equation f(x) = 0 has a Galois group with two as well as five systems of imprimitivity, but which cannot contain G'lo or G'2o, f(x) = 0 must be, according to the above scheme, an equation belonging to Glo if f(x) is irreducible. For the proof of the fact that G'io and G'20 is not a subgroup of the Galois group of f(x) = 0 we shall need the two substitutions al = (16) (20) (39) (48) (57) which is in G'io and a2 = (1748) (2936) (50) which is in G'20 These three substitutions s, eo, 02' already show that we have chosen as our two systems of imprimitivity [X1X2X3X4X5], [X6X7X8X9X10] and as our five systems of imprimitivity [X1X6], [X2X7], [X3Xs8, [X4X9], [X5X1o] Looking over types 1 and 2 for Abelian groups we may easily show that there is no other Abelian group. Regarding type 2, the order of an Abelian group of degree 10 cannot exceed the order 2. 52 = 50,* so that there is only one group to be considered, namely the group of order 50, which Colet indicates by G5o (XX2X3X4X5) 5(X6X 8xx9xloX) 5(Xl6 6'X2X7 X3X8 X4X9' X5X10) This group, however, contains the two substitutions (16) (27)(38)(49)(50) and [(12345)(16) (27) (38) (49)(50)]5 = (19)(74)(20)(85) (36) so that G50 would contain more than one substitution of order 2, which is impossible here by Sylow's theorem, hence there is no Abelian group of type 2. Regarding type 1 we note that the only bases for the construction of the thirteen groups in question are (16)(27)(38)(49)(50) and {(16)(27)(38) (49) (50) }po so that every group of type 1 contains the substitution a = (16)(27). Further, an examination of Cole's list shows that every group contains the substitution b.= (13579)(68024), hence every group of type 1 contains a-cba = (63529)(18074) $ b, and hence no group of type 1 is Abelian. * Netto, loc. cit., ~170. t Lne. cit. n. 41. and Miller. loc. cit.

Page 118 118 University of California Publications in Mathematics [VOL. 1 We have then the result: All irreducible Abelian equations of the tenth degree belong to the cyclical group of order ten, which we have chosen to be Glo 1 {s = (xlx7x3x9x5x6x2x8x4xl0) } Throughout the whole investigation we shall assume that there is given a certain rational domain, for which the necessary and sufficient conditions for all the irreducible Abelian equations are to be derived, as the reducibility of a polynomial depends on the rational domain employed. We shall not distinguish at all between the quantities lying in the natural domain, usually denoted by R(1), and quantities lying in the given domain. All quantities lying in the given rational domain shall be called "rationally known" or simply "rational quantities," for we do not restrict our investigation to the natural rational domain and have therefore to give to the expression " rational" a broader meaning than is done usually in mathematics. Whenever we mean specially a quantity lying in the natural rational domain we shall always write " quantity belonging to R(1)." THE IRREDUCIBLE ABELIAN EQUATIONS OF THE TENTH DEGREE. The corresponding group is Gio {s = (1739562840)}. Throughout these investigations let e be a primitive root of 10- 1 = 0, and co be a primitive root of 5- 1 = 0. Then we may write = = 2 - = 7 = [- 1 + -5 + /- 10- 2/5], o2 = E4 = - c9 = i[- 1 —d- + 4- 10 +2 45], 3= E6 = 6 - = 4[- 1 - - 10- 10 + 25], o4= E8 = - = [1 + - -4- 10 - 2/5], 05 = =10 = + 1, E5 = -1. Let us first consider the five systems of intransitivity and form the functions <Pi = xi + Xi+5 (i = 1 to 5) These five functions, while formally distinct, may be numerically equal. Assuming then that at least two of these five spi are equal, we have by applying our group Gio to the relation pi = fit (i $ i') the result that they are all equal to one another and whenever in our equation f(x) = 0 we have j=10 xi = o, '=1

Page 119 1914] Kuschke: Irreducible Abelian Equations of the Tenth Degree 119 then oi = 0 and hence xi = -xi+5 (i = 1 to 5), so that our equation f(x) = 0 is a quintic in x2. We note here that the transformation of f(x) = 0 into an equation in which 10 Exi = 0 is a very simple matter and does not change the character of f(x) = 0. Therefore without loss of generality we may assume that in our f(x) = 0 the sum of the roots vanishes, as we are then gaining much simplification in our investigations. This leads us to a division of our subject into the two parts A. =O ii= 1 to 5. B. c105O A. i =0 (i = 1 to 5). Here our given equation is a quintic in x2 with the roots xi (i = 1 to 5). This quintic can be taken again in the reduced form and hence we may write throughout this section i=5 i.2 = O. i=1 The following Lagrange's system is the skeleton for our investigations in this section,1 = Xi + W3X2 + OX3 + C4 X4 + C2XX5, t2 = Xi + &4X2 + w3X3 + H02 4 + WX5, t3 = X1 + x.X2 + 02X3 + w3X4 + C4X5, 44 = X1 + C2X2 + w4X3 + &X4 + 3X5, 45 = X1 + X2 + X3 + X4 + X5. Consider now for a moment the two systems of intransitivity by forming the function i=5 Y =Xi, t=l which is changed by G10 into i=5 Y = =1Xi+5. i=1 It follows then that y2 is rationally known, and we have, if y2 = c, the condition c is rational, but not a perfect square and - c is of course the constant term inf(x) = 0 throughout part A. Just as fc, so is i=5 E = Xi ~=1 a function which is changed only by the odd substitutions of G10; it follows then if = 5r, c then r is rationally known.

Page 120 120 University of California Publications in Mathematics [VOL. 1 Similarly /5' (i = 1 to 4) takes only two values under Go1, so that i5 is a quantity in our given rational domain if we adjoin to it c, and we must have, for instance, ~15 = c[k + l+5 + m - 10 + 2 +5 + n 4- 10 - 25], where k, 1, m, n are necessarily rational quantities. Replacing finally k, 1, m, n respectively by 55c2K, 55.c2L, 55c2M, 55c2N, then K, L, M, N are rationally known, and Lagrange's system becomes i = 54 c pi (i = 1 to 4), P1 = 4K+ L5 +M - 10 + 25 + N'- 10 - 2/5, P2 = K - L - N 10 + 245 + M - 10- 25, p3 = K- L 5 + N -10 +245-M -10-2 5, p4 =K+L -M M- 10+2 -N - 10-2 5, Combining the two forms of Lagrange's system, we get by additionthe roots of f(x) = 0, namely X1 = - X6 = + c[Pi + P2 + p3 + p4 + r], X2 = - X7 = + C[a2pl + wP2 + W4p3 + C3p4 + r], X3=- = -X = + c[w4pl + W2P2 + wp + Wp4 + r], X4 = - Xg = + [ oP1 + CO3p2 + 2p3 + - 4p4 +r], X5 = - X10 = + /c[W3pl + w4p2 + 0P3 + W2p4 + r]. It is seen now by direct computation that 14 = 25c VP + Q 5 = (w + w4)fi + (c2 + c3)f2 = 25cplp4, and V2.3 = 25c Vp - Q- 5 = (w2 + o3)fi + (w + w4)f2 = 25cp2p3, where P = K2+ 5L2 + 10(M2 + N2), Q = 2KL - 2(M2 - N2) - 8MN, fi = XlX3 + Xlx4 + x2X4 + X2X5 + X3X5, f2 = XlX2 + XlX5 + X2X3 + X3X4 + X4X5. Hence P and Q are rational and, as fi and f2 are unchanged by our Gi0, both fi and f2 are rational too. It follows then that P1P4 and P2P3, V14 and 4243 are rationally known if we adjoin 45 to our rational domain; for (W2 + w3) as well as (? + a4) are quantities in R( I5). Therefore P = Q 15 is a perfect fifth power of a rational

Page 121 1914] Kuschke: Irreducible Abelian Equations of the Tenth Degree 121 quantity in this extended domain and hence we may write P1P4 = R + S 5 and p2p3 = R - S 5, where then R and S are rationally known and defined by P = R5 + 50R32 + 1ORS4, Q = 5R4S + 50R2S3 + 25S5. From our condition E X2 = 0 follows by direct computation r2 + 2(p1P4 + P2P3) = 0, hence r2 R =Whenever our given rational domain would not include any imaginary quantity, then R = r = 0, for if R < 0 then under our condition P < 0 but P is the sum of squares. We take up this natural division here and write A. I. r = 0. A. II. r 0. A. I. r = R = P =,65 = 0.. * 4 = -P2P3 This case will be subdivided again, namely into 1. S = 0. 2. S 0. 1. If S = 0, then pip4 = P2P3 = 0 and we may take without losing any generality P1 = P2 = 0. From pi = 0 follows: K+L5+ MJ- 10 25 + N - 10 - 25 = 0, and, from p2 = 0 K - L5-N- 10 + 2+M + M- 10 - 25 = 0, hence 03 = 5 j2(K - L ) and 04 = 5c ~2(K + L 5). By addition we get further 2K + (M - N) - 10+2+ (M + N) - 10-2 = 0, or 2K + -10+245[ M-N +(M +N) - -1 = 0. 10 + 2 The condition i=5 nfi= 4 i=1 shows K= ( 1 \2 2ca)

Page 122 122 University of California Publications in Mathematics [VOL. 1 and hence K - 0, so that the coefficient of - 10 + 2 5 cannot vanish and w belongs to our rational domain for with ~- 10 + 2 5 is 5 rationally known and hence also - 10-25 - 4 5 4- 10 + 25 As K 5 0, both &3 and 14 cannot vanish; in fact, if Pi = P2 = P3 = P4 = 0 then X1 = X2 = X3 = X4 = X5 and f(x) is reducible. We may, however, very well have that three p's vanish and we again take a subdivision without losing any generality by writing a. p = 0; p4 O. - p3 O 0; p4 O 0. a. If p3 = 0, then K = L = 1 4c2 and 4 = V1/C2 In this case the ten roots of f(x) = 0 are Xl = -X6 = 'P4 1=;C X2 = -X7 = Wo21c, The corresponding equation is f(x) = - ccrr =, and hence c is not a perfect fifth power in our domain, for then f(x) splits up into five rational factors each of the second degree. Our next question is: Are the derived conditions also sufficient that f(x) = 0 represents an irreducible Abelian equation? In this simple case it is readily seen from the above roots that x10 - c = 0 is irreducible under the derived conditions. Adjoining 1-c to the given rational domain, then x10 - c splits up into two rational factors (x5 - fc) and (x5 + -c); adjoining W to our given domain, then x10 - c splits up into five rational factors, each of the type (x2 - w Fc ). We see therefore that the Galois group of f(x) = 0 can only be a group of type 3 in Cole's list.* In fact the group is Go0, for the ten roots in our cycle s = (1739562840) satisfy the condition, that every root is the same rational function of the preceding one in the cycle, namely X1 = - w3X1; X7 = - W3X1; X3 = - w3X7, etc., X being rational by condition. * Loc. cit.

Page 123 1914] Kuschke: Irreducible Abelian Equations of the Tenth Degree 123 Type I: If c is neither a perfect fifth nor a perfect second power of any quantity in the given rational domain and if w, c, are in this domain, then x10 - c = 0 is an irreducible Abelian equation having the above roots. An example is furnished by taking c = 2. Here ~- 10 + 21d must be adjoined to R(1). f(x) = x1o - 2 = ( - ( 2)(x5 + 2) = (X2 - 2)(- 2 - - ))(x 2 -- wt2 2)(x2 - w3 2)(x2 - o4 2) = 0 is an irreducible Abelian equation under the above conditions. i P3 5. 0; P4 5^ 0. The function r432 (X1 + WX2 + 02X3 + W3X4 + w4X5)2 424 X1 + W2X2 + W4X3 + Cx4 + W3X5 is a two-valued function under Gio just as -4c; hence, writing,32 — 3 = 5e, 44 -C then e is rationally known but not zero, and we have 25c 44(K - L 5)2 = 25e c /2(K + L ), or, as 1 K =4c2, 24.5L2c4 - 23L 45c2(1 + e5c2) + (1 - 2ec2) = 0, which determines L for L 7 0, as then e = E2K = V1/2c2 and (x - x -)(x -- x6) = X2 - 4ce2 would be a rational factor of f(x) = 0. Solving for L we get L = 2 [1 + e5c2 4 e2c ec2 + 4e]; as L is rationally known it follows e6c2 + 4e is a perfect square of a rational quantity. Observing that p32 22(+ K - L )2, we have to choose the negative sign in the expression for L and hence L = 2 [1 + e5c2 - e2c e6c + 4e]. So we have e5 e2 P3 - + - e6c2 +4e, 54 / e5 e2 P4 = - 2+ -- 2c 2 + 4e,

Page 124 124 University of California Publications in Mathematics [VOL. 1 and f(x) = 0 becomes f(x) x - 10c2p33p4x6 - 25p32p44c3x4 + (15p36p42 - 10p3p47)c4X2 - = 0. All coefficients of f(x) are rational at the same time when P32 e =P3 P4 is rational, for as co is in our domain p35 and p45 are rationally known; hence P3- p45 P4 and therefore p33p4 is rationally known; but then all powers of p33p4 are rational, so that p3p42; p34p43; p32p44 are in our given domain. Further p3 * p4 is irrational, for if it were rational P3P42 - P4, P3P4 and hence p3 are rationally known, but then (x - xl)(x - x6) would be a rational factor of f(x). It follows then that e5 e2 _2 2 e6c2 + 4e is not a perfect fifth power of a rational quantity. The roots of f(x) = 0 are now Xl = - X6 = 4c(p3 + P4), X2 = - X7 = 4(C4p3 + W3p4), X3 = - X8= -C(cW3p3 + CWP4), X4 = - X = -C((2p3 + W4p4), x5 = - X0 = -c(wp3 + W2p4), where P3= - + e6c2 +4e, si e5 e2 1 42 2e6c2+4e. Regarding the sufficiency of the derived conditions, we note first that the sum of any two roots is either zero or irrational as w is rational and both p3 and p4 are irrational. The sum of two roots can be zero only if their subscripts differ by 5, in which case however the product xi ~ xi+5 (i = 1 to 5) is irrational, for it is of the form C(Wip3 + Cjp4)2, and must contain the irrationalities p3 and p4 by conditions. It follows then, that f(x) cannot contain a rational factor of the second degree. Similarly it is seen that by means of our conditions there is no rational factor of the fourth degree, for the product of any four roots is a polynomial of the fourth degree in the p's with rational coefficients, and this cannot be rationally known, as the p's satisfy an irreducible quintic. There is certainly no rational factor of odd degree, for the constant term would contain V/c as a factor, which cannot be expressed rationally in terms of the p's. It follows then that f(x) is irreducible and its Galois

Page 125 Kuschke: Irreducible Abelian Equations of the Tenth Degree 125 group is transitive. Furthermore f(x) contains two as well as five systems of intransitivity, for the factors (x - xi)(x - xi+5) are free from -4c and the two factors i=5 i=5 I (x + x) and I (x - x+5) i=1 i'l1 are free from ps and p4 and contain only those combinations of p3 and p4 or those powers of p3 and p4 which have been seen to be rationally known; for we have i=5 EXi = 0, i=1 i=4 j=5,E ix = 0 (i <j), i=1 j=2 5 i, j, k Z XiXjXk = 5c JCp42p3, i<j<k 5 i, j, k, XiXjXkXi = - 5C2P33P4, i<j<k<l i=5 XII = ~c. i=l The Galois group of our above f(x) = 0 is therefore a group of type 3. Next we show that neither Glo' nor G20' can be a subgroup of this Galois group. Applying a1 = (16)(20)(39)(48)(57) as well as 02 = (1748)(2936)(50) to the function 5e_ (x1 + COX2 + w2X3 + 03X4 + c04X5)2 0 e c(X1 + W2Z2 + W4X3 + WX4 + W3x5) we change 5e to zero, which is contrary to condition. a, as well as 02 cannot belong to the group of f(x) = 0 under our derived conditions and hence the group of f(x) = 0 is G1o. Type II. If c, e, o, %e6c2 + 4e but not -6 22c +e e6c2 +4e and [ are quantities in our given domain and if e 7 O, then the above equation f(x) = 0 with the roots on page 124 is an irreducible Abelian equation. An example is furnished by taking e = 1, c = 4; then 4e + c2e6 = ( p= 41/3 p4 = V1/9 15. 754 105 3 (X) = x10 - 6 - - + 6 - 2 = 0. 2 8 16 2

Page 126 126 University of California Publications in Mathematics [VOL. 1 The roots are xl = - X6 = 3/2[ 3 + 57], X2 = - X7 = 4_3/2[,4 Vj3 + (c3 /9], * etc. As given domain may be taken R(w). 2. If S - 0 and hence pi 5 0 (i = 1 to 4), we shall find that there is no Abelian type. Actual computation shows i=5 3 + 3 W12~2 + M42&3 = -3 F - F2 + 2(1 + q-5)F3 + 2(1 - 15)F4 i=1 2 2 and 3= - 3-/5 3 + 5 32;1 + l-22%4 = ]xi3 -- F- F2 + 2(1 - /5)F3 + 2(1 + f5)F4, ==l 2 2 where F1 = X12x2 + x22X1 + x12x5 + x52xl + x22X3 + x32X2 + x42x3 + x32x4 + x42x5 + x52x4, F2 = X12X3 + X32X1 + X12x4 + X42X1 + X22X4 + X42X2 + X22x5 + X52X2 + X32x5 + X52X3, F3 = X1X2X3 + x1X2X5 + X1X4X5 + X2X3X4 + X3x4X5, F4 = X1X2X4 + XX3X4 + X1X3X5 + X2X3X5 + X2X4X5. All these four functions are double-valued under G1o just as fc. It follows, then, if X1 and X2 are in our given rational domain, that we may write 12P2 + p42p3 = Xi + X2 '5, (1) p22p4 + p32pl = X1 - X2 5. (2) Xi and X2 cannot both vanish at once, for then, observing that PP4 = + S 5, (3) P2P3 = -S 5, (4) we would have 1 p2 P44 = p23P3 and 2=; P42 p33 hence, by proper multiplication of the last two, -1 = p-5 or P2 = -P3, and hence (2) appears in the form pi + p4 = 0 so that f(x) would contain the rational factor (x - X1)(X - x6) = x2. Using equations (3) and (4) in order to eliminate P2 and p4 in (1) and (2) and eliminating finally p3, we get 1042 =! [(X- )- X215)2 _- 2 0 S35], (5)

Page 127 Kuschke: Irreducible Abelian Equations of the Tenth Degree 127 where G = Xi + X2 5 4 i(X1 + X225)2 + 20535. Eliminating p3 and p4 in (1) and (2) by means of (3) and (4) and finally between these two new equations (la) and (lb) eliminating p2 we get 4p1 Xi- X\245 4(X - 5)2- 2) 20S34 G2 10S2 (6) Comparing (5) and (6) we have G4 = 2000S6, (7) and hence there are two possibilities. If (X1 + X2 5)[X1 + X2z5 (X1 + X25)2 + 20S34] = 0, then X1+ X2 5 = 0, (8) for if the other factor were zero we would have S = 0. If (X1 + X25)[XI + X2 5 (X1 + X2X45)2 + 20S3^5] = - 20S3, then (X1 + X2 45)2 = - 20S3 5. (9) Doing with P2 exactly as has just been done with pi we are led to the following result: We must have either X1- X2-5= 0 (10) or (XI - X2 -4)2 = + 20S3 /5. (11) One of the two relations (8) and (9) must exist with one of the two relations (10) and (11). As we saw, X1 = X2 = 0 is impossible, hence we cannot pair (8) and (10). Also relation (9) cannot exist with relation (11), for we would get Xi2 + 5X22 = 0, which leads to S = 0. It follows that either (8) and (11) or (9) and (10) exist together. Both cases are identical in the abstract; they differ only in the sign of 4-. Assuming then i + X2 5 = 0, (8) we have from (11) X12 = 5S3 5, (lla) and 4 is rational and further Sa 5 is in our domain. Equations (1) and (2) appear now in the form p2p2 - p42p3, (12) and p22p4 + P32pl = 2X1, (13)

Page 128 128 University of California Publications in Mathematics [VOL. 1 which may be written, by observing (3) and (4), 4 + 5S3 = P2 P4 or, by means of (lla), X1 = P22P4; hence from (13) we have X1 = p22p4 = p32pl; hence p22p4 P22P4 ( p =~ (14) ~p32 From (12) we get, by means of (3), (4), and (lla) X12 P12P2 = + -; P12P2 hence P12P2 = X1, and we have finally X1 = P22P4 P32Pl = p12P2 = = p42p3, (15) so that P1 =- p3 (16) P2 and p421= (17) P3 Observing (14), (16), (17) we may write 2 p22p4 p32 P1i P2P4P.P1= (p42)2 p32 p44 P3 2 P 2 Multiplying the last two relations together we obtain P15 = P45 or P= p4, (18) so that P1P4 = P1 = P4 = is rationally known, as we saw above. With pi and p4 are P2 and p3 rationally known by means of (15) and hence (x - X1)(x - X6) is a rational factor of f(x). There is no irreducible Abelian type under A. I. 2. A. II. r 0. We had the relation r2 + 2(pip4 + P2P3) = 0, so that at most two p's can vanish, either pi = p4 = 0 or P2 = p3 = 0. Without

Page 129 1914] Kuschke: Irreducible Abelian Equations of the Tenth Degree 129 loss of generality we may take the following division of case II, pi id 0; p4 id 0 and (1) P2 = p3 = 0, (2) P2 =0; P3 0, (3) P2 0; p3 0. 1. If P2 = p3 = 0 then K = L45 and N M -2. Suppose for a moment K = L = 0 then pi = - p4 so that f(x) contains the rational factor (X2 - cr). It follows then K= L5# 0 and hence a-5 is in our given domain of rationality. We may now write pi5 =2K + M' - 10 + 225, p45 = 2K- M' 4- 10 + 2/5, where 2 is necessarily rationally known. (plP4)5 = - ) = 4K2 - M'(- 10 + 2j5), so that M' can be determined by c and r as K is found from the condition i=5 nXi =,c, i=I which leads to the relation 119 r5c2 1 - r K = 4 hence r5c2 1 as K 7 0 and c $ 0. 1 ]16 - 152r5c2 + 363r10c4 ^ -10+- 10 + 2 5 0, for if M' = 0, then M = N = 0 and p = - p4, so that PP4 = - p2 = - p42 and therefore pi, p4 would be rational and hence f(x) is reducible. It follows then ]16 - 152r5c2 + 363r'c4 0 - 10 + 2 15 -but is in our rational domain.

Page 130 130 University of California Publications in Mathematics [VOL. 1 The roots of f(x) = 0 are given on page 120, where the p's are defined by 11- 19 rSC2 Pi 2 + C2 16 - 152r5c2 + 363r1~c4, P2 = P3 = 0, 1 -19 r52 p4 = -- - -c 16 -152r5c2+ 363rl~c4. P4 = 2c2 8c2 The corresponding equation becomes 115 4 15 r \4/ 541 35 \ f(x) = + -r- x + 5 -r6 2 4 + r8 —r c4x2-c =0. 4 \2/ \ 16 / We ask again: Are the derived conditions also sufficient, that f(x) = 0 is an irreducible Abelian equation? Our f(x) cannot have any rational factor of odd degree, for the constant term would contain -c as a factor which is irrational by conditions. Ic and pi cannot be expressed rationally by each other. Further, as every root is of the form /[+w-kpi+ -kp4 + r] = [ p 2kpl r, the sum of any two roots assumes one of the two forms either ~~(1) ^ [~(..+~*)[ r2(2plWk+i ] or (2) 4 [p- 2pWi (W- O. The type (2) shows that, if k = j, then the sum of the two roots vanishes, which can happen only if the subscripts differ by five. In this case, however, the product xixi+5 contains the irrationality pi, for we have xixi+5 = - C [wP1 - 2wkp + r] 0. In all other cases the sum of any two roots does not vanish and hence both irrationalities remain in the expression of the sum, for if in (2) r2 pl =2plk then r = pliwc 42, which cannot happen, as pi satisfies an irreducible quintic, a12 a quadratic and c an irreducible quadratic for 5 is rational. If in (1) r2 ( k + Co j) pi(Wk + w) - = 0, 2piwk+i

Page 131 Kuschke: Irreducible Abelian Equations of the Tenth Degree 131 then ( / r 21 p~ = tAc 2' which would be again impossible, as pi is irrational and satisfies an irreducible quintic. Exactly the same thing can be shown in the case of a factor of the fourth degree, having now instead of pl(Ck + wi) and k+j, always pl(wk + Wj + 0oI +,m) and ok+i+l+m respectively. If ck + o(1 + Co + Om = - 1, then Ck+l+j+m = + 1, and the sum of the four roots vanishes; but then the product of the four roots is irrational for the same reason as in the case xixi+5. Whenever, then, pi is irrational, we see, that the above conditions are sufficient for the irreducibility of f(x). Hence we write 19 r 5 4 rc2 -1 J16 - 152c2r5 + 363c4r10 2c2 8 c2 -10 + 2 -is irrational. Under the above conditions, then, the Galois group is transitive and must be again of type 3 in Cole's list for the five factors (x - Xi)(x - xi+5) (i = 1 to 5) are rational after the adjunction of pi to the rational domain and the two factors i=5 i=5 II(x-xi) and II(x-Xi+5) i=1 i=l are rational after the adjunction of -c to the given domain of rationality for we find: =5 xi = 0, 25 i, j Xixi = -- r2c, i<j i* j, k XiXjXk = C i<j<k 5 45 ~1 4 *, j, k, t ~ XiXjXkXi = - ric2, i<j<k<l i=5 Exi = += 4. Now a- interchanges,=t and = a4, from which would follow that pi = = p4, if oa were in our Galois group, and hence K = 0 or M' = 0, which is contrary to our conditions. 02 changes 1/14 # 0 to zero which is impossible for any substitution in our

Page 132 132 University of California Publications in Mathematics [VOL. 1 Galois group, as 4144 is rationally known. It follows then that our f(x) = 0 is an irreducible Abelian equation. Type III. If c, a, r but not -Ic are in our given rational domain, and if 16 - 152r5c2 + 363r1c4 0 -10+2 5 but rational and 19 5 1 - 4 rc 1 16 - 152r5c2 + 363c4r10 2c2 +-2- -1+2 4 is irrational and r F 0, r5c2 7- then the above f(x) is an irreducible Abelian equa19 tion with roots as given above. An example is furnished by taking r = 1; c = 2. Then 16 - 152c2r5 + 363c4r10 16 * 163 - 10+ 25 -5 + 5 hence our rational domain must contain 163 -5+A5 -f(x) = x10 + 115x6 + 280x4 + 4012 -2 = 0 The roots are X1=-X:6=+= [ -9+S/326+ 9 —326 +l] + X2= - X7 = + 2 [2 - + 326 + w - -- 326+1] etc. 2. If P2 = 0, Pi F 0 (i = 1, 3, 4), we have P3 =;2(K- L(5) ~ 0, which follows from p2 = 0, that is, from K - L = N- 10+2 - M2 - 10 - 2s, so that o is rationally known. The functions and are seen to be rationally known, if applied to by G. Calling them 5e and 5d are seen to be rationally known, if applied to by Gio. Calling them 5e and 5d

Page 133 1914] Kuschke: Irreducible Abelian Equations of the Tenth Degree 133 respectively we have e= P3 and d= pl2 P4 P3 ed2r2 P1 = -- e2r4 P35 = + e' 4d ' r8 P45 = +2ed2 If pi were rational, then p4 and p3 are rational and vice versa; hence we have the condition e, d, are rationally known and 24ed2r2 is not a perfect fifth power. The last condition excludes e = 0, and d = 0 as well as r = 0. The corresponding equation becomes f(x) = x10 + 5A r2c2x6 + 5A2r3cx4 + A3r4c4x2 - c = 0, where 23 1 r2e 3 r3 Al = — r2+ de + 3re - -- d 2d e2r2 r6 15 5 r3e 5 5 r5 A2 =-2r3 + d2e — -5r2e + 2-rde - -4re2 d 8d2e 2 2 d 4 16 d2' 71 1 r2e3 r8 3d2e2 3r4e2 3r6 5 r4e 7 r7 AS = m r4.] _ - 7r2de 8A 2 Td 16 d3e 2 + 8d2 +4d2 -4 d 8 d2e' r7e2 7 r5 51 5 r5e + 7rd2e -_ 2 r3e + 5r2e2 5re2d - d d 2d 2 4d2 The roots are2 e r 51 4d 51 r86e Xl = - X6 = + C L- -2 + \r4d + \16ed2+ ] ' X2 = - X7 = + [- + o4 4d + 3 6 +r. etc. 4d + Ned2 etc. r, c, e, d are not independent of each other, for they have to satisfy the identity 1 19 r8 r4e2 ed2r2 15 r6 15 5 5 r5e 1 r 5 + _ 2] _ -. -_ - _ r4e _]_ r~de - C2 4 16ed2 4d 2 8d 2 4d ' which follows from the condition i=& fxi = 4. i=l Regarding the sufficiency of the derived conditions, we first note that f(x) cannot have a rational factor of odd degree, for the constant term would contain -fc as a factor and -c cannot be expressed rationally by pi (i = 1, 3, 4). Whenever the sum of any two or four roots vanishes, then these sums must be of the form Xi + xi+5 or i + xi+ + X5+ + xj+ (i ~ j),

Page 134 134 University of California Publications in Mathematics [VOL. 1 in which cases the product of the two or four roots is of the form - c[Wkpl + OCp3 + WCmp4 + r]2 or + c2[o'pl + o8p3 + Otp4 + r]2 * [ckpi + olp3 + Wop4 + r]2. In each bracket stands an expression satisfying an irreducible quintic and hence only a power of s = 0 (mod. 5) can make the product rational. Both products are of lower degree than five in the p's and hence they are irrational. Whenever the sum of any two or four roots does not vanish, it must contain A-fc as a factor. It follows, then, that f(x) is irreducible; therefore its Galois group is transitive. This group must contain again two as well as five systems of intransitivity, for the five products (x - xi)(x - xi+) (i = 1 to 5). do not show the irrationality /c, and the two products i=5 i=5 II(x-xi) and I (x-xi+5) i= - i=-1 do not show the irrationality p, for we have i=5 Exi = 5r, i=l i,j=5 25 xix = + — r2c, i<j jj, k58j 5^,cr2 r21 XiXZXZk -= - [ 14r - 2e + d, t,j, k=l i<j<k i, j, k, 1=5 5C2r2 r r3 er2 I E XiXjXkXI = 4 [11r2 + 2de +2 -e- 2re, i~i, ljA,4Ld d ' i<j<k<l i=5 - xI x = -c. i — Applying ao and a2 to e = P3, we come to the reductio ad absurdum e = 0. P4 It follows then that f(x) = 0 is an irreducible Abelian equation under the derived conditions. Type IV. If c, co, d, r, e but not V24ed2r2 and -c are in the given rational domain, and if the identity exists 1 195 r8 r4e2 ed2r2 15 r6 15 5 5 r5e =T 1r+lG +_ + d +Tre+ r3e de C2 4 i 16ed2+ 4d 2 +8d + 4 2 4 rde ' then the above f(x) = 0 with the given roots is an irreducible Abelian equation of the tenth degree. An example is furnished by supposing 4 r=d=e = 1, c=.1 ~ 1 8-3

Page 135 1914] Kuschke: Irreducible Abelian Equations of the Tenth Degree 135 Here f183 must be in our rational domain as well as w, but not V4183; further, the identity between r, d, e, c exists as given above and 24ed2r2 is not a perfect fifth power; hence J2 must be a quantity not in the given domain. The corresponding equation is 619 1019, 808 4 13714, 88 4 f(x) = X10+ 18 X6 - - 4 1+ x2 - 0. 183 183 -183 1832 /-183 The roots are: 2 5 5 Xl = - X6 = /183- + X2 = - X7 = 4 - + 24 + W3 + 1] etc. 3. pi 5 0 (i = 1 to 4). We proceed to show, that this case is impossible here just as case A. I, 2. We saw above that, if Xi and X2 are in our given rational domain, we may put P12P2 + p42p3 = X + X2 J5, (1) p22p4 + p32pl = X - X2 I5, (2) P1P4 = + R + S ]5 - T O0, (3) P2P3 = + R - S 5- Q $ 0. (4) Eliminating p2 and p4 in (1) and (2) by means of (3) and (4), and finally removing Pa in the new equation (2), we come to the result A2p5 Xi - X2\5 -(X - X245)2 - 4Q2T 4T4 2 (5) where A = Xi + X25: (X1 + X2/5)2 - 4T2Q. Removing p3 and p4 in (1) and (2) by means of (3) and (4), and finally P2, then equation (2) can be written 4pi5 X1 - X2~ 15 (X - X25)2 - 4Q2T (6 A2 2Q2 (6) Comparing (5) and (6) we have A2 = - 22QT2. (7) The last relation may be written (Xi + X2-5)[X1 + X2 - (X + X\2 - )2 _ 4QT2] = 4QT2 or = 0, (8) hence there are only two possibilities, namely X + X2 'N5 = 0, (9) or (X1+ X2 J5)2 = 4QT2 0. (10) Doing the same thing with P2, we find, that one of the following two relations must exist: X1 - X25 = 0 (11)

Page 136 136 University of California Publications in Mathematics [VOL. 1 or (X1- X2 5)2 = 4Q2T. (12) Pairing equations (9) and (11), then X1 = X2 = 0; hence we get from 1 P22 equation (2) that + = -p32( P42 -32(2R- P2P3) and equation (1) that + p44 = - 2(2R- P2P3)2 + = P44* P42 ) P2, or P2= P3 P35 and from (2) again we get, if P2 = p3, that pi = - p4. From P2 = P3 we have N - 10 + 2-5 - M /- 10 - 245 = 0, and from p = - p4 we have K+L 1/5 = 0. As now pi 7 0, it follows -6 is rationally known but then P1P4 = -p2 = - p42 as well as P2P3 = p22 = p32, and hence pi is rationally known, so that f(x) contains the rational factor (x - Xl)(x - x6). It follows then that X1 = X2 = 0 is excluded. Pairing (9) and (12), then -5 is rationally known, as X1 = -\2 F f 0 and we have from (12) Xi2 = Q2T, or Xi = (R - S5) R+ 5. From (2) we have Q2T p22p4 = 2X1- p32pl = 2X1- 2 P22P4' or (p22P4)2 - 2X1P22p4 + X12 = 0, P22P4 = X1. It follows again by (2) P22P4 = PlP32 = X1 and hence p22p4 p P 2 = (13) p32 eliminating Pi From (1) we havee g p12 P2 = — p42P3 (14) P24P42P2 **4 -= - P42P3, P3 or P2 = -P3 (15) and hence pi = p4 by (13). We have arrived at a similar impossible result as in the case X1 = X2 F 0.

Page 137 1914] Kuschke: Irreducible Abelian Equations of tne Tenth Degree 137 The case of pairing (10) and (11) is identical to the case just discussed; the only difference lies in the other sign of -5. It remains then to show that relation (10) and relation (12) cannot exist together. From (2) we have P22P4 = X - X2 -5 - p32p1 = X1 - X2 -2 P2 P4 or (p22P4)2 - (X - X2 5)P2P4 + Q2T = 0,.-. by (12) p22p4 = + -- 2 or by (2) again P22P4 p22p4 = p32pl or pi = (16) P32 Similarly from (1) follows exactly in the same way by means of (10) = + X2 = =2 p1 P2 = = PP3 (17) Eliminating pi between (16) and (17), we are led to the relation P2 = P3 and hence, by (16), Pi = P4. Now p12p2 + p22p4 = Xi and as P2 = P3 Pi2p3 + p22p4 = Xi = rationally known. Applying however Glo to p12p3 + p22p4, we see it is changed into wopl2p + p22p4 and, as pi ~ 0, we have the result: There is no Abelian equation corresponding to case A. II, 3. B. ALL pi = Xi + Xi+5 (i = 1 TO 5) ARE DISTINCT. The irreducible Abelian equations, left for consideration, do not reduce to quintics in x2. Let us consider first the five systems of imprimitivity by taking the function i=5 II (Xi - i+5), (i = 1 to 5) i=1 This function is a double-valued function under Glo; hence its square is a quantity in our given rational domain. Putting then i=5 = (Xi - Xi+5)2 (i = 1 to 5), i=1 y is rationally known, but not a perfect square. The Lagrange resolvent =X1 = Xi+ EX7 + 62X3 + E3X9 + E4X5 + 65X6 + 66X2 + EXg + e8X4 + e9X10 can be written in the form 1 = - X6 + W3(X2 - X7) + CO(X3 - X8) + W4(X4 - X9) + 2(X5 - X10).

Page 138 138 University of California Publications in Mathematics [VOL. 1 Similarly we may write the other values of,1 under G1o in the form: '2 Xi + X6 + w(X2 + X7) + o 2(X3 + X8) + c3(X4 + X9) + 04(X5 + X10), 13 = - X6 + o4(X2 - X7) + w3(X3 - X8) + 2(X4 - X9) + W(X5 - X1) 4 = X1 + X6 + W2(X2 + X7) + w4(X3 + X8) + W(X4 + X9) + W3(X5 + o10), 5 = X1 - X6 + (X2 - x7) + (X3 - xs) + (X4 - x9) + (X5 - x10), 46 = X1 + X6 + C3(X2 + X7) + w(X3 + X8) + W4(X4 + X9) + C02(X + X10), It7 = Xi - X6 +' (X2 - X7) + c2(X3 - X8) + 3(X4 - X9) + W4(X5 - X10), 8 = X1 + X6 + c4(X2 + X7) + W3(X3 + X8) + C2(X4 + X9) + O(X5 + X10), 9 = 1 - X6 + 2(X2 - X7) + C4(X3 - X8) + C(X4 - X9) + 3(X5 - X10), 10 = X1 + X6 + (X2 + X7) + (X3 + X8) + X4 + Xg) + (x5 + x10), Taking our f(x) = 0 again in the reduced form, then -1o i=5 ~5 xi= 1io = 0 and _xi=. i=5 i=5 Both functions, Exi and II (x - xi+5), are double-valued functions under Glo; i=1;i=l hence calling i=5 Exi 5r= t=5 1 n (i- xi+5) <== then r is in our given domain and '5 = 10 - * r. All ka5 are unchanged by Glo if a is even; and if a is odd then a must be rationally known if we adjoin o to the rational domain. Analogy to case A shows that we may write 1 = 10o +1+ L14+ Ml -10 + 25+ N1 - 10 - 245- = lOpli, 2 = 10 -K-L2+- N2 - 10 + 25-M2-10- 25 =10O p2, 13 = 104 jKL-L -NV - 10 + 25+ M1'- 10- 2-5 = 104yp3, {4= lO4K2+L2-M2- 10 + 2 -N2 -10-2~ = 10p4, = 10 lo 2 - -L 5- M2- 10 +- 25 + N2 10 - 2/5 = 1Op4, 6 = 10 K2 - +M2 - 10 + 24 - N2 - 10-25= 10p6, '67 = 10 <K+N, 0 2-10 2 M1 -=10 -2-0 p7, 18 = 10 K2- 2- N2- 10 + 25+ M2 4- 10 - 2'5 = lOps, 9 = lO i -Km+ L/5-M - 10 + 25- N - 10 -2r = 10-p9, 5 = 5r '/, o10 = 0.

Page 139 1914] Kuschke: Irreducible Abelian Equations of the Tenth Degree 139 As our (l = xi + x6 takes five distinct values under G(o, the quintic resolvent equation.=5.(( - ( i) = 0 is an irreducible Abelian equation of the fifth degree in (p and the corresponding group in the x's is Glo, which either leaves every pi fixed or interchanges them cyclically. We have now from our Lagrange system by addition <P1 = xI + x6 = 2[P6 + Ps + P2 + P4], (P2 = X2 + X7 = 2[C2p6 + wps + 04p2 + W3p4], P3 = X3 + X8 = 2[W4p6 + W2P8 + -3p2 + WP4], t4 = X4 + x9 = 2[cP6 + O3pS + W022 + t4p4], t5 = X5 + Xlo = 2[c3p6 + 04P8 + C-P2 + W2p4], and, defining (1i+5 =i - xi+5 (i = 1 to 5), then P6 = Xl - X6 = 2 y[ + pi + p3 + P7 + P9 + r], (7 = X2 - X7 = 2 4-[o2pl + Op3 + C?4p7 + W3p9 + r], s8 = X3 - Xs = 2 [wo4pl + -2p3 + O3p7 + cp9 + r], P9 = X;4 - X9 = 2 y[copl + W3p3 + c2p7 + 04p9 + r], (plo = X5 - 10o= 2 I[W3pl- + 04p3 + Wp7 + CW2p9 + r]. Knowing pi and pi+5, we know of course xi (j = 1 to 10). The roots are: X1, X6 = P6 + P8 + P2 + P4 - [P1 + P3 + P7 + P9 + r], X2, X7 = Wo2p6 + po8 + c4p2 + -cJ3p4 -.-[2pl + cop3 + W) 4p7 + W3p9 + r], X3, X8 = W4p6 + w2p8 + c3p2 + WP4 y =t[C4p1 + c2p3 + W 3p7 + cWp9 + r], X4, X9 = WP6 + W3ps + 2p2 + w 4p4 7[WP1 + w3p3 + 2p7 + 'c4p9 + r], X5, X10 = c3P6 + W4p8 + W+p2 + W2p4 - ~[c,3pl + w4p3 + wp7 + w2pg + r], where + - belongs to x, (i 1 to 5). and - 4y belongs to xi+5. Actual computation shows i=5 461 = 10 P2 + Q25 = (w + W4)F1 + (W2 + W3)F2 + Z i2 /=l and _=5 28s = 102 P2 - Q25 = (w2 + o)F1 + (w + o4)F2 + (i2, i=1 where P2 = K22 + 5L22 + 10(M22 + N22), Q2 = 2K2L2 - 2(M22 + N22) - 8M2N2, Fl = (P1(p3 + P2(P4 + "1(P4 + "P2P56 + 'P365, F2 = (PlP2 + (P1<P5 + -P2(P3 + <P3(P4 + 'P4P5.

Page 140 140 University of California Publications in Mathematics [VOL. 1 Both F1 and F2 are unchanged by Glo and hence they are rationally known. We may therefore write I4V6 = 100P4P6 = 100(R2 + S2 5), 82'08s = 100p2ps = 100(R2 - S24), where R2 and S2 are defined by P2 = R5 + 50R23S2 + 10R2S4, Q2 = 5R24S2 + 50R22S23 + 25S25. We have the relations: P4P6 + P2P8 = 2R2, P4P6 - P2P8 = 2S2 5. We shall divide case B into (I.) R2 = 0; (II.) R2 5 0. The case R2 = 0 will have two parts, (1) S2 = 0; (2) S2 i 0. B. I. (1) R2 = S2 = 0. Without loss of generality we may assume P6 = p8 = 0; hence we have here again the similar relation as under (A.) 2K2 + - 10+ 24f5 M2 -N2 + (M2 + N2) 10- 2 5-] =0. LR,+i-o~z~;[,-N,+( ~1 - 10 + 2^5 J Further from p6 = 0 follows p4 = /2(K2 + L25) and from ps = 0 follows P2 = /2(K2 - L5). It follows then K2 0, for then p2 = - p4 and <pj would be zero; hence the expression in the bracket cannot vanish and therefore w is in our rational domain. For the same reason, at most three pi can vanish and we shall divide the discussion into (a) P2 = 0; p4 0 ( P) P2 0; p4 0. (a) P2 = 0; hence K2=L 0 and p4 = ~4K2 4K2 is not a perfect fifth power, for then (pi would be rationally known. Putting ^ _ A3_ 17 _ 1. i_ f 4 - =; /,42 = —; 43; 44 - then 6', 0', ~', a' are quantities in our given domain, for numerators and denominators change at the same time the sign, or are left fixed by Gio. We get then P6 = Xl - X, = 2 47-[103,' V(4K2)4 + 102r' V(4K2)3 + 10A' (4K2)2 + ' /4-K2 + r]. Putting 1037' =; 100' = 7; 102' =; 5' = 56

Page 141 1914] Kuschke: Irreducible Abelian Equations of the Tenth Degree 141 4K2 = K is not a perfect fifth power, but rationally known }, 7, O, ' are rationally known we have x21, x7 = W 4 47[r + ^; + WK2 + 3 + + w t4], X2, X7 = W 3 VK- =t [r- + W,38 a + W,, -K2 +,,,4 j T3 + W2,q TK] * ~ ~ etc. The corresponding equation is f(x) = [x5 - Alx4 + A2x3 - Ax2 + A4x - As] x [x5 - B14 + B2x3 - B3x2 + B4x - B5] = 0, where Al, B1= 5r ^/, A2, B2 = + 10yr2 - 57K(6l + 6;): 57 K ', A3, B3 = - 15r7'YK + 10756K + 57KtK2 - 5 -Y[2yr - 3ryK(t- + t) + K + yK2(2 n) ( + K(2 + ) + (2 + ) A4, B4 = 5y2r4 + 572K2(682f2 + r2.2) - 572K(.36K + 63 +.32.K2 + _3rlK) + 5,y72K2 - 15r2y2(a7 + ti)K + 10y2rK(2lK + 2 + 2 72 + 2K + ) + 1OqyrK - 57216t'K2 - 157Y6 i 5 4[- 7-3K2 - tK - 3r2-y7/K + 27yrO2K - y7tvK2 - 3y782K + 27m2 K2 + 8r7yb$K], A5, B5 = K + 57234 K + 10y-2K - 5iy23K2 K 52y23K33 - 5y 2 - 15-2K2 62 + 572t2 2K3 - 572r3'K + 5y2r2-2K + 1Or2y26rK - 5r7y23K2 - 5YK - 5r'y2r7n0K2 - 15r7y,6K - 15ry2t62K + lOr$y2 a2K2 - Iy[7Y25SK4 + y72t5K2 + y2 5K3 + y2 5K + 72r + 56K + 10763K - 5y7263,K2 - 572 ~- K - 572773K33 - 5y72h73K3 - 157d-K2 + 572K2(q2-26K + 2a62- + 12-t2K + 62r02) + 57t2&K2 + 573t2K2 - 5r372K(C + 76) + 572Kr2(r2fK + 62 -+ q2- K + -26) + 57r2rK - 5r72K(33-K + 63- + 73 K2 + 3 cK) + 5r(y2K2(l262 + W292) - 5y2r r-lK2 + 5r772K2]. y, r, 5, 7, O, are not independent of each other, for the condition i=5 II (Xi - i+5) =. =1 leads to the identity - = 5K4 + 5K3 + 05K2 + 5K + r5 - 5[53t6K3 + 3t6K2 + ~37&K3 + 6 37K2] 25~, + 5[rt262Kc2 + ~t262K2 + rf772t2K3 + 67722K3] + 5rK2(fl252 + 2 2) + 5r2((2OK2 + t25K + r2nvK2 + 62tK) - 5r3(6 + 6~)K - 5r(3t.K3 + r3nK2 + 3.8K2 + 63VK) - 5K23ri..

Page 142 142 University of California Publications in Mathematics [VOL. 1 There is obviously another condition to be imposed-upon r, 5, r, A,,; namely not all five can vanish at once, for if r= = == =, then the irrationality -Iy disappears entirely and f(x) splits up into the two rational factors i=5 i=5 II (x- i) and II (x - xi+), i=l i=l as can be seen from the above Ai and Bi. We ask again: Are the derived conditions also sufficient, that the above f(x) = 0 is an irreducible Abelian equation? f(x) cannot contain a linear, rationally known factor, for -Y7 and K cannot be expressed rationally by each other; the former satisfies an irreducible quadratic, the latter an irreducible quintic. The sum of any two, three, or four roots is of the forms respectively ((A + WA') + 4sfl(, cK) (CA + W )+ ) f+ f2(0, (c), (w + W^ + w + W)') T + /Yf2(w, ). Hence, for the same reason that a-' and K cannot be expressed rationally by each other, the last three forms must be irrational, even if fi,(, K) = 0 W/Ccan disappear in a factor of the fifth degree, only if the roots of this factor are xi or xi+5 (i = 1 to 5) and in fact the above symmetric functions of xi and xi+5, which we called Ai and Bi, show -'y as the only irrationality. Our question is then simply: Can - disappear in all As (i = 1 to 5) without contradicting the derived necessary conditions? Al and A2 show that we must have (a) r = r = 0. Then we must have further, if l-y shall disappear, (b) + yr- 2 + 7y62 = 0 (by As) and (c) a + 37562 + y3K = 0 (by A4). Let us first see whether one of the quantities I, A, 5, 1 + 3y52, can vanish without contradicting the necessary conditions derived. (1) If 5 = 0 then by (b) 0 = 0 and hence by (c) a = 0. This case would require r = v = a = v = a = O and contradicts our conditions. (2) If v = 0 then by (b) 5 = 0 or = 0 as y $ 0. = 5 = 0 requires 0 = 0 by means of (c) and is therefore again impossible. If then r= 7== t==0 560 then A 5 shows that 7264 + 10752 + 5 = 0

Page 143 Kuschke: Irreducible Abelian Equations of the Tenth Degree 143 if l-7 shall disappear in A 5. Hence 72 - 5: 2g and, as our identity leads to = S5K, we must have 7 is not a perfect fifth power for otherwise K is a perfect fifth power which contradicts the conditions. (3) If 0 = 0 then ~ = 0, which is again case (2). (4) If 1 + 37b2 = 0 then 0 = 0, which is again case (2). Assuming then that (a), (b) and (c) are true and that v $ 0, a 0, 8 - 0, 1 + 3762 5 0, we must have =- K by (c) 1 + 3y82 and hence by (b) _ (1 + 7y2)(1 + 3y72)2 r36K2 so that from A 5 it must follow 1 + 15782 + 607y24 + 757y36 + 257468 $ 0. In this last case, that is r0,,3K (1 + yS2)(1 + 32)2 r = X=, a= _ _ 5= ( s + ) + 1 + 31752 2 y36 K2 our identity reduces to 7263 K 25[1 + ys2 + 14y264 + 48y736 + 20y468]' i=5 Under these conditions II (x - xi) must contain coefficients involving -Gy. i=1 This proves, that f(x) is irreducible, if all conditions are observed. Furthermore the group of f(x) = 0 must have two as well as five systems of intransitivity; for Ai and Bi show that f(x) splits up into two rational factors after having adjoined - to the rational domain; and having adjoined p4 to the rational domain then f(x) = 0 splits up into five equations with rational coefficients, namely into the five equations (x - xi)(x - xi+5) = 0. It follows then that the group of f(x) = 0 is a subgroup of the G240 above and it must be G1o, as ao as well as a2 change K = {46 - 0 into a quantity which is zero. Type V. If a, r, 7, 6, A, y, 7), K are quantities of the given rational domain, but not and if the identity exists as given on page 141 then f(x) = 0 with the roots on page 141 is an irreducible Abelian equation, if not all of the five quantities r, 6, r, r, 5 are zero

Page 144 144 University of California Publications in Mathematics [VOL. 1 with two special exceptions, namely (1) If r = X = = a =O then y2 - 5 [ 2 5, [.. = irrational] (2) If r = 7f = O, +5 = _ )(1 + 37y2)2 = - 3K 7S3K2 ' + 362 then 1 + 15y72 + 60y264 + 757y36 + 257468 S 0. Illustrations: (1) 7 = 2, = 2, r = r = a =0; then from the identity we have K = Hence the roots are: Xl, X6 = =2 ^4 X2, X7 4= 3 u 2 4 * etc. The corresponding equation is f(x) = 16x'1 - 60x6 - 45 -20 23 + 25x2 -5x - 0, 75 = 0. Our domain must include R(w). (2) v= = = =, K= 2, r= 2 then by the identity we get = 2-5. Our domain must include R(co). 55 45 5 5 210 - 1 f(x) = X10 + 556 - 4x5 + 45 x4 + 2 2 x + 5 = 0. f25 x 28 212 -24 215 =0. The roots are: X1, x5 = F2 2, X2, X7 = 2 -. etc. (3) P2 5 0, p4 # 0. As o is rationally known, we may put P25 = X, p45 = K where X, K are in our given domain. Similarly, as under (a), we may write, if v', U', 6', A' are in the given domain =7 =, = V' = ='4;a423 44 42

Page 145 Kuschke: Irreducible Abelian Equations of the Tenth Degree 145 Hence <6 = Xl - X6 = 2 -y[1023' 3 + 10t' VK2 + Vy' + t' + r] or, putting 102V' = ~; 10'' = a; v' = =,' then 6, 6, v, v are rationally known and not all five quantities, 6, 6,,j, v, r can vanish, as then V6 = 0. Our roots become X1., = - + = [yL3 + + vy + V a + r+], X2, X7 = -c4 /X + ca Vc Y W[c2 X3 + o 2 + cW4 V+ + w3 W + r]. * etc. * Both VX and W are not in our given domain, for otherwise (x - xi)(x - xi+5) is a rational factor of f(x) and as both VX and W are rational or irrational at the same time. Further, X 7 i K, for then /' = 37 421P42 would become '34'7= /o'. 7'23 Applying G1o to 13t7 we see that it is a rationally known function and hence 23 and therefore p2 = X would be rational. t2t42 is unchanged by G1o; hence if r = 5X- then r is a quantity in our rational domain and we have the relations: T5 T3 4 T4 X= -- 4K3 = -; K2 K K The quantities ', Ki 7, r 7,,, a, y are connected by the identity, based upon i=5 II (i - xi+) = y i=l1 1 5715 5 5 [76 T4 25 = y + 5K2 + 5 + 5 + r5 - 5 3 8+ 0KTvt + 3 + 6.["5223 "-22rl+ 5 + 2 d2 2S* + u2/ 2 + V + 5r 22 + 2T2 K2 K3 J K2 T6 T5 r0 r3 + + 5r 6[20 2 + 02aK + 2 2 + 2V] - 5r pv- + T3 + V- + 630 - 5r3 [#p —3-+ V -- 5,v 6r —. - [, 7K Jv] 6 CK The roots may now be written in the form X1, X6 + T; V+ K VK2 J +r X X;7 '\; [i + En KK2 + V 36; + ] 0=+4KT * * * etc.K 2 + a etc.

Page 146 146 University of California Publications in Mathematics [VOL. The corresponding equation is f(x) = (x5 - Ax4 + A2x3 - A3x2 + A4x - A5) x (x5 - B1x4 + B2x3 - B3x2 + B4x - B5) = 0, where.1 Al, B = - 5r ^7, A2, B2 = l0r2y - 5yT(i-/A+ - ) =F 5T (I-+ - ), K K A3, B3 = 5T + 57(62T + t2K) - 15ryT (-+ + 1O v-r + V6T)Sr T5 T56 54I [27yr3 + + VT + 26r + v27y - + y62v7 + my2 t + 26K -3 KKru- VK K] 3yry6- - 3yr6 V1 KK A4, B4 = - + 52r4 + 5 ) 5 ( AL T + 06 K K K2 / \ K + 40r6T 5 5 [ 2rr - T3 (-+ 4) - - _ 572Va3KT _ 582T \38-+ <C + K 3 ) q- M3,'- tzr3 q 5yr2 (V-+r q 22) -- 1 5rz - q2-yT ( T+, K 4 K7~~~~ K +- 1057r, 2 _ - (2 _ + t2 -) 10t (r + v- ) -- 57T (2- + -) + 27r52T -+ 522VT — +( 8 v 5) -( 15 K K2 K K K K - 5r2r 3K - 52- 15rv - 6K + V- -K - 57/<T (Vf^ + <y^) -l15w514 - l2IVT~rK K 4 ) 2LI + 4T ( T + (62' +1 OT2 (+v) K, K2 K K - 5y2r3T ( + x1 + 5yr2r + 5y2r2(62T + g/c) - 5r'y076 4 (V + 6) - 15r'yv- - 15r76 ( VZ+?) - 15r72 (_ 22 _ + '6K) + lOry2T2 (V2 + 8t2 T) ) [5 y4 T5 + 5 -2 +- K2 K — 5,~r( V3 0q-t 4 3g K K2 K K K 2 — i 5rEt- + 72 T- - 15r - V —T 5 - - K 4 /3 7 / T4 /:~-4-~ 5S + 5a~ + ~ W- + o~~- + 5+ + r

Page 147 Kuschke: Irreducible Abelian Equations of the Tenth Degree 147 v3'5Kr \ -3 75 `8 \ + 107 (- + 53 ) - 572r (V63- + KM5K + /VV aT + L aK K " \K xK K K 5A 54 _ 157;5VU4 a Ts 3 T1\ r4 - 53Kr 5- - 15 - 15762 KC \ K K K _ — 5722 ( v22 + v22 -+ -22v + T) +- 5t0r(1 + 2) K3 K2 + 57r2 ( 562 K + 652 + -2T ) + 207yvrT2 - 572r7 ( +, -) + 572r2 ( Vi r+ 52T + 2t+ aaK ) + 5'yr2 ( + ^) K2 /(2T3L T 3\ + lOr2Iy6r - 5r — 5r72 ( V3K-+ 4K + u/V-4 + I,3 ) K ' KI K + 5r72T2 (V212 + 62 2 - 5r7-62 + 5r72 (2 +7 r2 ) - 5r a- - 15r + 2 - r K K Regarding the sufficiency of the derived conditions, we first note that the group of the above f(x) = 0 must contain two as well as five systems of intransitivity; for, adjoining QK to the rational domain, f(x) splits up into the five rational factors (x - x1)(x - xi+5), as seen directly from the roots of f(x) = 0. The above A5 and B show, that f(x) is the product of two rational factors, if is adjoint to K K K 4 the rational domain. It follows then that the group of f(x) = 0 must be a sub-- 5ryt -- _ 15ryrV --. K K Regarding the suiiency of the derived conditions, we firs note that the groupible. oJust as under () the sum of any two, three, or four roots can not be rativityonal for, adjoining tounder the derived conditional domain, f() splits up into the fivexpressed rational factors (each other and as seerationalectly known. The rootsame is true for any singl. The root and henced B show, hat f(x) cannois the product ofain any rational factor of the first, if s, third or fourth the rational domain. It follows then that the group of f(x) = 0 must be a subgroup of the above G240, if f(X) is irreducible. Just as under (a) the sum of any two, three, or four roots can not be rational under the derived conditions, as 4~ and W]~ cannot be expressed rationally by each other and as co is rationally known. The same is true for any single root and hence f(x) cannot contain any rational factor of the first, second, third or fourth degree. The sum of any five roots is either irrational for the same reason or it is zero, which can happen, only when we pick out the five roots belonging to the same system of intransitivity. It remains then to see when the coefficients of '7y in all Ai may vanish. A1 and A2 can be rational only if respectively (a) r = o (b) / - + - = 0 K as r $ 0, y O70. A3 is rational under the assumption that (a) and (b) hold and if -- 72 I _ _ — LT2 23K (C) _ v + 2 + (- + 62V + 3 +- )=0 K K T/ Assuming the truth of (a) and (b), again A4 is rational, if + -t3T2 3 ) - (6 + -) +y2 ) -3K( + 3 3,Vr - 3 ~~~~~~~~~(d)~~- 2 2 + 2v2= 0, - 3r —52 + 227 62 + 27y V = 0,

Page 148 148 University of California Publications in Mathematics [VOL. 1 and A is rational, if 1 2( 65T3 05K2 5T3 55K\ VT3 6K 3 K 22( V2+ K-+ V3 +T-+6 ( - yr( y V2K v3_ 3-_ - 3 2_ r -_ + 2 + 3ry(v + + -2) (e) + 37y6T2r - K2 ( V,232 + V26 + V - )+ 7V2 7' y9( qK 2 62 2/) + 20tg 0. We have then the following condition: if r z 0, f(x) is irreducible and if r = 0 and. = - - K/72 then r, K, 8,, must not satisfy at the same time all three relations (c), (d), (e). If then all the above conditions are true f(x) is irreducible and the group is a subgroup of G240. If u = 6 = 8 = v = 0, then our identity reduces to 25y2r5 = 1 and hence in this case 7 is a perfect fifth power of a rational quantity. oa cannot be a substitution of the Galois group of f(x) = 0, for it produces, if applied to 7 = p2p42 0, P8P62 which is equal to zero. a2 applied to r = p2p42 produces P6P22 = 0. It follows then that the Galois group of f(x) = 0 can only be Glo and hence f(x) = 0 is an irreducible Abelian equation. Type VI. If w,, r, r, K, 8, 6,, v but not -, Kc are in our given rational domain, and if not all five quantities r, I, 6, v, v vanish, and if the identity of page 145 exists, then f(x) = 0 with the roots on page 145 is an irreducible Abelian equation with the special restriction, that, if r = 0 and gu = - A'K/r2, then v, 6, T, r, K must not satisfy all the three relations (c), (d), (e) of pages 147-148. Illustrations: (1) i = O = a = v = 0, r = 2. To satisfy the given identity 7 = 21, r, K may be chosen arbitrarily according to the given conditions. The roots are: 1 0047' + W1 c4T 5- 1 X2, X7 = - + 5 FC --- K22 2 3-2 etc. ^) 5x- -10xr~+5 (21 2T) x-(2 5 I+2 ) f(x) x - - x 8-1OTx + 5 X - - T + 2- X 8 2 2 -K 4K 2/ 9 25\ 1. 125 504 5 T5 5. + 25'T2 --- x4 + -2 '+ _ 2 - K 2 x (/8~ ~ ~ - 2 2 /

Page 149 1914] Kuschke: Irreducible Abelian Equations of the Tenth Degree 149 (2573 175 76 75 2 + 24K 12 +102 K2+235 — 2T, 75, 75 r4 5 5 T5 + 28 2-24 - - 25 K 25-+ 10r3 + 10 3 710 o 25 5 5 25 3 35 76 + +2 + 2 + 2 -2 K -2 5 + K - 2 0. K 26 K 22 K2 Assuming r = 1, K = 2 then 5 _ 10x7 155x6 13 5 175 4 725 3 (x)=x 10 -X8 - ~- -6 5 4 + x3 8 32 4 8 25 44085 x2 465 405457 + 212 +x + +-22-= The roots are: 1 1 X1, X6 = + 2 /2 4 1 X2, X7 + 2 ' * etc. * The given domain must include R(w) but not /2. (2) r = IA = a = = 0, v = 1, y = 2. As this is the case, OK r= 0 and u = 0, we first must see that not all three relations (c), (d), (e) on pages 147-148 are satisfied. In our case (c) reduces to v = 0 and hence the values chosen are admissible. Our identity reduces to 1 V5T5 5 27 K2 K2 Choosing K = 2, then;42 must not be in our domain and T/2 must not in the given domain, but c must be rationally known. We have then r = 4. ) = x + 35 + 295 25 x 4 15753 1727 2075 1165 f(x) =x1"+5x7 —x6+ x — x4+ —x3+ xX + 1..0. 8 2 6 4 24 28 2o 2" The roots are: 1 4+ 2 X1, X6= 254 X +,3 42 2 o 4 3 4W4/2 X2, X7- = a+ c o2 4' 25 -4/ 25 ' * etc. - B. I. (2) R2 =0, S2 $ 0. There is no irreducible Abelian equation of this proposed type. The proof is here omitted as it is entirely identical to the proof given under A. I. 2; we need

Page 150 150 University of California Publications in Mathematics [VOL. 1 only to replace Pl, P2, P3, P4, Xi, by, respectively, P6, P8, P2, P4, Pi. B. II. R2 0. From the relation P6P4 + P2P8 = 2R2 we can have at most two of the four p's equal to zero: either P6 = p4 = 0 or P2 = ps = 0. Both cases are identical in the abstract. We therefore cover all cases corresponding to B. II. by dividing B. II. into the three sub-cases (1) P8 = P2 = 0, (2) P8 =; P2 0, (3) P8 0; P2 0. B. II. (1) P8 =P2= 0, p4 0, P6 0. From the conditions: P2 = 0 follows K2 -L2 5 = -N2 /- 10+ 2 + M2 - 10 - 25, P8 = 0 follows K2 - L25 = + N2 - 10 + 25 - M2 - 10 - 25,.K2-L245 = 0 or K = L225. Suppose for a moment K2 = L2 = 0; then P6 = - p4 and Sip = Xl + x6 = 0, which is impossible; hence K = LX 7 0 and therefore 5 is in our given domain. It further follows from the above, that N2 _ - 10- 25 -2 4- 10 + 2o or N2 = - M2 + so that 20M2 P65 = 2K2 - 20M2 - 10 + 245 and p45 = 2K2 + 2 2 /- 10 + 2 hence NP 6 =2R2 54 2 i-400 P4P6 = 2R2 = 4K22 -10 + 2 M22 or, putting 20 M2 - 10 + 2 45 2 then M2' is rationally known and (2R2)5 = 4K22 - M2'(- 10 + 2 -/5) is a perfect fifth power. The following four functions 1i/A6, /3/442, 7/te62,,9/,4 are seen to be double-valued

Page 151 Kuschke: Irreducible Abelian Equations of the Tenth Degree 151 functions under Glo; hence, if we write P1= 7TP6, P3 = 5P42, P7 = 'P62, P9 = 5p4, then r, t, a, 6 are quantities in our given rational domain after the adjunction of w. By actual computation we have 5 1^9 = 102'yplp9 = / E 02i.+5 + (w + w4)Fi + (o2 + W3)F2, 1 '5 137 = 1027p3p7 = i E (p2+5 + (W2 + C3)Fi + (W + W4)F2, 1 where Fl = P6P8 + 6(P69 + (P7(P9 + -(P710 + 8(PO10, F2 = <6(P7 + (6<'10 + (P708 + <s8<(9 + (-9<o10. F1, F2 are seen to be rationally known, if applied to by Glo and as - is in our given domain, the two functions pip9 and p3p7 are rationally known and hence from PlP9 = 'rP4P6 = 2'r5R2 and P3P7 = aTO42p62 = 4o0R22 it follows that 7r8 and oa are in our given domain. Further, P15 + p95 = (K1 + Li 15)2 = 65p45 + P6g5W5 = 2K2(7r5 + 65) + M2' - 10 + 25(65 - r5). Similarly P35 + P75 = (K1 - L )2 = 295p410 + 0a5p10 = [4K22 + M2'2( -10 + 2 45)](5 + a5) + 22K2M2' 4- 10 + 2( - a5). Summarizing now the facts that pi5 + p95, p35 + p75, or, 8r are quantities in our given domain and that 8, 7r, a, 0 are also in our domain, if we adjoin w to this domain, then it is seen that we may write (1) r = a + b - 10 + 245, (2) 8 = a- b- 10+2 25, (3) O = a-+i- - 10- +2-, (4) a = a-, — - 10-+2~5, where a, b, a,, are in our given rational domain. The relation (5) (2R2)5 = 4K22 - M22(- 10 + 2 -5) which we established above and the identity 1,= r5p6 + 35p45 + 955410 + 5p610 + r5 + 5 * 2R2(a2R2 - 7r)[(2 (6) + 682)4R24 - (O2 p45 + a2rp65)] + 5r 4R22(7r262 + 2(r2) + 5r2[(7r2t + $82a)4R22 + (,2^p45 + (rPp65)] - 5r * 2R2(r8 + oa-'2R2) - 5rrba * 8R23 - 5r(73raP65 + ap45) - 5r * 2R2(3hrp45 + a36P 5),

Page 152 152 University of California Publications in Mathematics which is based upon the condition [VOL. 1 i=5 II (i - i+5) =, i=l enable us to express M2' and K2 in terms of R2, a, O, w, 6, y. The relation (6) does not involve any quantity lying outside of our rational domain, for we have r 6 = a2 - b2(- 10 + 25), 0a = a2 - 12(- 10 + 25), 7r2 + 62 = 2a[a2 + b2(- 10 + 2/5)] + 40ab(- 10 + 25), 026p45 + 27rp65 = 4K2[aa2 + aj2(- 10 + 2/5) - 2aab(- 10 + 25)] + 2M2'[2aac - - ba2 - b32(- 10 + 25)](- 10 + 2-5), 7r3^p65 + 63p45 = 4K2aa3 + (- 10 + 25)2[aabaK2 - (3a2b + b3[- 10 + 25])(23K2 + M2') + 1M2'(a3 + 3ab[- 10 + 2F])], 37rp45 + -r3p65 = 4K2aa + 2(- 10 + 25)[raaa2K2 + 6ba2iK2 + 2K2b#2(- 10 + 25) + M2'(ba - 3ba32[- 10 + 2]5] + 3aa23 + a/2(- 10 + 2L5)], r5P6e5 + 65p45 = 4K2[a5 + 10a3b[- 10 + 245] + 5ab4[- 10 + 2/5f2] - 2M2'(- 10 + 25)[5a4b + 10a2b3[- 10 + 2L5] + b5[- 10 + 245]2], 05p410 + ap610 = 2[4K22 + M2'2(- 10 + 2 5)][a5 + a33-2( 10 + 245) + a4(- 10 + 2'5)2] + 8K2 + M2'[a4j + a20f3(- 10 + 2V5) + -5(- 10 + 25)2](- 10 + 2f5). Having found K2 and M2' from (5) and (6), we know p4 and P6. The roots of f(x) = 0 are X1, X6 = P4 + P6 = 7[(a + b -10 + 245)p6 + (a + P 4- 10 + 25)p4 2 + (a - P- 10 + 245)p62 + (a - b - 10 + 2)p4 + r], X2, X7 = W2p6 + wp4 = [,2(a + b I- 10+2)p + 2()p a + ( + - 10 + 24)p42 + (a - - 10 + 2)4p62+ 3(a - b - 10+ 2)p4 + r], etc., where p4 = 2K2 + M2' - 10 + 2, P6 = 2K2- Ms' - 10+2, where K2 and M2' are found from the relations (5) and (6).

Page 153 Kuschke: Irreducible Abelian Equations of the Tenth Degree 153 The corresponding equation is f(x) = (x5 - A1x4 + A2x - Asx3 + A4x - A5) X (x5 - B1x4 + B2x3 - B3x2 + B4x - B) = 0, where A1, B1= 5r4, A2, B2 = lOr2 - 10r 2 - 10RR2y(7r + 22aR2) 1OR2 -(ir + 6), A3, B3 = 57(a2p65 + 92p45) - 15ry * 2R2(7r + - a) + 10y(irt + au)4R22 *- Jy5[2,yr3 + 4R22(oa + -) + 7y(a2p65 + -26p45) + 47R22(62o + -r20) - 6rR2 - 6rR2(7rq + a-2R2)], A4, B4 = + 20R22 + 5y2r4 - 30yr2R2 + 207,2R22(7r262 + 472A2R22) - 5,2[(o38p65 _+ T37rp45)2R2 + 63 9p45 + 7rP3a65)] + 20,R22(7r2 + 62) - 30r2y2R2('r- + 2arR2) + 10y2r[U2irp65 + t-2 p45 + 4R22(62a( + r2-t)] + 40yrR22(cr + 9) - 40ayOR23(1 + 7y'r) - 157y(7rop65 + q p45) + 80y7rSR22 ~-4 5-~[- 2^yR2(asp65 + 33p45) + 2yr(a2p65 + t-2P45) - (6p45 + ap65) - 6r2yR2(7r + -) - 87(7r + -6)aR23 - 37(7r2os65 + 626p45) + 8SyR22(r + 5)(1 + 7r6) + 32rTR22(rt + as)], A 5, B= 4K2 + 572(7(4p65 + 64p45) + 107(7r2p65 + 82p45) - 5y2[2R2(3 + r3) + 0 73t) p4 + (4 + P65)]4R22 - 407yR23(a + 86) - 120yR23(a + rn) - 1207y2R237r6(aS + t7r) + 10yR2(U2p65 + 2 2p45) + 10y2R2(a27r2p65 + -6232p45) - 1-y2rR2(W + 6) + 5,2r2(a,2p65 + O2p45) + 40r2y2R22(Ro( + Tri) - 10r-y2R2(a3p65 +?3p45) - 5r7y(p45 + oPp65) - 40rey20tR23(r + 8) - 15ry(6dp45 + r(TP65) - 15ry2(to2p45 + ar2p65) + 40ryR22(7r + 8) + 40ry2R22?7r(7r + 8): 4[y2(5p65 + 5p4 + 10 5p + p60 8545 + r5) + 5(pP65 + 8P45) + 10y(73p65 + 63p45) - 20y2R22[2R2i7r(a2 + 7r2) + #ta(2pP45 + tr2rp65)] - 40R23(o + t-) - 120YR2375r(o + - ) - 120yR23(ua62 + 6r2) + 10,2R2(76((27rp65 + t2,p45) + 8Ro23oa(uo2 + 7r2)) + 107R2(aO8R23(S- + O) + a2p65 + Tr22p45) - 10r3R2 - 10y2r3R2(7r5 + 2R20O) + 5Y2(u27-rP65 + 2ap45)r2 + 2072r2R22((82 + 7-r2k) + 207r2R22( + -) + 20rR22 - 5ry2(683p45 + 7r3Sp65) - 10ry2R2(a38p65 + -737rp45) + 207ry2R22(ir22 + 4R22a2t2) - 40r.y2R23a7r8 + 207yR22((r2 + 62) - 40ryR23aO + 80ryR228t]. Our next question is: Are the derived conditions also sufficient, that f(x) = 0 with the above roots is an irreducible Abelian equation? First of all we note that not all five quantities a, b, a, 3 and r can be zero at the same time, for then f(x) would contain the two rational factors x- - A1x4 + A2x3 - A3x2 + A4x - A and X - Bx4 + B2X3 - B3x2 + B4x - B5, as seen from the above functions Ai, Bi. Further P4 F: P6,

Page 154 154 University of California Publications in Mathematics [VOL. 1 for then P4P6 = == p42 = - p62 = 2R2 and hence <.i= X1 + X6 would be rationally known. It follows then that equations (5) and (6) must lead to K2 $ 0 and M2' 0 otherwise we have no irreducible equation corresponding to case B. co may be rationally known or not; it certainly cannot be expressed rationally by p4 or p6, for if co is not in our domain, then w satisfies an irreducible quadratic, as f5 is rational, whereas p4 satisfies an irreducible quintic. It follows then, as just explained under B. I, that the sum of any two, three, or four roots must always contain the irrationality p4 and p6 respectively and that f(x) cannot have a rational factor of the first, second, third, and fourth degree. Whenever the sum of any five roots does not vanish, this sum is irrational for the same reason; the first case, however, can happen only if we pick out five roots belonging to the same system of intransitivity. Our question of the irreducibility of f(x) simply amounts to the following question: Under what conditions can all five Ai be rationally known? Ai shows that we must have r = 0, if ~-y shall disappear. Further, from A2, we must have 7r + a = 0 as R2 $ 0; hence a = 0. Under these conditions we get further the relations from As 4aR22 _ 4K2bap- + M2'b(a2 + [2[- 10 + 2 a5]) (I) - - 10 + 2-45 1 + b2(- 10 + 2~5) from A4: 4-yR2K2[a3 + 3aI2(- 10 + 24)] + 27R2M2'(- 10 + 2~45)[3a23 (II) + 33(_ 10 + 215)] + [2K2a + 3M2'(- 10 + 25)] x [1 + 37b2(- 10 + 245 )]= 0, and from A5: (III) y2{2b5M2'(- 10 + 245)3 + [8K22 + 2M2'2(- 10 + 245)] x[a5 + 10a32( — 10 + 2 -5) + 5a4(- 10 + 25)2] + 8K2M2'(- 10 +2 '5)[5a43 + 10a2$2(- 10 + 2 ) + $5(- 10 + 25)} - 10M2'(- 10 + 245)(b + 2b3y) - 80R23ca[y2b4(- 10 + 25) - 1] + 10R2b(- 10 + 25)[8K220a + 2M2'a2 + 2M2'/2(- 10 + 245)][272R2(a2 - -2[- 10 + 25]) - b2(- 10 + 2-5) - y] + 160R24ya[a2 - -2(- 10 + 2q5)] x [1 + yb2(- 10 + 25)] = 0. If then r = a = 0 and equations I, II, III are true f(x) is reducible, having two rational factors each of the fifth degree. If, however, not all these five equations are satisfied, f(x) is irreducible and must have two as well as five systems of intransitivity, as seen directly from the A i, Bi respectively from the roots. Its group, being therefore a subgroup of G240, can only be Glo, as al and a2 are impossible in the Galois group of f(x) = 0 under our conditions, for we have

Page 155 Kuschke: Irreducible Abelian Equations of the Tenth Degree 155 oa interchanges p4 and P6, so that P4 = - P6, if oa were in our Galois group and hence M2' = 0 or K2 = 0, but this contradicts our conditions. Similarly a2 is impossible in the Galois group, as a2 changes p4 to o4P2 = 0, which contradicts R2 # 0. It follows then that our f(x) = 0 is an irreducible Abelian equation under the derived conditions. Type VII. If a, a, P, a, b, K2, M2', y, r, R2, but not iJ are quantities in our given rational domain and if not all five quantities a, (3, a, b, r are zero and if relations (5) and (6) on page 151 exist and if R2 - O, K2 ~ 0, M2' 0-then the above f(x) = 0 with the given roots is an irreducible Abelian equation, if \2K2 - M2M' - 10 +2~5 is not in our domain and if not all five equations r = a = 0, (I), (II), (III), are satisfied. Illustrations: (1) a = b = a = = 0, r = 2, M'= 1. Our identity (6) shows 1 -r5 25y72 hence Y = 2-5. Equation (5) shows (2R2)5 = 4K22 - (- 10 + 245), hence, taking K2= +we have R2 = pif. Here our rational domain cannot include ~f2, and V2 2 -10+2-,but must contain 5 -, 10 -.J3 As r 5 0, not all five equations r = a = 0, (I), (II), (III) are satisfied and hence all conditions are satisfied. The roots are x1,x6 = 2 + - 10+255+ /325- /- 10+2'5 =,= x2, x7 = W /2,5 + 4- 10 + 2 5 + w2 25 - 5 - 10 + 25 3 etc. f(x) = x10 - + 2 Wo10) 5x8 + ( -16 O1 + 25 o02) x6 -4 2 45x + 1 119 X4 ) )5x ao — 2~0 ) ~2~x

Page 156 156 University of California Publications in Mathematics [VOL. 1 ( 189 1 51 4l+ 45 wo2 -_ (743 ) X2-5 5.212 10 1-6 2 - ~+s^+ - +1 7 -+ - 7 - + 2 qf-a = 0. + 8 5+52. 212 5-.28 5.26 + (2) r =a= =b=0, a= 1, M2'= 1, R2 =- 2 i5, K2=!-10, = =. 2 232V- 10 By these values chosen all conditions are satisfied. The roots are: X1, x6 = 4- 10 + 4- 10 + 245 + - 10 --- 10 -- 2-5 2- 1 0[;I -10+4-10+24+ — 10- -10+2,], 1 _ X2,X 71 = iw / —0 + 4- 10+2-5+ 2- ~- 10 - -- 10 +2 2] 2 I, [|3 xi 10+ - 10+25+2 - -10+2d |, 1 1 30 1 f(x) = x+'O5+ 2 + o x8+ 245+ 26 --- - 234 5 5 +4 - [l[ 4)t-10+2- o] + 540 x [20x-^ + +2 45.- 26, + +10 2/ 10 27 3 ]5 _ 5 7 + 270 + 3 \3 + 100'24. 5L1 + 5 + 7 f2x) - 10+ 2 ~ 2 ~ - 16 ----0 -- -- -- 26 1 210 2'lO026 01U 1 1 5 x-2++40 ^ [10 ]20 1+ -'+ 4o 29 4 --- 10 1 [4]4/ —10 21-/- 10 1 1 15 55 +274y- 103 +5 213J+8 2[2Vi- 1+2 4 10 + 2'1 -2O3- 2 12 ] = 0 Here the given rational domain must not include 2 -, 1 - lO + 21, but it must contain - 10,;24. B. II. (2) P8 =0, P2 O0, p4 P- 01 P6 1 0. From P8 = 0 follows K2- L25 = + N2 4-0l +2-M2- 10-2 - 2 -2

Page 157 1914] Kuschke: Irreducible Abelian Equations of the Tenth Degree 157 hence p2 = -o2(K2-.L2 5) 0. It follows then w is rationally known. Putting then P1 P3 P9 P7 P6 P42 P4 P2 then ir, y, 6, a are in the given domain. Here again not all five quantities r, ir, i, 6, a can be zero at once, for then f(x) would be reducible. As co is rationally known, we may put p22/P4 = g and P62/p2 = h, where g, h are quantities in our given domain, g 57 O, h # 0. We have then P22\P62 2 P65 = P 64 +- 2gh2R2, P4 \ P2 / 5- P22. _ 492R22 P25 = (P2 P2 '(P4P6)2 = -4R22 P4 Pe2 h p4 = (P6P4)5 24R24 Ps65 gh2 If P6 were rationally known, all three pi would be rational and if p6 is not in our given rational domain, then p2 and p4 cannot be rationally known. Whenever pi is in our given rational domain, f(x) splits up into the two rational factors i=5 i=5 II(x-xi) and II(x-x~+s). i=1 *=1 It follows then that 2gh2R2 is not a perfect fifth power. The last condition includes the three conditions above, namely R2 0, h 0, g O. We have here again an identity, based upon i=5 ni ( -x+5) = 1=1 namely = 2r5gh2R2 + 4- g2R24 28_ 5R28 245 R24 h g2h4 h2 — 40R23 r( + 8h3agR23 + rig 3 Ora -' 20 R 23*262R23 + 4'a22R22 + 22 'r2'2R22 62 ~)2 gh2 h2 h h2 7rg + 4R22 1 /rR2 +2OrR 222_a + lOrR2 2r2aR + 8 R23 2 h 2 g gh2 2a36gR2 + 2s6sR2S) 40r~rS 63R23 5 h gh2 h -r.

Page 158 158 University of California Publications in Mathematics [VOL. 1 The roots of f(x) = 0 are here x1, x6 = 42gh2R2 + + 4g 2 244 V 2 2ir /2gh2R2 + ' 4 5/4g2R22 5 R24 ] + < ^r^-+ N +r etc. f(x) = (x5 - A1X4 + A2x3 - A3x2 + A4x - A5) x (x5 - Blx4 + B2x3 - B3x2 + B4x - B) = 0, where A1, B1 = 5rr, A2, B2 = + 10yr2 - lOR2 - 10yR2(7r6 + tro 2R2) =F;F1OR2 ( r + + 2 ) As, Bh = 1OR2 ( + 2R2 + 10+2 (2g + h2 + /j- ) - 307rR2 ( + 2h- + 8) + 20R( 7r2R2 + ag + a62R2 - o [lOr-3 ++ 20R2 (g +-2 2-R2) + 10yR2 ( 2rg + h -R2 + r22R2 + 8R 2 -30rR2 1 + 75 + ' h + 20R (h + ag) ] A4, B4 = - 1R2 (2R + hg) + 20R22 + 572r4- 30yr2R2 + 20y2R22( 252 + 4R2__ )-1Oy2R2 ( e62R2 + 8R2363h 2 1 h g h2 \4R- \ / \4(R2 g+8R2\ + 7r3orhg + 4h2 ) + 20yR22( 7r2 + 4Rh22 2) + 20yR2262 + n'(hg +24 ( gh2 h2 - 30r2^y2R2 (1r + 2R2a0) + 20y2r ( U2g + 262aR2 + 22R +23 2 R23) r 20^rR (r +2 U R R22) _ 40'2R23ra' _ 40R23 (r + o + 6) - 30^yR2hgir(a. + r) 0 R22( + o) 240 h2 + 807R227rb + 80ryR2 ( g + ) h g h2 h - 5[4rR (g + 2R2 2R2 (ffR29 + r3gh + 24R24 '3 - 2R2 (2R2u9 + 8R23- + gh)- 30r26R,- 30r27R2 (- + 2R2 ) +4R2 ( + 2 62R + 8R23 2 - 82R23 y_ R + ^ + 4R2a. + 4R 2R ( —g + h- + gh) +R(+h gh / h h

Page 159 1914] Kuschke: Irreducible Abelian Equations of the Tenth Degree 159 -67rR2hg(^y7r + 1) - 12R22 (ory + 1) - 4872R24 h gh2 32-yo72R24 + 86ryR22(r + 8) + 8R22(7r + 5) + 24 + 16ryR2( 2rR2 + 7r + 2R2)], h h A 5, B5 = 2R2 gh -+ h2 23R2 + 1022R2 2r gh2 + ly2a R2 + 23R23 6 2R2U2g2 23R2362 40R2 3 40y2R2(363 320y2R26 ( + ) _ 40R ( + 3h) h3 h 40,hR23 120(2R23 _ 40R2'2 (a+ + ' + bh) _- (aS + 2r + 'ag + 2r~h + 62) 120y2R3+ OR _ l20^y2R237 (a2 + fa2g + 8itrh) + 20R229(1 + 'a2 + r2y) 80+hR2 (1 + 2R2 + 20,2R2 (2 a27r2 + 7r2 ~2 * 22R 2 + 23R23 ~262 160R24r'r + 40gyR226(r + + ( + ) + 80y7raR22g - 10y2r3R2 - 1072r3R2 (r + 2R2 ) + 10r2gR2(1 + 2) + 20r2R22 (1 + 2+22R2 ) + 20.y2R2( rag + 2Red +2R2r) 1Ory 2 R2 39 + -gh + -1Or7R2 ( R k h gh2 /h -10r'f'R~ - +^+-10 2-r~R h hh h - 3yrRg (2R + rh - 30rR ( + 302 (2R2a + 2g + S R2 gh) + 40r7R22(r + 6) + 40r2RI22 ( r82 + 22R22 + ir2) (~y 28 + 2g2 + 2485R r i= { 2 (27r5gh2R2 + 2g2 h 2- 22 + + r) 91\g2h4 h gh2 + R2rh2 + 2ag2R2 + 23 R23) + 207R2 ( r3gh2 + 2a3g3R2 2363R2 3 or83ir u83s323R23 +o~ +6 + 23 402R23 ( h + h3 + h 9h2 h 3 40R23 23R23 _403R23 120R 23 ghk3 ( h2 / h (a + r + P+h)- h 120- R23 ( + ag + Sth + a- + 62 + — g + r h) h \r Tr

Page 160 160 University of California Publications in Mathematics [VOL. 1 a'2 - 2 ~r2o-2 627r22 + 202R22 ( a2r2g + 22R22 + 22R22. + 23R23 h2 \ h2 h gh2 +20R2 + 22R2 + 207R2296(oa2 + r2) + 80R24Sy (a2 + 62) h2 (h2 80 R2h o + rgh R + 40R22g(r + U) + 80oTCR22 irg + 22R22~ \ - 10r3R2 (1 + 2R2a- + ra) + 10y2r2R2rg(l + a2) 20yr2R 22 ( 22t26R22 + h q22 + i2 h + o ' + 'h + h ) + 20r2yR2 (09 + 2R2) - 10rR2 (2R2 + h ) + 20rR22 - 10ry2R2( 2R2a369 + 23R2363: + 3 h + 24R24i3r) i 2 2 B 22 R22a _ 2 2._2 40ryR23. + 20ry2R22 + (2aRy +- + + o-) + 20-yrR22 (1 + r262 + 22R22 )- 30r7R9( 2R22 + 2 h) + 80rT7r R22 Regarding the sufficiency of the derived conditions we first decide whether f(x) can be reducible under our conditions obtained so far. As P2, P4, P6 each satisfy an irreducible quintic, ~d7y an irreducible quadratic, 7 cannot be expressed rationally by p2, p4 or P6, and hence no root can be rational, for we may write e/p 62 W 2R2 0P2 + cokp4 + W-;p6 in the form - + -- 2 + o-kp6. If the last expression h P6 were rational, we must have necessarily that p6 is a rational quantity. It follows then, that no root is here rationally known and f(x) has no rational linear factor. The same theory holds for the sum of any two, three or four roots, as c is rationally known and we have here again, that the sum of any two, three, four or five roots must be irrational except when it is zero which happens only for 5 5,>xi and i, xi+5. 1 1 Looking into the above Ai and Bi, we see that they contain as only the irrationality 7. This irrationality drops out in A1 and B1 if (1) r = 0, in A2 and B2 if (2) r - + +,2 and hence in As and B3 if rg(1 + y2) + 2og-g - + 6)3h (3) - + 0. r- + [a + ' h + yr~ + ypr2rh + 2[] l'~~~~~~~~~~

Page 161 1914] Kuschke: Irreducible Abelian Equations of the Tenth Degree 161 in A4 and B4 if, in addition to (1) and (2), + 7 + [ ya-3g + 8g - 262 + 3og(1 + aoy)] + 2a2g - 7r3hgy - ahg - 37rhg(1 + yro-) (4) 5)3 (4) ( + 3h [Ty2 + 3y^2 - 27yg9 + 1 - 2r] + (7 + 5)2h [+ 1 - Y( + a + 37)]. Finally, assuming the truth of (1) and (2), A5 and B5 are rationally known, if 75gh2'2 + 5 2Sgh2 + 103gh -- 575 ' + 2 +g2 + 2y3g2 + 8gh(^y2a2r2 1 + 7yr2 + 772 + 4-77r) + 2gh(r + ) a - 5 (7r + )h [2 7273T + y%2'3'vg + 6Thy72 + y 63 + v (5) + ( +. + g) + 35O- 3- + 6 + 2 + 9)] (ir + ~ h [ - 5 ( +5 [ + ) 37rg + 5 72 + + 2753 + 72_262g + y2gr2ua2h + 'g + rgy(a2 + 62) + 'yagh + 4o6 g ].(rr + )4^ (~rr. ^'. >(+ 6)5f2 (1 + ~ ( ) 2 (+ )5 ) h2~r(2 + 1) + 5 (r + )h2 + ) (r + )7h32 (i 5 +-2 )h21 + 5.(2g 7(1 + 70-2) - 2- g Hence, to assure the irreducibility of f(x), we must have: Not all of the above five equations must be satisfied. If this condition is observed in addition to the conditions already found, then f(x) is irreducible, its group being therefore transitive, must be imprimitive with two as well as five systems of intransitivity, for f(x) contains the five rational factors (x - xi)(x - xi+5), if we adjoin P2, P4 or P6 to the given rational domain; and adjoining 7 then the above functions Ai, Bi show that f(x) splits up into the two rational factors i=5 i=5 I (x -xi) and tI(x - x+5). i=l i=l It follows then that the Galois group of f(x) = 0 is a subgroup of G240 and it can only be Glo under our derived conditions, as al and a2 are impossible substitutions in the group of f(x) = 0. al changes p25 7 0 to P85 = 0 and a2 applied to P65 produces ps5 = 0, so that under the conditions derived f(x) = 0 is an irreducible Abelian equation. Type VIII. If w, 7r, a, 6, j, g, h, y, r, R2 are quantities in our given rational domain and if 47y, 12gh2R2 are not in our given domain, and if r, 'r, ~, 6, a are not all at the same time zero and if the identity on page 157 exists and if not all five equations

Page 162 162 University of California Publications in Mathematics [VOL. 1 on pages 160-161 are satisfied, then f(x) = 0 of page 158 with the roots on page 158 is an irreducible Abelian equation. An example may be furnished by taking r = = a - = 0. Here our identity reduces to 2= 2gh2R2zr. Assuming then g =h = R= = 1, we have 1 1 72=2; hence =23. /2 and 42 cannot be in our rational domain, as +/ as well as /2gh2R2 are irrational. Further equation (2) on page 160 is not satisfied. 8971 9075 f(x) = x10 - 20x8 -6 607 + 65x6 + 8215 + 9 X54 = 0. 51755 65925 96475 433563 + X2 + +2 + x + 26 27 29 213 The roots are: x2, x, = W4 4 + W 2 + c3 16 2 -22 2 ~/2' x1, x6 = I4 + / + 2~ -- etc. The given rational domain must contain w. B. II. (3) pi O (i = 2, 4, 6, 8). There is no irreducible Abelian equation of the proposed type. The proof, which is here omitted for brevity, is identical to the proof given under A. II. (3) if we replace there P1, P2, P3, P4, Xi, respectively by P6, P8, P2, P4, 'Pi. C. CONCLUSION. There are eight types of irreducible Abelian equations of the tenth degree. Types I to IV are quintics in x2. There is no such equation in the natural rational domain R(1), for in all types Z must be a quantity of the given domain. Six types are possible, only if w belongs to the rational domain.

