Course in elementary physics. By Charles R. Cross.

('OURS 1IN EL EM ENTA LY P HYSIC(S. BY CHARLES R, CROSS, Assistan*t Professor of Physics in the Massachusetts Institute of Technology. B3 O STON: PRESS OF A. A. KINGMAN. 1878. i~~~~

COURSE IN ELEMENTAR Y PH1YSICS. BY CHARLES R. CROSS. Assistant Professor of Physics in the M;AassachLsetts Institute of Technology. BOSTON: PRESS OF A. A. KINGMAN. 1 878).

Entered according to Act of Congress, in the year 1873, by CHARLES R. CROSS, in the office of' the Librarian of Congress, at Washington.

PREFACE. The present pamphlet comprises the text of the first few chapters of an elementary work on Physics, designed to supplement the lectures delivered to the students of'. the Massachusetts Institute of Technology during the first two years of