Page 162 UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 7, pp. 163-169 February 16, 1915 ABRIDGED TABLES OF HYPERBOLIC FUNCTIONS. BY F. E. PERNOT In the calculation of the operating characteristics of long transmission circuits the most convenient and direct solution is afforded by the use of hyperbolic functions of complex variables. For lines consisting of large conductors in which the losses are small it is desirable to make computations to the degree of accuracy afforded by six-place tables of logarithms. Also, in such cases, the argument to which the hyperbolic functions are taken from the tables is small; for power circuits usually below 0.5. The most convenient table of these functions is that of the Smithsonian Institution, in which are tabulated both the logarithms and natural values for arguments from zero to 6.0, five decimal places being given. A table of Gudermannian functions is also appended, which makes it possible to find the values of functions (hyperbolic) to six places; not, however, without involving a double interpolation, first to the Gudermannian and then from a table of trigonometric functions. For small arguments the hyperbolic cosine is not varying rapidly, hence easy interpolations are assured. The hyperbolic sine, on the other hand, is varying with the same rapidity as the argument, and therefore the interpolations for log sinh x are cumbersome unless the tabular interval is very small. These considerations are immediately seen from the series expressing the two functions: 2 4 6X cosh x - 1 + + ++ + sinh x x + X + + - +.... 3! 7 In all cases the hyperbolic tangent is most easily derived from sinh x and cosh x by sinh x tanh x = (2) cosh x

Page 164 164 University of Califorlia Publications in Mathematics [VOL. 1 The series for sinh x in (1) can be written 2 4 at6 sinh x x3(1+)- 1~,+. sinhx —x (1 +-3 + -+ +.-...) (3) which immediately suggests the advisability of tabulating values of the quantity in brackets, 2 4 6 < =1 + 3 + xo+ +.... +(4) for this quantity is seen to have an even smaller rate of change than cosh x. The value of the hyperbolic sine is then given by sinh x= xy sinhx C (5) Since extended computations are most conveniently done by logarithms, the tabulation was made of log y. Log x naturally being at hand, the value of log sinh x is immediately found by simple addition of log x and log y, and, further, it is found without having to make any inconvenient interpolations. A glance at the values of log sinh x as tabulated in any table will immediately impress one with the impracticability of interpolating directly for log sinh x for values of x between 0 and 0.10, in which region a large proportion of the values of x fall for the particular work above referred to. This scheme of tabulating the ratio of a function of a variable to the variable itself is the same as is used in the 'S and T" tables found in Bremiker's logarithm tables, where for small angles the ratios of the trigonometric sine and tangent to the angle are given. CONSTRUCTION OF TABLES sinh x The following table contains values to base 10 of log and log cosh x, x together with the differences to be used in interpolating. The arguments progress in steps of 0.005 from zero to 0.600, giving 121 entries. Logarithms are given to seven places. sinh xc The first column, logs -, was computed by first evaluating the expression x for y in (4) and then taking out log y from tables. For the small values of x used, this series is very convergent, hence the labor involved was not excessive. The evaluation of the series was made for alternate entries, and the intermediate values obtained by interpolation. sinh x 1 Using this value of log ---- the value of log - was formed. Using this x sinh2 x as argument in Zech's table of addition logarithms, the value of logo1 cosh2 x is immediately obtained, from the relation cosh2 x -sinh2 x +- 1. Interpolations in the addition logarithm tables were made to the nearest even number in order that the resulting value of log cosh x might appear to the nearest unit in the last place.

Page 165 Pernot: Abridged Tables of Hyperbolic Functionsl 165 The complete set of values was checked by differences, and in a few cases the last unit was changed by one in order to give uniformity in the second differences, which in both tabulations are practically constant. In addition to the check by differences, every tenth entry was checked independently by calculating directly the values of sinh x Ex — e- Ex-t E-x Y — -— 2 x and cosh x - - x 2x using the tabulated values of the exponentials as given to nine decimals in tables of the exponential function by J. W. L. Glaisher, F. R. S., published in the Transactions of the Cambridge Philosophical Society, vol. XIII. From the above values, the logarithm to seven places was taken from tables, which agreed with the previously tabulated values to the nearest unit of the last place, except in two or three cases where a difference of one was noted. To facilitate interpolation, the values of the first derivatives of the function multiplied by the tabular interval (o( -0.005) are tabulated in units of the last place in the tabulated seven-place logarithm of the function. The second differences are also tabulated. For the first three columns: sinh x f (x) loglo = logioy d sinh x d of/ (x) -- ]oglo --- = -logilo dx x dx ~~~~2 4 ~(6) d x2 X4 o loglo(1++ + +....) dx 8 5. x1 2x' 3x~ 4x 2 / 1 x+ + + + 4 + Y _3 7 )9 M = modulus of the common logarithm system = 0.43429448. This series is very convergent, and was used in the form for the value of -, 0.005, ~f (x) -0 O1M x 2 x 3x + (7) Of course the above value was multiplied by 107 to reduce to units of the last place in the tabulation of log y. Values of log y to be used in the computation were taken directly from the table. 52 is the average of the differences in of'(x) immediately preceding and following a tabular value of log y.

Page 166 166 University of California Publications in Mathematics [VOL. 1 For the last three columns: f(x) = loglo cosh x f' (x) -= - logio (osh x dx (8) sinh x (osh x J Thus of'(x) = 0.005M1 tanh x (9) A2 is the same as defined for the previous case. It is t(, be noted that the second differences, or accelerations if x be considered equicresccent, are practically constant. Easy and accurate interpolations are thus assured, even to seven decimals and for tabular intervals in the argument as great as 0.005, which is five times as great an interval as is used in the Smithsonian Tables for the functions tabulated to five decimals only. TO INTERPOLATE USING SECOND DIFFERENCES Let it be required to determine the value of f(x) for an argument x - Xo a Xo is the value of the argument nearest that of x. a, then, is the distance over a which the interpolation has to be made, and - = =the fraction of the tabular (o interval. Using the second differences, we have f(x) =f(xo ~ no) =f(xo) ~ ni Wf'(x) + -A2 (10) f(xo) is the value of the function corresponding to the argument xo. Illustration: Required, log sinh 0.1163740 log cosh 0.1163740 x 0.1163740 xo0 0.115 a +=0.0013740 n = - + 0.27480.005 for sinh x for cosh x f(xo) 0.0009569 0.0028655 n wf'(x) 228.6 683.2 -n22 = 1.4 4.1 f(x) 0.0009799 0.0029342 log x - 9.0658560-10 log sinh x = 9.0668359-10 log cosh x = 0.0029342 log tanh x = 9.0639017-10

Page 167 1915]1 Pernot: Abridged Tables of Hyperbolic Functions 167 LOGARITHMS OF HYPERBOLIC FUNCTIONS OF A REAL VARIABLE From x 0 to x - 0.600 sinh x x.. logio ---. f(x ) 2 loglocosh x x 0.000 0.0000000 000.0 36.2 0.0000000.005.0000018 36.2 36.2.0000054.010.0000072 72.4 36.2.0000217.015.0000162 108.6 36.2.0000489.020.0000289 144.8 36.2.0000869.025.030.035.040.045.050.055.060.065.070.075.080.085.090.095.100.105.110.115.120.125.130.135.140.145.150.155.160.165.170.0000452.0000651.0000886.0001158.0001466.0001809.0002189.0002605.0003057.0003546.0004071.0004632.0005229.0005862.0006531.0007236.0007977.0008755.0009569.0010418.0011304.0012226.0013184.0014178.0015208.0016274.0017376.0018514.0019688.0020899 180.9 36.2 217.1 36.2 253.3 36.2 289.5 36.2 325.7 36.2 361.8 36.2 398.0 36.2 434.2 36.2 470.4 36.2 506.5 36.2 542.7 36.2 578.8 36.1 615.0 36.1 651.1 36.1 687.2 36.1 723.3 36.1 759.4 36.1 795.6 36.1 831.7 36.1 867.8 36.1 903.8 36.1 939.9 36.1 976.0 36.0 1012.0 36.0 1048.1 36.0 1084.1 36.0 1120.1 36.0 1156.1 36.0 1192.1 36.0 1228.1 36.0 1264.1 36.0 1300.1 36.0 1336.0 36.0 1-372.0 36.0 1407.9 35.9.0001357.0001954.0002659.0003473.0004396.0005426.0006565.0007813.0009168.0010632.0012203.0013882.0015670.0017565.0019568.0021679.0023897.0026222.0028655.0031194.0033841.0036595.0039456.0042423.0045496.0048676.0051962.0055354.0058852.0062456.0066165.0069979.0073899.0077923.0082052 Wf'(x) } 2 000.0 108.6 108.6 108.6 217.1 108.6 325.7 108.6 434.2 108.5 542.8 108.5 651.2 108.5 759.7 108.4 868.1 108.4 976.5 108.4 1084.8 108.3 1193.1 108.2 1301.3 108.2 1409.5 108.1 1517.5 108.0 1625.6 108.0 1733.5 107.9 1841.3 107.8 1949.1 107.7 2056.7 107.6 2164.3 107.5 2271.7 107.4 2379.0 107.3 2486.2 107.2 2593.3 107.0 2700.3 106.9 2807.1 106.8 2913.8 106.6 3020.4 106.5 3126.8 106.3 3233.0 106.2 3339.1 106.0 3445.0 105.8 3550.8 105.7 3656.3 105.5 3761.8 105.3 3867.0 105.1 3972.0 104.9 4076.9 104.7 4181.5 104.5.175.0022145.180.0023427.185.0024745.190.0026099.195.0027489.200.0028915 1443.8 35.9.0086286 4286.0 104.3

Page 168 168 University of California Publications in Mathematics [VOL. 1 sinh x x loglo - fl (x) A2 logio cosh x wf'(x) A2 x.200.0028915 1443.8 35.9.0086286 4286.0 104.3.205.0030377 1479.7 35.9.0090624 4390.2 104.1.210.0031874 1515.6 35.9.0095066 4494.2 103.9.215.0033407 1551.4 35.9.0099612 4598.0 103.7.220.0034977 1587.3 35.8.0104262 4701.6 103.5.225.0036582 1623.1 35.8.0109015 4805.0 103.3.230.0038223 1658.9 35.8.0113872 4908.1 103.0.235.0039900 1694.7 35.8.0118832 5011.1 102.8.240.0041612 1730.5 35.8.0123894 5113.7 102.5.245.0043361 1766.3 35.8.0129059 5216.2 102.3.250.0045145 1802.1 35.7.0134326 5318.3 102.1.255.0046965 1837.8 35.7.0139696 5420.3 101.8.260.0048821 1873.5 35.7.0145167 5522.0 101.6.265.0050712 1909.2 35.7.0150739 5623.4 101.3.270.0052639 1944.9 35.7.0156413 5724.5 101.0.275.0054602 1980.5 35.6.0162188 5825.4 100.8.280.0056600 2016.2 35.6.0168064 5926.1 100.5.285.0058634 2051.8 35.6.0174040 6026.4 100.2.290.0060704 2087.4 35.6.0180117 6126.5 99.9.295.0062809 2123.0 35.6.0186293 6226.3 99.7.300.0064950 2158.5 35.5.0192569 6325.8 99.4.305.0067126 2194.0 35.5.0198945 6425.0 99.1.310.0069338 2229.6 35.5.0205419 6523.9 98.8.315.0071586 2265.1 35.5.0211993 6622.5 98.5.320.0073869 2300.6 35.5.0218665 6720.9 98.2.325.0076187 2336.0 35.4.0225435 6818.9 97.9.330.0078541 2371.5 35.4.0232302 6916.6 97.6.335.0080930 2406.9 35.4.0239267 7014.0 97.2.340.0083354 2442.2 35.4.0246330 7111.1 96.9.345.0085814 2477.6 35.3.0253490 7207.8 96.6.350.0088309 2512.9 35.3.0260746 7304.3 96.3.355.0090839 2548.2 35.3.0268098 7400.4 96.0.360.0093405 2583.5 35.3.0275546 7496.2 95.6.365.0096006 2618.8 35.2.0283090 7591.7 95.3.370.0098643 2654.0 35.2.0290730 7686.8 95.0.375.0101315 2689.2 35.2.0298464 7781.6 94.6.380.0104022 2724.4 35.2.0306293 7876.1 94.3.385.0106764 2759.5 35.1.0314216 7970.2 93.9.390.0109541 2794.7 35.1.0322233 8064.0 93.6.395.0112353 2829.8 35.1.0330344 8157.4 93.3.400.0115201 2864.9 35.1.0338548 8250.5 92.9.405.0118083 2899.9 35.0.0346845 8343.2 92.6.410.0121000 2934.9 35.0.0355234 8435.6 92.2.415.0123952 2969.9 35.0.0363716 8527.6 91.8.420.0126940 3004.9 35.0.0372289 8619.2 91.5.425 0.0129963 3039.8 34.9.0380954 8710.5 91.1

Page 169 1915-1 I Pernot: Abridged Tables of Hyperbolic Fulnctions 169 sinh x x loggo f- (x) A2 logo cosh x of'(x) A2 x.425 0.0129963 3039.8 34.9.0380954 8710.5 91.1.430.0133020 3074.7 34.9.0389710 8801.4 90.7.435.0136112 3109.6 34.9.0398557 8892.0 90.4.440.0139239 3144.5 34.8.0407494 8982.2 90.0.445 0.142400 3179.3 34.8.0416521 9072.0 89.6.450.0145597 3214.1 34.8.0425638 9161.4 89.3.455.0148828 3248.8 34.7.0434844 9250.5 88.9.460.0152095 3283.5 34.7.0444139 9339.2 88.5.465.0155396 3318.2 34.7.0453522 9427.5 88.1.470.0158732 3352.9 34.6.0462993 9515.4 87.7.475.0162102 3387.5 34.6.0472552 9602.9 87.3.480.0165507 3422.1 34.6.0482199 9690.1 86.9.485.0168946 3456.7 34.6.0491932 9776.8 86.6.490.0172420 3491.2 34.5.0501752 9863.2 86.2.495.0175929 3525.7 34.5.0511659 9949.2 85.8.500.0179472 3560.2 34.5.0521651 10034.8 85.4.505.0183049 3594.6 3-.4.0531728 10119.9 85.0.510.0186661 3629.0 34.4.0541890 10204.7 84.6.515.0190307 3663.4 34.3.0552137 10289.1 84.2.520.0193988 3697.7 34.3.0562468 10373.1 83.8.525.0197703 3732.0 34.3.0572883 10456.7 83.4.530.0201452 3766.3 34.2.0583382 10539.9 83.0.535.0205235 3800.5 34.2.0593963 10622.7 82.6.540.0209053 3834.7 34.2.0604627 10705.1 82.2.545.0212905 3868.8 34.1.0615373 10787.1 81.8.550.0216791 3903.0 34.1.0626201 10868.6 81.4.555.0220711 3937.0 34.1.0637111 10949.8 81.0.560.0224665 3971.1 34.0.0648101 11030.6 80.6.565.0228653 4005.1 34.0.0659171 11111.0 80.2.570.0232675 4039.1 34.0.0670322 11190.9 79.7.575.0236731 4073.0 33.9.0681553 11270.4 79.3.580.0240821 4106.9 33.9.0692863 11349.5 78.9.585.0244945 4140.8 33.8.0704252 11428.3 78.5.590.0249103 4174.6 33.8.0715720 11506.5 78.1.595.0253294 4208.4 33.8.0727265 11584.4 77.7.600.0257519 4242.2 33.7.0738888 11661.9 77.3

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 8, pp. 171-186 February 18, 1920 A LIST OF OUGHTRED'S MATHEMATICAL SYMBOLS, WITH HISTORICAL NOTES, BY FLORIAN CAJOR1 There are two considerations which have induced the present writer to exhibit William Oughtred's symbols in a table: (1) The extraordinary emphasis laid by Oughtred upon the invention and use of mathematical symbols at a period when - the notation of modern elementary algebra was in a state of formation, and (2) the inaccessibility of Oughtred's books to most mathematicians of the present time. By means of this table it is hoped that some of the desired information is placed within easier reach and that some of the erroneous statements relating to his notation may be avoided hereafter. Omitted from the list are all astronomical symbols used by Oughtred, also a few very specialized and tentative symbols in mechanics which seem devoid of historical interest. The texts referred to are placed at the head of the table; the symbols, in the column at the extreme left. Each number in the table indicates the page of the text cited at the head of the column containing the number on which the symbol on the left occurs. Thus the notation 01500, 5 occurs on page 3 of the Clavis of 1652. The page assigned is not always the first on which the symbol occurs in that volume. Some desirable information which could not be inserted in the table itself is found in the accompanying notes. The table affords an interesting view of the growth and changes in notation in Oughtred's works during half a century. The following is a key to the abbreviated book titles used in the table: Clavis of 1631 =Arithmeticae inlnumeris et speci-|ebvs institvtio:lQvae tvm logisticae, tvm analyti!cae, atqve adeoltotivs mathematicae, qvasilclavislest. -Ad nobilissimvm spe|ctatissimumque invenem Dn. Gvilellmvm Howard, Ordinis, qui diciltur, Balnei Equitem, honoratissimi Dn. Thomae, Comitis Arvndeliae & Svrriae, Comitis Mareschallli Angliae, &c. filium. - Londini,i Apud Thomam Harpervm, M. DC. xxxr. Clavis of 1647=The Key I of the Mathematicks I New Forged and Filed:l Together with ] A Treatise of the Resolution of all kinde of Affected Aequaltions in Numbers.1 With the Rule of Compound Usury; And demonstration of the Rule of false Position. And a most easie Art of delineating all I manner of Plaine Sun-Dyalls. Geomeltrically taught I By I Will. Oughtred.1 London, Printed by Tho. Harper, for Rich.| Whitaker, and are to be sold at his I shop in Pauls Church-yard 1647.

Page 172 172 University of California Publications in Mathematics [VOL. 1 Clavis of 1648=C Clavis mathematica I denuo limata,I sive potius I fabricata. Cui accedit I Tractatus de Resolutione Aequatio I num qualitercunque adfectarum I in numeris:lf Et Declaratio turn decimi elementi Euclidis I de lateribus incommensurabilibus: tum I decimi tertii & decimi quarti elelmenti de quinque solidis I Regularibus.l Atque hic passim I Logisticae decimalis, & Logarithmorum i Doctrina intexitur.l Autore Gulielmo Oughtredo Anglo. I Londini, IExcudebat Thomas Harper, sumptibus Thomae I Whitakeri, apud quem venales sunt in I Ccemiterio D. Pauli. 1648. Clavis of 1652 = Guilelmi Oughtred I AEtonensis, I quondam Collegii Regalis in Cantabrigia Socii, Clavis mathematicae I denvo limata, I Sive potius I fabricata. Cum aliis quibusdam ejusdem Commentationibus, quae in sefquenti pagina recensentur.lEditio tertia auctior & emendatior.f Oxoniae, I Excudebat Leon. Lichfield, Veneunt I apud Tho. Robinson. 1652.1 Clavis of 1667 = Guilelmi Oughtred I AEtonensis, I quondam Collegii Regalis I in Cantabrigia Socii, Clavis mathematicae I Denvo Limata, Sive potius I Fabricata. I Cum aliis quibusdam ejusdem I Commentationibus, quae in selquenti pagina recensentur.l Editio quarta auctior & emendatior.1 Oxoniae, I Typis Lichfieldianis, Acad. Typog. Veneunt apud Joh. Crosley, &.f Amos Curteyne. 1667.1 Clavis of 1693=Guilelmi Oughtred AEtonensis,j Quondam Collegii Regalis in I Cantabrigia Socii, I Clavis Mathematicae denuo Limata, I Sive potius I Fabricata. I Cum aliis quibusdam ejusdem I Commentationibus, quae in I sequenti pagina recensentur.l Editio Quinta auctior & emendatior.| Oxoniae,j Excudebat Leon. Lichfield. I Anno Dom. MDCXCIII. A second impression of the fifth Latin edition appeared in 1698. The changes in the title-page come after the word "emendatior." In 1698 the closing part of the title-page is as follows:... emendatior. f Ex Recognitione D. Johannis Wallis, S.T.D.I Geometriae Professoris Saviliani.l Oxoniae:f Typis Leon. Lichfield: Impensis Tho. Lsigh I ad Insigne Pavonis juxta Ecclesiam S. Dun-lstani, Lond. 1698. Clavis of 1694 = Mr. William Oughtred's I Key f of the l Mathematicks.f Newly Translated from the Best Edition. With Notes, I Rendring it Easie and Intelligble to I less Skilful Readers. I In which also, I Some Problems Left Unanswer'd by the Author are Resolv'd. Absolutely necessary I For all Gagers, Surveyors, Gunners, Military- Officers, Mariners, &c.| Recommended by Mr. E. Halley, Fellow of the Royal Society.1 London:f Printed for John Salusbury, at the I Rising-Sun in Cornhill. MDCXCIV. An impression bearing the date 1702 is identical with the publication of 1694. No attempt was made to correct even the grossest oversights in proof reading. The title-page of 1702 is the only part set in type anew. Its wording is the same as in 1694, except for the part after the word "London;" in 1702 we read:... London;f Printed for Ralph Smith at the Bible I under the Piazza of the RoyalExchange,f Cornhil. 1702. We shall use the following abbreviations for the designation of tracts which were added to one or another of the different editions of the Clavis Mathematicae: Eq. =De Aequationum affectarvm resolvtione in numeris. Eu. = Elementi decimi Euclidis declaratio. So. = De Solidis regularibus, tractatus An. = De Anatocismo, sive usura composita. Fa. =Regula falsme positionis. Ar. =Theorematum in libris Archimedis de sphaera & cylindro declaratio Ho. = Horologia scioterica in plano, Geometrice delineandi modus.

Page 173 1920] Cajori: Oughtred's Mathematical Symbols, with Historical Notes 173 List of the tracts added to the CLAVIS MATHEMATICAE of 1631 in its later editions, given in the order in which the tracts appear in each edition, the numbers in parentheses indicating the pagination: Clavis of 1647: Eq. (121-169), An. (169-172), Fa. (173, 174), Ho. (1-30). Clavis of 1648: Eq. (113-160), An. (161-164), Fa. (164, 165), Eu. (166-187), So. (188-207). Clavis of 1652: Eq. (110-151), Eu. (1-22), S ). (23-41), An. (42-44), Fa. (45, 46), Ar. (1-10), Ho. (1 — 41) Clavis of 1667: Eq. (110-151), Eu. (1-21), So. (23-41), An. (42-44), Fa. (45, 46), Ar. (1-10), Ho. (1-41). Claris of 1693: Eq. (110-150), Eu. (1-20), So. (21-39), An. (40-42), Fa. (43, 44), Ar. (1-10), Ho. (1-43). Clavis of 1694: Eq. (158-208). I have seen a pamphlet, Oxford, 1662, containing the Eu., So., and Fa. I have seen also Ar. published as a separate pamphlet, Oxford, 1663. Cir. of Prop. = The I Circle I of I Proportion, and I The Horizontall I Instrument. Both invented, and the vses of both written in Latine by that learned Mathe-!matician Mr. W. 0.! Bvt Translated into English: and set forth for I the publique benefit by William Forster, louer I and practizer of the Mathematicall Sciences.l London I Printed by Avg. Mathewes,! dwelling in the Parsonage Court, neere I St. Brides. 1632.1 + An Addition I vnto the Vse | of the Instrvment [ called the circles of I Proportion, For the Working I of Nauticall Questions.l Together with certaine necessary Consi-l derations and Advertisements touching I Navigation. All which, as also the former Rules concerning I this Instrument are to bee wrought not onely I Instrumentally, but with the penne, by Arith-lmeticke, and the Canon of I Triangles.l Hereunto is also annexed the excellent Vse of two I Rulers for Calculation.l And is to follow after the 111 Page I of the first Part.! London, IPrinted by Avgvstine Mathewes, 1633. In our tables "Ad." stands for Addition vnto the Vse of the Instrument etc. Trigono. (Lat.) = Trigonometria:l Hoc est, Modus computandi Triangulorum I Latera & Angulos, ex Canone | Mathematico traditus & demonstratus.l Collectus ex Chartis Clarissimi Domini I Willelmi Oughtred I AEtonensis. Per I Richardum Stokesium Collegii Regalis inl Cantabrigia Socium.l et I Arthurum Haughton, Generosum.l Una cum Tabulis Sinuum, Tangent: & Secant, &c.] Londini,l Typis R. & L. W. Leybourn, Tmpensis Thomae Johnson, apud quem I vaeneunt sub signo Clavis Aureae in Coemeterio I S. Pauli, 1657. In our tables, "Ca." stands for Canones sinuum tangentium, secantium et logarithmorum pro sinubus et tangentibus, which is the title for the tables in the above work. Trigono. (Eng.) =Trigonometrie, or,l The manner of calculating the Sides I and Angles of Triangles, by the | Mathematical Canon, demonstrated. By William Oughtred Etonens.l And published by Richard Stokes Fellow of Kings Colledge in I Cambridge, and Arthur Haughton Gentleman.l London, Printed by R. and W. Leybourn, for Thomas Johnson at the I Golden Key in St. Pauls Church-yard.l M.DC.LVII. Opu. Posth. =Guilelmi Oughtred I AEtonensis, Quondam collegii Regalis I in I Cantabrigia Socii.l Opuscula Mathematica hactenus I inedita.l Oxonii,I E Theatro Sheldoniano,l Anno 1677.1

Page 174 174 University of California Publications in Mathematics [VOL. 1 Opu. Posth. contains the following nine tracts: Institutiones Mechanicae (pp. 1-54). De variis corporum generibus gravitate & magnittdine compzratis (pp. 55-67). Automata (pp. 68-86). Quaestiones Diophanti Alexandrini Lib. 3. (pp. 87-129 De Triangulis planis rectangulis. (pp. 130-138). De Divisione Superficierum. (pp. 139-153). Musicae Elementa. (pp. 154-170). De Propugnaculorum Munitionibus. (pp. 171-194). Sectiones Angulares. (pp. 195-222+207-212, irregular pagination). Ought. Explic. =Oughtredus explicatus,l sive I Commentarius I in I Ejus Clavem Mathematicam.l Cui additae sunt I Planetarum Observationes I & ] Horologiorum Con;tructio.l Authore I Gilberto Clark.! Londini,l Typis Milonis Flesher; Veneunt apud Ric.l Davis, Bibliopolam Oxoniensem, 1682.11

Page 175 1920] Cajori: Oughtred's Mathematical Symbols, with Historical Notes 175 I I 0[56 0.56.[56 0, -,' 0[500 a.b 2.314 2,314 2.314 R.S I Meanings of Symbols equal to separatrix2 separatrix separatrix3 separatrix same value as.5 ratio a:b, or a-b 4 }f separating5 J [ the mantissa - characteristic arithm. proportion 6 a: b, ratio7 given ratio geomet. proportion8 contin. proportion contin. proportion geom.9 proportion ( )10 ( ) ( )11 ( )11 ( )12 )12 therefore addition13 addition additioni4 subtraction plus or minus subtraction lessl4 negative 2 multipli cati on 5 I 1631 I 1647 I 38 1 5 21 5 13 45 40 2 49 2 51 7 34 1.... 8 Eq.136 Eq.167 28 8 18 57 58 115 58 65 3 3 3 57 3 1 10 P I Clavis Mathematicae 53 1.... 3 7 158 158 22 An. 162 33 7 16 107 99 106 58 3 3 3.... 3 106 66 9 10 1648 1) I 1652 1667 30 1.... 3 12 150 150 21 32 7 16 104 92 104 95 57 57 57 57 10 56 57 1 10 15 1.... 3 7 113 150 21 25 7 16 52 56 104 95 57 3 3 3 53 3 1 10 16 1 3 7 113 150 21 25 7 16 53 92 104 63 63 An. 42 3 3 3 17 3 73 2.... 17 25 175 207 32 24 49 11 25 149 119 95 122 95 97 89 4 4 4 4 140 4 4 1693 1694 I am~ 00 m1 (6 co kr eu 20 3.... 7 4 10 19 7 96 35 99 96 21 96 37 I cc, -4 0 cd U~c4 E —4 I - 3 13 235.... 3 235 36 3 34 32.... 3 3 16 ~; 0 o.t_ O 3 63 221 3 140 87 3 142 101 101 102 151 3 112 4 130 I 29 1 5 2 27 42 27 29 114 89 75 53 98 97 116 101 81 5 5 5 97 8 6 a 04 -+-600 co c u,:j pi mo mi le 2 X 5 16 10 13 32 143 I I I I I

Page 176 176 University of California Publications in Mathematics [VOL. 1 Meanings Clavis Mathematicae m H "3 I - __ - ____ O- 6 -+3bc cZ Symbols!.g - *. Sybl 1631 1647 1648 1652 1667 1693 1694 64.. 66 Hq bq X by juxtaposition 7 11 10 11 10 37 13.... 5 87 17 in multiplication'6 7 10 10 10 10 10.... 96.... 219 59 I fraction, division 8 12 11 11 11 11 23 21 16 9 5 a)b(c b — a =c 10 14 13 14 14 13 21........ 99 50 4 -l ~ --- 3 = -4.................................... 156 Aq AA 7 11 10 10 10 10 14........ 104 17 Ac AAA 7 11 10 10 10 10 14........ 105 25 Aqq AAAA 7 11 10 10 10 10 14........ 106 41 Aqc AAAAA 7 11 10 10 10 10 55............ 67 Ace AAAAAA 7 11 10 10 10 10 55............ 41 ABq AB 17.... 11 11 11 11 11 15 I 0 — 4th..10th power 23 37 35 34 34.... 53............ 55 [4] [10] 4th.. 10th power............ 35 35 34 52 a2.. a a. a.................................... 205 24 Q, qaesitum.... 17 16 16 16 16 25 Q_ square'8 38 33 31 57 30 30 47 28 5 100 75 Q~u square....... 105 Q, u squaire I.... I........ I............................ 105 C cube 38 33 31 30 30 30 47 28.... 62 53 Cu cube.... 136 128 123 123 123 175 QQ, 4th power 45 33 61 30 30 30 47........ 210 QC 5th power.... 33 31 30 30 30 47........ 210 D diameter........ 187 Eu. 21 Eu. 21Eu.20.... E 37 L, 1 latus, radix'9.... 121 113 110 110 110 158 37........ 139 Z _ _ angle................................ 19 192.... Z- angles................................. 16........ P perimeter............................ 37............ p Z A - A q 41........................................ R radius.... 120 111 109 109 109 154 37 32 211.... 1, R remainder.... 152 134 126 128 142................ 45 R rational........ 166 Eu. 1 Eu. 1 Eu. 1.................... ( superficies curva............ Ar. I Ar. 1 Ar. 1............ V root.... 33 31 30 30 30 47........ 102.... V square root.... 53 48 47 47 47 70........ 134.... V/q square root 35 49 48 46 46 46 65 96............ V/b latus binomii.... 33 31 30 30 30 47................

Page 177 1920] Cajori: Oughtred's Mathematical Symbols, with Historical Notes 177 Meanings Clavis Mathematicae Sof l s t---------— 1631 1647 1648 1652 1667 1693 1694 S Symbols 1631 1647 1648 1652 1667 1693 1694.~- *- c' 0 H 0 0 I Vr latus residui.... 34 31 30 30 30 47 V/ sq. rt. of polyno.20.... 55 53 53 52 52 96 /qq fourth root 35 52 47 46 46 48 69 Vc cube root 35 52 49 46 46 46 69 V/qc fifth root 35 49 47 46 46 46 65 Vc sixth root 37 52 49 48 48 48 69 /ccc ninth root Vccc.c twelfth root 37 52 49 48 48 48 69 Vqu square root 49 [12] or twelfth root 37 52 50 49 49 49 69 rq, Ire V ~........................................ 73 r, ru square root......................................... 74, 96 A, E nos.,A>E 21 33 31 30 30 30 47........ 87 53 Z A+E 21 33 31 30 30 30 47 19 16 87 53 X A-E 21 33 31 30 30 30 47.... 16 87 53 z A2+E241 33 31 30 30 30 47........ 98 54 X A2-E2 41 33 31 30 30 30 47........ 99 54 Z A3+E3 44 33 31 30 30 30 47............ 94 GX A3 -E3 44 33 31 30 30 30 47............ 94 a+e........ 167 Eu.1 Eu.1 Eu. I a-e............ Eu. 1 Eu. 1 Eu. I a2+b2 3.. 167 Eu......... 5 3 a2-b2........ 167 Eu.2 Eu. 2 Eu. I: majus21....Ho. 1716 4 166 145 Eu. Eu. I minus....Ho. 17 166 Eu. 1Eu. 1Eu.... 1 I non majus 166 Eu......... 4I __ non minus........ 166 Eu. 1Eu. 1Eu. I C minus22.... Ho. 30........ minus22............ Ho. 31 Ho. 29.... major ratio 166 Eu. 1 Eu. 1 Eu. I............ 11 - minor ratio........ 166 Eu. 1 Eu. I.................6 < less than2................................... > greater than........................ 4 commensurabilia........ 166 Eu. 1 Eu. 1 Eu. I................ - incommensurabilia........ 166I Eu. 1 Eu. 1 Eu. I I

Page 178 178 University of California Publications in Mathematics [VOL. 1 Meanings Clavis Mathematicae o I 1 0 of ___ __ _ 0Symbols 1631 1647 1648 1652 1667 1693 1694 ~................. 1 1-__ __15 1 commens. potentia........ 166 Eu. 1 Eu. 1 Eu. I ~IO J incommens. potentia........ 166 Eu. 1 Eu. 1 Eu. I W T rationale....r.... 166 Eu. 1 Eu. 1 Eu. I Vf irrationale........ 166 Eu. 1 Eu. 1 Eu. I TT medium........ 166 Eu. 1 Eu. 1 Eu. I linecut extr. & mean........ 166 Eu. 1 Eu. 1 Eu. I irratio. O major.ejus portio........ 166 Eu. 1 Eu. 1 Eu. I T minor ejus portio........ 166 Eu. 1Eu. 1Eu. I sim simile........ 166 Eu. 1 Eu. 1 Eu. I.... 33 proxime majus........ 166 Eu. 1 Eu. 1 Eu. I proxime minus........ 166 Eu. 1 Eu. 1 Eu. I _ aequale vel minus........ 166 Eu. 1 Eu. I aequale vel majus........ 166 Eu. 1 Eu. 1 Eu. I 0 rectangulum 51.... 167 Eu. 2 Eu. 2 Eu. I........ 17 149 O quadratum........ 167 Eu. 2 Eu. 2 Eu. I A triangulum....r.... 167 Eu. 2 Eu. 2 Eu. I....... 1147 latus, radix.... 167 Eu. 2 Eu. 2 Eu. I Tit media proportion......... 167 Eu. 2 Eu. 2 Eu. I 'n differentia23............ Eu. 2 Eu. 2 Eu. I I| parallel.................................. 197 log logarithm.... 172 158 150 150 122 207.... 17 log Q,: log. of square.... 135 127 122 122 122 174 S sine24.... H o. 29.................... 96 5 172 t tangent.... Ho. 29............... 96 3 174 se secant........ 1......... 14 sv sinus versus 76 107 99 98 98 98 140 s ver sinus versus25.......................... 5.... sin: com sine complement.......................... Ad. 69.... A s co cosine.......................... 96 3 174 t co cotangent.............................. 96 3 se co cosecant.........................I..... 4 sin sine H....4........ Ho. 1 1. A2 d. 69 35.... 37 tan tangent.... Ad. 69 Ca. 3... sec secant........................... ' A d. 41........ sec:parallI sum of secants _Ad. 41....

Page 179 1920] Cajori: Oughtred's Mathematical Symbols, with Historical Notes 179 Meanings Clavis Mathematicae O of 6_ _ _ o ul ____ - _ ________-____ -______oc^ ____ - *e Symbolsi 47 4~1631 1647 1648 1652 1667 1693 1694 m 20^00 tang tangent.... Ho. 29.... Ho. 41 Ho. 41 Ho. 42.... 12 235 C.01 of a degree.......... 236 Cent.01 of a degree............................ 235 ~'" 9 degr., min., sec..... 21 20 21 20 21 32 66........ 36 Ho. ' " hours, min., sec............................. 67 9 180 - angle I............................. 2 v equal in no. of degr................................. 7r - =3.1416.... 72 69 66 66 66 99 -- - cancelled26 68 100 94 90 90 90 131 M mean proportion........... Ar. Ar. 1 Ar. 1Ar. 1 Ar. 1 m minus 2 3 4 3 4 9 27 3-3 -=2,- -9_27- 20 32 30 29 29 29 45 2 3 2 3 8 Gr. degree.... 20 19 Ho. 23 Ho. 23 19 29.... 235 min. minute............................ Ad. 19.............JU differentia................................. 134 (-| aequalia tempore.................................... 68 Lo logarithm.................. Ca. 2 - \separatrix 24......... -1 separatrix I................................ 244 D differentia.................. 19 237 Tri, tri triangle.... 76 191 Eu. 26 70 69........ 24 M Cent. minute of are.................... a.......... Ca 2 X multiplication28.....-....................-.. * 5 101 16 Z cru Z sum, X diff............................ 17 Z crur of sides of................................ 16 X cru rectangle29................................ 17 X crur or triangle................................ 16 A unknown 38 53 51 50 50 50 72........ 113 84 altit. frust. of pyramid 77 109 101 99 99 99 141 or cone T altit. of part cut off 77 109 101 99 99 99 142 a first term 13 85, 18 80, 17 78, 16 78, 16 78, 16 116, 26 19........ 30, 116 last term..... 85, 18 80, 17 78, 16 78, 16 78,16 116, 26............ 30, 116 T no. of terms.... 85 80 78 78 78 116............ 116 X common differ..... 85 80 78 78 78 116............ 1-16 Z sum of all terms J.... 85,18 80, 17 78,16 78, 16 78, 16 116, 26 19.... 30,116

Page 180 180 University of California Plublications in Mathematics [VOL. 1 HISTORICAL NOTES2 1. All the symbols, except "Log," which we saw in the 1660 edition of the Circles of Proportion are given in the editions of 1632 and 1633. 2. In the first half of the seventeenth century the notation for decimal fractions engaged the attention of mathematicians in England as it did elsewhere. In 1608 an English translation of Stevin's well-known tract was brought out, with some additions, in London by Robert Norton, under the title, Disme: The Art of Tenths, or, De imall Arithmetike. Stevin's notation is followed also by Henry Lyte in his Art of Tens or Decimall Arithmetique, London, 1619, and in Johnsons 123 Arithmetick, 2.ed., London, 1633, where 3576.725 is written 35761725. William Purser in his Compount Interest and Annuities, London, 1634, p. 8, uses: as the separator, as did Adrianus Metius in his Geometriae practicae pars I et II, Lvgd. 1625, p. 149, and Rich. Balam in his Algebra, London, 1653, p. 4., Jonas Moore uses the inverted comma, 3'571, in his Arithmetick in two Books, London, first part, p. 211. The decimal point first appears in John Napier's Descriptio, (Edward Wright's translation into English, London, 1616) and Napier's Rabdologia, Edinburgh, 1617. Oughtred's notation for decimals must have delayed the general adoption of the decimal point. E. Wells's Elementa arithmetica universalis, Oxford 1698, uses Oughtred's notation. 3. This mixture of the old and the new decimal notation occurs in the Key of 1694 (Notes) and in Gilbert Clark's Oughtredus explicatus only once; no reference is made to it in the table of errata of either book. In Oughtred's Opuscula mathematica hactenus inedita, the mixed notation 128,57 occurs on page 193 fourteen times. Oughtred's regular notation 128 57 hardly ever occurs in this book. We have seen similar mixed notations in the Miscellanies:or Mathematical Lucubrations, of Mr. Samuel Foster, Sometime publike Professor of Astronomie in Gresham Colledge, in London by John Twysden, London, 1659, page 13 of the "Observationes Eclipsium;" we find there 32.466, 31.008 4. The dot. used to indicate ratio is not, as claimed by some writers, used by Oughtred for division. Oughtred does not state in his book that. signifies division. We quote from an early and a late edition of the Clavis. He says in the Clavis of 1694, p. 45, and in the one of 1648, p. 30, "to continue ratios is to multiply them as if they were fractions." Fractions, as well as divisions, are indicated by a horizontal line. Nor does the statement from the Clavis of 1694, p. 20, and the edition of 1648, p. 12, "In Division, as the Divisor is to Unity, so is the Dividend to the Quotient" prove that he looked upon ratio as an indicated division. It does not do so any more than the sentence from the Clavis of 1694, and the one of 1648, p. 7, "In Multiplication, as 1 is to either of the factors, so is the other to the Product" proves that he considered ratio an indicated multiplication. Oughtred says (Clavis of 1694, p. 19, and the one of 1631, p. 8): 'If Two Numbers stand one above another with a Line drawn between them, 'tis as much as to say, that the upper 12. 5 is to be divided by the under; as - and 4 12 In further confirmation of our view we quote from Oughtred's letter to W. Robinson:3 "Division is wrought by setting the divisor under the dividend with a line between them." 1 This is not a book written by Oughtred, but merely a commentary on the Clavis. Nevertheless, it seemed desirable to refer to its notation, which helps to show the changes then in progress. 2 Note 1 refers to the Circles of Proportion. The other notes apply to the superscripts found in the column "Meanings of Symbols." 3 Rigaud, Correspondence of Scientific Men of the Seventeenth Century, Vol. I, 1841, Letter VI, p.'8.

Page 181 1920] Cajori: Oughtred's Mathematical Symbols, with Historical Notes 181 5. In Gilbert Clark's Oughtredus explicatus there is no mark whatever to separate the characteristic and mantissa. This is a step backward. 6. Oughtred's language (Clavis of 1652, p. 21) is: "Ut 7.4: 12.9 vel 7.7-3: 12.12-3. Arithmetice proportionales sunt." As later in his work he does not use arithmetical proportion in symbolic analysis, it is not easy to decide whether the symbols just quoted were intended by Oughtred as part of his algebraic symbolism or merely as punctuation marks in ordinary writing. John Newton (Institutio mathematica or, a mathematical institution, London, 1654, p. 129) says: "As 8,5:6,3. Here 8 exceeds 5, as much as 6 exceeds 3." John Wallis (Mathesis universalis, Oxonii, 1657, p. 229) says: "Et pariter 5,3; 11,9; 7,5; 19,17. sunt i-n eadem progressione arithmetica." In Pavlini Chelvccii Institutiones anaylticae, editio post tertiam Romanam prima in Germania, Viennae, 1761, p. 3, arithmetical proportion is indicated thus, 6.8 '.' 10.12. Oughtred's notation is followed in the article "caractere" of the Encyclopedie methodique (Mathematiques) Paris, Liege, 1784. R. P. Bernard Lamy (Elemens des mathematiques ou traite de la grandeur en general, 3 6d., Amsterdam, 1692, p. 155) says: "Proportion arithmetique, 5,7. 10,12. c'est a dire qu'il y a meme difference entre 5 et 7, qu'entre 10 et 12." In the Nouveaux elemens de geometric, 2e ed., a La Haye, 1690, also in the edition issued at Paris, 1667, p. II., the same symbols are used for arithmetical progression as for geometrical progression, as in 7.3::13.9 and 6.2:: 12.4. 7. In the publications referred to in the table, of the years 1648 and 1694, the use of: to signify ratio has been found to occur only once in each copy; hence we are inclined to look upon this notation in these copies as printer's errors. W. W. Beman has pointed out in L'Intermedia 're des mathematiciens, Vol. 9, 1902, p. 229, that Oughtred's Latin edition of his Trigonometria, 1657, contains in the explanations of the use of the tables, near the end, the use of: for ratio. It is improbable that the colon occurring in those tables was inserted by Oughtred himself. In the Trigonometria proper, the colon does not occur and Oughtred's regular notation for ratio and proportion A.B::C.D, is followed throughout. Moreover, in the English edition of Oughtred's trigonometry, printed in the same year, 1657, but subsequent to the Latin edition, the passage of the Latin edition containing the: is recast, the new notation for ratio is abandoned and Oughtred's regular notation introduced. The: used to designate ratio in Oughtred's Opuscula mathematica hactenus inedita, 1677, may have been introduced by the editor of the book. We are able to show that the colon: was used to designate geometric ratio some years before 1657, by at least two authors, Vincent Wing the astronomer, and a schoolmaster who hides himself behind the initials "R.B." Wing wrote several works. In 1649 he published in London his Urania Practica, which, however, exhibits no special symbolism for proportion. But his Harmonicon Coeleste, London, 1651, contains many times Oughtred's notation A.B:: C.D, and many times also the new notation A: B:: C: D, the two notations being used interchangeably. Later there appeared from his pen, in London, three books in one volume, Logistica astronomica (1656), Doctrina spherica (1655) and Doctrina theorica (1655), each of which uses the notation, A: B:: C: D. The book by the second author is entitled, An Idea of Arithmetick at first designed for the use of the Free Schoole at Thurlow in Suffolk... by R.B., Schoolmaster there, London, 1655. One finds in this text 1.6:: 1.24 and also A: a:: C: c. It is worthy of note also, that in a text entitled, Johnsons Arithmetick; In two Bookes, second edition, London, 1633, the colon: is used to designate a fraction. Thus 3 is written 3: 4. If a fraction be considered as an indicated division, then we have here the use of: for division at a period 51 years before Leibniz first employed it for that purpose. However, dissociated from the idea of a fraction, division is not designated by any symbol in Johnson's text. In dividing 8976 by 15 he writes the quotient 598 6: 15. 8. Oughtred's notation A.B::C.D, is the earliest serviceable symbolism for proportion. Before that proportions were either stated in words as was customary in rhetorical modes of

Page 182 182 University of California Publications in Mathematics [VOL. 1 exposition, or else was expressed by writing the terms of the proportion in a line with dashes or dots to separate them. This practice was inadequate for the needs of the new symbolic algebra. Hence Oughtred's notation met with ready acceptance. It was used in Wingate's Arithmetick made easie, edited by John Kersey, London, 1650, p. 378, in Seth Ward's In Ismaelis Bullialdi astr. philol. fundamenta inquisitio brevis, Oxford, 1653, p. 7; in Isaac Barrow's Euclidis data, Cantabrigiae, 1657, p. 2; in John Wallis's Operum mathematicorum pars altera Oxonii, 1656, p. 181; in Samuel Foster's Miscellanies (posthumous), London, 1659, p. 1; in Jonas Moore's Arithmelick in two Books, London, 1660, second part, p. 61; in N. Mercator's Logarithmotechnia, London, 1668, p. 29; in Thomas Brancker's Introduction to Algebra (trans. ot Rhonius), London, 1668, p. 37; in R. P. Bernard Lamy's Elemens des mathematiques, Amsterdam, 1692, p. 208. John Collins in his Mariners Plain Scale New Plain'd, London, 1659, and James Gregory in his Appendicula ad veram circuli & hyperbolae quadraturam, 1668, use:: for proportion and: for ratio. See Note 7. 9. We have seen this notation only once in this book, namely in the expression R.S. =3.2. 10. Oughtred says, Clavis of 1694, p. 47, in connection with the radical sign, "if the Power be included between two Points at both ends, it signifies the universal Root of all that Quantity so included; which is sometimes also signified by b and r, as the /b is the Binomial Root, the /r the Residual Root." This notation is in no edition strictly adhered to; the second: is often omitted when all the terms to the end of the polynomial are affected by the radical sign or by the sign for a power. In later editions still greater tendency to a departure from the original notation is evident. Sometimes one dot takes the place of the two dots at the end; sometimes the two end dots are given, but the first two are omitted; in a few instances one dot at both ends is used, or one dot at the beginning and no symbol at the end; however, these cases are very rare and are perhaps only printer's errors. We copy the following illustrations: Q: A-E: est Aq-2AE+Eq, for (A-E)2=A2-2AE+E2. From Clavis of 1631, p. 45. BAq ) - I BCq=/q: i BCqq-CMqq.=CAq, for ~ BC2d - (-BC4-CM4) =BA2 or CA2. From Clavis of 1648, p. 106. Vq: BA+CA=BC+D, for -/(BA +CA) =BC +D. From Clavis of 1631, p. 40. AB ABq CXS AB / AB2 CXS\ -+V/q R -: = A., for - + \ V -2- - =A. From Clavis of 1652, p. 95. Q.Hc+Ch: for (Hc+Ch)2. From Clavis of 1652, p. 57. Q.A-X=, for (A-X)2=. From Clavis of 1694, p. 97. B Bq B //B2 - +r.u. -- CD. =A, for _- + li - -CD =A. From Oughtredusexpticatus, 1682, p. 101. 2 4 2 2 4 Oughtred's notation to designate aggregation did not originate with him. In a slightly different form it occurs in writings from the Netherlands. In 1603, C. Dibvadii in geometriam Evclidis demonstratio numeralis, Leyden, contains many expressions of this sort V.136+V/2048, signifying V/(136+\/2048). The dot is used to indicate that the root of the binomial (not of 136 alone) is called for. This notation is used extensively in Ludolphi a Cevlen de circvlo, Leyden, 1619, and in Willebrordi Snellii De circuli dimensione, Leyden, 1621. In place of the single dot, Oughtred used the colon:, probably to avoid confusion with his notation for ratio. To avoid further possibility of uncertainty, he usually placed the colon both before and after the algebraic expression that was being aggregated. We have noticed the use of the colon to express aggregation, in the manner of Oughtred, in the Arithmetique made easie, by Edmund Wingate, 2. edition by John Kersey, London, 1650, p. 387; in John Wallis' Operum mathematicorum pars altera, Oxonii, 1656, p. 186, as well as in the various parts of Wallis' Treatise of Algebra, London, 1685, and also in Jonas Moore's Arith

Page 183 1920] Cajori: Oughtred's Mathematical Symbols, with Historical Notes 183 metick in two Books, London, 1660, second part, p. 14. The 1630 edition of Wingate's book does not contain the part on algebra, nor the symbolism in question; these were probably added by John Kersey. 11. These notations to signify aggregation occur very seldom in the texts referred to and may be simply printer's errors. 12. Mathematical parentheses occur also on pages 75, 80 and 117 of G. Clark's Oughtredus explicatus. Perhaps the earliest use of parentheses to indicate aggregation has been found by G. Enestrom in a work of Tartaglia. In Bibliotheca mathematica S. 3, Vol. 7, 1906-1907, p. 296, Enestr6m says: "Als Erganzung einer friiheren Notiz (BM. 63, 1905, S. 405) fiber Klammern als Zeichen der Zusammengehorigkeit verschiedener Ausdriicke zum Zwecke der Ausftihrung einer neuen Operation, bemerke ich, dass meines Wissens gewohnliche (runde) Klammern fur diesen Zweck zuerst von TARTAGLIA im 2. Bande (1556) seines General trattato di numeri e misure angewendet worden sind. Sehr oft kommen solche Klammern bei Wurzeln aus zusammengesetzten Ausdrucken vor (vgl. B1. 167b, 169b, 170b, 174b, 177a usw.); so z. B. druckt TARTAGLIA (B1. 167b) V/ 28- V 10 auf folgende Weise aus: 1' v. (1 28 men R 10); 1 v. bedeutet,,radiz universalis". Ausnahmsweise benu zt TARTAGLIA (B1. 168b, 169a) die erste Klammernhalfte um zu bezeichnen, dass zwei vor einem Minuszeichen stehende Monome als eieininziger Term betrachtet werden sollen; so z. B. bedeutet (Bl. 168b): men (22 men 1R 6 nicht-22 —/6 sondern-(22-w/6). Dagegen hat TARTAGLIA meines Wissens die Klammern nie als Multiplikationszeichen gebraucht." Parentheses had been used by Clavius, as appears from his Operum mathematicorum tomus cecundus, Mogvntiae, M.DC.XI., algebra, p. 39, and by Albert Girard in his Invention nouvelle en lalgebre, Amsterdam, 1629, p. 17, but did not come into general use until after the time of Leibniz and the Bernoullis. However, they were used before the time of Leibniz in some elementary books, as for instance, in La geometric et practique generate d'icelle par I. Errard de Batle-Duc, Ingenieur ordinaire de se Majeste, 3' ed., reveue par D.H.P.E.M., Paris, 1619, p. 216; and in the Novae geometriae clavis algebra, authore P. Jacobo de Billy, Paris, 1643, p. 157; as also in an Abridgement of the Precepts of Algebra, Written in French by James de Billy, London, 1659, p. 346, and in the Miscellanies:or Mathematical Lucrubations, of Mr. Samuel Foster, Sometime publike Professor of Astronomie in Gresham Colledge in London, London, 1659, p. 7. 13. In the Clavis of 1631, p. 2, it says "Signum additionis siue affirmationis, est+plus" and "Signum subductionis, siue negationis est-minus." In the edition of 1694 it says simply "The Sign of Addition is + more" and "The Sign of Subtraction is - less," thereby ignoring, in the definition, the double function played by these symbols. 14. In the "Errata," following the preface o? the 1694 edition it says, for "more or mo. r. [ead] plus or pl.", for less or le. r.[ead] minus or mi." 15. Oughtred's Clavis mathematicae of 1631 is not the first appearance of X as a symbol for multiplication. In Edward Wright's translation of John Napier's Descriptio, entitled, A Description of the Admirable Table of Logarithms, London, 1618, the letter X is given as the sign of multiplication in the part of the book called "An Appendix to the Logarithmes, shewing the practise of the calculation of Triangles, etc." The authorship of this "Appendix" is not indicated. It has been surmised by Augustus De Morgan, as Dr. J. W. L. Glaisher has informed me, that this "Appendix" was written by Oughtred. The fact that it contains this symbol for multiplication and the then unusual word "cathetus" for the perpendicular leg of a right triangle (a term occurring in Oughtred's later works) are named by Glaisher in support of this conjecture. This first printed appearance of the symbol hardly explains its real origin. The author of the "Appendix" may have written the St. Andrew's Cross X, while the printer may have substituted the letter X as the nearest available type for it. Studies as to the origin of X have been made by C. Le Paige, "Sur l'origine de certains signes d'operation," in Annales de la societe scientifique de

Page 184 184 University of California Publications in Mathematics [VOL 1 Bruxelles, seizieme annee, 1891-1892, second partie, pp. 79-82, and Gravelaar, "Over den oorsprong van ons maalteeken (X) " in Wiskundig Tijdschrift, 6e annee. In 1551 E. O. Schreckenfuss placed the cross between two factors, one above the other. The use of the letters x and X for multiplication is not uncommon during the seventeenth and beginning of the eighteenth centuries. We note the following instances: Vincent Wing, Doctrina theorica, London, 1656, p. 63; John Wallis, Arithmetica infinitorum, Oxford, 1655, pp. 115, 172; Moore's Arithmetick in two Books, by Jonas Moore, London, 1660, p. 108; the Novveaux elemens de geometrie, Paris, 1667 ["par M.D.M.G.B. "], p. 6; Lord Brounker in Philosophical Transactions, II, p. 466 (London, 1668); Exercitatio geometrica, auctore Laurentio Lorenzinio, Vincentii Viviani discipulo, Florence, 1721. The Dioptrica nova by William Molyneux of Dublin, London, 1692, uses * as the symbol of multiplication. John Wallis used the X in his Elenchus geometriae Hobbianae, Oxoniae, 1655, p. 23. 16. in as a symbol of multiplication carries with it also a collective meaning: For example, the Clavis of 1652 has on page 77, "Erit -Z+~B in 2Z-4B= Zq -Bq". 17. That is, the line AB squared. 18. These capital letters precede the expression to be raised to a power. Seldom are they used to indicate powers of monomials. From the Clavis of 1652, p. 65 we quote: "Q: A+E: +Eq=2Q: ~A+E: +2Q.1A." i.e., (A+E)2 +E2 =2(A+E)2 +2 (A)2 19. L and 1 stand for the same thing: side or root, 1 being used usually when the coefficients of the unknown quantity are given in Hindu-Arabic numerals, so that all the letters in the equation, viz., 1, q, c, qq, qc, etc. are small letters. The Clavis of 1694, p. 158, uses L in a place where the Latin editions use 1. 20. The symbol Vu does not occur in the Clavis of 1631 and is not defined in the later editions. The following throws light upon its significance. In the 1631 edition, chapt. XVI, sect. 8, p. 40, the author takes V/q BA +B = CA, gets from it V/q BA = CA - B, then squares both sides and solves for the unknown A. He passes next to a radical involving two terms, and says: "Item Vq vniuers: BA+CA: -D=BC: vel per transpositionem /q: BA+CA=BC+D;" he squares both sides and solves for A. In the later editions he writes " \/u" in place of "V/q vniuers: " The u evidently stands or "vniuers," a word meaning collective; i.e. \/u means the square root of all terms taken collectively. 21. Harriot's symbols > for greater and < for less were far superior to the corresponding symbols used by Oughtred. While Harriot's symbols are symmetric to a horizontal axis and asymmetric only to a vertical, Oughtred's symbols are asymmetric to both axes and therefore harder to remember. Indeed some confusion in their use occurred in Oughtred's own works, as is shown in the table. Isaac Barrow used ~ for majus and _ for minus in his Euclidis Data, Cantabrigiae, 1657, p. 1. Seth Ward, a pupil of Oughtred, writes in his In Ismaelis Bullialdi astronomiae philolaicae fundamenta inquisitio brevis, Oxoniae, 1653, p. 1, C" for majus and - for minus. Another pupil, John Wallis, writes in his Treatise of Algebra, London, 1685, p. 127 J for majus and __ for minus. For further notices of discrepancy in the use of these symbols, see Bibliotheca Mathematica, 123, 1911-12, p. 64. Harriot's > and < easily won out over Oughtred's notation. Wallis follows Harriot almost exclusively; so does Thomas Gibson in his Syntaxis mathematica, London, 1655, p. 20; as does also Thomas Brancker in his Introduction to Algebra (translation of Rhonius' algebra), London, 1668, p. 76. 22. This notation for "less than" in the Ho. occurs only in the explanation of "Fig. EE' In the text (Chapt. IX) the regular notation explained in Eu. is used. 23. The symbol - so closely resembles the symbol co which was used by John Wallis in his Operum mathematicorum pars prima, Oxford, 1657, pp. 208, 247, 334, 335, that the two symbols were probably intended to be one and the same. It is difficult to assign a good reason why Wallis

Page 185 1920] Cajori: Oughtred's Mathematical Symbols, with Historical Notes 185 who greatly admired Oughtred and was editor of the later Latin editions of his Clavis mathematicae should purposely reject Oughtred's % and intentionally introduce c\ as a substitute symbol. In fact, the symbol c\ occurs twice, in the place of <, in the Clavis of 1698 (edited by Wallis), page 18 of the Elementi decimi Euclidis declaratio, while in the list o? symbols collected at the beginning of this Declaratio Oughtred's usual co is given. 24. Von Braunmiihl, in his Geschichte der Trigonometrie, 2. Teil, Leipzig, 1903, pp. 42, 91, refers to Oughtred's Trigonometria of 1657 as containing the earliest use of abbreviations of trigonometric functions and points out that half a century later the army of writers on trigonometry had hardly yet reached the standard set by Oughtred. This statement must be supplemented in several respects. Our table shows that Oughtred used such abbreviations a quarter of a century before the publication of his Trigonometria. Moreover, as early as 1624, the contractions sin and tan appear on the drawing representing Gunter's scale, but Gunter did not use them in his book, the Description and Vse of the Sector, the Crosse-staffe and other Instruments, London, 1624, except in the drawings of his scale, as in "the second book," p. 31. A competitor for the honour of first using abbreviations of the trigonometric functions is Richard Norwood, who, in his Trigonometrie, London, 1631, First Book, p. 20, and also in the second edition of it, 1651, p. 18, declares that " in these examples s stands for sine: t for tangent: sc for sine complement: tc for tangent complement: sec: for secant." Another trigonometry that was published before Oughtred's and uses abbreviations is the Idea Trigonometriae demonstratae, Oxoniae, 1654, by Seth Ward, Savilian Professor of Astronomy at Oxford and at one time a pupil of Oughtred. On page 1 Ward writes: s. Sinus Complementum S' Sinus Complementi Z Summa t. Tangens X Differentia (T Tangens Complementi Cr Crura Anthony Wood, in his Athenae Oxonienses (Ed. Bliss) Vol. IV. London, 1820, p. 249, says in connection with Ward's trigonometry: "The method of which, mention'd in the preface to this book, Mr. Oughtred challenged for his." Vincent Wing, in his Logistica as'ronomica, London, 1656, p. 8, uses s and t for "sine" and "tangent," and in his Doctrina spherica, London, 1655, p. 28, he uses cs for "cosine." John Ca well writes in his Trigonometry at the end of John Wallis's Algebra, London, 1685, s for sine E for cosine, f for secant T for tangent, and cT for cotangent. For example, p. 3, Y,A' B:: j,B'f, A/stands for cos A: cos B=sec B: sec A. Mention should be made of abbreviations used even earlier than any of the preceding, given in "An Appendiz to the Logarithmes, shewing the practice of the Calculation of Triangles, etc." in Edward Wright's edition of Napier's A Description of the Admirable Table of Logarithmes, London, 1618, where we read on p. (4): "For the Logarithme of an arch or an angle, I set before it (s) for the antilogarithme or complement thereof (s*) and for the Differential (t)." In further explanation of this rather unsatisfactory passage, the author (Oughtred?) says, "As for example: s B+BC =CA. that is, the Logarithme of an angle B. at the Base of a plain right-angled triangle, increased by the addition of the Logarithm of BC. the hypothenuse thereof, is equall to the Logarithme of CA the cathetus." Here "logarithme of an angle B" evidently means "log. sin B," just as with Napier, "Logarithms of the arcs" signify really "Logarithms of the sines of the angles." In Napier's table, the numbers in the column marked "Differentiae" signify og. sine minus log cosine, that is the logarithms of the tangents; this explains the contraction (t) in the "Appendix," the notation (s*)' for cosine resembles somewhat Seth Ward's notation. In view of the fact that Ward was an early pupil of Oughtred, we have here a new argument in favor of the hypothesis mentioned in Note 15 that Oughtred is the author of the "Appendix" mentioned above. Observe also that in the Trigonometria p. 2 Oughtred indicates 1800- angle, by (').

Page 186 186 University of California Publications in Mathematics [VOL. 1' 25. This reference is to the English edition, the Trigonometrie of 1657. In the Latin edition there is printed on page 5, by mistake, s instead of s versus. The table of errata makes reference to this misprint. 26. The horizontal line was printed beneath the expression that was being crossed out. Thus, on page 68 of the Clavis of 1631 there is: B Gqq-B GqX2 BKXBD+BKqXBDq=BGqXBDq+ BGqXBKq-BGqX2BKXBD+BGqX4 CAq. 27. This notation, says Oughtred, was used by ancient writers on music, who "are wont to 3 4 connect the terms of ratios, either to be continued" as in -X=2, "or diminished" as in 3.4 9 2 3 2 3 8 28. See Note 15. 29. Cru and crur are abbreviations for crurum, side of a rectangle or right triangle. Hence Z cru means the sum of the sides, X cru, the difference of the sides. January 18, 1915.

Page 186 UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 9, pp. 187-209 February 17, 1920 ON THE HISTORY OF GUNTER'S SCALE AND THE SLIDE RULE DURING THE SEVENTEENTH CENTURY BY FLORIAN CAJORI TABLE OF CONTENTS PAGE I. Introduction............................................................................................................................. 187 II. Innovations in G unter's Scale............................................................................................... 188 Changes introduced by Edm und W ingate........................................................................ 188 Changes introduced by M ilbourn..................................................................................... 189 Changes introduced by Thomas Brown and John Brown............................................ 190 Changes introduced by W illiam Leybourn...................................................................... 192 III. R ichard D elam ain s "G ram m elogia"................................................................................... 192 D ifferent editions or im pressions........................................................................................ 192 D escription of D elam ain's instrum ent of 1630................................................................ 195 Delamain's later designs, and directions for using his instruments.............................. 196 IV. Controversy between Oughtred and Delamain on the invention of the circular slid e ru le.......................................................................................................................... 1 9 9 V. Independenceand priority of invention....................................................................... 203 V I. O ughtred's "G auging Line," 1633.......................................................................................... 206 VII. Other seventeenth century slide rules................................................................................... 207 I. INTRODUCTION In my history of the slide rule, and my article on its invention2it is shewn that William Oughtred and not Edmund Wingate is the inventor, that Oughtred's circular rule was described in print in 1632, his rectilinear rule in 1633. Richard Delamain is referred to as having tried to appropriate the invention to himself3 and as having written a scurrilous pamphlet against Oughtred. All our information 1 F. Cajori, History of the Logarithmic Slide Rule and Allied Instruments, New York, 1909, pp. 7-14, also Addenda i-vi. 2 F. Cajori, "On the Invention of the Slide Rule," in Colorado College Publication, Engineering Series Vol. 1, 1910. An abstract of this is given in Nature (London), Vol. 82, 1909, p. 267. 3 F. Cajori, History etc., p. 14.

Page 188 188 University of California Publications in Mathematics [VOL. 1 about Delamain was taken from De Morgan,4 who, however, gives no evidence of having read any of Delamain's writings on the slide rule. Through Dr. Arthur Hutchinson of Pembroke College, Cambridge, I learned that Delamain's writings on the slide rule were available. In this article will be given: First, some details of the changes introduced during the seventeenth century in the design of Gunter's scale by Edmund Wingate, Milbourn, Thomas Brown, John Brown and William Leybourn; second, an account of Delamain's book of 1630 on the slide rule which antedates Oughtred's first publication (though Oughtred's date of invention is earlier than the date of Delamain's alleged invention) and of Delamain's later designs of slide rules; third, an account of the controversy between Delamain and Oughtred; fourth, an account of a later book on the slide rule written by William Oughtred, and of other seventeenth century books on the slide rule. II. INNOVATIONS IN GUNTER'S SCALE We begin with Anthony Wood's account of Wingate's introduction of Gunter's scale into France.5 In 1624 he transported into France the rule of proportion, having a little before been invented by Edm. Gunter of Gresham Coll.and communicated it to most of the chiefest mathematicians then residing in Paris: who apprehending the great benefit that might accrue thereby, importun'd him to express the use thereof in the French tongue. Which being performed accordingly, he was advised by monsieur Alleawne the King's chief engineer to dedicate his book to monsieur the King's only brother, since duke of Orleans. Nevertheless the said work coming forth as an abortive (the publishing thereof being somewhat hastened, by reason an advocate of Dijon in Burgundy began to print some uses thereof, which Wingate had in a friendly way communicated to him) especially in regard Gunter himself had learnedly explained its use in a far larger volume.6 Gunter's scale, which Wingate calls the "rule of proportion," contained, as described in the French edition of 1624, four lines: (1) A single line of numbers; (2) a line of tangents; (3) a line of sines; (4) a line, one foot in length, divided into 12 inches and tenths of inches, also a line, one foot in length, divided into tenths and hundredths. 4Art. "Slide Rule" in the Penny Cyclopaedia and in the English Cyclopaedia [Arts and Sciences ]. 5 Anthony Wood, Athenae oxonienses (Ed. P. Bliss), London, Vol. III, 1817, p. 423. 6 The full title of the book which Wingate published on this subject in Paris is as follows: L'Vsagelde lalReigle de!Proportionlen l'Arithmetique &lGeometrie.lPar Edmond Vvingate,j Gentil-homme Anglois.] E ev stX6oAea Tras, ganrj oXvuxa T7s. In tenui, sed no tenuis v fufve, laborne.l A Paris, Chez Melchior Mondiere,[demeurant en l'Isle du Palais,'a la[rue de Harlay aux deux Viperes.lM. DC. XXIV.]Auec Priuilege du Roy.f Back of the title page is the announcement: Notez que la Reigle de Proportion en toutes fayons se vend a Paris chez Melchior Tauernier, Graueur & Imprimeur du Roy pour les Tailles douces, demeurant en 'Isle du Palais sur le Quay qui regarde la Megisserie a l'Espic d'or.

Page 189 1920] Cajori: History of Gunter's Scale and Slide Rule 189 The English editions of this book which appeared in 162' and 1628 are devoid of interest. The editions of 1645 and 1658 contain an important innovation.7 In the preface the reasons why this instrument has not been used more are stated to be: (1) the difficulty of drawing the lines with exactness, (2) the trouble of working thereupon by reason (sometimes) of too large an extent of the compasses, (3) the fact that the instrument is not readily portable. The drawing of Wingate's arrangement of the scale in the editions of 1645 and 1658 is about 66 cm. (26.5 in.) long. It contains five parallel lines, about 66 cm. long, each having the divisions of one line marked on one side and of another line on the other side. Thus each line carries two graduations: (1) A single logarithmic line of numbers; (2) a logarithmic line of numbers thrice repeated; (3) the first scale repeated, but beginning with the graduations which are near the middle of the first scale, so that its graduation reads 4, 5, 6, 7, 8, 9, 1, 2, 3; (4) a logarithmic line of numbers twice repeated; (5) a logarithmic line of tangents; (6) a logarithmic line of sines; (7) the rule divided into 1000 equal parts; (8) the scale of latitudes; (9) a line of inches and tenths of inches; (10) a scale consisting of three kinds, viz., a gauge line, a line of chords, and a foot measure, divided into 1000 equal parts. Important are the first and second scales, by which cube root extraction was possible "by inspection only, without the aid of pen or compass;" similarly the third and fourth scales, for square roots. This innovation is due to Wingate. The 1645 edition announces that the instrument was made in brass by Elias Allen, and in wood by John Thompson and Anthony Thompson in Hosier Lane. William Leybourn, in his The Line of Proportion or Numbers, Commonly called Gunter's Line, Made Easie, London, 1673, says in his preface "To the Reader:" The Line of Proportion or Numbers, commonly called (by Artificers) Gunter's Line, hath been discoursed of by several persons, and variously applied to divers uses; for when Mr. Gunter had brought it from the Tables to a Line, and written some Uses thereof, Mr. Wingate added divers Lines of several lengths, thereby to extract the Square or Cube Roots, without doubling or trebling the distance of the Compasses: After him Mr. Milbourn, a Yorkshire Gentleman, disposed it in a Serpentine or Spiral Line, thereby enlarging the divisions of the Line. On pages 127 and 128 Leybourn adds: Again, One T. Browne, a Maker of Mathematical Instruments, made it in a Serpentine or Spiral Line, composed of divers Concentrick Circles, thereby to enlarg the divisions, which was the contrivance of one Mr. Milburn a Yorkshire Gentleman, who writ thereof, and communicated his Uses to the aforesaid Brown, who (since his death) attributed it to himself: But whoever was the contriver of it, it is not without inconvenience; for it can in no wise be made portable; and besides (instead of compasses) an opening Joynt with thirds [threads] must be placed to move upon the Centre of the Instrument, without which no proportion can be wrought. 7 The title-page of the edition of 1658 is as follows: The Use of the Rule of Proportion in Arithmetick & Geometrie. First published at Paris in the-French tongue, and dedicated to Monsieur, the then king's onely Brother (now Duke of Orleance). By Edm. Wingate, an English Gent. And now translated into English by the Author. Whereinto is now also inserted the Construction of the same Rule, & a farther use thereof... 2nd edition inlarged and amended. London, 1658.

Page 190 190 University of California Publications in Mathematics [VOL. 1 This Mr. Milburn is probably the person named in the diary of the antiquarian, Elias Ashmole, on August 13 [1646?]; "I bought of Mr. Milbourn all his Books and Mathematical Instruments."8 Charles Hutton9 says that Milburne of Yorkshire designed the spiral form about 1650. This date is doubtless wrong, for Thomas Browne who, according to Leybourn, got the spiral form of line from Milbourn, is repeatedly mentioned by William Oughtred in his Epistle0t printed some time in 1632 or 1633. Oughtred does not mention Milbourn, and says (page 4) that the spiral form "was first hit upon by one Thomas Browne a Joyner,... the serpentine revolution being but two true semicircles described on severall centers." ' Thomas Brown did not publish any description of his instrument, but his son, John Brown, published in 1661 a small book,'2 in which he says (preface) that he had done "as Mr. Oughtred with Gunter's Rule, to a sliding and circular form; and as my father Thomas Brown into a Serpentine form; or as Mr. Windgate in his Rule of Proportion." He says also that "this brief touch of the Serpentineline I made bold to assert, to see if I could draw out a performance of that promise, that hath been so long unperformed by the promisers thereof." Accordingly in Chapter XX he gives a description of the serpentine line, "contrived in five (or rather 15) turn." Whether this description, printed in 1661, exactly fits the instrument as it was developed in 1632, we have no means of knowing. John Brown says: 1. First next the center is two circles divided one into 60, the other into 100 parts, for the reducing of minutes to 100 parts, and the contrary. 2. You have in seven turnes two inpricks, and five in divisions, the first Radius of the sines (or Tangents being neer the matter, alike to the first three degrees,) ending at 5 degrees and 44 minutes. 3. Thirdly, you have in 5 turns the lines of numbers, sines, Tangents, in three margents in divisions, and the line of versed sines in pricks, under the line of Tangents, according to Mr. Gunter's cross-staff: the sines and Tangents beginning at 5 degrees, and 44 minutes where 8 Memories of the Life of that Learned Antiquary, Elias Ashmole, Esq.; Drawn up by himself by way of Diary. With Appendix of original Letters. Publish'd by Charles Burman, Esq., London, 1717, p. 23. 9 Mathematical Tables, 1811, p. 36, and art. "Gunter's Line" in his Phil. and Math. Dictionary, London, 1815. 10To the English Gentrie, and all others studious of the Mathematicks, which shall bee readers hereof. The just Apologie of Wil: Ovghtred, against the slaunderous insimulations of Richard Delamain, in a Pamphlet called Grammelogia, or the Mathematicall Ring, or Mirifica logarithmorum projectio circularis. We shall refer to this document as Epistle. It was published without date in 32 unnumbered pages of fine print, and was bound in with Oughtred's Circles of Proportion, in the editions of 1633 and 1639. In the 1633 edition it is inserted at the end of the volume just after the Addition vnto the Vse of the Instrument etc., and in that of 1639 immediately after the preface. It was omitted from the Oxford edition of 1660. The Epistle was also published separately. There is a separate copy in the British Museum, London. Aubrey, in his Brief Lives, edited by A. Clark, Vol. II, Oxford, 1898, p. 113, says quaintly, "He writt a stitch't pamphlet about 163(?4) against... Delamaine." 11 Thomas Browne is mentioned by Stone in his Mathematical Instruments, London 1723, p. 16. See also Cajori, History of the Slide Rule, New York, 1909, p. 15. 12 The Description and Use of a Joynt-Rule:... also the use of Mr. White's Rule for measuring of Board and Timber, round and square; With the manner of Vsing the Serpentine-line of Numbers, Sines, Tangents, and Versed Sines. By J. Brown, Philom., London, 1661.

Page 191 1920] Cajori: History of Gunter's Scale and Slide Rule 191 the other ended, and proceeding to 90 in the sines, and 45 in the Tangents. And the line of numbers beginning at 10, and proceeding to 100, being one entire Radius, and graduated into as many divisions as the largeness of the instrument will admit, being 10 to 10 50 into 50 parts, and from 50 to 100 into 20 parts in one unit of increase, but the Tangents are divided into single minutes from the beginning to the end, both in the first, second and third Radiusses, and the sines into minutes; also from 30 minutes to 40 degrees, and from 40 to 60, into every two minutes, and from 60 to 80 in every 5th minute, and from 80 to 85 every 10th, and the rest as many as can be well discovered. The versed sines are set after the manner of Mr. Gunter's Cross-staff, and divided into every 10th minutes beginning at 0, and proceeding to 156 going backwards under the line of Tangents. 4. Fourthly, beyond the Tangent of 45 in one single line, for one Turn is the secants to 51 degrees, being nothing else but the sines reitterated beyond 90. 5. Fifthly, you have the line of Tangents beyond 45, in 5 turnes to 85 degrees, whereby all trouble of backward working is avoided. 6. Sixthly, you have in one circle the 180 degrees of a Semicircle, and also a line of natural sines, for finding of differences in sines, for finding hour and Azimuth. 7. Seventhly, next the verge or outermost edge is a line of equal parts to get the Logarithm of any number, or the Logarithm sine and Tangent of any ark or angle to four figures besides the carracteristick. 8. Eightly and lastly, in the space place between the ending of the middle five turnes, and one half of the circle are three prickt lines fitted for reduction. The uppermost being for shillings, pence and farthings. The next for pounds, and ounces, and quarters of small Averdupoies weight. The last for pounds, shillings and pence, and to be used thus: If you would reduce 16s. 3d. 2q. to a decimal fraction, lay the hair or edge of one of the legs of the index on 16. 3~ in the line of 1. s. d. and the hair shall cut on the equal parts 81 16; and the contrary, if you have a decimal fraction, and would reduce it to a proper fraction, the like may you do for shillings, and pence, and pounds, and ounces. The uses of the lines follow. As to the use of these lines, I shall in this place say but little, and that for two reasons. First, because this instrument is so contrived, that the use is sooner learned then any other, I speak as to the manner, and way of using it, because by means of first second and third radiusses, in sines and Tangents, the work is always right on, one way or other, according to the Canon whatsoever it be, in any book that treats of the Logarithms, as Gunter, Wells, Oughtred, Norwood, or others, as in Oughtred from page 64 to 107. Secondly, and more especially, because the more accurate, and large handling thereof is more then promised, if not already performed by more abler pens, and a large manuscript thereof by my Sires meanes, provided many years ago, though to this day not extant in print; so for his sake I claiming my interest therein, make bold to present you with these few lines, in order to the use of them: And first note, 1. Which soever of the two legs is set to the first term in the question, that I call the first leg always, and the other being set to the second term, I call the second leg... The exact nature of the contrivance with the "two legs" is not described, but it was probably a flat pair of compasses, attached to the metallic surface on which the serpentine line was drawn. In that case the instrument was a slide rule, rather than a form of Gunter's line. In his publication of 1661, as also in later

Page 192 192 University of Catifornia Publioations in Mathemnatics [VOL. 1 publications,13 John Brown devoted more space to Gunter's scales, requiring the use of a separate pair of compasses, than to slide rules. The same remark applies to William Leybourn who, after speaking of Seth Partridge's slide rule, returns to forms of Gunter's scale, saying:14 There is yet another way of disposing of this Line of Proportion, by having one Line of the full length of the Ruler, and another Line of the same Radius broken in two parts between 3 and 4; so that in working your Compasses never go off of the Line: This is one of the best contrivances, but here Compasses must be used. These are all the Contrivances that I have hitherto seen of these Lines: That which I here speak of, and will shew how to use, is only two Lines of one and the same Radius, being set upon a plain Ruler of any length (the larger the better) having the beginning of one Line, at the end of the other, the divisions of each Line being set so close together, that if you find any number upon one of the Lines, you may easily see what number stands against it on the other Line. This is all the Variation.... Example 1. If a Board be 1 Foot 64 parts broad, how much in length of that Board will make a Foot Square? Look upon one of your Lines (it matters not which) for 1 Foot 64 parts, and right against it on the other Line you shall find 61; and so many parts of a Foot will make a Foot square of that Board. This contrivance solves the equation 1.64x=1, yielding centesimal parts of a foot. James Atkinson15 speaks of "Gunter's scale" as "usually of Boxwood... commonly 2 ft. long, 1~ inch broad" and "of two kinds: long Gunter or single Gunter, and the sliding Gunter. It appears that during the seventeenth century (and long after) the Gunter's scale was a rival of the slide rule. III. RICHARD DELAMAIN'S GRAMMELOGIA We begin with a brief statement of the relations between Oughtred and Delamain. At one time Delamain, a teacher of mathematics in London, was assisted by Oughtred in his mathematical studies. In 1630 Delamain published the Grammelogia, a pamphlet describing a circular slide rule and its use. In 1631 he published another tract, on the Horizontall Quadrant.16 In 1632 appeared Oughtred's Circles of Proportion"7 translated into English from Oughtred's Latin manuscript by another pupil, William Forster, in the preface of which Forster makes 13 A Collection of Centers and Useful Proportions on the Line of Numbers, by John Brown, 1662(?), 16 pages; Description and Use of the Triangular Quadrant, by John Brown, London, 1671; Wingate's Rule of Proportion in Arithmetick and Geometry: or Gunter's Line. Newly rectified by Mr. Brown and Mr. Atkinson, Teachers of the Mathematicks, London, 1683; The Description and Use of the Carpenter's-Rule: Together with the Use of the Line of Numbers commonly call'd Gunter'sLine, by John Brown, London, 1704. 14 William Leybourn, op. cit., pp. 129, 130, 132, 133. 15 James Atkinson's edition of Andrew Wakely's The Mariners Compass Rectified, London, 1694 [Wakely's preface dated 1664, Atkinson's preface, 1693]. Atkinson adds An Appendix containing Use of Instruments most useful in Navigation. Our quotation is from this Appendix, p. 199. 16 R. Delamain, The Making, Description, and Use of a small portable Instrument... called a Horizontall Quadrant, etc., London, 1631. 17 Oughtred's description of his circular slide rule of 1632 and his rectilinear slide rule of 1633, as well as a drawing of the circular slide rule, are reproduced in Cajori's History of the Slide Rule, Addenda, pp. ii-vi.

Page 193 't I Ih I;:. <s, 2 2 rZ-,Z. N o pz~z *.~ 0Q ^ h * > Z2, CO -+~>;;21 0z +-fS11 * sI. 1 *I~ 0Z The two title-pages of the edition of the GRAMMELOGIA in the British Museum in London which we have called 'Grammelogia IV."

Page 194 194 University of California Publications in Mathematics [VOL. 1 the charge (without naming Delamain) that "another... went about to preocupate" the new invention. This led to verbal disputes and to the publication by Delamain of several additions to the Grammelogia, describing further designs of circular slide rules and also stating his side of the bitter controversy, but without giving the name of his antagonist. Oughtred's Epistle was published as a reply. Each combatant accuses the other of stealing the invention of the circular slide rule and the horizontal quadrant. There are at least five different editions, or impressions, of the Grammelogia which we designate, for convenience, as follows: Grammelogia I, 1630. One copy in the Cambridge University Library.17 Grammelogia II, I have not seen a copy of this. Grammelogia III, One copy in the Cambridge University Library.18 Grammelogia IV, One copy in the British Museum, another in the Bodleian Library, Oxford.19 Grammelogia V, One copy in the British Museum. In Grammelogia I the first three leaves and the last leaf are without pagination. The first leaf contains the title-page; the second leaf, the dedication to the King and the preface "To the Reader;" the third leaf, the description of the Mathe17 The full title of the Grammelogia I is as follows: GrafelogiaIor, The Mathematicall Ring. Shewing (any reasonable Capacity that hathi not Arithmeticke) how to resolve and workelall ordinary operations of Arithmeticke. And those which are most difficult with greatest|facilitie: The extraction of Roots, the valuation oflLeases, &c. The measuring of Plainesland Solids.lWith the resolution of Plaine and SphericalllTriangles.lAnd that onely by an Ocular Inspection, land a Circular Motion. Naturae secreta tempus aperit.lLondon printed by John Haviland, 1630. 18 Grammelogia III is the same as Grammelogia I, except for the addition of an appendix, entitled: De la MainslAppendixlVpon hislMathematicalllRing. Attribuit nullo (praescripto tempore) vitae vsuram nobis ingeniique Deus. London, I... The next line or two of this title-page which probably contained the date of publication, were cut off by the binder in trimming the edges of this and several other pamphlets for binding into one volume. 19 Grammelogia IV has two title pages. The first is Mirifica Logarithmoru' Projectio Circularis. There follows a diagram of a circular slide rule, with the inscription within the innermost ring: Nil Finis, Motvs, Circvlvs vllvs Habet. The second title page is as follows: GrammelogialOr, the Mathematicall Ring.lExtracted from the Logarythmes, and projected Circular: Now published in thelinlargement thereof unto any magnitude fit for use: shewing any reason-lable capacity that hath not Arithmeticke how to resolve and worke, all ordinary operations of Arithmeticke:lAnd those that are most difficult with greatest facilitie, the extracti-jon of Rootes, the valuation of Leases, &c. the measuring of Plaines and Solids, |with the resolution of Plaine and Sphericall Triangles applied to theJPracticall parts of Geometrie, Horologographie, GeographielFortification, Navigation, Astronomie, &c.lAnd that onely by an ocular inspection, and a Circular motion, Invented and first published, by R. Delamain, Teacher, and Student of the Mathematicks.l Naturae secreta tempus aperit.l There is no date. There follows the diagram of a second circular slide rule, with the inscription within the innermost ring: Typus proiectionis Annuli adaucti vt in Conslusione Lybri praelo commissi, Anno 1630 promisi. There are numerous drawings in the Grammelogia, all of which, excepting the drawings of slide rules on the engraved title-pages of Grammelogia IV and V, were printed upon separate pieces of paper and then inserted by hand into the vacant spaces on the printed pages reserved for them. Some drawings are missing, so that the Bodleian Grammelogia IV differs in this respect slightly from the copy in the British Museum and from the British Museum copy of Grammelogia V.

Page 195 Cajori: History of Gunter's Scale land Slide Rule 195 matical Ring. Then follow 22 numbered pages. Counting the unnumbered pages, there are altogether 30 pages in the pamphlet. Only the first three leaves of this pamphlet are omitted in Grammelogia IV and V. In Grammelogia III the Appendix begins with a page numbered 52 and bears the heading "Conclusion;" it ends with page 68, which contains the same two poems on the mathematical ring that are given on the last page of Grammelogia I but differs slightly in the spelling of some of the words. The 51 pages which must originally have preceded page 52, we have not seen. The edition containing these we have designated Grammelogia II. The reason for the omission of these 51 pages can only be conjectured. In Oughtred's Epistle (p. 24), it is stated that Delamain had given a copy of the Grammelogia to Thomas Brown, and that two days later Delamain asked for the return of the copy, "because he had found some things to be altered therein" and "rent out all the middle part." Delamain labored "to recall all the bookes he had given forth, (which were many) before the sight of Brownes Lines." These spiral lines Oughtred claimed that Delamain had stolen from Brown. The title-page and page 52 are the only parts of the Appendix, as given in Grammelogia III, that are missing in the Grammelogia IV and V. Grammelogia IV answers fully to the description of Delamain's pamphlet contained in Oughtred's Epistle. It was brought out in 1632 or 1633, for what appears to be the latest part of it contains a reference (page 99) to the Grammelogia I (1630) as "being now more then two yeares past." Moreover, it refers to Oughtred's Circles of Proportion, 1632, and Oughtred's reply in the Epistle was bound in the Circles of Proportion having the Addition of 1633. For convenience of reference we number the two title-pages of Grammelogia IV, "page (1)" and "page (2)," as is done by Oughtred in his Epistle. Grammelogia IV contains, then, 113 pages. The page numbers which we assign will be placed in parentheses, to distinguish them from the page numbers which are printed in Grammelogia IV. The pages (44)-(65) are the same as the pages 1-22, and the pages (68)-(83) are the same as the pages 53-68. Thus only thirty-eight pages have page numbers printed on them. The pages (67) and (83) are identical in wording, except for some printer's errors; they contain verses in praise of the Ring, and have near the bottom the word "Finis." Also, pages (22) and (23) are together identical in wording with page (113), which is set up in finer type, containing an advertisement of a part of Grammelogia IV explaining the mode of graduating the circular rules. There are altogether six parts of Grammelogia IV which begin or end by an address to the reader, thus: "To the Reader," "Courteous Reader," or "To the courteous and benevolent Reader...," namely the pages (8), (22), (68), (89), (90), (108). In his Epistle (page 2), Oughtred characterizes the make up of the book in the following terms: In reading it... I met with such a patchery and confusion of disjoynted stuffe, that I was striken with a new wonder, that any man should be so simple, as to shame himselfe to the world with such a hotch-potch.

Page 196 196 University of California Publications in Mathematics [VOL. 1 Grammelogia V differs from Grammelogia IV in having only the second titlepage. The first title-page may have been torn off from the copy I have seen. A second difference is that the page with the printed numeral 22 in Grammelogia IV has after the word "Finis" the following notice: This instrument is made in Silver, or Brasse for the Pocket, or at any other bignesse, over against Saint Clements Church without Temple Barre, by Elias Allen. This notice occurs also on page 22 of Grammelogia I and III, but is omitted from page 22 of Grammelogia V. In his address to King Charles I, in his Grammelogia I, Delamain emphasizes the ease of operating with his slide rule by stating that it is "fit for use... as well on Horse backe as on Foot." Speaking "To the Reader," he states that he has "for many yeares taught the Mathematicks in this Towne," and made efforts to improve Gunter's scale "by some Motion, so that the whole body of Logarithmes might move proportionally the one to the other, as occasion required. This conceit in February last [1629] I struke upon, and so composed my Grammelogia or Mathematicall Ring; by which only with an ocular inspection, there is had at one instant all proportionalls through the said body of Numbers." He dates his preface "first of January, 1630." The fifth and sixth pages contain his "Description of the Grammelogia," the term Grammelogia being applied to the instrument, as well as to the book. His description is as follows: The parts of the Instrument are two Circles, the one moveable, and the other fixed; The moveable is that unto which is fastened a small pin to move it by; the other Circle may be conceived to be fixed; The circumference of the moveable Circle is divided into unequall parts, charactered with figures thus, 1. 2. 3. 4. 5. 6. 7. 8. 9. these figures doe represent themselves, or such numbers unto which a Cipher or Ciphers are added, and are varied as the occasion falls out in the speech of Numbers, so 1. stands for 1. or 10. or 100., &c. the 2. stands for 2. or 20. or 200. or 2000., &c. the 3. stands for 30. or 300. or 3000., &c. After elaborating this last point and explaining the decimal subdivisions on the scales of the movable circle, he says that "the numbers and divisions on the fixed Circle, are the very same that the moveable are,.." There is no drawing of the slide rule in this publication. The twenty-two numbered pages give explanations of the various uses to which the instrument can be put: "How to performe the Golden Rule" (pp. 1-3), "Further uses of the Golden Rule" (pp. 4-6), "Notions or Principles touching the disposing or ordering of the Numbers in the Golden Rule in their true places upon the Grammelogia" (pp. 7-11), "How to divide one number by another" (pp. 12, 13), "to multiply one Number by another" (pp. 14, 15), "To find Numbers in continuall proportion" (pp. 16, 17), "How to extract the Square Root," "How to extract the Cubicke Root" (pp. 18-21), "How to performe -the Golden Rule" (the rule of proportion) is explained thus: Seeke the first number in the moveable, and bring it to the second number in the fixed, so right against the third number in the moveable, is the answer in the fixed. If the Interest of 100. li. be 8. Ii. in the yeare, what is the Interest of 65. li. for the same time. Bring 100; in the moveable to 8. in the fixed, so right against 65. in the moveable is 5.2. in the fixed, and so much is the Interest of 65. li. for the yeare at 8. li. for 100. li. per annum.

Page 197 1920] Cajori: History of Gunter's Scale and Slide Rule 197 The Instrument not removed, you may at one instant right against any summe of money in the moveable, see the Interest thereof in the fixed: the reason of this is from the Definition of Logarithmes. These are the earliest known printed instructions on the use of a slide rule. It will be noticed that the description of the instrument at the opening makes no references to logarithmic lines for the trigonometric functions; only the line of numbers is given. Yet the title-page promised the "resolution of Plaine and Sphericall Triangles." Page 22 throws light upon this matter: If there be composed three Circles of equal thicknesse, A.B.C. so that the inner edge of D [should be B] and the outward egde of A bee answerably graduated with Logarithmall signes [sines], and the outward edge of B and the inner edge of A with Logarithmes; and then on the backside be graduated the Logarithmall Tangents, and againe the Logarithmall signes oppositly to the former graduations, it shall be fitted for the resolution of Plaine and Sphericall Triangles. After twelve lines of further remarks on this point he adds: Hence from the forme, I have called it a Ring, and Grammelogia by annoligie of a Lineary speech; which Ring, if it were projected in the convex unto two yards Diameter, or thereabouts, and the line Decupled, it would worke Trigonometrie unto seconds, and give proportionall numbers unto six places only by an ocular inspection, which would cbmpendiate Astronomicall calculations, and be sufficient for the Prosthaphaeresis of the Motions: But of this as God shall give life and ability to health and time. The unnumbered page following page 22 contains the patent and copyright on the instrument and book: Whereas Richard Delamain, Teacher of Mathematicks, hath presented vnto Vs an Instrument called Grammelogia, or The Mathematicall Ring, together with a Booke so intituled, expressing the use thereof, being his owne Invention; we of our Gracious and Princely favour have granted unto the said Richard Delamain and his Assignes, Privilege, Licence, and Authority, for the sole Making, Printing and Selling of the said Instrument and Booke: straightly forbidding any other to Make, Imprint, or Sell, or cause to be Made, or Imprinted, or Sold, the said Instrument or Booke within any our Dominions, during the space of ten yeares next ensuing the date hereof, upon paine of Our high displeasure. Given under our hand and Signet at our Palace of Westminster, the fourth day of January, in the sixth yeare of our Raigne. In the Appendix of Grammelogia III, on page 52 is given a description of an instrument promised near the end of Grammelogia I: That which I have formerly delivered hath been onely upon one of the Circles of my Ring, simply concerning Arithmeticall Propo tions, I will by way of Conclusion touch upon some uses of the Circles, of Logarithmall Sines, and Tangents, which are placed on the edge of both the moveable and fixed Circles of the Ring in respect of Geometricall Proportions, but first of the description of these Circles. First, upon the side that the Circle of Numbers is one, are graduated on the edge of the moveable, and also on the edge of the fixed the Logarithmall Sines, for if you bring 1. in the moveable amongst the Numbers to 1. in the fixed, you may on the other edge of the moveable and fixed see the sines noted thus 90. 90. 80. 80. 70. 70. 60. 60. &c. unto 6.6. and each degree subdivided, and then over the former divisions and figures 90. 90. 80. 80. 70. 70. &c. you have the other degrees. viz. 5. 4. 3. 2. 1. each of those divided bv small points.

Page 198 198 University of California Publicatiowns in Mathematics [VOL. 1 Secondly, (if the Ring is great) neere the outward edge of this side of the fixed against the Numbers, are the usuall divisions of a Circle, and the points of the Compasse: serving for observation in Astronomy, or Geometry, and the sights belonging to those divisions, may be placed on the moveable Circle. Thirdly, opposite to those Sines on the other side are the Logarithmall Tangents, noted a like both in the moveable and fixed thus 6.6.7.7.8.8.9.9.10.10.15.15.20.20. &c. unto 45.45. which numbers or divisions serve also for their Complements to 90. so 40 gr. stands for 50. gr. 30. gr. for 60 gr. 20. gr. for 70. gr. &c. each degree here both in the moveable and fixed is also divided into parts. As for the degrees which are under 6. viz. 5.4.3.2.1. they are noted with small figures over this divided Circle from 45.40.35.30.25. &c. and each of those degrees divided into parts by small points both in the moveable and fixed. Fourthly, on the other edge of the moveable on the same side is another graduation of Tangents, like that formerly described. And opposite unto it, in the fixed is a Graduation of Logarithmall sines in every thing answerable to the first descrition of Sines on the other side. Fifthly, on the edge of the Ring is graduated a parte of the -Equator, numbered thus 10 20. 30. unto 100. and there unto is adjoyned the degrees of the Meridian inlarged, and numbred thus 10 20.30 unto 70. each degree both of the Equator, and Meridian are subdivided into parts; these two graduated Circles serve to resolve such Questions which concerne Latitude, Longitude, Rumb, and Distance, in Nauticall operations. Sixthly, to the concave of the Ring may be added a Circle to be elevated or depressed for any Latitude, representing the zEquator, and so divided into houres and parts with an Axis, to shew both the houre, and Azimuth. and within this Circle may be hanged a Box, and Needle with a Socket for a staffe to slide into it, and this accommodated with scrue pines to fasten it to the Ring and staffe, or to take it off at pleasure. The pages bearing the printed numbers 53-68 in the Grammelogia III, IV and V make no reference to the dispute with Oughtred and may, therefore, be assumed to have been published before the appearance of Oughtred's Circles of Proportion. On page 53, "To the Reader," he says:... you may make use of the Projection of the Circles of the Ring upon a Plaine, having the feet of a paire of compasses (but so that they be flat) to move on the Center of that Plaine, and those feet to open and shut as a paire of Compasses... now if the feet bee opened to any two termes or numbers in that Projection, then may you move the first foot to the third number, and the other foot shall give the Answer;... it hath pleased some to make use of this way. But in this there is a double labour in respect to that of the Ring, the one in fitting those feet unto the numbers assigned, and the other by moving them about, in which a man can hardly accommodate the Instrument with one hand, and expresse the Proportionals in writing with the other. By the Ring you need not but bring one number to another, and right against any other number is the Answer without any such motion.... upon that [the Ring] I write, shewing some uses of those Circles amongst themselves, and conjoyned with others... in Astronomy, Horolographie, in plaine Triangles applyed to Dimensions, Navigation, Fortification, etc.... But before I come to Construction, I have thought it convenient by way introduction, to examine the truth of the graduation of those Circles... These are the words of a practical man, interested in the mechanical development of his instrument. He considers not only questions of convenience but also of accuracy. The instrument has, or may have now, also lines of sines and tangents. To test the accuracy of the circles of Numbers, "bring any number in the moveable to halfe of that number in the fixed: so any number or part in the fixed shall give his double in the moveable, and so may you trie of the thirds, fourths &c. of num

Page 199 1920] Cajori: History of Gunter's Scale and Slide Rule 199 bers, vet contra," (p. 54). On page 55 are given two small drawings, labelled, "A Type of the Ringe and Scheme of this Logarithmicall projection, the use followeth. These Instruments are made in Silver or Brasse by John Allen neare the Sauoy in the Strand." IV. CONTROVERSY BETWEEN OUGHTRED AND DELAMAIN ON THE INVENTION OF THE CIRCULAR SLIDE RULE Delamain's publication of 1630 on the 'Mathematicall Ring' does not appear at that time to have caused a rupture between him and Oughtred. When in 1631 Delamain brought out his Horizontall Quadrant, the invention of which Delamain was afterwards charged to have stolen from Oughtred, Delamain was still in close touch with Oughtred and was sending Oughtred in the Arundell House, London, the sheets as they were printed. Oughtred's reference to this in his Epistle (p. 20) written after the friendship was broken, is as follows: While he was printing his tractate of the Horizontall quadrant, although he could not but know that it was injurious to me in respect of my free gift to Master Allen, and of William Forster, whose translation of my rules was then about to come forth: yet such was my good nature, and his shamelessnesse, that every day, as any sheet was printed, hee sent, or brought the same to mee at my chamber in Arundell house to peruse which I lovingly and ingenuously did, and gave him my judgment of it. Even after Forster's publication of Oughtred's Circles of Proportion, 1632, Oughtred had a book, A canon of Sines Tangents and Secants, which he had borrowed from Delamain and was then returning (Epistle, page (5)). The attacks which Forster, in the preface to the Circles of Proportion, made upon Delamain (though not naming Delamain) started the quarrel. Except for Forster and other pupils of Oughtred who urged him on to castigate Delamain, the controversy might never have arisen. Forster expressed himself in part as follows:... being in the time of the long vacation 1630, in the Country, at the house of the Reverend, and my most worthy friend, and Teacher, Mr. William Oughtred (to whose instruction I owe both my initiation, and whole progresse in these Sciences.) I vpon occasion of speech told him of a Ruler of Numbers, Sines, & Tangents, which one had be-spoken to be made (such as it vsually called Mr. Gunter's Ruler) 6feet long, to be vsed with a payre of beame-compasses. "He answered that was a poore invention, and the performance very troublesome: But, said he, seeing you are taken with such mechanicall wayes of Instruments, I will shew you what deuises I have had by mee these many yeares." And first, hee brought to mee two Rulers of that sort, to be vsed by applying one to the other, without any compasses: and after that hee shewed mee those lines cast into a circle or Ring, with another moueable circle vpon it. I seeing the great expeditenesse of both those wayes; but especially, of the latter, wherein it farre excelleth any other Instrument which hath bin knowne; told him, I wondered that hee could so many yeares conceale such vseful inuentions, not onely from the world, but from my selfe, to whom in other parts and mysteries of Art, he had bin so liberall. He answered, "That the true way of Art is not by Instruments, but by Demonstration: and that it is a preposterous course of vulgar Teachers, to begin with Instruments, and not with the Sciences, and so in-stead of Artists, to make their Schollers only doers of tricks, and as it were Tuglers: to the despite of Art, losse of precious time, and betraying of willing and industrious wits,

Page 200 200 University of California Publications in Mathematics [VOL. 1 vnto ignorance and idlenesse. That the vse of Instruments is indeed excellent, if a man be an Artist: but contemptible, being set and opposed to Art. And lastly, that he meant to commend to me, the skill of Instruments, but first he would haue me well instructed in the Sciences. He also shewed me many notes, and Rules for the vse of those circles, and of his Horizontall Instrument, (which he had proiected about 30 yeares before) the most part written in Latine. All which I obtained of him leaue to translate into English, and make publique, for the vse, and benefit of such as were studious, and louers of these excellent Sciences. Which thing while I with mature, and diligent care (as my occasions would give me leaue) went about to doe: another to whom the Author in a louing confidence discouered this intent, using more hast then good speed, went about to preocupate; of which vntimely birth, and preuenting (if not circumuenting) forwardnesse, I say no more: but aduise the studious Reader, onely so farre to trust, as he shal be sure doth agree to truth & Art. While in this dedication reference is made to a slide rule or "ring" with a "moveable circle," the instrument actually described in the Circles of Proportion consists of fixed circles "with an index to be opened after the manner of a paire of Compasses." Delamain, as we have seen, had decided preference for the moveable circle. To Oughtred, on the other hand, one design was about as good as the other; he was more of a theorist and repeatedly expressed his contempt for mathematical instruments. In his Epistle (page (25)), he says he had not "the one halfe of my intentions upon it" (the rule in his book), nor one with a "moveable circle and a thread, but with an opening Index at the centre (if so be that bee cause enough to make it to bee not the same, but another Instrument) for my part I disclaime it: it may go seeke another Master: which for ought I know, will prove to be Elias Allen himselfe: for at his request only I altered a little my rules from the use of the moveable circle and the thread, to the two armes of an Index." All parts of Delamain's Grammelogia IV, except pages 1-22 and 53-68 considered above, were published after the Circles of Proportion, for they contain references to the ill treatment that Delamain felt or made believe that he felt, that he had received in the book published by Oughtred and Forster. Oughtred's reference to teachers whose scholars are "doers of tricks," "Iuglers," and Forster's allusion to "another to whom the Author in a loving confidence" explained the instrument and who "went about to preocupate" it, are repeatedly mentioned. Delamain says, (page (89)) that at first he did not intend to express himself in print, "but sought peace and my right by a private and friendly way." Oughtred's account of Delamain's course is that of an "ill-natured man" with a "virulent tongue," "sardonical laughter" and "malapert sawsiness." Contrasting Forster and Delamain, he says that, of the former he "had the very first moulding" and made him feel that "the way of Art" is "by demonstration." But Delamain was "already corrupted with doing upon Instruments, and quite lost from ever being made an Artist." (Epistle page (27)). Repeatedly does Oughtred assert Delamain's ignorance of mathematics. The two men were evidently of wholly different intellectual predilections. That Delamain loved instruments is quite evident, and we proceed to describe his efforts to improve the circular slide rule.

Page 201 1920] Cajori: History of Gunter's Scale and Slide Rule 201 The Grammelogia IV is dedicated to King Charles I. Delamain says:...Everything hath his beginning, and curious Arts seldome come to the height at the first; It was my promise then to enlarge the invention by a way of decuplating the Circles, which I now present unto your sacred Majestie as the quintessence and excellencie there of... His enlarged circular rules are illustrated in the Bodleian Library copy of Grammelogia IV by four diagrams, two of them being the two drawings on the two title-pages at the beginning of the Grammelogia IV, 4 inches in external diameter, and exhibiting eleven concentric circular lines carrying graduations of different sorts. In the second of these designs all circles are fixed. The other two drawings are each 10a4 inches in external diameter and exhibit 18 concentric circular lines; the folded sheet of the first of these drawings is inserted between pages (23) and (24), the second folded sheet between pages (83) and (84). All circles of this second instrument are fixed. Counting in the two small drawings in Grammelogia III, there are in all six drawings of slide rules in the Bodleian Grammelogia IV. On pages (24) to (43) Delamain explains the graduation of slide rules. He takes first a rule which has one circle of equal parts, divided into 1000 equal divisions. From a table of logarithms he gets log 2=0.301; from the number 301 in the circle of equal parts he draws a line to the center of the circle and marks the intersection with the circles of numbers by the figure 2. Thus he proceeds with log 3, log 4, and so on; also with log sin x and log tan x. For log sin x he uses two circles, the first (see page (27)) for angles from 34' 24" to 5~ 44' 22", the second circle from 5~ 44' 22" to 90~. The drawings do not show the seconds. He suggests many different designs of rules. On page (29) he says: For the single projection of the Circles of my Ring, and the dividing and graduating of them: which may bee so inserted upon the edges of Circles of mettle turned in the forme of a Ring, so that one Circle may moove betweene two fixed, by helpe of two stayes, then may there be graduated on the face of the Ring, upon the outer edge of the mooveable and inner edge of the fixed, the Circle of Numbers, then upon the inner edge of that mooveable Circle, and the outward edge of that inner fixed Circle may be inserted the Circle of Sines, and so according to the description of those that are usually made. In addition to these lines he proceeds to mention the circle giving the ordinary division into degrees and minutes, and two circles of tangents on the other side of the rule. Next Delamain explains an arrangement of all the graduation on one side of the rule by means of "a small channell in the innermost fixed Circle, in which may be placed a small single Index, which may have sufficient length to reach from the innermost edge of the Mooveable Circle, unto the outmost edge of the fixed Circle, which may be mooved to and fro at pleasure, in the channell, which Index may serve to shew the opposition of Numbers" (p. (31)). From this it is clear that the invention of the "runner" goes back to the very first writers on the slide rule. After describing a modification of the above arrangement, he adds, "many other formes might be deliverd, about this single projection" (p. (32)).

Page 202 202 University of California Publications in Mathematics [VOL. 1 Proceeding to the "enlarging" of the circles in the Ring, to, say, the "Quadruple to that which is single, that is, foure times greater," the "equall parts" are distributed over four circles instead of only one circle, but the general method of graduation is the same as before (p. (33)); there being now four circles carrying the logarithms of numbers, and so on. Next he points out "severall wayes how the Circles of the Mathematicall Ring (being inlarged) may be accommodated for practicall use:" (1) The Circles are all fixed in a plain and movable flat compasses (or better, a movable semicircle) are used for fixing any two positions; (2) There is a "double projection" of each logarithmic line "inlarged on a Plaine," one fixed, the other movable, as shown in his first figure on the title-page, a single index only being used; (3) use of "my great Cylinder which I have long proposed (in which all the Circles are of equall greatnesse,) and it may be made of any magnitude or capacity, but for a study (hee that will be at the charge) it may be of a yard diameter and of such an indifferent length that it may containe 100 or more Circles fixed parallel one to the other on the Cylinder, having a space betweene each of them, so that there may bee as many mooveable Circles, as there are fixed ones, and these of the mooveable linked, or fastened together, so that they may all moove together by the fixed ones in these spaces, whose edges both of the fixed, and mooveable being graduated by helpe of a single Index will shew the proportionalls by opposition in this double Projection, or by a double Index in a single Projection" (p. (36)). Next follows the detailed description of his Ring "on a Plaine, according to the diagramme that was given the King (for a view of that projection) and afterwards the Ring it selve." The diagram is the large one which we mentioned as inserted between pages (23) and (24). The instrument has two circles, one moveable, upon each of which are described 13 distinct circular graduations. The lines on the fixed circle are: "The Circle of degrees and calendar," E. "Circle of equall parts, and part of the Equator, and Meridian," TT. "The Circle of Tangents," S. "'The Circle of Sines," D. "The Circle of Decimals," N. "The Circle of Numbers." The lines on the movable circle are: N. "The Circle of Numbers," E. "The Circle of equated figures, and bodies," S. "The Circle of Sines," TT. "The Circle of Tangents," Y. "The Circle of time, yeares, and monethes." On pages (84)-(88) Delamain explains an enlargement of his Ring for computations involving the sines of angles near to 90~. On page (86) he says: I have continued the Sines of the Projection unto two severall revolutions, the one beginning at 77.gr. 45.m. 6.s. and ends at 90.gr. (being the last revolution of the decuplation of the former, or the hundred part of that Projection) the other beginning at 86.gr. 6.m. 48.s. and ends at 90.gr. (being the last of a ternary of decuplated revolutions, or the thousand part of that Projection) and may bee thus used. He explains the manner of using these extra graduations. Thus he claims to have attained degrees of accuracy which enabled him to do what "some one" had declared "could not bee done." It is hardly necessary to point out that Delamain's Grammelogia IV suggests designs of slide rules which inventors two hundred or more years later were endeavouring to produce. Which of Delamain's designs

Page 203 1920] Cajori: History of Gunter's Scale and Slide Rule 203 of rules were actually made and used, he does not state explicitly. He refers to a rule 18 inches in diameter as if it had been actually constructed (pages (86), (88)). Oughtred showed no appreciation of such study in designing and ridiculed Delamain's efforts, in his Epistle. Additional elucidations of his designs of rules, along with explanations of the relations of his work to that of Gunter and Napier, and sallies directed against Oughtred and Forster, are contained on pages (8)-(21) of his Grammelogia IV. V. INDEPENDENCE AND PRIORITY OF INVENTION The question of independence and priority of invention is discussed by Delamain more specifically on pages (89)-(113); Oughtred devotes his entire Epistle to it. It is difficult to determine definitely which publication is the later, Delamain's Grammelogia IV or Oughtred's Epistle. Each seems to quote from the other. Probably the explanation is that the two publications contain arguments which were previously passed from one antagonist to the other by word of mouth or by private letter. Oughtred refers in his Epistle (p. (12)) to a letter from Delamain. We believe that the Epistle came after Delamain's Grammelogia IV. Delamain claims for himself the invention of the circular slide rule. He says in his Grammelogia IV. (p. (99)), "when I had a sight of it, which was in February, 1629 (as I specified in my Epistle) I could not conceale it longer, envying my selfe, that others did not tast of that which I found to carry with it so delightfull and pleasant a goate [taste]..." Delamain asserts (without proof) that Oughtred "never saw it as he now challengeth it to be his invention, untill it was so fitted to his hand, and that he made all his practise on it after the publishing of my Booke upon my Ring, and not before; so it was easie for him or some other to write some uses of it in Latin after Christmas, 1630 and not the Sommer before, as is falsely alledged by some one..." (p. (91)). Delamain's accusation of theft on the part of Oughtred cannot be seriously considered. Oughtred's reputation as a mathematician and his standing in his community go against such a supposition. Moreover, William Forster is a witness for Oughtred. The fact that Oughtred had the mastery of the rectilinear slide rule as well, while Delamain in 1630 speaks only of the circular rule, weighs in Oughtred's favour. Oughtred says he invented the slide rule "above twelve yeares agoe," that is, about 1621, and "I with mine owne hand made me two such Circles, which I have used ever since, as my occasions required," (Epistle p. (22)). On the same page, he describes his mode of discovery thus: I found that it required many times too great a paire of Compasses [in using Gunter's line], which would bee hard to open, apt to slip, and troublesome for use. I therefore first devised to have another Ruler with the former: and so by setting and applying one to the other, I did not onely take away the use of Compasses, but also make the worke much more easy and expedite: when I should not at all need the motion of my hand, but onely the glancing at my sight: and with one position of the Rulers, and view of mine eye, see not one onely, but the manifold proportions incident unto the question intended. But yet this facility also

Page 204 204 University of Californiac Publications in Mathematics [VOL. 1 wanted not some difficulty especially in the line of tangents, when one arch was in the former mediety of the quadrant, and the other in the latter: for in this case it was needful that either one Ruler must bee as long againe as the other; or else that I must use an inversion of the Ruler, and regression. By this consideration I first of all saw that if those lines upon both Rulers were inflected into two circles, that of the tangents being in both doubled, and that those two Circles should move one upon another; they with a small thread in the center to direct the sight, would bee sufficient with incredible and wonderfull facility to worke all questions of Trigonometry... Oughtred said that he had no desire to publish his invention, but in the vacation of 1630 finally promised William Forster to let him bring out a translation. Oughtred claims that Delamain got the invention from him at Alhallontide [November 1], 1630, when they met in London. The accounts of that meeting we proceed to give in double column. DELAMAIN'S STATEMENT Grammelogia IV, page (98) "... about Alhalontide 1630. (as our Authors reporteth) was the time he was circumvented, and then his intent in a loving manner (as before) he opened unto me, which particularly I will dismantle in the very naked truth: for, wee being walking together some few weekes before Christmas, upon Fishstreet hill, we discoursed upon sundry things Mathematicall, both Theoreticall and Practicall, and of the excellent inventions and helpes that in these dayes were produced, amongst which I was not a little taken with that of the Logarythmes, commending greatly the ingenuitie of Mr. Gunter in the Projection, and inventing of his Ruler, in the lines of proportion, extracted from these Logarythmes for ordinary Practicall uses; He replyed unto me (in these very words) What will yov say to an Invention that I have, which in a lesse extent of the Compasses shall worke truer then that of Mr. Gunters Ruler, I asked him then of what forme it was, he answered with some pause (which no doubt argued his suspition of mee that I might conceive it) that it was Archingwise, but now hee sayes that hee told mee then, it was Circular (but were I put to my oath to avoid the guilt of Conscience I would conclude in the former.) At which immediately I answered, I had the like my selfe, and so we discoursed not a word more touching that subject... Then after my coming home I sent him a sight of my Projection drawne in Pastboard: Now admit I had not the Invention of my Ring before I discoursed... it was not so facil for mee... to raise and compose so complete, and absolute an Instrument from so small a principle, or glimpse of light..." OUGHTRED'S STATEMENT Epistle, page (23) "Shortly after my gift to Elias Allen, I chanced to meet with Richard Delamain in the street (it was at Alhallontide) and as we walked together I told him what an Instrument I had given to Master Allen, both of the Logarithm:s projected into circles, which being lesse then one foot diameter would performe as much as one of Master Gunters Rulers of sixe feet long: and also of the Prostaphaereses of the Plannets and second motions. Such an invention have I said he: for now his intentions (that is his ambition) beganne to worke:... But he saith, Then after my comming home I sent him a sight of my projection drawne in pastboard. See how notoriously he jugleth without an Instrument. Then after: how long after? a sight of my projection: of how much? More then seven weekes after on December 23, he sent to mee the line of numbers onely set upon a circle:... and so much onely he presented to his Majesty: but as for Sine or tangent of his, there was not the least shew of any. Neither could he give to Master Allen any direction for the composure of the circles of his Ring, or for the division of them: as upon his oath Master Allen will testify how hee misled him, and made him labour in vain above three weeks together, until Master Allen himselfe found out his ignorance and mistaking, which is more cleare then is possible with any impudence to be outfaced."

Page 205 Cajori: History of Gunter's Scale and Slide Rule 205 Oughtred makes a further statement (Epistle, p. (24)) as follows: Delamain hearing that Brown with his Serpentine had another line by which he could worke to minutes in the 90 degree of sines... gave the [his booke to Browne: who in thankfulnesse could not but gratify Delamain with his Lines also: and teach him the use of them, but especially of the great Line: with this caution on both sides, that one should not meddle with the others invention. Two dayes after Delamain... because he had found some things to be altered therin,... asked for the booke... but as soone as he had got it in his hands he rent out all the middle part with the two Schemes & put them up in his pocket & went his way... and... laboureth to recall all the bookes he had given forth... And shortly after this he got a new Printer (who was ignorant of his former Schemes) to print him new: giving him an especiall charge of the outermost line newly graven in the Plate, which indeed is Brownes very line: and then altering his book... This and other statements made by Oughtred seem damaging to Delamain's reputation. But it is quite possible that Oughtred's guesses as to Delamain's motives are wrong. Moreover, some of Oughtred's statements are not first hand knowledge with him, but mere hearsay. One may accept his first hand facts and still clear Delamain of wrong doing. There is always danger that rival claimants of an invention or discovery will proceed on the assumption that no one else could possibly have come independently upon the same devices that they themselves did; the history of science proves the opposite. Seldom is an invention of any note made by only one man. We do not feel competent to judge Delamain's case. We know too little about him as a man. We incline to the opinion that the hypothesis of independent invention is the most plausible. At any rate, Delamain figures in the history of the slide rule as the publisher of the earliest book thereon and as an enthusiastic and skillful designer of slide rules. The effect of this controversy upon interested friends was probably small. Doubtless few people read both sides. Oughtred says:20 "this scandall... hath with them, to whom I am not knowne, wrought me much prejudice and disadvantage.." Aubrey,21 a friend of Oughtred, refers to Delamain "who was so sawcy to write against him" and remembers having seen "many yeares since, twenty or more good verses made" against Delamain. Another friend of Oughtred, William Robinson, who had seen some of Delamain's publications, but not his Grammelogia IV, wrote in a letter to Oughtred, shortly before the appearance of the latter's Epistle: I cannot but wonder at the indiscretion of Rich. Delamain, who being conscious to himself that he is but the pickpurse of another man's wit, would thus inconsiderately provoke and awake a sleeping lion... he hath so weakly (though in my judgment, vaingloriously enough) commended his own labour.. 22 Delamain presented King Charles I with one of his sun-dials, also with a manuscript and, later, with a printed copy of his book of 1630. A drawing of his improved slide rule was sent to the King and the Grammelogia IV is dedicated to him. 20 Epistle, p. (8). 21 Aubrey, op cit., Vol. II., p. 111. 22 Rigaud, Correspondence of Scientific Men during the 17th Century, Vol. I, Oxford, 1841, p. 11.

Page 206 206 University of California Publications in Mathematics [VOL. 1 The King must have been favorably impressed, for Delamain was appointed tutor to the King in mathematics. His widow petitioned the House of Lords in 1645 for relief; he had ten children.23 Anthony Wood states that Charles I, on the day of his execution, commanded his friend Thomas Herbert "to give his son the duke of York his large ring-sundial of silver, a jewel his maj. much valued." Anthony Wood adds, "it was invented and made by Rich. Delamaine a very able mathematician, who projected it, and in a little printed book did shew its excellent use in resolving many questions in arithmetic and other rare operations to be wrought by it in the mathematics."24 VI. OUGHTRED'S GAUGING LINE, 1633 It has not been generally known, hitherto, that Oughtred designed a rectilinear slide rule for gauging and published a description thereof in 1633.25 In his Circles of Proportion, chapter IX, Oughtred had offered a closer approximation than that of Gunter for the capacity of casks. The Gauger of London expostulated with Oughtred for presuming to question anything that Gunter had written. The ensuing discussion led to an invitation extended by the Company of Vintners to the instrument maker Elias Allen to request Oughtred to design a gauging rod.26 This he did, and Allen received an order for "threescore" instruments. On page 19 Oughtred describes his 'Gauging Rod:' It consisteth of two rulers of brasse about 32 ynches of length, which also are halfe an ynch broad, and a quarter of an ynch thick... At one end of both those rulers are two little sockets of brasse fastened on strongly: by which the rulers are held together, and made to move one upon another, and to bee drawne out unto any length, as occasion shall require: and when you have them at the just length, there is upon one of the sockets a long Scrue-pin to scrue them fast. There are graduations on three sides of the rulers, one graduation being the logarithmic line of numbers. He says (p. 39), "the maner of computing the Gaugedivisions I have concealed." W. Robinson, who was a friend of Oughtred, wrote him as follows:27 23 Dictionary of National Biography, Art. "Delamain, Richard." See also Rev. Charles J. Robinson, Taylors' School, from A.D. 1562 to 1874, Vol. I, 1882, p. 151; Journal of the House of Commons, Vol. IV., p. 197b; Sixth Report of the Royal Commission on Historical Manuscripts, Part I, Report and Appendix, London, 1877. In this Appendix, p. 82, we read the following: Oct. 22 [1645] Petition of Sarah Delamain, relict of Richard Delamain. Petitioner's husband was servant to the King, and one of His Majesty's engineers for the fortification of the kingdom, and his tutor in mathematical arts; but upon the breaking out of the war he deserted the Court, and was called by the State to several employments, in fortifying the towns of Northampton, Newport, and Abingdon; and was also abroad with the armies as Quartermaster-General of the Foot, and therein died. Petitioner is left a disconsolate widow with ten children, the four least of whom are now afflicted with sickness, and petitioner has nothing left to support them. There are several considerable sums of money due to the petitioner, as well from the King as the State. Prays that she may have some relief amongst other widows. See L. J., VII. 6. 657. 24 Anthony Wood, Athenae Oxonienses (Edition Bliss) Vol. IV., London, 1820, p. 34. 25 The New Artificial Gauging Line or Rod: together with rules concerning the use thereof: Invented and written by WILLIAM OUGHTRED, etc., London, 1633. The copy we have seen is in the Bodleian Library, Oxford. The book is small sized and has 40 pages. 26 Oughtred, op. cit., p. 11. 27 S. J. Rigaud, Correspondence of Scientific Men of the 17th Century, Oxford, Vol. I, 1841, p. 17.

Page 207 1920] Cajori: History of Gunter's Scale and Slide Rule 207 I have light upon your little book of artificial gauging, wherewith I am much taken, but I want the rod, neither could I get a sight of one of them at the time, because Mr. Allen had none left... I forgot to ask Mr. Allen the price of one of them, which if not much I would have one of them." Oughtred annotated this passage thus: "Or in wood, if any be made in wood by Thompson or any other." Another of Oughtred's admirers, Sir Charles Cavendish, wrote, on February 11, 1635 thus:28 I thank you for your little book, but especially for the way of calculating the divisions of your gauging rod. I wish, both for their own sakes and yours, that the citizens were as capable of the acuteness of this invention, as they are commonly greedy of gain, and then I doubt not but they would give you a better recompense than I doubt now they will. On April 20, 1638, we find Oughtred giving Elias Allen directions29 "about the making of the two rulers." As in 1633,30 so now, Oughtred takes one ruler longer than the other. This 1633 instrument was used also as "a crosse-staffe to take the height of the Sunne, or any Starre above the Horizon, and also their distances." The longer ruler was called staffe, the shorter transversarie. While in 1633 he took the lengths of the two in the ratio "almost 3 to 2," in 1638, he took "the transversary three quarters of the staff's length,... that the divisions may be larger." VII. OTHER SEVENTEENTH CENTURY SLIDE RULES In my History of the Slide Rule I treat of Seth Partridge, Thomas Everard, Henry Coggeshall, W. Hunt and Sir Isaac Newton.31 Of Partridge's Double Scale of Proportion, London, I have examined a copy dated 1661, which is the earliest date for this book that I have seen. As far as we know, 1661 is the earliest date of publications on the slide rule, since Oughtred and Delamain. But it would not be surprising if the intervening 28 years were found not so barren as they seem at present. The 1661 and 1662 impressions of Partridge are identical, except for the date on the title-page. William Leybourn, who printed Partridge's book, speaks in high appreciation of it in his own book.32 In 1661 was published also John Brown's first book, Description and Use of a Joynt-Rule, previously mentioned. In Chapter XVIII he describes the use of "Mr. Whites rule" for the measuring of board and timber, round and square. He calls this a "sliding rule." The existence, in 1661, of a "Whites rule" indicates activities in designing of which we know as yet very little. In his book of 1761, previously quoted, Brown gives a drawing of "White's sliding rule" (p. 193); also a special contrivance of his own, as indicated by him in these words: A further improvement of the Triangular Quadrant, as I have made it several times, with a sliding Cover on the in-side, when made hollow, to carry Ink, Pens, and Compasses; then on the sliding Cover, and Edges, is put the Line of Numbers, according to Mr. White's first Contrivance for manner of operation; but much augmented, and made easie, by John Brown. 28 Rigaud, loc. cit., p. 22. 29 Rigaud, loc. cit., pp. 30, 31. 33 Oughtred, An Addition vnto the Vse of the Instrument called the Circles of Proportion, London, 1633, p. 63. 31 F. Cajori, History of the Slide Rule, New York, 1909, pp. 16-22, Addenda, pp. vi-ix 32 W. Leybourn, op. cit., 1673, Preface, and pp. 128-29.

Page 208 208 University of California Publications in Mathematics [VOL. 1 He gives no drawing of his "triangular quadrant," hence his account of it is unsatisfactory. He explains the use of "gage-points." His placing logarithmic lines on the edges of instrument boxes was outdone in oddity later by Everard who placed them on tobacco-boxes.33 In Brown's publication of 1704 the White slide rule is given again, "being as neat and ready a way as ever was used." He tells also of a "glasier's sliding rule." William Leybourn explains in 1673 how Wingate's double and triple lines for squaring and cubing, or square and cube root, can be used on slide rules.34 Beginning early in the history of the slide rule, when Oughtred designed his "gauging rod," we notice the designing of rules intended for very special purposes. Another such contrivance, which enjoyed long popularity, was the Timber Measure by a Line, by Hen. Coggeshall, Gent., London, 1677, a booklet of 35 pages. Coggeshall says in his preface: For what can be more ready and easie, then having set twelve to the length, to see the Content exactly against the Girt or Side of the Square. Whereas on Mr. Partridge's Scale the Content is the Sixth Number, which is far more troublesome then [even] with Compasses. One line on Coggeshall's rule begins with 4 and extends to 40, these numbers being the "Girt" (a quarter of the circumference), which in ordinary practice of measuring round timber lies between 4 inches and 40 inches. This "Girt line" slides "against the line of Numbers in two Lengths, to which it is exactly equal." A second edition, 1682, shows some changes in the rule, as well as an enlargement and change of title of the book itself: A Treatise of Measures, by a Two-foot Rule, by H. C. Gent, London, 1682. In this, the description of the rule is given thus: There are four Lines on each flat of this Rule; two next the outward edges, which are Lines of Measure; and two next the inward edges, which are Lines of Proportion. On one flat, next the inward edges, is the Square-line [Girt-line in round timber measurement] with the Line of Numbers his fellow. Next the outward, a Line of Inches divided into Halfs, Quarters, and Half-Quarters; from 1 to 12 on one Rule; and from 12 to 24 on the other. On the other flat, next the inward edges, is the double Scale of Numbers [for solving proportions]. Next the outward on one Rule a Line of Inches divided each into ten parts; and this for gauging, etc. On the other a foot divided into 100 parts. Later further changes were introduced in Coggeshall's rule.34 It is worthy of note that Coggeshall's slide rule book, The Art of Practical Measuring, was reviewed in the Acta eruditorum, anno 1691, p. 473; hence Leupold's description35 of the rectilinear slide rule in his Theatrum arithmetico-geometricum, Leipzig, 1727, Cap. XIII, p. 71, is not the earliest reference to the rectilinear rule found in German publications. The above date is earlier even than Biler's reference to a circular slide rule in his Descriptio instrumenti mathematici universalis of 1696. 33 Cajori op. cit., Addenda, p. ix. 34 William Leybourn, op. cit., 1673, p. 35. 34 See Cajori, op. cit., pp. 20, 28, Addenda, p. ix. 35 See F. Cajori, "A Note on the History of the Slide Rule," Bibliotheca mathematica, 3 F., Vol. 10, pp. 161-163.

Page 209 1920] Cajori: History of Gunter's Scale and Slide Rule 209 Two noted slide rules for gauging were described by Tho. Everard, Philomath, in his Stereometry made easie, London, 1684. He designates his lines by the capital letters A, B, C, D, E. On the first instrument, A on the rule, and B and C on the slide, have each two radiuses of numbers, D has only one, while E has three. The second rule is described in an Appendix; it is one foot long, with two slides enabling the rule to be extended to 3 feet. Everard's instruments were made in London by Isaac Carver who, soon after, himself wrote a sixteen-page Description and Use of a New Sliding Rule, projected from the Tables in the Gauger's Magazine, London, 1687, which was "printed for William Hunt" and bound in one volume with a book by Hunt, called The Gauger's Magazine, London, 1687. This appears to be the same William Hunt who later brought out descriptions of his own of slide rules. The instrument described by Carver "consists of three pieces, two whereof are moveable to be drawn out till the whole be 36 inches long." It has several non-logarithmic graduations, together with logarithmic lines marked A, B, C, D, of which A, B, C are "double lines," and D a "single line" used for squares and square roots. It is designed for the determination of the vacuity of a "spheroidal cask lying," a "spheroidal cask standing," and a "parabolical cask lying." Another seventeenth century writer on the slide rule is John Atkinson, whom we have mentioned earlier. He says:36 "The Lines of Numbers, Sines and Tangents, are set double, that is, one on each side, as the middle piece slides: which middle piece is so contrived, to slip to and fro easily, to slide out, and to be put in any side uppermost, in order to bring those Lines together (or against one another) most proper for solving the Question, wrought by Sliding-Gunter." The data presented in this article show that, while the earliest slide rules were of the circular type, the later slide rules of the seventeenth century were of the rectilinear type.37 January 12, 1915. 36 John Atkinson, op. cit., 1694, p. 204. 37 Probably the oldest slide rule now in existence is owned by St. John's College, Oxford, and is in the form of a brass disc, 1 ft. 6 in. in diameter. It was exhibited along with other instruments in May, 1919. According to the Catalogue of a Loan Exhibition of Early Scientific Instruments in Oxford, opened May 16, 1919, the instrument is inscribed with the name of the maker ("Elias Allenfecit") and with the name of the donor, Georgius Barkham. It is dated 1635, which is only three years after the first publication of Oughtred's description of his circular slide rule. It is stated in the Catalogue: " Unfortunately all the movable parts but the base-plate and a couple of thumb-screws are missing. The face of the instrument is engraved with Oughtred's Horizontal Instrument. The back is engraved with eleven Circles of Proportion as described in Arthur Haughton's book, a copy of which was presented to St. John's College by George Barkham, to explain the use of the instrument." As Arthur Haughton's Oxford edition of Oughtred's Circles of Proportion did not appear until 1660, it would seem that the instrument was probably not presented to the College before 1660. As far as is known, the next oldest slide rule is of the year 1654, kept in the South Kensington Museum, London, and is described in Nature of March 5, 1914. It is a rectilinear rule, "of boxwood, well made, and bound together with brass at the two ends. It is of the square type, a little more than 2 ft. in length, and bears the logarithmic lines first described by Edmund Gunter. Of these, the num, sin and tan lines are arranged in pairs, identical and contiguous, one line in each pair being on the fixed part, and the other on the slide." The instrument is inscribed, "Made by Robert Bissaker for T. W., 1654." Nowhere else have we seen reference to Robert Bissaker. His slide rule seems to antedate the "Whites rule." mentioned above. [This foot-note was added on October 15, 1919.]

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 10, pp. 211-222 February 17, 1920 ONf A BIRATIONAL TRANSFORMATION CONNECTED WITH A PENCIL OF CUBICS BY ARTHUR ROBINSON WILLIAMS PREFATORY NOTE When I prepared the following paper I was acquainted only with the paper by Cayley, cited on page 212, and the articles therein mentioned. The problems of the construction of the cubic through nine points and the construction of the ninth point of intersection of all cubics through eight points, virtually the same problem, have been treated by Weddle, by Hart, by Chasles, by Cayley, and of course by later writers. Weddle's paper may be found in the Cambridge and Dublin Mathematical Journal, Vol. VI (1851), page 83, and Hart's in the same volume, page 181. Chasles' memoir appeared in Comptes rendus lebdomadaire des seances de l'Academie des Sciences, Vol. XXXVI (1853), page 942. It does not seem to have occurred to these writers to inquire concerning the locus of the ninth point when seven remain fixed and the eighth moves in a given manner. However, Professor H. S. White, to whom this paper was referred, pointed out that the subject had been well covered by Geiser and Milinowski. The paper by Geiser, " Ueber Zwei Geometrische Probleme," is in Crelle's Journal, Vol. 67 (1867), page 78. Starting from the well-known fact, which follows from the elementary properties of the cubic, that if seven points be fixed and an eighth be taken on the line joining two of them, the ninth intersection of all cubics through the eight lies on the conic through the other five, and conversely; and making use of a theorem of Steiner's regarding the locus of the double points of all the cubics through seven points, he derives by synthetic and very simple methods practically all the results which I have obtained, including the special cases arising from relations among the seven fixed points. Naturally neither he nor Milinowski, whose method is also synthetic, mentions the system of lines which I consider in section 12. Milinowski, whose paper on this subject is found in Crelle's Journal, Vol. 77 (1874), page 263, sets up a one-to-one correspondence between the cubics through seven points and the points of the plane. The same end is served by making cor

Page 212 212 University of California Publications in Mathematics [VOL. 1 respond to a given cubic of the net its polar line with respect to a fixed point. To obtain the locus of the ninth point when seven are fixed and the eighth moves, I have expressed its co-ordinates directly in terms of those of the eighth. This would not be feasible of course for curves of higher order than the third. Thus in case of a net of quartics, if twelve points are fixed, there correspond to a thirteenth not one but three others. Milinowski's methods are equally applicable to this and to curve nets of higher order. The second and longer portion of Geiser's paper is given to the discussion of an analogous problem in space. All quadric surfaces through seven fixed points have an eighth in common. Hesse, then privatdocent at Konigsberg, has given a construction for this eighth point in Crelle's Journal, Vol. 26 (1843), page 147. Geiser discusses the locus of the eighth point when six are fixed and the seventh moves in a given manner. The problem is rather more complex than that of the net of plane cubics, and the results obtained are very curious and interesting. The theorem that all the cubics through eight points have a ninth in common, suggests the following transformation, which is evidently birational. Fix seven points, and make correspond to a variable eighth the remaining common point of all the cubics through the eight. 1. Construction of the ninth intersection of the cubics through eight points. It follows from the properties of the cubic regarded as the locus of intersections of corresponding elements of a pencil of conies and a pencil of lines projectively related, that if any four of eight points be taken, the ninth point of intersection of all the cubics through the eight projects to the four selected points in four rays whose anharmonic ratio is the same as that of the four conics which have the four remaining points in common, and are determined respectively by the four chosen points.1 Therefore 9 is seen to lie on a certain conic through the four chosen points. In particular, if we take two sets of four points which have three points in common, 9 will be determined as the fourth intersection of two conics three of whose intersections are known; or, in other words, as the point which projects simultaneously to two sets of four points (which have three points in common) in two sets of four rays, each four having a given anharmonic ratio. Thus 9{4 6, 7 8} -(1 2 3 5) {4 6, 78}, and 9{56, 7 8} =(1 2 3 4){5 6, 78}; where (12 3 5) 4 6, 7 8} means the anharmonic ratio of the four conics which have 1 2 3 5 in common, and which pass through 4 6 7 8 respectively. In these anharmonic ratios the order of elements is arbitrary, but once chosen must be consistently maintained. In the above notation, primary and secondary elements are separated by a comma. Cayley, in the paper cited, has given the following construction due to Dr. Hart for a ninth point so determined: Let (1 2 3 5) {4 6, 7 8} be denoted by p, and (1 2 3 4) {5 6, 7 8} by s. The anharmonic ration of four conics of a pencil is given by that of their four points of inter1 Cayley "On the Construction of the Ninth Point of Intersection of the Cubics which Pass through Eight Points." Collected Papers, Vol. 4, p. 495.

Page 213 Williams: On a Birational Transformation 213 section with any line passing through any one of their four common points, or by that of the four tangents at one of the common points, or again by that of the po'ars of any point with respect to the four conies. After determining p and s, Dr. Hart proceeds thus: Let 74 and 65 meet in M, and on 74 take Q so that { 4M, 7Q } = p. Let 85 meet 64 in N, and on 85 take R so that {5N, R8} = s. Let QR meet 65 in K and 64 in L. Then 7K and 8L give 9. The whole construction may be accomplished with the ruler. 2. Analytic treatment of this construction shows that the co-ordinates of 9, X'/' v', are octavic functions of the co-ordinates of 8, of the form X' = C2C3K1, ' = CC1K2, v' = C1C2K3, where the Cs are of degree three and the Ks of degree two. Now introducing coordinates, let 4 5 6 be the vertices of the triangle of reference; 6 being (1: 0: 0), 5being(0:0: 1), and 4(0:1: 0). Then45isthelinex=0, 56 is y=0, and 64 is z=0. Let 1 2 3 7 be (X1: ul: i), (X2::2: V2), etc., and let 8, which is to be regarded as variable, be (X: v: v). The quantities p and s are now themselves functions of X gu v. p is (1 2 3 5) {4 6, 7 8}. The four conies all pass through 5, and it will therefore suffice to find the anharmonic ration of their four other points of intersection with the line 45, i.e., the line x=0. The conic 1235-4 meets x=0 at 4, i.e. (0: 1: 0). The equation of 1235-6 may be written thus: y2 xy xz yz jC12 Xl/1t1 X11 V, lV1 = L122 X2/12 X2 2 X 2 2 V2 A32 X3/ 3 X3P3 /3 V3 Setting x=0, we have y: z=a: b, where a and b are the minors of yz and y2 respectively. Similarly the equation of 1235-7 may be written in the determinant form, and, setting x=0, we have y: z=f: g where f (Xi12. X322/3 U.X7V7) and g = (X12. X2U v 3 V3 ' 7 V7). For 1 23 5-8 we have the same equation except that the coordinates of 7 are replaced by those of 8. Setting x=0, we have y: z = Fs: G8, where Fs and G8 are obtained on replacing X7/7V7 by X/ v in f and g. The capital letters and the subscript will indicate the presence of the coordinates of the variable point. Hence the four points whose anharmonic ratio we desire are respectively (0: 1:0), (0: a: b), (0: f: g), (0: F: Gs). Following the order of elements indicated above, we have the anharmonic ratio of the four conies g (aGs-bFs)/ Gs(ag - bf) =p. In the same way the anharmonic ratio (12 3 4) {5 6, 7 8} is found to be g(bH -cGs)/Gs(bh-cg). Only three new determinants have been introduced, C- (v12 X212 X3V3), h=X2. 22.X3/3 'X7Y7), and H8, which is obtained on replacing X7~/7V7 by X/1v in h. Thus a, b, c are of the third order and involve only the co6rdinates of 123. The other determinants are of the fourth order and related as indicated. In accordance with the construction outlined, we take Q on 7 4 so that {4M, 7Q} = p, where M is the intersection of 74 with 65, that is, of 7 4 with y = 0. The

Page 214 214 University of California Publications in Mathematics [VOL. 1 coordinates of Q are thus found to be;(X7: P7: V7). Similarly the intersection of 8 5 with 6 4 is N, and taking R on 8 5 so that{5N, R8 =s, R is (sX: us: y). The'intersection of QR with y=0 gives K, and with z=0 gives L. The intersection of 7K and 8L is 9. Thus for X'//'v', the coordinates of 9, we obtain X' = ( 7XS - 7 V) (X7A - A7XP) (p - s) ' = (\71 —17\Xp) (/7p V- V7S) (S-1) - 1) V' = (/7p V - PV7S) ( V7XS - 7 v) (1 -) The following identities are at once evident X\(17p V- V1V71S) +-A( v7\S -7 V) + V(X7/ -/L7Xp) = 0 ( (p - s) + (s —1)+(1- p) =0 To obtain X'u'yv' as homogeneous expressions in X/up it is necessary only to substitute the values of p and s in (1). After removal of a common denominator and the factor g from each numerator, we have X'= {v7X(bHs-cGs)g-X7 v(bh-cg)G8s} X { X7/(ag-bf)G8 - 7X(aG8-bFs)g} X { (bh - cg) (aG8- bF8) - (ag - bf) (bH8- cG8) }. /'= { X71(ag-bf)Gs8- 17X(aG8- bF)g } X {Piv(bh-cg) (aGs-bFs)- v71u(ag-bf)(bHs-cG )}X ) (2) {(bHs-cGs)g- (bh-cg)Gs8. /= {/U7v(bh-cg)(aGs-bFs)- v7/1(ag-bf)(bHs-cGs) } X { 7X(bH8-cGs)g-\X7 (bh-cg)Gs} X { (ag-bf)Gs-(aGs-bF8)g}. J Recalling that Fs Gs Hs are determinants homogeneous and of degree two in X / v, it is evident that X',' v' are homogeneous of the eighth degree, and of the form X' = C2C3K1 A1'=C3C1K2 (2') I' = C1C2K3 where the Cs are of the third degree and the Ks of the second. The identities connecting the Cs and Ks are XgC + - (ag -bf)C2+ v(bh - cg) C = 0 )( gK1+ (ag -bf)K2+ (bh - cg)K3 -=O 3. The factors C and K may be replaced by determinants proportional to them. From the form of Fs Gs Hs it follows that they all vanish when the coordinates, X, v, of 8 are replaced by those of any one of the points 1 2 3. Then, expanding in terms of X /u v, it is easily seen that Gs = 0, aGs- bFs = 0, bH8- cGs = O represent respectively the conics 1 2 3 4 5, 12 3 5 6, 12 3 4 6. Noting that the left-hand mem

Page 215 1920] Williams: On a Birational Transformation 215 bers of these conics become g, ag - bf, bh - cg when X7 7 V7 are put for X y v, it appears immediately that C2 =0, C3 = 0, C = 0 represent three nodal cubics, each of which passes through all the seven fixed points, and which have their double points at 4 5 6 respectively. Similarly K1=0, K2=0, K3=0 represent three conics which have the points 1 2 3 7 in common and which pass through 6 4 5 respectively. The symmetry of X',u' v' is quite striking. Thus the two cubics that form part of X' have their double points at (xy) and (xz), while the accompanying conic, K1, passes through 6, i.e., the vertex opposite to x=O. Similarly for the factors of g' and v'. Thus the form of each factor C and of each factor K is determined down to a constant multiplier by the character of the curve that it represents when equated to zero. For example K2=0 gives the conic 1 2 3 7 4, and therefore K2 differs at most by a constant factor from the determinant V2 X2 J /v V\X Xh V12 X12 P1Xi Y 1X1 Xl,1 V22 X22 /2 V2 V2X2 X2Y/2 V32 X32 /3 v3 V3X3 X31i3 72 X72 7/7 V7 }7X7 X7-7 Expanding this determinant by the elements of the first row, and expanding also K2- (bHs- cG8)g- (bh- cg)G8, i.e. b(gH8- hGs), comparison of the coefficients shows that K2 is identical with the above determinant multiplied by -bJ. J is the determinant (X12 X22 X3 3) -X1X2X3(X1 '2 v3). Similar relations hold for the other Cs and Ks. Thus K1- (2 Vy12. L2 V2 ' Y3X3 X7 V7) J2b K2 -- ( V. *12. y2V2 ' 3X3 X7/17) ( - Jb) K3 ' (X2 * 12 */2 V2 ' 3X3' X7 V7)Jb C2 = (X2 * X12 ' X212 V2 ' X3 V32 ' /7 72) (- JJ1) C3 _ (X2 * X2 V1 X2122 * X3 3 V3 /' 172 V7) JJ1 C1 = (XP2 ' Xl11 V1' X2 V22 132 V3 * /7 72) J2J1 J1-(Xl P'12 ). It is seen, therefore, that X' u' v as given by (2) contain the common factor - J4J12 - X-X24X3tJ1i.-Removing this common factor, we may use for the Cs and Ks the determinants indicated in the last set of equations. 4. Geometric character of the transformation. In (2) X' /' v' represent curves of the eighth order which have seven triple points in common, that is sixty-three (=82-1) common intersections. Since the possession of a triple point at a given point imposes six conditions, it follows that a curve of order eight having seven fixed triple points is subjected to forty-two conditions, two less than the number sufficient to determine a curve of that order. This accords with the general theory of Cremona transformations which demands that in a birational transformation of degree n the expressions of the nth degree

Page 216 216 University of California Publications in Mathematics [VOL. 1 shall represent curves which have n2-1 common intersections, while these intersections must represent n (n+3) 2 2 conditions, two less than the number required to determine a curve of order n. Therefore if 8 moves on a line not passing through any of the seven fixed points, 9 will describe a proper, i.e., undegenerate, curve of order eight having a triple point at each of the seven fixed points. If 8 moves on a conic which does not pass through any of the seven fixed points, 9 will describe a curve of order sixteen having a sextuple point at each of them. According to the general theory there will correspond to each of the seven fixed points a curve of order three. The nature of this correspondence appears immediately from (2). For example, if 8 is any point on C1, 9 is (1: 0: 0). But C1 is the cubic which has its double point at (1:0: 0). For 8 on C3, 9 is (: 0: 1), and for 8 on C2, 9 is (0: 1: 0), which are the double points of C3 and C2 respectively. Considerations of symmetry enable us to conclude immediately that in this transformation the curve of degree three corresponding to any one of the seven fixed points is the nodal cubic passing through all of them and having a double point at the one in question. These seven cubics taken together constitute the Jacobian of L M N; where L M N are the functions of X' A' v' got by solving (2) for X, for Aj, and for v respectively. Since in this transformation the coordinates of 8 are evidently the same functions of the coordinates of 9 as the coordinates of 9 are of those of 8, the same seven cubics constitute the Jacobian of X' y' v'. If 8 is on K1 (2) shows that 9 is on x = 0. But K1 is the conic 12376 and x=0 is the line 4 5. Similarly for the conics K2 and K3 and the lines y=0 and z=0. It thus appears from (2), as is obvious from the general properties of the transformation and from those of the cubic, that to the line joining two of the fixed points corresponds the conic through the other five. Or, in other words, the curve of order eight corresponding to such a line consists of this conic and the two cubics which correspond to the two fixed points. With the exception of the seven special cubics just mentioned, all cubics through the seven fixed points are self-corresponding. For the points of any one of them correspond in pairs. To a sextic which has a double point at each of the original points corresponds another sextic which itself has a double point at each of them. It will be seen that there is one such sextic all of whose points are invariant. It is in fact the locus of the invariant points of the transformation. 5. There are infinitely many invariant points, and their locus is a sextic. To ascertain the invariant points, we may put X u v for X' /' v' in (1). Then dividing the first equation by the second, the first by the third, and the second by the third, and cross-multiplying, we have the three equations, two of which are independent, X(iU7p- v7s) (s-) (-)-/ (v7Xs-X7 V)(p-s) = 0 p( V7XS-X7V) (1-p)- (X7/a-/ 7Xp)(s-1) =0 v (X7/ - 7Xp)(p - s) - X (7p v - V7EIs) (1 - p) = 0

Page 217 1920] Williams: On a Birational Transformation 217 Any invariant points must be common to the three curves represented by these equations. But all three reduce to Xps(Y7 V- V7+) + IS( 7X\-7 V) + Vp(X7j, - 7X) = 0 (4) Hence, instead of a finite number of invariant points there are infinitely many; in fact all the points of the curve (4). This curve is evidently a sextic, and, referring to (2'), may be written in any of the three forms XC1K2 - IC2K1 = 0, JLC2K3- vC3K2 = 0, vC3K1-XC1K3=0. Any of these equations shows that the sextic has a multiple point which is at least a double point at each of the seven fixed points. But no one of them can be of higher order than two, for in that case there would correspond to the sextic a curve of lower order than six. An interesting property of the invariant points appears when they are considered in connection with the original construction. It will be recalled that 9 is the fourth intersection of two conics one of which is the locus of points that project to 4 6 7 8 in the anharmonic ratio p, while the points of the other project to 5 6 7 8 in the anharmonic ratios. Let the tangents to these conics at 8 be lp and 1,. Then the equations of these tangents may be easily obtained. For the anharmonic ratio of the four rays 84, 86, 87 and lp is p, while that of 85, 96, 87, Is is s. Making use of the fact that the anharmonic ratio of four lines of the form L, M, L-rM, L-rllM is r/rl, and keeping the same order of elements as in calculating p and s, the equations of lp and Is are found to be respectively p vx-r vy + (r- Xp)z = six - (k v + Xs)y+kIz = 0 V7X- X7 V where r - - /7 - V-V7 and k=7 — h7X /7 V- V7,1 The conics will have their fourth intersection at 8 if lp and Is coincide; that is if pv. rv rIu-Xp siu kv+Xs ku the variables being X 1u v, the coordinates of 8. Forming the three possible equations and bringing all the terms to the left hand member, we have in each case the

Page 218 218 University of California Publications in Mathematics [VOL. 1 factor Xps-ysr+vpk. Hence all points of the curve that correspond to this common factor are invariant. Replacing r and k by their values we have Xps(g7 V - V7A1) + iS ( V7X - 7 V) + VP (X7 - 17X) = 0 which is (4) above. Substituting the values of p and s in (4), we obtain X (7 v- V71) (aG8 - bF8) (bH8-c G8)g + ( 7X-X7 ) (ag - bf) (bH8 - cG8) Gs + (X7, — U7X) (bh - cg) (aG- bFs)Gs = 0. (5) It will be observed that at points 8 for which lp and ls coincide they have two intersections with all the cubics of the one parameter family through the seven fixed points and the point 8 in question. In other words the cubics of the family determined by the seven fixed points and an invariant point have a common tangent at the latter, whose equation is given equally by lp and I,. There is, however, one cubic of the family which has a double point at the invariant point, and for which the line Ips is not in general a tangent. The envelope of this set of lines Ips will be sought after a little further examination of the sextic. In (5), note that 17 -- V7/1, v7 -X7V, X7i - 7XA, when equated to zero are the lines 67, 47, 57 respectively; while G8=0, aG8- bF8=O, bHs-cG8=O are the conies 12345, 123 5 6, 12346. Thus it appears that the sextic passes through the intersections of the conic 12345 with the line 67, of 12356 with 4 7, and of 12346 with 57. It is thus obvious from symmetry that the sextic passes through the intersections of the conic determined by any five of the seven fixed points with the line joining the other two. There are twenty-one such pairs making a total of forty-two points. By a simple regrouping of the terms of (5) with change of signs, we obtain XGsCl+/u(aGs- bF8)C2+ v(bH8 - cG8)C3 = (6) where C1 C2 C3 are defined as in (2). C2 has a double point at 4, C3 at 5, and C1 at 6. Using (6), it is very easy to show by means of the second partial derivatives that at 4 the tangents to the sextic coincide with those of C2, at 5 with those of Cs, and at 6 with those of C1. It is necessary only to indicate the differentiation. To tell the terms that vanish, it suffices to remember through what points the conics pass and the double points of the cubics. Hence it appears immediately from symmetry that the sextic has a double point at each of the seven fixed points whose tangents coincide with those of the cubic that has the same double point and passes through the other six fixed points. Of the eighteen intersections of the sextic with any of the cubics of this system six are at their common double point and the remaining twelve at the other six points. The conic through any five of the fixed points has ten of its twelve intersections with the sextic at those points, and the other two are on the line determined by the two remaining points. The line joining any two of the fundamental points meets the sextic four times at those points and twice where it meets the conic through the other five. Thus the sextic fulfills seventy-seven conditions; twenty-one by reason of its seven fixed

Page 219 Williams: On a Birational Transformation 219 double points, fourteen by reason of the determination of the tangents at those double points, and forty-two others corresponding to the twenty-one point pairs noted in the preceding paragraph. One other property is easily obtained. C1C2C3 are three linearly independent cubics through the seven fixed points. Hence any of the o 2 cubics through them is of the form C= —C1i+C2+-C3=0 where i, r, v are constants. It follows immediately from the form of the first partial derivatives of C, that if C have a double point, it must lie on the Jacobian of C1C2C3. But from the character of the latter it follows that their Jacobian is a sextic which must have a double point at each of the seven fixed points, and must have at 6, at 4, and at 5 the same tangents as C1, C2, C3 respectively. Therefore it fulfills twenty-seven conditions which determine it uniquely as coincident with the invariant sextic. Summarizing the results of this section we have the following theorem: Given seven points in a plane, the (forty-two) points in which the line determined by any two meets the conic determined by the other five lie on a sextic which has a double point at each of the seven given points, and whose tangents at any one of these points coincide with those of the cubic that has the same double point and passes through the other six given points. This sextic is the locus of the double points of all nodal cubics that pass through the seven given points. It is also the locus of points 8 such that the single infinity of cubics determined by 8 and the seven given points have a common tangent at 8. 6. The single infinity of cubics determined by the seven fixed points and an invariant point have a common tangent at the latter; and the envelope of the system of lines thus defined is of class four and order twelve. These common tangents are, as has been seen in section 5, paragraph 2, the lines lp, or Is, which coincide at points on the sextic. Their envelope, therefore, is the envelope of lp, or, which is the same thing; that, of Is, when 8, whose coordinates determine their coefficients, moves on the sextic. Difficulties of elimination render the more direct methods inapplicable. The class is, however, easily obtained. For if in lp we fix xyz, the number of lines of the system through (x: y: z) will depend on the intersections of the sextic with lp regarded as a locus in X/tv. The latter is a quartic only four of whose intersections with the sextic depend on xyz. Hence the class is four. The same is obtained from 1,. It is not feasible, however, to express the condition that two of these four intersections coincide. But the lines whose envelope we seek join 8 and 9, which for points on the sextic are infinitely near. From (1) the equation of the line connecting 8 with 9 is X(/i7 vp - V7/8) I +y( V7Xs - X7 v) + Z (X7/A — 7Xp) = 0 where 2 is the left hand member of the sextic given by (4). Removing this common factor the coefficients of xyz are easily shown to be proportional to the corres

Page 220 220 University of California Publications in Mathematics [VOL. 1 ponding coefficients in lp and ls for points on the sextic. Substituting the values of p and s in the equation of 8 9, we have xgC1+y(ag-bf)C2+z(bh-cg)C3 = 0 (7) which, for X iv variable, is a cubic through the seven fixed points, and through (x: y: z) by the first of identities (P), end of section 2. It has therefore, as expected, four intersections with the sextic which depend on xyz, and at each of them it is tangent to the corresponding line of the system. Hence the four lines of the system through an arbitrary (x': y': z') are the four tangents to the corresponding cubic (7) from (x': y': z'). If (x': y' z') is an inflection, there are only three such tangents; but in that case (x': y': z') is a point P on the sextic and the line of the system corresponding to P makes the fourth. For this line is the common tangent at P to all the cubics (7) determined by points (x': y': z') upon it, and in particular it is the inflectional tangent to that determined by P itself. And there are no other points (x': y': z') which have this property. For they must lie on the Hessian of (7), which is a sextic in x'y'z' when the latter are put for X/v, and hence must be the same sextic. If the cubic (7) has a double point, it must be on the sextic. Thus the envelope contains the points (x': y': z') whose corresponding cubics (7) have double points. For the latter count for two intersections with the sextic in each case, and there will be only three lines of the system through (x': y': z'). And the envelope in general contains no other points. For the cubic (7) determined by an arbitrary x'y'z' is tangent at its four intersections with the sextic that depend on x'y'z' to the corresponding line of the system, and in general there are no points on the sextic at which this line coincides with the sextic. The condition that (7) have a double point is the vanishing of the eliminant of the first partial derivatives of (7) and of its Hessian. This is of degree twelve in xyz. 7. The introduction of relations among the seven original points leads to transformations of lower order, each of which has an invariant curve. Something should be said of the special cases produced by introducing relations among the seven original points. It was noted in section 3 that the coordinates of 9, X'Iu'v', as given by (2) contain the common factor (X'!2' V3) which vanishes when 1 2 3 are collinear. This does not mean that 9 is indeterminate if three of the seven points are on a line. The difficulty is obviated by numbering them suitably. It is very easy to deal with the special cases if we recollect in connection with the elementary properties of the cubic the geometric character of the Cs and Ks in the equations of transformation (2) or (2'). If three points are collinear they may be numbered 23 7. Then all the steps taken in deriving (2) may be retraced and no equation becomes illusory. But K1 K2 K3 all contain a linear factor corresponding to the line 237. This linear factor is therefore common to X' a' v' and appears in the invariant sextic. We have, therefore, a transformation of degree seven and an invariant quintic curve, whose properties are immediately deduced by regarding it as part of the sextic.

Page 221 1920] Williams: On a Birational Transformation 221 Thus the quintic will have double points at 145 6, and will pass through 23 7. Since 2 3 7 are collinear, the cubic through the seven fixed points that has a double point at 2 must consist of the line 2 3 7 and the conic 14 5 6 2. Hence this conic and the quintic have the same tangent at 2. Corresponding relations hold at 3 and 7. Proceeding thus, the quintic is seen to fulfill fifty-three conditions. Removing the common factor, X' u' v' represent curves of order seven, each of which has a triple point at 14 5 6 and a double point at 2 7 3. This gives forty-eight common intersections and imposes thirty-three conditions, thus satisfying the requirements for a birational transformation of degree seven. Other cases are dealt with in similar fashion. If four of the seven points are collinear, 9 is necessarily indeterminate. If six are on a conic they may be numbered 45 6 1 27. Then C1 C2 C3 all contain a corresponding quadratic factor, which appears to the first degree in the sextic, and to the second degree in X' u' v'. We have thus a quartic transformation with an invariant quartic curve. This quartic has a double point at 3, and its tangents at the other six fixed points are the lines joining them respectively to 3. Hence the class is at least ten; and since 3 is known to be a double point, it is exactly ten, and the deficiency is two. The same is true if 3 should be a cusp; for from a cusp three less tangents can be drawn than from an arbitrary point. This indicates that in the general case the only singularities of the non-degenerate sextic are those at the fixed points, and that its deficiency is three. If the six points lie on a proper conic, the quartic fulfills forty-seven conditions, relatively the largest number satisfied by any of the invariant curves connected with this transformation. If the six points lie on two lines, three points on each, the quartic satisfies thirty-five conditions. The coordinates of 9 represent quartic curves which have a common triple point at 3 and single intersections common at each of the other six points-a well known type of birational transformation of degree four. Since 4 5 6 1 2 7 are on a conic, 9 must lie on 3 8. This line meets the quartic twice at 3, and 9 is the harmonic conjugate of 8 with respect to the other two intersections of 3 8 with the quartic. If 8 is at 3, 3 8 is any line through 3, and thus to 3 will correspond a curve which is in fact the polar cubic of 3 with respect to the quartic, since any line through a double point of a quartic is divided harmonically by the quartic and the polar cubic. But this polar cubic has a double point at 3 and passes through the other six fixed points, since the tangents to the quartic at those points are the lines joining them to 3. To each of these six points corresponds the line joining it with 3. If 4 5 6 1 2 7 are on a conic and 3 is on 2 7, the transformation is of degree three, and the invariant curve is a cubic of deficiency one. For it passes through 3 1 4 5 6, and its tangents at the last four points are 3 1, 3 4, 3 5, 36. The cubic fulfills twenty-nine conditions if the fixed conic does not degenerate, and twenty if it does. X',' v' represent cubic curves having a common double point at 3 and single intersections common at 1 4 5 6. 9 is again the harmonic conjugate of 8 with respect to the other two intersections of 3 8 with the cubic. Thus to 3 corresponds its polar conic with respect to the cubic. To 1 4 5 6 correspond the lines which join them respectively to 3.

Page 222 222 University of California Publications in Mathematics [VOL. 1 The case when 456 127 are on a conic and 3 is the intersection of 2 7 and 1 6, is particularly interesting. The transformation is quadratic, and the invariant curve is a conic which, if the fixed conic does not degenerate, satisfies sixteen conditions. For it is tangent to 3 4 at 4, and to 3 5 at 5, and passes through the intersections of 1 5 with 4 6, of 75 with 24, of 14 with 5 6, of 74 with 5 2, of the conic 3 24 5 6 with 17, of 3 2 145 with 67, of 3 4 5 1 7 with 62, and of 34 5 6 7 with 12. 9 is again the fourth harmonic of 8 with respect to the intersections of 38 with the invariant conic, that is, the intersection of 3 8 with the polar of 8. The canonical form of this transformation is x' = yz, y' = xy, z' = zx. Inversion with respect to the unit circle is a special case. All these curves exist apart from the transformation that revealed them. The arbitrary numbering in each case simply serves to adapt it to the original procedure. Thus in the last case we may say that if six points are taken on a proper conic, and if any vertex of the complete quadrangle of any four is taken as a seventh point, there is a conic which has the properties just described. Finally if 456127 are on a conic, and if 27, 16, and 45 are concurrent at 3, the transformation is a collineation. The invariant line is the polar of 3 with respect to the fixed conic, and 9 is the harmonic conjugate of 8 with respect to 3 and the intersection of 3 8 with the invariant line. 3 is a self-corresponding point. Whenever six of the fixed points are on a conic, 9 must lie on the line joining 8 and the remaining point, unless 8 is on the fixed conic, in which case 9 is any point on that conic. It was observed in the case of the quartic, the cubic, and the quadratic transformations that the line connecting 8 with this remaining point had two intersections with the invariant curve that depended on 8. Hence in those cases the envelope considered in section 6 is indeed of class four; but it is simply the fixed conic and the remaining point (a curve of class one) counted twice. The cubics Ci, C2, C3 are not linearly independent when 456127 are on a conic, but the results just stated may be verified in certain simplified cases by use of lp r Is as in section 6, paragraph 1. Transmitted May 2, 1919.

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 11, pp. 223-240 February 17, 1920 CLASSIFICATION OF INVOLUTORY CUBIC SPACE TRANSFORMATIONS BY FRANK RAY MORRIS In a paper "On the Rational Transformation between two Spaces' 1 Cayley gives a general treatment and two special cases of what he calls a cubo-cubic transformation. The correspondence is between the points of two spaces which may or may not be superimposed upon one another. The term rational is used in the sense of birational, i.e., the ratios of the homogeneous coordinates of either space can be expressed as rational functions of the coordinates of the corresponding point in the other space. If these rational functions are ratios whose terms are cubic functions of the coordinates the transformation is said to be cubo-cubic, or simply cubic. Let x, y, z, w be the coordinates of a point in the first space and x', y', z', w' those of the corresponding point in the second space. Then the transformation is x: y:: zw=X': Y: Z: W' where X', Y', Z', W' are certain cubic functions of x', y', z', w'. It is evident that a point in the second space determines a corresponding point in the first space. However, a point in the first space determines three cubic surfaces in the second and these in general determine 27 points. Hence the cubic surfaces must be so related that only one of their common points depends upon the point in the first space. Cayley shows that this condition is satisfied if the cubics have in common a twisted sextic curve with 7 apparent double points. This curve is called:'. Any cubic of the form aX'+bY' +cZ'+dW' = 0, and in particular X' = Y' = Z'= W = 0, passes through '. With this restriction on the cubic functions it is possible to express the transformation in the form x': y': z': w'=X: Y: Z: W, 1 Collected Math. Papers, vol. VII, pp. 189-240. Also Proc. London Math. Soc., vol. III (1869-1873), pp. 127-180.

Page 224 224 University of California Publications in Mathematics [VOL. 1 where X, Y, Z, W are cubic function of x, y, z, w having a similar relation to that existing between those of the second space. They determine a sextic curve 2 with the same properties as 2'. It is assumed that the most general transformation which gives rise to the cubo-cubic transformation is given by the equations x'Al+y'A2+z'A3+w'A4 = 0, x'B1+y'B2+z'B3+w'B4 = O, x'Ci+y'C2 +-'C3+w'C4 = 0, where the capital letters are linear functions of x, y, z, w. The points on 2 and 2' are exceptional. To a point on 2 corresponds a line in the second space meeting 2' in three points. As the point moves on 2 the corresponding line generates a surface of degree 8 with 2' as a triple curve. The equation of this surface is got by equating to zero the Jacobian of any four of the cubic surfaces through 2'. We shall call it J'=0. There is a similar surface, J=0, in the first space corresponding to the curve A'. The transformation defined above is not in general involutory and neither of the special cases which Cayley gives is. But the involutory case is especially interesting. It is easily treated from the projective point of view if the two spaces are thought of as superimposed upon one another in such a way that Z coincides with 2' and J=0 with J'=0. The transformation is given by three quadric surfaces and their polar planes. To any point in the space corresponds a polar plane with respect to each of the three quadrics. These three planes in general determine a single point which, by the same process, determines the original point. Thus there is a one-to-one correspondence, with certain exceptions, between the points of the same space. The exceptional points are those for which the three polar planes have a common line. Such points lie on the curve 2 and, as in the general case, to Z corresponds the surface J=0. A discussion of this case is given later. A purely synthetic treatment of the involutory case is given by D. N. Lehmer in a paper, "On Combinations of Involutions."2 Without the use of a system of coordinates or a single equation he discovers the properties of the curve Z and the surface J=0. The properties are identical with those given by Cayley for the general case except that they do not include the number of apparent double points. This point of view leads to a number of special cases of particular interest where 2 breaks up in such a way as to contain one or more lines.3 This paper is an analytic treatment of the involutory cubic transformation. Case 1 gives the theory of the general case in which Z is a proper curve.4 It also gives the notation and plan of treatment used throughout the paper. The remainder of the paper develops the transformation and properties of Z and J=0 for the cases in which Z breaks up into lower degree curves one or more of which 2 Amer. Math. Monthly, vol. XVIII, No. 3, Mar. 1911. See also Steiner's Gess. Werke, vol. II, p. 651. 3 Elizabeth J. Easton gave a synthetic treatment of six of these cases in her thesis for the degree of M. A. at the Univ. of Cal. 1917. They occur in this paper as cases 2, 3, 5, 9, 11, 12. 4 This case is given in a number of text books on solid geometry. See Salmon's Geometry of Three Dimensions, Art. 238a, p. 247 (fifth edition); Snyder and Sisam's Analytic Geometry of Space, pp. 170-172.

Page 225 1920] Morris: Classification of Involutory Cubic Space Transformations 225 are straight lines including double lines. It is hoped that these special cases may prove valuable in the study of cubic surfaces and in resolving noninvolutory transformations into the product of involutory ones. 1. Proper sextic curve Capital letters are used to denote functions of the homogeneous coordinates, and small letters, except the coordinates, to denote constants. An unaccented letter and the same letter accented denote the same function of x, y, z, w and x', y', z', w' respectively. The subscripts 1, 2, 3, 4 denote the partial derivatives with respect to x; y, z, w or x', y', z', w' respectively, except for the functions A, B, C, in which case they denote half the partials. Let the three quadric surfaces be A = anx2 + a22y2 + a33z2 + a44w2 + 2a12xy + 2al3xz + 2al4xw + 2a23yz + 2a24yw + 2a34zw = 0, B = blx2+ b22y2+ b33z2+ b44w2+2bl2xy+2bl3xz+2bl4xw+2b23yz+2b24yw+2b34zw = 0, C = CllX2 + c22y2 + C332 + C44W2 + 2 C12X x + 2 c13xz + 2 c14xw + 2 c23z yw 2 24 + 234zw = 0. The three polar planes of a point P whose coordinates are x, y, z, w are x tA4+y'A2+z'A3+w'A4 = 0, x'Bl+y'B2+z'B3+w'B4 = 0, X' Cl+y' C2+z C3+fw' C4=0. These equations determine a point P' which corresponds to the point P. They may be written in the form - xA' +yA'2+zA'3+wA'4 = 0, xB'l+yB'2+zB'3+wB'4 = 0, XC'l+y C'2+zC'3+wC'4 = O. The last three equations represent the polar planes of a point P' and they determine the point P. Hence the transformation is involutory. The equations are linear in either set of coordinates so that the transformation is what Cayley calls a cubo-cubic. It may be expressed as x': y': z': w'=X: Y: Z: W where X, Y, Z, W are the determinants, with the proper sign, of the matrix A1 A2 A3 A4 B1 B2 B3 B4. C1 C2 C3 C4 Each determinant is a cubic function of x, y, z, w. Similarly, we get: y: z: w=X': Y': Z': W'. If for a point P the four determinants of the above matrix are each equal to zero, then the three polar planes no longer determine a point P', for this is the

Page 226 226 University of California Publications in Mathematics [VOL. 1 condition that they have a line in common. Then P is on each of the four surfaces X= Y= Z = W= and its locus is the sextic curve.5 Cayley shows that to the curve 2 corresponds the Jacobian surface of the four cubic surfaces. It is X1 X2 X3 X4 J= Y1 Y2 Y3 Y4 =0 Z1 Z2 Z3 Z4 W1 W2 W3 W4 It is a surface of degree eight and contains 2 as a triple curve. Both the curve and the surface play an important part in the special cases which we are now ready to consider. 2. Sextic curve made up of a line and a quintic The simplest form of this case is that in which two of the quadrics are unlimited, the third is a pair of planes and the three are unrelated. The equations of the first two are the same as in the general case while the third may be taken as C = 2xy= 0. The matrix is A1 A2 A3 A4 B1 B B3 B 4 y x 0 o from which we get the cubic surfaces X=xD=O, Y=yD=O, Z = x(AB4- A4B1) -y(A2B4-A4B2) =0, W=x(A 1B3-A3B1) -y(A2B3-A3B2) =, where D is the determinant A3B4-A4B3. The Jacobian surface is D+xD, xD2 xD3 xD4 = yD1 D+yD2 yD yD4 Zi Z2 Z3 Z4 W1 W2 W3 W4 To reduce it multiply the first and second columns and divide the first and second rows by x and y respectively, subtract the second row from the first and add the first column to the second. The result is D+xD-+yD2 D3 4 D D D4 J=D xZ1+yZ2 Z3 Z4 = 3D Z3 Z4 =0. xW1+YW2 W3 W4 W W3 W4 5 For a proof that these three cubics have a common sextic curve see footnote 4 and Salmon's Lessons on Higher Algbra, Art. 271.

Page 227 1920] Morris: Classification of Involutory Cubic Space Transformations 227 The last step is obtained by adding z times the second and w times the third column to the first, and the applying Euler's theorem with regard to partials of homogeneous functions to the elements of the first column. With these equations before us we are now in a position to examine the sextic curve and the Jacobian surface. S is made up of the line x = y =0 and the quintic curve which is the intersection of D =0 and Z=0 except the line A4=B4 =0, or of D=0 and W=O except the line A3=B3=0. Since the line of S is not on D=0, is on Z=0 and does not meet the line A4=B4 =0, it must meet the quintic curve in two points and have three apparent double points with it. Hence the quintic must have four apparent double points in order to complete the required number seven.6 The Jacobian surface is made up of the quadric D=0, which corresponds to the line of Z, and a sixth degree surface corresponding to the quintic of I. The line is a triple line on the sixth degree surface, while the quintic is a double curve on the sixth degree surface and a single curve on the quadric. The same results could have been obtained without having the quadric C=0 become a pair of planes, by choosing it as a general quadric so related to the other two that the three polar planes of a point on the line x = y = 0 meet in a line. The quadrics are so related if ci3= hai3+kb3i, where i = 1, 2, 3, and ci4= hai4+kbi4, where i=1, 2, 3, 4. As a point moves on the line x=y=0, the polar planes with regard to the three quadrics generate three axial pencils whose axes are rulings of the same set on D =0. Any three planes corresponding to the same point or the line meet in a ruling of the other set of D = 0. If in addition to the conditions already imposed on C = 0 we choose h = 0, then we have three general quadrics, one independent and the other two related in such a way as to give a line on 2. This means that two of the axes mentioned above coincide and that the polar planes of a point on the line with regard to B =0 and C=0 coincide. If the planes of reference are appropriately choosen the cubic surfaces of this case and the one of the preceding paragraph become identical with those where one quadric is a pair of planes so that I and the Jacobian surface are also identical. 3. Sextic curve made up of two skew lines and a quartic As in the preceding case the work is much simplified by choosing two of the quadrics pairs of planes. Then the three are A=O, B=2zw=O, C=2xy=0. The matrix is A1 A2 A3 A4 0 0 w z. y x O O 6 This particular quintic is discussed by Salmon, Geometry of Three Dimensions, Art. 352, also in Cam. and Dub. Math. Jour., vol. v, p. 39.

Page 228 228 University of California Publications in Mathematics [VOL. 1 And if we put zA3-wA4=D, xA1-yA2=E the cubic surfaces are X=xD=O, Y=yD=O, Z=zE=O, W=wE=O. The Jacobian of these surfaces is D+xD, xD2 xD3 xD4 J yD1 D+yD2 yD3 yD4. zE1 zE2 E+zE3 zE4 wE1 wE2 wE3 E+wE4 By a process similar to that used in the preceding case, it may be reduced to ^J-DED+xD1+yD2 zD3+wD4 =3DE D zD3+wD4 _ xE[+yE2 E+zE3+wE4 E E+zE3+wE4 Thus we see that the Jacobian is made up of the two quadric surfaces, D =0, E=0, and a quartic surface, while Z is made up of the two skew lines, z=y=0, z=w=0, and the quartic curve D=E=O. The first line is a ruling on E=0, is a double line on the quartic surface and meets the quartic curve in two points and has two apparent double points with it. The second line bears a similar relation to D=0, the quartic surface and the quartic curve. The two lines have one apparent double point and the quartic two making the correct total of seven. Salmon has called this quartic one of the first family. It is a single curve on each of the quadrics and the quartic of the Jacobian. To it corresponds the quartic surface, to the line x=y=O corresponds D=O and to z=w=O corresponds E=O. The same form of the sextic and the Jacobian may be obtained by choosing a special relation between the three quadrics. This relation is defined by the equations cii=haiij+ckb, where the combinations of ij are 13, 23, 33, 14, 24, 34, 44, hij=gaii, where ij= 13, 23, 14, 24, and gaij+fci,=f(haij+kbij) +bi, where ij= 11, 12, 22. The theory with this choice of the quadrics is similar to that with the general related quadrics in the preceding case. If in addition to the above conditions on the general quadrics we assume h =g =0, we then have A =0 and B =0 unrelated, but C =0 has a special relation to the other two. The polar planes of a point on x=y=O with respect to B=0 and C = 0 coincide and those of a point on z =w = 0 with respect to A = 0 and C =0 coincide. As in the preceding case, the reference planes can be so chosen that the cubic surfaces derived from the general related quadrics are the same as when two of the quadrics are pairs of planes. Hence there is no change in 2 and the Jacobian. 4. Sextic curve made up of a double line and a quartic This form of 2 is given if two of the quadrics are tangent to each other along one of their rulings. Let x = y = 0 be the line of tangency of B = 0 and C =0. The

Page 229 1920] Morris: Classification of Involutory Cubic Space Transformations 229 reference planes can be so chosen that the tetrahedron of reference is a selfpolar tetrahedron of A = 0. Then the quadrics are of the form A =x2 +y2 +z2 +w2 = 0, B = blx2 + 2b2xy + b22y2 + 2bl3xz + 2b4xw + 2b23yz + 2b24yw = 0, C = CllX2 + 2cl2xy + C22y2 + 2bl3xz + 2bl4xy + 2b23yz + 2b24yw = 0. The matrix may be simplified by subtracting the second row from the third to x y z w B1 B2 B3 B4. (c1 -bl)x+ (C12-bl2)y (c2 -b12)x+ (C22-b22)Y 0 0 Let D, H, K be defined by the following equations, D=zB4-wB3, H = (c12x - b12X + 22y - b22y)Bi - (c12y - b12y +c 1 - blx)B2, K = (C2x - b12x + C22y - b22y) x - (C2y - b2y + c11x - blx) y. Making use of these equations we have from the matrix X = { (12-b12)x+(C22-b22)y D =0, Y= (cnl-bn)x+ (c12-b12)y} D=O, Z =wH-B4K=0, W=zH-B3K=O. As in case 2, the Jacobian may be reduced to D D3 4 J=3D Z Z3 Z4 =0. W W3 W4 By the introduction of the values in this case, it becomes D B4 -B3 J=3D (wH-B4K) wH3 wH4+H =0. ( zH-B3K) zH3+H zH4 By expanding this determinant we obta'n J =D2 2K(B4H3+B3H4)-3H(zH3+wH4)-H2 = 0. } It is easy to see that x =y =0 is a double line on X=0 and Y=0, and it can be shown that the other two cubics are tangent along this line. Hence it counts as a double line of I. The remainder of 2 is the intersection of D=0 and Z=0 except the lines x=y=0, w=B4=0. Since the last two lines lie in the plane B4 = 0 they intersect, and the quartic is again of the first family with two apparent double points. The double line meets the quartic in two points and has two ap

Page 230 230 University of California Publications in Mathematics [VOL. 1 parent intersections with it, counting for four apparent double points. The double line absorbs one double point, making a total of seven. The Jacobian is made up of a double quadric and a quartic surface. The double line is on the double quadric and a double line on the quartic surface so that it has four sheets of the Jacobian through it. The quartic curve is a single curve on both parts of the Jacobian thus having three sheets through it. To the double line corresponds the double quadric, and to the quartic curve corresponds the quartic surface. This case may be derived from the preceding one by letting the two skew lines coincide, which would cause the two quadrics D=0 and E =0 to coincide. 5. Sextic curve made up of three skew lines and a twisted cubic This form of 2 is obtained by choosing for the quadrics three unrelated pairs of planes. We may choose the system of reference in such a way that their equations are A=2xy=0, B=2xw=0, C=2MN=0, where M=mlx+m2y+m3z+m4w, N =nlx-n2y+n3z+n4w. The matrix is y x 0 0 0 0 w z mlN+nlM m2N+n2M m3N+n3M m4N+n4M from which we get X=xD=0, Y=yD=0, Z=zE=0, W=wE=0, where D = z (m3N + nM) - w(m4N + n4M) = M(n z- n4w) + N(m3z-m4w), E= x(mlN+niM)- y(m2N+n2M) = M(nx-n2 y) +N(mlx-m2 y). The Jacobian surface is D+xD1 xD2 xD3 xD4 J yD1 D+yD2 yD3 yD4 zE, zE2 E+zE3 zE4 wE1 wE2 wE3 E +wE4 This determinant can be factored and reduced, as in case 3, to the form J=DE D+xD1+yD2 zD3+wD4 xE+ yE2 E+zE3+wE4

Page 231 1920] Morris: Classification of Involutory Cubic Space Transformations 231 If this second order determinant be written in full it may be factored into m3 z-m4w nz-n4W N+ -nx+rn2y N+n3z+ n4w m1x-m2y nix-n2y M+mlx+m2y M+m3z+m4w And the second factor can be reduced to mmlx+m2y m3z +m4w n1x+ n2y n3z + n4w Let these two factors be called F and G respectively. Then the Jacobian J=DEFG=O is composed of four quadric surfaces. 2 is in this case the three single skew lines x = y = 0, z = w = 0, M = N = O and the twisted cubic D = E = F = 0. The line x = y= 0 is on the three quadrics E =0, F =0, G =0, and to it corresponds the quadric D=0; the line z=w=0 is on D=0, F =O, G=O0, and to it corresponds E=O; the line M=N=0 is on D=0, E=O, G-=0, and to it corresponds F=O; and to the cubic curve corresponds the quadric G =0. The three lines each meet the cubic in two points and give one apparent double point with it. They themselves give three apparent double points and the cubic gives one making a total of seven. 6. Sextic curve made up of a double line, a single line skew to the double line and a twisted cubic The analysis of this case is made simple by choosing one general quadric, the second quadric a pair of planes and the third quadric a pair of planes with the double line a ruling on the first. The second is independent of the other two. The equations are A = a11x2 + a22y2 + 2a12xy + 2a13xz + 2a14xw + 2a23yz + 2a24yw = O0 B=2zw=0, C=2xy=0. The matrix is IA1 A2 A A4 0 O w z ly x O 0 from which we get X=xD=O, Y=yD=O, Z=zE=0, W=wE=0, where D and E are defined as in case 3. Due to the fact that A3 and A4 are functions of x, y alone, we see that zD3+wD4 = D, and the Jacobian of case 3 is J=D2E(zE3+wE4) =0.

Page 232 232 University of CaliforniaP Publications in Mathematics [VOL. 1 The Jacobian is composed of the double quadric D = 0, and the single quadrics E=0, zEs+wE4=O. S is composed of the double line x=y=0. the single line z = w =0, and the twisted cubic common to D =0 and E =0. Each line meets the cubic in two points and has one apparent intersection with it, giving three apparent double points. The two lines give two apparent double points, the cubic one and the double line absorbs one, making a total of seven. The double line is on each of the three quadrics of the Jacobian and hence has four sheets through it. * It is easy to see that the remainder of S has three sheets through it and to determine the correspondence between the parts of 2 and the parts of the Jacobian. This same transformation may be obtained by choosing two general quadrics tangent along a line and the third a pair of planes. A third method of obtaining it is to choose two general quadrics tangent along a line and the third related to these two in the same manner as was given in the second part of case 2. In either case the analysis is the same as given above. 7. Sextic curve made up of three concurrent lines and a plane cubic To obtain this form of the transformation let one of the quadrics be unrestricted and the other two be pairs of planes with their double lines intersecting. The equations are A = a11x2 + a22y2 + a33z2 + w2 + 2a12xy + 2a3xz + 2a13yz = 0, B = x2-by2=O. C=x2-cz2=0. The matrix is A1 A2 A3 u x -by 0 0 x 0 -cz O from which we get X= bcyzw = O, Y= cxzw = 0, Z= bxyw = 0, W = bcyzA i + cxzA2+bxyA3 = 0. The Jacobian is 0 zw ywyz _ zw 0 xw xz yw xw O xy W1 W2 W3 0 To factor and reduce the determinant multiply the columns in order by x, y, z, w and apply Euler's theorem. The result, with the exception of a constant, is J = xyzw2W = 0. The surface W=0 is a cubic cone with the vertex at (0, 0, 0, 1). Thus the Jacobian surface is made up of the four planes forming the tetrahedron of reference

Page 233 1920] Morris: Classification of Involutory Cubic Space Transformations 233 with the w = 0 plane as a double plane and cubic cone containing three edges of the tetrahedron. 2 is composed of the lines x = y =0, x = z =0, z = w =0, and the plane cubic W=w=0. It is easy to see that each part of / is a triple curve on J=0. Each line meets the plane cubic in one point and gives two apparent double points with it, making a total of six. Cayley remarks (op. cit. page 227, footnote 1) that he has ascertained that an actual triple point on 2 counts as an apparent double point. Since the three lines give an actual triple point, the total is raised to 7. To each of the lines, with the exception of the triple point, corresponds the plane through the other two. To the triple point corresponds the double plane and to the cubic curve the cubic cone. The same transformation is given if one of the quadrics is a general quadric and the other two are cones with a common vertex. It is evident that one of the cones may degenerate into a pair of planes. The double plane in this case is the polar with respect to the general quadric of the common vertex. This plane cuts the cones in two conics which have a selfpolar triangle. The sides of this triangle with the vertex of the cones determine the three single planes of the Jacobian surface. 8. Sextic curve made up of a single line, a double line and a plane cubic This case is obtained from the preceding case by letting two of the lines of 2 fall together. This happens if a plane of one of the two pairs of planes contains the double line of the other pair. Let A=0 and B=0 be unchanged, and let C = 2xz = 0. The matrix is A1 A2 A3 w x -by 0 0 x 0 x 0 from which we get X=bxyw=O, Y=x2w=0, Z=byzw=0, W = byxA +x2A2 -byzA3 = 0. The Jacobian surface is yw xw 0 xy 2xw 0 0 x 0= 0. 0O zw y Wi W2 W3 0 When the determinant is reduced by a process similar to that used in the preceding case, the result is J = x2w2yW = O.

Page 234 234 University of California Publications in Mathematics [VOL. 1 In this case 2 is made up of the single line x=y=0, the double line x=z=O, and the plane cubic w=W=0. The lines intersect and each meets the cubic in one point. The single line gives two apparent double points with the cubic and the double line four, and the double line itself counts for one making a total of seven. The Jacobian surface is made up of the single plane y = 0, the two double planes x2 =0, w2 = 0, and the cubic cone W= 0. The single plane is tangent to the cone along the double line of 2. The double plane x2=0 also passes through this line, so that it has four sheets of the Jacobian through it. It is easy to analyze the remainder of S and the Jacobian surface and to trace the correspondence between the parts of the two. This transformation may be obtained by taking, instead of two pairs of planes, two cones with a common vertex and a common tangent plane along one of their common elements. Also we may choose one pair of planes and a cone having the double line of the pair of planes for an element. 9. Sextic curve made up of three concurrent lines, a conic and a line in the plane of the conic This case is a special form of case 7 in which the plane cubic has broken up into a line and a conic. It is obtained by the quadrics as in case 7 with the additional requirement that A=0 be a pair of planes also. The equations are A = (nlx+n2y +n3z)2-W2= N2-W2 =0, B=x2-by2=0, C= X2-_2=0. The matrix is n1N n2N n3N -w x -by 0 0 x 0 -cz 0 from which we obtain X= bcyzw =O, Y= cxzw =0, Z= bxyw =0, W = (nlbcyz+n2cxz+n3bxy)N = 0. The reduction of the Jacobian is the same as in case 7. The result is J = xyzw2N(n +bcyz +ncxz+nbxy) = 0. The Jacobian is composed of four planes, a double plane and a quadric cone. 2 is composed of the three concurrent lines x=y=0, x=z=0, y=z=0, the line w=N=0, and the conic determined by the double plane and the cone. It is easy to see that each part of 2 is a triple line on J =0. The plane N=0 passes through the triple point formed by the three concurrent lines and through the line of A = 0. Each of the first three lines meets the conic and has one apparent double point with it and one apparent double point with the fourth line. With the triple

Page 235 1920] Morris: Classification of Involutory Cubic Space Trainsformations, 235 point this makes a total of seven. The three lines x = w = 0, y=w=0, z=w=0 are triple lines on J =0 but are not a part of 2. The correspondence between the parts of 2 and the Jacobian is the same as in case 7 except that we must consider the parts of the plane cubic. To the line in the plane of the conic corresponds the quadric cone, and to the conic corresponds the plane determined by this line and the tr'ple point. 10. Sextic curve made up of a single line meeting a double line, a conic and a line in the plane of the conic Two of the three concurrent lines of the preceding case now coincide. The analysis may be derived from case5 by choosing the plane M=O through the line x=y=O. Without loss of generality it may be taken as M=x+y=O. Then from case 5 we obtain the cubics X = x(x+y) (n3z-n4w) = 0, Y = y(x +y) (n3z-n4w) = 0, Z=Z(x+y)(nlx-n2y) +(x-y)N=0, W = w(x+y) (njx-n2y) + (x-y)N = 0. And the Jacobian is J = (x+y) (n3z-n4w)(x - y) (n3z+n4w) (x+y) (nlx-n2y) + (x-y)N } =0. In this case 2 is composed of the double line x = y = 0, the single lines z = w = 0, x+y=N=O, and the conic n3z-n4w =(x+y)(n1x-n2y) + (x-y)N = 0. The last line is skew to the first two but lies in the plane of the conic. The seven apparent double points may be accounted for as in case 8, since the only change in 2; is that the plane cubic is now a line and a conic. The Jacobian is made up of two double planes, two single planes, and a quadric cone. The double line of 2; has four sheets of the Jacobian through it for the same reason as given in case 8. The correspondence between the parts of 2; and the Jacobian may be traced as in cases 8 and 9. 11. Sextic curve made up of six lines. One line has four double points on it and a skew line to it has two triple points The double line of the pair of planes MN= 0, used in case 5, may be so situated that it meets each of the double line of the other pairs of planes. This condition exists if m = nl, m2= n2, m3= kn3, m4= kn4. With this restriction the cubics determining: are X'= x(m3z - m4w) (N + kM) = 0, Y = y(m3z - m4w) (N+kM) = 0, Z = z(mlx- m2y) (N+M) = 0, W = w(mx - m2y) (N+M) = 0.

Page 236 236 University of California Publications in Mathematics [VOL. 1 The Jacobian surface is J = (mix- m2y) (m3z - m4w) (mx +m2y) (m3z + m4w) (N+M) (N+kM) = 0. It is easy to see that 2 is made up of the six lines M=N=0, mx - m2y = m3z - m4w=0, x=y=0, z=w=0, mlx-m2y=N+kM=0, m3z-mw =M+N=O. The first and second are skew lines with one apparent double point. The third and fifth meet the second in different points, and the first in the same point forming a triple point. The fourth and sixth meet the second in separate points, and the first in the same point, which is a triple point different from the first. The third and fifth are skew to the fourth and sixth, thus accounting for four apparent double points and making a total of seven triple points and apparent double points. The six planes of the Jacobian are located as follows: four of the planes pass through M = N= 0 and are determined by the four lines which meet this line. The two double planes pass through the line skew to M = N =0 and are determined by the two triple points. Although the line M = N = 0 has four planes of the Jacobian through it and the line Mlx-m2y =m3z-m4w=O has two double planes through it they are only single lines on the sextic curve. Each of the other four lines are triple lines on the Jacobian. One of the triple points is common to the four planes of A = 0 and C =0. Its polar with respect to B = 0 is the double plane of the Jacobian which does not pass through it, i.e., m3z-m4w=0. The plane mx+m2y= 0 is the plane of the double lines of A=0 and C=0 while M+N=0 is its conjugate with respect to C=0. A similar theory holds for the other triple point. 12. Sextic curve made up of six lines which are the edges of a tetrahedron If the three quadrics have a selfpolar tetrahedron the edges of this tetrahedron are the sextic curve. If this tetrahedron is used for the tetrahedron of reference the equations of the quadrics are A = anx2 +a22y2 +a33z2+a44w2 = 0, B = b11x2+ b22y2+ b33z2+ b44w2 = 0, C = Cll2 + C22y2 + C332+ C44W2 = 0. The matrix is anx a22y a33z a44w bllx b22y b33z b44w C11X C22Y C33Z C44wI By dividing out constant factors we get X=yzw=0, Y=xrw=0, Z=XYW= 0, W=xyw=O0.

Page 237 1920] Morris: Classification of Involutory Cubic Space Transformations 237 The Jacobian is 0 z w ywyz w 0 xw yz = yw xw O xy yz xy 0 Therefore the Jacobian is made up of the four faces, each counted twice, of the selfpolar tetrahedron, and S is the six edges of the tetrahedron. Each of the six lines of 1, being the intersection of two double planes, is a fourth order line on the Jacobian. There are four triple points on I, viz., the four vertices of the tetrahedron. Each of the three pairs of opposite edges of the tetrahedron gives an apparent double point. This makes a total of seven triple points and apparent double points. The correspondence between the parts of ~ and the Jacobian is evident. A special form of this case is obtained by choosing all three of the quadrics pa'rs of planes situtated so that their double lines lie in the same plane. Where they are so related they have a selfpolar tetrahedron of which the three double lines of the pairs of planes are edges. The cubic equations of the transformation are identical with those for general quadrics having a selfpolar tetrahedron. 13. Sextic curve made up of two skew double lines and two skew single lines each meeting each double line The simplest form of the quadrics which gives this form is that in which they are all pairs of planes with the double lines of the first and second lying in separate planes of the third. Their equations are A=2xy=O, B=2zw=O, C=2(x+y)(z+w)=0. The cubics which determine 2 are X=x(x+y)(z-w) =0, Y= y(z+y)(z+w) =0. Z= z(x-y)(z+w)=O, W=w(x-y)(z+w) =0. And the Jacobian is J = (x+y) (x-y) (z+w) (z-w) =0. The sextic ~ is made up of the double lines x=y=0, z=w=0, and the single lines x+y=z-w=O, x-y=z+w=O. Each of the single lines meets each of the double lines. The single lines are skew and give one apparent double point, the double lines are skew and give four apparent double points, and each of the double lines absorbs one double point making the correct total. The Jacobian surface is made up of the four double planes determined by the four lines of I. Thus each of the four lines has four sheets of the Jacobian through it.

Page 238 238 University of California Publications in Mathematics [VOL. 1 This same transformation is obtained if any three quadrics, no matter whether they are general quadrics, pairs of planes, or cones, are so chosen that the third is tangent to the first and second along different rulings of the same set on the third. This is a special case of Cayley's transformation in which Z is made up of six lines. The four skew lines coincide in pairs making the transformation involutory. 14. Sextic curve made up of a double line and four skew lines cutting it. The two skew lines of Cayley's transformation in which 2 is made up of six lines will fall together if the quadrics have a line -in common. Although it is much easier to study this case with three pairs of planes, it is interesting to see that three general quadrics with a common line give the same form of I. For this reason the latter is given. Let the equations be A = a1lx2 + a22y2 + 2a12xy + 2a13xz + 2a14xw + 2a23yz + 2a24yw = 0, B= bllx2+ b22y2+2bl2xy+2bl3xz+2bl4xw+2b23yz+2b24yw = 0, C=2(xw+yz) =0. The matrix is A1 A2 a13x+a23y aux+a24y B1 B2 b13x+b23y b14x+b24q w z y x In order to simplify the cubics of the matrix use the following definitions: D = (a3x+ a23y) (b14+ b24y) - (a14x +a24y) (b3x+ b23y), F = A2(bl3x+ b23y) - B2(alx + a23y), G = A2(bl4x+ b24y) - B2(a14X+ a24y), H = Ai(bl3x+b23y) - B1(al3x+a23y), K = A (b14x + b24y) - B (a4X + a24y), E=A1B2-A2B1. Then the cubic surfaces are X=xF-yG+zD=O, Y=xH-yK+wD=0, Z=xE-zK+wG=O, W=yE-zH+wF=0. It is easy to see that w=y=0 is a double line on the first two cubic surfaces It is not so easy to see that the last two are tangent along this line, but this can be shown to be true. Therefore the line is a double line on the sextic curve. The remainder of the curve may be examined from another point of view. Salmon gives a formula7 for the number of points common to three surfaces absorbed by a curve common to the three surfaces. It shows that a line common to three quadric surfaces absorbs four of the eight points common to the quadrics. 7 Solid Geometry, Art. 355.

Page 239 1920] Morris: Classification of Involutory Cubic Space Transformations 239 He also statess that the sextic curve is the locus of the vertices of the cones passing through the eight points of intersection of the three quadrics.9 Since four of these points lie on the line x = y = 0, a quadric cone through all eight points must either have its vertex on this line or break up into a pair of planes. If the cone is a pair of planes one plane is determined by the line and one of the four points and the other plane is determined by the remaining three points. The common line of the pair of planes is a line of the sextic curve. It intersects the double line x = y = 0. There are four such pairs of planes and four such lines meeting the double line. The four lines are in general skew to each other, and give six apparent double points. The double line absorbs one apparent double point making a total of seven. In order to study the Jacobian surface take the simple case in which the three quadrics are pairs of planes. We need make only a slight modification of the work done in case 5 so that the quadrics are A = 2xy=O, B= 2zw=O, C= 2(x+z)N = 2(x+z)(nlx+n2y+n3z+n4w) =0. As in case 5 X=xD=O, Y=yD=0, Z=zE=O, W=wE=0, where now D = (x+z) (n3z-n4w) +zN, E= (x+z)(nlx-n2y) +xN. From these equations we find that the sextic is made up of the double line x = z = 0 and the four single lines x=y=O, z=w=O, x+z=N=0, n xx+n3z=n3(nlx+n4w) + nl(n2y+n3z) =0. The Jacobian surfaces is J = DEFG =0, where F and G are now given by F= z(nlx-n2y)-x(n3z-n4w), G=x(n3z+n4w)-z(nlx+-n2y). Each of the four quadrics of the Jacobian passes through the double line of the sextic and through three of the single lines. To each single line corresponds one of the quadrics and the double line corresponds to itself. 8 Ibid., Art. 238a. 9 See reference to Snyder & Sisam, footnote 4.

Page 240 240 University of California Publications in Mathematics [VOL. 1 RECAPITULATION 1. Sextic: a proper curve with seven apparent double points. Jacobian: a proper eighth degree surface. 2. Sextic: a line and a quintic. The line meets the quintic in two points. The quintic has four apparent double points. Jacobian: a quadric and a sextic surface. 3. Sextic: two skew lines and a quartic. Each line meets the quartic in two points. The quartic has two apparent double points. Jacobian: two quadrics and a quartic. 4. Sextic: double line and a quartic. The line meets the quartic in two points. The quartic has two apparent double points. Jacobian: a double quadric and a quartic. 5. Sextic: three skew lines and a twisted cubic. Each line meets the cubic in two points. The cubic has one apparent double point. Jacobian: four quadrics. 6. Sextic: a single line, a double line and a twisted cubic. The two lines are skew and each meets the cubic in two points The cubic has an apparent double point. Jacobian: two single quadrics and a double quadric. 7. Sextic: three concurrent lines and a plane cubic. Each line meets the cubic in one point. Jacobian: three single planes, a double plane and a cubic cone. The common point of the planes is the vertex of the cone. 8. Sextic: a single line, a double line and a plane cubic. The two lines intersect and each meets the cubic once. Jacobian: a single plane, two double planes and a cubic cone. 9. Sextic: four lines and a conic. One line lies in the plane of the conic. The other three are skew to it but meet in a point. Each of the three concurrent lines meets the conic once. Jacobian: four single planes, a double plane and a quadric cone. The four single planes pass through the vertex of the cone. 10. Sextic: two single lines, a double line and a conic. One of the single lines lies in the plane of the conic. The other meets the double line and the conic. The double line also meets the conic. Jacobian: two single planes, two double planes and a quadric cone. 11. Sextic: six lines. The first and second lines are skew. The other four meet the first in four distinct points and the second in pairs in two points. Jacobian: four single planes and two double planes. 12. Sextic: six lines which are the edges of a tetrahedron. Jacobian: four double planes which are the faces of the tetrahedron of the sextic curve. 13. Sextic: two skew single lines and two skew double lines. Each single line meets each double line. Jacobian: two double quadrics. 14. Sextic: four skew lines meeting a double line. Jacobian: four quadrics having a common line.

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 12, pp. 241-248 April 12, 1920 A SET OF FIVE POSTULATES FOR BOOLEAN ALGEBRAS IN TERMS OF THE OPERATION "EXCEPTION" BY J. S. TAYLOR INTRODUCTION There are three binary operations between classes which have come into general use in Boolean algebras. These three are "logical addition," "rejection," and "exception," and are expressed respectively by the symbols " +," "," and "-. Simple and elegant sets of postulates already exist for the logic of classes in terms of "logical addition,"' and in terms of "rejection."2 The set of postulates by B. A. Bernstein,3 however, to whom the third operation is due, is somewhat involved and it is therefore the purpose of this paper to present a comparatively simp'e set in terms of "exception." I A SET OF FIVE POSTULATES FOR BOOLEAN ALGEBRAS IN TERMS OF THE OPERATION "EXCEPTION" Let us take as undefined ideas a class K of elements a, b, c,...... and an operation "-," a-b reading "a except b." Then the logic of classes may be defined as a system 2 (K,-) which satisfies the following five postulates: Postulates I. K contains at least two distinct elements. II. If a and b are elements of K, a-b is an element of K. III. If a, b, and the combinations indicated are elements of K, a-(b-b) =a. 'E. V. HUNTINGTON, "Sets of Independent Postulates for the Algebra of Logic," Transactions of the American Mathematical Society, Vol. V (1904), pp. 288-309. 2B. A. BERNSTEIN, "A Set of Four Independent Postulates for Boolean Algebras," Transactions of the American Mathematical Society, vol. XVII, pp. 50-52. 3B. A. BERNSTEIN, "A Complete Set of Postulates for the Logic of Classes in Terms of the Operation 'Exception', and a Proof of the Independence of a Set of Postulates Due to Del Re," Univ. Calif. Publ. Math., vol. I, pp. 87-96 (May 15, 1914).

Page 242 242 University of California Publications in Mathematics [VOL. 1 IV. There exists a unique element 1 in K such that, if a, b, and the combinations indicated are elements of K, a-(l-b) =b-( -a). Definition 1. a' = 1 - a. V. If IV holds, and if a, b, c, and the combinations indicated are elements of K, a-(b-c) =[(a-b) - (a-c')]'. Definition 2. a'' = (a')'. Theorem 1. Proof. but and Theorem 2. Theorem 3. Corollary 1. Corollary 2. Theorem 4. Proof. and Theorems a =a a-(1 —1) =1-(1-a) a-(1-1)=a 1 —(1-a) = 1-a' =(a ') =a a' is unique (for any a in K). a-b' =b-a' a'-b=b'-a a-b=b' -a (a-a) = (b-b)' (a -a)' -(b -b) = (a-a)' (a-a) -(b-b) = (b-b) -(a-a) =(b-b)' by IV. by III. by Def. 1. by Def. 1. by Def. 2. by IV, II. by IV, Def. 1. by III. by Th. 3, Cor. 1. by III. Theorem 5. 1 = (e-e)'. Proof. First, (e-e)' is unique. by II; Th's 2 and 4. Secondly, (e-e)' satisfies the equation of IV, in other words, a-[(e-e)' -b] =b-[(e-e)' -a],l ~ a-[(e-e) -b]=a-[b' -(e-e)] by Th. 3, Cor. 1. for and =a-b' b-[(e-e)' -a]= b- [a' -(e-e)] =b-a a-b' =b-a but Theorem 6. a-a' =a Proof. Set b = c = a' and a = a in V. The left then becomes a-(a - a')=a The right becomes [(a-a')' -(a-a' ')]' =[(a-a' )' -(a-a)]' =[(a-a')']' =a-a Corollary. a'-a=a by III. by Th. 3, Cor. 1. by III. by Th. 3. by III. by Th. 1. by III. by Th. 1.

Page 243 1920] Taylor: A Set of Five Postulates for Boolean Algebras 243 but an( her her an( Theorem 7. a-(b-c)=[(b'-a')'-(c-a')]' by V; Th. 3; Th. 3, Cor. 2. Corollary. a -(b -c) =[(b'-a') -(c '-a')]' by Th. 1. Theorem 8. (b' -a)' -(b-a) =a Proof. Set a=a', c=b in Th. 7, which then becomes a' -(b-b) =[(b'-a ') '-(b-a ')]' t a' - (b -b) =a' by III. d [(b'-a')'-(b-a')] =[(b'-a)'-(b-a)]' by Th. 1. ace [(b'-a)' -(b-a)] =a ace [(b' -a)' -(b-a)]" =a by Th. 2. d (b' -a) -(b-a) =a by Th. 1. Corollary. (b' -a)' -(b' -a) = a Definition 3. Theorem 9. Definition 4. Theorem 10. Theorem 11. Proof. but Theorem 12. Proof. but and a b=a'-b a aa=a a' =a a a' =a (b I a) I (b' a)=a (b' -a)' -(b -a) =a (b -a) (b -a) = ( b a)(ba) - (b' a) = (b | a) ( (b' a) a' (b' I c)=[(b | a') I (c' I a')]' a -(b -c)=[(b'-a) -(c' -a')]' a -(b" -c) = a - (b' c) =a | (b' I c) =a' (b' I c) [(b'-a')'-(c -a')]'=[(b a') '-(c' a')]' = [(b | a') (c' l a')]' = [(b a') (c' I a')]' by Th. 6, Cor. and Def. 3. by Def. 4, Th. 9. by Th. 8, Cor. by Def. 3. by Def. 3; Th. 10. by Th. 7, Cor. by Def. 3. by Def. 3. by Th. 10. by Def. 3. by Def. 3. by Th. 10. Sufficiency That postulates I-V are sufficient is now evident. In the light of Definition 3 postulates I and II give P1 and P2 of B. A. Bernstein's set of four postulates in terms of "rejection" referred to in an earlier part of this paper; while P3 and P4 of that set are here exhibited as theorems 11 and 12. That postulates I-V may likewise be derived from P1-P. of Bernstein's set is also easily shown. Thus the two sets of postulates are equivalent. Consistency The consistency of the set of postulates I-V is demonstrated by the following system composed of two distinct elements el and e2 which satisfies all five postulates. As in succeeding examples, ei-ej will be given by means of a table; so that if, as

Page 244 244 University of California Publications in Mathematics [VOL. 1 in the present instance, el -e = e2, el-e2 = el, e2-el=e2, and e2-e2 = e2, this will be stated in the form: — I el e2 el e2 el e2 e2 e2 That this system 2 satisfies all the postulates the reader may verify without difficulty. Independence The independence of the five postulates is demonstrated by exhibiting five systems each satisfying all but one of the postulates, the unsatisfied postulate being I-V in turn. The system ~ failing to satisfy the ith postulate will be designated 2". ei-ej=x means that ei-ej does not give an element belonging to K. E-1; K a class of one element e1, with el-el=el. ~-2; K a class of two distinct elements el and e2, with ei-ej defined by the accompanying table: e1 e2 el el el e2 e2 x S"3; K a class of three distinct elements, with ei- e defined by the accompanying table: -| el. e2 e3 el el el e e2 e2 el e2 e3 el el el III fails for a=e3. IV holds, for e2, and e2 only, satisfies the conditions imposed on 1. V holds. It holds obviously for a, b, and c limited to el and e2, for that part of K is identical with the system used to demonstrate the consistency of the postulates with e1 and e2 simply interchanged. The other possibilities may be disposed of as follows: (1) a = e or e3, b = ei, c=ei (i,j = 1, 2, 3). el or —(ei-ej)=el, [(el or3-ei)' -(el or3-ej')]' =[e2-el]' =el (2) For a =e2 we have the following as yet undisposed of cases: e2- (e3-ei) =e2, [(e2-e3)'-(e2-ei')]' = [el-e] '=e2 e2- (el-e3) =e2, [(e2-el)'-(e2-e3')]' = [el-el] =e2 e2- (e2- e3) = el, [(e2 - e2)' - (e2 - e3 )]' [e2 - el] = el Z4; K a class of two distinct elements el and e2 with ei-e6 defined by the table: - el e2 el el el e2 e2 e2 Neither e1 nor e2 satisfies the conditions imposed on 1 by IV. V is satisfied vacuously.

Page 245 Taylor: A Set of Five Postulates for Boolean Algebras 245 275; K a class of three distinct elements with e i-e defined by the table: -- el e2 e3 el el el ei e2 e2 el e3 e3 e3 el e III is obviously satisfied. IV is satisfied, for e2 satisfies the conditions imposed on 1. V fails for a= b = c=e3. II COMPLETE EXISTENTIAL THEORY As has already been shown, postulates I-V are independent in the ordinary sense that no one of the postulates is implied by the other four. Professor E. H. Moore,4 however, has suggested the question in connection with sets of postulates of determining not only the implicational relations existing among the postulates as they stand, but also all the implicational relations which exist among properties defined either by the postulates themselves or by the negatives of the postulates. A set of postulates is said to be completely independent if, and only if, no such implicational relations exist. For example, I have shown in an earlier paper5 that while Bernstein's set of four postulates in terms of "rejection" already referred to are independent in the ordinary sense, they are not completely independent, since the negative of the first postulate implies the third and fourth. Any system 2 (K, -) of the type prescribed earlier in this paper has with respect to the five postulates there stated one of the 25= 32 characters: (1) (+++++),(++++ —),(+++-+),... (+ ----),( --- -); the ith sign of the character being plus or minus according as 2 does or does not satisfy the ith postulate. The body of thirty-two propositions stating for the various characters represented in (1) that there exists or does not exist a system having the character in question constitutes what Professor Moore has called "the complete existential theory" of the five postulates. For the five postulates in question the complete existential theory consists of fourteen propositions of existence and eighteen propositions of non-existence. The eighteen non-existencies arise from the fact that the negative of; I implies III, IV, and V, and that also the negative of IV implies V.6 4E. H. MOORE, "Introduction to a Form of General Analysis," New Haven Mathematical Colloquium, Yale University Press, p. 82. 5J. S. TAYLOR, "Complete Existential Theory of Bernstein's Set of Four Postulates for Boolean Algebras," Annals of Mathematics, Second Series, vol. XIX, No. 1, pp. 64-69 (September, 1917). 6The question naturally arises as to whether it might not be possible to modify the postulates in a way such that they would become completely independent. The writer has investigated this possibility in considerable detail but has been unable to make such a modification without a considerable loss of simplicity. The simplest change found to bring about the desired results is as follows:I'. K contains at least four distinct elements. V'. There exists an element e in K such that, for each a, b, c choice for which there exists any element in K satisfying the condition imposed upon 1 by the equation of IV for each pair of elements in the group of elements obtained by combining a, b, and c in all possible ways, the element e does so,

Page 246 246 University of California Publications in Mathematics [VOL. 1 Propositions of Non-Existence The eighteen propositions of non-existence, as implied above, may be expressed by the two following propositions:7 (2) -1 D345 (3) -43;5 The truth of these two propositions is readily perceived from the following considerations. First, the hypothesis that I is not satisfied necessitates either Knull (a class without any elements) or Ksingular (a class with only one element). But if K contains no elements, postulates III, IV, and V are satisfied vacuously. And if K contains only one element, then they are satisfied either evidently or vacuously, according as 2 does or does not satisfy postulate II. Secondly, the hypothesis that IV is not satisfied obviously results in the vacuous satisfaction of V. Propositions (2) and (3) render impossible the existence of systems with the following eighteen characters. (-+++-), (-++-+), (-+-++), (-++ —), (-+-+-), (-+ —+), (-+ ---), ( —++-), ( —+-+), ( ---++), ( —+-), ( ---+-), ( - +), ( - -), (+++ —), (++ ---), (+-+ —), (+ ----). Propositions of Existence The fourteen propositions of existence are established by the exhibition of fourteen systems having the remaining fourteen characters; there are two examples for Ksinular seven for Kd",u and five for Ktripl. In each case K contains the least number of elements possible. Examples for K singular Systems having the characters (-++++) and ( —+++) respectively are the following: System I1. Character (-++++); class composed of single element el. with el- el = el. System I2. Character ( —+++); class composed of single element el, with e- el F el. and such that, for such an a, b, c choice, if e' be defined as e-e, and if a, b, c, and the indicated combinations are elements of K, a-(b-c) =[(a-b) -(a-c')]' Since considerable space would be occupied by a proof of the fact that the postulates thus modified are completely independent and since the reader should meet with no very serious difficulty in establishing this fact for himself, such proof is here omitted. 71-i )Zjh' Ii means, "If a system Z does not satisfy the ith postulate, then it does satisfy postulates j, k,...., and n."

Page 247 1920] Taylor: A Set of Five Postulates for Boolean Algebras 247 Examples for K dual System II1 -I e e2 (+++++), el e2 el e2 e2 e2 System II3 - |el e2 (+-+++), el el ei e2 e2 x System 115 -| el e2 (+ + — +),e el e e e2 el ei System 112 -I el e2 (+++-+), el ei ei e2 I e2 e2 System II4 -- el e2 (++-+-), el el e2 e2 i el; System II6 - I el e2 (+-+-+), el j x x e2 I x System II7 - el e2 (+ ---+), 6el x el e2 I el e Examples for K triple System III1 - e e2 e3 (++++-), ei I ei ei e2 I e2 el e3 e3 e3 el e System II13 -| el e2 e3 (+-++ - -), ei i el el x e2 e2 el e3 e3 e3 el el System III2 - el e2 e3 (++-++), el el e e e2 Ie e e e2 e3 I e e1 e; System III4 - el e2 e3 (+ —++), el | el el el e2 I e2 e e2 e3 I e 1 el x System III5 -| e e2 e3 (+ —+-), el 1 el e2 x e2 I el el x e3 ei e2 x

Page 248 248 University of California Publications in Mathematics [VOL."I III THE ELEMENT 1 AND NEGATION It is interesting to note that it is impossible to express either the element 1 or negation, "not-a," directly in terms of the operation of " + " or "-," although this can be done in terms of "rejection." It has been found necessary in all sets of postulates in terms of "logical addition" or "exception," therefore, to postulate one or both of these two ideas. Curiously enough, although there are several sets in which only 1 is postulated and "not-a" then defined, there has been no set formulated in which "not-a" is postulated and the element 1 defined. This might lead one to believe that the element 1 plays a more important r6le than negation, but that this does not follow in the case of "exception," at least, is demonstrated by the following set of five postulates in which only "not-a " is postulated. I. K contains at least two distinct elements. II. If a and b are elements of K, a-b is an element of K. III. If a, b, and the combinations indicated are elements of K, a-(b-b)= a IV. For every element e in K there exists another element e' in K, unique for each e, such that, if the combinations indicated are elements of K, (1) e-e =e and (2) a-(b-c) =[(c-a) '-(b'-a')]', where each dotted element is an element satisfying (1). Definition 1. a = (a) Theorem 1. a = a Proof. Set b = c = a in IV (2) the left becomes a- (a- a) = a while the right becomes [(a-a')'(a-a ')' =[(a-a ')']' = [(a) ]' by III, by III, by IV (1), by Def. 1. by IV (1), Th. 1. Theorem 2. Theorem 3. Proof. whence a -a=a b'-a' =a-b Set a=a, b=b, c=b' in IV (2) a-(b-b')=[(b -a') '-(b'-a')]' a-b=[(b'-a')]' =b' -a a' -b=b' -a a-(b-c) =[(b -a) '-(c-a')]' (b' -a)' -(b-a) =a Set a=a', c=b in Th. 5. 1= (a - a) by IV (1), Th. 2. Corollary. Theorem 4. Theorem 5. Proof. Definition 2. by Th. 3, Cor. The rest of the development and the proof of the sufficiency of the set of postulates follow so closely those of the set for which it has already been explained in detail that the reader is left to complete the work for himself.

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS VOLUME 1, NO. 13 (Flow of Electricity in a Magnetic Field, Four Lectures, by Vito Volterra, pp. 249-320, 40 text figures. Issued May 28, 1921.) ACKNOWLEDGMENT Page 253. The method of obtaining equations (2) by the use of the electron theory is due to Professor Corbino.

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 13, pp. 249-320, 40 text figures May 28, 1921 FLOW OF ELECTRICITY IN A MAGNETIC FIELD FOUR LECTURES BY VITO VOLTERRA FIRST LECTURE Section 1-Theory of electrical phenomena. Section 2-Electrons. Section,lectron theory of metals. Section 4-The problem of flow of electricity in a e, homogeneous, isotropic metal plate kept at a constant temperature and ect to the action of a uniform magnetic field perpendicular to it. Section 5 -hematical theory in relation to the preceding problem: General formulae. ion 6-Characteristics of the problem. Section 7-Point electrodes. Section Dther general formulae. Section 9-The principle of reciprocity. SECTION 1 in historical order, the first theory of electrical phenomena may be characterized Dy the name of Coulomb. This theory assumes the real existence of particular, imponderable substances called electricity, and the instant action at a distance (activ in distans) of such substances, an action analogous to that between masses in the theory of gravitation. A new theory of electricity corresponds to the general energetical tendencies of the second half of the last century; it is characterized by the names of Faraday, Maxwell, Hertz, Heaviside. This theory does not admit the real existence of particular substances forming electricity; it denies the possibility of immediate action at a distance. According to this theory, the actions between electrified bodies are exercised by means of ether, in which the presence of these bodies determines particular deformations and perturbations; electrical actions propagate themselves in ether with the velocity of light. In the Maxwell-Hertz theory the electric charge has a purely mathematical definition; it loses the physical concrete meaning attributed to it by the old theory. Among the advantages of Maxwell's theory, must be remembered that of having connected two parts of physics, electricity and optics, which before were always considered completely distinct. Already in the last years of the nineteenth century the deeper study of already known phenomena and of new ones, led physicists to a new theory of electrical

Page 250 250 University of California Publications in Mathematics [VOL. 1 phenomena, i.e., the electron theory (1) which has been considerably developed in these first years of present century. This new theory takes from the older one the notion of the real existence of a particular substratum of electrical phenomena to which it attributes an atomistic structure, and on the other hand it holds from Maxwell's theory the principle of action by means of a medium and the general form of equations. The modifications that these undergo in the new theory, though of an important intrinsic meaning, are externally very slight. SECTION 2 We shall recall some experimental facts that led physicists to the notion of elementary electric charges, or electric corpuscles, or electrons. Helmholtz was the first to observe that the laws of electrolysis given by Faraday lead to the conclusion, that to the atomistic structure of matter corresponds an atomistic structure of electricity. In fact, according to these laws, the electric charge that is carried by a monovalent atom in electrolysis always retains the same absolute value; and the one carried by a bivalent atom always retains the double absolute value, etc. This law is very easily explained, if one considers that each atom of matter transfers in electrolysis a whole number, equal to the valency, of elementary electric charges. Researches on the electrical conductivity of gases have demonstrated in a clear and suggestive manner, the existence of atoms of electricity. (2) For instance, under the action of R6entgen rays, or of radio-active substances, gases acquire, together with conductivity, the property of condensing more easily the supersaturated vapor of water; the condensation takes place in the form of electrified drops, of which it has been possible to measure the charge. For this latter either the same value of the charge carried by a monovalent atom in electrolysis has been found or a multiple of that value (between 4 and 5X10-10 electrostatic units C.G.S.). Among the other experimental researches that led to the acceptance of the granular structure of electricity and the determination of the value of the elementary electric charge, we note those on the cathode rays, those on the emission of electrified particles from bodies brought to an incandescent state or exposed to ultra-violet radiations, those on the phenomena of radio-activity and on the Zeemann effect. It will be observed, that while in a whole series of phenomena the existence of negative, free electrons (not bound to ordinary matter) has been demonstrated, the same can not be said for positive electrons. Lastly we note that the hypothesis of bodies containing electrons of different distribution and mobility, according to the substance of which they are formed, has been able to eliminate the difficulties that arose in Maxwell's theory concerning the propagation of light in material media. SECTION 3 To explain the different phenomena, different properties have been attributed to electrons (concerning especially their degrees of freedom) according to the nature of the substances in which they are found. It will suffice to give a short explanation of the electronic theory of metals.

Page 251 Volterra: Flow of Electricity in. a Magnetic Field 251 To explain the conductivity of metals, it is assumed that their atoms emit with facility electrified corpuscles. According, then, to the electron theory, in metals there are not only, as in insulators, bound electrons, that follow the molecules or the atoms in their thermal oscillations and that only separate from them under particular actions, but also free electrons that move in the inter-molecular or inter-atomic spaces, with a movement analogous to that of the molecules of a gas. Concerning the movement of these electrons, there are two different ways of viewing the question: The electron theory of metals may be developed on the assumption of Riecke, Drude, Lorentz (3) that a free electron effects a great number of collisions with molecules or with the neutral atoms before uniting again with a charge of contrary sign; -or one may assume, as J. J. Thomson proposes (4) that the electron emitted by an atom is instantly captured by the nearest atoms. We shall retain the first point of view, which is more generally adopted. We shall thus be able to assume that in a metal an equilibrium of temperature between molecules and electrons is reached and to these latter the formulae of the ordinary kinetic theory may be applied. Let us suppose that at the absolute temperature T, the positive and negative electrons per cubic centimeter are respectively N1 and N2, U1 and U2 are the average velocities of agitation, in all directions, of the two kinds of electrons; e stands for the absolute value of the charge of a positive or negative electron. The average free path 11 of the positive electrons will be the average of the free paths traveled between two collisions, by these electrons; the average interval of time between two collisions will be represented by T1 = 1; 2, and T2 will have a similar meaning U1 for the negative electrons. Having once reached the thermal equilibrium, the average kinetic energy of all the electrons will be equal to that of a gas molecule at the same absolute temml U12 m2 U22 perature; we shall then have according to the kinetic theory U= m2 = aT. (a is a universal constant; mi, m2 the masses of the electrons of the two kinds). Let us now consider a conductor in which an electric field X parallel to the axis x has been created; let us suppose that X does not appreciably effect U1, U2,11,12. This field will communicate the accelleration - to a positive electron in the direcmi tion of the said axis. At the end of its free path, therefore, the electron will have acquired, in the direction x, an excess of velocity (above the velocity of agitation U1 eX eX1, existing even before the creation of the electric field) -T =. in the collision m1 mlU ' that ends the free path the electron loses this excess of velocity. We may, therefore, say that owing to the field X during a free path a positive electron acquires 1 eXll 1 U in the direction x an average excess of velocity W1== - eX-. The excess i2 ml\ l 4aTr of velocity that an electron acquires in a given direction when subject to a unit force having the same direction, is called the mobility of the electron. Calling V1 toe1 11 U1 the mobility of a positive electron, we shall have V1 = 2m 1 UT In the same 2m U1 4aTX

Page 252 252 University of California Publications in Mathematics [VOL. 1 1 e X 12 12U2 12 way, for a negative electron we shall have W2= - -I -eX14; V2=2 — 2 m2 U2 4aT 2m2 U2 12U2 =4aT. Under the action of the electric field, there will be added to the disorderly movement of the electrons a movement of the positive electrons in the direction of the field and of the negative electrons in the opposite direction. Because of this movement, the unit of surface perpendicular to the axis x will be crossed in the unit of time by the quantity of electricity: eN1W- eN2W2 = e2X(N1V 1+N2V2). Denoting by j the density of current we shall have: j = e2(NV1+N2V2)X. This equation expresses Ohm's law. Denoting by a the electric conductivity of the metal we obtain: a = e2(N1V +N2V2= e2ml U+ m2 N212 U ) 2\mUi m2U2 4aT"1 We will limit ourselves to these short notes on the general part of the electronic theory of metals, in order to consider some particular problems that, owing to this theory, it has been possible to solve. SECTION 4 Among the phenomena which the electronic theory has been able to explain and often to foresee, there is a whole class of phenomena that take place when a conductor is traversed by an electric or thermal current and is subject to the action of a magnetic field. These lectures will treat especially of the results of researches on conductors traversed by an electric current and subject to the action of a magnetic field. Let us consider a plane, homogeneous, isotropic, metal plate subject to the action of a uniform magnetic field H normal to it. We shall refer the points of the plate to the axes x, y, which form together with the direction H an orthogonal right-handed system. Let us denote with X, Y, the components of the electric force (parallel to the axes) of potential V; with jx,jy, the components of the density of current; with dt d- d the components of the velocities of the electrons dt' dt' dt' dt' of the two kinds (these velocities are added to those of agitation that exist also without the action of the electric field and of the magnetic field). e, N1, N2, V1, V2, will keep the meaning they had before. The components of the density of current will now be given by the formulae /(1 (d~ A d 2. dni dn2\ (1) ix = e N dt N2 d; = c N d —jN2 d-t A positive electron that moves in the direction I with the velocity V is equivalent to an element of current (having the same direction) if the product of i, intensity of current, by dl, length of the element, be equal to ev. If, therefore, the direction I form with the magnetic field H an angle 0, the electron is subject to an electromagnetic force, perpendicular to L and to H and having the direction indicated by the well known rule of the three fingers of the left hand; the intensity of this force is given by evH sin 0. If the plate be kept at a constant temperature, every electron is subject only to the electric force depending on the distribution of the potential and to the electromagnetic force produced by the movement of the electron in the magnetic field.

Page 253 1921] Volterra: Flow of Electricity in a Magnetic Fielld 253 The velocities of electrons will therefore be determined by the formulae: d=l eV(X-Hddnt1 d 2 =eV2(X-Hd2 dt -=dt dt - dt dnl pv(, dn2j d! v d \ eVlY-H dt) d-t2 eV2(Y+Hd2) dV dV In these formulae in place of X, Y, we can put- dx' dy dj1 d-1 d2 dr2 Let us solve the equations (2) with respect to d dt' dt' d' and let us substitute the expressions thus obtained in the equations (1); then we shall have: /dV _dV\ V _ KdV+XdV (3) jx= -K dx dy- ' j = Kdy+dx d where K-2 N1V1 N2V2 \ N12 N2V2 N2 1 l+ e2V12H2~ 1+e2V22H2' K e3H +e2H2V12 1 +e2H2V22} The flow of current through every closed line that forms the complete boundary of a portion of the plate free from electrodes, must be zero; we therefore have djx+djy d+I = o. dx dy We deduce this equation from (3), remembering that the potential V must (as is the case when the magnetic field is missing) satisfy the condition 2V =0. If the plate be insulated, the components of the density of current in the direction of the normal to the free boundary must be zero, likewise the component of the density of current in the direction of the normal to a line of flow. Bearing in mind (3) and indicating with t and n the tangent and the normal respectively to a line of flow, or to the insulator boundary (t, n and the direction H form a right-handed system) we deduce from the boundary conditions dV dV +- =0. dn dt Therefore, if I be a direction forming with n an angle 3, so that tg =X, (the positive direction I should be included in the right angle formed by the positive dV direction n and ) we have, d =-a. (Fig. 1). y1 Fig. 1

Page 254 254 University of California Publications in Mathematics [VOL. 1 Therefore, the condition to which V must answer in the interior of the plate, dV is the same one we should have without the magnetic field; for the condition d=, dn that we should have along the insulated boundary without the field, we must dV substitute instead the condition -d = a, if the field exist. If along parts of the boundary there are electrodes of a negligible resistance, the potential V will have to assume determined constant values along them. We may also conclude, that under the action of the magnetic field either the lines of flow, or the equipotential lines, or both the two systems of lines, are deformed so that the line of flow and the equipotential line passing through a given point of the plate, are no longer perpendicular to one another, as is the case when the magnetic field is missing, but form instead an angle - 3. The value of: depends on the nature of the metal and on the intensity of the magnetic field. For bismuth 3 can exceed 10~. Limiting our researches to the variation of potential produced by the magnetic field, i.e., to the determination of the Hall effect, it will be sufficient to consider the function v=V-Vo. (Vo represents the potential when there is no magnetic field). We have already seen that by indicating with t and n the tangent and dV dV the normal to a line of flow or to the insulator boundary, we find n= X as dn dt dVo dV dV -~ =; we shall also have =-X- -. Bearing in mind also (3) we obtain: dn dn dt K1dV j= - K(1+X2) dt dv X From this equation and from the preceding one we deduce -=K(1+X2) it Let us consider the case in which the lines of flow do not change under the action of the magnetic field; in this case, whatever the value of the field may be, the normal n to the line of flow that passes through a point of the plate, coincides with the tangent to the equipotential line, corresponding to the zero field, that passes through the same point. If s be an equipotential line, when the magnetic field does not exist and A and B be two points along it, by integrating the last equation X X along s, between A and B, we shallhave VA- VB= ( +2) fABjtds = K+X2)J in which J is the ratio between the intensity of the total current flowing in the plate and the thickness of this latter. From the last equation we see that between two points of the boundary that are at the same potential when the magnetic field is zero, we must observe, under the action of a magnetic field, a constant Hall effect, however the said points may be chosen. In the case we are considering, the last equation determines completely the Hall effect. SECTION 5 Starting from the laws of which we have been speaking, I shall develop the mathematical theory of the propagation of electric currents in a plate under the action of a magnetic field. (6).

Page 255 1921] V olterra: Flow of Electricity in a Magnetic Field 255 We shall now begin, and in the following lessons continue, the explanation of such a theory. The theoretical results have led to several experimental researches, of which we shall also speak. Let us begin by establishing some fundamental formulae, which are to be used through all the following exposition. (Fig. 2). Let s be any curve whatever, in the interior or including part or the whole of the boundary of the plate; let n be its normal so directed as to make the pair of orthogonal directions s, n, congruent in the plane to the pair x, y. y \ J y S x x Fig. 2 Fig. 3 From the formulae (3) of Section 4, it follows that the flow of electricity through ds will be expressed by: /, _ /8V -V\ -K K 6V j,ds = ( +j cos nx+j, cos ny)ds= -K -+n X ds= -ds 6 6 S C cos3 61l where I is a direction forming the angle A with n. The flow across MN, that is to say across the whole line s, will be: (4) f, jq,, d -K dds - XK(VN- VM)= s J ds, We find analogously, the other formula: (5) f. Vj,, ds= -Ki VV ds- 2K N2V V ds. For the validity of these formulae we have only to admit V to be finite all along s; and its first derivatives to be finite, or infinite of an order inferior to a number smaller than one. (Here and henceforward in defining the order of infinity of a function in a given point, we shall take as fundamental infinity the inverse of the distance to the same point.) (Fig. 3).

Page 256 256 University of California Publications in Mathematics [VOL. 1 Let us suppose that the curve s is closed (Fig. 3). Then V being single valued, since we assume that in no region of the plate electromotive forces are acting, the formulae (4) and (5) become (A) ~., ~ fdV - K 6 1V (A) fJj, ds=-K -V ds= - K 3V ds. j6 s n cos 3Js 61 B61 - K 6V (B) JfVjn ds=-K V^ ds=Cos V6- ds. 6n cos 3J, 61 ' x S n Fig. 4 SECTION 6 Let us suppose that the current enters through the portion A B of the boundary, by means of an electrode of negligible resistance, and goes out through the portion CD of the boundary, which also forms an electrode of negligible resistance, whereas the portions BC and AD are insulated. (Fig. 4). We shall have, then, along AB and CD respectively, V=C1, V =C2 (C1, C2 6V being constants); and along BC and AD =0. 61 For points of the plate in the interior of the curve a, V will be regular and harmonic. Now, if we call s the whole boundary and assume as verified the above mentioned condition about the order of infinity of the derivatives of V along s, we shall have, by reason of (B) fABVjn ds+fcDVjnd6V dsK ds J8 6n because -V-=jn C is null on B C and A D. Then 61 -K C1f ABjnds+C2fcDj ds=-KJ V6V ds. 5n

Page 257 Volterra: Flow of Electricity in a Magnetic Field 257 Let us call J the ratio between the intensity of the total current flowing in the plate and the thickness of this latter; we have: J = f cD ds = fAB nds and also (C2-C1)J=-K V-n ds and according to the well known divergence theorem in harmonic functions: (C1-C2)J = Kf, A VSa. It follows from this equation that, if C1 = C2 = 0, V must be zero at every point of a; and, consequently, that only one solution for V may correspond to given values of C1, C2. We, therefore, have the theorem: When the constant values of the potential along the electrodes AB and CD are known, the distribution of currents in the plate is fully determined. ( (I I Fig. 5 In like manner we see that if J =0, V must be constant; and therefore the distribution of current is determined also if the intensity of the total current flowing in the plate be known. The preceding theorem may easily be extended. Let ao be an area (Fig. 5) in the interior of which the harmonic function V is regular, and let s be its boundary. In consequence of (B) we shall have (if on the boundary V be finite and its derivatives be finite also or infinite of orders inferior to a number smaller than 1): V V ds = cos3 V d = cos OfJ Vda. V1 n Hence, if V be zero in some portions of the boundary and -V zero in (Fig. 5) the other portions of it, V must be zero in a; and therefore if V be known in certain aV portions of the boundary and -- be known in the other portions of it, V must be sV determined in a. If only - be known along the whole boundary, V is determined aV except for an additive constant. (A) gives us the condition with which must comply: (Al) - ds = 0. aoln~l~: (u~l~sal

Page 258 258 University of California Publioations in Mathematics [VOL. 1 SECTION 7 Let us now suppose that the current enters into the plate and comes out of it through two point electrodes A and B (Fig. 6). Let us then go back to the formula (A) and suppose at first that the curve s is the curve sa which surrounds the point A (Fig. 6). If we call J the ratio between the intensity of the current and the thickness of the plate, and suppose n to be directed externally to the area which sa encloses, we shall have: J= f sajndSa =-KJ n dSa whereas, if we call sb a curve surrounding the point B and n the normal external to the area which sb encloses, we shall have: J= f saindSa =-K - dSa sa an n n b~ Fig. 6 whereas, if we call sb a curve surrounding the point B and n the normal external to the area which sb encloses, we shall have: J= fS bjndSb=-K Sb. Jb sn If, however, we take a line sc including both A and B or excluding both, we shall have: r _v O = fsjndsc= -K } V dsc. In the above-mentioned hypothesis, according to what is done in the ordinary case when there is no magnetic field, we shall assume: (6) V= 2K(log rB- log rA)+ W W being harmonic and regular, and rA, rB, being the distances between the generic point xy and A and B respectively. On the boundary of the area of the plate the condition: sV (7) — = O

Page 259 1921] Volterra: Flow of Electricity iI a Magnetic Field 259 must be fulfilled; which is the same as the condition: 6W b/ JA (71) -= 2 (log rB-log rA)/. Thus the problem of the distribution of currents is brought back to the deter3W mination of the harmonic regular function W, the values- of which are known 61 on the boundary, being given by (71). It is easy to see that the condition (A1) /6V of Section 6, 3I ds= 0, is fulfilled. W will therefore be determined but for an arbitrary additive constant, which had evidently no influence on the law of the distribution of currents. A Fig. 7 Let us now suppose that the current enters through a point electrode A and comes out through an electrode BC, placed on the boundary and of negligible resistance; and let us give to J the usual meaning. In this case (Fig. 7) according to what we have already seen in Section 6, we have: -J V= log rA+W where W is a function harmonic and regular in the interior of a. Along the free and insulated portion BCD of the boundary we must have: sl l iL2rK log rA and along BC, W=C+ 2-K log rA where C is the constant value that V is to assume along the electrode BC. Hence, if the function W be finite and its derivatives along s be finite or infinite of an order inferior to a number smaller than 1, this function will be determined within a (see Section 6) and therefore the law of the distribution of currents will be determined also. We shall assume as is generally done, that if a current of intensity I comes out through an electrode, the fact may also be described by saying that a current of intensity -I enters through it. Let J be the ratio between the intensity of the

Page 260 260 University of California Publications in Mathematics [VOL. 1 current (positive or negative) that penetrates into the plate through the point electrode and the thickness of the plate. In the neighborhood of the electrode, the potential will be: v=-J V= 2K log r+W where r is the distance from the generic point xy to the electrode and W is a function harmonic and regular in the neighborhood of the same point. The portion of potential: -J J 1 log r log - 2 7rK 2rK r will be called potential of the electrode and will be the logarithmic potential of a point, the mass of which is 2 K' 2wK n / / Fig. 8 SECTION 8 Let us now fix some other fundamental formulae, to be added to those which we found in Section 5. Let us see what happens when we invert the sense of the magnetic field, that is to say change H into -H. In order to distinguish one case from the other, we shall say that in the former the magnetic field is direct, in the latter the magnetic field is inverted. From the expressions given in Section 4 for K and X, it follows that, when the field is inverted, K remains unchanged, whereas the sign of X is changed, which is the same as saying that the sign of the angle f is changed; consequently, instead of the direction I we must consider the direction symmetric to it with (Fig. 8) respect to the normal n, which new direction we shall call 11. If we give an index 1 to all the elements corresponding to this case, we shall have the formulae: jl,=_K( 6KVli_6Vl jly =-K( 6V1 _ V1I Jlx= aK + 1 =-KK -x,x by by 5x and also sV1 -0 all along the portions of the boundary that are free and insulated.

Page 261 1921] Volterra: Flow of Electricity in a Magnetic Field 261 Let now s be any closed line whatever, in the interior or reaching the boundary of the plate; let n be the normal to s, directed as we explained in Section 5. We shall have: Vljn- Vjln = V(jx cos nx+jy cos ny)- V(jlx cos nx+jly cos ny)= -a -K I a-+xav - aV/- l -K vv aVl -XK W = [ An + as ) an a s a -( n an ) as cos V 1 al ) and integrating along the whole line s: (C) fs(Vjn-Vj n)ds=-K (VI — V- ) ds= sVil 6v rV 6l )a J a n an/ CosJs\ 8 all which is one of the formulae that we wanted to establish. This formula may be extended. In fact, let us denote by S not merely one closed curve but a number of closed curves bounding a field a' internal to the area a of the plate. As the above formula is true for every closed curve, it will be true also if we substitute the curves S for the single curve s; hence if V and V1 be regular in the interior of the area a', we shall have, by taking n external to the field a' (D) f,(V 1j-Vn)af V1 ( - 1l) ds= f~(~V~Cjos- Val) all since according to Green's theorem, we have f(V1 a_ V d = 0. J\1 an an Let us suppose S to be formed by the two systems of closed curves S' and S"; then the formula (D) may be written also as: (D') co (V1 ( -V al)dS' + ft (V1- V6Vf)dS" =0. Cos 61 611 sI n An The formulae (D) and (D'), which we have thus found, are extensions of Green's theorem. Let us now proceed to some applications of the preceding formulae. Let us take V as given by (6) with the condition (7); let us suppose S' to be formed by the boundary s of a, and S" to consist of two circles sa and sb having their centres in A and B. Let us suppose (Fig. 9) V regular in the interior of a. The area a' will be obtained by subtracting from a the areas enclosed within sa and sb; a-' will therefore be limited by s, sa, sb. In V and V1, being regular, we may apply(D'); and aV remembering that, on s, - =0O, we shall have: -1 fVT; v1s(7 aV a V V dsa (V a V ^)d = O. Cos / Vaan V n / dsa+ a Vl

Page 262 262 University of California Publications in Mathematics [VOL. 1 But if we make the circles sa and sb grow indefinitely smaller, we see that lim - ( V ' d-V — )da= KVA; m lim V V dsb = VB F 6ln 8n ) a Vn 8n ) K where VA and VB denote the values of V at the points A and B; hence K f _ VA-V B= V 'ds. J cosf 3 8l Sa >~Sb Fig. 9 From this formula the following proposition is deduced: If the distribution of currents in a plate be known, when the current enters through A and goes out through B and the magnetic field is direct, the difference of the values of a harmonic regular function in the points A and B may be determined provided its derivatives be known at the boundary in the direction 1. It is evident that also another proposition subsists with it, i.e.: If the distribution of currents in a plate be known, when the current enters through A and goes out through B, and the magnetic field is inverted the difference of the values of a harmonic regular function in the points A and B may be determined provided its derivative be known at the boundary in the direction 1. In other words, the potential V, corresponding to the direct magnetic field and to the pair of point electrodes A and B, is a function analogous to Green's in the case when the values of the derivative of a harmonic regular function are known on the boundary in the direction 11; whereas the potential corresponding to the inverted magnetic field and to the pair of point electrodes A and B, is a function analogous to Green's in the case when the values of the derivative of a harmonic regular function are known on the boundary in the direction 1. Let us now consider the case when the current enters through a point electrode A and goes out through an electrode BC placed on the boundary and of negligible resistance (Section 7, Fig. 7); let us suppose the electrode BC to be at potential zero. Let V1 be a function harmonic and regular within a. Let us take the boundary s of the plate as S', and a circle sa having its centre in A as S". Applying the formula (D') we shall obtain: 1co V l ds- 1 V6V ds- (Vi-V VV1)dsa=0 cos I3 BCD l cos BC A isa An an and making the circle sa grow indefinitely smaller V K C O 0V _ K d J 6V ds 1A= J cos s61, J cos pBC 61

Page 263 Volterra: Flowl of Electricity i,. a Magnetic Field 263 Hence, if the distribution of currents be known when the current enters through A and goes out through the electrode BC and the magnetic field is direct, the value of a harmonic regular function in A may be determined provided its value be known along BC and the value of its derivative be known along ADC in the direction 1. The analogous proposition subsists when the magnetic field is inverted. n S a S /t3b, j b Fig. 10 SECTION 9 Let us now take V as given by (6) with the condition (7), and V as given by V1 =2 (log rB- log rA1)+ W1 where W is a function harmonic and regular within a; let Al and B1 be two new points chosen in this field (Fig. 10); along the boundary s of a let us have -0 =0. Let us suppose S' to be the boundary s and S" the system of four circles sa, sb, sal, sb1, having their respective centres in A, B, Al and B1. The area a' will be obtained by subtracting from ao the areas enclosed within the four circles (Fig. 10). In a', V and V1 are regular; then applying the (D') and remembering that on s - — W = 0, we shall obtain: J aK|V1n Vn a )dsa~J Vn l)dsb a (Vn nda1+ +~ (v1 -VbV1)dsbl = 0 Jb\ n An Now, making the four circles grow indefinitely smaller, we easily have fimj. a aVVn.J, f t 5 6V -- J._ lim an ( V a n-V dsa=RVA; lim V (Vi -VVdsb= - V1B Ja bnan n K b J, n \n K. -V 815sv,\A -Ji,. [ (.r~v ^yA, J lim f ( V ai-V V]dsal= - VA1; 1m h VI -- V dS bl= -VBI Ja\ n an K Jm an an K hence J(V1B-VlA) =Jl(VB1-VAl); and, if J and J1 be equal: V1B —VIA = VB1- VA1.

Page 264 264 University of California Publications in Mathematics [VOL. 1 From which the following law of reciprocity is deduced: If under the action of a certain magnetic field a current of given intensity passes through a conducting plate, entering through the point A and going out through the point B, and a difference of potential be obtained in two points Al, B1, the same difference will be obtained between the potentials of the points A and B when a current of the same intensity is caused to enter through A1 and to go out through B1 and the magnetic field is inverted. We shall demonstrate in the third lecture that, by the same principle of reciprocity now established, the condition of homogeneity may be omitted. AA B 1 B, Fig. 11 Let us suppose that on the boundary of the plate, there are the electrodes A1Bj, A2B2, A3B3, A4B4,...... of negligible resistance and that the other portions of the boundary are free and insulated. Let us suppose moreover that no electrode exists in the interior. Let J(1), J(2), J(3), J(4)...... be the ratios between (Fig. 11) the intensities of the currents entering through these electrodes and the thickness of the plate C(1), C(2, C(3) C(4)...... the values of the potential V along the electrodes when the magnetic field is direct. Let J1(1), J1(2), J1(3), J1(4)...... C1(), C(2), C1(3) C1(4), V1 be the corresponding quantities when the magnetic field is inverted. It is obvious that we shall have: 2kJ(k) =ZkJ1(h) =0 a result which we might also have obtained from the formula (A). Let us now apply the formula (D), taking the boundary s of the plate as S. We shall have 0 = zhfSAhBh( Vljn- Vjl,)ds = zn(C( h)J1( h)- C h)J( h)) hence ZhC( h)J( h) = h0C( h)J( h) If the Jh's are all zero, except J(1 and J(2); and the Jl(h) be also all zero except J1(3) and J1(4), we shall have: J(1) = J(2) = I J1(3) = J1(4) = I, hence: (C() - C(2))J= (C(3) -C(4))J1 and if J = J1, C, (1 - C1(2) = C-(4) These formulae express new theorems of reciprocity, the interpretation of which is easy.

Page 265 Volterra: Flow of Electricity in a Magnetic Field 265 Let us finally suppose that, the magnetic field being direct, there are besides the electrodes Al, B1, A2, B2..... the boundary, some internal point electrodes M (1, M(, M......; let us call J(1), J(2), J(3), the ratios between the intensities of the currents which these electrodes admit into the plate and the thickness of this latter. And let us give an analogous meaning to M1('), M1(2), M1(3)....... J, J1(2), J1(3, in the case of the inverted magnetic field. We shall have then: z2hJ h) +ZkI(k) = 2hJi( h) +fklIl(lkl) = 0 and ZhC ( h)J(h) +ZkVl ()( k ) = ZhC( h)J1( h) +Zkl V(kl)Il(k) where V1(k) is the value of the potential V1 in the point M(k) and V(kl) the value of the potential V in the point M(kl). From the last formula theorems of reciprocity analogous to that already stated may be likewise deduced. SECOND LECTURE Section 1-The fundamental function. Section 2-Circular plate: Two principles of images. Section 3-The principle of point electrodes at the boundary. Section 4-Cyclic plates. Section 5-The problem of linear electrodes of a negligible resistance, situated at the boundary. SECTION 1 Let us denote by V' the conjugate function of V: then we have 5V aV' 5V aV' =7 bV1. =V=-V x by ' 8y x ' The equations (3) of Section 4 of the preceding lecture will then become -K(V+XV') (3)' o= -K-6(+V; j- =-K -6(V+V-) If we put V+XV'= U, we shall have (3)" jx -K u K -KU j x' jy= y. The function U will satisfy first, the condition A2 U= 0 at every point of the plate eU and second, the condition = 0 at every point of the insulated boundary. Therefore the distribution of current in the plate happens as if there were no magnetic field, but as if the potential were U instead of V, and the conductivity were K. As K for a given value of the field is constant the lines of flow are independent of K. We shall call U the fundamental function of the distribution of current in the plate or, more simply, the fundamental function. U coincides with the potential only

Page 266 266 University of California Publicatioqs in Mathematics [VOL. I if the magnetic field be null. When we know the potential V we can have U by the operation (E) ~ U= V +;= X -V cos O+ V' sin f (E) U=V+xV = cos 3 Now let us resolve the problem of the calculation of the potential when the fundamental function is known. Let U' be the conjugate function of U; we shall have (8) U' = V' - XV. Then, from the formula (E), we have U-XV (E)' V= - = (U cos 0- U' sin f)cos f. 1; )X2 S~ C Fig. 12 Fig. 13 The proposed problem is thus solved. The preceding formulae easily provide us with the condition which the fundamental function must fulfill all along the edge of the electrodes of negligible resistance. Let s be a portion of one of these edges. At every point of s we shall have V = (U cos - U' sin,8)cos j = constant V asU bUl b) aU s )i 0= - cos j- sin fco os( cos +n sin )cos. as as as )6n Assuming then 3 <, the condition will be 2' sU sU. SU, cos + -sin = =0; as an 5/ A is a line which forms with the tangent t and with the normal n to the arc s the same angles which 1 forms respectively with n and with t. (Fig. 12). From the preceding consideration it follows that in the case of the homogeneous plate placed in a uniform magnetic field the distribution of currents depends on four harmonic functions V, V' U, U' connected by the relations previously established. The lines V= const. give us the equipotential lines, the lines U'= const. U' give us the lines of flow; therefore the function K may be considered as the function of the currents. We can easily relate among themselves the functions U, U', V, V' by means of functions of a complex variable.

Page 267 Volterra: Flow of Electricity in a Magnetic Field 267 Indeed, if we put x+iy=z, U+iU'=f(z), V+iV'= o(z) we shall have (E)f" Af(z)= os P(z). cos 3 The conjugate function of log rAis 0A, where we denote by 06 the angle that the radius vector which has its origin in A, forms with a fixed direction; 7 GA is the 2 r potential of a vortex situated in A and having the moment m. (We suppose that the axes x, y, are situated as in figure 2 of Section 5 of the first lecture and that the angle OA is measured in the sense of the rotation by which the positive direction x may reach the positive direction y; also we assume that a vortex having a positive moment rotates in the sense opposite to that just mentioned). Then, if we have in A a point electrode through which the current I enters, putting I= vJ(v thickness -J of the plate), the corresponding potential will be 2rK log rA and the corresponding - J fl,,, J I XJ fundamental function will be 27 log rA+OA = log - 2KA (log AA, = 2 K logrA 2 Therefore we can say that the fundamental function corresponding to a point electrode A, through which the current of intensity I enters, is the logarithmic potential J XJ of a mass 2 and of a vortex having the moment -, both situated in the point A. This result may be stated by saying that the action of the magnetic field on the distribution of currents consists in modifying the conductivity, which becomes equal to K, and in modifying the fundamental function by adding to each point electrode, through which the current of intensity I enters, a vortex having the XJ moment -; the normal derivative of the fundamental function remains zero K at every point of the insulated edge of the plate. (The addition of the vortex by the action of the magnetic field appears quite natural if we think of the deflection of the movement of electrons, in the neighbourhood of the electrode, produced by the same field). The function OA is many valued and therefore the fundamental function U is also many valued; U has the electrodes as branch points. Thus, the potential is a single valued function becoming logarithmically infinite at the point electrodes, whereas the fundamental function has the same points as points of logarithmic 'infinity and as branch points. SECTION 2 Now, let us suppose that the plate is circular and that the current enters and goes out through internal point electrodes. We shall calculate the effect of the boundary C (that consists in annulling the normal derivative of the fundamental function), by adding masses and vortices situated in the inverse points (images) of the electrodes, with respect to the circumference C. (Fig. 13). J Let A1 be the inverse point of the electrode A. If we put in Al the mass 2wK the logarithmic potential of the two masses 2K situated in A and A1 will have Z7rK

Page 268 268 University of California Publications in Mathematics [VOL. 1 along C the normal derivative equal to - KR' R being the radius of C; and, if we 27rKR ' -xJ -xJ put in Al a vortex having the moment K the potential of the two vortices K XJ and - situated in A1 and A will have the normal derivative zero at the boundary. Bearing in mind that the algebraic sum of the intensities of the currents which enter through the various internal electrodes is zero, we shall obtain for the fundamental function the following expression: (9) U =-2K { log rA+log rAl +XOA-XOA1{; 27rK the sum being extended to all the internal electrodes. The same expression can be written: U= - 2 (log rA+XOA) + o where s = - ~ K2(log rAl- XOA1): evidently p is regular within the area occupied by the plate, because all the points where it becomes infinite and the branch points are external of this area. The result now obtained can be thus expressed: The distribution of the currents, which enter and go out through point electrodes in a circular plate subject to a magnetic field, is identical with the distribution that we should have if the plate were indefinitely extended, if the magnetic field were suppressed, and if the following xJ conditions were fulfilled: the addition of a vortex having the moment K to each electrode through which the current I= vJ enters (v thickness of the plate); the addition in the inverse point of each internal electrode, of an image electrode traversed by the same current I and of a vortex, the image of the internal vortex -xJ and having the moment K The formula (9) gives us the fundamental function of the distribution of currents; and from this, bearing in mind the rule (E'), we can obtain the expression of the potential. The conjugate function of U is U'= -2 K(OA + A1-X log rA +X log rAl); therefore we shall have (10) X2 = U -'(log rA+ log rAl- 1 2X A) (10)- - = 109o rA- X2 log rA-l 9A1

Page 269 Volterra: Flow of Electricity in a Magnetic Field 269 Now, as X=log / we have 1 -X2 2X - = cos 21 - 2= sin 2 hence (10') V= -- (log rA+cos 2/ log rAl-sin 2/0A1). 2rK This formula can be written in the following manner: (10") V=-2; log rA + 2irK where (10a) S= - 2 (cos 2/. log rAl-sin 213. OA1) 2 rK The first term of (10") is the potential of the electrodes; the second term sp is a function regular within the area occupied by the plate, because the infinities and branch points are external. The result now obtained can be stated in this way: If in a circular plate, subject to a magnetic field, the currents enter and go out through point electrodes, we shall obtain the potential by adding to that of each electrode where the intensity of current is I= vJ the potential of an image electrode, situated in the inverse point and traversed by the current I cos 23, and the potential of a vortex situated in the same inverse point -J sin 2/ and having the moment - J n 2 K' Thus, in the case in which the plate is subject to the magnetic field, we have two different principles of images; one of these concerns the distribution of currents and therefore the fundamental function, the other concerns the electric potential. The same results can be easily obtained by using the functions of complex variables introduced in section 1. Let us suppose that a and a' are the values of the complex variable z=x+iy in two points inverse to each other with respect to a circle; that M and m are real constants and that M M= 0; then along the circumference that limits the said circle the real part of the function (11) eim2M log (z - a) - e -imM log (z - a') and the imaginary part of the function (11') eimfM log (z-a) +e-imfM log (z-a') are constant. Now, if we denote with a the values of the complex variable z corresponding to the point electrodes, the potential of these latter is the real part of the function =-2 r log (z-a); according to the formula (E") the corresponding fundamental function will be -e- r J the real part of f-= c: log (z-a). cos / 27KK

Page 270 270 University of California Publications in Mathematics [VOL. 1 According to what we have said before, since along the circular boundary of the plate U' must be constant (see section 1), the fundamental function U will be the real part of F= —2rK cos [e-i log (z-a)+e-iO log (z-a')]; and, owing to the formula (E"), the electric potential will be the real part of = — K[lOg (z-a)+-ei log(z-a')] From the two last formulae we can easily deduce the formulae (9) and (10). Hitherto we have supposed the electrodes to be internal, now let us suppose that they are at the boundary. It will be sufficient to suppose, in the formulae (9) and (10), that the points A and Al coincide, that is it will be sufficient to put log rA = log rAll OA = OA. Therefore we shall have U=-J log rA; V -— K(COS f log rA-sin f OA)cos. IrK rK These formulae show us that, if the point electrodes be at the boundary, the magnetic field does not alter the distribution of currents, but alters the electric potential. In the case of two electrodes A and B the formulae (9) and (10) become J [logr BrB 2K rArA — ] V=2-K - log rBrm J sin( sin +AIB cos t 27-K rTATA1 rK -i Alo i where Q1, Q2,.are the angles under which the points A, B, and Al, B1 are seen from a generical point x y. If the two electrodes are on the boundary, we have J rB J rB U=K log -, V= -K (cos f. log — sin /fAB) cos f. rnK rA rK \rrA We have thus completely solved the problem of the flow of electricity under the action of the magnetic field, in the case of a circular plate; therefore we are able to solve the same problem in all the cases in which the area occupied by the plate can be conformally represented in a circle. From this result we can immediately deduce a consequence; in fact, if we know the law of the distribution of currents in an acyclic plate, provided only with point electrodes and not subject to the magnetic field, we are able to make the conformal representation of the area occupied by the plate in a circle, and therefore we are also able to determine the distribution of currents and the potential when the currents enter and go out through any point electrodes and the plate is subject to the magnetic field.

Page 271 Vollterra: Flow of Electricity in a Magnetic Field 271 SECTION 3 Let us consider the case in which the point electrodes are on the boundary, then, provided the plate be acyclic whatever form it may have, the distribution of currents is not altered by the action of the magnetic field. This result, obtained in the case of the circle (Section 2) and extended to the case of a generical acyclic field by means of the conformal representation, can be also deduced from the conditions that U must fulfill. (Section 1) In fact, if the plate be acyclic and the point electrodes are on the boundary, U is a single valued function, at the boundary n =0, in the neighborhood of an electrode U must be the sum of a logarithmic term and of a regular part, and lastly if, by means of an arc of any curve, we separate from the plate the region occupied by the electrode, we must have,s U dz =- (where J is the ratio between the intensity of the current which flows through the electrode and the thickness of the plate). Then, if the intensity of the total current be not altered, U can differ from the electric potential corresponding to the case in which the magnetic field is zero, only by a constant factor of proportionality, equal to the ratio. between the conductivity of the plate before the action of the magnetic field, and K We will call this proposition the principle of the point electrodes at the boundary. Evidently, if the plate be not acyclic, we cannot represent its area conformally in a circle and we cannot say that U must be a single valued function; therefore the demonstration which we have given does not hold good in this case, and in fact, as we shall see in the following paragraph, the preceding proposition generally is not true, when the plate is not acyclic. From what we have previously said, we can deduce that, if we know the law of the distribution of currents in a generical acyclic plate, provided with an arbitrary number of point electrodes situated at the boundary when the magnetic field does not exist, we have immediately U and, by applying the formula (E') of Section 1, we can calculate the electric potential V, when the magnetic field exists. For instance, if a rectangular plate be not subject to a magnetic field, we can express (according to Betti's calculations) (1) the distribution of currents when the electrodes are in the middle points of two opposite sides, by means of the elliptic function Aam; then, by applying the preceding results we are able to solve the analogous problem, when the plate is subject to a magnetic field. Let us consider an acyclic plate traversed by currents of given intensities entering and going out through point electrodes situated at the boundary, let us suppose that these intensities remain constant before and after the creation of the magnetic field. Let us denote respectively with u and with U the fundamental function when the magnetic field does not exist and when it exists; then we have U=su, where s is the ratio between the conductivity of the plate, (without the magnetic field) and K. Let us denote with u' the conjugate function of the harmonic function u; by applying the formula (E') we shall have: V=. 1+X2

Page 272 272 University of California Publications in Mathematics [VOL. 1 Without the magnetic field, let MN and QP be lines of flow, MQ and NP. \Fig. 14 equipotential lines. We shall have UM = U2, U U Upl UM' = U2= Up' and hence VM= VN+VP-V2=O we can then state the following theorem: When the point electrodes are on the boundary, if we consider a quadrangle formed by lines of flow and equipotential lines corresponding to the value zero of the magnetic field, and if we determine the values of the electric potential in the four vertices when the magnetic field exists, the difference of the values in two adjacent vertices is equal to the difference in the other two vertices. This proposition, which we will call the theorem of the four vertices, can be easily verified experimentally. Let us consider the particular case of a circular plate in which the currents enter and go out through the point electrodes A and B situated at the boundary. Let M N ' Fig. 15 MN and QP be circles passing through A and B; MQ and NP circles orthogonal to the preceding. Whatever the intensity of the magnetic field may be for the quadrangle formed by these circles we have VM-VN+ VP-V2 = SECTION 4 Various results among those that we have considered, hold good in both the cases of an acyclic and of a cyclic plate; others hold good only in the case of an acyclic plate (we have explicitly declared this, when we stated those results).

Page 273 1921] Volterra: Flow of Electricity in a Magnetic Field 273 The formulae and the theorems of reciprocity, stated in the last paragraph of the first lecture, evidently hold good also in the case of a cyclic plate; but in this case they may assume a different aspect, because the portions of the boundary where the potential is constant may be constituted by some of the whole closed curves which form the total boundary of the cyclic area occupied by the plate. Let us suppose the plate to be bounded by the closed curves s/, 2',......, s,' and Si", S2",..... s, "; let us assume the first contours to be insulated, the second kept at a constant potential. We shall call S' the system of all the contours sr', S" the system of all the contours, st"; we shall exclude, for the sake of simplicity, the existence of point electrodes. With the direct magnetic field let C(), C(2,...... C(m be the values of the potential V along si", s2",....... Sm"; with the inverted magnetic field let C(1i), C1(2)...... Cl(m) be the values of the potential V1, along the same curves. Let us denote by J(1), J(2),....... J(m) the ratios between the intensities of the currents which enter, through the lines si("), S2",....... Sm", in the plate and the thickness of this latter; let J(1), J1(2),......, Jl(m) have the same meaning when the magnetic field is inverted. By applying the results of Section 9 of the first lecture, we shall have: ZkJ( k)C l( k) = kJl(k)C(k) If we consider Si", s2,..., Sm" as edges of electrodes of negligible resistance, we can easily deduce, from the preceding formula, the principles of reciprocity (analogous to those stated in the first lecture) in the case in which the currents enter into the plate through internal electrodes having a finite area and a negligible resistance. Now let us consider the case of the direct magnetic field and let us apply the formula (E) (Section 1) in order to pass from the potential V to the fundamental function U. We shall have: (12) U=V+xV'. If we draw a closed contour s in the area occupied by the plate and denote by Sat sa/"..... Ss " the curves s" within s, we have FsV' j )v 1 [ ds=- V ds= I-(J(a)+J(3)+...... +J( ); as jsn K therefore we can say that V' is a many valued function, that the closed curves J(a) enclosing curves s" are the cycles, and that the numbers - are the cyclic constants. As V is a single valued function, (see first lecture, Section 5) U will have the same sort of multiplicity as V', except that the cyclic constants will be changed in the ratio X. By applying the formulae (A) and (B) obtained in the Section 5 of the first lecture, we deduce (see first lecture, Section 6) ZkJ(h) =0 ZkJ(k)C(k) = Kf AVdoa. From the last equation we deduce that if J(k) be zero along some of the lines Slh and Cl(h) be zero along the remaining lines S"k, V must be zero; and if all the J's be zero, V must be constant; We can therefore state that the knowledge of

Page 274 274 University of California Publications in Mathematics [VOL. 1 J(h) determines V except for a constant and the knowledge of some values of J(h) and of the remaining values of C(h) determines V completely. Let us suppose that in the plate there are no insulated edges s'h and that all values of C h) are known; then V will be determined and independent of K. Whereas if all values of J(h) be known, V will vary inversely as K. Lastly if C(h) be known for some of the lines s"h and J(k) be known for the other lines s"h, V will depend on K. From what we have said we can deduce that if each of the various curves which form the boundary has a constant and unalterable potential, the electric potential V will not depend on the magnetic field, but this latter will alter the fundamental function U and therefore also the distribution of the currents. It follows that, in this case, if we know the distribution of currents without the magnetic field, and therefore the potential V (with or without the field), we shall be able to calculate, by means of the formula (E), the alteration of the distribution of the currents caused by the magnetic field. Likewise we shall have similar results by supposing known and inalterable the Jh These results agree perfectly with those obtained by Professor Corbino in the case of a plate bounded by two closed curves kept at constant potentials.(2) Let us consider the particular case in which the two closed curves are concentric circumferences having the radii R1 and R2. Then we shall have V (C(1)-C(2))log r+C(2) log R1-C()C log R2) (C(-C(2)) log R1-log R2 log Ri-log R2 U (C(') -C(2(log r+X0)+C log R1-log R2 where r and 0 are the polar coordinates of the points of the circular ring which constitutes the plate (having taken as origin the centre of the ring) and C is a constant. If the current enters through the external circle of radius R1 and its intensity be I= vJ (v thickness of the plate) we shall have V = log r, V1= U= (log r+X). 2 rK 2 U 2l rK ) The results obtained at the beginning of this paragraph can be extended. Let us suppose we have a cyclic plate bounded by the closed curves sl, S2...... SnWithout making any hypothesis about the values of the potential corresponding to these curves, or the way in which point or linear electrodes are situated along them, we shall indicate with J1, J2,...... Jn the ratios between the intensities of the currents which across the said curves penetrate into the plate and the thickness of this latter. Let M1, M2,...... Mm be internal point electrodes; let I1, I2......, Im be the ratios between the intensities of the currents which enter through these n m electrodes and the thickness of the plate. We shall have 2kJh+2mKIK=O, and, 1 1 if a closed line s within the plate enclose the lines sa, sg,...... sT and the points MA, MB,......,Mt f ds =- ds= (J+Ja+...... +J, +Ia+Ib+...... +It)

Page 275 1921] Volterra: Flow of Electricity in, a Magnetic Field 275 This proves that V', and therefore U, is a many-valued function, unless J1, J2,...... Jn, Il, I2,...... Im be all zero. We can then enunciate the theorem: The necessary and sufficient condition that the fundamental function be uniform is that no internal electrodes exist and that the total quantity of electricity which enters through the electrodes distributed along each closed curve making part of the boundary be zero. Whereas, in the case in which the area occupied by the plate is acyclic, it is sufficient, for the uniformity of the fundamental function, that no internal point electrodes exist; while owing to the presence of these electrodes the fundamental function becomes many-valued. Let us make an application of the results now obtained. Let us suppose that the area is cyclic and that the electrodes are punctiform and distributed along the curves Sl, S2,......, Sn which form the boundary. Let ip(1), i(2)..... (hs) be the ratios between the intensities of the currents which enter through the electrodes distributed along the curve Sp and the thickness of the plate; then the necessary and sufficient condition, in order that the fundamental function be singlep valued, is that tip(t)= 0, (p = 1, 21....n). Now if the fundamental function be 1 single-valued, by repeating a reasoning made at the beginning of the preceding paragraph, we shall be able to demonstrate that the magnetic field does not alter the distribution of currents, while, if the same function be many valued, we shall see that the action of the magnetic field will have to change the distribution of currents, because without the field the fundamental function coincides with the potential and therefore is single-valued. We shall have thus the following theorem: If the plate be cyclic and all the electrodes be point electrodes and situated on the boundary, the action of the magnetic field changes the distribution of currents unless the sum of the intensities of the currents, which penetrate through the electrodes placed on each closed curve that forms part of the boundary, is zero. When the preceding condition is fulfilled and one knows the law of the distribution of currents without the magnetic field, one can, by means of the formula (E'), calculate the potential corresponding to a given magnetic field and therefore one can completely solve the problem. It remains to consider the case in which the above mentioned condition is not fulfilled. Let us denote by s1 the closed curve which forms the outer boundary of the cyclic plate, with s2, 3s,...... Sn the closed curves that form the inner boundaries. Let us trace a curve I which joins st with Sh without meeting other parts of the boundary and let us imagine that along I there is an electromotive force having the value I and directed in the sense in which the arc Sh increases. Supposing that the plate is not subject to the magnetic field, the electric potential ph will be a regular and harmonic function within the area occupied by the plate and will have a discontinuity I along 1. Moreover we shall have A6nh 'Ph =' h. dsh — f. dSsg =, 1 g 7 h. 6n 6n2... Js'h s

Page 276 276 University of California Publications in Mathematics [VOL. 1 We can also consider Ph as a finite, continuous and many-valued function, in the area a when there is no cut l; the cycles of this function enclose Sh and its cyclic constant is -1. The functions 02, ps3,...... will only depend on the geometrical form of the field a, and we will call them the elementary cyclic potentials of a. We will indicate by p2', (P3,....... the conjugate functions of (p2, (p3- (3). Fig. 16 Now let us suppose that every electromotive force in the interior of the plate is taken away; let us call W the electric potential without the magnetic field (but reduced in the ratio between the conductivity of the plate before the action of the same field and K), when the currents enter and go out through point electrodes disposed along the various closed contours which form the boundary; let us call V the electric potential with the magnetic field, U the fundamental function. Let Ih be the algebraic sum of the intensities of the currents which enter and go out through the electrodes distributed along sh; let us put Ih= vJh (v thickness of the plate). If Sh be a line which encloses only Sh we shall have JhK | dSh =K f - dSh =-K dSh = - K dSh S JNh Sh3Nh JShh J ShSh where V' and W' are the conjugate functions of V and W. Hence follows (see formula (E)) f 5U - -(_+x__) _ XJh j UdSh= | -+- V) dSh =Sh Sh iSh 6Sh K We can therefore take U= W+-K hJh(Ph K2 (3) The elementary cyclic potentials are considered in classical hydrodynamics in order to obtain the irrotational motions of a fluid in a cyclic space bounded by rigid walls. They correspond in the theory of elasticity to the distortions. Let 2, *k3,...., kn be regular harmonic functions; let each of these functions,h be zero along all the lines sl, s2,...., S excepted, Sh where *' k has the value 1. If we know,2,,J3,....,,in we shall be able to obtain (by combining them linearly with constant coefficients) the function ('2 and then we shall be able to calculate the functions <PhThe knowledge of the <Ph or of the,h is therefore analytically equivalent.

Page 277 Volterra: Flow of Electricity in a Magnetic Field 277 because the second member of this equation satisfies all the conditions which U must fulfill. In order to obtain W we have only to apply the rule (E'); we shall obtain thereby w -hJhfh- W -shJh (h) V K2 1+X2 Therefore if one know the elementary cyclic potentials of the cyclic area occupied by the plate, one can determine the perturbation produced by the magnetic field on the electric currents whatever they may be, provided they enter and go out through point electrodes situated on the boundary. The two last formulae express the principle of point electrodes on the boundary modified for the case of the cyclic plates (see the preceding paragraph). Let us consider the particular case in which the plate is a ring bounded by two concentric circles. In this case the elementary cyclic potential is, where 0 2w' Fig. 17 \A AI \ A, Fig. 18

Page 278 278 University of California Publications in Mathematics [VOL. 1 is the angle which the vector radius, having its origin in the center of the plate, forms with a fixed direction; the conjugate function of this elementary cyclic potenlog r tial is 2 27r' Therefore the two last formulae become W-2 J2 log r-X (W'+- 0) XTW A 7 2, 7rK.v 27rK U=W-2KJ20, V==- 2 If the two circles be not concentric we take the points A and Al, images of each other with respect to the two circles contemporaneously, and the elementary cyclic potential will be 7 where 0 is the supplement of the angle under which 2ir from the generical point B of the plate one sees the segment A Al; the conjugate 1 AB function will be I log. AB 2r AIB' SECTION 5 We shall now pass to the solution of the problem in the case in which the electrodes, supposed to be of negligible resistance, constitute portions of the boundary. Therefore let us go back to the conditions examined in Section 6 of the first lecture. Let us suppose what we have been able to make the conformal representation of the acyclic area a in a parallelogram abcd of the plane 4y in such a way that A r bad=- -. Let us suppose also that the sides ab and cd are parallel to the axis 2 y and that db and AB, be and BC, cd and CD, da and DA correspond respectively to each other. Let us take the function V=MS+N where M and N are two constants, and let us consider V as function of 4 and y. 8V V will be constant along the sides ab and cd and will satisfy the condition =0 67 Ua a p c Fig. 19

Page 279 1921] Volterra: Flow of Electricity in a Magnetic Field 279 along be and ad. It is easy to see that along be and ad the directions y and - form respectively the angle A with the external normals n to the same sides. Let us now consider 5 as function of x and y and let us refer the function V to points of the area a in the plane xy. V will result harmonic and regular, will be constant along the portions AB and CD of the boundary and along the portions sV BC and AD will satisfy the condition =0. 81 Making use of the arbitrariness of the constants M and N, we shall be able to reduce the values of V along AB and CD to the given values, and therefore V will be the required potential. It is easy to recognize the order of infinity of the derivatives of 5 (with respect to x and y) in the angular points of the boundary. Let us now suppose ao to be a square. Let us begin by taking on the real axis of the complex plane z two points a and -a and let us put (13) Z= foZ(a2-z2)v-ldz (13') Z1 = f Oz(a2 - 2)- 1dz While z moves in the half plane corresponding to the positive coefficient of the imaginary part of z, Z and Z1 move respectively in the two isosceles triangles ABC, A1B1C1 the angles at the base of which have respectively the amplitudes V7r and u7r. Therefore, applying the principle of symmetry, we see that, while -Io -a +a +oO C A T r B D Cl Fig. 20 z moves in the whole plane, sectioned by the two cuts -a-oo and +a+ o,Z and Z1 move respectively in the two rhombs ABCD and A1B1C1D1. Taking 1 h the first rhomb becomes a square, and taking =- the second rhomb becomes 7 2 a parallelogram having an angle equal to 2-f. By means of the rotation Z1'= Zie (-) 7 one transforms the second rhomb so that it has a coup'e of sides parallel to an axis, that is one reduces this rhomb to be in the condition of the parallelogram represented in the figure 19. The conformal representation of the square in the parallelogram which we have thus obtained solves the problem of

Page 280 280 University of California Publications in Mathematics [VOL. 1 determining the potential and the distribution of currents in a square plate subject to a magnetic field, when two opposite sides of the plate are electrodes of negligible resistance through which the current enters and goes out, and the other two sides are free and insulated. It is obvious that, having taken v=,, the integral (13) is elliptic, and in fact, if we put a2 z2=X4 and we take a= 1, we have z f | dz 2fx dx (a_ r2) 4 1 - x4 In the first paragraph of the following lecture we shall study the case of linear electrodes of negligible resistance, situated at the boundary of a circular plate. THIRD LECTURE Section 1-The problem of the flow of electricity, under the action of magnetic field, in a circular plate, provided with a linear electrode of negligible resistance, situated on the boundary and with point electrodes. Section 2-The problem of the flow of electricity in a curved and non-homogeneous plate subject to the action of a non-uniform magnetic field; general formulae and results. Section 3-Comparison between the problem relative to the plane homogeneous plate and uniform magnetic field and the problem relative to the curved and non-homogeneous plate and the non-uniform field. Section 4.-The theorem of reciprocity in the case of the curved and non-homogeneous plate subject to the action of a non-uniform magnetic field. SECTION 1 Let us consider a circular plate in which the current enters through an internal point electrode and goes out through a linear electrode, of negligible resistance, situated along the arc AB of the boundary. (See Fig. 21). In this case, for studying the flow of electricity under the action of the magnetic field, it will be sufficient to determine a regular, harmonic function W, its value 6W along the arc BC and the value of 5, along the arc CDB being known. (See Section 7 of the first lecture). Let us represent conformally the circle in an angle of the plane ~y having the amplitude 2 - (Fig. 21), in such a way that the side be corresponds to the arc BDC, the vertex b corresponds to B and the point at infinity C- to the point C. Let us transfer to the angular figure the values of 6W W; then- a will be known on the two sides of the angle and, as it is harmonic, we 1Betti, Opere, vol. II, p. 267. 2Corbino, Nuovo Cimento, 1911, vol. I, p. 397.

Page 281 Volterra: Flow of Electricity in a Magnetic Field 281 shall be able to determine its values within the angle and therefore we shall be also able to calculate W. Evidently, we should have the same result if a side of the angle were parallel to and corresponded to the arc CC and the other inclined side corresponded to 6W 3W the arc BDC, and if we considered - instead of -. B Fig. 21 But, in the case that we are considering, it is useful to employ the functions of a complex variable which we introduced in Section 1 of the preceding lecture. n-2 The case in which = -- is particularly interesting, because in this case, by a convenient conformal representation and by the application of the two principles of images (see Section 2 of the preceding lecture) we can obtain the solution more simply. By means of the function io io ze-2-e2 Z= Z-1 we can represent the area within a circle situated in the plane z, having the centre in the origin and the radius equal to 1, (Fig. 22), in the half-plane Z=X+iY corresponding to the positive values of Y. If we consider the points of the circumference of the circle (for which z=e "i), we find for Z. ( —0 sin 2 Z= W; (0 sin - therefore these points correspond to the points of the real axis in the plane Z, and more precisely, the portion of the circumference for which 0 < < 0 corresponds to the negative real semiaxis and the portion of the circumference for which < 0< 27r corresponds to the positive real semiaxis.

Page 282 282 University of California Publications in Mathematics [VOL. 1 Let us suppose 0< u<1 and let us put S=Zu; thus we obtain the conformal representation of the semiplane Z, corresponding to the positive values of Y, in an angle having the amplitude 7ru, of the plane S (Fig. 24), in such a way that the y X Fig. 22 y y POO > -;_- > -- x P oo D C E Bo0 Fig. 23 Fig. 24 positive real semiaxis of Z corresponds to the positive real semiaxis of S and the negative real semiaxis of Z corresponds to the side jf co (Fig. 24) of the angle riu; then, by putting (14) \ ze-1 ~= Ze-i e1) we shall represent the circle (Fig. 22) in the angle (Fig. 24). By putting z = reiw, S = +iy = sei'

Page 283 Volterra: Flow of Electricity in a Magnetic Fieldd: 283 for 0 <w < 0, r= 1, we shall have. w-0 A sin - 2 S=( ) e sin 2.w-0 /sin \ that is=( 2 ), p= tr sin and for <w <27r, r = 1 we shall have. w- 0 sin )S=( - \. w / sin 2 Differentiating the two equations. 0-w. w sin - 2 sin w0 tha i. wsin 2 2 Y=O. (sin w ) V. w sin 2 with respect to S and 4 respectively, we find.* -w d sin 2 A. w-0 d,sin 2 I =dw\. w sin - 2 dw_ A = 2-(in - -w) dS 2 (. 6-w\ (sin j^ 2 0 sin - 2 I- (sin L) + dw dS I 0 C in - -7 2i111 r~ dw 2 d~2 w- i s dw. d ' that is,-w (1-)- ^(s in l z dw -2 (sin 221 = — SinO -, O<w<0 dSin 2 a^ /x o- ^~~ /. w-6\ /. w\ ~ dw 2 (sin w-) l(sin 2)1+ 2 P sin2 sin -2 O<w<27r. Now, let w (z) = u+iv be a function regular within the circle and along its boundary. By inverting (14) we can have z as function of S, and by substituting the expression thus obtained in the preceding function, we shall obtain w(z(S)) = w(S) which will be regular in the interior and on the boundary of the angle j (Fig. 24). Let us calculate dwl du d v (16) u2+iv2 =w2(S)= dS =d +i d and for this purpose let us suppose that u is known along the arc ceb as function of w. Let us denote by Z (w) this function and by Z'(w) its derivative with

Page 284 284 University of California Publications in Mathematics [VOL. 1 respect to w. Further let 1 be a direction which forms the angle — = -7r with du the external normal n to the circle and let,u(w) be the derivative l. Then, owing to (15') along the positive semiaxis E, that is along the side jS3 co of the angle 7ru (Fig. 24) we shall have Su b- u dw sin ( w'(2 2sin ) sin 2 and along the side jaso of the same angle, denoting with X a direction which forms the angle: with the normal v, we shall have i/. O-w) l- (. w\ 1 5u 6u but dw\ 2 2 )AY) 2sin 2) 26E=T - -Ts1 \dS} P(w sin 2 If in (16) we substitute for S the value (14), we obtain W2(S(z)) = u2(X1Y) +iv2(X1Y); u2 along the arc bdc will be equal to /. 6-W) 1-( /. 2 \ l +2 (sin — w2 (sin -- Y(W) 2 sin and along the arc ceb will be equal to 2 sin (w-)- (sin W)1+ Z '(w)6 sin Thus we know at the boundary of the circle the values of the real part u2 of the function w2(S(z)) and therefore, by applying a well known formula we can easily calculate w2 in the circle. The formula which we employ is 2,r e1 iw+ d-Z W2(S(z)) -= 2u2(w) iw dw+iC (w) the values of u at the boundary of the circle which we denote with u2 and with C a real constant. Then, since for z = 1 w2 must be zero, we shall have /. 6-W\ 1-# /. W\ i+,x 0 (sin -— w ( W2(S(z)) = M(w) (ei + z _ eiw+ 1) d sin 1 12 s. nW-0\ l-~ (. w\ l+/ + -r Z '(w) 2 i2 (e:_z eiw+ )dw. +/ z ~ 0 ei"w —z ew — ~I sin - 2

Page 285 1921] Volterra: Flow of Electricity in a Magnetic Field 285 But from formula (14) we deduce. 0 dS io si2 = 2/_ie Z _io___ dz (z-eni) 1 -A) (z -1)1+ and therefore with single operations we are able to calculate dw dw dS 2( dS dz dS dz dz' we obtain O /. 6-w\ l- f. W\i w _dw 2 e_ (1-) 2 (sin 4W (sin W)\e (17) dZ (z-e) ) M(w) zw dw dZ- r (z - 0) -JA ( - J (W ) z - eiw /. w-0\ei 1W\ iw 27 (sin -) (sin 2) - '(w) i dw Now, by an integration, we shall calculate w except for a constant, and, by separating the real part from the imaginary part, we shall have the harmonic function u, leaving a constant that will be easily determined. Therefore, if along the arc bdc we know -, and along the arc ceb we know u, we are able to calculate the harmonic function u within the circle. According to what we have stated in Section 7 of the first lecture, we can say that the problem of the distribution of currents in a circular plate, when the current enters through one or even through several, internal point electrodes and goes out through a linear electrode, of negligible resistance, situated along an arc of the boundary, can be completely solved by the preceding formulae. The integration with respect to z that we have to make for obtaining from (17), 1 w, can be calculated by elementary function when = -, n being a whole number. But in this case the method of the images permits us to obtain the solution more easily, as we shall now see. Let al be a point in the interior of the angle dj2 (Fig. 24) having the amplitude 7ru =- (n, whole number). n Let us complete the division of the plane in angles all equal to - and let us conn sider the successive images a2, as..... an of the point al with respect to the various sides of the angles (see Fig. 25); in this figure we have taken n = 6, =60~. The smallest value of n is 2, in which. case we have 3 = 0). Let us consider the two functions of the complex variable S n -2(h - 1) ri -2ri (18) 4 = he ) [log (S-a2h -1)-e e- log (S-a2h)] (18') F=2he r [ log (S-a2h-1)-e — log(S-a2h)] 1 Le

Page 286 286 University of California Publications in Mathematics [VoL. 1 which are connected by the relation n (18") F=e 9 and let us study their properties. We can easily recognize that each term of these functions does not vary by changing h in h+n. In fact, since after 2n reflections the images reproduce themselves, we have a = ag+2nl and also -2(h +n-l)7ri -2(h-l),i e n =e n We can therefore substitute in any term h' for h, without altering its value, provided we have h' =h (mod. n). I "\ i 6'. I0./ / K. - - I 8 /, ~ / Or- /i ( I I e' / oc,\ I" x - 0 I % \. '. * 0So \' I k i! i/ 9 Fig. 25 I Fig. 25 Now let us observe that a2h-1 and a2k are images each of the other with respect to the side j2 when 2h-1+2k= 2n+1 and that a2h-1 and a2k' are images each of the other with respect to the side jd when 2h-1+2k'= 2n+3. From the two last equations we obtain k=n-h+l, k'=n-h+2. Let us consider now the two following terms of the sum (18); -2(h-1) 7ri e log(z- a2h-1) and -2(k -1) ri -2ri -2 (h -1)ri -e e n log (z- a2k) = -e n log(z-a2k) their sum will be (h -i -2(h-1) -2-1) ri e n log (z- a2h) -)- e n log(z- a2k) and will have its real part constant along the side j2 (see Section 2 of the second lecture).

Page 287 Volterra: Flow of Electricity in a Magnetic Field 287 Thus, the terms of 0 can be coupled in such a way that the real part of the sum of each couple is constant along j2; therefore along this side the real part of 0 is also constant. Let us consider the two terms of the sum (18') -2(h-1) 7ri 7rt e n e n log(S-a2h-1) and -2(k'-1)ri -ri -2(k —1) ri -7ri e n e n log(S-a2k =e n e n log (S- a2k) their sum will be - (2h -3) 7i - (2h -3) 7i e — T" — log(S- a2h -i)- e n log (S- a2k') and will have this real part constant along the side j6. We can thus couple the terms of F in such a way that the real part of the sum of each couple is constant.along j; therefore also F will have the real part constant along this side. Now let us take -J4 <19) 2 2 rk -J (19') f=- F. 7F 27rk i sin - n Evidently sp will have the real part constant along the side j2 and f will have the imaginary part constant along j6, that is the normal derivative of the real part of f will be zero along j6. e - i But we can easily verify, owing to (18") that we have f= s < when we take cos f = (n -2 )r. Therefore, if we assume the real part of p as electric potential, the real part of f will be the corresponding fundamental function (Section 1 of the second lecture). Let us separate in f and ~p the real from the imaginary part, by writing gp= V+iV', f= U+ixU'; then V and W are respectively the electric potential and the fundamental function corresponding to the distribution of the current in an indefinite plate bounded by the two radii j: and j and subject to the action of a magnetic field, when the current of intensity I= vJ (P thickness of the plate) enters through the point al and goes out through an indefinite electrode of negligible resistance placed along the side j', when the side bdc is insulated and the angle A is equal to ( -2 r. Now let us put in (19) and (19') (see formula (14)) S= ze- — e -

Page 288 288 University of California Publications in Mathematics [VOL. 1 we shall thus obtain the conformal representation of the angle /2j5 (Fig. 25) in the circle (Fig. 22). To the point al will correspond a point al in the interior of the circle; al and al will be connected by the relation iL i20 1 ale- 2- e2 n=al al-1 / and therefore the complex numbers al, as, a5,..... a2n-1 will be the n values of n io io ale- - e~ a-1 while a2, a4,...., a2n will be the number conjugates of the preceding, that is the n values of n - i is aIle2 - e- 2 a a'- 1 having denoted with al' the complex number conjugate to al. Therefore: If we have / ilO ilo n-2 z i io l = r S= ze- - e 2M \ z-1 and if the (P2h-1 and the a2h denote respectively the n values of n l i i n io io ale — e-2 and ale - e- 2 al-1 al'-1 (a' being the conjugate of a), the real parts of - Jfn -2(h-1)i -2 (P= K —Jh og 2h )-Zhe n log(-a2 )-e log(S-a2h) and — J n -2(h-1)iF r? -7ri 7f. i Ze ( [e log(S-a2h -)-e -e; logS-a2h) 2irKi sin- 1 e S n are respectively the electric potential and the fundamental function of the distribution of currents under the action of the magnetic field in a circle having the radius equal to 1 (Fig. 22), when the current enters through an internal point electrode a and goes out through a linear electrode, of negligible resistance, placed along the arc bec of the boundary, while the part bdc of the same boundary is insulated. Now let us suppose that the point electrode al coincides with a point of the arc bdc of the boundary; then we shall have a2h-= a2h and if we put a1=eiw with w< 0, these numbers a will be the n values of n e/ i (ae -2 - ei.. eitv - 1 that is the n values of. -w 1 /sin 2 n (2h-1) 7ri -. w } e n \ sin / 2

Page 289 1921] Volterra: Flow of Electricity in a Magnetic Field2 289 for h =1, 2,....., n, therefore we shall have -w -_J -2(h — - 1) Isn 2 (2h -1)ri = K e n logS- - e 2 nns sin sin -. J Sin n -(2h-1)r-i 2 (2h-1)2r I..0-W rK ^he;n logs- e/-sin sin - where we have to take for _ _2 the positive value and we must always sn 2 sin w_ suppose the expression ze 2- e2 n substituted for S. z-1 SECTION 2 We will now study the case in which the plate is not plane and the magnetic field is not uniform. Let us subdivide the plate into infinitesimal elements. We can consider each of these elements as plane and subject to a uniform field, and therefore we can apply to it the fundamental formulae found in the case of a plane plate subject to the action of a uniform magnetic field; we may conveniently develop these formulae by using curvilinear coordinates. Then, by passing through an element to the contiguous element, we shall be able to state the relations which are to be verified on the whole surface and from these relations we shall deduce the general differential equations and the boundary conditions. Let us consider an infinitesimal, plane element of the plate, adjacent to a point A, and let us remember the equations (3) jX K/ V _X 6V; b s xV sV\ 6x=-K 6 j iy-K - +b -. Let ds be a linear element passing through A and dn the corresponding element of normal; let ds and dn be situated as we have said in Section 5 of the first lecture; then we shall have: dx dy dy x ds dn' ds dn J = dJx. dy K6V 8V\+ n =Jxd+JY -+F dn dn 6n 6s

Page 290 290 University of California Publications in Mathematics [VOL. 1 where jn denotes the density of the current normal to ds. By employing a system of curvilinear coordinates u and v, we shall have (20) jn' -K 7 V a8 8V v 178V mu +Vv\ ] (20) jn= LKLu an+ a n as +v as/' (Let us regard positive directions of the curves V=const. and u=const., as the directions in which u and v increase respectively. We shall make the convention that the rotation of the positive direction of the line v = const. towards the positive direction of the line u= const., through the angle smaller than 3, has the same sense as the rotation of 90~ by which the positive direction of the line s can reach the positive direction of the line n; moreover we shall here and in what follows, take for /EG-F2 the positive value). Now, if the square of the linear element be ds2 = Edu2 +2Fdudv+Gdv2 we have u_ 1 __u m v\ ( u 1 m u v\ F +=v F +G —!m 1 (j —GF —+G -1 an 4EG - F2 as s/ ( as EG - F2\ An (a) = E - \ 'a E ~u vv\) IS +F=-~C~Ev l u EG - F A Es As F ( 8n and the equation (20) becomes 8V 8FV 8V 8V TE F- G- - FF_______ vu l V m u mu 6v 87 6v (20') jn =-K iv 8 ]+X -+KI a _xA - L iEG-F2 buJ 6s LS/EG-F2 - v js' Let us suppose the line s to coincide with the line u=const.; then the preceding equation will become 2G - _F 8V F8u 8-V -1 1 (21) jin =K E -XS U where jnu is the density of the current normal to the element of the line u= const. Analogously we shall obtain 87V 817 6_o -F V1 (211) nv =-K EG-F X 4E (4G and jE have to be taken with the positive sign.) All the preceding formulae hold good for an infinitesimal plane element: now if the plate be curved and be placed in an uniform or non-uniform magnetic field, the preceding formulae hold good for each infinitesimal element of the surface, we have only to suppose that K and X = tg3 change from element to element; in other words, the preceding formulae will hold good in the more general case of a metallic plate, of any form, situated in any magnetic field, provided K and X be considered as known functions of u and v. In order to obtain the values of these functions in a

Page 291 1921] Volterra: Flow of Electricity in a Magnetic Field 291 point of the plate it is sufficient to substitute in the formulae (see Section 4 of the first lecture): K=e2 (Nviv N2v2 V = N112 N22 2 2e \1+e2v22 1+ e222H/ K' eH 1 +e2 H2 1+e2v22H for H the component of the magnetic field in the direction of the normal to the plate; this component will vary from point to point of the plate because the angle that the normal forms which the direction of the magnetic field and the intensity of this latter vary. (We do not take into account, here, the secondary actions by which a plate loses the character of isotropy, with respect to its electrical conductivity, when it is subject to the action of a magnetic field, the direction of which does not coincide, in every point, with the normal to the plate [see fourth lecture, Section 1.] We observe only that the perturbations caused by these secondary actions do not take place when the magnetic field is perpendicular to the plate at every point of this latter). From the formula (20') which holds good on all the surface, we obtain E aV F_ V s 6 V (22) I nds= {-K 6V - - +Xu] +K -a = 6V X6U -ds. <EG]F2 6uj L u_s EG-F2 6v j6sj If s be a closed line which encloses no electrode, the first member of the preceding equation is zero and therefore V FV &GV F6V av 6n17, J 6u _ v -VF (23) -K LEG F F2 +X- udu+K L -LE — - dv L EG F2 u L jEG -F2 6V must be an exact differential; calling it dW, we shall have G V 6V [ a\ ^ + ^ 6 5 1av 6W - x/Gju =-K - JEG - F2Xsv W (24) ( I-G-j _ U K[6v -6u 61 _ V 8W KEi -K - - K_ -' K+ XI, - IEG - F2 6u3] u Therefore V must fulfil the differential equation Gu 6V 5V E V F + V ~(F)!67- 617 Kv E -F u 3m 6v^-x 6V + GL F + - 6 =~j SV SV By solving the equations (24) with respect to - and - we obtain 6W sW E6W F6W 1 G u 5v sW] 8SV — 1 6v 6u - l ~W 6V K(1+2) EG -F2 6v v ' K(+X2)L G -F2 mu u I d 2E - +2 wEG = ~ ~ - F ~

Page 292 292 University of California Publications in Mathematics [VOL. 1 therefore W fulfills the differential equation I G 1 E F w = aGF au a6V aw) a 1 [ v au aw? -. K(I )! EG - F2 + v +6K(~+X2) LIEG- F u Now let us suppose that in the formula (22) the integration is extended to an open line s. Owing to (24), we shall have f jnds = s dW = W2-W where W1 and W2 are respectively the values of W at the origin and at the extreme of the arc s. Therefore along the insulated portions of the boundary (as along each line of flow), W will be constant, while along all the electrodes of negligible resistance V will be constant. If we suppose that in the interior of the plate there are no electromotive forces, V will be a single-valued function, while W will become many-valued if we go along any closed curve which encloses some electrodes through which a total quantity of electricity different from zero enters into the plate. (We shall say that some electrodes are enclosed by the curve if they be on the same side of the said curve). If the plate be acyclic and all the electrodes be at the boundary, W will be evidently single-valued. We shall call W the function of currents. In the case in which K and X are constant, for a given value of H, we have W=KU', where U' is the conjugate function of the fundamental function U. (see Section 1 of the second lecture). The expression for jn (formula (20')) can be transformed in various ways. In fact, by using (24), we can write aW (20") J- 6 whereas, by applying (a'), we have X V -FsV X( sV Fav (20"') jn= -K 6V \p 6 u au V 6 )+_ V A an EG - F2 an v EG - F2 an Lastly, bearing in mind the formula (20"), that is, jn= — s- + av and (a'), 6u 6s av 6s we find (20/1"") in E - F2 { (F Wc bE, +-i (Gaw F 6W) EG-F 2 amu 6v an au av ) an Bearing in mind (24) the formulae (21) and (21') can also be written in this way 1 6W. 1 8W (21') jnu-= xG Sv' "= nu- E 6u

Page 293 1921] Volterra: Flow of Electricity in a Magnetic Field 29,3 Let us call j, and jv the orthogonal projections of the density of the current in the directions of the lines u= const. and v=const.; we may obtain their values by supposing that n coincides successively with the directions of the lines u=const. and v = const.; we shall have (25) 6V 6V.-K/S6V C n - Fv G~-v 1+EG -F2 ' v 6 -F 6 -K8V 6 8-v -Fa KE ~EG —F By employing the function W, the preceding formulae become (25') sV 8FW 1 f 6n 6v 3 -G\ EG-F FsW sEW F - -E - 1 bu 6v iv =E EG - F2,' Lastly, let us c )nsider the components of density of the current parallel to the directions of the lines u = const. and v = const.; we obtain formulae K<G /61V 61V - 6V\ u- EGF E — F +X/EG — F2v; EG - 6v au 6n (26) v EG-F2=6n 6v EG F2 6v which, by employing the function W, become (26') = G bW u EG 2 6' vu EG-F2 &t ' -4E 6W V=EG- F2 v ' From any one of these groups of formulae we obtain the square of the density of current, that is I2=K2(l +X2)\ sv / u 2 G Su ] (27) ~(27) 6W 5\2 SW 6W /6W\2 E(W- _ 2F-W - +6 )2 I2 6v 2 6u 6v au 3 ~ EG-F2 These formulae, by using the symbol of the differential parameter of the first order, become (27') 2 = K2(1 +X2) A1V j2=AW From the expressions that we have obtaineed for jt we can deduce other forms for the condition jn= 0 which has to be fulfilled along the insulated portions of the boundary. For instance, by using the formula (20'") we can write the said condition in this way: E a -F5 — G - F (6s 6 v 6u\u 6 v 6u v\8v -x I -)- +X: ) =0. \6u <EG - F2 6n 6v JEG-F2 6n

Page 294 294 University of California Publications in Mathematics [VOL. 1 SECTION 3 When the plate is homogeneous and the magnetic field H is zero, we have K=const. and X=0; and the equations (F) and (G) of the preceding paragraph become G _F E SV Fv 6{= EG-F> 66vt EG-F2 =0, WFW EW6W_ sW GG -F E -F 6 u 6v a 6v 6u b6u{ EG-F2 \ s + E 1EG-F2 } and can also be written, by using the symbol of the differential parameter of the second order, in this way (A') A2V=0, A2W=0. Therefore V and W, in this case, are two harmonic functions on the surface, as we could also have foreseen. Moreover the equations (24) become GV 6V W 6V 6V W G —F 6- E -F 6 bu 6v K 6v bu K JI EG -F2 6' EG- F2 u' The conditions (P') are the necessary and sufficient conditions in order that V and W be harmonic on the surface; the conditions (j) are the necessary and sufficient conditions in order that V+i- be a complex variable on the surface (1). K Nov, when KX is not constant, the equations (F) and (G) are essentially different from (3) and therefore V and W are not harmonic on the surface. In the case of the plane and homogeneous plate subject to the action of a uniform magnetic field, we saw in the preceding lectures that the potential V was harmonic, as it was without the magnetic field; for only the condition which V must fulfill along the insulated portions of the boundary changes. Whereas, in the case of the curved and non-homogeneous plate, subject to the action of a non-uniform magnetic field, the action of the magnetic field alters (as we have just now seen) not only the condition which the potential must fulfill along the free and insulated boundary, but also the nature of the potential in all the area occupied by the plate. We can say, summarizing the results obtained, that when we pass from the case of the plane plate and of the uniform field to the case of the curved homogeneous or non-homogeneous plate of the uniform or non-uniform magnetic field, we pass, (from the analytical point of view from Laplace's equation to new equations of different character (the equations (F) and (G)). Only two of the four functions considered in the first case (potential, fundamental function and their conjugate functions) remain also in the second case: the potential and the function that we have called function of the currents. But the one and the other lose the character of harmonic functions on the surface occupied by the plate, and this is the reason

Page 295 Volterra: Flow of Electricity in la Magnetic Field 295 for which the other two functions cease to subsist; in fact, as the potential V and the function of the currents W are not harmonic (that is have not zero the second differential parameter) but instead fulfill the equations (F) and (G), the conjugate functions of V and W, in the sense of the theory of the functions of complex variables on a surface, cannot exist. From all we have now said, we can deduce that the principle of the point electrodes at the boundary exists no more in the general case of the curved nonhomogeneous plate, subject to the action of a non-uniform magnetic field; the same can be said for the principle that the potential remains unaltered, under the action of the field, in the case of a cyclic plate, when the lines which form the boundary are electrodes of negligible resistance, kept at a constant potential. On the contrary, we shall see, in the following paragraph, that the principle of reciprocity (see the Section 9 of the first lecture) holds good also in the general case just said. SECTION 4 From the formula (20') of Section 2, denoting by V1 an arbitrary function and by S the boundary (formed by one or several lines) of a part of the plate, and supposing V and V1 to be regular, we deduce EV -FV VilndS= I -KV1[ - - U+X6 -U+ _ 6V 6 V V 6V18V 8V1 ~~~~~=l ~~C J,'c FF 6 8 8 /5 V u \6u vu 6u 8v = d\.V1 j + ) I8V YV G-FV, 8iV 8V IvA 8V 8v 8 u 8u 8u 8u v 8u 8v 8v 8v J^iEG-F — 2EG_ F +X(6V 6 1 av, \ 6)= (We suppose the normal n drawn towards the interior of a and thus we fix implicitly the direction s). Let us denote with A1VV1 the intermediate or mixed differential parameter of the functions V and Vi, that is let us put G6V 5V1 F 5V1 6V +V 5V1 +E V 5V, Gu V u a i3u 6V au V V 3V 6V_ A1VV1= _a EG-F2 and let us consider the determinant 8V 5V av sv u' 6v _d(V1V1) av v1 V1 d(ulv) u ' 3v

Page 296 296 University of Californiam Publications in Ma~thematics [VOL. 1 We can immediately recognize that A1VV1 is symmetrical with respect to V and 6(VV1) V1 and that the determinant d(uv) changes sign by exchanging V with V1. The equation (28) can be written thus (L) fVtljndS=f K (AVV1~+ EG F2 d(V —1) )d. Now let us suppose V1 to be the electric potential when the magnetic field is inverted; let us denote with jin the corresponding density of current normal to S: since on reversal of the field K remains unaltered and X changes only in sign, together with the formula (L) we shall have the other Hence (M) f(Vjln Vjn)dS = O. If we compare the formula (D) (see Section 8 of the first lecture) with the formula (M), we recognize that from this latter we can deduce the same consequences which we have deduced from the former, and in particular the theorems of reciprocity (see Section 9 of the first lecture). For instance, let us suppose that with the direct magnetic field the current of intensity I= vJ (v thickness of the plate) enters through the point electrode A and goes out through the point electrode B, and with the inverted field the current Ii= vJ1 enters and goes out respectively through the two point electrodes A1 and B1. Let us suppose S to be formed by boundary s of the plate and by the four geodetical circumferences Sa, Sb, Sal, Sbl, having the center s in A, B, A1 and B1. We shall suppose that, at least when these circumferences are sufficiently small, if jn be drawn from the inside towards the outside of the same circumferences, jn is positive along Sa, negative along Sb, and jln is positive along Sa, and negative along Sb; moreover we shall suppose that, along Sa, Sb, Sal, and Sbl, V and V1 are finite or infinite of an order smaller than 1. Since on s jn and j lnare void, we shall have fs ajln Vds a + f sbjln Vdsb+ f s aljlnVdsal+ f s bljl nVds bl- f sajnVildsa- fsbjnVldSb- Vd- f sa - S,bljdsa jnVldbl = 0. If we pass the limit, making the four geodetical circles grow indefinitely smaller, we obtain J1(VA1- VB1) = J(V1A- V1B) and hence, if J1 =J, VA1- VBI = V1A — VIB. In an analogous manner, all the theorems of reciprocity (relative to internal electrodes of finite areas and negligible resistances or to electrodes situated at the boundary and also of negligible resistances) can be extended from the case of the plane and homogeneous plate situated in a uniform field, to the case of the curved homogeneous or non-homogeneous plate, situated in a uniform or non-uniform field. The equation (M) can also be written in this way s S 5sbS

Page 297 1921] Volterra: Flow of Electricity in a Magnetic Field 297 therefore VdW1- V1dW will be an exact differential: this will be another manner of expressing the theorem of reciprocity. From the formula (20'), denoting with S the boundary (formed by one or several curves) of a part a of the plate, we can deduce the other formula (H) f8s VjndS = faK A Vda which can also be written (H') fv a^dS= fKAVdo is or also (H") fVjndS= K(I +X2) 2d or also (H/"') IV K( +2)1Wda. The first members of the preceding equations are proportional to the quantity of energy which, in the unity of time, penetrates in the area a through its boundary S. (The factor of proportionality is the thickness v of the plate). If a be infinitely small, K1 +X2) j2 is the quantity of heat that, for Joules' effect, is developed by the small, K (I + X2) current in each element of surface of the plate. From the formula (H') we can also deduce that, if V be zero along a certain part of the boundary S and W be constant along the other parts, V is zero within the area a. Therefore, if some portions of the boundary be electrodes of negligible resistance, where either the value of the potential or the intensity of the current which penetrates in the plate is known, and if the other portions of the boundary be free and insulated, the distribution of the currents will be determined. (See first lecture, Section 6). FOURTH LECTURE Section — The problem of the flow of electricity in a non-homogeneous anisotropic conductor, kept at constant temperature and subject to the action of a nonuniform magnetic field. Section 2-Generalization of the principle of reciprocity. Section 3-On the change of resistance of a conductor subject to the action of a magnetic field. Section 4-The electromagnetic phenomena that can be observed when a disc traversed by a radial current is placed in a uniform magnetic field. Section 5-An reversible generator for steady currents founded on the action of magnetic field on the electrons of conductors placed in the same field. Section 6 -Experimental verifications of the theorem of reciprocity in the different cases. Section 7-A consequence of the theorem of reciprocity. Section 8-An applica1Beltrami, Dele variabili complesse sopra una superficie qualunque. Opere, vol. I, p. 318.

Page 298 298 University of California Publications in Mathematics [VOL. I tion of the principle of the point electrodes at the boundary. Section 9-Experimental tests of the theory of flow of electricity in a circular plate perpendicular to the lines of force of a uniform magnetic field. SECTION 1 We shall now speak of the generalization to the case of a three-dimensional conductor of the problem previously considered, of the flow of electricity in a plate under the action of a magnetic field. This generalization was made, in 1916, by Miss Freda. (1). The experimental results that have been obtained by studying the properties of a conductor placed in a magnetic field have led physicists to make the hypothesis that the field determines a transitory alteration of the specific properties of the substances subject to its action. According to this hypothesis the substance also if homogeneous and isotropic without the field, under the action of the latter acquires electrical properties different in different points, if the field is not uniform, and different in the different directions that pass through a point according to the angle that the same directions form with that of the field. If we do not exclude this hypothesis, in order to study analytically the flow of electric currents in a three-dimensional medium subject to any magnetic field, it is necessary to consider media that are heterogeneous and anisotropic. (The said hypothesis on the other hand does not introduce difficulties in the case of a plate coincident with an equipotential surface of the magnetic field, because then, for reasons of symmetry, the plate will remain homogeneous and isotropic under the action of the field if it was such before the creation of this latter; we have only to consider, in the formulae which we have already seen, N1, N2, v1, v2, as functions of the absolute value of H.) Let x, y, z, be three axes that form an orthogonal right-handed system. Let d~l d'l dS~ d42 dv~ dS... d, d- 1 dS1' d 2 d2' dS2 be the components of the velocities of a positive electron dt'dt ' dt' dt ' dt it and of a negative electron the charges of which have both the absolute value e. Let N1, N2, be the numbers of positive electrons and of negative electrons per cubic centimeter of the conductor. The components jx, jy, jz, of the density of current are expressed in terms of the components of the velocities of the electrons by the equations: (29) jx e (N 1-N2; j =e e(Nd -N2 d jz e (N -dtN d). Let eElx, eEly, eElz, eE2x, eE2y, eE2z be the components of the total electromotive forces acting respectively on a positive electron and on a negative electron; let sV sV Hx, Hy, Hz, be the components of the magnetic field. Let X= -; Y=- -; Z=- Vbe the components of the electric force which admits the potential V. 0'z

Page 299 1921] Volterra: Flow, of Electricity in a Magnetic Field 299 According to Ohm's law for anisotropic media, the components of the velocities of the electrons of the two kinds are expressed, by linear equations, in terms of the components of the total electromotive forces which act on the same electrons. We have therefore (30) d_ = e(allElx+al2Ely+al3E1z) dt I- = e (a2Elx+a22Ely+a23Elz) dt dS1 = e(aslElx+ as32Ely a33Ez) (31) d 2=- e(b2lE2 + b22E2y + b23E2z) dI =d- - e(bllE2x + b2E2y+ bl3E2z) Idt dS2 dS2 =- e (b31E2-J+ b32E2y+ b33E2z) dt The experimental results that induce us to infer an alteration of the specific properties of a conductor caused by the magnetic field induce us also to infer that this alteration does not depend on the sense of the field. We shall therefore assume that the coefficients N1, N2, amn, bmn (which characterize the specific properties of the metal) can eventually depend, not only on x, y, z, but also on Hx, Hy, Hz, but do not vary when we change the sign of all three components of the field. As the conductor is kept at a constant temperature, the total electromotive force that acts upon an electron is the resultant of the electric force depending on the distribution of the potential and of the electromotive force caused by the movement of the electron in the field. We therefore have (32) zdt dt E^ = Y+ Hd- H E1Z = Z+Hx d —Hy d, dt Ydt (33) E2y= Y+HZ- 2 dS2 I ~ dt `dt E2= X+ HdS -H d2 Ydt dt d7y2 d 42 E2Z = Z + HX — H dt Y dt By means of the equations (30) (31) (32) (33) d - d, dS1 d 2 can be exp sdt ' dt ' dti o X tn th te exat pressed as linear functions of X, Y, Z. We obtain thus the equations: (34) d 4, d = e(atX+ Ya + a23Z) dt 1 = e(aX Y + 22Y+ a23Z) dS1 =e(ax+ + ) dt =e(aclX+a32Y+ a33Z) (35) d =-eX+ +Z) (350)i - -=-e(021X-k022Y+023Z) I3) dt I - -- _e(311X+012Y+013Z) dt dS2 d = - e(031X+-32 Y+-33Z) I1dt The coefficients ars, ts are functions of the components of the magnetic field and of the coefficients ars or of the coefficients brs respectively; for the sake of brevity we do not transcribe their expressions.

Page 300 300 University of California Publications in Mathematics [VOL. 1 By means of the equations (23) (34) (35), jx, jy, jz, can be expressed as functions of the derivatives of the potential V. We have I (5V 6V 5V~ jx= -e2[ (Nlal+N2i1,) + (Niai2+N2012)-+ (N+ NaI+ N13) - ox by 6z (36) - aV 8V 3z (36) jy= - e2 (Nia2l+N2i21)+ + (N1a22 +N222) + (Ni1a23 +N2f23) jI i= -e2[(Nia3l+N2F3l) + (Nila3+N2i32) + (Nya33+N2I33)z The condition (37) a+ 3+ z = 6x by -z gives us the differential equation that V must fulfill. If n is the normal at any point of the free surface of the conductor, from the condition j, =0 we deduce a boundary condition for V. If along the surface of the conductor there be electrodes of negligible resistance kept at constant potentials V1, V2, we have also for V the conditions V = V1 along a part of the boundary V = V2 along another part of the boundary. Now let us suppose that the conductor is homogeneous and isotropic when the field does not exist; that the field is uniform and has the lines of force parallel to the axis z. Then, if the magnetic field does not alter the specific properties of the conductor, the equations (36) become, / a(V V\ a 5VaV\ a 8V (36') Jx=-K x- )>; jy=-K Xa ^; jz= (K and X have the same values they had in the formulae (3) of the first lecture; a is given by the formula a = e2(Nvi+N2v2). SECTION 2 From the formulae (36) and (37) of the preceding paragraph, Miss Freda has deduced that, if the coefficients ar, brs of the equations (30) and (31) satisfy the conditions (38) ars= asr brs= bsr we have (39) f8(Vljn-Vjln)dS = 0. (We denote with V, j, V1, jl, the potential and the density of current when the field is direct and when the field is inverted; with S the complete boundary of the conductor or of a Dart of it). The formula \39) is perfectly analogous to the formula (D) of the first lecture. From the formula (39), by specifying conveniently the boundary S, Miss Freda has stated (analogously to what we have already seen in the case of a plate) that my theorem of reciprocity holds good also in the case of a non-uniform magnetic

Page 301 1921] Volterra: Flow of Electricity in a Magnetic Field,301 field and of a non-homogeneous anisotropic conductor for which the conditions (38) are satisfied. The four electrodes of the conductor can be point electrodes (in the interior or at the boundary) or else laminar, situated at the boundary and of negligible resistance, or else three-dimensional, situated in the interior and of negligible resistance. If the magnetic field be zero, the necessary and sufficient condition in order that the equation (39) may be fulfilled is (38') Niars,+N2brs = Niasr+N2br If we consider the physical meaning of the coefficients ars brs of the formulae (30) and (31) we see that the formulae (38) express some elementary laws of reciprocity. For instance, the property expressed by the formula a12 = a21 can be thus translated in words: The component, parallel to the axis y, of the velocity that a positive electron acquires when subject to an electromotive force equal to 1 and parallel to the axis x, is equal to the component, parallel to the axis x, of the velocity that the same electron acquires when subject to an electromotive force equal to 1 and parallel to the axis y. My theorem of reciprocity in the case of anisotropic conductors is, then, a consequence of these elementary laws of reciprocity. Maxwell, speaking of the flow of electricity in an anisotropic medium (not subject to the action of a magnetic field) says that in every case of aelotropy (with the possible exception of magnets) we can grant that the determinant of the coefficients, in the equations that express the components of the electric current linearly in terms of the components of the electromotive force, is a symmetrical determinant (2). If we refer to the electronic theory, this hypothesis of Maxwell's is equivalent to the conditions (38'). If not only the conditions (38') but also (38) be satisfied in every case of aeolotropy, the same generality holds good for my theorem of reciprocity. We can however easily determine a case in which the conditions (38) are certainly satisfied. Among others there is the case, the most interesting here, of a conductor that is isotropic without the magnetic field and that acquires a temporary aeolotropy only under the action of the field. To state this it is sufficient to bear in mind that for evident reasons of symmetry, all the directions, passing through a point P of the conductor and tangent to the equipotential surface that passes through P, are equivalent from the electromagnetic point of view. SECTION 3 As is known, the conductors show, under the action of a magnetic field, a change of resistance. The experimental researches on this phenomenon are very numerous; but the results obtained and the empirical formulae proposed by the various experimenters to represent the phenomenon are often discordant and even incompatible among themselves. This depends in part, as will be seen from what we shall say in this paragraph, on not having clearly defined what we mean by change of resistance produced by the field.

Page 302 302 University of California Publications in Mathematics [VOL. 1 One result with reference to which the said experimental researches may be said to be concordant is the following: the ferromagnetic substances (iron, nickel, cobalt) show an increase of resistance parallel to the magnetic lines of force and a decrease in the perpendicular direction; the diamagnetic substances, among which bismuth has the first place, show an increase of resistance in all directions (different in the different directions); for all the other substances the change of resistance is very slight. To explain these experimental results the hypothesis has been made that the field causes a real and true temporary alteration of the specific properties of the conducting substances subject to its action. The theoretical researches that have been made in order to give an explanation of the said results, taking as basis not the hypothesis above alluded to, but the laws of the movement of electricity in a conductor subject to the action of a magnetic field, are few compared with the experimental researches. Miss Freda after having briefly examined these theoretical researches proposed to herself the problem of stating what the electronic theory, to which in these lectures we have referred, predicts about the experimental results that can be interpreted as a change of electric resistance caused by the magnetic field; whether there are or not phenomena that the theory does not foresee and to explain which it will be necessary to introduce the hypothesis of an alteration of specific properties produced by the field in the conductors subject to its action. (3) We shall briefly resume the results of this research. Let us again consider the formulae (36)' of the first paragraph, formulae that hold good in the case of a conductor which remains homogeneous and isotropic also under the action of a uniform magnetic field H. Let P be a point of the conductor; when this conductor is not subject to the action of a magnetic field, let lo be the line of flow passing through P, let j0 be the density of current, Vo the potential in P, co the specific conductivity of the substance (Co and a have the same value). )V0 We shall have jo= -Co ao Let now I be the line of flow passing through P when the conductor is subject to the action of the magnetic field; let j and V be the values of the density of current and of the potential at the point P. From the equations (36)' we can easily deduce K(1+X2)cos2 lz 6V _V (40) j==[K(l+X)+[o-K(l+X)]XK( x2) z (-cos lz) 6 =-c -. It will be plainly seen, that unless cos2 lz = o, in which case we have c = Co, we always have c <co. Therefore the present theory predicts for a homogeneous isotropic conductor kept at a constant temperature and subject to the action of a uniform magnetic field, an apparent increase of specific resistance in all directions, that of the magnetic lines of force excepted; in this direction the theory does not foresee an apparent alteration of the specific resistance. K, a, X2 do not vary when we change H to-H; then the apparent specific conductivity c does not vary by inverting the magnetic field, if cos2 lz does not vary. (This condition is satisfied, for instance, in the case of a plane plate situated in a

Page 303 1921] Volterra: Flow of Electricity in a Magnetic Field 303 transverse magnetic field and in the case of a wire situated any way in the field.) c, for a conductor of arbitrary form, will be different generally in the different points, because it depends on cos2 lz. The dependence of c on cos2 lz shows us that the theory foresees an apparent aelotropy of the conductor. (That is shown moreover also by the equations (36)'). Let us consider a conductor in which, with or without the action of the field, the lines of flow remain the same, and at every point of these lines the value of the density of current remains the same if we leave unaltered the total current flowing in the conductor. From what we have previously seen we can deduce that, if the conditions now stated be fulfilled, the difference of potential between two points of a same line of flow 1 increases under the action of the field, if I be not parallel to the axis z; only in this latter case the difference of potential remains constant. The increase of the said difference of potential can be interpreted as an increase of the total resistance of the conductor. The said conditions (invariability of the lines of flow and of the density of current) are satisfied, as we can easily prove in the case of a conductor in which the lines of flow are straight lines parallel to the axis z, when H= O. Then the theory does not foresee an apparent increase of the total resistance of such a conductor. The same conditions are also verified in the case of a plate provided, along the boundary, with point electrodes and subject to the action of a uniform magnetic field normal to it. (Second lecture, Section 3). Besides in this case the apparent specific conductivity c, corresponding to a given value of H, is constant at every point of any line of flow (because at every point we have cos lz = o), and does not depend on the sense of the field. In this case the ratio of the values that the difference of potential between two points, of a same line of flow and not coincident with the electrodes assumes before and after the creation of the magnetic field, is equal to the ratio of the specific resistance - that the plate has without the field Co and of the apparent specific resistance - determined with the field. Therefore, c for a plate placed in a transverse magnetic field and provided with point electrodes at the boundary, the theory predicts an apparent increase A- of the total resistance; the ratio between Aa and the resistance o- without the field depends only on the intensity of this latter and on the specific properties of the conducting substance; this ratio does not vary by changing H to -H. The same can be said for a rectilinear metallic wire, inclined with respect to the lines of force of the magnetic field. (Also in this case cos lz is constant at every point of the conductor). In a plate placed transversally in the field but not provided with point electrodes situated at the boundary (for instance in a rectangular plate provided along two opposite sides with electrodes of negligible resistance) under the action of the field not only the distribution of the potential changes, but also that of the current. If we consider, in this case, the ratio of the value that the difference of potential, between two points of the same line of flow or between two electrodes of negligible resistance, assumes for (H)=0, and for (H) >0 we find that the ratio depends not only on the apparent increase of specific resistance of which we have already

Page 304 304 University of California Publioatiowns in Mathematics [VOL. 1 spoken, but also on the altered distribution of currents; in this case the ratio 0o will depend generally not only on the specific properties of the conducting substance and on the intensity of the field, but also on the form and on the dimensions of the plate and on the position of the electrodes. The preceding observations may partly explain the divergences of the results obtained by the physicists who have measured in different conditions the change of resistance of a conductor in the field. Another fact, to explain which has been introduced the hypothesis of an alteration of specific properties caused by the field in the conductors subject to its action, is the law of dependence of the Corbino effect (see the next paragraph) on the intensity of the field. But Miss Freda has remarked that the dependence of the Corbino effect on H is measured by the factor X = tg3; that X increases more slowly than H, and that, therefore, theory and experiments (as far as the metals that conduct themselves like bismuth are concerned) agree, at least qualitatively. From all that we have seen in this paragraph it follows that by confining ourselves to the consideration of the substances only which conduct themselves like bismuth, we can assert that the theory of flow of electricity perpendicularly to the magnetic lines of force agrees, at least qualitatively, with the phenomena interpretable as a change of resistance that the experiments have verified. To be able to say whether there is or is not the said agreement also from the quantitative point of view and therefore whether the hypothesis that we have already several times mentioned in this paragraph is necessary or not would require having the exact values of v1, v2, N1, N2, but for the present we can not say that we have them. Whereas, if we consider the case in which the flow of electricity takes place parallel to the lines of force of the field, we must admit that the theory does not foresee the increase of resistance that the experiments have verified. It is not easy to decide whether that depends on a real alteration of the properties of the conductors determined by the field, or on the fact that the conditions supposed in the theory are verified only approximately in the experiments, or on the superposition of other phenomena to those that the theory considers. We shall, lastly, observe that all these results hold good only on the hypothesis that the mobility of the positive electrons is not zero. SECTION 4 We shall now speak of various experiments which are applications or verifications of some points of the theory previously stated. We shall begin by remembering in this paragraph some results (obtained already in 1911 by Professor Corbino) relative to the flow of electricity in a metallic plate bounded by two concentric circles, provided along these circles with two electrodes of negligible resistance and subject to the action of a uniform magnetic field (4). The same results could also be deduced from what we have said in the second lecture on the cyclic plates. Professor Corbino found that, if the plate is perpendicular to the direction of the magnetic field, under the action of the latter the equipotential lines remain

Page 305 Volterra: Flow of Electricity in a Magnetic Field 305 circles having their centres in the centre of the plate; whereas the lines of flow, which are radial without the magnetic field, become, under the action of this latter, logarithmic spirals. We can therefore consider the disc as traversed by radial currents and by circular currents. These circular currents enable the disc to induce a current in a concentric circular coil, in the instant in which we open or close the circuit of which the disc itself forms a part. The inductive action is evidently proportional to the intensity of the circular current; this intensity will be easily determined if we bear in mind the theory explained in the preceding lecture. As the lines of flow form the angle 3 with the concentric circles which coincide with the equipotential lines, the intensity of the circular current will be Itg3=-I, if K' I be the intensity of the total current flowing in the plate. Then the inductive measured by Professor Corbino for bismuth, and by Adams and Chapman (5) for many other metals. It we assume that the current is carried only by negative / | 7 r electrons, we have ev2H = - (v2 is the mobility, n2 the mass of the negative Fig. 2 6 electon Corbino effect) is proportionterval to (the factor of proportllisions). In this hypothesis fon the dmensions of the plate). This action has been effect, thively proved 2 can be deduced. That has measured by Professor Corbino for bismuth, andy metals by Adams and Chapman and by other physicists. many other metals. It we assume that the current is carried only by negative electrons, we have K= ev2H = -2-2H (v2 is the mobility, me the mass of the negative electrons; 72 average interval of time between two collisions). In this hypothesis from the measure of the Corbino effect, the value of r2 can be deduced. That has been done for many metals by Adams and Chapman and by other physicists. A metallic disc traversed by a radial current, as we have seen, under the action of the magnetic field is transformed into a particular magnetic shell perpendicular to the lines of force of the field and produces thus a slight alteration of this latter; reciprocally the creation of the field should create a radial electromotive force in the disc.

Page 306 306 University of California Publications in Mathematics [VOL. 1 Let us suppose, lastly that the disc traversed by a radial current is suspended in such way that its plane forms with the magnetic lines of force an angle a. In such conditions an electromagnetic couple tends to turn the disc; it can easily be verified that the moment of this couple is proportional to Itgf sin 2a. These two last theoretical predictions were also experimentally verified in 1911, by Professor Corbino. SECTION 5 With the phenomena mentioned in the preceding paragraph are connected some other experiments made in 1915 by Professors Corbino and Trabacchi. (6) Let us consider a rectangular prismatic surface (Fig. 27) the faces of which are thin metallic plates; a is of bismuth, 3, -y, and a of copper. H Fig. 27 Two copper wires are soldered in the middle points M and N of the bismuth plate; through these wires a current I can be sent in the plate. Let us suppose that the prismatic surface, which can rotate about the axis Y Y', is placed in a uniform magnetic field perpendicular to the bismuth plate. Then the current I will be partially distorted and a part of it will flow in the prismatic surface afbj transforming this surface into a magnetic shell which will tend to turn in the field about the axis YY'. If 0 be the angle that the axis of the prism bounded by a3jS forms with H in a phase of the rotation, the component of the field perpendicular to the bismuth plate will be H sin 0; the intensity i of the distorted current can be said to be proportional approximately to IH sin 0. The moment of the couple acting on the prismatic surface will be proportional to iH sin 0; its approximate value will be then (except for a constant factor) IH2 sin2 0.

Page 307 1921] Volterra: Flow of Electricity in a Magnetic Field 307 If another identical prismatic surface can be rotated about the same axis YY' and is turned 90~ from the first prismatic surface, this second surface will be acted on by a couple the moment of which will be approximately proportional to IH2 cos 2. Then the rigid system of the two prismatic surfaces will be acted on by a couple the moment of which will be proportional to IH for any value of 0. The figure 28 represents the intersection of the two prismatic surfaces with a plane which cuts them in half and is perpendicular to the axis. YY',NM,N'M', m e A r B lN MS.0 Fig. 28 Fig. 29 are the intersections with the two bismuth plates. The electrical connections are made by the wires AM, NM', M'B; the ends A and B are soldered respectively to the upper half m and to the lower half n of the axis of the apparatus (Fig. 29); the said wires are situated in a plane perpendicular to the axis of rotation YY'. The two halves mn of this axis are metallic, joined together mechanically but insulated electrically from each other; they are connected through the points of support PQ to the source of electricity. According to what we have said, the system of the two prismatic surfaces represents a model of an armature which turns uniformly (with a constant couple proportional to the square of the magnetic field) when we send in it a steady current through two "fixed" contacts. (P and Q). Vice-versa by turning the armature in the field with a constant velocity we can produce a steady constant electromotive force between

Page 308 308 University of California Publications in Mathematics [VOL. 1 the points P and Q without sliding contacts. Lastly, by holding fast the armature in a Ferraris rotary field we obtain between the same two fixed points P and Q a steady and constant electromotive force. The apparatus acts, therefore, as an armature provided with Pacinotti collector and with brushes situated in such a way that the straight line passing through them is perpendicular to the exterior field however this field or the armature may turn. For the details of the experiments which have made it possible to verify these properties we refer to the memoir cited at the beginning of this paragraph. The apparatus described cannot have any practical utilization because of the slight magnitude of the described phenomena, but is interesting on account of its properties which we have noted. SECTION 6 We shall speak now in these last paragraphs of the experiments which have been made with the actual object of testing some points of the theory developed in Fig. 30 the preceding lectures. We shall begin with the experimental verifications of my theorem of reciprocity. Dr. Tasca Bordonaro, in 1915, verified this theorem first in the case of a bismuth plane plate situated in a uniform magnetic field normal to it (7) and afterwards in the case of a bismuth plane plate coincident with an equipotential surface of a non-uniform magnetic field. (8) We shall briefly relate the conditions in which these experiments were made and the results obtained. The magnetic field was produced by the large Weiss electromagnet of the Physical Institute of Rome. In order to obtain a uniform magnetic field, the electromagnet was provided with cylindrical pole pieces having a diameter of 10 centimeters. (When the pole faces were 1.4 centimeters apart, by sending in the electromagnet a current of 4 amperes a field of about 6700 gausses was obtained. The plate had the form indicated in Fig. 30; it was provided, in the points indicated in the figure, with point electrodes obtained by soldering at the said points four thin copper wires. (These wires were protected by rubber tubes to prevent their touching the pole faces of the electromagnet). To avoid the thermoelectric forces, the plate was enveloped in cotton. By means of two conveneient interrupters it was possible to connect two electrodes A and B of the plate with the

Page 309 1921] Volterra: Flow of Electricity in a Magnetic Field 309 poles of a battery of accumulators and two other electrodes C and D with a Hartman and Braun galvanometer provided with movable coils, and vice-versa. In the circuit of the battery and of the plate was inserted a high resistance in order to maintain constant the current when the electrodes A and B were exchanged with the electrodes C and D. As the current in the plate was interrupted soon after the reading of the impulsive deflection of the galvanometer, it was possible to avoid the influence of the thermoelectric phenomena which accompany the Hall effect. The zero of the galvanometer was observed after the creation of the magnetic field. The following are the results obtained by Dr. Tasca with a field of 6000 gausses and a current in the plate of 0.2 amperes: Direct field Inverted field VII - VIv 420 104 VI -VIII 104 420.5 P A B C D/ /////// Q Fig. 31 (The differences of potential were measured by means of the corresponding deflections of the galvanometer expressed in millimeters. It is understood that when the difference of potential was measured between II and IV the current entered and went out through I and III, and vice-versa). These results show clearly the law of reciprocity to be verified, as theory requires, only when, by exchanging the electrodes the field is also inverted; the asymmetry of the effect without the inversion of the field is very remarkable. Dr. Tasca afterwards substituted for the plate represented in the Fig. 30 a rectangular bismuth plate (Fig. 31) provided with copper electrodes of negligible resistance soldered along the two longest opposite sides and with two thin copper wires soldered in the middle points of the two free sides. The results obtained are the following Direct field Inverted field VE- VF 51.5 54.5 VP-VQ 54 51 Also in this case the theorem of reciprocity is completely confirmed.

Page 310 310 University of California Publications in Mathematics [VOL. 1 Dr. Tasca has made the following observation on the mechanism by which the law of reciprocity is maintained in the case of wide copper electrodes: the copper electrodes P and Q, when the current enters and goes out through them, cause the recombination of the charges which owing to Hall effect, reach the free edges; P and Q thus diminish this effect. Whereas, when the current enters and goes out through the point electrodes E and F, the copper electrodes P and Q absorb a good part of the current; the current which flows in bismuth is therefore only a part of the total current sent into the plate and thus the Hall effect is diminished also in this case. Theory demonstrates and experiments confirm that the diminution is identical in the two cases; we could not have foreseen intuitively this result. Dr. Tasca has lastly tested my theorem of reciprocity in the case of a plane bismuth plate situated along an equipotential surface of a non-uniform magnetic field. (See third lecture, Section 4); this latter was obtained by providing the Fig. 32 Weiss electromagnet with pole pieces having the form of truncated cones and terminating with circles of 5 millimeters diameter; in those conditions the intensity of the magnetic field assumed values extremely different in the different points, from some hundred of units to about 15000 units. The plate, being circular in form and provided with point electrodes asymmetrically soldered (fig. 32), was situated along the equatorial plane of the electromagnet; for evident reasons of symmetry, this plane is an equipotential surface of the magnetic field. In this case the following experimental results were obtained: Direct field Inverted field Vi -V Vi 256 21 Vii - Vv 20 256 The action of the field is very remarkable; in fact the value of the difference of potential by exchanging only the electrodes passes from 256 to 20, but by inverting the field we have again the original value. In the study of the problem of which we have spoken in the preceding lectures I had not taken into account the secondary actions for which a conductor loses

Page 311 Volterra: Flow of Electricity in a Magnetic Field 311. the character of isotropy, with respect to its electrical conductivity, when it is subject to the action of a magnetic field; but I remarked that the perturbations caused by these secondary actions do not take place when the magnetic field is perpendicular to the plate at every point of this latter. (Third lecture Section 2). It is for this reason that Dr. Tasca, in his experiments, tested my theorem of reciprocity only in the case of a plate placed along an equipotential surface of the field. In 1916, Miss Freda, starting from the theoretical results obtained by her (see Section 1 and 2 of this lecture) was able to remove this restriction, and wished to verify experimentally that my theorem of reciprocity holds good also when the plate is situated in any way in a uniform or non-uniform magnetic field. For this verification (as for those of which we shall speak shortly) the same electromagnet was used that had already served for Dr. Tasca, with the same pole pieces, having the form of cylinders or of truncated cones, capable of producing respectively a uniform or a non-uniform field. The experimental arrangement adopted was the same of which we have already spoken with reference to the experiments of Dr. Tasca. For the said verification a bismuth disc was used, of diameter cm. 4.5 and of thickness 2 mm, provided with four point electrodes ABCD asymmetrically situated. Let us denote with d the least distance between the pole pieces, with I the intensity of the current in the electromagnet, with i the intensity of the total current in the conductor. With a uniform field, for d=5 cm., I=5 amperes, i=0.2 amperes, having placed the disc in such a way that its plane forms an angle of about 45~ with the lines of force of the field, the following results were found: Direct field Inverted field Without field VA-VB 26 53 38 VC-VD 53.5 26 38.5 With a non-uniform magnetic field, for d= 2.6 cm., I = 6.3 amperes, i =.2 amperes, having placed the disc in such manner that its plane forms an acute angle with the equatorial plane of the electromagnet, these results have been obtained Direct field Inverted field Without field VA-VB 18 71.5 38 VC-VD 71.5 18.5 38.5 According to the theoretical results summarized in the Section 2 of this lecture, my theorem of reciprocity can be extended to the case of a three-dimensional conductor, isotropic without the magnetic field. Miss Freda wished to verify experimentally this result obtained by her. She employed for this verification a bismuth prism having as base a square, with the side of 1.5 cm. and having the height of 6 cm. The prism was provided with electrodes, situated in any way along its surface

Page 312 312 University of California Publications in Mathematics [VOL. 1 (fig. 33) or with point electrodes and laminar copper electrodes of negligible resisttance(fig. 34). The following results were obtained by using the four point electrodes ABCD (fig. 33) with a uniform magnetic field, for d= 3.4 cm., I= 7.8 amperes i = 0.43 amperes, the prism having been placed with its lateral faces inclined with respect to the lines of force of the field: Direct field Inverted field Without field VA-VB 411 421 395.5 VC-VD 421 411.5 395.5 With a non-uniform magnetic field, for d= 1.8 cm., I=7.8 amperes, i=0.58 amperes, by using the four electrodes H'B'C'D' (fig. 33) the differences of potential had these values: Direct field Inverted field Without field VA'-VB' 20 13.5 15 VC- VD' 13.5 20.5 15.5 B" -----------— D —aC L X C ";.- - -— / --- —--- \ D-D' \ --- — A!A **"\ r \1......- c_. ' ti X - Fig. 33 Fig. AAM Fig. 33 Fig. 34 (In this experiment the face provided with the electrodes B'C' and the opposite face provided with the electrode D' were parallel to the equatorial plane of the electromagnet). After having soldered to the prism the two laminar electrodes A" B", the theorem of reciprocity was tested by using these electrodes and the point electrodes C"D". (fig. 34) Then with a non-uniform magnetic field, for d= 1.8 cm., I= 7.7 amperes, i=0.59 amperes, having placed the prism in such a manner that the faces provided with the laminar electrodes were inclined with respect to the equatorial plane of the electromagnet, the following results were obtained: Direct field Inverted field Without field VA,'-VB" 383 387 367.5 VC'-VD"' 387 383 367 As we see, in the case of the prism the action of the field is less remarkable than in the case of the disc, but the theorem of reciprocity is always verified. Miss Freda wished then to show if this theorem held good for a piece of bismuth crystal (obtained artificially) of a quite irregular form and provided with four point electrodes asymmetrically soldered along its surface. (It is to be remarked that if a negative result had been obtained with or without the action of the magnetic field the existence would have been proved of cases of aeolotropy for which the

Page 313 1921] Volterra: Flow of Electricity in a Magnetic Field 313 conditions (38) or (38') of the Section 2 are not fulfilled). The following are the values obtained for the differences of potential with the cylindrical pole pieces, for d=4.8 cm., 1=7 amperes, i=0.31 ampere and for any position of the crystal: VA-VB VC- VD Direct field 104 156.5 Inverted field 157 104 Without field 119 119 The values obtained with the non-uniform field, for d= 2 cm., I=7 amperes, i=0.37 ampere, the crystal being placed in an arbitrary position in the field are the following: VA -VB Vc-VD Direct field 115 215 Inverted field 215 114.5 Without field 136 136.5 Fig. 35 As we see, also in the case of the crystal the theorem of reciprocity is completely verified. In all these tests of the theorem of reciprocity not only the experiments of which we have shown the results have been made, but many other experiments by either exchanging the electrodes with each other, or by changing the position of the conductor in the field. The action of this latter was more or less remarkable in the various cases, but the agreement with the theory was always equally good. SECTION 7 We shall now speak of a consequence of my theorem of reciprocity stated and verified by Professors Corbino and Trabacchi (9). Let us consider a disc in which the current enters and goes out through two concentric circular electrodes of negligible resistance; the lines of flow will be radial without the magnetic field, but will become logarithmic spirals under the action of this latter. Let us supose the disc cut along two of these spirals; then the distribution of the currents is not altered and therefore we can limit ourselves to consider only the plate that we have detached (fig. 35) and that has its free boundary formed by spirals. If the magnetic field be inverted the lines of flow will

Page 314 314 University of California Publications in Mathematics [VOL. I undergo a marked deformation, whereas the specific resistance of the metal will maintain the same value. It would seem that the remarkable alteration of the lines of flow which takes place on reversal of the field had to be accompanied by a change of the total resistance of the plate, that is by a change of the ratio between the difference of potential of the two electrodes and the intensity of the current flowing in the plate. When the experiment was made, however, a negative result was obtained. This result could neither have been foreseen nor explained, intuitively; whereas we can easily demonstrate that it is a consequence of my theorem of reciprocity. In fact, let ABCD be four point electrodes placed on the wide copper electrodes of the plate (fig. 35). The theorem of reciprocity, which holds good also for large electrodes of negligible resistance, can be applied to the points ABCD; therefore the difference of potential, that we have between C and D, when the field is direct and the current enters and goes out through A and B, must be equal to the difference of potential that we have between A and B, when the field is inverted and the same current enters and goes out through C and D. But the said differences of potential (between C and D, between A and B) both coincide with the difference of potential that we have, respectively in the two cases, between the large electrodes of negligible resistance; therefore the inversion of the field (however the distribution of the lines of flow may be altered by this inversion) can not change the total resistance of a plate in which the current enters and goes out through electrodes of negligible resistance. Miss Freda has extended this result to the case of a three dimensional conductor provided with electrodes of negligible resistance and has verified it in the case of the bismuth prism provided with laminar electrodes, represented in fig. 34 (1). SECTION 8 Professor Corbino in his memoir cited in the first lecture, has observed that, if a rectangular plate be provided along two opposite sides AB and CD with electrodes of negligible resistance P and Q (fig. 31), owing to the presence of these electrodes, the free sides AC and BD of the plate are short-circuited. Then the Hall effect, owing to the presence of electrodes of negligible resistance, undergoes a remarkable diminution which rapidly increases as one goes from the center of the plate (points E and F) towards the electrodes. Dr. Tasca (7) measured this diminution using the same rectangular bismuth plate (fig. 31) with which we had tested the theorem of reciprocity (Section 6). The plate had the side AB 52 mm. long and the side AC 22 mm. long. With a uniform field of 6000 units, sending the current through P and Q, Dr. Tasca obtained VE-VF=52.5; taking off P and Q and sending the current through the point electrodes M and N, he obtained VE-VF=318. As we see the depression of the Hall effect caused by the electrodes of negligible resistance was in this case very remarkable. With a square bismuth plate Dr. Tasca observed a smaller, but always observable diminution. In Section 4 of the first lecture we have stated that, if the lines of flow do not vary under the action of the field, by considering two points A and B at the boundary

Page 315 Volterra: Flow of Electricity ir al Magnetic Field 315 and at the same potential when the magnetic field is zero, we have VA - VB = = K(1+X2); then the Hall effect in this case depends only on the intensity of the field, on the intensity of the current flowing in the plate, and on the nature of the metal by which this latter is formed. Therefore, bearing in mind the principle of the point electrodes at the boundary, we can state that if the current enters and goes out through point electrodes situated at the boundary of the plate, the Hall effect (difference of potential between two points of the boundary at the same potential when H =0) depends neither on the form nor on the dimensions of the plate, nor on the position of the electrodes, nor of the points between which the effect is measured, but only on the intensity of the field, on the intensity of the current and on the nature of the metal. On the other hand, for a plate provided with electrodes of negligible resistance the Hall effect depends on the form and on the dimensions of the plate, on the nature and on the position of the electrodes and on the position of the points between which the effect is measured. This observation can partly explain the divergences of the results obtained by the various physicists in the measure of the Hall effect. On this basis Dr. Tasca has proposed to designate by normal Hall effect the effect measured using point electrodes at the boundary. He has observed that from the historical point of view it is to be remarked that the condition of the point electrodes on the boundary corresponds to the primitive arrangement of Hall. This arrangement was afterwards abandoned by physicists because they thought the conditions were theoretically simpler with the rectangle provided with large electrodes. SECTION 9 In Section 2 of the second lecture we have studied the flow of electricity in a circular plate subject to the action of a uniform magnetic field normal to it. According to the results then stated, we can say that in the case in which the plate is provided with two point electrodes A and B the potential V is given by the formula -J/log rA r V= 2 (log-+cos 23 log -sin 2a); 2rK\ rB rB1 / A1 and B1 are the inverse points of A and B with respect to the circumference that forms the boundary of the plate; Q is the angle under which the points A1 and B1 are seen from a generical point. In 1915 Miss Alimenti considered the problem of constructing the equipotential lines corresponding to the preceding formula, in the case in which the electrodes are situated in the middle points of two perpendicular radii of the plate. She observed, therefore, that the electric field corresponding to the potential V, can be considered as resulting from the superposition of three electric fields, the potentials V1, V2, V3, of which are given (neglecting a constant factor common to all three) by the formulae: V = log A; V2= cos 20 log rA V3= sin 2Q2. rB rBl1

Page 316 316 University of California Publications in Mathematics [VOL. 1 By combining the curves Vi = const. (circles orthogonal to that passing through A and B) with the curves V= const. (circles orthogonal to that passing through A1 and B1), Miss Alimenti obtained the curves V1+V2=const. represented in the figure 36; by combining these latter with the curves V3=const. (circles passing through A and B) she obtained the curves V1+V2+ V3= const., that is the curves V=const. represented in the figure 37. As other phenomena (Ettingshausen, Nernst, etc.) superpose themselves on the phenomena which the theory considers, it would have been difficult to verify experimentally the form of the curves V= const. Therefore Miss Alimenti verified otherwise some points of the theory. Fig. 36 The difference of the potential between two points I and II of a circular plate, perpendicular to the lines of force of a uniform magnetic field H and provided with two point electrodes A and B, is given by the formula: =V1-Vll=27K logr -log 11+cos23 log r o1 g rAr - sin2f(-11) 2K rB1 grBlo rBll rBlI Let us denote with V' the potential corresponding to the inverted field; let us put 5= V' -V'1; since on reversal of the field only the sign of changes, we shall - 6' -J have 2 2 K sin 2 (Q21-i - ). (The values of vi, v2, N1, N2, in the expressions of / and K may depend on the absolute value of H but not on its sign; therefore these values do not change on reversal of the field (see Section 1 of this lecture).

Page 317 1921] Voilterra: Flow of Electricity in a Magnetic Field,317 a -a' Miss Alimenti has remarked that 2 must have the following properties: 2 s-It must have a maximum corresponding to the two points at the extremes of the diameter perpendicular to the straight line which passes through A and B; for two points, however situated on this diameter, '6 and a' must have equal absolute values but contrary signs. ca-It must be zero for two points however situated on the same arc of a circle passing through A1 and B1; i.e. the inversion of the field must leave unaltered the difference of potential between two points situated in the said manner. 'A.V,'A~~ B 'IV' A E _ Fig. 37 — It must be constant for two points however chosen on two circles both passing through A and B. Miss Alimenti verified experimentally the two last properties (a and 0) and then the theorem of the four vertices (see second lecture Section 3) in the particular case of a circular plate. For making the experimental tests, she used a bismuth disc situated perpendicularly to the lines of force of a uniform magnetic field, obtained by means of an electromagnet. The difference of potential between two points was measured by means of the corresponding deflexion (expressed in mm.) of a galvanometer connected with the same points. In the first experiment six thin copper wires were soldered to the plate in the points A,B,C,D,E,F, (fig. 38); the points A and B, through which the current entered in the plate, were situated in the middle points of two perpendicular radii; the points C,D, and E,F, coincided respectively with two generical points of two circles

Page 318 318 University of California Publications in Mathematics [VOL. 1 passing through the inverse points of A and B. In these conditions the following results were obtained: Points connected with the 6 2 galvanometer CD 532 532 0 D E 325 280 22.5 C F 420 375 22.5 The values of the first line confirm the property a; that of the second and of the third line confirm the property 0. Fig. 38 Fig. 39 In a second experiment the points C and D were placed on another circular are passing through the inverse points of A and B (Fig. 39) then for 65' and 2 these values'were obtained: Points - 5' connected with the - galvanometer C D 458 458 0 E F 253 253 0 D E 342 246 48 C F 435 339 48 C E 227 108 59.5 D F 116 -3 59.5 The values of the two first lines confirm also in this case the property a; the values of the third and of the fourth line, of the fifth and of the sixth line, confirm the property 0.

Page 319 Volterra: Flow, of Electricity in a Magnetic Field 319 Let us observe that, when the point electrodes A and B are at the boundary, the points Al and B1 coincide with A and B; the circles passing through A and B coincide with the lines of flow (with or without the magnetic field. Now let us consider the results of the third experiment made by Miss Alimenti. She placed the points A, B, C, D,E,F as is indicated in figure 40, that is, the points A and B were at the boundary, the points C, D, E, F coincided with the vertices of a quadrilateral formed by two circles passing through A and B (lines of flow) and by two other circles orthogonal to the first (equipotential lines corresponding to Fig. 40 the value zero of the magnetic field); thus she could also test the theorem of the four vertices. (See second lecture, Section 3), in the case of a circular plate. These are the results obtained: Points Difference of - ( - 5' connected with the potential with- 5 6' - -^ galvanometer out the field corrected values 2 2 C E -7 41 55 41+7=48 -55+7 =-48 48 D F 2 50 46 50-2=48 -46-2=-48 48 The slight differences of potential existing between C and E, and between D and F without the magnetic field, show that the equipotential lines were not constructed with a perfect exactness; therefore the values of 6 and 6' have been corrected, taking into account the said differences of potential and the sense of the deflections of the galvanometer. The corrected values of 6 and 6' confirm the theorem of the 6- 61 four vertices; the values of - confirm the property 0; we have the confirmation 2 of this latter also without correcting the values of 6 and 5', and it must be so because this property is independent of the position of the points C, D, and E, F, situated respectively on two circles passing through A and B. All these experimental results, together with that of which we have spoken in the preceding paragraphs, constitute a completely satisfying confirmation of the theory developed in these lectures.

Page 320 320 University of California Publications in Mathematics [VOL. 1 BIBLIOGRAPHY (1) Elena, Freda: Rendiconto Accademia dei Lincei, 2~ sem. 1916, p. 28, p. 60. (2) Maxwell: A Treatise on Electricity and Magnetism, sec. 297, p. 346; sec. 303, p. 350. Oxford, Clarendon Press, 1873. (3) Elena Freda: Rend. Acc. dei Lincei, 20 sem. 1916, p. 104, p. 142; Nuovo Cimento, 1916, XII, p. 177. (4) Corbino: Nuovo Cimento, 1911, t. I, p. 397. (5) Adams and Chapman: Phil. Mag., 1914, Vol. XXVIII, p. 692. Keith K. Smith, Physical Review, 1916, Vol. VIII, p. 402. (6) Corbino and Trabacchi: Nuovo Cimento, 1915, t. IX, p. 80. (7) Tasca Bordonaro: Rend. Acc. dei Lincei, 1~ sem. 1915, p. 336. (8) Tasca Bordonaro: Rend. Acc. dei Lincei, 1~ sem. 1915, p. 709. (9) Corbino and Trabacchi: Nuovo Cimento, 1915, t. IX, p. 118. (10) Adalgisa Alimenti: Nuovo Cimento, 1915, t. IX, p. 109; 1916, t. XI, p. 217.

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 14, pp. 321-343 November 11, 1920 THE HOMOGENEOUS VECTOR FUNCTION AND DETERMINANTS OF THE P-TH CLASS BY JOHN D. BARTER CONTENTS PAGE I. The Hom ogeneous Vector Function............................................................................. 321 1. G eneral O utline................................................................................................ 321 2. Survey of A pplications.................................................................................... 322 3. Notations and Definitions.................. 323 4. Homogeneous Vector Function of the mth Degree.................................................... 324 5. First Addition Theorem................................................................................. 325 6. First Addition Theorem in case m = 2.................................................................. 327 7. Transformation of Matrix Expression............................................................ 328 8. Derivation of Alternative Expression for Vector Operator..................................... 329 9. Products of Extensive Magnitudes...................................................................... 331 10. Transformation of Matrix Expression................................................................... 331 II. Determinants of P-th Class................................................................... 332 1. Introductory................................................................................................... 332 2. Fundamental Properties of Determinants of Even Class............................................ 334 3. Note on Determ inants of Odd Class..................................................................... 335 4. Transform ation of Form s................................................................................... 335 5. Products of Determinants of Higher Class.............................................................. 336 6. Exam ple of Binary Quartic............................................................................... 338 7. Coefficient of a Given Term in a Symmetric Determinant........................................... 341 8. Differential Invariants and Covariants................................................................ 341 I. THE HOMOGENEOUS VECTOR FUNCTION 1. General Outline The object of the present paper is to present the properties of a vector function, which may be considered as the generalization of the linear vector function. At the same time we shall attempt to develop the various analytical representations, and to a certain extent indicate the applications. 321

Page 322 322 University of California Publications in Mathematics [Vol. 1 Our homogeneous vector function of the mth degree is defined as satisfying the relation: kme. (r) = p(kr), where r is any vector, and k is a constant. Rather more general would be a vector function satisfying the relation: f(k) p(r) = (p(kr). We shall not in this part specifically occupy ourselves with the latter, beyond indicating that it may in general be expressed in terms of the former. Moreover we may, and shall, confine ourselves to the case when m is a positive integer. The case when m is a negative integer involves the investigation of the inverse function, and will be treated under that head. In this present part we further confine ourselves to a method of representation analogous to that of matrices or dyadics, employed in the case of the linear vector function. It will appear that there is a marked contrast between the properties of the latter and those of the general vector function. However in this case also there present themselves singular vectors which play much the same r6le as the axes of the linear vector function, and are in consequence assigned the same term. On the other hand those characteristic numbers which are termed the latent roots of the linear vector function, will readily be seen to have no analogues in the general case. We shall by this method, partly for their own sake, partly for the sake of later comparison, when a different mode of representation, that by means of quarternions, is studied, develop certain of the most fundamental properties, which we believe to have an interest of their own. In the second part will be given the algebraic theory leading to the determination of the axes of these vector functions. The method will be found closely analogous to that employed in the case of the algebraic theory of eliminants, and its results are, we believe, of some value in the theory of differential equations. The third part is to be confined to a survey of the geometrical applications, the vector function being regarded as determining a transformation of space. The fourth part is to be devoted to a brief development of the theory by way of quaternions, and to a comparison of results. 2. Survey of Applications The concept of the vector function enters into three branches of mathematics. In mathematical physics are encountered two distinct types. The first is that determined by two localized vector sheafs, functionally related. As examples of this type may be quoted the relation between the stress and strain vectors in the theory of elasticity, and which, it may be stated in passing, is completely and adequately represented by the self-conjugate linear vector function. Further, there present themselves such relations as that existing between the vectors of electric force and displacement, or between magnetic force and induction. In

Page 323 1920] Barter: The Homogeneous Vectorfunction 323 these latter cases a more general form of representation is possibly called for than that provided by the linear vector function. The second type is that of the general vector field in which the characteristic vector is to be regarded as a function of the vector of position. This form can probably be most satisfactorily treated by means of quaternionic functions, as we hope to show at a later point. In geometry the general vector function represents, as already stated, a very general transformation of space. In consequence of the property it may be employed to present many geometric relations. The transformation is a collineation only in the case of the linear vector function. In the general case only lines through the origin, which remains fixed, are transformed into straight lines which likewise pass through the origin. It will furthermore appear that in the case of vector functions of odd degree the surface defined is closed, while in the case of even degree the surface is entirely confined to the positive semispace. In purely analytic applications this function can be caused to play an auxiliary role, in virtue of the relations between quantics which are uncovered by an investigation of its properties. In the second part this aspect will be given some consideration. 3. Notation and Definitions In the present paper the symbolism plays such an important part that it may be considered as an example of the applications of the method, capable, we hope, of much wider extension. We shall accordingly proceed with a precision of the terms employed. Let there be given a system of nm+l distinct terms, and let each be represented by a symbol of the form aplp2... Pm+l in which each of the m + 1 suffixes may assume all values from 0 to n, there being obviously the requisite number of such distinct symbols (nm+l). The aggregate of these symbols will be denoted: (an...(m+l)...) = (a(m+l)), (1) and will be termed a matrix of the nth order and class m + 1. Then the product of a matrix of class m + 1 and one of class 1, will be defined thus: (an...(m+l)...n) (bln) = (lan,...(m).5bmp1), being evidently a matrix of class m and order n. We may accordingly again multiply this matrix by a matrix of class 1, obtaining a matrix of class m - 1: (aa...(m+l)...n) (blA) (b2n) = ( a an... (m)...anpblp (b2n) = ( E...pl2blp ) (2) pl=l plp2=l By proceeding in this manner we may successively reduce the class of the matrix

Page 324 324 University of California Publications in Mathematics [Vol. 1 at each multiplication by unity, obtaining after m such multiplications a matrix of class 1: (an...(ml)...n)(bln)(b2A )... (bmn) = ( apl...pmblplb2p2 ' ' bm-lpm-lbmpm) (3) where the summation is to be extended as before, each subscript separately being ascribed all the values from 1 to n. We may, and in the sequel frequently shall, denote a vector by a matrix of class 1, in which the constituents are simply the components of the vector. If required, the fundamental vectors of reference may be introduced by means of another matrix of class 1, whose constituents are these unit vectors. Then any vector may be expressed as the product of two matrices of class 1, one of which has numerical constituents, and the other unit vector constituents, thus: = (an) (ai). In general we shall consider the vector matrix as understood, and denote a vector simply by the corresponding numerical matrix. Now it is evident that by the process above indicated, we are able to define a matrix of class m - n as a function of n given vectors merely by multiplying a matrix of class m by these n vectors, which may or may not be distinct. In general of course the order in which the multiplication is considered as executed is essential. In the special case when n = m - 1, we have thus defined a vector as a function of n given vectors. This is the case specifically treated in this paper, but it may be pointed out, at this stage, that under certain circumstances it may be advisable to consider the more general standpoint. Thus for example we may consider n points in space as defining at each point a set of n vectors, which in turn may by multiplication with a matrix of class m, in the manner indicated, be considered as defining a matrix or vector operator of class m - n. It is not difficult to see that in some problems such a device may be of real service. 4. Homogeneous Vector Function of the mth Degree We are now in a position, by means of the symbolic resources developed in last section, to set up the expression for the homogeneous vector function of the mth degree, as this term was defined in the first section, as follows: Pm(g ) = (Pm {(bi)(aan)} - pm(bin), mn (4) = (a, (m+l)) (bna,),)m (aa(ml)) (b)m. This expression is evidently a matrix of class 1, and accordingly, as already pointed out, may be regarded as a vector, and functionally related to the given vector as follows: (Pm(kbj) = km. m ((bn), which, by definition, is characteristic of the homogeneous vector function of the mth degree. We shall accordingly employ this notation to develop its properties.

Page 325 1920] Barter: The Homogeneous Vectorfunction 325 We shall in the first place develop the addition theorems, that is to say, set up the expressions for the sum of the vector functions of a set of vectors, and also the expression for the vector function of a sum of vectors. Each of these relations will be developed first for the special case of m = 2, not because the general case presents any additional difficulty, but because in the latter certain features, which are prominent in the former, become more subsidiary, and less interesting. We shall also determine an alternative form for this function, analogous to, and to be regarded as a generalization of the m-adic of J. W. Gibbs. 5. First Addition Theorem We shall give the proof of this theorem in its most general form, since from this standpoint its essential properties are more sharply defined than would be the case if we made a more special assumption regarding m. We have then: cpm(bll + b2n) = (aj(m+l)) (bl, + b2,)m, _n (5) = ar1...rm(bl + b2)r(bl + b2)r *... (b + b2) r, rl...rm=l where the subscripts outside the brackets are the second, those inside the first subscripts to be ascribed the b's. Let m = qlalq22... q,, where each of the q's is a prime. If, as customary, we define the special mth roots of unity to be those' mth roots of unity which are not also nth roots of unity, where n < m, then the following propositions are well known, or may be readily deduced: I. There are m( 1 — ) ( 1i) ( 1 -) special mth roots of unity. II. If o is any special mth root of unity, then w2 when r is prime to m, is likewise a special mth root and all the special mth roots are of this form. III. If o is a special mth root, then all the roots are given by the series: co0 l1 2 (3... m-1 IV. The sum of all the mth roots is zero, whence it follows immediately that if co is a special mth root: m-1 iE A = 0, /L=O V. More generally, if e is any mth root of unity then: EK = 0. K=1 VI. It therefore follows immediately that, if co is any mth special root, then: E ( ) = o, fK= for all values of q < m.

Page 326 326 University of California Publications in Mathematics [Vol. 1 VI. The sum of all the special mth roots is 0. VII. If w, be any special yuth root, then: Up E ( wq = o, (6) K=1 for all values of q, prime to u. We shall now apply these properties of the special mth roots to the problem in hand. Let w be such a special mth root of unity. Consider the product: n (agm+l)(blf, + b2nWK)ln = E ai r,....rm(b +,b2)... (bl + oKb2)r. (7) rl...rm-=l In this product any term which involves b2 q times, and consequently bl (m - q) times, has the factor wc, in addition to a vectorial factor which we may reserve for future consideration. If then we form all the distinct products obtained by ascribing to k all values from 1 to m, and take their sum, then in the latter such a term as above indicated will be multiplied by the factor: E (WKq) cK=1 which in virtue of the properties of the special mth roots of unity, as above developed, is zero. Accordingly this sum simply reduces to the sum of the vector functions of bin and b2n, multiplied by m. That is to say: m E (an(m+l))(bl + cKb2n) = m{ (a.(ml))[(bl,) + (b2n)]m}, K=l = m{(pm(bln) + (pm(b2i)}, (8) = (ani(m+)) (Kbln + b2), K=l where the latter relation follows from considerations of symmetry, and requires evidently: (Pm(Cpobin + b2n) = <Pm(bln + coKb2n). This latter condition is satisfied, for: m{ "m-q(bl + Wqb2n)} =- (pm(m-qbln + b2n) = mm(m- q) 'pm(bln + q1qb2n) = Pm(bl n + b2nC3 q). By a repetition of the procedure we obtain: m m m Z m<(bi + Ab2~ n + Aob3n) = m * E (b1i + wA"b2.n) + m2w2m(b3,), A2=1 Ai=l Ai=l (9) = m2{fpm(bin) + (Pm(b2n) + pm(b3in)} It is evident that the general expression is obtained by successive repetitions

Page 327 1920] Barter: The Homogeneous Vectorfunction 327 of the above process, in the form m m O P E * * * E (b ln + Z -bsn ma-1. (pm(bsn), Ap=l A2=l1 s=1 s =1 representing an identical relation between two sums of vector functions of the same order and degree. It may be pointed out that these are not the only identities which may be obtained by an analogous process. 6. First Addition Theorem in Case m = 2 As a special case of the above theorem, when m = 2, we have since then co = - 1, o2 = 1, the following: (P2(bln + b2n) + (P2(bln - b2n) = 2. { 2(bl) + P22(b2n)}. Now on each side of this equation occurs a sum of two vectors. Consequently if this is regarded as giving an identical expression for the sum of the vector functions of the vectors (bn), (b2n), on reading from right to left, then the application of this same theorem to the sum of the vector functions of (bin + b2n), (bln - b2), on the left must give an expression reducible ultimately to that occurring on the right of the above equation. That in this simple instance this is as a matter of fact the case, is obvious. For we have: (b1i + b2n) + (b1n - b2n) = 2bli, (b 1 + b2n) - (b1i - b2n) = 2b2 and therefore: 'p2(bln + b2n) + ( 2(bln - b2n) = 12 i{2(2bln) + 2(2b2fi)}, = 2 {2(bl) + (P2(b2n)} which is the same result as above obtained. Consider similarly the more general case of this theorem when m = 2: (p2(bln + b2n + b3t) + (2(bin + b2i - bil) + (P2(bln - b2n + b3n) (10) + (P2(bln - b2n- b3) = 22. { P2(b1n) + (2(b2n) + (p2(b3n)} If we consider this as giving an identity for the vector function sum on the right, then conversely the application of this addition theorem to the sum on the left into which enter the vectors: (bin + b2n + b3n), (bin + b2n - b3n), (bln - b2n + b3n), (bin - b2n - b3n) an expression must be obtained which ultimately reduces to the expression on the right. The reader may readily convince himself that such is the case, though the formal proof is too tedious to be given here.

Page 328 328 University of California Publications in Mathematics [Vol. 1 Finally consider the most general case of the theorem for m = 2: P 22(YZWbsi) = 2P-1. Z 2(bsn), (11) where the first summation i1, is to be extended over the b's, ascribing bi the positive sign and to the remaining b's either the + or - sign, and then extending the summation 22 over all such expressions without repetition, obtained by taking all possible combinations of sign. There are evidently 2p-1 such terms. Now those relations which have been indicated in the preceding simple cases must also hold here. That is to say, the application of the general law obtained by reading the equation from left to right or right to left must ultimately yield the same expression. That such is the case is not self-evident, in fact it gives rise to a theorem, which however does not appear to be of sufficient interest to state explicitly. 7. Transformation of Matrix Expression In the preceding we have based the definition of the vector function operator spm on a matrix expression (an(m+l)). We shall now transform latter into an expression which is the generalization of the dyadic expression of Gibbs. Evidently cPm(bn) = (aA(m+l))(bln)m(en) = E aplpl, pl=l where we assume that the matrix (a,(m+l)) represents the vector function <pm with respect to the set of independent vectors (e,), and the vector (,n)(bn) is written (. Further in this equation we have written: m-1 c o pl = (apln....).)(bn)(n) = ( E p2aplp2a(m_).. ) (ba)m-)1 p2=1 = ( E sPP2) * P2=l and m-2 aPlP2 = (aplp2n(m-l)...n) (bn) (en), (12) = ( E P1P2P3) p)' (1 P3=1 and m-3 taPlP2P3 = (aplP2P3n (m-2)...n) (bn) (e~n), == [ P4CtP W'r ).EP \ P4=l * etc., etc. Finally: a1...Pm-l = (apl... Pm, n) (b) (Ef), ( V-\ Em,.. )e(13) - [ 2Pw aPl"'.*Pm ' PM pm=l

Page 329 1920] Barter: The Homogeneous Vectorfunction 329 and =(-=(~ ~ ~~~~~-~) t alp..Pm= ( E2ai P...mP+l ) Consequently we obtain: m fW o (a (m+l)) (b-) (c)) = E ~i EP2 E p3 pl=1 p1 pa3=1 (4)! 2 (14) X { * p {m+laplp. },-1 ' ' 3 I J Consider now the special case that each of the forms (arn......(..n) (bn)m, P = 1 ** n, is resolvable into linear factors, so that we have: (apln...(m)....n) (bn)m (Cpe 2n) (a) (Cpp2n) (b,)... (cpm,) (bN) = (pll 1)(*p12' ) '' (Y^pl1 m), (15) where Tplr = (Cpri) (e). Then the above expression reduces to the simpler form: (pm(5) = (El1T1112T13 * * Tlm + * + ~nYniYn2Yn3. * ymn) ' (). (16) This is the direct generalization of Gibbs' Dyadic, and will be termed the m-adic. 8. Derivation of Alternative Expressions for Vector Function Operator In the above we showed that a vector function which was represented by a matrix of the (m + 1)th class was as a matter of fact a homogeneous vector function of the mth degree. In this section we shall show conversely that every homogeneous vector function may be expressed as such a matrix. Let I be a vector function operator operating distributively, but not necessarily commutatively, so that we have: ()- (a1 + 12 + * * * + an) = 4(a1) + I(a12) + + (an), each summand being again a vector operator. Whence: +. (. + a2 + Q3 + * * + an)m = * [aI + Q2 + * + A]n, symbolically, where we assume the multinomial on the right to be expanded, the order of the a's being assumed as retained in every case, and then each term in that form operated on by b. By making use of the notion of a gap expression, due to Grassmann, we may without loss of generality, replace the operator J, by an operator I, which is to be assumed as operating commutatively. Consider an expression into which enter p symbols 111213.. Ip, each in a perfectly distinct manner, and which we shall denote L. (111213... lp).

Page 330 330 University of California Publications in Mathematics [Vol. 1 Thus of course any algebraic expression into which enter both stated quantities or numerical coefficients and general symbols, is such a gap expression. The number of gaps in such an expression is that number of general symbols which it is necessary to define before the expression for the purpose in hand, is determined. Thus the number may vary, as in the following case of a polynomial in a set of variables. If the form of the polynomial alone be in question, the gaps are the constant coefficients, the variables being left undefined. If however the value of the polynomial be considered, then in addition it is necessary to define the values of the variables. Then we define as the product of L. and a set of p values (or at least domains of values), that value or expression which is obtained by replcaing all the gaps 111213.. * p, by the given values in all possible distinct combinations without repetition, and dividing the sum of the results by their number. It is then evident that the order of the operands is immaterial, that is to say, the operation is commutative. Replace now Z- ( rl W. ^2'. r8') (17), where the indices al'aa * * *a a ', are to be considered as constant for each sum, and the summation is to be extended over all possible sequences of the 0's, by the gap product operator I. Operating on the symbolic product ( a", a2... an) in which now the factors may be considered as commutative, multiplied of course by the multinomial coefficient m! a!a! *... a Then we may write for above expression: '- (01 + 02 + 53 * * + O)m = ' [01 + 52 + 3 * * + n], where the expression on the right is to be assumed as expanded by the multinomial theorem, and then each term operated on by I. It then follows from the property characteristic of the homogeneous vector operator that IC (clai + C2L2 + C3a3 + ''' + CnOn)m = [CIat + C2a2 + '. + cnan]m, (18) = (A1A2A3... A.Cn)m in the ordinary notation for an n-ary m-ic. Here the coefficients A,,,..., (18a) etc., are the products T. (11'52 2.. n") (19) as above defined. Now when we write for the above form the matrix representation, (An(m)) (C)m, in which the coefficients of the first matrix are vectors (or, as we also consider them, simply extensive magnitudes), and which is to be considered as symmetric, we have obtained an expression precisely equivalent to that which we have hitherto made the basis of our presentation. Thus if we assume, as may be done without loss of generality, that the matrix expression referred to the set of vectors

Page 331 1920] Barter: The Homogeneous Vectorfunction 331 (an) be (aA(m+l)), and symmetric with respect to all the subscripts except the first, then: (an(l)...(o,))(an) = A,,... n, etc., (20) where an()...(n) signifies a term in which wc of the subscripts are equal to 1, W2 equal to 2, etc., all terms of this common form being by assumption equal. 9. Products of Extensive Magnitudes We may also obtain the above results by considering products of extensive magnitudes, as follows. Write A = (a(+l)) = Z alpp,...pm+Elp~2p2 * * * m+lPml p=l (21)' B = (bn) = p = bp 2 = * * Pl= P2=1 where lp denotes as usual the complement of rp and accordingly Er~ = 1, or = 0, according as r + s, or r = s. Then evidently the product is of the same form as the products of the corresponding matrices, as above defined. Suppose that we multiply A by B y times, where the class of A (which is equal, by definition, to the class of the corresponding matrix) is >,y. Then we have for A By an algebraic expression, or homogeneous form, of the uth degree in n variables, whose coefficients are extensive magnitudes of class m -, + 1. We may treat this form as equivalent to a system of forms with extensive coefficients of lower class, in various ways, according to the requirements of the problem. The spirit of the method consists then in this, that a theorem which holds for any form, or system of forms, with coefficients which are extensive magnitudes of any class, may be readily, with certain limitations, which will become obvious in the sequel, generalized for all such forms. By this means the invariant relations are exhibited most naturally. Before investigating this side of the subject it is however necessary to develop to some extent the theory of determinants of the pth class. 10. Transformation of Matrix Expression Suppose that with reference to a set of n independent vectors (an), the matrix of the vector function operator be (a(m+l)). It is required to determine the transformation which this matrix undergoes when we pass to a second set of independent vectors (5n), related to the former by the relation: (22) (an) = (d)-l( n). Evidently ~. (K) = (a,) (a,(m+l))(kh)m, (23) = (n)(d)-: (a(m+l)) (d,,)(k') = () (bn(m+l)) (kC.)m

Page 332 332 University of California Publications in Mathematics [Vol. 1 Consequently (b,(m+l) = (dAAt)-l(a,(m+l))(dA )m (24) = (Dnn') (a,(m+l)) (d )m, (24) = (plapm+...pdplndp2n *P' * dpmnDp+nf) P=1 The result is evidently again a matrix of class (m + 1), in which the second suffixes of the d's and of D are the determining suffixes. II. DETERMINANTS OF THE P-TH CLASS 1. Introductory Let e1ne2ne3... ep_ln, be p -1 unrelated sets of independent extensive magnitudes, or alternate numbers, which accordingly, for all values of the first subscripts satisfy the relations: [e,eIerx2erA3... erx4] = 0, when an element is repeated, = 1, when no element is repeated, the + or - sign being ascribed according as the number of inversions of the second subscripts is even or odd. Then an extensive magnitude of the (p - l)th class and nth order is defined as the product: A = (an(m+l)) (el) (e2in) * (el). The product of n such magnitudes is evidently a scalar, and will be termed the determinant of the matrix (An) = (an...(p-l)...n), determined by these magnitudes, with respect to the first suffix. Thus no n (25) aa... (p)...n| = [A1A2 * *. An] = II aq l,....KpelcKleK2 ep-lKp_1 q=1 K=1 -In a similar manner may be defined the determinants of the matrix with respect to the remaining suffixes. Each of these determinants will consist of the algebraic sum of products of the form: ~- aal...apapl...Op *.. avl...vp there being in each product n factors, supposed ordered according to the suffix of reference, in this case the first, and every value from 1 to n being represented in case of each suffix, once and once only. The + or - sign is to be taken in the case of any given product, according as the number of inversions of the suffixes when, as stated, the suffixes of reference are ordered naturally, is even or odd. The summation is then to be extended over all possible products of this type, without repetition.

Page 333 1920] Barter: Determinants of the P-th Class 333 It is then evident that the products which enter into the expressions of any two distinct determinants are, except possibly for sign, the same in each case. We now propose to investigate the change of sign which any particular term undergoes when we pass from a determinant with respect to any given suffix to that with respect to any other. Let the first determinant be that with respect to the ath suffix, the second that with respect to the rth. The sign of any particular term in the former case will be determined by the number of inversions by which each of the remaining suffixes is affected, when the oth suffixes are ordered naturally; while in the latter case it will be determined by the corresponding sum of inversions when the rth suffixes are ordered naturally. Denote former sum thus: Ia = ql + q2 + q3 + * + q2-l + q,+l + * + ' + qp. Now if ql'q2, etc., be the corresponding number of inversions (increased or decreased by an even number, which does not affect the present question, which is simply to determine whether I is even or odd) in for the same term, in the case of the second determinant, i.e. when the suffixes are ordered naturally, then: q/ = qr + qr, etc., q = qT. Consequently IT = I + (p- 2)qT, and therefore the sign of this term in the two suffixes differs by (- 1)pqT = (- 1)pq. Whence: Theorem.-Determinants of even class are independent of the suffix of reference. In the case of determinants of odd class, the determinants are all distinct, the only term which has the same sign for all such determinants being the principal term a1...1a2...2a3...3.* an...n. From these facts alone, it will be inferred that the determinants of the matrices of forms of even degree, which of course are determinants of even class, will play a much more important role than the determinants of forms of odd degree. We shall see indeed that they are invariants of these forms. On the other hand, for systems of forms, whose matrix will have one of the suffixes distinguished from the remainder, determinants of odd class will be significant, and will indeed also be found to have the invariant property. The theory of forms of even degree will, from this standpoint, be seen to differ essentially from that of forms of odd degree. This is in harmony with the results of other methods of investigation. However it will be seen later that the theory of forms of odd degree can also be treated by this method. It will however first be necessary to develop the theory of determinants of even class, which we now proceed to do. We shall first determine certain algebraic properties of these determinants, following immediately from their definition in terms of alternate numbers, as

Page 334 334 University of California Publications in Mathematics [Vol. 1 already given. Then we shall proceed to the consideration of the multiplication of determinants, and the relation which exists between the determinant of the product of matrices and the product of the determinants of the separate factors. From this will flow readily the invariant property of determinants of even forms. 2. Fundamental Properties of Determinants of Even Class We shall set up the following: Definition.-The matrix of class (p - 1) obtained by ascribing to the rth suffixa certain value, 7rl say, and ascribing to the remaining suffixes, as before the values 1 to n, will be termed a first face of the matrix, and denoted (ril | ). Similarly a first face of this matrix will be termed a second face of the original matrix, and so forth. Theorem.-If in the case of matrices of even class, the significance of two faces with respect to the same suffix be interchanged, then the absolute value of the determinant is unchanged, but it changes sign. By this is simply meant, that if we assume the determinant as expressed in terms of the general symbols a&... p, then we shall obtain two quantities when we replace these symbols by definite values, first so that the symbols of a pair of faces with respect to the same suffix assume a given set of values, and secondly when these faces assume these same values, but in the reverse order. Then the theorem states that the absolute values of these determinants will be the same in each case, but that they will have opposite signs. The theorem is now evident from this explicit statement. For such an interchange is equivalent to an interchange of two alternate numbers in each product, which effects a change of sign. Thus interchange of the values to be ascribed to the corresponding letters of the faces (r I T) (r IT2), is equivalent to the interchange of the alternate numbers er,, and e7T2 throughout, which evidently changes the sign of every term, provided the quantities thus attributed to the letters are extensive magnitudes of even class (under this term of course being included those of zero class, that is, ordinary numbers.) The permutation of such numbers does not affect the sign of the product. If the face be one with respect to the suffix of reference, then the interchange of the values of two such faces, in the case of determinants of even class evidently also effects a change of sign, as may be inferred either from the equivalence of the different determinants, or from the fact that interchange of two such faces is equivalent to the interchange of p - 1 sets of alternate numbers, which effects a change in sign. When p is odd, however, the latter argument shows that in this case no change in sign is effected. Consequently we have: Theorem.-In the case of determinants of odd class, interchange of two faces which are not faces of reference, effects a change in sign, interchange of two faces of reference leaves sign unchanged.

Page 335 1920] Barter: Determinants of the P-th Class 335 3. Note on Determinants of Odd Class It has already been pointed out that the determinants of matrices of odd class are in general all distinct. We propose to investigate the nature of the sum of all these determinants, confining ourselves in the first place to cubic determinants. Adopt the following symbolism: For the sum of the terms, which are such that when the first suffix is taken as suffix of reference, have an even number of inversions of the second suffix, and likewise an even number of inversions of the third suffix, write T(ree), and similarly for the other terms T(reo), T(roe), T(roo). Then we have: D = T(ree) - T(reo) - T(roe) + T(roo), = T(eee) - T(eeo) - T(eoe) + T(eoo), as it may also be written more symmetrically. Further, D1 + D2 = 2T(eee) - 2T(eeo), D1 + D3 = 2T(eee) - 2T(eoe), and D1 + D2 + D3 = 3T(eee) - T(eeo) - T(eoe) - T(oee). This result may be readily generalized. These sums are then completely determined by the matrix in question, without ambiguity, but it may be easily seen that they are not invariants of the corresponding forms. 4. Transformation of Forms Consider the n-ary p-ic (aA(j))(x,) whose matrix is (a,...(p)...). If for the variables x, be substituted the variables yn, related to the former by the equation xn = yAn'rnA', then the quantic is transformed into the n-ary p-ic in these y's, given by: (an...(p)..n) (rnniYny)p, whose matrix is (an...(p)...) n) (,n)p = a q q..... qP * * * T qp, (27) where the determining suffixes are now the second suffixes of the r's. The same relations hold of course in the case of a system of m n-ary forms, whose matrix is (amn...(p)...n) and whose transformed matrix is (a-m.. (p)....n) (7ra)P More generally, suppose in the above quantic that the variables xA be replaced by quantics not necessarily linear, in the variables ya. That is, we write: (Xn) = (rnA(+l))(ya)yn. Then the original matrix is transformed into an n-ary (p a)ic. Its matrix is given by (n...(.....n): (Tn...()...n = ( aql q2 p... q * * Tq... n. qp... ), (28) ql...qp=l in which the last a- suffixes of the r's are now the determining suffixes.

Page 336 336 University of California Publications in Mathematics [Vol. 1 As before, the same relations hold if, instead of considering a single quantic, we consider a system of such quantics, the resulting expression being obtained by merely adding another subscript to the a's. Further it may be convenient to generalize the relation between the original variables, and the substituted variables in a different manner. Thus instead of substituting for the former a single definite system, we may consider them as substituted by a set of such systems, thus: (XAn) = (an= + ))(n) which, of course is merely equivalent to considering, and grouping in a single expression, the various equations determined by a set of substitutions. Nevertheless such a treatment is on occasion valuable. The set of forms, thus obtained, is given by the expression: (an...n): (7n... (e )n... (~)...n)P. (30) 5. Products of Determinants of Higher Class (1) First expression for product. Let n | an... I 1= I { (ar...A)(nlin)(n2n).. (Epn) r=l I bn...n Il = {(brn...,)(nn) (n2n).. (n fn), r=l each being the determinant of the corresponding matrix with respect to the first suffix. Then the expression: n |an...An| 1 |bf...n 1 = E { (arA...nbrh...n) (En) * * * (epn) (nin) * * * (n n) } (31) r=l for the product is obtained by combining corresponding pairs of factors in the two determinants into a single factor of the resulting determinant; the special selection of pairs in this case being those which are characterized by the same values of the first suffix. Any other combination of factors might also have been adopted. The different expressions, so obtained, have of course all the same value, except possibly for sign. The class of the product, expressed in this manner as a determinant, is p + 7r + 1. The process is evidently applicable to determinants of both odd and even class, in which respect it differs from the second expression for the product, next to be considered, which is applicable only in the case of determinants of even class in its full generality. (2) Second expression for product. This is the expression which will be of most value in the sequel. Let En be a system of extensive magnitudes of odd class so that: Er' Es = - E. E,.

Page 337 1920] Barter: Determinants of the P-th Class 337 Further if (En) = (bn...n)(nl) * * (7rn), where the n's are assumed to be simple extensive magnitudes, then the product of the E's given by [E1E2 En] = |bn...n is a determinant of even class. More generally, on one hand: II {(aa...n) (Eln )(e2) ~* * (pfn) } -= I { (ar...): (bn...n) (nil) * (tln) ' } =- H {(ar F... qb q,...n,) (nla) * (nn)(el) ''' (e~) }, (32) and on other hand: = II { (a rn...)(~(n) (~2) * * (pn) } [ElE2. En], r=l I= an-...n l | bn...n| I Consequently we are able to express the product of two determinants of classes p + 1 and,r + 1, where one at least of the determinants is of even class, as a determinant of class p + ir. Consider now the determinant of the matrix of the transformed quantic: (aAn...(...(.. n) ' ( (b (.. )a) p = a. qpbiq...b nb 2..n * bqpAn... (34) (where we shall assume the matrix in the b's to be even) taken, for simplicity, with respect to the first suffix. Then by preceding, the determinant 1 ( Adarn...Abrq...n ' (bA...nA)P-1 -(el) (~p-lA) (nlAn) *. (n -A)().* (35) r= — r=l1 (where it may be recalled that the first suffixes of the b's are to be considered as connected with the remaining suffixes of the a's in succession) is equal to: H ( arn...nbrqn...n (tin) (p-ln),(nl) *' (nA-ln) b.... n-1. (36) q=l r-=1 If now the first of the above determinants be even, which implies that the determinant in the a's is even, then it will be equal to the determinant of its matrix with respect to any other suffix. On this supposition above determinant will be equal to: i {(a rn... (p-2)... ): (br... )(E2n) * (p-ln) (lln) * (.n -ln) } 0r= (37 = H {(aAn r... )(l)C( 2a) * * * (~pln) } * |bn...n r=1 Consequently the determinant of the transformed matrix, as above defined, is, under the assumption that both the original matrix and the matrix of transforma

Page 338 338 University of California Publications in Mathematics [Vol. 1 tion are of even class: |I a n...n * bn...n P. Therefore: Theorem.-The determinant of any even p-ic is an invariant with respect to any transformation whose matrix is of even class, that is to say, whereby the original variables are substituted by quantics of odd degree. A particular case is that of linear transformation, which is that generally employed in geometry. The advantage of the present method is that the more general transformation indicated presents substantially no additional difficulties, while the scope of the theory is greatly increased, as will, for instance, become evident when Tschirnhausen's transformation is considered. It is further immediately evident that the weight of these invariant determinants is equal to the degree of the quantic. 6. Example of Binary Quartic Consider the binary quartic (abcdenx_)4 - (a2-2) (x4)4, the former of these two expressions being the familiar one classical since Cayley, while the latter alternative is in harmony with the symbolism employed in this paper. The value of its determinant is found to be a2222 = a1111a2222 - a1112a2221 + a1221a2112 - a1121a2212 + a1122a2211 - a1211a2122 _+ a1212a2121 - a1222a2111 =ae - 4bd + 3c2. This, by our general theorem, is an invariant of the quartic with respect to any transformation of even class. In the particular case of linear transformation this fact, under somewhat different form, has been known for a long time, this being in fact the invariant generally designated by the symbol I. It furthermore follows from our general considerations that its weight is two. From the above binary quartic we derive the ternary sextic by multiplying through by 2. In the determinant of this sextic, all the terms which do not involve exactly 2 suffixes of value 3, are zero. Consequently in the expansion of this determinant, any summand in which all three factors are not of this description, will be zero. Now, if we denote the number of suffixes equal respectively to 1 and 2, in the first and second factors of such a summand by n1', n2"; nl', n2"; nl', n2"; then it is evident that the following equations must be satisfied

Page 339 1920] Barter: Determinants of the P-th Class 339 by positive integral values of the n's: n1' + n2' = 4, nl n2'1 ni' + nl" -- nl"' = 6, nl" + = 4, nl = 4nn2 = 4. n2' + n21 + n2/" = 6, ni"'l + n2f' = 4; ni n2/ 6" These equations are evidently consistent, and if, for example, the first three, and one of the last two are satisfied, the remaining equation is likewise satisfied. The diagram on the right merely summarizes the relations, indicating that the sum of the n's in rows is 4, in columns 6. The following are all the possible solutions: 40 40 31 31 22 22 13 22 31 22 04 13 13 04 22. Their interpretation in terms of the coefficients is evidently: a a b b c c d c b c e d d e c each multiplied possibly by a numerical factor, whose value is, as a matter of fact, seen from the following to have in each case the value 1/15: ()a = ( )(40) or (40) = 1/15 a ()b = ( (31) or (31) = 1/15b (4) = (2)(22) or (22) = 1/15 c (38) ( d =( (13) or (13) = 1/15d (o)e = (0 (04) or (04) = 1/15e Now it is evident that if this ternary sextic be transformed by any substitution of the variables which leaves the third variable v unchanged, then it will pass into a sextic of exactly the same form, bearing the same relation to the corresponding transformed quartic, as the original sextic bore to the original quartic. The modulus of transformation of the sextic is equal to that of the quartic.

Page 340 340 University of California Publications in Mathematics [Vol. 1 Further, the determinant of the sextic being an invariant of weight six this syncopated determinant is an invariant of weight six of the quartic, provided of course that it does not vanish identically. The latter however is readily seen not to be the case, even without determining the actual numerical coefficients of the several terms in this determinant, since it involves terms, of the same form, in odd number. By the preceding it will then consist of a sum of numerical multiples of ace, ad2, bcd, b2e, c3, and the complete solution then requires the determination of these numerical coefficients in the above determinant, which is to be assumed as symmetric. In the next section this problem will be attacked, in a more special case. In the meantime we shall anticipate the result sufficiently to state that the invariant so obtained is ace + 2bcd - ad2 - b2e - c3. It is in fact the invariant which has been designated by the symbol J. These two invariants, I, J, are the only two independent invariants of the quartic, as is known from other considerations, a fact which forms an illustration of a principle to be developed at a later stage. If we attempt to derive another determinant, say by multiplying the quartic through by 3'x, obtaining a quarternary sextic, there results the following. In the first place each summand in this determinant will have four factors, and will be zero unless each factor does not contain one and only one subscript of value 3 and 4. Using same symbolism and diagram as before we shall have accordingly: nil + n2' nl" + n2" nl/"' + n2/" nliv + n2 iv 6 which would require 4.4 = 2.6, and consequently the equations are inconsistent, and there is no summand in the determinant of this form. This is a special case of the following: Theorem.-In order that the K-ary r-ic, derived from the q-ary p-ic by the process above indicated of addition of auxiliary variables by multiplication, have a determinant in which all the summands are not zero, it is necessary that the relation K'p = T'.q be satisfied. The truth of this theorem is evident from the above special cases, since their generalization involves no new principles. In the case under discussion, we have, since p = 4, q = 2, for any such derived K-ary 7r-ic: 2'K = r. This condition is satisfied by the ternary sextic already investigated in detail,

Page 341 Barter: Determinants of the P-th Class 341 and furthermore this sextic or ternary form is the only one that satisfies it. The next quantic which also satisfies this condition is a quaternary octavic, which may be derived either by multiplication by 3 or ~2o2. 7. Coefficient of a Given Term in a Symmetric Determinant Consider any term in the expansion of a symmetric determinant of the second class and order n. It will consist of the product of n letters each with 2 subscripts. It is required to find the coefficient of such a term in the expression for the determinant. Let 1 occur in such a factor as first suffix in combination with pi as second suffix. Then pi as first suffix in combination with p2 as second suffix, and so on. Write down the resulting sequence as a cycle of a substitution: llpi2P3..' PaIf this cycle does not exhaust all the numbers from 1 to n, repeat the operation as often as required to do so. Then we say: Theorem.-The coefficient of this term is 2", where K is the number of cycles of order > 2, with either - or + sign, according as the number of inversions is odd or even. (The cycle qpq is termed of order 2.) If two terms are equal, without being identical, they can differ only in the order in which one or more of their cycles is to be taken; that is to say according as we consider the operation of substitution to be from first to second suffix or inversely. Since each cycle may be reversed independently of the remainder, the number of such equal terms is obtained by considering the number of all possible such pairs of cycles. Furthermore each of these terms has the same sign in the expansion of the determinant. The theorem consequently follows. In the case of determinants of higher class this method is evidently inapplicable. 8. Differential Invariants and Covariants (1) Hessian.-Let d2U ( aU )(dxA)2 and let the variables be transformed as follows: dxn = (rnn)(dyn), so that d2U= OaxfOx ' (7n) (dyn) (dy )(2U (yoy ) (dy),

Page 342 342 University of California Publications in Mathematics [Vol. 1 where aWU 92U dyndy = dx dax (ax n) and consequently y2U a2U Oyyn yn OxOx,n 1 Therefore: Theorem.-The Hessian of a function of n independent variables is a covariant of weight 2. (2) Generalized Hessian.-Let dmU = (^x U )(dxn), and dxn (rn) (dyn), so that dmU = x... x (r)m(dy )m y 9mu = ( edy^ U ): (dy,)m, and consequently: amU OmU aYn..~.yn dXn * **' X ITn ax Whence: dmU Theorem.-General Hessian - of a function U of n independent aXn * * * aXn variables, is a covariant of weight m. (3) Jacobian.-Let (dff)) x= (dxA), where the arrow signifies order in which subscripts are to be taken, and (dxn) = (Tnn)(dyn), so that (df,) = ( (y -,) (dy) =(f (dy) and consequently afy, afXnf Whence: ax,1 Theorem.-Jacobian df of n independent functions of n independent af,1 variables, is a covariant of weight 1. (4) Generalized Jacobian.-Type I.

Page 343 1920] Barter: Determinants of the P-th Class 343 Let (dm-lf,) = (. dxa2 — f Ixa: (dxn), x= (a:: adx: (\ rdy)m-2 = y ('' d y ' (dy)m-l. Accordingly am-iA -1 am-i a *m = 1 m-l a-lf n dye... O9yf 4x... Oxf and consequently: am-lf Theorem.-Generalized Jacobian I is a covariant of weight m - 1. Oxf... ax (5) Type II. In a similar manner it may readily be shown that the generalized Jacobian am-qf l axn *** aXn, is a covariant of weight m - q. Note.-For a different treatment and for extensions along certain lines, the reader may consult E. R. Hedrick, Annals of Mathematics, Vol. I, p. 49. The above results were obtained independently, and I am indebted to Professor E. B Wilson for this reference. The significance of the above results in connection with the theory which we are now investigating, that of the general vector function, lies in the fact that any vector field, or rather the corresponding vector function of the position vector r, may be expressed as a series of homogeneous vector functions of r, whose determinants are of the form of these differential covariants and invariants, being consequently independent of the axes of reference. We shall have occasion to treat this point at length at a later stage.

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 15, pp. 345-358 November 8, 1923 INVOLUTORY QUARTIC TRANSFORMATIONS IN SPACE OF FOUR DIMENSIONS BY NINA ALDERTON ~1. Cayley in his paper "On the Rational Transformation between two Spaces"' gives a general discussion of the quadric transformation between two planes and the cubo-cubic transformation between two spaces. The cubic transformation in space was further studied by F. R. Morris2 who gives an analytic treatment of the cases in which the Jacobian, the sextic curve whose points go into lines by means of this transformation, breaks up into curves of lower degree, one or more of which are straight lines. A synthetic treatment of the general case of the cubic space transformation is given by D. N. Lehmer3 in his paper "On Combinations of Involutions," and of six of the special cases by Elizabeth J. Easton.4 ~2. A discussion of the general involutory quartic transformation in space of four dimensions has been given by P. H. Schoute5 in an article, "La Surface de Jacobi d'un systeme lineaire d'hyperquadriques Q32 dans l'espace E4 a quatre dimensions." The writer was n6t acquainted with this article when work upon the present paper was begun and consequently began with a general consideration of the involutory transformation by means of four hyperquadrics. Schoute makes his transformation with respect to a pencil of hyperquadrics and defines the transform P' of the point P as being the intersection of the polar spaces of P with respect to the triple infinitude of hyperquadrics of the pencil. This is equivalent, however, to using four hyperquadrics, for four independent hyperquadrics determine the pencil. The present paper gives the discussion of the general involutory quartic transformation with respect to four hyperquadrics, as originally planned, before going on to a consideration of the cases in which the Jacobian, now a surface, breaks up into 1 Proc. London Math. Soc., vol. 3 (1869-1873), pp. 127-180. 2 " Classification of Involutory Cubic Space Transformations," Univ. Calif. Publ. Math., vol. 1, pp. 223-240. 3 Am. Math. Monthly, vol. 18, no. 3 (March 1911). Also Steiner's Ges. Werke, vol. 2, p. 651. 4 Ms., Master's Thesis, "Certain Special Cubo-cubic Space Transformations," 1917, in Univ. Calif. Library, Dept. of Mathematics. 5 Archives du Musee Teyler, serie 2, 7, 1900-01.'

Page 346 346 University of California Publications in Mathematics [VOL. 1 surfaces of lower degree including at least one plane. Although it has seemed advisable to examine the subject analytically as well as synthetically to a considerable extent, the synthetic treatment only will be presented in most cases in this paper. ~3. NOTATION The exponent of the symbol of a surface will be used to denote the infinitude of points on the surface and the subscript to denote the degree of the surface; thus S23 designates a quadric hypersurface. ~4. DEFINITIONS 1. All of the 3-spaces through a line form what we shall call an axial pencil of 3-spaces. All of the 3-spaces through a plane form a plane pencil of 3-spaces. There are co 2 3-spaces in an axial pencil and oo in a plane pencil. 2. Harmonic 3-spaces of a plane pencil are four 3-spaces which are cut by any line in four harmonic points. 3. The polar 3-space of a point P with respect to a hyperquadric is the locus of a point which is the fourth harmonic to P and the two points in which any line through P cuts the hyperquadric. 4. The simplex of reference in 4-space is a figure bounded by five 3-spaces. The five 3-spaces intersect two at a time in ten planes, three at a time in ten lines, and four at a time in five points which are the vertices of the simplex. ~5. PRELIMINARY THEOREMS 1. In a plane pencil (~4,1) of 3-spaces, if four planes are cut by one line in four harmonic points they are cut by every line in four harmonic points. Proof.-Upon any line p take four harmonic points. These points together with the plane of the plane-pencil determine four harmonic 3-spaces; for, cut across the plane-pencil by any 3-space through p. We then have an axial pencil of planes and we know that planes corresponding to four harmonic points of the line are four harmonic planes which are cut by any line in four harmonic points. Since this is true in any 3-space through p, we see that any line cuts the four 3-spaces corresponding to four harmonic points of the line in four harmonic points. Hence as a point P moves along p, the polar 3-spaces of P with respect to four hyperquadrics will form four projective plane-pencils. Similarly, as P moves over a plane, the polar 3-spaces of P will form four projective axial pencils. 2. There are o2 lines cutting four planes in 4-space. Proof: Call the planes al, a2, a3, a4. If we. pass a 3-space through al, it will cut a2, as, and a4 in three lines. There will be co lines in the 3-space cutting these three lines, and these lines will also cut al, since they lie in the same 3-space with al. Hence in every 3-space through al, there will be co lines cutting al, a2, as and a4. But there are o 3-spaces about al (~4, 1) and consequently oo2 lines cutting al, a2, as and a4.

Page 347 1923] Alderton: Involutory Quartic Transformations 347 CASE I GENERAL TRANSFORMATION BY MEANS OF FOUR HYPERQUADRICS ~6. We may set up an involutory one-to-one correspondence between the points of 4-space by means of four arbitrarily chosen hyperquadrics. To a point P corresponds a point P', the intersection of the four polar 3-spaces of P with respect to the four hyperquadrics. Since the polar 3-spaces of P' must all pass through P and since four 3-spaces can intersect, in general, in only one point, the point P also corresponds to the point P' and we have an involutory, one-to-one correspondence. ~7. Thus, in general, to a point P will correspond a point P', but there are certain points to which correspond a whole line of points. The locus of such a point is the Jacobian. We shall show that Theorem I. The locus of all points whose transform is a line is a surface of the tenth degree in 4-space, J21o. (2, ~3). Proof: Let the four hyperquadrics be A = aijxixj=O i=1..51 B=bixixj=Oj =1...5 C = 2ciXxi= O D = dixixj = O The polar 3-spaces of a point P with respect to A, B, C and D are then x'lA 1+x'2A2+x'3A3 +'4A 4+x'5A5 = 0 X'1B1 +x'2B2+X'3B3 + x'4B4+x'5B5 = 0 XtlC1 + x'2C2+ x'3C3 + X'4C4+ xt5C5 = 0 x'D1 +x'2D2 +x'3D3 +x'4D4-x'D5 = 0 Ordinarily these four 3-spaces will intersect in a point. If, however, the equations are linearly dependent they will intersect in a line. The condition for this is that the matrix of the coefficients be of rank three; i.e. that all of the four-rowed determinants of the matrix vanish. Hence from the matrix A A2 A3 A4 A5 B1 B2 B3 B4 B5 C1 C2 C3 C4 C5 D1 D2 D3 D4 D5 we shall have five four-rowed determinants equal to zero. Each of these equations represents a quartic hypersurface. The Jacobian is the locus of points lying on all five of these hyperquartics. The hyperquartic we get by omitting the fourth column and the one we get by omitting the fifth column will intersect in a surface of degree sixteen, S216. But they have in common the matrix formed of the first three columns which represents a surface of the sixth degree through which the other hyperquartics do not pass. Hence the Jacobian, the surface through which all five hyperquartics pass, is a J210.

Page 348 348 University of California Publications in Mathematics [VOL. 1 ~8. Theorem II. The lines which are the transforms of the points of J21o form a ruled hypersurface, j315 Proof: If we denote by X=0, Y=0, Z=0, W=0, and V=0 the hyperquartics of the matrix of ~7 which we get by omitting the first column, then the second, etc., we know that the equation of the Jacobian hypersurface, which is made up of the lines which are the transforms of points of J210, may be found by equating to zero the determinant of the partial derivatives of X, Y, Z, W and V with respect to X1, X2, X3, X4 and x5; thus X1 X2 X3 X4 X51 Y1 Y2 Y3 Y4 Y5 j= Z1 Z2 Z3 Z4 Z5 =0 W1 W2 W3 W4 'W5 V1 V2 V3 V4 V5 Each element of this determinant is of degree three in X1, x2, x3, x4 and x5 and hence the equation represents a hypersurface of degree fifteen which we shall designate as j315. ~9. Thus we see that the points of 4-space go by this transformation into other points with the exception of points on a J210 whose points transform into the lines of a ruled j315. ~10. It seems at first thought as though there might be points in 4-space whose polar 3-spaces meet in planes, but this is not true in general. The condition for this would be that the matrix of the coefficients of the four polar 3-spaces of ~7 be of rank two. The forty cubic hypersurfaces obtained by setting each three-rowed determinant equal to zero would all have to pass through the points which transform into planes. Taking the first three rows of the matrix, A1 A2 A3 A4 A5 B1 B2 B3 B4 B5 C1 C2 C3 C4 C51 the three cubic hypersurfaces represented by columns 1, 2, 3; 1, 2, 4; and 1, 2, 5 intersect in a cubic surface and a curve. The other seven cubic hypersurfaces represented by this matrix will not pass through the cubic surface but will pass through the curve. Similarly, taking the matrix B1 B2 B3 B4 B51 C1 C2 C3 C4 C5 D1 D2 D3 D4 D5 we find that the t.en cubic hypersurfaces whose equations are the different threerowed determinants of the matrix set equal to zero pass through another curve. Two curves do not, in general, intersect in 4-space and therefore the twenty cubic hypersurfaces so far examined have no points in common and hence the forty have none. Consequently there are no points whose transforms are planes when the hyperquadrics are unrelated.

Page 349 1923] Alderton: Involutory Quartic Transformations 349 ~11. Theorem III. If a point P moves along a line p, its corresponding point P' moves along a quartic curve in 4-space. Proof: As the point P moves along the line p, its polar 3-spaces with respect to the four hyperquadrics revolves about four planes forming what we shall call plane pencils of 3-spaces. These four plane pencils are projective pencils for there is a one-to-one correspondence between the points of p and the 3-spaces of the four plane pencils, and to four harmonic points of the line correspond four harmonic 3-spaces of the pencils. (2, ~4, and Theorem I, ~5.) The locus of the intersection of corresponding 3-spaces of the four projective plane pencils will be the transform of the line p. If we cut across the plane pencils by a 3-space, we have four projective axial pencils of planes in a 3-space. Four and only four sets of corresponding planes of these pencils meet in points, (Reye, Geometrie der Lage, vol. 2, XII, p. 93), so there are four points of the locus in every 3-space and hence the transform of the line p is a quartic curve in 4-space. The single infinitude of points of the curve corresponds to the single infinitude of points of the line p. ~12. Theorem IV. If the point P moves over a 3-space, the corresponding point P' moves over a quartic hypersurface. Proof: If the point P moves over a 3-space, the polar 3-spaces of P with respect to the four hyperquadrics revolve about four points forming what we shall call four points of 3-spaces. There will be a triple infinitude of points P' corresponding to the triple infinitude of points P and hence points P' lie on a hyper-surface. In order to find the degree of this hypersurface, cut across it by a line. Transforming, the line goes into a quartic curve, as we have just seen, and the hypersurface back into the 3-space, giving off also the j315. The hypersurface must give off the j315 when transformed, for it contains j21o, since the 3-space of which it is the transform cuts all of the lines of j315. The points of the hypersurface which do not transform into lines but into points will go back into the points of the 3-space on account of the involutory relation between the points under this transformation. But the quartic curve cuts the 3-space in four points. Therefore the line cuts the hypersurface in four points and it is a quartic hypersurface. ~13. Theorem V. If P moves over a plane, its corresponding point P' moves over an S62. Proof: The plane may be considered as the intersection of two 3-spaces, R1 and R2. When we transform, these two 3-spaces go into two quartic hypersurfaces which intersect in a surface S216. But we have seen (~12) that every quartic hypersurface which is the transform of a 3-space must pass through J210. Hence J21o is a part of S216 and the remaining part is an S62 which is then the transform of the plane. ~14. Theorem VI. The multiplicity of lines of j315 through points of J210 and the multiplicity of points of J210 on lines of j315 is four. Proof: If p' is a line which is the transform of a point P of J21o, then the polar 3-spaces of all points of p' will pass through P. Ordinarily the four polar 3-spaces

Page 350 350 University of California Publications in Mathematics [VOL. 1 of points of p' intersect in only one point and this must be the point P. Since P is always a common point of four polar 3-spaces of points of p', in order for p' to transform into a quartic curve it must go into four lines passing through P. But P is any point of J20o and hence the Jacobian must be a surface of j315 of multiplicity four. The points of p' all go into the point P except four points whose transforms are the four lines through P. Hence there are four points of J210 on every line of j315, or the lines of j315 cut J210 in four points. ~15. Summary.-The principal facts which have been established concerning the general involutory quartic transformation in 4-space are: that lines go into quartic curves, planes into surfaces of the sixth degree, and 3-spaces into quartic hypersurfaces; also, that the locus of points whose transforms are lines is a surface of the tenth degree, and the lines which are the transforms of these points form a hypersurface of the fifteenth degree. CASE II ONE FUNDAMENTAL HYPERQUADRIC IS A SPACE-PAIR ~16. (a) In Case I the four hyperquadrics of the transformation were perfectly general. We shall now consider the case in which one of the hyperquadrics A is a space-pair whose 3-spaces intersect in a plane al. It is evident that we may still set up a one-to-one correspondence between the points of 4-space, for ordinarily to a point P will correspond a point P', the intersection of polar 3-spaces of P with respect to B, C and D and the 3-space conjugate with respect to the 3-spaces of A to that determined by P and the plane al. ~17. The polar 3-space with respect to A of a point P on al is indeterminate and hence to such a point corresponds the line of intersection of the polar 3-spaces with respect to B, C and D. Hence al is a part of J210 and Theorem VII. The Jacobian is a plane and an S29 when one of the fundamental hyperquadrics is a space-pair. ~18. Theorem VIII. The Jacobian hypersurface is an S3. and an S312 when one of the fundamental hyperquadrics is a space-pair. Proof: The transforms of points of al are lines of j315. As P moves over al, the polar 3-spaces of P revolve about three lines forming three projective axial pencils. (1, ~5). Now three projective axial pencils of 3-spaces intersect in a hypersurface of the third degree, for if we cut across them by any 3-space we have three points of planes intersecting in a cubic surface. Hence j315 breaks down into a hypersurface of the third degree and one of the twelfth degree. ~19. (b) If the two 3-spaces of A coincide; i.e. if A is composed of two coincident 3-spaces we cannot set up an involutory relation between points of 4-space, for the polar 3-space of any point will be A itself and the points of 4-space will transform into those of a 3-space.

Page 351 Alderton: Involutory Quartic Transformations 351 CASE III TWO FUNDAMENTAL HYPERQUADRICS ARE SPACE-PAIRS ~20. (a) Suppose A and B are the two space-pairs and ai and a2, the planes of intersection of the pairs of 3-spaces of A and B respectively, have only a point in common. It is evident that we may still set up an involutory one-to-one correspondence between the points of 4-space. The planes ai and a2 are part of J21o and we have the theorem Theorem IX. The Jacobian is composed of two planes and an S82 when two of the fundamental hyperquadrics are space-pairs. ~21. The planes al and a2 both go into cubic hypersurfaces of j315. Hence Theorem X. The Jacobian hypersurface is composed of two S33's and an S39 when two of the fundamental hyperquadrics are space-pairs. ~22. There is a plane lying on both of the cubic hypersurfaces and hence forming part of their intersection; namely the transform of the point of intersection of the two planes al and a2. This point transforms into a plane since its polar 3-spaces with respect to A and B are indeterminate, and its transform is then the intersection of its polar 3-spaces with respect to C and D. ~23. A line I cutting either al or a2 will transform into a line and a cubic curve. Suppose it cuts al. The line 11 is the transform of the point where the given line 1 cuts a1. To get the transform of the rest of 1, allow the point P to move along 1. The point P and the plane al always determine the same 3-space and hence all points of 1 have the same polar 3-space with respect to A. The polar 3-spaces of points of I with respect to B, C and D form the three projective plane pencils of 3-spaces and these intersect in lines of a ruled S23, for if we cut across them by a 3-space we have three projective axial pencils whose corresponding planes intersect in points of a cubic curve. The polar 3-space of points of I with respect to A will cut this S32 in a twisted cubic. Hence I transforms into a line and a twisted cubic lying in the polar 3-space with respect to A of points of 1. Similarly, a line cutting a2 will transform into a line and a twisted cubic lying in the polar 3-space with respect to B of points of the line. ~24. A line I cutting both a1 and a2 will transform into two lines 11 and 12 and a conic C2 lying in the plane of intersection of the two polar 3-spaces of points of 1 with respect to A and B. ~25. Other lines which do not transform into quartic curves are those lying on al (or a2). A line lying on al (or a2) and not passing through P1, the intersection of al and a2, becomes an S32, the locus of intersections of corresponding 3-spaces of three projective plane pencils of 3-spaces. If the line passes through this point P1, also, the cubic surface breaks up into a plane, the transform of P1, and an S22 lying in the polar 3-space with respect to B (or A) of points of the line.

Page 352 352 University of California Publications in Mathematics [VOL. 1 ~26. The surface S62 which is the transform of a plane has two lines lying on it, the transforms of the points of intersection of the plane with al and a2. If these two points coincide at P1, the S62 breaks down into a plane which is the transform of P1 and an S25. Further degeneration of the S62 occurs when the plane intersects al (or a2) and both al and a2 in lines. ~27. A 3-space containing al (or a2) will transform into a ruled cubic hypersurface which is the transform of al (or a2) and the 3-space which is the polar of points of the 3-space with respect to A (or B). (b) Now suppose A and B are so related that al and a2 intersect in a line L3; i.e. that all of the 3-spaces of A and B pass through L3. Then we have the theorem Theorem XI. The Jacobian is composed of three planes and a S27 when spacepairs A and B have a line in common. Proof: The points of L3 transform into the planes of a planed hyperquadric which are the intersections of two plane pencils of polar 3-spaces of points of L3 with respect to C and D. Hence the plane al (or a2) will transform into the planes of a planed hyperquadric and oo2 lines of the 3-space which is the polar of points of al (or a2) with respect to B (or A). Hence ai and a2 are still parts of J21o and have a line lying on them whose points transform into planes. Now al and a2 determine a 3-space since they have a line in common. Points of this 3-space will have a single polar 3-space with respect to A and a single polar 3-space with respect to B and these two 3-spaces will intersect in a plane O3 passing through L3. The polar 3-space of points of 03 with respect to both A and B will be the 3-space determined by al and a2, and this 3-space will cut the polar 3-spaces with respect to C and D in lines. Hence /3 is also a part of J21o, as are the planes al and a2. ~28. Theorem XII. The Jacobian hypersurface is composed of a planed hyperquadric counted twice, three 3-spaces, and an S83 when the four 3-spaces of A and B have a line in common. Proof: The plane ai (or a2) goes into a hyperquadric which is the transform of L3 and the 3-space which is the polar 3-space with respect to B (or A) of points of al (or a2). The plane /3. goes into the 3-space determined by al and a2. Hence three 3-spaces and a hyperquadric counted twice will be part of the j315. It should be noted that a line on J0o whose transform is counted twice is a triple line on J2xo; in this case the three planes al, a2, and /3 pass through L3. ~29. Again, lines which have a special position with respect to al and a2 will not transform as usual into quartic curves. Any line cutting L3 will transform into a plane and a conic lying in the plane of intersection of the two polar 3-spaces of points of the line with respect to A and B. The transform of a plane cutting L3 or passing through L3 will be a degenerate S62. The 3-space determined by al and a2 will go into the planed hyperquadric and the two 3-spaces which are the remainder of the transforms of al and a2. All points of the 3-space not on al or a2 go into points of /3. ~30. (c) Suppose one of the 3-spaces of B (or A) passes through the plane of intersection of the 3-spaces of A (or B). If A and B are so related that one of the

Page 353 1923] Alderton: Involutory Quartic Transformations 353 3-spaces, R3 of B, passes through al, then a, and a2 determine R3 and must intersect in a line L3 as in (b). The plane 33 will now coincide with al, the intersection of the conjugate 3-space of Rs with respect to A and R3 itself, since it is self-conjugate with respect to B. Hence Theorem XIII. The Jacobian is composed of a single plane, a double plane, and an S72 when one of the 3-spaces of B (or A) passes through a, (or a2). ~31. Theorem XIV. The Jacobian hypersurface is composed of a planed hyperquadric counted twice, a single 3-space, a double 3-space and an S83. The single 3-space is the conjugate with respect to A of Rs and the double one is R3 itself. ~32. (d) If ai and a2 coincide in al, the points of 4-space go into points of ai and we no longer have a one-to-one correspondence. CASE IV THREE FUNDAMENTAL HYPERQUADRICS ARE SPACE-PAIRS ~33. (a) Suppose A, B and C are space-pairs and their planes, al, a2, and a3 respectively, have only points in common. Theorem XV. The Jacobian is composed of three planes and an S72 when three of the fundamental hyperquadrics are space-pairs. The planes al, a2 and as which form part of the J21o intersect two at a time in three points Pi, P2 and P3. These points transform into planes. There are two such points on each of the three planes. ~34. Theorem XVI. The Jacobian hypersurface is composed of three hypercubics and an S63 when three of the fundamental hyperquadrics are space-pairs. The three hypercubics are transforms of the planes al, a2 and as. ~35. If al and a2 intersect in P3, a2 and as in P1, and as and al in P2, then a line such as the one joining P1 with any point of a1 transforms into a plane and two lines. A plane through two of the points P1, P2, P3, say P1 and P2, will go into two planes which are the transforms of P1 and P2, another plane which is the transform of the other points of the line P1 P2, and a cubic surface lying in the 3-space conjugate to that determined by the plane to be transformed and as. The plane P1, P2, P3, goes into six planes. ~36. (b) Suppose two of the three planes al, a2 and as, say a1 and a2, intersect in a line L3. Theorem XVII. The Jacobian is composed of four planes and an S26 when two of the three fundamental space-pairs A, B and C have a line in common. The four planes are the planes al, a2 and a3 and the plane 3s which is the intersection of the two polar 3-spaces of points of the 3-space determined by al and a2 with respect to A and B.

Page 354 354 University of California Publications in Mathematics [VOL. 1 ~37. Theorem XVIII. The Jacobian hypersurface is composed of three 3-spaces, a planed hyperquadric counted twice, a hypercubic and an S53 when two of the three fundamental space-pairs A, B, C have a line in common. ~38. (c) Suppose the three fundamental space-pairs A, B, C are so related that a 3-space R3 of B passes through al. This implies that al and a2 have a line L3 in common. * Theorem XIX. The Jacobian is composed of two planes, a double plane, and an S62 when three of the fundamental hyperquadrics are space-pairs and a 3-space of one of them passes through the plane of another. This is true since two of the four planes of Theorem XVII now coincide in ai. (Compare Theorem XIII.) ~39. Theorem XX. The Jacobian hypersurface is composed of one single 3-space, one double 3-space, a planed hyperquadric counted twice, a hypercubic and an S53 when three of the fundamental hyperquadrics are space-pairs and a 3-space of one of them passes through a plane of another. ~40. (d) The three fundamental space-pairs may be so related that A and B have a line L3 in common and B and C have a line L1 in common. In this case Theorem XXI. The Jacobian is composed of six planes and an S42 when A and B have a line in common and B and C have a line in common. The six planes are al, a2, a3, the plane /3 which is the intersection of the two polar 3-spaces of points of the 3-space R' determined by al and a2 with respect to A and B, the plane /i which is the intersection of the two polar 3-spaces of points of the 3-space R" determined by a2 and a3 with respect to B and C, and the plane /2 which is the intersection of the polar 3-space of points of R' with respect to A and the polar 3-space of points of R" with respect to C. ~41. The lines L3 and L1 intersect since they both lie in a2 and this point P which lies on each of the three planes a1, a2 and a3 transforms into a 3-space. It should be noted that the planes /1, 02 and O3 also pass through P and hence a point on J210 whose transform is a 3-space of j315 counted three times has six sheets of the surface passing through it. The 3-space into which all points of al except those lying on L3 transform is the 3-space determined by a2 and a3, and similarly for a3. This may be shown by taking A, B and C as space-pairs through three planes of the simplex of reference. (4, ~4.) The 3-spaces determined by a1 and a2 and by a2 and a3 are then two of the five faces of the simplex. Hence in this case Theorem XXII. The Jacobian hypersurface is composed of one triple 3-space, four double 3-spaces and an S43 when the three fundamental space-pairs A, B and C are so related that A and B have a line in common and B and C have a line in common. Two of the double 3-spaces are of course the 3-spaces determined by al and a2 and by a2 and a3. ~42. Any line lying in a2 and not passing through P transforms into three planes since it cuts both L1 and L3. If it passes through P it transforms into a 3-space and a plane.

Page 355 1923] Alderton: Involutory Quartic Transformations 355 ~43. (e) Let one of the 3-spaces R3 of B pass through al while a2 and as still have a line L1 in common. Theorem XXIII. The Jacobian is composed of four single planes, a double plane and al S42 when one of the 3-spaces of B passes through a, and B and C have a line in common. This is due to the coincidence of two of the planes of Theorem XXI, the double plane being the plane al. ~44. Theorem XXIV. The Jacobian hypersurface is composed of two triple 3-spaces, two double 3-spaces, a single 3-space and an S43 when one of the 3-spaces of B passes through al and B and C have a line in common. The 3-space R3 is one of the triple 3-spaces and the plane determined by a2 and a3 is the single 3-space. ~45. (f) Suppose one of the 3-spaces R3 of B passes through ai and one of the 3-spaces R5 of C passes through a2. Then Theorem XXV. The Jacobian is composed of two single planes, two double planes, and an S42 when one of the 3-spaces of B passes through al and one of the 3-spaces of C passes through a2. The two double planes are al and a2 while a3 is one of the single planes. ~46. Theorem XXVI. The Jacobian hypersurface is composed of a triple 3-space, four double 3-spaces, and an S43 when one of the 3-spaces of B passes through al, and one of the 3-spaces of C passes through a2. The 3-spaces R3 and R5 are double 3-spaces. ~47. (g) Suppose one of the 3-spaces, R3, of B passes through ai and the other, R4, passes through as. Theorem XXVII. The Jacobian is composed of two single planes, two double planes and an S42 when one 3-space of B passes through al and the other through as. The planes al and a3 are double planes while a2 is now a single plane. ~48. Theorem XXVIII. The Jacobian hypersurface is composed of a triple 3-space, four double 3-spaces, and an S43 when one 3-space of B passes through ai and the other through a3. R3 and R4 are double 3-spaces. ~49. (h) Let the planes al, a2, and as intersect in lines two at a time. Theorem XXIX. The Jacobian is composed of six planes and an S42 when the planes al, a2 and as intersect two at a time in lines. Call the intersection of al and a2 L3, the intersection of a2 and a3 L1, and the intersection of a3 and al L2. The planes al, a2 and a3 now lie in a 3-space R and intersect in a point P so the lines L1, L2 and L3 are concurrent. The 3-space R will have conjugate 3-spaces with respect to A, B and C which will intersect two at a time in planes 03, P1 and O2 which together with planes a, a2 and a3 are the six planes of the J21.

Page 356 356 University of California Publications in Mathematics [VOL. 1 ~50. Theorem XXX. The Jacobian hypersurface is composed of one triple 3-space, four double 3-spaces and an S43 when the planes al, a2 and as intersect two at a time in lines. This is true since the plane al transforms into lines of 01 and the three 3-spaces which are the transforms of L2 and L3, a2 into lines of 02 and the three 3 -spaces which are the transforms of L1 and L3, and as into lines of /3 and the three 3-spaces which are the transforms of L1 and L2. ~51. The transform of points of R with respect to A, B and C will be three 3-spaces intersecting in a line L4. Hence the planes i1, 02 and /3 all transform into R, the transform of points of L4, which is then one of the double 3-spaces of j315. (~28.) Points of this line L4 will transform into planes of R. Hence there are four lines, L1, L2, L3 and L4, whose transforms are 3-spaces of planes. ~52. Any line in al, a2 or a3 will transform into three planes if it does not pass through P, for it will cut two of the lines L1, L2 and L3. If it does pass through P it will transform into a 3-space and a plane through one of the lines L1, L2 or L3. The planes /1, /2 and /3 all pass through L4 and hence any line in /1, /2 or /3 will transform into two planes since it cuts L4 and either L1, L2 or Ls. ~53. (i) Suppose one of the 3-spaces R3 of B in (h) passes through al. The planes al, a2 and as now all lie in R3 and hence Rs passes through a3. Hence part of J210 is al taken twice, as taken twice, a2 and /2 or Theorem XXXI. The Jacobian is composed of two double planes, two single planes, and an S42 when a 3-space Rs of B passes through al and as. ~54. Theorem XXXII. The Jacobian hypersurface is composed of one triple 3-space, four double 3-spaces, and an S43 when a 3-space R3 of B passes through al and as. The 3-space R3 is one of the double 3-spaces. ~55. (j) If L2 coincides with L1, then L3 also coincides with L1. In this case all of the points of 4-space go into points of L1 and we no longer have a one-to-one correspondence. CASE V THE FOUR FUNDAMENTAL HYPERQUADRICS ARE SPACE-PAIRS ~56. (a) Let A, B, C and D be four space-pairs any two of which have only a point in common. The planes al, a2, a3, and a4 of A, B, C and D respectively are planes of the J21o. Hence Theorem XXXIII. The Jacobian is composed of four planes and an S62 when the four fundamental hyperquadrics are space-pairs intersecting in pairs in points. ~57. Theorem XXXIV. The Jacobian hypersurface is composed of four hypercubics and an S33 when the four fundamental hyperquadrics are space-pairs intersecting in pairs in points.

Page 357 1923] Alderton: Involutory Quartic Transformations 357 ~58. There are 002 lines cutting al, a2, a3 and a4. (2, ~5.) Any one of these lines will transform into four lines which are rulings of the j315. It cuts J210 four times so the point into which the remainder of the line transforms is the point through which the four lines pass. Hence the four lines are concurrent. Lines cutting only three of the planes al, a2, as and a4 also transform into four lines but these lines are not concurrent. ~59. (b) Suppose the 3-spaces of A and B have a line L3 in common. Theorem XXXV. The Jacobian is composed of five planes and an S52 when two of the four fundamental space-pairs have a line in common. The planes are al, a2, as and a4 and the plane /3 which transforms into lines of the 3-space determined by ai and a2. ~60. Theorem XXXVI. The Jacobian hypersurface is'composed of three 3-spaces, a planed hyperquadric counted twice, two cubic hypersurfaces and an S23. ~61. (c) Suppose a 3-space R3 of B passes through ai. Theorem XXXVII. The Jacobian is composed of three single planes, a double plane, and an S52 when a 3-space of B passes through al. The double plane is the plane al. ~62. Theorem XXXVIII. The Jacobian hypersurface is composed of a single 3-space, a double 3-space, a planed hyperquadric counted twice, and an S23, when a 3-space of B passes through al. The double 3-space is R3. ~63. (d) Suppose A and B have a line L3 in common and C and D a line L2 in common. Theorem XXXIX. The Jacobian is composed of six planes and an S42 when A and B have a line in common and C and D have a line in common. The planes are ai, a2, as, a4 and also /3 and /2 which transform into the 3-spaces determined by al and a2 and by a3 and a4 respectively. ~64. Theorem XL. The Jacobian hypersurface is composed of two hyperquadrics counted twice and seven 3-spaces when A and B have a line in common and C and D have a line in common. ~65. (e) If the lines L2 and L3 of (d) intersect in a point P, then every point in 4-space transforms into the point P and we no longer have a one-to-one correspondence. ~66. (f) Suppose the space-pairs A, B, and C are so related that A and B have a line L3 in common and B and.C have a line L1 in common. Theorem XLI. The Jacobian is composed of seven planes'and an S42 when A and B have a line in common and B and C have a line in common. The planes are al, a2, a3 and a4 and the planes /1, /2 and a3 of Theorem XXI. ~67. Theorem XLII. The Jacobian hypersurface is composed of a triple 3-space, four double 3-spaces, a single 3-space, and a hypercubic.

Page 358 358 University of California Publications in Mathematics [VOL. 1 ~68. (g) Suppose the space-pairs A and B have a line L3 in common, B and C a line L1 in common, and C and D a line L2 in common. Theorem XLIII. The Jacobian breaks down into the ten planes of the simplex of reference (4, ~4) when the four fundamental hyperquadrics are space-pairs and three pairs of them intersect in lines. This is more easily seen analytically. Take as the fundamental space-pairs A = x21-bx22 = O B=x21- cx23 = C =X2 - dx24 = O D = x2 — ex25 = O D=x%-ex%=O The Jacobian turns out to be, in addition to the planes al, a2, a3 and a4, the planes x2= f O X= x=O { =O =O X2=O and X2= X3=O0 X4=0 X5=0 X5=0 X4=O0 X4=O ~69. Theorem XLIV. The Jacobian hypersurface breaks down into the five faces of the simplex of reference counted three times when the four fundamental hyperquadrics are space-pairs and the three pairs of them intersect in lines. The equation of the Jacobian hypersurface turns out to be X31'X32' X33-X34 X35 = 0

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 16, pp. 359-369 December 19, 1923 ON THE INDETERMINATE CUBIC EQUATION x'+Dy +D2z: - 3Dxyz = 1 BY CLYDE WOLFE 1. The object of the following paper is to develop and to discuss some processes for solving the equation x3 + Dy3 +D2z3- 3Dxyz = 1. 2. In a letter to M. Liouville, Dirichlet' sets forth this beautiful theorem: If f(s)-sn+asn-l+... +gs+h-(s-a) (s —f)... (s-w)=0 has at least one real root, then the equation F(xy... z) = (a) (3). (.. (w)=m has an infinite number of integral solutions, where 4(a)=x+aay+.. +an-lZ, and similarly for f... To establish this theorem, Dirichlet shows that there is at least one value of m for which the equation F(xy... z)=m has an infinite number of solutions; in particular, two solutions, F(xy...z) =m and F(x'y'... z')=m, such that x-x', y-y',... zz', (mod m), whence multiplying the fraction x'-] - J ay+ -. an-1lz x +ay + +an-lz by 1(03).... (w) one obtains X+aY+... +a"-.Z, X +ay + +a~ n1Z where X Y=... — ZO, (mod m), whence X, M_ - is an integral solum m m tion of the equation F (xy... z)=l. In the case of the binary quadratic, Euler2 had shown that if X, Y is a particular solution of the equation f(xy) = ax2+ 2bxy+cy2 =m of discriminant D =b2-ac, and if T, U is a particular solution of the equation F(tu) = t2- Du2= 1, then an infinite set of solutions of f(xy) = m is obtained from the equation ax+(b+/VD)y=[aX+(b+xD)Y](T+U D)n by giving to n different positive and negative integral values, and equating the rational and irrational parts on the two sides of the equation. Lagrange3 showed that this process gives all the solutions of the equation f(xy)=m corresponding to the particular solution X, Y when, and only when, T, U is the fundamental solution of F(tu) = 1, that is, the numerically least solution, excepting of course the solution, 1, 0. 3. Dirichlet4 points out that this relation divides the totality of solutions of f(xy) =m into sets, all the solutions of any set being obtained from any solution of the set by Euler's method with Lagrange's restriction, whence the field to be investigated for particular solutions is limited to a <ax+ (b+ /-D)y < (T+ UV/D),

Page 360 'j60 University of California Publications in Mathematics [VOL. 1 where a is any positive number whatsoever. Generalizing to the third degree, as a special case of the general theorem set forth to Liouville, Dirichlet4 shows that if f(s) =s3-+as2+bs+c-(s-a) (s-3) (s-y) =O, a, 3, and y, being irrational, and only a real, and if X, Y, Z is a particular solution of F(xyz) (x+ay+a2z) (x+- y+ 32z) (X+yy+Y2z) = m, and finally if T, U, V is one of the two fundamental solutions of f(tuv) = 1, then the equation x+ay+a2z= (X+aY+a2Z) (T+aU+a2V)' gives all the solutions of F(xyz)=m corresponding to the particular solution X, Y, Z by giving to n different positive and negative integral values and equating the coefficients of like powers of a on the two sides of the equation. Here again Dirichlet4 shows that this relation divides the totality of solutions of F(xyz)=m into sets, all the solutions of any set being obtained from any solution of the set by Euler's method with Lagrange's restriction, whence the field to be investigated for particular solutions is limited to ao<x+ayy+a2z<oa(T+aU+a2V), o being any number whatever. 4. Indeed, in the case of the binary quadratic form, Fermat5 had already announced that the equation t2 -Du2= 1 has an infinite set of solutions all of which may be derived as powers of the fundamental solution, that is, if T1U1 is the fundamental solution, the others are obtained as Dirichlet has pointed out in the general case from the equation Tn+U, /D=(T1+U1 V/D)n by giving to n different positive and negative integral values, and equating the coefficients of the rational and irrational parts on the two sides of the equation. In the case of the ternary cubic, as Mathews6 points out, there are two reciprocally related fundamental solutions of F(tuv)=l, namely T1U1V1 and T-_lUlVl such that T_l+aU_l +-a2V_l= (Ti+ aUl+a2V)-1. The solution T_1U_1V_i will be called the reciprocal of the fundamental solution, while the fundamental solution will hereafter signify that one for which T1U1V1 is the set of least positive integers satisfying the equation F(tuv)= 1. ToUoVo will signify the solution, 1,0,0, and TnUnVn the nth derived solution obtained by putting T+ aUn+a2Vn = (T+ aUi = a2Vl)n and equating like powers of a on the two sides of the equation. 5. The generalization of the binary quadratic form FD(xy)-(x+ty) (x —ty) -X2 -DY2 =IDy x. where t and -t are the square roots of D, becomes in the case of the ternary cubic form, FD(xyz) —(xty+ty+z) (x+toy+t2co2) (x+tco2y+t2z) I x y l -x3+Dy3+D2z3-3Dxyz- I Dz x y, where t is the real cube root of D, and c jDy Dz xI and co2 are the imaginary cube roots of unity. The solution of the quadratic equation depends upon the expansion of the square root of D into a simple continued fraction, which Lagrange7 first proved must become periodic, and this periodic continued fraction expansion is closely connected with the reduction of the general binary quadratic form. Hermite8 raised the question as to whether these relations held for forms of higher order, and Jacobi9, in 1868, devised an extension of the continued fraction process for determining approximations to the ratios of three numbers involving a cubic irrationality, each approximation being expressed

Page 361 Wolfe: On the Indeterminate Cubic Equation 361 linearly in terms of the three preceding ones, the expansion in some cases becoming periodic, and giving a solution of FD(xyz)= 1 just as in the case of the ordinary continued fraction. Jacobi's fractions may be called proper, regular, ternary continued fractions-"proper" because in his expansion the multiplier of the last set of convergents at any point is never less than the multiplier of the next to the last set of convergents; "regular" because the multiplier of the second from the last set of convergents is always +1. In semi-regular fractions, the third multiplier may be +1 or -1, which always keeps the determinant of three successive sets of convergents equal to + 1 or - 1. In irregular fractions, the third multiplier may be any positive or negative integer whatsoever, which corresponds to the general binary continued fraction where the numerators are not unity, and where variety of form is obtained at the expense of uniqueness. 6. Bachmann'1 in 1872 showed that periodicity does not occur in Jacobi's expansion unless the two inequalities, x- D2 < k andY - D< are satisZi ZiZ' Zi Zi'" fied for all values of i>m, where m is a number depending upon an arbitrarily chosen k, xi: yi: zi being the convergents to 4 D2: D: 1 obtained by Jacobi's method. 7. Mathews6, in 1890, applying Dirichlet's method of investigation, showed in more detail than does Dirichlet, that, if an integer m can be represented by the form F(xyz) at all, it can be so represented in an infinite number of ways, from which he proved, as did Dirichlet, that the equation FD(xyz)= 1 can be satisfied by integral values of x, y, z in an infinite number of ways, all of which may be derived from one of two reciprocally related fundamental solutions. 7a. Meissel" in 1891 derived the form of the reciprocal solution of FD(xyz) = 1, cross-product and linear recursion formulas for successive solutions, one solution being known, exhibited a few special forms of D for which the equation is solved by simple algebra, and obtained by tentative combinations of solutions of FD(XlylZl)=mi, FD(X2Y22) =m2, and so on, solutions for all non-cubic integral values of D<82. He admits the uncertainty of his method, and suggests and illustrates the use of congruences in determining whether or not a given solution is fundamental. He does not, however, give any solutions except in illustrating his method as applied to the solution of the equation for D= 29, 71, 73, and 69. The x for D =69 is a 45-digit number (the greatest for any D<101), and Meissel remarks that the ratio x/y or y/z gives the cube root of 69 correct to 65 places of decimals. 8. Berwick'2 in 1913 developed from geometric considerations a process for deriving the expansion of a cubic irrationality, for the case of an equation having but one real root, so that periodicity ensues in every case, whence in particular from the solution of t3-D=O, a solution of FD(xyz)=l is given by the last set of convergents in the period, just as in the case of the binary quadratic. In spite of the fact that Berwick's expansion is neither proper nor regular, his method leads to a unique set of convergents.

Page 362 362 University of California Publications in Mathematics [VOL. 1 9. Lehmerl3 has shown recently that certain cubic irrationalities which are not periodic as proper continued fractions become periodic when expanded into improper continued fractions, and in particular, if the solution of the equation FD(xyz)= 1 is known, that a periodic expansion in an improper fraction follows at once. Berwick further shows that an ideal in a cubic corpus of negative discriminant can be linked up with an equivalent ideal by a substitution derived from the coefficients of his expansion, that every ideal is equivalent to one of a finite number of reduced ideals, which equivalence is shown in a finite number of operations. Finally he shows that the reduced ideals of each class are linked up in one closed cycle which gives rise to the primitive unit of the corpus. 10. Mathews points out in detail some of the applications of the theory of the form FD(xyz) = x3+Dy3+D2z3-3Dxyz to the solution of related problems in number theory. All numbers representable by the form FD(xyz) are cubic residues of D; the product of any number of such numbers is representable by the same form; all of the infinite set of representations of a given number may be obtained from a fundamental representation as soon as the solution of the equation FD(xyz)= 1 is known; the relation by which these representations are obtained is the very equation that Dirichlet had used a half a century before, and gives also an infinite set of linear transformations of the form into itself; and finally the successive sets of values of x: y: z which satisfy the equation FD(xy) = 1 furnish rapidly converging approximations to the ratio of X D2: / D: 1. Mathews concludes his article by giving a table of actual solutions of the equation FD(xyz) = 1 computed, as he says, by Jacobi's method, for D = 2, 3, 4, 5, 7, and 11. Berwick in illustrating his method derives solutions for D = 23, and 65. 11. In view of the importance of the equation FD(xy) = 1 in the theory of cubic irrationalities, a table of fundamental solutions for successive integral values of D is as highly desirable as is such a table of solutions of the so-called Pellian equation of the second degree, and this paper aims to supply such a table to D = 100, together with a discussion of the methods used in constructing the table. Some theorems will be proved that will establish a method of solution for an infinite set of values of D, some 60 of which lie below 100, and some 300 below 1000: other theorems will be proved that will expedite processes already known. 12. Theorem.-Among the solutions of the equation FD(xyz) = x3+Dy3+D2z3 -3Dxyz = 1 there exists one solution, whence an infinite set of solutions, such that y is divisible by n, and z by n2, n being any positive integer whatsoever. For if xjyjz\ is a solution of F3D(xyz)= 1, then x2=x1, y2=ny1, z2=n2z1, is one of the solutions of the equation FD(xyz)=1, for FD(x2Y2z2) —F3D (x1ylzl)=1, and since there is an infinite set of solutions of Fn3D(xyz) = 1, there is a corresponding infinite set of solutions of FD(xyz) = 1 of the form xl, nyl, n2zi. By this theorem, if FD(xlylzl)=1, yj being divisible by n, and zl by n2, then FnD ( nX11 =1 and " and are integral. Conversely, if Fn3D(xDy1z1)=1, then FD(x1ny1n2z1)=l. n nT

Page 363 1923] Wolfe: On the Indeterminate Cubic Equation 363 Again if FD(XlylZi) =1, yI being divisible by D, then FD2 (x2 = 1 and Y is integral. Conversely, if FD2(XilZl) = 1, then FD(XDZlYl) =1. The solution of the equation FDi(xyz)= 1 being known, combinations of these two processes lead to a solution of the equation FD(xyz) = 1, where D =m3D, m and n being any positive integers whatsoever, or the reciprocals of positive integers. In particular, if the solution of Fmn(xyz) =1 is known, then the solution Fmn (x m ) = 1 follows, and is integral when that solution of Fmn(xyz) =1 is chosen in which y is divisible by n. For example: If m=3 and n=2, the first solution for D=m2n=18 in which y is divisible by 2 is x =9073, y=3162, z= 1321, whence the solution for D=mn2= 12 is x = 9073, y = 3z = 3963, z = = 1731. 2 13. Theorem. —Fp3p2 (p2q2~3pqg+l, pq2~2q, q2~) = 1 where p and q are positive integers, p> 1, and q an exact divisor of 3p2, gives positive integral values for all of the arguments involved, except (a) when p = 2, or 3, q = 1, and D = p3- 3- q or (b) when z is fractional on account of q not being exactly divisible by p. Exception (a) is trivial, and the difficulty of exception (b) can be removed. For it can be shown by mathematical induction that the only term containing the fraction q in the nth derived solution of Fp3P2 (pV2q23pq+1, pq12~2q, q2+ -)=1 occurs P q P in Zn, and has the numerical coefficient n: whence by taking n equal to the denominator of q reduced to lowest terms, a solution is obtained, all of whose terms, includp ing z, are positive integers. For example: Putting p=3, q=l, D=p3+3 —=54 q = p2q2+3pq+=19, y = pq2+2q =5, z= q2+q = 1, whence 3= 61561, y3= 16287, Z3= 4309. 14. Several methods of obtaining sets of convergents, x: y: z, to the ratio, t2: t: 1, which are variations of Jacobi's method have been tried out in an endeavor to obtain those sets for which FD(xyz) = 1, but in every instance the method applied only to a limited number of cases. Jacobi's ternary fraction method applied directly to the numerical approximations to t2 and t as obtained from Barlow's tables gave immediately solutions for every value of D below 21 except D= 17, in which case and for D=21, 22, and 24, the solutions of FD(xyz)=1 were obtained only indirectly as linear combinations of two sets of convergents. Jacobi's method alone does not give solutions of FD(xyz)= directly for even such small values of D as 4 and 6, and fails to satisfy the condition imposed by Bachmann with increasing frequency as D becomes larger, though sets of values of x, y, z which do satisfy FD(xyz) = 1 are frequently found among simple linear combinations of neighboring sets of convergents obtained by Jacobi's process.

Page 364 364 University of California Publications in Mathematics [VOL. 1 15. In Berwick's method alone is found a direct non-tentative process for obtaining the successive sets of convergents for which FD(xyz) has its smallest values. Even this excellent method becomes laborious if the values of x, y, z exceed 8 or 10 digit numbers, and necessitates the development of some theorems for shortening the calculation. These theorems depend upon certain relations between the arguments of the equation FD(xyz) =x3+Dy3+D2z3-3Dxyz = m whereby the solution of FD(xyz) = 1 may be derived. A more general investigation would consider relations between the arguments of the set of equations (a) FD(xyz)=FD (x'y'z')=,.. =m, or (b) FD(xyz)=m, FD(x'y'z') =m',... An illustration of set (a) is given by Mathews6 (p. 283) where he shows that if two solutions FD(xyz) = FD (x'y'z') =m can be found, such that x-x', y-y', z-z', (mod m), then a solution of FD(xyz) = 1 is easily derived. This is the theorem that was used by Dirichlet in his proof of the existence of a solution of F (xy... z) = 1 in the general case, but is occasionally useful in actually obtaining numerical solutions. A numerical illustration of set (b) occurs in the solution of F38(xyz) =1. Among the sets of convergents discovered, F38(34, 10, 3) =12, and F38(1718, 511, 152) =18, whence F38(x'y'z')=216 where x'+ty'+t2z'=(34+10t+3t2) (1718+511t+152t2). Moreover, x' y'- z' - O, (mod 6), whence, as is shown in the next paragraph, F38 ( XY )=1, which are the values tabulated. The investigation of relations (a) and (b) is not yet completed. 16. Theorem.-If FD(xyz)=m, then FD(X3Y3Z3) =M3, where x3+ty3+t2z3= (x+ty+t2z)3. This is only a special case of the general relation between solutions of the equation F(xy...z)=m, proved by Dirichlet, that if F(xiyi... z)=m, and if F(xjy... j) =mj, then F(xkyk... Zk)=mk, where Xk+ayk+. +an —lZk =(Xi+ayi+.. +an-lZi) (Xj+ay +... +an-lZj), and mk=mim. 17. Corollary.-If x- y=z- 0. (mod m), then Y - is an integral solution of FD(xyz) = 1, since FD(xyz) is homogeneous of the third degree in x, y, z. In particular, if FD(xyz)=3, then X3YZ33 always gives an integral solution of FD(xyz)=l, since expanding (x+ty+t2z)3 it is easily shown that x3=FD(xyz) +9Dxyz= 9Dxyz+3, y3=3D(y2z+z2x)+3x2y, and z3=3Dyz2+3(zx2+xy2), whence xyz-0, (mod 3). The condition imposed in the general case of this corollary is one of frequent occurrence, and easily tested. When applicable, the use of this device reduces the labor of computation at least two-thirds. For example: In the case of D=29, F29(324876, 105743, 34418)=3, whence expanding (324876+105743429 +34418 4 292)3, and dividing the coefficients of 1, 4 29, and 4 292 by 3, the solution as tabulated was obtained. Other numerical cases where this theorem applies are D=4, 6, 10, 15, 23, 25, 34, 46, 51, 74... Several other special cases of the general theorem are of interest and follow herewith.

Page 365 1 923] Wolfe: On the Indeterminate Cubic Equation 365 18. Theorem.-If FD(xyz) =p, a prime, and D=kp, then x=0, (mod p). For putting D=kp, FD(xyz) Fkp(xyz) x3-kp(+kpy+k2p2z3-3kpxyz-p=0, whence -X3=-O, (mod p), since p is prime. 19. Corollary.-If FD(xyz) = p, a prime, and D = kp, then X3y3Z3 is an integral solution of FD(xyz)=l where x3+ty3+t2z3= (xty+t2z)3. For since x=-D-O, (mod p), [x3 = 9Dxyz+p] [y3 = 3D(y2z+z2x) +3x2y]- [3 = 3Dyz2+3(zx2+xy2)] 0, (mod p), whence X3Y3-3 are integers. In particular, if F (xyz)=D, a prime, then x=0 (mod D), and y, z, D is an integral solution of F (xyz) = 1. 20. Theorem.-If FD(xyz) =p2, p a prime, and D=kp lp2, then x-y-0, (mod p). For putting D = kp, FD(xyz) -Fkp(xyz) = x3 + kpy+ lk2p23 - 3kpxyz- p = 0, whence x-X3-0, (mod p), since p is prime. Putting x=np, Fkp (npyz) =n3p3+ kpy+k2p2z3-3kp2nyz-p=O, whence y-y3-O, (mod p), since p is prime, and k O0, (mod p). 21. Corollary.-If FD(xyz)=p2, p a prime, and D=kp#lp2, then X_3Y33 isan integral solution of FD(xyz) = 1, where x3+-ty3+t2z3= (x+ty+t2z)3, t= 4 D. For since x —y=O0, (mod p), [x3= 9Dxyz+p]- [y = 3D(y2z+z2x)+3x2y]-=[z3 3Dyz2+3 (zx2+xy2)]-O, (mod p), whence -33-3 are integers. P P P 22. Theorem.-If FD(xyz)=p2, p a prime, and D = kp2,then x-0, (mod p). For putting D = kp2, FD(xyz) = Fkp2(Xyz) = X3 + kp2y3 + k2p4z3- 3kp2xyz -p2 = 0, whence x-X3-O, (mod p), since p is a prime. Putting x =np, Fkp2(npyz) -n3p3-kp2y3+ k-p4z3-3kp3nyz-p2=0, whence y-0, (mod p). 23. Corollary.-If FD(xyz) = p2, p a prime, and D = kp2, then X3Y33 is a soluXs~~~Y3Z3 ~p222 tion of FD(xyz)= 1, where p3Y-33 are integral. For since x-0, (mod p), and D=0, (mod p2), [x3 = 9Dxyz + p2] = [y3 = 3D(y2z+ 2x) +3x2y] 0, (mod p2), and [z = 3Dy2 + 3 (zx2+xy2)]0O, (mod p). Putting X3 =' y ' F (X Y y = 1, whence x'p y'p 'p is an integral solution of FD(xyz) = 1 where x'p+ty'p+t2z'-, =x'+ ty'+ t )as shown in paragraph 13. 24. Two methods of solution herein described may be illustrated by the same numerical example. In the case of D= 75=3X52, the solution for D=45 = 5 X 32 having been obtained as tabulated, the solution for D = 75, by the method of paragraph 12, is x =1477441, y=350340, z=830744, which raised to the fifth power as pointed out in paragraph 13, eliminates the fraction in z, giving the result as tabulated. Or otherwise: Having discovered that F75 (160,38,9) =25, cubing F75 (36936025,8758500,2076870) = 253, by paragraph 16, whence dividing by 25, by paragraph 17, F75 (1477441,350340.83074) = 1, the same result as obtained by the first method.

Page 366 366 University of California Publications in Mathematics [VOL. 1 25. To determine whether a given solution is fundamental or not, one makes use of the fact that if the solution is the nth derived solution of the fundamental solution, that is, if Xn+ty,+t2zn=(xl+tyl+t2zl)n, then x1ylzi may be determined by solving the equation xl+tyl+t2z1= Vxn+tyn+t2zn. In practice, n is unknown, so that different integral values are tried in order, beginning with n =2. If x2+ty2 +t2z2= (Xl+tyl+t2zl)2, then x2=xl2+2Dylzl 3xl2, Y2=Dz12+2x1yl 3Dz2, z2= yl2+2zlxl=3y12: or X2-3x12, 2 2= Z2x2, where t= / D. The application of this t t2 test is further facilitated by the fact that if FD(xlylZl) = 1, then only values of x1 such that x13- 1, (mod D), need be tried. Having thus discovered that the given solution is, or is not, the square of a simpler solution, either repeat the test on the simpler solution, or test the given solution to see whether or not it is the cube of a simpler solution. The relations to be used in the test as to whether a given solution is the nth power of a simpler solution arise from the equation x,+ty,+t2, = (xl+tyl+ t2z1)n, and the approximate equalities, xl- tyl t2z. In practice the values of x soon become so small that separate tests may be made for smaller values of x, such that X3-1, (mod D). Not all the tabulated solutions have been tested, owing to lack of time, but there is reason to believe that most of them are fundamental. As an illustration of this method of testing a solution to see whether it is fundamental or not, consider the solution x=9073, y=3963, z=1731 for D= 12. If this is the square of a fundamental solution, x'y'z', then x' -=903 55, y'- =24, z' x' /44 1031. As a matter of fact, F12(x'y'z') = 1, but z' is fractional. The only values of x<55 such that x3=1, (mod 12), are 1, 13, 25, 37, 49, and none of these give solutions of F12(xyz) = 1, excepting of course the solution, 1, 0, 0. For large numbers, of course, this testing is done congruentially, as Meissel11 suggests and illustrates. 1 1 0 0 2 1 1 1 3 4 3 2 4 5 3 2 5 41 24 14 6 109 60 33 7 4 2 1 8 1 0 0 9 4 2 1 10 181 84 39 11 89 40 18 12 9073 3963 1731 13 94 40 17 14 29 12 5

Page 367 1923] Wolfe: On the Indeterminate Cubic Equation 367 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 5401 16001 324 55 64 361 1705 793 2166673601 649 9375075001 9 1 9 102866541757601689 811 101209 768096001 15270674074129 334153 25776 109 100 29071 529 5041 21169 49 5351941 1477441 16449049 562944292769 3810843073 8786992 12867751 107846641 209 113015453598 61561 32947340560201 2190 6350 126 21 24 133 618 283 761875860 225 3206230550 3 888 2520 49 8 9 49 224 101 267901370 78 1096515424 1 0 0 3 1 33481749309704842 10897883001448120 261 84 32218 10256 241935080 76204775 4760876269140 1484279131362 103146 31839 7880 2409 33 10 30 9 8647 2572 156 46 1474 431 (931197095781897587447729 270051748734525954034260 l 78316338533401657636358 6090 1752 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 14 1515981 415374 4590798 155990973316 1048593882 2401273 3492845 29081484 56 30087022392 16287 8663621462574 4 429414 116780 1281255 43224852030 288531726 656210 948104 7841994 15 8009779969 4309 2278130361072

Page 368 .68 University of California Publications in Mathematics [VOL. 1 56 765073 199974 52269 57 1460968 379620 98641 58 929 240 62 21618361237973511050873 59 ( 5553141829215933501576 1426444115533632242954 60 2161 552 141 61 3905 992 252 62 8929 2256 570 63 16 4 1 64 1 0 0 65 16 4 1 66 9505 2352 582 67 4289 1056 260 68 2449 600 147 404886837053487091694212951195653956127452401 69 I 98715184393700556938337454013404500951638820 [ 24067681974543893805323831567684099602695630 70 1121 272 66 71 1788355606552816482 431884645684316172 104299361097095425 72 1270801 303738 73011 73 99928 23910 5721 74 1025641 244297 58189 f570211212113607886743089699946001 75 t 135212029500803659084176761891820 [ 32062317494528092279877193074334 76 305 72 17 77 3752144035327073 881960753382348 207309411148166 78 9818137169569 2297898779424 537814730970 79 370454 86336 20121 80 28977465601 6725074038 1560751428 81 2460229201 568609218 131417204 (621336111092869050144359 82 / 143017322810145255729601 [ 32919307696127305290702 (7512137677415067474067653515153 83 A 1722149465980481177969345011132 394800908946784666615761008466 84 82988024365 18949117863 4326757632 1302427709068833776364750689951004721 85. 296219732439518970205699942411045068 [ 67371209377351163649079349355266292 86 565 128 29 f 86965911738730294017560094302901018001 87 19626489977891548853457921605649509148 [ 4429311452624360573996059841563029246

Page 369 Wolfe: On the Indeterminate Cubic Equation 369 88 89 90 91 92 93 94 95 23761 421204668989443920 58321 81 34447365811144441 90139756288 961302655531 3576393871601 5342 94340138120314920 13014 18 1201 21130016630426161 2904 4 96 97 7630624358730583 1690301325891661 19895524677 4391313206 211422331357 46498781564 783797980668 171776179178 100428638580128172512127484936339201 21933124190915540772279199097914760 4790087205954739309459432490919945 1040068758302891753134 226362669236418190113 [ 49266029399288296684 823541096 178625415 f2370504989476787061889 512423622402913328862 110768789756683516940 3389917630695 730335613992 3796883757 99 15734603821501 LITERATURE CITED 1 Dirichlet, G. L.: Werke, vol. 1, pp. 621-623. 2 Euler, L.: Corresp. Math. Phys., vol. 1 (1843), pp. 629-631. 3 Lagrange, J. L.: Mem. Acad. Berlin, vol. 24 (1770), (Oeuv., vol. 2, p. 74). 4 Dirichlet, G. L.: Werke, vol. 1, pp. 627-632. 5 Fermat, Pierre de: Oeuv., vol. 2, pp. 333-335. 6 Mathews, G. B.: Proc. London Math. Soc., vol. 21 (1889-90), pp. 280-287. 7 Lagrange, J. L.: Mein. Akad. Berlin, vol. 24 (1770), (Oeuv., vol. 2, pp. 603-615). 8 Hermite, Ch.: Oeuv., vol. 1, pp. 185-187. 9 Jacobi, C. G. J.: Ges. Werke, vol. 6, pp. 385-426. 10 Bachmann, P.: Crelle, vol. 75 (1873), pp. 23-24. Meissel, E.: Beitrag zur Pell'schen Gleichung hoherer Grade, Progr., Kiel, 1891. 12 Berwick, W. E. H.: Proc. London Math. Soc., (2) vol. 12 (1913), pp. 393-429. 13 Lehmer, D. N.: Manuscript. For a history of the Pellian equation, the reader is referred to E. Konen, Geschichte der Gleichung, t2-Du2=l, Leipzig, 1901, and to L. E. Dickson, History of the Theory of Numbers, vol. 2, chap. 12. For further references to the literature of the cubic form x3+-Dy3+D2z3-3Dxyz, see L. E. Dickson, History of the Theory of Numbers, vol. 2, chap. 21, pp. 588-595, esp. 593-595.

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 17, pp. 371-387 November 8, 1923 A STUDY AND CLASSIFICATION OF RULED QUARTIC SURFACES BY MEANS OF A POINT-TO-LINE TRANSFORMATION BY BING CHIN WONG 1. The purpose of this paper is to study and classify ruled quartic surfaces by means of a point-to-line transformation. The paper will consist of four sections: Section I. A historical discussion of the subject. Section II. The setting up of the machinery of transformation. Section III. The study and classification of the ruled quartics of the species from I to IX. Section IV. A slight modification of the machinery of transformation given in Section II and the classification of the remaining three species. SECTION I 2. Ruled quartic surfaces have been studied and classified by Cremona,1 Sturm,2 Cayley,3 Salmon,4 and others.5 These authors classify these surfaces according to their deficiency,6 and reclassify them according to the nature of the double curves they have on them, and once more reclassify them according to the kinds of surfaces into which they are reciprocated. 3. There are two species of ruled quartics of deficiency unity and ten species of deficiency zero. The double curve on those of deficiency unity is a pair of straight lines which, if distinct give species I, and if coincident, give species II.7 The 1 Bologna Accad. Sci., Mem., vol. 8 (1868) 2 Liniengeometrie, Bd. 1, pp. 52-61 3 Phil. Trans., vols. 154 (1864) and 159 (1869). 4 Analytic Geom. of Three Dim. (ed, 5), vol. 2, chap. 16. 5 A discussion of ruled quartic surfaces may also be found in Jessop, The Line Complex, Cambridge, 1903, chap. 5 and in Basset, The Geometry of Surfaces, Cambridge, 1910, chap. 6. 6 The deficiency of a surface is defined as the difference between the maximum and the actual number of double points on any plane section of the surface. 7 This discussion is based upon the works of Cremona and Sturm, but the order of the species is Sturm's. Different authors number the classes differently. Owing to the difference in the point of view, the order in this paper is yet a different one.

Page 372 372 University of California Publications in Mathematics [VOL. 1 double curve on those of zero deficiency is either a space cubic, or a straight line and a conic, or three straight lines. Those ruled quartics with a double space cubic are of species III if their reciprocal surfaces are of the same kind, and of species IV if their reciprocal surfaces have a triple line. Those whose double curve is a straight line and a conic belong to species V if they reciprocate into surfaces of the same kind; to species VI if they reciprocate into surfaces with a triple line. Those with three double straight lines make up the remaining six species: if the three lines one of which must be a generator are distinct, the surfaces belong to species VII; if two are coincident, the distinct one being the generator, the surfaces belong to species VIII. If the three double lines are all coincident, we have species IX if the surfaces reciprocate into those with a double place cubic; species X if they reciprocate into those of the same kind; species XI if the triple line is itself a generator counted once; and finaly, species XII -if the triple line is itself a generator counted twice. Species XI reciprocate into surfaces with a straight line and a conic, and species XII into those of the same kind.8 4. The table9 given (p. 373) presents a view of the different orders given to the species by Sturm, Cremona, Cayley, and Salmon, and also exhibits the special features of the different classes. In this table k3 stands for double space cubic; c2 stands for double conic; e stands for double generator; d stands for double directrix; d+d stands for two coincident double directrices; d+d' stands for two distinct double directrices; d+d+d stands for a triple line, d+d+de stands for a triple line which is itself a generator counted once; d+de+de stands for a triple line which is itself a generator counted twice. The double curves on the reciprocal surfaces are denoted by the corresponding Greek letters. It is obvious that species IV and VI reciprocate into species IX and XI respectively, and vice versa; while the other species are their own reciprocals. 5. The classification given in the following paragraphs is from a different point of view. Here we start with certain plane curves10 which, by a point-to-line transformation, go into quartic surfaces, and, it will be shown, all the different classes of ruled quartics can be obtained in this manner. It will also be shown that there exists a close relation between the singularities" on the surfaces and the position and class of the plane curves. 8 Salmon divides those surfaces with a triple line into five classes, I-V according to his enumeration, making thirteen species in all, while Sturm divides them into four, IX-XII as given above. As a matter of fact, V of Salmon is only a subform of IV, that is, a subform of XII of Sturm. See ~ 47 below. 9 This table, with a slight modification and an addition of Salmon's enumeration, is the combination of the two found in Jessop, The Line Geometry, p. 80. 10 For our purposes, conies and plane cubics only need be employed. Plane quartics may be used but they offer nothing new. No quintic curve or curve of higher degree can give quartic surfaces. But if one were to study and classify ruled surfaces of higher degree from this point of view, one would have to resort to curves of higher order, both plane and space. 1 Including double curves and pinch-points.

Page 373 1923] Wong: A Study and Classification of Ruled Quartic Surfaces 373 6. This method has three interesting features: (1) the one-to-one correspondence between the generators of the surfaces and the points on the plane curves; (2) the dependence of the singularities on the surfaces upon the position and class of the plane curves; and (3) the possibility of the method in studying and classifying scrolls of higher order. Double Double curve Def. curve on rec. s. Sturm Cremona Cayley12 Salmon13 d+d' s+5' I 11 1 11 p=l.1.. -- d+d 6+6 II 12 4 13 k3 k3 III 1 10 6 k3 5+-+- IV 7 8 7 c2+d k2+5 V 2 7 8 c2+d +-+s VI 4 (11) 9 d+d'+e 5+5'+e VII 5 2 10 p=O — d+d+e 5+6 VIII 6 5 12 d+d+d k3 IX 8 9 1 d+d+d 5+5+6 X 9 3 2 d+d+de k2+6 XI 3 (12) 3 d+de+de a5+e+5e XII 10 6 4 (5) SECTION II 7. Take two fixed quadrics 4 F1 2zai2i = 0, 4 F2 - -bix2i = 0. 1 Any point Y(yl, Y2, Y3, y4) in space is transformed into a line f4 1aiyixi =0, -t I biyixi =0, 12 Cayley does not consider (11) and (12) as separate species, but as subforms of 8 and 9 respectively. 13 Salmon begins with surfaces having the highest singularity, that is, a triple line, and divides them into five classes, and then takes up those having a proper space cubic, and then those whose cubic degenerates, etc. See note 8.

Page 374 374 University of California; Publications in Mathematics [VOL. 1 which is the intersection of the polar planes of Y with respect to F1 and F2. To the c 3 of points in space there c 3 of polar planes with respect to F1 and also c 3 of polar planes with respect to F2 corresponding. The space consisting of the o 3 of polar planes with respect to F1 and the space consisting of the a 3 of polar planes with respect to F2 are clearly collineated to each other, for to every plane of the one space there corresponds a plane of the other space, both being polar planes of the same point. Then every line 1, being a line of the intersection of the corresponding planes of the two collineated spaces, is a line of a tetrahedral complex14 whose fundamental tetrahedron A1A2A3A4 is the self-polar tetrahedron common to F1F2. To show this algebraically 15, write the six homogeneous coordinates of 1: ql2 ' ql3 q14 q34 ' q42 q23 = 712Y1Y2: 7I13Y1Y3 ' 14lY14 7'34Y3Y4 7'42Y4Y2 7'23Y2Y3 where 7ik= aibk-akbi. Then q42ql3 742713 q14q23 7T14723 q14q23 7r147r23 q12q34 7127T34 ql2q34 -7127r34 and q13q42 7r13742 Clearing fractions and adding, we have Aql2q34+Bql3q42+Cql4q23 = O, where A = 7T137r42 - 147r23, B = 7r14723 - 7127r34, C = 7127r34 - 7r137r42, which is the equation of the tetrahedral complex. Since the coordinates of the point Y are not involved in this equation, every point in space goes into a line belonging to the above complex determined by the fixed quadrics. 8. If a point describes a straight line in space, its two polar planes with respect to F1 and F2 describe two projective axial pencils whose corresponding planes meet in the generators on a ruled quadric. Algebraically, let (Y1+Xz1, y2+XZ2, Y3+Xz3, Y4+XZ4) be the coordinates of any point on a given line g through Y and Z. Then the intersections of the corresponding planes of the two projective axial pencils determined by the given line g are given by 4 2ai(yi+-Zi)xi = 0, 4 0bi(yi+Xzi)i = 0. 14 Reye, Geometrie der Lage, Abth. 2, erste Auflage (1868), 15. Vortrag. An exposition of this complex is also given by Sturm in his Liniengeometrie, Bd. 1, pp. 332-382. 15 Jessop, The line Complex, Cambridge, 1903, chap. 7.

Page 375 1923] Wong: A Study and Classification of Ruled Quartic Surfaces 375 Eliminating X, we have 4 I 7rkPikPiXk — 2pl2XlX2+1l 3p13Zl13+ 714Pl4X1X4+7 34p34X3X4+r-42p42X4X2+7r23p23X2X3 i=1 k= = 0, [where pik = YiZk-ykZi] which is the equation of a quadric surface. But the pik are the six homogeneous coordinates of the given line, and therefore, we conclude that when the pik of any line are given we can at once write down the equation of its corresponding quadric in the above form. 9. Now if we take three straight lines p'ik, p/ik, p'I/ik forming a triangle in space whose corresponding quadrics are, respectively, 4 Q - 7 1 ikPikXiXk = 0, 4 1 4 Q -fTzkp i k - 0 Q -=7 ikp..X ikk = 0k any conic Allp'2ik+A22P"2ik+A33p"'2ik+2AlA2pikp'kP"ik2A2A3p" ikp'ik+2A3A p"ikpik = in the plane of this triangle will go into a ruled quartic surface whose equation is A llQ'2+A22Q"2+A33Q"'22+2A12Q'Q"/-2A 3Q'Q"'+2A23Q"Q"/ = 0 or 4 Twikwji[ZArsp(r)ikp(8) il]XiXkXijXi [r, s= 1, 2; Ars=Asr]. A plane cubic in the plane of the same triangle will go into a sextic surface whose equation is 2ArstQ(r)Q(s)Q(t);7rikrT jilmn[2A rstP (r) ikP(s) ilp(t) mn]XikX XjXXmXn = 0 [Arst=Arts=Asrt=Astr=Atsr=Atrs; r, s, t=l, 2, 3]. Similarly, a plane curve of degree n goes into a ruled surface of degree 2n. All the surfaces thus obtained have the four vertices of the fundamental tetrahedron in common. 10. Corresponding to the a 2 of points in a plane u is a congruence of lines all cutting across a cubic k3 twice. To prove this, when u is given, we have two points Z and Z' fixed which are the poles of u with respect to F1 and F2. Z and Z' can be considered as two projective point systems, and the locus of the intersections of the corresponding rays which do meet is a space cubic. Every point in u will have its two polar planes, one through Z and the other through Z', intersecting in a line cutting across the cubic twice. Since every line in u has its corresponding quadric

Page 376 376 University of California Publications in Mathematics [VOL. 1 through the four vertices Al, A2, A3, A4 of the tetrahedron of reference, this cubic k3 must go through them also, and is therefore a cubic of the complex. Thus, every plane in space determines a cubic of the complex uniquely. Every plane curve goes into a surface on which lies the cubic determined by the plane of the curve. 11. A pair of intersecting lines is transformed into a pair of quadrics having in common a generator which comes from the point of intersection of the two given lines and cubic k3 which is determined by the plane of the two given lines. To obtain the parametric equations of k3, let (Ul, U2, U3, U4) be the coordinates of a given plane u, and let p'ik and pik be any two lines lying in it. The two quadrics corresponding to these two lines 4 Q' - T7rikP'ikXiXk = 0, 4 Q= _ 7rik p ikXiXk =, 1 have, besides the common ruling into which the intersection of p'ik of p"ik goes, a common cubic k3 whose parametric equations are easily found to be, with X as the parameter, pX1=ul(a2X-b2) (a3X-b3) (a4X-b4), px2=U2(a3X-b3) (aX-bl) (alX-bl), px3=U3(a4X-b4) (aiX-bi) (a2X-b2), pX4=U4(aiX-bi) (a2X-b2) (a3X-b3), or 4 pxi=uuP(aj,-bj) [i=l, 2, 3,4; iZj]. 1 These equations are independent of the coordinates of the lines but not of those of the given plane; therefore, when the coordinates of any plane are given, the equations of its corresponding cubic can be easily written. 12. Now if ul=0, i.e., if the plane u passes through the vertex Al of the fundamental tetrahedron, k3 degenerates into a conic c2 lying in the plane xl and a straight line d which passes through Al and cuts across the conic. c2 and d must have a point in common, for, if not, the quadric which corresponds to a line in u (0, u2, U3, u4) and therefore contains c2 and d, would be intersected by a straight line lying in xl through P in which d pierces xl in three points, one being P itself and the other two being on c2. This is impossible. 13. We can get the parametric equations of the conic c2 by putting ul =0 in the parametric equations of k3 (Art. 11), or we can get them as follows: Let (o, U2, U3, U4) be the coordinates of the plane u and let any other plane Ut(U'1, u'2, u'3, u'4) intersect it in the line qik(ulU2: U'u3: U4: q34: q43: 23).

Page 377 1923] Wong: A Study and Classification of Ruled Quartic Surfaces 377 Then corresponding to this line we have the quadric 7rl2q34XlX2 + 7rl3q42XlX3 + 714q23lXX4 - '(7r23U4X2X3+ 742U3x2x4+7r34u2x3x4) =0. Putting x1 = 0, we have the conic in which this quadric is intersected by the plane x1: X1 = 0, 7r23U4X2X3+-742U3X2X4+ 7r34U2X3X4 = 0. These equations are independent of the coordinates of the plane u', and therefore represent the conic common to all the quadrics corresponding to all the lines in the plane u. This conic is none other than c2, part of the degenerate cubic k3. 14. Now to obtain d: Take two lines through A1: U(O, U2, U2, U4), = U(O, U 2, U 3, U14), _ U(O, U2, U3, U4), 2 U (O, UL 2, U 3, U 4), whose corresponding quadrics (pairs of planes) Q1- =X1[7Tr1234X2+713q42x3+L714q23X4] = 0, Q 2 -X2[7rl2'34X2 + l3q 42X3 +T7r14q 23X4] = 0, have, besides the plane x1, the line 71 12q 34X2 + 7r13q 42X3 + 7r 14q23X4 = O0 7r12q 34X 2 +7r 13q' 42x3 + 7r1 4q23x4 = 0, in common. This line has its homogeneous coordinates: 0: 0: 0: 7137T14U2: 7I127711U3: 712713U4 and has its parametric representations: pxl = X7r127T143, PX2 = 7rl37r142, PX3 7- 12714U3, pX4 = 7r12713U24. Both the coordinates and the representations of this line are independent of the coordinates of the plane u' and those of the plane u", and therefore this line is the line d common to all the quadrics coming from all the lines in the plane u. 15. Now if u1=u2=0, i.e., if the plane u passes through the edge A1A2 of the fundamental tetrahedron, we have its k3 broken up into three straight lines d+d'+e. d lies in the plane x1, d' in x2, while e being the intersection of x1 and x2 cuts across the other two. Let any plane U (U'1, U 2, U'3, U 4) intersect the plane u (0, 0, U3, u4) in the line whose transform is the quadric 7r12q34XlX2 +7rl3UIUf2X1X3 - 714U3U 2X1X4 - 7r42 1U3X2X4 - 7r23U'tU4X2X3 = 0.

Page 378 378 University of California Publications in Mathematics [VOL. I When xl = 0, we have X2=0, and r23U4X3 + r42U3X4 =; and when x2 =0, we have 1 = 0, and 7rI3u4x3- 7r4u3x4 = O. The equations of these three lines d I xl=0, 7r23U4X3 + 743U3X4 0; d X2=0, d' 7r13U43-7T14U3X4 0; fXl=0, e x2 = 0, being independent of the co6rdinates of the plane u', represent the degenerate k3 common to all the quadrics which correspond to all the lines in u. e is the common generator, while d and d' are the common directrices. 16. Every plane u, besides determining uniquely a space cubic k3, contains a conic 72 of the complex, the conic to which all the complex lines in the plane are tangent. All these lines, co in number, correspond to the points on k3; or in other words, all the points on a cubic curve of the complex determined by a plane go into lines tangent to the conic of the complex in that plane. 17. To obtain the equations of this conic 72, write the equations of the polar planes of the points on k3 (Art. 11) with respect to F1 and F2, respectively, 4 4 2 aiui P (ajX-bj)xi=O, i=1 j= 1 4 4 Z biui P (ajX-bj)xi=O [i~j]. i= 1 j= 1 These two planes intersect in a line which we shall designate by L. For every value of X, there is a point on k3 and there is corresponding to it a line given by L. All these lines of the system L lie in the plane u, for the matrix alulB2B3B4 a2u2B3B4B1 a3u3B4B1B2 a4u4B1B2B3 blulB2B3B4 b2u2B3B4B1 b3u3B4B1B2 b4u4BlB2B3 I U1 U2 U3 - U4 [where Bi=aiX-bi] of the coefficients of the two equations of L and those of the equation of the plane u is of rank 2. To show this is merely algebraic work which any one may easily verify. 18. The line L pierces the plane xi and x2 respectively in Rl- (0: 7r34u3u4B2 w42u4u2B3: r23u2u3B4), R2 -(7r34u3u4B2: 0: 7r41u4u1B: 7rl3ulu3B4).

Page 379 1923] Wong: A Study and Classification of Ruled Quartic Surfaces 379 The plane determined by R1, R2 and A(0: 0: 0: 1) is given by p 7r41UlB2B3x1 7r42u2B3Bx2 + 7r43u3B1B2x3 = 0 or 3 3 r4iUi P (ajX-bj)xi=0 [ij]. i=1 j=i This equation involves X in the second degree and is therefore a system of planes enveloping a quadric cone whose vertex is the point A4(0: 0:: 1). Therefore, the equations p=0, u=0 represent the system of lines tangent to the conic y2 of the complex in the plane u and these lines are lines of the complex. 19. This system of rays is none other than the system L given by the equations of Art. 17. Thus, when a plane is given we can easily write down the equations of the system of lines of the complex lying in it. 20. The system of complex lines in the plane u (0, u2, U3, U4) is made up of two flat pencils of the first class, one of which corresponding to the points on the conic c2 (Art. 13) in xl has its center at Al and the other corresponding to the line d (Art. 14) has its center at the point A'1i(0: 7T127r34U3U4: 71r37r42U4U2: 7r147r23U2U3) whose transform is the line d itself. To obtain the equations of the first pencil, one needs only to put u = 0 in p=O, u=O given in Art. 18 or in the equations given in Art. 17. The equations of the other pencil are those of the polar planes of points of d whose coordinates are given in Art. 14 with respect to F1 and F2 and are given by XalU37rl27rl4Xl+balu27r37rl4X2+la3u37rl27'rl43+ - 4u47rl27rl3X4= 0 Xblu3yi2ri4xl + b2u2ir137r14x2 + b3u3rpl27i,14x3 +- b4u4rr12wi-13x4 = 0. For every value of X there is a ray of the pencil, and every ray of this pencil pierces the plane xl in the same point A1 whose coordinates are given in Art. 20. The polar planes of this point A'1 with respect to F1 and F2 are a2U3U47r27r34X2 +-a3U2U7r137r42x3 -a4u2U37lr147r23x4 = 0, b2U3U47r127r342 + b3u2u47r 13742x3 + b4U2U37r147r23x4 = 0 which intersect in a line which is no other than d itself. 21. Similarly, we can show without any difficulty that the system of complex lines in a plane containing an edge, A 1A 2, of the tetrahedron of reference is made up of two flat pencils, one with center at Al and the other with center at A 2. 22. Now we examine further the relations between the lines and points in any plane u and the points and bisecants of the space cubic k3 which u determines. Since every point on k3 transforms itself into a line of the complex in u and every

Page 380 380 University of California Publications in Mathematics [VOL. 1 point in u transforms itself into a line cutting across k3 twice, let 11 and 12 be two lines in u corresponding to the points L1 and L2 on k3, and then the point P in which 11 and 12 intersect goes into the line p cutting across k3 in L1 and L2. Thus, a oneto-one correspondence exists between the points in u considered as the intersection of two tangents to the conic 72 of the complex and the bisecants cutting across the cubic k3 in the two points to which the tangents to 72 correspond. If L1 and L2 approach coincidence, the rays 11 and 12 also approach coincidence. The p becomes a tangent to k3 while the point P in u moves up to the conic 72. We can easily conclude that A point without 72 goes into a real bisecant of k3; on 72, a tangent; within 72, a line with no point in common with k3. 23. We have seen that every conic y'2 by our method of transformation, goes into a quartic scroll. In particular, if 7'2 is coincident with 72 every point of which goes into a tangent to k3, the surface is a developable with k3 as the cuspidal edge. But in general, y'2, when not coincident with 72, is cut by every tangent to 72 in two points; therefore, every point on k3, in general, is a double point on the corresponding quartic surface. In other words, k3 is a double curve. We can easily see that the ruled sextic surface coming from a plane cubic contains k3 as triple curve, and that, in general, the ruled surface of degree 2n coming from a plane curve of degree n contains k3 as n-fold curve. 24. From the figure of the preceding article we see that corresponding to Pi on the conic 7'2, which is the intersection of the complex lines 11 and 12, is the ruling pi on the corresponding quartic, cutting across k3 in L1 and L2. But the ray 12 intersects 7'2 again in P2 which is on another complex line 13; therefore, on the quartic surface, from the point L2 on k3 issues another ruling p2 meeting k3 again in the point L3. Continuing the process, we see that from every point on k3 go two generators of the quartic meeting the curve again in two different points, from each of which goes another generator. But if a ray of the complex is tangent to the given conic 7'2, i.e., cuts it in two coincident points, then corresponding to the two coincident points P4 and P5 common to 7'2 and 15 are two coincident generators from the point L5 on k3 on the quartic. Then L5 is a pinch-point on the surface. We may thus conclude that the existence of a line of the complex tangent to a given plane curve means the existence of a pinch-point on the corresponding surface, and that, in particular, a ruled quartic with a double space cubic which does not degenerate can have no more than four pinch-points. If 712 intersects y2 in two real points, two of the pinch-points become imaginary, for the two conics can have only two real common tangents.

Page 381 1923] Wong: A Study and Classification of Ruled Quartic Surfaces 381 SECTION III 25. The most general equation of the ruled quartic from any conic is given in Art. 9. Let the lines p'ik, p'ik, p"'ik be three lines of the complex in the plane u, i.e., form a circumscribed triangle to y2, and further let P', P", P"' be the three points on k3 to which p'ik, pltik, pif/ik correspond respectively. Finally, let plik and pllik intersect in R"', p"ik and p'ifk in R', p'ik and p'ik in R". Without loss of generality, let a given conic in the plane pass through R" and R"' but not R'. Then the corresponding quartic surface is given by A 1Q'2 +2A 12Q'Q +2A 13Q"/Q' +2A23Q"Q'" = 0 where the Q's are the left members of the equations of the quadric cones corresponding to the lines p'ik, P"ik, P"'ik (Art. 9). This equation can be written 4 3 rtkTj; [Arsp()ikPil XiXkXjXl= 0 [A22=A33 = 0] 26. Now corresponding to the points R', R", R"' are three lines rik, r/'ik, r'fik forming a triangle with its vertices P', P", P"' on k3. Let the plane of this triangle be denoted by u' and its complex cubic by k'3. The above quartic surface is intersected by u' in a conic 7"2 through P" and P"' and two straight lines which are rulings of the surface corresponding to the points R" and R"'. Note that R', R", R"', being three points whose transforms are three lines in u', are three points of the space cubic k'3 which u' determines. Then the conic y '2 in u' will go into a quartic surface which is intersected by u in the conic y'2 and the lines p"ik and p'"ik which are rulings of the surface. Therefore, we conclude: 27. Any conic through two vertices of a triangle whose sides are complex lines (or through two points on any complex space cubic) goes into a quartic surface all of whose generators intersect another conic through two vertices of another triangle whose sides are complex lines (or through two points on another complex cubic). The quartic corresponding to the latter conic will have its rulings all going through the former conic. This class of quartics we designate as class I (III of Sturm). 28. But if the given conic /'2 goes through R' also, we have class II (IV of Sturm). The equation of this quartic surface is A12Q'Q" +A13Q'Q'" +A 3Q"IQ'" = 0 which is the same as the equation of the quartic surface given in Art. 9 after having imposed the condition that r s. This surface is intersected by the plane u' in four straight lines three of which are r'ik, r"ik, r" ik and the fourth which we designate by sik is the very line whose corresponding quadric is intersected by u in the given conic 7'2. Every plane through sik cuts the surface in three generators and this is to be expected, for there are an infinite number of triangles with vertices on 7'2 and sides tangent to 72 [C. Smith, Conic Sections, p. 275].

Page 382 382 University of California Publications in Mathematics [VOL. 1 29. These two classes have a space cubic for double curve. Class I has all its generators going through a conic with two points in common with the double curve and class II has all its generators going through a straight line which has no point in common with the double curve. The former reciprocates into one of the same kind, while the latter reciprocates into one with a triple line [see below, class VIII]. 30. Now let the plane u pass through A1, a vertex of the fundamental tetrahedron. The k3 of this plane is made up of the conic c2 in the plane xl and a straight line d, and the system of complex lines is made up of two flat pencils A and A'1 (Art. 20). Every conic y'2 not going through AI in u transforms itself into a quartic with c2 as double conic and d as double line. From every point on d issue two generators meeting c2 in two different points and vice versa. It can be easily seen that there are in general two pinch-points on c2 and two on d, for there can be in general two tangents from A1 and two from A'1 to 7'2. 31. Let p'ik and p"ik be two rays of A'1 and p"'ik(x'2: /x"3: x"'4: 0 0 0 ) be a ray of A1, meeting p'ik in R" and p"ik in R'. Without loss of generality we can let a given conic y2 pass through R' and R" (but not through AI nor A'1) and the equation of its corresponding quartic is the same as that of class I given at the end of Art. 25 after making the necessary changes in the pik's. This quartic is intersected by the plane u' determined by the line d (corresponding to A'1) and the lines r'ik and r"ik (corresponding to R' and R") in a conic 7'2 and two generators (r'ik and r"ik). This conic 7'2 goes back into a quartic whose section by u is no other than the given conic Y'2 and the two lines p'ik and p"it. Therefore, this quartic has all its generators going through a conic. This is class III (V of Sturm). If 7'2 is tangent to the line A A'I, one of the pinch-points kn c2 and one of the pinch-points on d fall together at the point common to c2 and d. 32. If 7'2 goes through A'1 also, its transform is of class IV (VI of Sturm) whose equation can be easily obtained by putting A33=0 in the equation of class III. The plane section by u' is made up of four straight lines r'ik, r"ik, the line d, and a fourth line Pik whose corresponding quadric is cut by u in the given conic y'2 itself. This quartic, every one of whose generators, besides meeting c2 and d, meets this line Pik, reciprocates into one with a triple line [see class IX below]. Note that there is one pinch-point on d and two on c2, for there is only one ray of the pencil A'i and two of the pencil A1 tangent to y'2. 33. If the edge A1A2 of the fundamental tetrahedron lies in the plane u, any conic y'2 in it not going through A1 and A2, the centers of the two flat pencils making up the complex conic 72 (Art. 21), transforms itself into a quartic surface whose double cubic is d+d'+e. Letting p'ik and p"ik be two rays of pencil A2 and p'"ik be a ray of pencil A 1, we have p'I13==P'14 = P34 = 0, p I13=P "14 = P34=O, p 23 = p"'42 = p 34 = 0;

Page 383 1923] Wong: A Study and Classification of Ruled Quartic Surfaces 383 and putting these in the equation of the quartic in Art. 9, we have the resulting equation representing the quartic in question. This is class V (VII of Sturm). There are in general two pinch-points on each of the lines d and d'. e, to which every point on A1A2 but not A1 and A2 is transformed, for y'2 has two points, real or imaginary, in common with A1A2. But if the given conic y'2 is tangent to A1A2 e is a cuspidal ruling and d and d' have each only one pinch-point. 34. To obtain the next class, VI (I of Sturm), of quartics which have only two double lines and no double generator, we take a cubic curve without a double point in the plane u through the points Al and A2. The corresponding surface is a sextic made up of the two planes x1 and x2 corresponding to Al and A2 respectively and a quartic. 35. Let the equation of a cubic cone with its vertex at A4(0: 0: 0: 1) and with two elements one through A1 and the other through A2 be 3 2ArstXrXsXt = 0 [A 1ll = A222 = 0]. The plane section of this cone by the plane u(0: 0: U3: u4) is a cubic curve through Al and A2. Letting p'ik, p"iik, p'"ik be the coordinates of the lines in which the planes xl, x2, xs intersect u, we have p'ik(O 0: 0: 0: P'42: p23) [where p42 = U3; p'23=-4], P"ik(O: p"13: p"14: 0: 0: 0) [where p"13 = U4; p"14= -u3], P"''ik(p"'12: 0: 0: 0: 0:: ) [where p"' =U4]. Then the quadrics corresponding to these lines are respectively Q' -X2[Tr42p'43X4 + -23p'23x3] = O, Q" - Xl[w13p"13X3-+7 14P"14X4] = 0, Q" -7r12p' 12X X2 = 0 Therefore, the sextic surface corresponding to the cubic curve is given by 3 2ArstQ(r)Q(s)Q(t) = 0 [A 1 = A2 22 = 0]. which, when expanded, is x1x2 [a quartic factor] =0. 36. The factor enclosed in the bracket equated to zero gives a quartic surface. If the cubic has a double point, this surface is of class V, for it has d and d' for double directrices and has a double generator which, however, is different from e. But if we impose upon the Arst's the condition that the cubic be without a double point, the surface will be without a double generator. 37. This surface has four pinch-points on each of the two double directrices, for, the class of the cubic curve without a double point being six, there can be drawn from a point on it four tangents exclusive of the one at the point itself.

Page 384 384 University of California Publications in lMathematics [VOL. 1 38. Note that the equation of the degenerate sextic surface given in Art. 35 could have been obtained from the sextic equation in Art. 9 by putting Alll, A222, pl12, pl13, P'14, P134, Pf12, p"134, P"42, p"23, P'll13, pll14, P"l34, P"I42, p"'/23 all equal to zero [Art. 35]. 39. Now if we put in the equation of the cone in Art. 35 All1 =A112= A113 = 0, the result is that of a cone with x2=0, X3=0 as double element, thus giving a plane curve in u with a double point at AI. With Al1, A112,, 113, pt12, p113, P'14, P'34, p/t12, p"34, P"42, P"23, p'l13, "'ll, P'3, p"1142, p"/23 all put equal to zero, the left member of the sextic equation in Art. 9 factors into x21 [a quartic factor]. x2l=0 represents a double plane corresponding to the double point A1. The quartic factor equated to zero represents a quartic surface which has the line d' as a triple line and d as a simple line. From every point on d' issue three rulings corresponding to the three points in which every line of the pencil A2 intersects the given cubic; and from every point on d issues only one generator corresponding to the point in which every ray of the pencil A1 meets the cubic besides A1 itself. This class of quartics is class VII (X of Sturm). Since from the point A2 which is not on the curve only four tangents can be drawn to the curve, the triple line has four pinch-points, i.e., points at which two of the three generators become coincident. 40. To get class VIII (IX of Sturm), we must return to the case where the plane u passes through the vertex A1 of the fundamental tetrahedron. Remembering that the coordinates of A'1 are [Art. 20]. (0: 7r127r34U3U 4: rl37r42U4U2: r147r23U2U3) and letting A'1A"1 and AA"1 be the lines of intersection of u with xl and x2 respectively, thus giving (0: 0: u4: -u3) for the coordinates of A"1, we have A'iA"-'ik(O: 0: 2: U3: u4), AiA"l-pik(O: U4: -U3 0 0 0: ), A A'l —p'"ik(l2r34U3U4: 7r13T742U4U 2: T147T23U2U3: 0: 0: 0). To these three lines correspond the following quadrics, respectively: Q 7r34U2X3X4 + 7r42U3X4X2 + 7r23U4X2X3 = 0, Q Xl((7rl3U4U3-7rl4U3x4) =0, Q Xl1(712127r134U3U4X2+7r2137r42U22u43+7r2147r23u2U3X4) = 0. Then the equation of the sextic surface corresponding to any cubic in u with a double point at A1 is given by 3 ZArstQ(r)Q(s)Q(t) = 0. [A III = A 112 = A 113 = 0]. 1

Page 385 1:23] Wong: A Study and Classification of Ruled Quartic Surfaces 385 The result of expanding this equation which is the same as that obtained from the sextic equation in Art. 9 by putting Alll=A112=A113=0 and by assigning to the pik'S their respective values given above, is an equation whose left member is made up of two factors, x21 and a quartic factor. x21 0 gives a double plane corresponding to the double point Al, and the quartic factor equated to zero represents a quartic surface which has a triple line coincident with d from every point of which issue three generators meeting the conic c2 in x1 in three different points. Quartics of this class have in general four pinch-points on d, and their reciprocals are of class II already considered. 41. If the cubic goes through A'1, then A222=0, and the corresponding quartic surface has one of its generators coincident with the triple line d. This is of class IX (XI of Sturm). SECTION IV 42. So far we have been able to obtain nine classes of ruled quartics; to obtain the remaining classes we need to alter the machinery of transformation. Let the fundamental quadrics touch each other along a straight line, and, for the sake of simplicity, let their equations be S1 -allx21+a22x22+2a23x2x3+2al4xlX4 = 0, S2bllx21-+b22x22+2b23x2x3+2b14xx = O [a23: b23=a14: b14], the line of tangency being x = 0, x2 = 0. Any point Y (yi, Y2, y3, y4) in space goes into the line (allyll+a14y4)l + (a22y2+a23y3)x2+a23y33+- a14ylx4 = 0, (blll+ b14y4)xl+ (b22y2+ b23y3)x2+ b23y2x3+ b14y1x4 = 0, intersecting the line xl = 0, x2 = 0. Therefore, all the lines thus obtained cut across the line of tangency of the two given quadrics. To the point (0: 0: 1: m) on this line of tangency correspond the common tangent planes to Si and S2 at the point. Every point in xl goes into a ray through A4(0 0: 0: 1) and every point in X2 goes into a ray through A3 (0: 0:: 0). Since there is nothing special about the plane x4 except that it is a face of the tetrahedron of reference, we shall find the remaining classes of quartics corresponding to curves in this plane. 43. The system of common tangent planes along x = 0, x2 =0 is intersected by the plane x4 in a flat pencil of rays with center at A3. Since any point in x4 is on one of these lines and hence in one of these common tangent planes, its corresponding ray must cut across the line X1x2 at the point at which the plane containing the given point is tangent to the two quadrics; but every point on xi = 0, x4 = 0 goes into the same line X2=0, X3=0. Therefore, every conic 7'2 in x4 goes into a quartic surface with a double ruling x2 =0, x3 = 0 which corresponds to the two points which 7'2 has in common with xi =0, x4=0. From every point of the line of tangency of S1 and S2 issue two generators of the quartic surface, corresponding to the two

Page 386 386 University of California Publications in Mathematics [VOL. 1 points in which every ray of the pencil A3 cuts y'2; therefore, this line is a double line. Furthermore, it is a line along which two double lines coincide, for every ray of the pencil As is a line in which two coincident planes tangent to S1 and S2 intersect X4. This quartic is of class X (VIII of Sturm), and its'equation is obtained as follows: Let 3 ZAijxjij=O be the equation of any cone whose plane section in X4 is a conic 7'2. Any line UXI+U2x22+U3X3=O, X4=0 goes into a quadric (U3(22, 11+U2W11, 23)XlX2+U3(P23, llXlX3+U2023, 22X2+U3(22, 14X2X4=0 where 'pij, kl=aiibkl-aklbij and Pij, kl= -Pkl, ii. Putting u = 1, u2= 3 =, we have Ql —<P22, 23X22 0, the quadric corresponding to the line xl==0, X4=0. Putting u2=1, Ul=U3=0, we have Q2 — 11, 23X1X2 = 0, the quadric corresponding to x2=0, x4=0. Finally, putting U3=1, Ul=U2=0, we have the quadric Q3=(11, 23XlX2+(Pll 23xlx3+<(14, 22X2X4=0 corresponding to the line x3 = 0, x4 = 0. Then the equation of the quartic is given by 3 EAi jiQ Qi= 0 or, when expanded, A 11P22, 23x42+2A13(P11, 22P22, 23xlx32+2A12(P11, 23(P22, 23X1X33+2A13(22, 23P14, 22x32x4 +(A22(211, 23+2A231(P1, 22(11, 23+A33P211, 22)x21x22+-A23?P211, 23X21X23+A33'P214, 22X22X24+ -2(ll, 23(A33P11, 22+A23<(11, 23)X21X2X3+2(P42(A33Pll, 22+-A23<11, 23) XlX22X4+2A33P11, 23(P14, 22XlX2X3X4=0. 44. Putting x2 =, we have x23 =0, and vice versa, showing that the line x2 =0, X3 = 0 is a double generator. Putting x4 = 0, we have the curve of intersection of the surface in x4, having a node at A 1 and a tacnode at A3 with the line A2A3 as tangent. 45. The quartics of class XI (II of Sturm) which have two coincident double directrices but no double generator, can be obtained from a cubic in x4, tangent to the line A2A3 [xrL=0, x4=0] at A3. This curve must not have a double point, for then the corresponding surface would have a double generator; nor must it go

Page 387 1923] Wong: A Study and Classification of Ruled Quartic Surfaces 387 through the point A((l: o: o: o) whose transform is the plane xl, for then it would be impossible to obtain a quartic surface. Then to the given cubic we have corresponding a sextic surface which is made up of the plane x2 counted twice corresponding to the point As of contact between the curve and A2A3, and a quartic which has no double ruling. 46. Let 3 2AiikXiXjXk==O, X4=0 [A333=A233=0] 1 be such a cubic curve, which transforms itself into the sextic surface 3 zAiikQiQ Qk = [A 333= A 233 = 0] where the Q's have the same meaning as those of Art. 43. This equation, if expanded, would be x22[a quartic factor] =0. The bracketed factor equated to zero is the equation of the required quartic surface. 47. If, in addition, A133 is put equal to zero, we have a cubic curve with a node at A3, and the corresponding surface is of class XII (XII of Sturm, which has the line x1=0, x2=0 for triple line, that is, a line through which three sheets of the surface pass. If the double point at A3 becomes a cusp, only one of the sheets passes through the triple line, while the other two unite into a cuspidal sheet. This is a special case of the above but Salmon made a separate class (V according to his enumeration) of it. 48. This concludes the study and classification of ruled quartic surfaces. It may be added that classes could have been obtained from plane quartics with their singular points at one or two of the vertices of the fundamental tetrahedron, from space cubis not belonging to the complex through one vertex of the fundamental tetrahedron and also from space quartics through all the four vertices of the fundamental tetrahedron, but none can come from quintic curves or curves of higher order, plane or space. Many thanks are due to Professor D. N. Lehmer, without whose kind and patient guidance this work would have been an impossibility.

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 18, pp. 389-400, 1 figure in text December 19, 1923 A SPECIAL QUARTIC CURVE BY ELSIE JEANNETTE McFARLAND The special quartic curve whose equation is (1) F= ax4+by4+cz4 +2fy2z2+2gx2z2+2hx2y2 = 0 was studied by Hilda Fay Webb, of the University of California, and the results of her investigation are to be found in her thesis for the Master's degree, submitted in May, 1915. Miss Webb found the quartic to be a non-singular curve with four double tangents passing through each vertex of the fundamental triangle. The six pairs so obtained form a "syzygetic sextuple." She also found that when the general quartic is represented by UW= V2, the complete condition for the reducibility of the general quartic equation to the special form F = is as follows: Two vertices of bitangent pairs of the same set (of six pairs) must coincide; and the conics U, V, and W must cut the polar of the coincident vertices with respect to the contact conic in pairs of an involution. It is my purpose in this paper to investigate still further the properties of this special quartic curve, and to set up other conditions for the reducibility of the general equation of the fourth degree to the special form F=0. THE BITANGENTS The equation of the quartic being given as ax4+by4+cz4+2fy2z2+2gx2z2+ 2hx2y2=0, the four double tangents through the point (0, 0, 1) will be of the form (2) x= ~AAly x= ~A2y where ~-A1 and iA2 are the roots of the equation: (3) (g2 - ac)A4+2(fg-ch)A2+ (f2 -bc) =0 Those through the point (1, 0, 0) are (4) y= 4Blz y= -B2z

Page 390 390 University of California Publications in Mathematics [VOL. 1 where -B1 and i-B2 are the roots of the equation (5) (h2 -ab)B4+2(gh-af)B2+ (g2-ac) = 0 Those through the point (0, 1, 0) are (6) z = -~Clx z = ~ C2x where - C1 and C2 are the roots of the equation (7) (f2 bc) C4+2(fh - bg)C2+ (h2-ab) = 0. We see from equations (2), (4), and (6) that each set of four double tangents through a vertex contains two pairs, the lines of which are harmonically divided by the sides of the reference triangle through that vertex. For example, the lines z= Cx and z= -Clx are harmonically divided by x =0 and z= 0. The quartic F =0 is transformed into itself by the following collineations: (a) (b) (c) (d) x=x' x=-x' x= x' X= —X Y=Y' y=-yl y=_y, y= y, z= zf z= zt z= zt z= zt If z=px+qy touches F=O twice, then, since (a), (b), (c), and (d) transform F into itself, the three lines z=-px-qy, z=px-qy and z= -px+qy, obtained by applying (b), (c), and (d) to z=px+qy, are likewise double tangents of the quartic. Similarly, if z=px+qy is an inflexion tangent, z=-px-qy, z=px-qy, and z - px+qy are likewise inflexion tangents. Let us now take as a double tangent the line z=px+qy. The equation (8) (a+cp4+ 2gp2)x4+4(4cp3q+gpq)x3y +6 cp2q2+-P +- + h)x2y2+4(cpq3+fpq)xy3+ (b+cq4+2fq2)y4 = 0, obtained by substituting px+qy for z in (1) must be a perfect square. If (8) is written: Ax4+4Bx3y+6Cx2y2+4Dxy3+Ey4=O, it will be a perfect square if ( AC-B2 CE-D2 (9) AC-B= CE-D, or AD2-EB2 = 0 AC-B2 AD-BC and A C -= 2 A- or3ABC-2B3-A2D=0 A 2B The resulting conditions on p and q are: (10) (ac2-cg2)Q2+ (2acf-2fg2)Q+ (cf2- bc2)P2+(2f2g-2bcg)P+ (af2- bg2) =0. (11) (ac2-cg2)P2Q (ag2-a2c)Q+ (c2h-cfg)P3+ (3cgh - acf- 2fg2)P2+ (ach + 2g2h -3afg)P+(agh- a2f) = 0 where P=p2 and Q= q2. (12) pq=O. This condition merely gives us the double tangents through (1, 0, 0) and (0, 1, 0). Eliminating Q between (10) and (11) we arrive at an equation of the sixth degree in P.

Page 391 1923] McFarland: A Special Quartic Curve 391 (13) (c5h2-2c4fgh + ac4f2-abc5 + bc4g2)P6 + (6ac3f2g-8c3fg2h-4ac4fh + 6c4gh2 - 2abc4g+2bc3g3)P5+ (2a2c3f2+ 13c3g2h2+ 13ac2f2g2 +2ac4h2 - 20ac3fgh - 102fg3h - 3abc3g2 + bc2g4-+ 2a2bc4)P4+ (Sa2c2f2g - 4a2c3fh - 32ac2fg2h + 8ac3gh2+ 12acf2g3 - 4cfg4h+ 12c2 g3h2+4a2bc3g -4abc2g3)P3+ (10a2cf2g2+-a2c3h2+4cg4h2 - lOa2c2fgh - 20acfg3h+ 10ac2g2h2 +a3c2f2 +4af2g4+3a2bc2g2- 2abcg4- a3bc3)P2+- (2a3cf2 - 8a2cfg2h + 2a2c2gh2 + 4acg3h2 +4a2f2g3 -4afg4h - 2a3bc2g+2a2bcg3)P+ (a2cg2h2 +a3f2g2 - 2a2fg3h -a3bcg2+a2bg4) = 0. This equation, however, contains the extraneous factor c2P2+2cgP+g2. Accordingly the equation in P reduces to' (14) c2(ch2-2fgh+af -abc+bg2)P'+4c(af2g -fg2h-acfh+cgh2)P3 + 2(a2cf2 + 2cg2h2 - 6acfgh+a2bc2 - abcg2 + 2af2g2 + ach)P2 + 4a(af2g -fg2h - acfh +cgh2) P a2 (ch2-2fgh + af2 -abc+bg2) =0. This equation can, of course, be solved, giving us four values of P: P1, P, P3, and P4. We have then four corresponding values of Q: Q1, Q2, Q3, and Q4. Sixteen double tangents are given by the equations: (15) Z = z pix~qiy z=:p2X~4q2y Z= ~ p3X+qy Z = z+p4x~q4y where pi = /P1 and ql = VQ1, etc. Let us represent the four double tangents z= p ix qiy by the numbers 1, 2, 3, and 4, and the intersection of any two, say 1 and 2, by (1, 2). Then the points (1, 2) and (3, 4) lie on one side of the reference triangle, points (1, 3) and (2, 4) on another side, and points (1, 4) and (2, 3) on the third side. xOA ARRANGEMENT OF THE BITANGENTS IN STEINER COMPLEXES

Page 392 392 University of California Publications in Mathematics [VOL. I If we pair together the double tangents x=Aly, x —A y, and set up the identical relation F ax4 + by + c4 + 2fy2z2+ 2gx2z2 +2hx2y2. (16) (x2 - A 2y2) (1x2+my2+nz2+rxy+sxz+tyz) - ('x2+m'y2+n'z2+r'xy+s'xz +t'yz)2, we find by equating coefficients and expressing the quantities 1, m, n, r, s, t, 1', m', n', r', s', t' in terms of 1' that we have: ( =a+1'2 f 1' is arbitrary. f A 22 f ^2 f___ Ai2 _ n=2g+21'V/-c 1 n'=V/-c r=s=t=O r'=s'=t'=O We then get F J f (f~+A12+ 22' _2 (18) F [x2-A12y2] (a+l'2)x2- I V-c +( A y2+ (2g+21'/-c)z2 A12 -[irx2 — (fi+ t +A121) y2+ (x-c)z2 where ' is arbitrary and takes the place of X as used by Weber throughout his chapter on the double tangents of a quartic.* Rearranging according to powers of 1': f2+ 2fA 2g +A 1492 F [x2-A 2y2T ax2- - - (c )+ 2gz2 A12 (19) +2' - A12 +gA 1 )y2 (V/C)z2 +l 2{x2-A1y2] [ { -(f+' )2+( v(/c) 2 +l'{2-AIy2}]2 = [x2- A 2y2][V+2XU+X2xy1] - [U+XX1Y1]2 We have expressed F in canonical form with 1' taking the place of X. If the variable conic V+2XU+X2x1y1a'x2+b'y2+c'z2+2f'yz+2g'xz+2h'xy = 0, a' h' g'! it will degenerate when h' b' f' =O. g' f' c'I Since the variable conic contains only x2, y2, and z2 terms, the determinant la'0 0 becomes 0 b' 0 = a'b'c'=0. 10 0 c' When a' =0 we get the lines y = iBlz, y= iB2z. When b' =0 we get the lines z= ~Cix, z= -C2x. When c' =0 we get only x= +A2y, since c' is of the first degree in 1'. We see, therefore, that the twelve double tangents through the vertices of the fundamental triangle are paired as follows to form a Steiner complex: * Weber, H., Lehrbuch der Algebra, Vol. 2.

Page 393 1923] McFarland: A. Special Quartic Curve 393 20(1) x=iAly (2) x=-A2y (3) zy=Clx (20) (4) z= C2x (5) y= Blz (6) y=4B2z. x+Aiy=0 x-A y=0 If, now, we pair x+A2y=O and x-A2yx+A2y=0 x-A2y-0 we shall get a second complex, and if we pair x+Aly= and x-Aly=O x-A2y=0 x+A2y=O we shall get a third, and these three complexes will contain all twenty-eight double tangents. Let us now obtain the complex determined by the pair 21 x+A IY0. Let ax4+by4+cz4+2fy2z2+2gx2z2+2hx2y2 (21) x+A2y=O. [x2+ (A 1+A2)xy.+A IA2y2] [Ix2+my2 +nz2+rxy+sxz+tyz] - ['x2+m'y2+n'z2+r'xy+s'xz+t'yz]2 We find that: l=a f2+2fm'/_-c-m'2c 2fg 2gm'\/-c g2 - R2c Rc Rc c m=b+m'2 Rm=b —M2 m' is arbitrary. R (22) n=2+2 n' = /-c 2Sfm' 2Sm'2 Sb Sm'2, Sf+Sm'V-c r= R1 _cc_ r - R2V/-c~ R2 R2 R2 RV/-c s=t=0 s'=t'=0 11 = f+m c _ c RV/-c - /-c S=A1+A2 f 2 -bc where =A +, A if R= - R =A1A2 ' g -ac f2 -be 2 (gf -hc) and S = 2 b 2 --- — h) which may be verified from equation (3). 92 - ac g2 -ac ' (23) F=[x2+Sxy —+Ry2]F -f 2fg g)x 2 by2+f Sb } +2m a(a -c_ Rc + R — 2+ YR 2 R2 xy (2,g f-c\ 2/V- sf x,x2 Ry2 Sxy ( - Rc R2C /XR Z R2 )2+R-Y f-iR2 R2 R -I(R v/-c- +/- Vc 2+ x — c. z v '+m' +y- -+.xyY (24) [= [+S+R[y2] a —f2 2fg g2+ b 2 2 Sb xy (24) = SxyNc C) +7-Ll+a-z-j R2 c Rc c/ R R R 2'

Page 394 394 University of California Publications in Mathematics [VOL. 1 (24) +2X (V -fV )x2+V/-c Z2+f^y - +X2{x+Sxy+Ry2] g[ Xc c f\ -- x{/_CcC. z2 + + 2+SxyRy}] -C R-] R.x — vV-c.z -cc. XY 1 +Xhx2~Sxy-fR Y2j which is in canonical form. (X= R If the variable conic is to degenerate, the following determinant must vanish: (a- f2 +2fg_ +2XgVc_ 2Xf - c + (2 - Sb XSf X2S/ 2R2 R/V-c 2 R O =(-f+2+/c ( St X4 )(2a C _2f/cS2f \I=C ) (25) +(R-+aR-Rc+-c - c + R~c+2R2) ( 2bg-\ c 2bfX-\c S2bf \/c-c Rc R2c R 3-c ( ab bf2 2bfg bg2 S2b2\ l = 0, or (x-A y) (x-A2+) = If the second factor is put equal to zero, we shall get four values of X for which the variable conic will degenerate. If we pair (x+-Ay) and (x-A2y) we replace the quantity S=A1+A2 by D=A1, -A2 and R=A1A2 by -R. Hence it will be seen at once that the remaining double tangents can be found by replacing S by D and R by -R in the expression for the variable conic and in equation (25). Since the variable conic is of the form a'x2+b'y2+c'z2+2h'xy=0, it can degenerate into (z-px-qy) (z+px+qy) =0, or (z-px+qy) (z+px-qy) =O. { x- A iy = 0 To show that the complex determined by x-+A2y = contains either four z= pz+qy = px-qy letussetupthe copairs of type z- -px+qy or four pairs of type z -px= qy let us set up the comz= px+qy The variable conic in this case is plex determined by one of the pairs, z-= P q The variable conic in this case is again of form a'x2+b'y2+c'z2+2h'xy = 0, as can be determined by carrying out the process of page seven. It will degenerate if c'=0, or if a'b'+h'2=O.

Page 395 1923] ' McFarland: A Special Quartic Curve 395 The coefficient c' is identical with X2+c, so that the complex contains two pairs of lines of the form x=Ay. If a'b'+h'2 =, the conic degenerates into (z-px-qy) (z+px+qy)=0, where p and q must satisfy conditions (10) and (11). It will be seen then that the same complex is determined by any one of the four pairs z= plx+qly z= p2x+q2y z= p3x+q3y Z= p4x+q4y z=-pix-qiy, z=-p2x-q2y, z-p3-, z =-p4x-q4y. x —A ly = O x+A ly = O Hence one of the two complexes determined by x+A y=O and x —Ay=O must contain four pairs of type z= px+qy and the other four pairs of type s c pi - qy z= px-qy z = -pxq- y. z+C=x=0O If we pair z — C2x=O and set up the identity ax4+by4+-cz4+2fy2z2 +2gx2z2+ 2hx2y2- (z2-Sxz+Rx2) (2+-my22+nz2+rxz+syz+-txy) - (1x2+Mlm'y2+n'z2+r'xz +-s'yz+t'xy)2, we find that t=t'=s=s'=0. Hence the variable conic is of the form a'x2 +-b'y2+c'z2+-2g'xz = O. It must then degenerate into a pair of straight lines (z+px+qy) (z+px-qy)=O, or else into a pair (z-px+qy) (z-px-qy)=O. z - Cx = O The complex then contains the pair Z-C2x=O1 and four pairs of the form z= -px- qy or of the form p - qy z= -px+qy z=px+qy. z=-px 1qy oroftheform z2px+qy.;=0. z-0i-=0 and The complex determined by Z+Ci x~ will contain the pair z-+Cx=O an the four pairs of double tangents which do not appear in the preceding complex. In like manner we can show that of the two complexes determined by y+Bz = 0 y+B2Z~=O= y+Blz=O z=-px-qy and the and y-B2z=0 one will contain four pairs of the type and the Y-B2Z= -~z= px-qy other four of the type z p-x+qy z= px+qy. Although other groupings may be made in the same manner, those already given are the most striking. They may be summarized as follows: xA ly=O x+A A -y=O To the complexes determined by x+A2Y= 0 and x-A2ly= belong the pairs z= px+qy d z=-px+qy z=-px-qy z= px-qy. y+Blzz=O y+Blz=O To the two determined by y+Bz= 0 and +-B2=0 belong the pairs z =-px-qy z = -px+qy z= px-qy z= px+qy. z+Cx=0O z+Cix = O To the two determined by z+Clx=0 and z-C2x =0 belong the pairs z=-px-qy z=px-qy z=-px+qy z= px+qy.

Page 396 396 University of California Publications in Mathematics [VOL. 1 REDUCTION OF THE GENERAL QUARTIC TO THE SPECIAL FORM ax4 + by4 + cz4 + 2fy2z2 + 2gx2z2 +2hx2y2 = 0 In his Higher Plane Curves, Salmon refers to this quartic, stating that its equation contains implicitly eleven independent constants. This is shown by replacing x by Ix'+my'+nz', y by l'x'+m'y'+n'z', and z by l"x'+m"y'+n"z'. F can then be written be subject to three conditions in order that it may be reducible to the special form 4In ordr tt te qrtc w e e n contn m the full fen t s be of the type discussed in this paper, its double tangents must be such that four of them pass through some point of the plane, four more through another point, If we 21represent two different double tangents of the fifteen-ter quartic (which we sbmay call Q) by =onit in wherer ij and where i and j assume values from ~f+ x+ —f +W2hl2'l2 xf+ f+ ~ f+- m' f =0 an expression which evidently contains eleven independent constants. Thus we see that the fourteen independent constants in the equation of the general quartic must be subject to three conditions in order that it may be reducible to the special form F = 0. CONDITIONS ON THE COEFFICIENTS OF THE GENERAL QUARTIC In order that the quartic whose equation contains the full fifteen terms shall be of the type discussed in this paper, its double tangents must be such that four of them pass through some point of the plane, four more through another point, and four more through a third point. If we represent two different double tangents of the fifteen-term quartic (which,we may call Q) by zain bi where irej antd where i and j assume values from 28x27 x 1 to 28,t the equation of degree -pos giving the value of for the points of inter2 Y section of the twenty-eight double tangents, will have for coefficients the elementary symmetric functions of the various quantities q i-q. But since any interchange pi -i j of the subscripts of the q's requires a corresponding interchange for the p's, it can be shown that the symmetric functions of qi-qi are also symmetric functions of the quantities p and q; hence that they are rational functions of the coefficients of the quartic Q. If Q is to be of type F, the equation of degree 14x27 (which may be denoted by V(x,y),) must contain a cubic factor repeated six times. For since four double tangents go through a point, six points of intersection fall together. The same thing occurs for two other sets of four double tangents. If V has this cubic factor occurring to the sixth degree, its first derivative V' will contain the cubic factor to the fifth degree. Hence V and V' must have a common factor of degree fifteen, which is a perfect fifth power. This can be found

Page 397 1923]. McFarland: A Special Quartic Curve 397 by the rational process of division. When this factor is found its fifth root can be extracted. This fifth root is a cubic which can be solved by radicals. The roots of it give - for the three points through each of which pass four double tangents. We can then determine the lines joining these three points, and taking them as sides of a new reference triangle we can find the equation of Q referred to this new triangle. If the transformed equation now contains only even powers, Q was a special quartic of the type under consideration. If we wish to set up conditions under which Q will be reducible to form F, we must first impose the condition that V and V' shall have a common factor of degree fifteen. This means that when sufficient steps have been taken in the process of finding the greatest common divisior, the remainder of degree fourteen must vanish identically. This implies fifteen conditions, which are not, of course, independent. This common factor of degree fifteen must be a perfect fifth power and conditions for that can be set up. These conditions are not, however, sufficient-or have not been proved so-since the pairs of double tangents through a vertex of a certain triangle must also be harmonically divided by the sides of the triangle through that vertex. Also the remaining sixteen double tangents are related in a special way to the sides of this same triangle. In order to determine the additional conditions we should transform to the new reference triangle, and equate the coefficients of odd powers, if such occur, to zero. The very large number of conditions obtained in this way must reduce ultimately to three. GEOMETRICAL CONDITIONS ON THE CURVE If Xlyl, x2y2, and x3y3 are three pairs of double tangents belong ing to a Steiner complex, the general quartic can be written in the form V/xi. y + /x2 Y2+ /x3 y3 = 0. If we take the points of intersection of x1 with yi, x2 with y2, and x3 with y3 for the vertices of our fundamental triangle, we can write the equation of the quartic as: p/(x-ay) (x-by)+qV(y-cz) (y-dz)+rV/(z-fx) (z-gx)=O, or, in rational form: (27) (p4_ 2fgp2r2+f2g2r4) x4 + (q4 -_ 2abp2q2+a2b2p4)y4 + (r4_ -2cdq2r2 + c2d2q4)z4 + (a2p4 + b2p4 + 4abp4-2abfgp2r2 - 2fgq2r2 - 2p2q2)2y2 + (c2q4 + d2q4 + 4cdq4 -2abcdp2q2- 2abp2r2- 2q2r2)y2z2 + (f2r4 + g2r4 + 4fgr4- 2cdfgq2r2- 2cdp2q2- 2p2r2)x2z2 + 2(-ap4 - bp4 + afgp2r2 + bfgp2r2)x3y + 2(-cq4- dq4 + abcp2q2 + abdp2q2)y3z + 2(-fr4-gr4 + cdfq2r2 + cdgq2r2)z3x + 2(-a2bp4-ab2p4+ap2q2 + bp2q2)xy3 + 2(- C2dq4 -cd2q4 + cq2r2 + dq2r2)yz3 + 2( - f2gr4 - fg2r4 + fp2r2 + gp2r2)zx3 + 2 (- afp2r2 - agp2r2- bfp2r2 - bgp2r2 + cfgq2r2 + dfgq2r2 + cp2q2 + dp2q2)x2yz + 2 (- acp2q2 —adp2-bp2q2 -bdp2q2 + abfp2r2 + abgp2r2 + fq2r2 + gq2r2)xy2z + 2 (-cfq2r2-dfq2r2_cgq2r2-dgq2r2 + acdp2q2 + bcdp2q2 + ap2r2 + bp2r2)yz2 = 0. Let us now set up the conditions that through each vertex of the reference triangle there shall pass a third double tangent to the quartic.

Page 398 398 University of California Publications in Mathematics [VOL. 1 A line through the point (1, 0, 0), of the form y=mz, will meet the quartic in four points given by: (28) (p4_2fgp2r2 + f2g2r4)x4 + 2 {(-ap4-bp 4 + afgp2r2 - bfgp2r2)m + (-f2gr4-fg2r4 + fp2r2 + gp2r2)} X3z + { (a2p4 + b2p4 + 4abp4 -2abfgp2r2-2fgq2r2 -2p2q2)m2 + 2(-afp2r2-agp2r2-bfp2bfr2-bgp2r2 + cfgq2r2 + dfgq2r2 + cp2q2 + dp2r2)m + (f2r4 + g2r4 + 4fgr4 - 2cdfgq2r2 - 2cdp2q2 - 2p2r2)} x2z2 + 2 {(-a2bp4 - ab2p4 + ap2q2 + bp2q2)m3 + (abfp2r2 + abgp2r2 + fq2r2 + gq2r2-acp2q2 -adp2q2-bcp2q2 -bdp2q2)m2 + (acdp2q2 + bcdp2q2 + ap2r2 + bp2r2 -cfq2r2-dfq2r2 -cgq2r2- dgq2r2)m + (-fr4- gr4 + cdfq2r2 + cdgq2r2) }xz3 + { (q4- 2abp2q2 + a2b2p4)m4 + 2 (- q4 - dq4 + abcp2q2 + abdp2q2)m3 + (c2q4 + 4cdq4 + d2q4 - 2abp2r2- 2q2r2 - 2abcdp2q2)2n2 + 2 (- c2dq4 - cd2q4 + cq2r2 + dq2r2)m + (r4- 2cdq2r2 + c2d2q4)4 = 0, which may be written: A 4 +4Bx3z+6Cx2z2 +4Dxz3 +Ez4 = 0. If the line y - mz is to be a double tangent, the above equation must be a perfect square, which will be the case if (1) AD2-EB2=0 and (2) 3ABC-2B3-A2D+O, or (1) m {m2 —(c + d) m + cd} {(f + g) (a + b) (q2-abp2)m2 + [(f + g)2abr2 -(f + g) (a + b) (c + d)q2 + (a + b)2p2]m + (f + g) (a + b) (cdq2-r2)} =0. (2) {m2-(c + d)m + cd} {(a b) (-fg)m + (f + g)} =0. Since m is an extraneous factor, and m=c or d gives us the original double tangents y - cz = and y - dz =, we may disregard the first two factors of (1) and the first of (2), and concern ourselves with the two equations: (f+ g) (a + b) (q2-abp2)m2 + {(f + g)2abr2-(f + g) (a + b) (c + d)q2 + (a + b)2p2} m + (f + g) (a + b) (cdq2-r2) =0. and -fg(a + b)m + (f + g) = 0. The eliminant of these equations is: (29) E2=(f-+ g)2(q2-abp2) fg {(f + g)2abr2-(f + g) (a - b) (c + d)q2 + (a + b)2p2} + f2g2 (a + b)2(f + g) (cdq2-r2) =0. If we had chosen x = ly or z = nx, we should have got two other eliminants: E,-(c + d)2(p2-fgr2) + cd {(c + d)2fgq2-(c + d) (f + g) (a + b)p2 + (f + g)2r2} -+ c2d2(f + g)2(c + d) (abp2-q2) =0. E3=(a+b)2(r2-cdq2) + ab {(a + b)2cdp2-(a + b) (c + d (f + g)r2 + (c + d)2q2} + a2b2(c + d)2(a + b) (fgr2-p2) =0. It is evident from an inspection of the three eliminants that they will vanish if a - b, c = -d, and f= -g. But the condition a = -b is just the condition that the two double tangents through the vertex (0, 0, 1) of the reference triangle shall form a harmonic pair with the axes x=0, y=0. Similarly, if c= -d, the double

Page 399 1923] McFarland: A Special Quartic Curve 399 tangents through (1, 0, 0) are harmonically divided by y= 0, z=0, and finally if = -g, the double tangents through (0, 1, 0) are harmonically divided by x=0, z=-0. Hence it will be possible to pass a third double tangent to the quartic through each vertex of the reference triangle (chosen as indicated on page 397), if the two original double tangents through each vertex are harmonic conjugates of the sides of the triangle meeting in that vertex. Let us now set up the conditions that our general quartic curve as given by equation (27) shall contain only even powers of the variables. The nine conditions are as follows: (1) p2(-ap2-bp2+afgr2 +bfgr2) - p2(a+b) (fgr2-p2) = 0 (2) q2(-cq2-dq2+ap2abp+abdp2) q2(c+d) (abp2-q2) = 0 (3) r2(-fr2-gr2+cdfq2+cdgq2) r2(f+g) (cdq2-r2) = 0 (4) p2(-a2bp2-ab2p2A+aq2A+bq2) p2(a+b) (q2-abp2) = 0 (5) q2(-c2dq2-cd2q2+-cr2A+dr2) q2(c+d) (r2-cdq2) = 0 (30) (6) r2( f2gr2 -fg2r2+fp2+gp2) r2(f+g) (p2-fgr2) =0 (7) (- afp2r2 - agp2r2 - bfp2r2 - bgp2r2 +cfgq2r2 +dfgq2r2 +cp2q2 +dp2q2) - -p2r2(a+b) (f+g)+(c+d) (p2+fgr2)q2 = (8) (- acp2q2 - adp2q2 - bcp2q2 bdp22 abfp2r2 abgp2r2 +fq2r2 +fq2r2 + q2r2) = -p2q2(a+b) (c+d)+(f+g) (q22+abp2)r2 = (9) (- cfq2r2 - dfq2r2 _ cgq2r2- dgq2r2 + acdp2q2+ bcdp2q2 + ap2r2 +bp2r2) - -q2r(c+d) (f+g) +(a+b) (r2+cdq2)p2=0 From inspection we see that the nine equations are satisfied if a+b =0, c+d =0, f+g=O, or if p = q = r =, and in no other way. But we must reject the system p =q = r = 0, since if these relations were true we should have no quartic at all. Therefore the nine conditions for the reduction of the general quartic to the special form we are studying are equivalent to three, a+b=O, c+d=O, f+g=O. But these conditions, as we have previously noted, are the conditions that each pair of double tangents through a vertex of the reference triangle shall be harmonically divided by the sides of the reference triangle through that vertex. Let us first select as reference triangle one whose vertices are the intersections of xl with yl, x2 with y2, x3 with y3, where x1y1, x2y2, x3y3 are three pairs of double tangents of the general quartic belonging to a Steiner complex. If, then, we can draw through each vertex a third double tangent to the quartic, or if-an equivalent condition-we can show that the two double tangents through each vertex are harmonic conjugates of the sides of the reference triangle through that vertex, the given quartic is reducible to the form ax4+-by4+cz4+-2fy2z2+-2gx2z2 + 2hx2y2 = 0. NOTE.-We can express a, b, c, d, f, and g.in terms of the moduli of the class of curves, and so obtain the conditions they must satisfy if the quartic is of this variety. Riemann, "Zur Theorie der Abelschen Functionen fiir den Fall p =3, Gesammelte Werke, p. 456.

Page 400 400 University of California Publications in Mathematics [VOL. 1 SUMMARY The quartic curve whose equation is ax4 + by4 +cz4+ 2fy2z2+ 2gx2z2+ 2hx2y2 = 0 has the following properties: Four double tangents pass through each vertex of the reference triangle, the four consisting of two pairs which are harmonically divided by the sides of the triangle. The remaining sixteen are grouped by fours, each four consisting of the lines z= ~px~qy. The quadrilateral formed by any one of these four-groups has for diagonals the sides of the fundamental triangle of reference. The actual equations of the double tangents can be determined by the solution of a biquadratic equation. The six pairs of double tangents through the vertices of the reference triangle x+Aly=O belong to a Steiner complex. To the complexes determined by x+A22 = 0 and x+Aiy=O z+-px+-qy = z+px-qy = x-A2=O belong the pairs y=o and z-px+qy=0. To those deteri z- Cx = 0 z-CX =,0 z +px +qy = a mined by z-Cx=O and z+C= belong the pairs z+pxqy=O and z-px+qy=0 y-Biz=O y-Biz=O z-px-qy=0. And finally to the two determined by y-B2z=O and y+B2z=O z+ px+qy = O z+px-qy-O belong the pairs z-px+qy=O and z-px-qy=0. Under certain conditions the general quartic can be made to reduce to the special form F =0. Let three pairs of double tangents be selected from a Steiner complex and let their intersections be taken as the vertices of a new reference triangle. Then if it is possible to pass a third double tangent through each vertex, or if the two original double tangents are harmonically divided by the sides of the reference triangle through that vertex, the general quartic equation is reducible to the form considered in this paper.

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 19, pp. 401-423 January 24, 1924 A STUDY OF CUBIC SURFACES BY MEANS OF INVOLUTORY CUBIC SPACE TRANSFORMATIONS BY JOHN FREDERICK POBANZ 1. Cayley in his paper On the Rational Transformations between Two Spaces' was the first to call attention to birational transformations in three dimensions, as an immediate extension of Cremona's work in two dimensions. In particular he investigated a cubo-cubic transformation by means of which the coordinates of any point in one space can be expressed as rational cubic functions of the coordinates of a corresponding point in a second space, and vice versa. To the planes in either space correspond cubic surfaces in the other space. If in this transformation certain relations exist among the coefficients, the transformation is involutory. 2. The purpose of the present investigation is to obtain by involutory cubic transformations the 23 varieties of cubic surfaces as given in Cayley's memoir2 and also the cubic cone omitted by him. 3. This paper is divided into three parts: Part I. A discussion of the singularities and classification of cubic surfaces; Part II. (A) A general consideration of the cubo-cubic and involutory transformations; (B) the notation used; Part III. The special transformations necessary to produce each of the 24 varieties of cubic surfaces. PART I 1. The order of a cubic surface without any singularities is 3 and its class is 12. The surface has upon it 27 distinct right lines lying by threes in 45 planes, which are triple tangent planes. The surface may also possess point or line singularities. Point singularities reduce the class of the surface and the number of distinct right lines. In order to account for the 27 lines, a certain multiplicity must be given to the lines through the singular point or points. The point singularities are cnicnodes (conical points), binodes and unodes. 1 Collected Math. Papers, vol. 7, pp. 189-240. Also Proc. London Math. Soc., vol. 3 (1868 -1873), pp. 127-180. 2 "Memoir on Cubic Surfaces" Phil. Trans. Royal Soc., vol. 153 (1863), pp. 193-241.

Page 402 402 University of California Publications in Mathematics [VOL. 1 2. At a cnicnode C2 the tangent plane is replaced by a proper quadric cone. Such a point reduces the class of the surface by 2, and, therefore, a cubic surface can at most have 4 C2. The 6 lines through a C2 have a multiplicity of 2. 3. At a binode B the quadric cone breaks up into a plane-pair, which are called the biplanes and their line of intersection the edge. There are 4 varieties of binodes: a) At an ordinary binode B3 the edge does not lie on the surface. A B3 reduces the class of a surface by 3, and, therefore, a cubic surface cannot have more than 3 B3. The lines through a B3 have a multiplicity of 3. b) If the edge is a line of the surface, and if the tangent plane at each point of the edge is distinct from one of the biplanes, except at two points, the binode is a B4. A B4 reduces the class of a surface by 4. Such a point may be considered as resulting from the union of 2 C2. c) In a B5 the tangent plane at each point of the edge coincides with one of the biplanes. A B5 reduces the class of a surface by 5. Such a point may be considered as formed by the union of a C2 and a B3. d) If one of the biplanes becomes oscular, the class of the surface is reduced by 6, and the binode is a B6. A B6 may be considered as resulting from the union of 3 C2. 4. At a unode U the quadric cone becomes a coincident plane-pair, the uniplane. There are 3 varieties of unodes: a) If the 3 lines in which the uniplane intersects the surface are distinct, the class of the surface is reduced by 6, and the U is a U6. A U6 may be considered as the union of 2 B3. b) If 2 of the 3 lines in the uniplane fall together, the class of the surface is reduced by 7 and the U is a U7. A U7 may be considered as resulting from the union of a B3 and 2 C2. c) If all 3 lines in the uniplane fall together, the class of the surface is reduced by 8 and the U is a Us. A Us results from the union of a C2 and 2 B3. 5. A cubic surface with a line singularity (nodal line) is necessarily a ruled surface. There are but 2 varieties: a) the nodal line distinct from the directrix and, b) the nodal line coinciding with the directrix. At every point of the nodal line there are two tangent planes, which fall together at a pinch point. The nodal line of the first variety has 2 pinch points on it, while the nodal line of the 2nd variety has but one. 6. Cubic surfaces were first classified by Schlafli.3 He finds 22 varieties depending upon the number and character of the point singularities which they possess. Each variety is further divided into species, depending on the reality of the lines on the surface. 3 On the Distribution of Surfaces of the Third Order into Species, in reference to the absence or presence of Singular Points. Phil. Trans. Royal Soc., vol. 153 (1863), pp. 193-241.

Page 403 1924] Pobanz: A Study of Cubic Surfaces 403 7. Cayley's4 classification is based upon, and is in a measure supplementary to Schlifli's memoir, but he disregards altogether the division into species depending on the reality of the lines. The surfaces are divided into 23 varieties depending upon the nature of the singularities. PART II (A) 1. It is assumed with Cayley that the most general transformation which gives rise to a cubo-cubic transformation is given by the three bilinear equations A - Zaikxiyk = 0, B _ -:bikxiyk -0, C- 2 CikXiYk=O, i, =1, 2,3,4. where the xi are the homogeneous coordinates of a point in the space X, and yk those of a corresponding point in the space Y. The transformation is then expressed by (2) pXi=-pi(yl,y2,y3,Y4), (2) l 7Yk = k(X1,X2,X3,X4), where the spi and 'k are rational cubic functions of (y) and (x) respectively. Cayley has shown that the spi have in common a twisted sextic curve J(y) and the Tk a similar curve J(x). To a point on J(y) corresponds a line in the space X meeting J(x) in three points. As the point moves on J(y) the corresponding line generates a surface R(x) of degree 8 with J(x) as a triple curve. There is a similar surface R(y) in the space Y corresponding to the curve J(x). 2. If in the equations (1) aik=aki, bik= bki, Cik = Cki, the transformation is involutory. Equations (1) may then be considered as the polar planes of the point (y) with respect to the three quadrics surfaces A 2-aikXiXk O, B- bikxiXk =0, C =CikXiXk = 0 i, k=l, 2, 3, 4. The spaces X and Y may be thought of as superimposed upon one another in such a way that J(x) coincides with J(y) and R(x) with R(y). To any point (y) corresponds (y'), the intersection of the three polar planes of (y) with respect to the three quadrics A, B and C. To the point (y') corresponds back again the point (y). There are, however, exceptional points, namely, points for which the three polar planes meet in a line. Such points lie on J and to it corresponds the surface R of degree 8. As (y) describes a line, its corresponding point (y') describes a twisted cubic curve. As (y) describes a plane, (y') describes a cubic surface. 4 Op. cit., p. 1.

Page 404 404 University of California Publications in Mlathema(tics [VOL. 1 3. J is the locus of vertices of cones in the general bundle X1A +X2B +X3C = 0.5 Those values of Xi, X2, X3 which satisfy the equation Xlaifk+X2bik+X3cik = 0, define the cones in the bundle. If this determinant is expanded and X1, X2, X3 interpreted as trilinear coordinates of the points in the X-plane, there results a plane quartic C4(X) which is in one-to-one correspondence with J. Since a general bundle does not contain any composite quadrics, C4(X) has no singular points. C4(X) has a deficiency of 3, hence J has the same deficiency.6 PART II (B) 1. a denotes the plane which is transformed and C3 the cubic surface corresponding to a. 2. Subscripts denote the number by which the class of a surface is diminished. 3. C2 is a cnicnode, where the tangent plane is replaced by a proper quadric cone; B = (B3, B4, B5 or B6) is a biplanar node, where the quadric cone is a plane-pair (the biplanes); U= (U6, U7 or U8) is a uniplanar node, where the quadric cone becomes a coincident plane-pair (the uniplane). 4. The numbers 1, 2, 3, 4, 5, 6 denote the intersections of a with J; 12...... 56 denote the lines connecting the points 1...... 6. 5. Roman numerals refer to conics; I, is the conic through the points 2, 3, 4, 5, 6, etc. 6. On the cubic surface the numbers denote the lines which are the transforms of the points, lines, and conics, respectively. 7. This investigation has three outstanding features: 1) the point-to-point correspondence between J and C4(X); 2) the dependence of the form of J upon the form of C4(X); and 3) the dependence of the singularities of the cubic surface upon the form of J. PART III I. General Cubic Surface. In this case is given the general cubic surface corresponding to any arbitrary plane a. C4(X) is an unrestricted plane quartic, and J a proper sextic curve. This case also gives the general plan of treatment followed throughout the remaining cases. 5 See Snyder and Sisam, Analytic Geometry of Space, pp. 170-172. Also Salmon, Geometry of Three Dimensions, art. 238a, p. 247. 6 The deficiency of a space curve is here defined as the number by which the number of double points, real or apparent, falls short of the maximum. Compare Salmon, Geometry of Three Dimensions, art. 353.

Page 405 1924] Pobanz: A Study of Cubic Surfaces 405 Let the three quadrics be A - ZaikXiXk = 0, B-bbikXiXk =0, C CikXiXk = 0, i, k= 1, 2, 3, 4.7 The three polar planes of a point (y) are 2 aikxiyk = 0, (1) bikxiyk =0, L Cikiyk = O. From equations (1) we get pxi, which are the determinants, with the proper sign attached, of the matrix kalkk Yak a2k a3kyk 2 a4kyk blhkyk Zb2kyk Zb3kyk 2b4kyk 2Clkyk YC2kyk YC3kYk YC4kYk C4(X) — = IXaik+X2bik+X3Cik =0 is an unrestricted quartic and, therefore, J a proper sextic curve. The plane a -2aixi = transforms into a cubic surface C3 which can be expressed in the following form C3- = ai(arbmscnt - altbmscnr)ylymyn, i, 1, m, n, r, s, t= 1, 2, 3, 4. i~ r, s, t. rz sF t. C3 has 27 lines on it: 8 1) 6 lines which are the transforms of the points 1, 2, 3, 4, 5, 6. 2) 6 lines corresponding to the conic sections through each 5 points I, II, III, IV, V, VI. 3) 15 lines corresponding to the lines 12, 13,...... 56. II1. Surface with 1 C2. If the self-polar tetrahedron of A and B is taken as the tetrahedron of reference, the equations of the three quadrics are A aiix2ii = 0, B-biix2ii =0, C = CikXiXk =0. The matrix is a lly a22y2 a33y3 a44Y4 bllyl b22y2 b33y3 b44Y4 cClkyk c C2kYk y C3kyk 2C4kyk 7All summations hereafter are from 1 to 4, unless otherwise stated. 8 See Table I, pp. 38a, 38b.

Page 406 406 University of California Publications in Mathematics [VOL, 1 from which we get pxl = d34y3y42C2Ayk - d24Y2Y4:C31,yk +d23y2y3LC4,kyl, PX2 = - d34Y3Y42ClkYk + d14Y1y42c3kyk- d13ylY32c4 kylk, px3 = d24Y2y42clkyk- dl4Yly 4c2kyk+ d12yly2c 4kyk, PX4 = - d23y2y32clkyk + d13y1y3c:c2kyk - dlIyly2;c3kyk, where (li =aiibij-a.ibii. [i, j= 1, 2, 3, 4]. C4(X)- XIaI,+X2b6k+X3cik = 0, [aik=bik=0, if k i] which is a proper quartic curve. J, therefore, is a proper sextic going through the four vertices of the tetrahedron of reference. To get a C3 with one conical point now take for a the plane x4=0. This plane contains the ruling of R corresponding to the point P1(0: 0: 0: 1). a goes into the following C3: C3 — d32 [c 1lYY2Y3 + c12y22y3 + C13y2y23 + C1 4Y2Y3Y4] + d13 [c12y21y3 + C22Y1y2y3 + C23yly23 + C2 4YlY3Y4] +d2l [c13y21Y2 + c23yly22 + c33YYl2y3 + c34yy2y4] = 0. This C3 has a C2 at Pi(0: 0: 0: 1). The tangent plane at this point is replaced by a proper quadric cone, the equation of which may be obtained by writing y4 = 1 in the terms containing y4 in the above equation. On C3 we have the lines: A. Lines through the C2= ( 0: 0: 1). a) Lines corresponding to the 3 points lying on the ruling of R. 1, 2, 3. b) Lines 56, 46, 45. B. Free lines.9 a) 4, 5, 6. b) I, II, III. c) 14, 15......3. II2. A C3 with 1 C2 may also be obtained by letting one of the quadrics be unrestricted and the other two be pairs of planes with their double lines intersecting. C4(X) is then a line and a cubic. J is made up of three concurrent lines and a plane cubic. As any plane a will cut the plane cubic of J in three points lying on a right line, the 6 points will have the same relative positions as given in II1. The C2 on C3 is the triple point of J. To it corresponds in the X-plane any point on the line of C4(X) except the 3 intersections of this line and the cubic of C4(X). The lines on the surface are the same as in II1. 9 A free line is a line on C3 which does not go through the singular point or points.

Page 407 1924] Pobanz: A Study of Cubic S1urfaces 407 II3. A C3 with 1 C2 may also be obtained by taking a tangent at any point of J. The lines on C3 are as follows: A. Lines through the C2. a) Line corresponding to (11). b) 12, 13, 14, 15. c) Line corresponding to conic section I. B. Free lines. a) 24, 25, 23, 34, 35, 45, 11. b) II, III, IV, V. c) 2, 3, 4, 5. II4. For a fourth way to obtain a surface with 1 C2, let A be unrestricted and B and C quadrics tangent along a ruling. C4(X) is a quartic with a cusp. J is made up of a double line and a quartic. Any plane a will transform into a C3 having a C2 on the double line. The distribution of the six points in a and the corresponding lines on C3 are as given in I3. IIIl. Surface with 1 B,. Let the three quadrics A, B and C be taken as in II, with C3 = 0. Then the plane a = 4 = 0 goes into the surface C3 = d2cl3y21y2 +d21c23yly22 + (d21C33+d13C22 +d32Cll) Y1Y2Y3 + d13cy21y d323 + d123y2 d32c12Y223 + d32Cl3yy23 + d3c24yly2y4 +d32Cl4yly3Y4 = O.10 Pl(0: 0: 0: 1) is a singular point on C3. Since the line of intersection of the biplanes y = 0, d13c24y2+d32C14y = 0 is not on C3 the point is a BK. On C3 are the lines: A. Lines through B3. (1) 2, 3,14,15, 45. B. Free lines. 4, 5, 24, 25, 34, 35 (11) and II, III. 1112. A surface with a B3 may also be obtained in the following manner. Letting A, B and C be unrestricted, and taking a osculating J at any point, the corresponding C3 will have a B3. The B3 may be found by getting the point of intersection of the line from the point of osculation and the line from the tangent line at the point of osculation.1l The lines on C3 are A. Lines through B3.. a) Line corresponding to (1). b) 12, 13, 14, (11). c) Line corresponding to conic (11) 234. 10 dij has the same meaning as in II1. 11 In general, if a curve goes through n-1 or n consecutive points of J, its transform will go through the singular point on C3.

Page 408 408 University of California Publications in Mathematics [VOL. 1 B. Free lines. a) 2, 3,4. b) 23, 24, 34. c) Lines corresponding to conies (111) 34; (111) 24; (111) 23. IVi. Surface with 2 C2. Let the three quadrics be A aalx21 +a22x22 +a33x23 +a44x24 = 0, B- bllx2 + b22x22 + b332 + b44x24 = 0, C = C11X21 + C22X22 + c33X23 + 44X24 +2C12XlX2 = 0 The matrix is allyl a22y2 a33y3 a44y4 bllyi b22y2 b33y3 b44Y4 Cllyli+C12Y2 C12y1 c22y2 C33y3 c44Y4 pxl = (c22d34+ c33d42 + c44d23) Y2Y3Y4+cl2d34yly3y4, PX2 = (clld43 +c33dl4+c44d3l)ylY3Y4 + 12d43y2y3y4, PX3 = (Clld24 +C22d41 + c44d12)ylY2y4 + c12d41y21y4 +Cl12d24y22y4, PX4 = (Clld32 +C22d13 +C33d2l) Y1Y2Y3 + C12d31y21Y3 +C12d23y22y3, C4(X) in this case is made up of the two lines a44X1+b44X2+c44X3 = 0, a33X+b33X2+ C33X3=0 and the conic (allXl+bllX2 +11X3) (a2X~l+b22X2 + 22X3) -c212X23 =0. The two lines each intersect the conic in two distinct points. J is made up of 6 lines. One has two triple points and the other skew to it has four double points. Any plane a -=aixi =0 goes into the surface C3 -[al(c22d34+c33d42 +44d23) + a2cl2d43] y2y3Y4+ [a2(Clld43-+C33d14+C44d3l) + alel2d34] Y1Y3Y4 -+ a3(Clld24 + C22d41+ —44d12) Y1Y2Y4+ a3C12d41y21Y4+a3cl2d24y22y4+a4(Clld32 +C22dl3+c33d21)ylY2Y3 + a4c12d31y21y3 + a4Cl2d23y223 = 0. It is easily seen that the C3 has a C2 at P1(0: 0: 0: 1) and one at P2(0: 0: 1: 0). On C3 we have the lines: Through P1 the lines: 1, 2, 3, 45, 56. Through P2 the lines: 1, 5, 6, 34, 24. Free lines: 4, 14, 25, 26, 35, 36, I. IV2. A C3 with 2 C2 may be obtained by taking A, B and C as in I and a through 2 rulings of R. The positions of the 6 points in a and the corresponding lines on C3 are the same as in IV1. IV3. As a third way to obtain the above C3, let A be unrestricted, B and C be two pairs of planes with the plane of one of the two pairs containing the double line of the other. C4(X) will then be made up of a cubic and a line tangent to the cubic. J consists of a single line, a double line and a plane cubic. Any arbitrary plane a

Page 409 19'24] Pobanz: A Study of Cubic Surfaces 409 will transform into a C3 having a fixed double point and another one on the double line. As the plane a changes its position, one of the C2 remains fixed while the other moves on the double line. The lines on C3 are as follows: Through the fixed C2: 1, 2, 3, 45, 55. Through the moving C2: (5), 15, 25, 35, 45. Free lines: 4, 14, 24, 34, I, II, III. V. Surface with a B4. Let the quadrics A, B and C be taken as in II1, but let C34=c24=0. The plane a-=4=0 then osculates J, and corresponding to it we have the surface C3 -d21c13y21y2 + d21c23yly22 + (d21C33+d13c22+ d32c11)l1Y2Y3 + d13cl2y21y3 + d13c23yly23 + d32C12y22y3+d32Cl3yly23+d32c14y1y2y4=O, where dij=a ib jj-ajjbii. P1(0: 0: 0: 1) is a B4, since the line of intersection of the biplanes yl = y2= 0 lies on the surface. 12 On C3 are the lines: A. Through B4 a) (1), 2, 3, b) 14, I. B. Free lines, 4, 24, 34, III, II. VI. Surface with B3+C2. To get this surface, let A be an unrestricted quadric, B a pair of planes, and C a pair of planes with the double line a ruling on A. The equations are A - aikXiXk = 0, B 2X3X4 = 0, C 2xlx2 =0. i, k=1, 2, 3, 4. a33=a44=a34=0. The matrix is 2alkyk Za2kyk 2a3kyk 2;a4kyk 0 0 Y4 Y3 Y2 Y 0 0 from which we get px1 = Y1 [Y3Za3kyk - Yy4a4kyk], pX2 = y2 [y4Za 4kyk - y3a3kyk], pX3 = Y3 [y2Ea2kyk - ylZalkyk], pX4 = Y4[Y1lalkyk - y2Ya2kyk] C4(X) -alX41+a2X31X2+a321X2X3 +a4X21X22 +a5X1X22X3+22X23 = O, where a, (j= 1; 2, 3, 4, 5) are functions of the coefficients of A, B and C. This quartic has a node and a cusp. J, therefore, is made up of a double line, a single line skew to the double line, and a twisted cubic. 12 It is easily shown that the edge is torsal.

Page 410 410 University of California Publications in Mathematics [VOL. 1 Any plane a —aixi=0 transforms into a C3 having a double point on the double line. The equation written out in full is C3-(ala3- a3al)y2ly3 — (alal4- a4all)Y2ly4 — (a2a23 - a3a22)y21y3 (a2a24 - a4a22)y22y4 - a3al3yly23 + a3a23Yly23 + a4a14Y1Y24 - a4a24y2y24 + (ala23 - a2a13)ylY2Y3 - (alae4 - a2al4)yly2Y4 - (a3a14 - a4al3)yly3y4 + (a3a24 - a4a23)y2y3y4 = 0. If the ai have such values that a is tangent to the twisted cubic, C3 has 2 C2. If a osculates the twisted cubic at any point, C3 has a B3 and a C2. On C3 the lines are arranged as follows: Through B3: 1, 12, 11, I. Through C2: 12, (2), 23, II. Free lines: 3, 22, III, making a total of 11 distinct lines. VII. Surface with a B5. Let A be unrestricted, B and C pairs of planes, one plane of C containing the double line of B. The equations are A 2aikXiXk =0, B -x - bx22 =0, C= 2xx3=0. aik=1, 2, 3, 4. a44=1, a14=a24=a34=0. The matrix is oalkyk Sa2kyk a3kyk Y4 Yi -by2 0 0 y3 0 yl 0 from which we get pxi= -byly2y4, pX2 = - 2Y4, px3 = by2y3y4, pX4 = (allb + a22) Y21Y2 +- a2by1y22 + a12y31 - a332y23 - a23by22 + a23y21y3. C4(X) X1 [a1X3l + a2X21X2 + a3X1X22 + a4X21X3 + a5X1X23 + a6X2X23 + a7XlX2X3] =0, where the ai (i= 1..... 7) are functions of the aik. The line intersects the cubic in the point 0:0: 1 and is tangent at 0: 1: 0. J, therefore, is made upof a single line, a double line, and a plane cubic. Corresponding to the plane a- al1 + a4x4 =0, we have the surface C3 — albyY2y4- a4 (allb+a22)y21y2- a4a12Y1Y22- a4Y31+ a4a33y2y23+ a4a23y22Y3- a4a23y21Y3= 0. Putting y4=1, we have yl=y2=0 as the biplanes and P1(0: 0: 0:1) is a B5. It is also easily shown that the tangent plane coincides with one of the biplanes. This surface has the lines: Through P(0: 0: 0: 1), (1), (2), 3. Free lines: 11, 22, III.

Page 411 1924] Pobanz: A St'tdy of Cubic Stcrfaces 411 VIII. Surface with 3 C2. Let A == 2a12xx2 + 2a13lx13 + 2a23x2x3 = 0, B 2b12xlx2 + 2b14xlx4 + 2b24x2x4 = O0 C- 2c23x2x3 + 2c24x2Z4 + 2c343X4 = 0, from which we get the matrix a12y2 + a13y3 bl2y2 +b4y4 0 al2yl +a23y3 bl2y-+b24y2 c23y3 + C24y4 al3syl +a23y3 0 C23y2 + C34Y4 and px1 = (a13yl-+a23Y2) (b14yI+b24y2) (c23y2+c24y4) - (al3yl+a23y2) ( c34y3)- (a212Yl+a23Y3) (b12yl+b24y2) (c23y2+C34y4), px2 =(al3yl+a23y2) (b12y2+b14y4) (c24y2+c34Y3) + (al2y2+al3y3) ( c34y4), px3 =(al2y2+al3y3) (b12y1+b24Y4) (c24y2+c34y3) - (al2y1+a23y3) ( c34y3)- (al2y2+aa3y3) (c23y2+c24y4) (bl4yi+b24y2), PX4= (a12y1-+a23y3) (b122+b14y4) (C23y2+c34y4) - (al2q2+-al33) ( c34y4)-(al3-l+-a23y3) (b12y2+b14y4) (c23y2+c24y4). 0 V bl4yI + b24y2 c24y2 + c34y3 (bl2y1+ b24Y4) (c24Y2 + (b14yl+ b24y2) (c23y2 + (bl2y2 +bl4y4) (c24y2 + b12y1 + b24y4) (c23y2+ C4(X)- aikXl+b6kX2+cikX3 |=0, [i, k=1, 2, 3, 4. i'k, a4k=b3k=clk=O], is a quartic with three double points. J, therefore, consists of three skew lines and a twisted cubic. The skew lines are bisecants of the cubic. Corresponding to aX2 = 0, we have the cubic surface C3- (a13b12c24 + a12b14c23) Y1Y22 + (a23b12c24 + a12b24c23) y32 + (a13b14c24 + a126b4c34) YlY2Y4+ (a23b14C24 - a12b24c34) y22y4 + (a13b12c34 + a13b14c23) Y1Y2Y3 + (a23b12c34 + a13b24c23) y22y3 + 2a13b14c34Y1y3y4 + (a23b14C34 + al3b24c34) Y2y3Y4 = 0. On C3 are the lines: A. a) (1), (2), (3), b) 12, 13, 23, c) I, II, III. B. Free lines: a) 11, 22, 33. IX. Surface with 2 B3. To get this surface let the quadrics be taken as follows: A - a11x2 + a2222 + x24 + 2a12x1x2 + 2a13X1x3 = 0, B - x1-bx22 = 0 C= 2X1X3 =0.13 13 The above equation is of the same form as Cayley's except that he has replaced the singular tangent planes which touch the surface along the axes by the planes yl =0, ya =0, y4 =0 giving him the form y3l+(y2+ya3+-y4)y2l+4aylY3Y4 =0. a must not be equal to -1, or the surface will have 4 C2.

Page 412 412 University of California Publications in Mathematics [VOL. 1 The matrix is ally + a12y2 +-al3y3 al2yl+ a22y2 al3yl Y4 Yl — by2 0 0 Y3 0 yi 0 PX= - byly2y4, px2 =- 21Y4, px3 = by2y3y4, p4 = (allb + a22)y21y2 + a12byly22 +al2y31. C4(X) =X1[a22Xl- bX2] [a13X1+X3]2=0, which is a double line and two single lines. To the point 0: 0: 1 corresponds the line X1=X3=0, a single line. To the point 0 1: 0 corresponds the double line x1=x2=0. To the line a22X1-bX2=0 corresponds x1 =x4=0, and to the double line al3X1+X3=0 the conic in the x4=0 plane. The conic is tangent to the line x1=x4=0 at the point 0 0: 1: 0. The plane a- a3X3+ a4X4 =0 goes into the surface C3 - a3by2y3y4 + a4 [(allb+a22)y2y2 + al2byiy22 +a12y31] = 0. P(0: 0: 0: 1) and P2(0: 0: 1:0) are biplanar points since the lines of intersections of their biplanes are not on the surface. C3 has 7 lines on it; viz: (1), (2), 3, 4, 13, 14, 22. X. Surface with B4+C2. Let all three quadrics be pairs of planes, and let the double lines of A and B each intersect the double line of C in a distinct point. The equations of the quadrics are A 2x1X2 = 0, B 2X3X4 = 0, C- 2(alxl + a2x2) (a3Xs +a4x4) = 0. Y2 Yi 0 0 0 0 Y4 Y3 aa3y3 + ala4y4 a2ay3 + a2a4y4 ala3yl + a2a3y2 ala4y + a2a4y2 from which we have pxl = yly4 (ala4yl + a2a42) - yly3 (ala3yl + a2a3y2), PX2 = - y2y4(ala4yl+a2a4y2) +y2Y3(ala3y l+a2a3y2), PX3 = Yly3(ala3y3 + ala4y4) - Y2Y3(a2a3y3 + a2a4y4), pX4 = - ylY4(ala3y3 +ala4y4) +y2Y4(a2a3y3 +a2a4y4). C4(X) =-1XX2(XlX2-4ala2a3a4X23) =0, which is a conic and two lines tangent to it. J, therefore, is made up of two skew double lines and two skew single lines each meeting each double line. a = alXi + a2X2 + a3X3 + a4X4 = 0 goes into the surface C3 aalcla4y21y4+ (aa2a4 - a2ala4)yy2y24 - alala3y1y3 - (ala2a3 - 2ala3) lyy3 - C2a2a4y22y4 + a2a2a3y22y3 +ala23yly2 - a2a23y2y3 - ala24yl4 4 +a2a24Y2y24 = 0.

Page 413 1924] Pobanz: A Study of Cubic Surfaces 413 It is easily shown that P1(0: 0: a4: a3) is a B4 and P2(a2: al: 0: 0) a C2. On C3 are the lines: (1), 2, 3, 12, 23, 22, III. X1. A second way to obtain the above surface is to let A, B and C have a twisted cubic in common. C4(X) is then a double conic and J the twisted cubic of A, B and C counted twice. Any plane through a tangent line of the twisted cubic will transform into a cubic surface with a B4+C2. The distribution of the lines on the surface is: Through B4: the lines (1), 12, 11, I. Through C2: the lines 12, (2) II, and the free line 22. XI. Surface with a B6. Let the equations of the three quadrics be A x21 + x22 + 2a3x1x2 + 2a4xlx3 + 2a6x2x3 + 2a7x2x4 = 0, B 2x3x = O, C 2xX2 = 0, from which we have the matrix yl + a3y2 + a4y3 a3y + 2 + a6Y3+ a7Y 4 a4yl+a6y2 a7y2 0 0 y4 Y3 Y2 Y1 0 0 and pxl = y [a4yly3 + a6y2y3 - a7y2y4], px2 = - y2 [a4ylY3 + a6y2y3 - a7y2y4], px3 = y3 [y32 + a6y2y3 + a7y2y4 - y31 - a4yly3], p4 = - Y4 [y32 + a6y2y3 + a7y2y4 - y3 - a4ylY3 ] In this case C4(X) =a24a7X41+ (2a6a7- 2a3a4a7)X1X2 - 2a4a7X1X2X3 +a23 2X22 +2a3XlX22X3 -XX3+XX23 = 0, is a quartic with a double point at 0:: 0 and a cusp at 0: 0: 1. J is made up of a double line, a single line skew to the double line and a twisted cubic. Taking a —xl-X2-a6X3=0 tangent at P1 where the single line cuts the cubic and through P2 where the double line cuts the cubic, we have the cubic surface C3 — (a4+a6) 21y3 - a7y22y4+ a4a6yly23 - ay22 3+ (a4+ a6) yy2Y3 - a7Y1y2y4 - a6a7y2Y3Y4 = 0. P1(0: 0: 0: 1) is a B6, the biplanes are y2=yl+y2+a6y3=0. On C3 we have the lines through P1, (1), (2), 12. There are no free lines. XII. Surface with a U6. To get this surface, let us take the quadrics as in VII. Then a= a2X2 + a4X4 =0 transforms into the surface C3 a2y21y4 - a4(allb + a22)y21y2 - a4al2byly22 + a4a33by2y23 + a4a23by22Y3 - a4al2y3 - a4a23y21y3 = 0.

Page 414 414 University of California Publications in Mathezmaltics [VOL. 1 Putting y4= 1, we have the y1=0 plane counted twice and PI(0: 0: 0: 1) is a U6. On C3 are the lines: A. Through U6: (1), 2, 3. B. Free lines: 4, 24, 34. XIII. Surface with B3+2 C2. If in VIII (Surface with 3 C2) the following relations exist among the coefficients of the quadrics, viz.: (a23b14 - a1324) (a2C34- a13C24) = 0, P1(0: 0: 0: 1) will go over into a B3. In VIII C4(X) was a quartic with three double points and the corresponding sextic J consisted of three skew lines and a twisted cubic. But if the relations as given above are satisfied, the quartic has two double points and a cusp. J is a double line, two single lines and a conic. The equation of the surface is C3-(a13b12c24 + a12b14c23) Y22 + (a23b12c24 + a12b24c23) 32 + (al3b14c24 + als2b1C34) Y1Y2Y4 + (a23b14c24 +- a2b24C34)y22y4 + (al3b12c34 + al3b14c23)yly2y3 + (a23b12C34 + a13b24C23)y 2y3 + 2a13b14C34Yly3Y4 +- (a23b14C34 + al3b24C34) Y2Y3Y4 = 0. On the C3 are the lines: (1), (2), (3), 13, 23, 22, 33, I. XIV. Surface with B5+C2. Let two of the quadrics be cones and the third a pair of planes with the double line through the vertex of one of the cones. Their equations are A a11x22 + a33x23 + 2a24x2x4 = 0, B -2x1x2 =0, C = 2222 - 2C13X3 = 0. The matrix is 0 a11Y2+ a24y4 a33y3 a24y2 Y2 Yl 0 0 - 13Y3 C22y2 - Cl13Y 0 from which we get pxl =-a24cl1y2ly2, px2 = a24cl3yly22, px3 = a24C22y32 + a24C13Yly2Y3, pX4 = -a33c22y2Y3 - a33c13lY2 - all113yly22 - a24C13YlY2Y4. C4(X) y21X23 = 0, namely two intersecting double lines. To the point of intersection corresponds the 4-fold line of J x1 = X2 = 0; to X1 = 0, the double line 2 = 3 = 0; and to X3 =0 a family of cones with vertex at 0: 0: 0: 1. a=x4=0 goes into the surface C3= a33c22Y22y3 + a33c13Yly23 + allcl31y122 -+ a24cl3YlY2y4=0. This cubic surface has a B., at 0: 0: 0: land a C2 at 1: 0: 0: 0 as can easily be shown. On the surface are the lines: (1), (2), 12, I.

Page 415 1924] Pobanz: A Study of Cubic Surfaces 415 XV. Surface with a U7. Let the quadrics be taken as follows: A =a33x23+x24+2a12x1x2 = 0, B x21-bx22 = 0, C-2xx3 = 0. The matrix in this case is a12y2 al2Yl a33y3 Y4 yl -by2 0 0 y3 0 yl 0 from which we get pxl= - byly2y4, px2 = -1y4, px3 = by2y3y4, pX4 = a12byy22 - a33by223 +a12y31. C4(X) X —I[a33bXl22- bX23+a212a33X31] =; the line intersects the cubic in the point 0: 0: 1, to which corresponds the single line of J, x1 =X3 =0, and is tangent at 0:1: 0. To the point of tangency corresponds the double line x1= 2=0. The other points of the cubic are in point-to-point correspondence with a cuspidal cubic in the 4 =0 plane. a — a2X2-+ a4X4 =0, through the cusp, goes into the surface C3 = a221Y4 - a4a2byly22 + a4a33by2y23 - a4al2y31 = 0. This surface has a U7 at 0: 0: 0: 1. On C3 we have the lines: Through U7, (1), 12, and the free line 2. XVI. Surface with 4 C2. The simplest way to obtain this surface is to let the three quadrics be pairs of planes with their double lines lying in the same plane. The equations are A — x2-ax24 = 0, B x22-bx24 = 0, C =X23 -cx24 =0. The matrix is Y1 0 0 -ay4 0 y2 0 -by4 0 0 Y3 - CY4 from which we have pxl = -ay2Y3Y4, px2 = - byly3y4, pX3 = -CY1Y24, pX4= -Y1Y2Y33

Page 416 416 University of California Publications in Mathematics [VOL. 1 C4(X) —X1X2X3[aXX1+bX2+CX3] = 0 is composed of the four sides of a quadrilateral. To the six points of intersection correspond the six edges of a tetrahedron. To the other points on the four lines correspond the 4C2 respectively. Any plane aaixi = 0 goes into a surface having a C2 at each vertex of the tetrahedron. The equation of the surface is C3 alay2y3y4 + a2bylY3y4+ a3cy1Y2Y4+ a4Yly2Y3 = 0. On C3 are the six edges of the tetrahedron and the lines 15, 24, 36. XVII. Surface with 2 B3+C2. If in VIII (Surface with 3 C2) relations a) and b) given below, or what amounts to the same thing, in XIII (Surface with B3+2 C2) relation b) exist among the coefficients of the three quadrics, namely: a) (a23b14- a3b24) (a12c34-a13c24) = 0, b) (b12c34-b14c23) (a13b24 -a23b14) = 0, VIII or XIII will go over into a C3 with 2 B3+C2. C4(X), which is a quartic with a cusp and two double points in XIII, goes into a quartic with two cusps and one double point. J, therefore, is made up of two double lines, corresponding to the two cusps, a single line corresponding to the double point and a single line corresponding to the other points on the quartic. The equation of the surface is C3- (a13b12c24 + a12b14C23)Y1y22 + (a23b12C24 + a12b24C23)Y32 + (a13b14C24 +- a12b1434)Y1Y2Y4 + (a23bl4C24 + al2b24c34)Y22Y4 + (a13b12c34 + a13bl4c23)Y1Y2Y3 + (a23b12c34 + a13b24c23)Y22Y3 + 2a13b14c34ysy3y4 + (a23b14c34 + a13b24C34) Y2Y3Y4 = 0. On C3 P1(0: 0: 0: 1) and P20: 0: 1: 0) are B3 if conditions a) and b) are satisfied. P3(1: 0: 0: 0) is a C2. The surface has the following lines: (1), (2), (3), 13, 33. XVIII. Surface with B4+2 C2. Let A be a cone, B and C each a pair of planes with their double lines intersecting. The equations are A - a22x22- 2a13x3x4 = 0, B x21-bx2 = 0, C2X2X3 = 0. The matrix is in this case 0 a22y2 - al3y4 - a3y3 Y1 -by2 0 0 0 Y3 y2 0

Page 417 1924.] Pobanz: A Study of Cubic Surfaces 417 from which we get pxl = al3by22y3, px2 = al3yl2y3, px3 = - al3yly23, PX4 = a3Yly3Y4 + a22yly22. C4(X) X-21X2[a22Xl- bX2]= 0, a double line and two single lines intersecting in the point 0 0: 1. To this point corresponds the triple line of J, x2=X3=0. To the point 0 1 0 corresponds the double line of J, x = x2=0, and to the line a22X1- bX2 = 0 the line xi = X3 = 0. a- alX1+ a4X4 =0 transforms into the surface C3 = -la13by22y3 + aC4al3ylyY4 + a4a22yly22 = 0. This equation is the same, except for the coefficients, as that which Cayley gives for a C3 with a B4+2 C2. The singular points are P1(0: 0: 0: 1) a B4, P2(0: 01::0) a C2, P3(1: 0: 0: 0) a C2. On C3 are the lines (1), (2), (3), 13, and the free line II. XIX. Surface with B6+C2. Let two of the quadrics be cones and the third a pair of planes with the double line through the vertices of the cones. Then A — a11x21+ a33x23 + 2a12x1x2 = 0, B =2xlx2= 0 C = C222 - 2cxIx4 = 0. The matrix is alyl + a12y2 a12yl a33y3 0 Y2 Y1 0 0 - 14Y4 C22Y2 0 - C14Y px = a33c14Y21y3, px2 = -a33C14y1y2y3, pX3 = - anC14Y31, p= - a33c22 22Y3- a33c14ylY3Y4 -C4(X) -X1X33 =0. J is made up of a 5-fold line and a single line cutting it. To the plane a- a3X3+ a4X4 =0 corresponds the surface C3 -- a3alal4y3 + a4a33c22y2Y3 + a4a33C14Yly3y4 = 0. This equation is equivalent to Cayley's. C3 has a B6 at 0 0: 0: and a C2 at 0:0: 1: 0. The two lines are (1) and (2).

Page 418 418 University of California Publications in Mathemlatics [VOL. 1 XX. Surface with a U8. *Let the quadrics be A -allx2+ a44x24+2a23x2x3 = 0, B — 22- cx2 = 0 C-2X1x3 =0, from which follows the matrix allyl a23Y3 ay23y,2 a44y4 0 y2 -cy3 0 2y3 0 yl 0 and pxl = a44yly2y4, px2 = -a44cy23y4, pX3= -a44y32y34, PX4 = - allY21y2 + a23cy33 + a23y2Y3. a - a2X2 + a44 = 0 transforms into the surface C3 = a2a44cy23y4+ a4ally212 - a4a23cy23 - a4a23y22y 3 = 0. This surface has a Us at 0 0: 0: 1 and but one line on it, y2 = y3 = 0, corresponding to the point (1). XXI. Surface with 3 B3. If in VIII (Surface with 3 C2) the following relations exist among the coefficients of the three quadrics, viz.: a) (a23bl4-al3b24) (al2c34-a13c24) = 0, b) (b12C34 - b1423) (al3b24 - a23b14) = 0, ) (bl4c23 - b1234) (a13c24- a12C34) = 0, each of the 3 C2 goes over into a B3. C4(X), which in VIII has 3 nodes, now has 3 cusps. J, therefore, is made up of three double lines intersecting in a point. The equation of the surface is C3- (a13b12C24 + a12bl4c23)y1y22 + (a23bl2c24 + a12b24c23)y3l + (a13b14C24 + al2b14C34)yly2y4+ (a23b14c24 +- a12b24c34)y22y4 + (a13b12c34 + aa3bl4c23)yly2y3 + (a23bl2c34 + a13b24C23)y22y3 + 2a13bl 4C34Y1Y3Y4 + (a23b14C34-+ al3b24c34) YlY3Y4 = 0. The three lines on C3. are (1), (2), (3). XXII. Ruled Surface. Nodal line of the first kind. To obtain this surface in the simplest manner, let A 2xlX3 = 0, B 2x2x4 =0, C=-X21-bx%2 = O.

Page 419 Pobanz: A Study of Cubic Szrfaces 419 The matrix is Iy/ 0 Yi 0 0 Y4 0 Y2 I Yl -by2 0 0 from which we get pxl = - byly22, pX2 = - Y2y2, px3 = by22y3, pX4 = y21Y4. C4(X) reduces to XI=0, X2=0, each counted twice. They intersect in the point 0 0: 1; to this corresponds the 4-fold line x1 = x2 =0 of J. To the point 0: 1: 0 corresponds the line X2 = x4 =0 and to 1: 0: 0 the line xI = 3 = 0 of J. a=a3:-+ a4X4 =0, a plane through the directrix, transforms into the ruled surface C3 a3by22y + a4y21Y4 = 0, which is Cayley's equation except for the coefficients. x1 = 2=0 is a nodal line of the first kind. 0: 0: 0 1 and 0: 0:: 0 are the two pinch points, i.e., the two tangent planes fall together at each of these points. XXIII. Ruled Surface. Nodal line of the second kind. Let the three quadrics have a twisted cubic in common. For the sake of simplicity they may be taken as follows: A 2xX4 - 2x2x3 = 0, B 2 22 2 - 2 x2x3- 23 + 2X1X4 = 0, C- 23 - 2x2X3 + 2x1X4 - 2x2X4 = 0 The matrix is Y4 -y3 Y-2 Y1 -3 + 4 - 2-y3 -Yl-Y2 Yi | 14 -Y3 -Y2-+Y3 Y1-Y2 from which we get pXi = 3Y1Y2Y3 - y21y4- 2y3, px2 = - yly2Y4 - 22Y3 + 2Yly23, PX3 = Y1Y3Y4 + y1y23 - 2y224, pX4 = 2y32 + yly 24- 3y2y3Y4. C4(X)-[X21+X22+X23+2X1X2+2X1X3+X2X3]2=0, which is a double conic. This conic is in one-to-one correspondence with the twisted cubic X: X2: X3 4=X 3: X2: X 1.

Page 420 420 University of California Publications in Mathematics [VOL. 1 This cubic is the common curve of intersection of A, B and C. a-= O=0 osculates the cubic at 0: 0: 0: 1 and to it corresponds the surface C3- 3yy23 - 21y3 - y32 = 0. yl = Y2 = 0 is a nodal line of the second kind having a pinch point at 0: 0: 0: 1. XXIV. Cubic Cone. To obtain the cubic cone, let one of the quadrics be unrestricted and the other two be pairs of planes with their double lines intersecting. The equations are A _ a1x21 + a22x22 + a33x23 + x24 + 2a12xlx2 + 2a13x1x3 + 2a23x2x3 = O0 B =x21-bx22 =0, C-x21-cx23 = 0. The matrix is allyl + a12y2 + a13y3 a12yl + a22y2 + a23y3 al3yl + a23Y2 + a33Y3 Y4 Yi -by2 0 0 Yi O - cy3 0 from which we get pxi = bcy2y3y4, PX2 = cyly3y4, pX3 = byl1y24, pX4= (allbc + a33b + a22c)yly2y3 + a12bcy22y3 + al3bcyy23 + a13by21y2 + a23byly2 + a12cy21y3 + a23cyly23. C4(X) is made up of a cubic and a line. J, therefore, consists of three concurrent lines and a plane cubic. a=x4=0, the plane of the plane cubic transforms into the surface. C3-(allbc+a33b+a22c)y1y2y3 + al2bcy223 + a13bcyy23 + a13by21y2 + a23byy23 + a12cy213 +a23cyly23 = 0. C3 is a cubic cone with vertex at 0: 0: 0: 1. The points of the plane cubic go into the rulings through 0: 0: 0: 1. All other points in the plane go into this vertex. This concludes the study of the 24 varieties of cubic surfaces. The preceding results are summarized in the following table in which the surfaces are given in the same order as they appear in this paper. Column 2 gives the plane quartic C4(\) and column 3 the space sextic J in point-to-point correspondence with C4(X). In column 4 the selection of the plane a is given; column 5 gives the six intersections of J with a; in column 6 the singularity or singularities on the corresponding cubic surface are given and column 7 gives the number of distinct right lines on the surface.

Page 421 TABLE I Surface C4(X) J a-plane Position of the 6 points._ Dist. Lines, Sing. I tUnrestricted quartic Proper sextic curve Arbitrary.2 None 27.5.4.3 III Unrestricted quartic Proper sextic curve Through a ruling of R 4.2 C2 21.1.6.5.. (11).2 II2 Unrestricted quartic Proper sextic curve Tangent to J.5 C2 21.3.4..(11) III, Unrestricted quartic Proper sextic curve Through a ruling of R.2 B3 15 and tangent to J.5.3.6 -(111) III2 Unrestricted quartic Proper sextic curve Osculating J.4 B3 15.2.3.3 IVi 2 lines and a conic J made up of 6 lines. One has 4 Arbitrary.2 2 C2 16 double points and the other 2.6.5.1 triple points.4 IV2 Unrestricted quartic Proper sextic curve Through 2 rulings of R Same as IVi.3 Cubic and a line tangent J made up of a single line, a.2 IV3 to it double line and a plane cubic Arbitrary.1 2 C2 16.. (44) _________ _________________________ _______________________________ _______________________.5 _____________________

Page 422 TABLE I-(Continued) Surface C4(X) J a-plane Position of the six points Sing. Dist. Lines..(111) V Unrestricted quartic Proper sextic curve Through a ruling of R.2 B4 10 and osculating J.1.4 Quartic with one double J made up of a double line, a Osculating the twisted. (111) VI and one cuspidal point single line skew to the double cubic.3 B3+C2 11 line and a twisted cubic. (22) Cubic and a line tangent Single line, double line and a Through 0 1 0:0. (111) VII to it plane cubic and 0: 0: 1 0.3 B5 6 (22) *(11) VIII Quartic with 3 double 3 skew lines and a twisted cubic Through vertices of..(22) 3 C2 12 points cones. (33) A double line, and 2 sin- A double line, a single line, a Through 1 0: 00.. (22) IX gle lines conic, and a line tangent to and 0: 1: 0: 0.3 2 B3 7 the conic.4.(11) A conic and 2 lines each Two skew double lines and two Through the point. (111) Xi tangent to the conic skew lines each meeting each where a single line.3 B4+C2 7 double line meets the double line. (22) X2 Double conic Double twisted cubic Tangent to cubic::(1111) B4+C2 7 (22) Double line, a single line skew to Plane tangent where single XI Quartic with a cusp and a the double line and a twisted line cuts the cubic and.'(111). (222) B6 3 node cubic through the point where the double line cuts the cubic Through 1:0:0:0.(111) 2 3 XII Same as VII and 0: 0: 1: 0 U ' 6.4 XIII Quartic with 2 nodes and Double line, single line and a Same as VII. (11): (22) B3+2 C2 8 a cusp conic. (33) I I......

Page 423 TABLE I-(Concluded) Surface C4(X) J a-plane Position of the 6 points Sing. Dist. Lines XIV Two double lines 4-fold line and a double line cut- 4=0.. (22) B5 +C2 4 ting it (1111) XV Cubic and a line tangent Single line, double line and a Through cusp *. (11111) U7 3 to it cuspidal cubic..2.3 XVI 4 sides of a quadrilateral J made up of the 6 edges of a Arbitrary.4 4 C2 9 tetrahedron.2.6.5.1..(33) XVII Quartic with one node Two double lines and two single Same as VIII (2) 2 B3+C2 5 and two cusps lines (11) * Double line and 2 single A triple line, a double line and a Through 1: 0: 0: 0 (111) XVIII lines intersecting in one single line intersecting in one and 0: 1: 0: 0 B4+2 C2 5 point point.3 (22) XIX Triple line and a single 5-fold line and a single line cut- Through 0:1 0: 0. (11111) B6+C2 2 line ting it and 0 0: 1: 0.. 2 XX Cuspidal cubic and in- Triple line and a cuspidal cubic Through cusp. (111111) Us 1 flexional tangent (33) XXI Quartic with 3 cusps Three double lines Same as VIII. (22) 3 B3 3 (11) XXII Two double lines 4-fold line and 2 single lines cut- (1) Nodal line and directing it Through directrix.3 trix distinct.2 XXIII Double conic Double twisted cubic Osculating cubic.(1) Nodal line and directrix coincide XXIV Cubic and a line Three concurrent lines and a 3 concurrent lines and co number of points of Cone with vertex at II____~__ _ -plane cubic a plane cubic plane cubic 0: 0: 0: 1

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 20, 425-443, 1 figure in text May 17,1924 THE HYPERSPACE GENERALIZATION OF THE LINES ON THE CUBIC SURFACE. BY DANIEL VICTOR STEED I. INTRODUCTION Of all general algebraic surfaces in space of three dimensions the cubic surface is unique in one respect, namely, there exists upon it a finite number of straight lines. The discovery of this property of the cubic surface was made by Cayley1 in 1849, and the number of the lines was determined by Salmon in the same year. Since that time the remarkable configuration which these lines form has been the subject of much important research not only in pure and analytical geometry but also in the theory of groups. Considering the very extensive mathematical literature which has grown up around this configuration, the existence of which can from a certain point of view be looked upon as a property of three dimensional space, it would seem that geometers interested in hyperspace would very naturally inquire: Is there an analogous property for space of four dimensions? Or to put the question more fully: Is there in space of four dimensions an algebraic hypersurface which possesses the unique property of containing upon it a finite number of straight lines? If so, what is the order of this hypersurface, and how many lines are there on it? Finally, what are the positions of these lines relative to each other? The first two of these questions were answered by Schubert, who proved that there are 2875 lines on the general hypersurface of order five in space of four dimensions. Schubert's result seems, however, to have been obtained not as a result of a direct attempt to answer the questions asked above, but rather as incidental to another investigation in which that mathematician was engaged, namely, the determination of the number of lines having six point contact with the general hypersurface of order m in space of four dimensions.2 Now a line having six points in 1 The Cambridge and Dublin Mathematical Journal, vol. 4, p. 132. 2 Schubert, "Die n-dimensional Verallgemeinerungen der Anzahlen fir die vielpunktig beriihrenden Tangenten einer punktallgemeinen Flache n-ten Grades," Mathematische Annalen, vol. 26, pp. 26-52.

Page 426 426 University of California; Publications in Mathematics [VOL. 1 common with a hypersurface of order five lies entirely on it. Hence for m=5 the formula for the number of six point tangents gives the number of lines on the quintic hypersurface. In 1919, however, Professor D. N. Lehmer, being then unacquainted with Schubert's result, made a more direct inquiry into the question of what in space of four dimensions corresponds to the existence of lines on the cubic surface, and came to the conclusion that, as far as the existence of lines upon it is concerned, the hypersurface of order five was the analogue of the cubic surface. The method by which Lehmer arrived at this result was as follows: Let (Xl, x2, x3, x) be the cartesian co-ordinates of a point in space of four dimensions. The equations of any line can be put into the form (1) xi=alx+a2, X2=blX+b2, X3=ClX+C2. If these values of xi, x2, and x3 be substituted in the general equation of the algebraic hypersurface of order m, there results an equation in x, (2) Alxm+A2xm -l +.................................................... +A m+ = whose roots are the x values corresponding to the m points of intersection of the line (1) with the hypersurface, and whose coefficients are polynomials of degree m in the quantities al, a2, bi, b2, cl, c2. If now the line (1) is to lie entirely on the hypersurface, these six quantities must satisfy the m+1 equations (3) A,=O (i=1, 2, 3,.... m 1). This system of equations, containing as it does six unknowns, will in general have a finite number of solutions when and only when there are six equations, i.e., when m+1 =6 or m =5. Thus the only general hypersurface containing a finite number of lines on it is that of order five. For m=5 (3) consists of six equations each of degree five. The number of lines will equal the degree of the resultant of this system of equations. Professor Lehmer proposed to me the problem of enumerating these lines, his discovery and suggestion thus leading to the present investigation. The results given in this paper were all obtained before I discovered that Schubert had already arrived at the result in the manner briefly described above. They constitute an independent verification of Schubert's work, the method differing from Schubert's in that no use is made of the notion of tangency. It is conceivable that there may be values of n and r for which the general hypersurface of order r in space of n dimensions contains on it a finite number of linear spaces of dimension d. The question whether such values of n and r actually exist is considered in Part II, where a relation between r, n, and d is obtained, the existence of which is a necessary condition for the existence of a finite number of linear spaces of dimension d on the general hypersurface of order r in space of n dimen

Page 427 1924] Steed: Hyperspace of the Lines on the Cubic Surface 427 sions. The determination of the values of n and r for which this condition is also sufficient is a problem requiring further investigation, which might lead to interesting results. In Part V recursion formulae are developed, successive applications of which make possible the determination of the number of lines on the general hypersurface of order 2n-3 in space of n dimensions. These formulae are then applied in Part VI in actually determining the number of lines on the corresponding hypersurfaces for spaces of dimensions four to seven, inclusive. II. GENERALIZATION The problems of the existence of a finite number of lines on the general cubic surface in S3, and of lines on the general quintic hypersurface in S4, are special cases of a more general problem for space of n dimensions which may be formulated as follows: To determine whether or not there exists on the general hypersurface of order r in space of n dimensions a finite number of linear spaces of dimension d. In this section we shall derive an equation connecting r, n, and d, the existence of which is a necessary condition for the existence of spaces of dimension d satisfying the condition stated in the problem as formulated above. Let the equation of the general hypersurface of order r in space of n dimensions be (1) f(Xl, X2, X3,.... Xn)=O. The equations of any space3 Sd of dimension d consist of n-d linear equations in the n cartesian coordinates xi, x2, X3,.... Xn of n dimensional space. These can be solved for n-d of the variables in terms of the remaining d, and the equations can thus be put in the form (2) xi = ailn-d+l+ai2Xn-d+2+ai3Xn-d+3+... +aidXn+aid+l(i =1, 2,.. n-d). The result of substituting these values of xi into (1) is an equation of degree r in the d variables Xn-d+k (k =1, 2, 3,.... d). If the space defined by equations (2) lies entirely on the hypersurface, then this equation in the d variables must vanish identically, and hence each of its coefficients must vanish. The vanishing of each of these coefficients imposes a condition upon the coefficients in (2)( which may be called the coordinates of Sd). If these coordinates be thought of as unknowns, then the number of unknowns is evidently (n-d) (d+l). And since the general equation of degree r in d variables contains ( rd ) or (r+d) (r+d-1)..... (r+2) (r+l) d! terms the number of equations of condition is (rd ) In order that the number 3 For the sake of brevity the word 'space' will be used hereafter to denote a 'linear space.'

Page 428 428 University of California Publications in Mathematics [VOL. I of solutions of these equations be finite, it is necessary that the number of equations equal the number of unknowns. Hence the Theorem. A necessary condition for the existence of a finite number of linear spaces of dimension d on the general hypersurface of order r in space of n dimensions is that r, n, and d satisfy the equation ( -d)= (n-d) (d+l). It follows immediately from the theorem that the general hypersurface of order r in space of n dimensions does not contain upon it a finite number of lines unless r = 2n- 3. By taking d = 2 it will be seen that a necessary condition for the existence of planes on the hypersurface is that r satisfy the equation r2+3r+ 14 = 6n, and hence that either r= 1 (mod 3) or r- 2 (mod 3). When r= 1, n = 3, which is the case of the plane in space of three dimensions, and the number of planes on the surface is obviously finite, being one. One is naturally lead to inquire whether the condition of the theorem is not also a sufficient one. That it is not can be seen by taking r=2, n=4, values which satisfy the equation above. That the general quadric hypersurface Q in space of four dimensions does not contain upon it a finite number of planes can be easily shown as follows: Suppose that Q contains upon it the plane wr. Then any hyperplane through ir meets Q in a quadric surface, which contains 7r and therefore consists of ir and a second plane ir' which is also on Q. Thus for every hyperplane through ir there is a plane on Q distinct from wr. Hence if there is one plane on Q there must be an infinity of them. Of course the assumption that there is one plane on the hyperquadric involves assuming a special hyperquadric.4 For d = 1, n =4, r = 2, then, we can conclude that the six equations are in general inconsistent, their resultant taking the form (3) 0 x4+c=0 where c is a function of the coefficients of the equation of the hypersurface. Now if these coefficients are so selected that c vanishes, then (3) will be satisfied by any finite value of x4. Hence (3) expresses the condition that the hyperquadric contain an infinity of planes on it. This property of the hyperquadric might be described as a "poristic property of space of four dimensions." For although the number of equations necessary to determine the coordinates of a plane on the hyperquadric equals the number of these coordinates, there exists in general no plane on it, and, secondly, if we take a hyperquadric containing one plane on it, the hyperquadric will then necessarily contain an infinity of planes. 4 This special hyperquadric need not be degenerate, i.e., consist of a pair of hyperplanes. Such a hypersurface would be a still more special case than the one here considered.

Page 429 1924] Steed: Hyperspace of the Lines on the Cubic Surface 429 III. THE ENUMERATIVE METHOD OF SCHUBERT FOR SPACE OF n DIMENSIONS It has been pointed out in Part I that the enumeration of the lines on the quintic hypersurface is equivalent to the determination of the degree of the resultant of a system of six equations, each of degree five. The equations of this system, however, are special, and hence we cannot say that the degree of the resultant is 56. All that we can conclude without further investigation is that 56 is an upper limit for the degree of the resultant. The writer has been unable to find in the theory of elimination a method for the determination of this degree. It seems that its determination by purely algebraic methods would be a more difficult task than the solving of the original problem by geometric methods. The solution of the three dimensional analogue of the problem now under consideration, i.e., of the problem of determining the number of lines on the cubic surface, is easily obtained by determining the number of double tangent planes that can be drawn to the cubic surface from an arbitrary fixed point P not on the cubic surface. For any such plane meets the surface in a cubic curve having two double points, and therefore consisting of a line and a conic. Conversely every plane through P and a line on the surface is a double tangent plane, and hence the number of lines on the surface is equal to the number of double tangent planes from P. Now in space of four dimensions, any plane through a point P and a line on the quintic hypersurface F meets the hypersurface in a quintic curve having four double points, namely, the four points of intersection of the line with the residual quartic. Hence such a plane is a fourfold tangent plane. But the converse is not true. A fourfold tangent plane does not necessarily contain a line on F. For a quintic curve can have four double points without degenerating. Hence, although from a point not onF a finite number of fourfold tangent planes can be drawn to the hypersurface, the number of such planes exceeds the number of lines on the quintic hypersurface. Having come to the conclusion that a more powerful tool would be necessary for the solution of the problem for four dimensions than was needed for the corresponding problem in space of three dimensions, I began the study of Schubert's Kalkiil der Abzdhlenden Geometrie.5 For an understanding of the following development, a knowledge of the fundamental concepts of enumerative geometry for space of three dimensions made use of by Schubert and explained in the textbook just mentioned, is essential and will be presupposed in this paper. The remainder of this discussion will be devoted to an extension of this method to n- dimensional space and its application to the particular problem of the present investigation. The method is that of pure rather than analytic geometry, and what follows will there5 Leipzig, 1879.

Page 430 430 University of California Publicatiolns in Mathematics [VOL. 1 fore require and presuppose in addition to what has been mentioned above a knowledge of n- dimensional geometry equivalent to what will be found in Schoute, Mehrdimensional Geometrie, vol. 1, part 1. In what follows, the word 'space' will always refer to a 'linear space,' i.e., a space which, if it contains two points of a line, contains the entire line. The word 'hyperplane' will denote a space of dimension of n-1 where n is the dimension of the space of operation. A hypersurface of order r is a manifold of dimension n-1 which in general has r points of intersection with an arbitrary line. Analytically it is the locus of a single equation of degree r in the coordinates of the space of operation. The word 'variety' will be used for a point locus of dimension greater than one and less than n-1. Analytically it is defined by means of a set of not more than n-2 equations, the dimension of the variety being n-d where d is the number of defining equations. A Notation for Line Conditions The adoption of a suitable notation is of fundamental importance in the extension we are about to make. A notation which will be found to have several advantages and which will be used throughout this paper is given in the following: Definition 1. The symbol g [i k1k2k3.... k (n)] where n>i o and i-2<kx<n (X= 1, 2, 3,.... j) denotes the condition that the line g lie in a space Sni of dimension n-i and meet j linear spaces of dimension n-kx-1, these spaces all being contained in the space S,_i; and the same symbol will denote also the number of lines satisfying this condition. The letter enclosed in parenthesis and indicating the dimension of the space of operation may be omitted when its value is n or is known from the context. The letters i and kx will be called arguments. When the number of arguments is small, they will sometimes be written as subscripts and the brackets omitted. Thus the condition g [1 2 (3)] will be written gl 2 (3). The following rule will be seen to be applicable to the above notation. If for any X, kx = i or i-1, then the argument kx can be omitted; and conversely, an argument kx where kx = i or i- 1 can be supplied without the numerical value of the condition being thereby altered. For if kx = i the presence of the argument kx expresses the condition that g meet a space of dimension n-i-1 contained in the space Sni which contains g. If the space Sni be thought of for the moment as the space of operation, then the argument expresses the condition that g meet a given hyperplane. This condition, however, being satisfied by any line in Sni, does not impose any restriction upon g, and the number of lines satisfying the original condition is not affected by the omission or presence of the argument. In similar manner the rule for the case when kx = i- 1 can be verified.

Page 431 1924] Steed: Hyperspace of the Lines on the Cubic Surface 431 Fundamental Theorems on Line Conditions Theorem 1. When i<ki1+k22+1-n, the condition g [i k1k2] is equivalent to the condition g[j k1k2], where j=k1+k2+ 1-n. For let M and N be the two linear spaces of dimension n- k1 -1 and n - k2-1 respectively which the line g meets. Then g has two points in common with the space R determined by M and N, and therefore lies entirely in that space. Now if M andN intersect the dimension x of the intersection6 is given by 2n-2-ki-k2-x=n-i and hence x = n+i- (kl+k2+2). By the hypothesis of the theorem, x is consequently negative, that is M and N do not intersect. Hence the dimension of R is n-j, where n-j= n-ck- 1+n-k2-1+ 1, whence j= kl+k2+1-n. Since then any line that satisfies the condition of meeting both M and N must lie in the space R of dimension n-j, the argument i can be replaced by the greater number j. Theorem 2. The product of two line conditions g [i k1k2k3.... ka] and g[j 111213.....] is g[i+j i+ll i+12... i+l, j+kl j+k2.... j+ka]. A line satisfying both of the factor conditions lies in one space of dimension n-i and in another of dimension n-j and hence lies in their intersection, which will be denoted by R. If the dimension of R is n-x, then n-i+n-j-(n-x) =n and hence x=i+-j, and the first argument of the product is accounted for. To any argument kx corresponds the condition that g meet a space M of dimension n-kx -1 contained in the space Sni of the first factor condition. But since, as has just been shown, g lies in a space R of dimension n-i-j, g must meet M in a point which is in the intersection of M and R. M and R are however both contained in Sni and hence the dimension of their space of intersection is n-y-1, where n-i-j+n-kx- 1-n+y+- =n-i and consequently y=j+kx. The presence of the subscripts i+lx in the product condition is accounted for in like manner. 6 Cf. Schoute, Mehrdimensionale Geometrie, vol. 1, p. 14, et seq.

Page 432 432 University of California Publications in Mathematics [VOL. 1 If definition 1 only is assumed, then the application of theorem 2 will sometimes lead to a symbol which is undefined, since we may have either i+j >n-i, j+kx > n or i+lx>n. In the second and third cases the spaces M and R have no point in common, as is evident from the fact that n-y-i is negative. Hence, since there can be no line in R which meets M, there is no line satisfying both factor conditions, i.e., their product is zero. In the first case it is evident that the dimension of R does not exceed zero, and hence no line can satisfy the condition of being in both Sni and Sn-i. Thus in this case also the product of the given conditions is zero, and the following definition is justified. Definition 2. 'The symbol g[i kk2k3.... kj (n)]=0 when any one of the arguments i or k exceeds n-1. In addition to the above theorems we shall make use of the relation g[i kik2k3.... ka(n)]=-g[O ki-i k2-i k3-i.... ka-i(n-i)] which follows immediately from the definition of line conditions. The Dimension of a Line Condition The order of infinity of lines in space of n dimensions is 2n-2. If c is the order of infinity of lines satisfying a given line condition, then the dimension7 of the condition is defined as being 2n-2-c. It will be seen that the dimension of the line condition g[i kjk2k3.... ka] can be expressed in terms of the arguments of the condition. On account of the rule stated in a preceding paragraph we can and will assume that no k is less than i+ 1. By theorem 2 g[ikil] g[O k2-i] g[O k3-i].... g[O ka-i]-=g[iklk2k3.... ka]. Let M be the space of dimension n - k -1 of the condition g[iki] contained in the space Sni. Then through each point of M an n-i-1 fold infinity of lines can be drawn, each of which satisfies the condition g[ikl]. The order of infinity of lines satisfying the condition g[ikl] is therefore n-kl-1+n-i-1=2n-kl-i-2, and hence the dimension of the condition is 2n-2 —(2n-kl-i-2) =i+kl. Similarly the dimension of g[0 kx-i] is kx-i. Hence, since the dimension of a product is equal to the sum of the dimensions of the factor conditions,8 the dimension 3 of the line condition g[ikk2k3...... k] is i+kl+k2-i+k3-i+. +ka-i or 6 = z kx-(a-2)i (X=l, 2, 3,.... a). When a =2 the dimension is simply kI+k2, and when i =0 the dimension is S kx. Since in an equation between conditions the terms must all be of the same dimension, this simple means of determining the dimension of a line condition will be found useful. 7 Schubert, loc. cit., p. 7. 8 Schubert, loc. cit., p. 9.

Page 433 1924] Steed: Hyperspace of the Lines on the Cubic Surface 433 As an example consider the condition g[1222(4)]. The condition is that in space of operation of four dimensions a line g lie in a given hyperplane and intersect three given lines in that hyperplane. By the formula for 5 given above, the dimension of the condition is 5. Since there is a sixfold infinity of lines in space of four dimensions, there is a single infinity of lines satisfying the given condition, a fact otherwise obvious since the lines satisfying the condition belong to a regulus. A Formula for the Evaluation of Line Conditions Consider the line condition g2[O k(n)] where k<n/2. Let the two spaces of dimension n-k- 1 of the condition be denoted by M and N. If the positions of the spaces M and N are specialized by taking them in the same hyperplane, then the dimension d of their intersection R is given by 2n-2k-2-d=n-1 and hence d = n-2k-1. Consequently either R is a point or d exceeds zero, and the lines that satisfy the condition of meeting both M and N are (1) any line which intersects R, and (2) any line which is in the hyperplane and meets both M and N. The first of these conditions is expressed by the symbol g[O 2k] and the second by g[1 kk]. Hence I g2[0 k] = g[l kk]+g[O 2k], (k<n/2). By Schubert's Princip von der Erhaltung der Anzahl (hereafter referred to as the principle of the permanence of number), this formula which is true for the special position of M and N is true in general, provided that the order of infinity of lines satisfying the given condition has not been increased by virtue of the assumption of a special position for M and N. But the formula for finding the dimension of a condition shows that each condition of the formula is of dimension 2k. Hence if an algebraically defined system of lines of dimension 2k is given, a finite number of lines will satisfy each of the conditions of the formula. Thus by the principle just mentioned the formula is true for an arbitrary choice of the spaces of the condition in the left-hand member of the equation. It may be worth while to note here a case in which by specializing the positions of the spaces of a condition the order of infinity of the lines satisfying the condition is increased. Consider the condition that in space of four dimensions a line intersect two given lines M and N, a condition expressed by g2[0 2(4)]. Clearly there is a twofold infinity of lines satisfying this condition. Let the position of M and N be specialized by supposing them to intersect in P. Then the lines meeting M and N are 1~ any line in the plane MN and 2~ any line through P. The lines of 1~ and 2~ are the lines satisfying g[2] and g[O 3] respectively. But a threefold infinity of lines satisfies g[O 3]. Hence we cannot say that g2[0 2] is equal to the sum of g[2] and g[O 3]. That such a relation does not exist follows immediately from an application of the formula for the dimension of the line condition which shows that the dimensions of the three conditions are 4, 4, and 3, respectively.

Page 434 434 University of California Publications in Mathematics [VOL. 1 It is to be noticed that we have taken a case where k does not satisfy the provision k<n/2 of formula I. In such a case, however, theorem 1 is applicable. For suppose 2k = n+j, (j O). Then by theorem 1, g2[0 k]=g[2k+l-n kk] =g[j+l kk(n)] =g[O k-j-1 k-j-1 (n-j-1)] =g[O k-j-1 (n-j-1)]. Formula I is now applicable to this last condition, since k-j 1=n j3 <n-j-1 22 2 The Incidence Formulae It is necessary for our purposes to develop a formula for the enumeration of the geometric forms which consist of a line and a point on the line. In this paragraph the word 'form' will be used to designate this figure. The point will be called 'the point of the form' and denoted by g. p will also express the condition that the point p lie in a given hyperplane, and denote the number of forms whose points satisfy this condition. In accordance with these definitions and the notation for a line condition, pg[O k] denotes the number of forms each of which has its point in a given hyperplane H, and its line intersecting a given space M of dimension n-k-1. Let the space M be taken in the hyperplane H. Then the lines satisfying the condition pg[O k] consist of two groups. First, there are those forms whose points lie in M. These are the forms which satisfy the condition pk+l, since in general k+ 1 hyperplanes intersect in a space of dimension n-k-1, and hence the condition that a point lie in k+ 1 hyperplanes is equivalent to the condition that it lie in a space of dimension n-k-1. The second group of forms consists of those whose lines lie in H and intersect M, and which consequently satisfy the condition g[l k]. Thus for this special position of the spaces M and H II pg[O k] = pk++g[l k] and consequently by the principle of the permanence of number this relation holds for an arbitrary choice of the spaces of the condition. It will be noticed that if this formula is applied to space of three dimensions it becomes for k = 1, and k = 2, the incidenz-formeln I and II respectively of Schubert.9 Thus the notation here introduced makes it possible to express these two formulae in one. For space of n dimensions the formula above includes the n-1 formulae for which k= 1, 2, 3,.... n-1. 9 Loc. cit., pp. 25 and 26.

Page 435 1924] Steed: Hyperspace of the Lines on the Cubic Surface 435 As an application of the above theorems and formulae consider the n-dimensional analogue of the problem in space of three dimensions of determining the number of lines which intersect four given lines. In the present notation the number for space of three dimensions is denoted by g401, and its value is well known to be two. Now in space of n dimensions the condition g2n-201 is of dimension 2n-2, as shown by the formula for the dimension of a line condition. Hence a finite number,of lines satisfy this condition. Consider the case n = 4. g601 = (g201)3 = (g111 +g02)3 = (g1+g902)3 Formula I and the rule preceding theorem 1. = g3 + 3g21g02 202 + gig20+g302 = g3+ 392go2+ 3g022 + g302 Theorem 2. But g2g2 = g24 by theorem 2 and g24 =0 by definition 2. By theorem 2, 91g022 = g133 = 9022(3) = 1. g302 = g02 (g122 + 04) = g1322 + g06 = 1 by definitions 1 and 2. Hence g60 =1+3 - 0+3 1+1=5, or the number of lines which in space of four dimensions intersect six planes is five. For space of five dimensions g801 = (gl+902)4 = g41+4g310g2+6g21922 2+4g31l02+402. Now g41 = 4 =, 93192 = g3902 = g35 =0 by definition 2, and 2lg2o2 = g2g022 = 202(3) = 1 91g302 = g1333 = g302(4) = 1 (See above.) g402 = (9122 + 904) 2 = 923333 + 291522 + 9044 =9401(3)+0++4 (by Theorem 1). =2+1. Hence gso0= 1+0+6+4+3 = 14, or the number of lines which in space of five dimensions intersect eight given linear spaces of dimension three is fourteen. The same method is applicable to the analogous enumerative problems for space of any dimension. IV. THE LINES ON THE HYPERSURFACE OF ORDER 2n-3 In space of n dimensions, the order of infinity of forms consisting of a line and a point on the line is 2n-1. For any line contains a single infinity of these forms, namely, any point on the line can be taken as the point of the form and the line itself is the line of the form. Thus, since there is a 2n-2 fold infinity of lines in space, the order of infinity of the forms under consideration is 2n- 1. Denoting by p the condition that the point p lie in a given hyperplane, then if p is the point of the form mentioned above, this condition is a condition imposed upon the form. It is required to determine the dimension of the condition p. Let P be

Page 436 436 University of California Publications in Mathematics [VOL. I any point of the hyperplane H of the condition p. Then the form which consists of the point P and any line through P satisfies the condition. Hence there is an n-l fold infinity of forms whose point is P. The order of infinity of the points P being n-1, it follows that the order of infinity of forms satisfying the condition is 2n-2, and consequently the dimension of the condition is one. Consider the form which consists of a line and the r points of intersection of the line with a general hypersurface of order r. The word 'form' will in the remainder of this article refer exclusively to this figure, excepting that later r will assume the particular value 2n-3. The line of the form will be denoted by g, and the r points by pi, p2, P3,... pr. In any line in space the number of forms is r!, there being one form for each of the r ways that the letters can be assigned to the r points of intersection of the given line with the hypersurface. Hence the order of infinity of forms in space is the same as that of lines, namely, 2n-2. The product plp2p3.... pr expresses the condition that the point pi lie in a given hyperplane H1, that p2 lie in a given hyperplane H2, etc., that Pr lie in a given hyperplane Hr, and that the r points be distinct. Consider in particular the case in which the given hypersurface, which will be denoted by F, is of order 2n-3. Then the dimension of the condition above being r, there is a single infinity of forms satisfying the condition P1p2P3.... p2n-3. The lines of the forms satisfying this condition are therefore the rulings of a surface10 which will be denoted by V. The order of V is the number of points of intersection of V with a space of dimension n -2, and is equal to the number of rulings of V which meet a space of that dimension. The order of V is therefore the number of forms which satisfy the condition goiplp2p3.... P2n-3 and is itself expressed by this symbol. Let H be any hyperplane distinct from the hyperplanes H1, H2, H3,.... H2n-3. Then H intersects the hypersurface F in a variety f of order 2n-3, and intersects V in a curve C of order go0plp2p3.... 2n-3. By Bezout's theorem the curve C has (2n-3)golplp2p3 P. 2n-3 points in common with f. Each of these points is a point on the given hypersurface F, and through each of them there is a ruling of V. Let P be any one of these points and let 1 be the ruling that goes through it. Then I meets F in the 2n-3 distinct points pl, p2, P3,.... p2n-3 and also in P. Thus the line has 2n-2 distinct points in common with the hypersurface of order 2n -3 and will therefore be entirely on it, provided that P is distinct from each of the distinct points px(X= 1, 2, 3.... 2n-3). Now if P coincides with pi, then pi satisfies the condition (expressed by p21) of lying in both H1 and H. And the form whose line is I must therefore satisfy the condition p21P2P3.... p2n-3. Similarly if P concides with p2 the form must satisfy the condition plp22P3.... p2n-3, and so on. Hence if Xn is the number of lines on F, then Xn = (2n-3) go1plp2p3.... P2n-3-P21P3p3.....P2n-3 - pp 2 p3.... P2n- 3-plP2P3.... 2n-3. 10 The word 'surface' will be used for a variety of dimension two, whether or not the variety is in a hyperplane.

Page 437 1924] Steed: Hyperspace of the Lines on the Cubic Surface 437 Since in this equation the terms following the first are all equal to each other, Xn= (2n- 3)gOlplP2P3.... p2n-3- (2n- 3)p21p2p3.... P2n-3. Now for k = 1 the incidence formula II becomes goipl = p21 +gl And when both members of this equation are multiplied symbolically by p2p3.... P2n-3, we have gOlPlP2P3... P2n-3=P21P2P3.. P2 n-3+glP2P3P4.... P2n-3. On account of this relation the expression for Xn becomes X= (2n-3)p2 p2p3.... P2n —3+ (2n-3)g9p2p3p4... * P2n-3 -(2n-3)p2p2p3.....p2n-3 or Xn=(2n-3)glplp2p3. *. P2n-4. Denote glpip2p3.... P2n-4 by Pn and the hyperplanes of the conditions gl and p(K = I, 2, 3,.... 2n-4) by Sn-_ and HK respectively. Then the hyperplanes HK intersect Sn- in spaces of dimension n-2 which are hyperplanes in the space Sn,_ considered as a space of operation. On account of the general nature of the investigation, the intersection of Sn-_ with F is a general hypersurface, f, of order 2n-3 in space of n- 1 dimensions. And pn is the number of forms (each consisting of a line and the 2n-3 points in which it intersects f) which in Sn-_ have 2n -4 points p, in 2n-4 given hyperplanes hK respectively. Hence from the last equation above follows the

Page 438 438 University of California Publications in Mathematics [VOL. 1 Theorem. On the general hypersurface of order 2n-3 in space of n dimensions there exist a finite number of lines, the number of these lines being (2n-3) pn where p, is defined as follows: In a space of operation of dimension n-1 take a general hypersurface of order 2n-3, then Pn is the number of lines which meet 2n-4 given hyperplanes in 2n-4 distinct points, each of which is also a point on the hypersurface. By means of this theorem the problem of determining the number of lines on the cubic surface is reduced to a simple problem in plane geometry. Let the line p meet the non-singular plane cubic curve C in the points Pi, P2 and P3, and let the line q meet C in Qi, Q2 and Q3. Then p3 is the number of lines which can be drawn to meet p and q in points which are also on C. But the 9 lines PiQ, where i, j= 1, 2, 3, and only these lines satisfy this condition. Hence p3 =9 and the number of lines on the cubic surface is X3 = (2n-3)p3=3X9=27. V. RECURSION FORMULAE FOR THE FUNCTION Ln (r) The number of lines which meet 2n-2 hyperplanes in points which are also on the hypersurface F of order r is finite and is a function of r. This function will be denoted by Ln(r), or more briefly by Ln. On account of the theorem of the preceding section, (2n- )Ln(2n- 1) = Xn+ and (2n-3)Ln-_(2n-3) = X, where Xn is the number of lines on the general hypersurface of order 2n-3 in space of n dimensions. The remainder of this paper will be devoted to the determination of recursion formulae by means of which the function Ln can be determined for successive values of n, and to the actual determination of these functions for particular values of n. The word 'form' from now on will denote the figure consisting of a line g and 2n-3 of its intersections pi, p2, pa.... pr with a general hypersurface F of order r, where r>2n-4. In any given line there are Pr, 2n-3 or r(r-1) (r-2) (r-3).... (r-2n+4) such forms, namely, one for each of the ways that the letters pi (i= 1, 2, 3,.... 2n-3) can be assigned to the r points of intersection of the given line with F. The following abbreviations will be used: giplp2p.... p2-2i —k = qn-i (2i+k > 1) gikplp2p3.... p2n —i-k-2 = qn-i k-i (i+k >0).

Page 439 1924] Steed: Hyperspace of the Lines on the Cubic Surface 439 The reasoning of the preceding section carried out for the hypersurface of order r instead of 2n-3 gives Ln= rqn_l+ (r-2n+3)qn or (1) Ln=r(r-2n+4)Ln_+ (r-2n+3)qn. We shall now derive an expression for qk (k > 1). In any line g which contains a form satisfying the condition qn, there are Pr-2n+k+1, k-i forms satisfying the same condition, namely, one for each of the ways that the k- 1 points, which in addition to the points pl, 2, P3.... p2n-k-1 constitute the points of the form, can be selected from the r-2n+k+ 1 remaining points of intersection of g with F. Hence the number Qk of distinct lines which contain forms satisfying qk is Qk -qn k Pr-2 n+k+l, k-1 Similarly the number of lines containing forms satisfying the condition qnk is qnk Qnk =- nk Pr-2n+k+2, k Let the linear space of intersection of the k hyperplanes of the condition pk be denoted by Sn-k, and let Hi denote the hyperplanes of the conditions pi (i= 1, 2, 3,.... 2n-k-1) respectively. The lines satisfying the condition p2p3p4.... P2n-k-1 are the k fold infinity rulings of a variety V, whose order is the number of rulings which meet a space of dimension n-k-1, and is therefore equal to Qnk. Sn-k intersects V in a curve C. For the rulings of V which meet Sn-k satisfy the condition go k-i p2p3p4.. P2 n-k-1, and since the dimension of this condition is 2n-1, a single infinity of forms satisfies it. The points in which these lines meet Sn-k are the points of C. The order of C, being the same as that of V, is Qnk. Hence by Bezout's theorem, C intersects F in rQnk points. Let pi be any one of these points. Through pi there goes a ruling of V which meets Hi in pi (i = 1, 2, 3,. 2n-k- ), and which is the line of a form satisfying the condition qk, provided that pl does not coincide with p2, or with p3, etc. If pi coincides with p2 then g is the line of a form satisfying the condition p +lp3p4.... p2n-k-2; if pi coincides with p3 then g is the line of a form satisfying the condition Pk+lp2pP4.... P2n-k-2, etc. The number of lines containing forms satisfying qnk is therefore equal to rQnk decreased by 2n-k-2 times the number of lines containing forms satisfying q+l. Hence the equation Qk = rQnk-(2n-k 2) Q+1. If both members of this equation be multiplied by Pr-2n+k+l, k the result is (r-2n+k+2)qk= rqnk- (2n-k-2)qk+l.

Page 440 440 University of California Publications in Mathematics [VOL. 1 On account of the incidence formula k r -m.l/+1 90gokpl = glkpl +Pl, we can replace qnk by q(n-l)(k-l)+ q +and obtain (r-2n+k+2)qk= rq(n-l)(k+-1) (r-2n-k —2)qk+1. Successive applications of the formula k-i +1 q(n-i) (k-i) = q(n-i —) (k-i-1) -+ qn=-1 lead to the result (3) (r-2n+k+2)q= rq(n-d) (k-d)+ (r-2n+k+2)qkn+r n — + (i=d-1, d-2,.... 3, 2, 1) For d = n- 2 this becomes (4) (r-2n+k+2)qk= rq(2) (k-n+2)+ (r-2n+k+2)qk +l +rq -+,+l, (i=n-3, n-4,.... 3, 2, 1). In particular placing k = n-1, we have i (5) (r -n+ 1)qn- = rq2l+ (r-n+ l)qn+r z qnn-3 Now if P is the point of intersection of the n hyperplanes of the condition pI, then P is not in general a point of F, and hence n=0, and similarly -i = 0. = I n-i Referring to the definition of the abbreviated notation (p. 438) it will be seen that q21 is the condition that the form lie in a given plane, have its line passing through a given point in that plane, and have one of its points in a given hyperplane H, and therefore in the line of intersection of H with the plane. Denote the plane by a, the given point by P, and the line Ha by a. a meets F in a curve f of order r. If A is any one of the points of intersection of a with F, then the line PA is a line containing Pt-i, 2n-4 forms satisfying the condition. There being r such lines, it follows that q2 = rPr-1, 2n-4 = Pr, 2n-3. Hence from (5) (6) (r-n+ 1 )q- = rPr, 2 n-3. In similar manner by means of equation (4) the value of qn-2 can be found in terms of r and n. For k=n-2, (4) is (7) (r - n) q — rq2+(r-n)q-+r - qn-i n-3 Now q2 is the number of forms which lie in a given plane and have one point in a given line in that plane and a second point in a second given line in that plane. Hence q2=r2Pr2, 2n-5. If the symbol n —i- were defined for the form consisting of a line and 2n-2i-3 of its intersections with F, (5) would be applicable, and the value of the symbol would be rP, 2n-2i-3- (r-n-i+ 1). But for every form satis

Page 441 Steed: Hyperspace of the Lines on the Cubic Surface 441 fying the condition defined in this way there are Pr-2n+2i+3, 2i forms satisfying the condition expressed by the same symbol but defined for the form consisting of a line and 2n-3 points of intersection with F, namely one for each of the ways that the 2i additional points can be selected. Hence (r -n+i+ 1)q n=- = rPr, 2n-2i-3 Pr-2n+2i+3, 2i =rPr, 2n —3 Applying this result and the value for q2 to equation (7) we obtain 1 (8) (r-n)qn-2=r3P,-2, 2-5+ r(r- Prn-3+r2 P 2-3 r-n+1 r-n+i+l n-3 1 =r3Pr-2, 2n —5+ Pr 2n-3+rPr, 2n-3 r-n+1 r-n+i+l n-3 Proceeding in a similar manner it would be possible to obtain formulae for q-3, for qn-4 and indeed for q~-k for any particular value of k. It is evident, however, that the expressions become quite complicated as k increases. For the determination of the functions Ln for particular values of n it will perhaps be simpler to revert to the recursion formula (3). In applying the above formulae it will be found necessary to distinguish between the symbol qk when defined for space of n dimensions and the same symbol when defined for space of any other dimension say of d dimensions. We adopt the notation Q[d for the condition qk defined for a space of operation of dimension d, the letter or number following the bracket indicating the dimension of the space of operation. When no bracket occurs in the symbol it will be understood that the symbol is defined for space of n dimensions. It is to be noted that the symbol qd refers to the form consisting of a line and 2n-3 of its intersections with a general hypersurface of order r, whereas the symbol qk[d has reference to the form which consists of a line and 2d-3 of its intersections with the general hypersurface of order r. Referring to the definitions of the abbreviated notation (p. 438) it will be seen that qd = gn-dP1P2P3.... 2d-k —i (k > 1). The forms satisfying this condition therefore lie in a space of dimension d. Now for each form satisfying the condition qk[d there are Y =Pr-2d+3, 2n-2d forms satisfying the condition qk, namely one for each of the 7 ways that the 2n-3-(2d-3) =2n-2d additional points can be selected from the r-2d+3 points other than p, P2, p3.... p2d-3 in which the line of the form intersects F. Hence (9) qd=Pr-2d+3, 2n-2d q[d (k > 1). In similar manner it can be shown that qd = Pr-2d+2, 2n-2d —1 Ld(r).

Page 442 442 University of California Publications in Mathematics [VOL. 1 VI. THE DETERMINATION OF Ln(r) FOR n=3, 4, 5, AND 6 The Function L3 (r) On account of (1) and (6) respectively, L3(r) =r(r-2)r2+(r-3)q23 and (r-2)q23=r2(r-1)(r-2), and hencel, and hence~" L3(r) = 2r4-6r3+3r2 The Function L4(r) For n=4 (1) becomes L4(r) =r(r-4) (2r4- 6r3+3r2) + (r - 5)q24 For the determination of q24, (8) can be used with the result q24 = 3r5-16r4+23r3 -8r2. Consequently12 Consequently2 L4(r) = 5r6 -45r5+ 130r4- 135r3+40r2. The Function L5(r) L5(r) =r(r-6)L4(r) + (r-7)q25. For d = k = 2, n = 5, equation (3) becomes (r - 6)q25 = rq3 (r - )q35 +rq24 where each of the q's is to be interpreted for space of operation of five dimensions. The value of q35 obtained by means of (8) is q35 = 4r7- 59r6 +310r5- 705r4 +660r3 - 180r2 By means of (9) and (10) q24 and q3 can be found from the values of q24[4 and L3(r) respectively, which are given above. The results are rq3 = (2r7- 24r6+ 101r5 - 183r4+ 140r3) (r-6) and and rq24 = (3r7-31r6+ 103r5 - 123r4+40r3) (r -6). Substituting these values, together with the value of L4 in the equation above for L5, we obtain L5(r) = 14r8 - 252r7+ 1712r6 - 5524r5+8767r4- 6300r3 + 1260r2. 11 Cf. Schubert, Abzdhlenden Geometrie, p. 236. 12 The expression here denoted by L4 was given by Schubert without proof in a footnote to p. 72 of Math. Annalen., vol. 26.

Page 443 1924] Steed: Hyperspace of the Lines on the Cubic Surface 443 The Function L6(r) For the determination of L6 equation (1) gives L6(r) = r(r- 8)L5(r) + (r- 9)q26. Equation (3) becomes for n = 6, k = d = 2, (r- 8)q26 = rq4+ (r- 8)q3+rq25 and for n = 6, k=d = 3, (r — 7)q36 = rq3+ (r - 7)q46+rq24+rq35 and from these it follows that (r-7)(r-8)26 = r(r-7)q4+r(r-8)q-+ (r- 7) (r- 8)q46+ r(r- 7)q25 r(r — 8)q24 +r(r-8)q35 where each of the q's is to be interpreted for space of six dimensions. Applying formulae (9) and (10) the equation for q26 becomes 26 = r(r -6) (r-7)L4+r(r - 4) (r -5) (r -6) (r- 8)L3+q46+r(r- 7)q25 [5 +r(r-5) (r-6) (r-8)q24[4+r(r- 8)q35[5 The values of all the q's in the right member of this equation have been determined above, with the exception of q46 the value of which by equation (8) is q4 = 5r9- 141r+ 1585r7- 9081r6+28114r5-45714r4+34304r3- 8064r2. Hence q26 = 28r9 — 644r8+ 5814r7 - 26300r6-+63078r5 - 77952r4+44492r3 - 8064r2 and consequently L6(r) = 42r10 - 1260r 9 + 15338r - 97846r7+ 352737r6 - 722090r5 + 797720r4 -418572r3+72576r2. On account of the theorem of Part IV, 4 = 5L3(5) =23 ~ 53 = 2875 X5= 7L4(7) =2035 ~ 73 =698005 X6= 9L5(9) =419949 ~ 93 =306142821 X7= 11L6(11) = 158557045 113 = 211039426895. The results may be stated as follows: On the general hypersurface of order five in space of four dimensions there are 2875 lines; on the general hypersurface of order seven in space of five dimensions there are 698,005 lines; on the general hypersurface of order nine in space of six dimensions there are 306,142,821 lines; and on the general hypersurface of order eleven in space of seven dimensions there are 211,039,426,895 lines.

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS IN MATHEMATICS Vol. 1, No. 20, pp. 425-443, 1 figure in text May 17, 1924 THE HYPERSPACE GENERALIZATION OF LINES ON THE CUBIC SURFACE BY DANIEL VICTOR STEED UNIVERSITY OF CALIFORNIA PRESS BERKELEY, CALIFORNIA THE

Page [unnumbered] UNIVERSITY OF CALIFORNIA PUBLICATIONS Note.-The University of California Publications are offered in exchange for the publications of learned societies and institutions, universities and libraries. Complete lists of all the publications of the University will be sent upon request. For sample copies, lists of publications or other information, address the Manager of the University of California Press, Berkeley, California, U. S. A. All matter sent in exchange should be addressed to The Exchange Department, University Library, Berkeley, California, U. S. A. MATHEMATICS.-Derrick N. Lehmer, Editor. Price per volume, $5.00. Cited as Univ. Calif. Publ. Math. Vol. 1. 1. On Numbers which Contain no Factors of the Form p(kp+ 1), by Henry W. Stager. Pp. 1-26. May, 1912.................................................. $0.50 2. Constructive Theory of the Unicursal Plane Quartic by Synthetic Methods, by Annie Dale Biddle. Pp. 27-54, 31 text-figures. September, 1912.............................................................................................................50 3. A Discussion by Synthetic Methods of Two Projective Pencils of Conics, by Baldwin Munger Woods. Pp. 55-85. February, 1913.......50 4. A Complete Set of Postulates for the Logic of Classes Expressed in Terms of the Operation "Exception,' and a Proof of the Independence of a Set of Postulates due to Del R6, by B. A. Bernstein. Pp. 87-96. M ay, 1914.....................................................................................10 5. On a Tabulation of Reduced Binary Quadratic Forms of a Negative Determinant, by Harry N. Wright. Pp. 97-114. June, 1914...............20 6. The Abelian Equations of the Tenth Degree Irreducible in a Given Domain of Rationality, by Charles G. P. Kuschke. Pp. 115-162. June, 1914.............................................-....................................50 7. Abridged Tables of Hyperbolic Functions, by F. E. Pernot. Pp. 163 -169. February, 1915....................................................................................... 10 8. A List of Oughtred's Mathematical Symbols, with Historical Notes, by Florian Cajori. Pp. 171-186. February, 1920......................25 9. On the History of Gunter's Scale and the Slide Rule during the Seventeenth Century, by Florian Cajori. Pp. 187-209. February, 1920.35 10. On a Birational Transformation Connected with a Pencil of Cubics, by Arthur Robinson Williams. Pp. 211-222. February, 1920...................15 11. Classification of Involutory Cubic Space Transformations, by Frank Ray Morris. Pp. 223-240. February, 1920............................................25 12. A Set of Five Postulates for Boolean Algebras in Terms of the Operation "Exception," by J. S. Taylor. Pp. 241-248. April, 1920...15 13. Flow of Electricity in a Magnetic Field. Four Lectures, by Vito Volterra. Pp. 249-320, 40 figures in text. May, 1920.................................. 1.25 14. The Homogeneous Vector Function and Determinants of the P-th Class, by John D. Barter. Pp. 321-343. November, 1920..............................35 15. Involutory Quartic Transformations in Space of Four Dimensions, by Nina Alderton. Pp. 354-358. November, 1923.........................................25 16. On the Indeterminate Cubic Equation x3 + Dy3 + D2z3 -3 Dxyz =1, by Clyde Wolfe. Pp. 359-369. December, 1923.........................25 17. A Study and Classification of Ruled Quartic Surfaces by means of a point-to-line Transformation, by Bing Chin Wong. Pp. 371-387. November, 1923......................................................25 18. A Special Quartic Curve, by Elsie Jeannette McFarland. Pp. 389-400, 1 figure in text. December, 1923................................................................25 19. A Study of Cubic Surfaces by Means of Involutory Cubic Space Transformations, by John Frederick Pobanz. Pp. 401-423. January, 1924.25 20. The Hyperspace Generalization of the Lines on the Cubic Surface, by Daniel Victor Steed. Pp. 425-443, 1 figure in text. May, 1924........25 PHILOSOPHY.-George P. Adams and J. Loewenberg, Editors. Volumes 1, 2, 3, and 4 are completed. Volume 5 is in progress. Vol. 1. Studies in Philosophy prepared in Commemoration of the seventieth birthday of Professor George Holmes Howison, November 29, 1904. 262 pages. 1. McGilvary, The Summum Bonum..................................................................... $0.25 2. Mezes, The Essentials of Human Faculty......................................................25 3. Stratton, Some Scientific Apologies for Evil..............................................15 4. Rieber, Pragmatism and the a priori............................................................. 20 5. Bakewell, Latter-Day Flowing-Philosophy..-........................ —...20 6. Henderson, Some Problems in Evolution and Education............10 7. 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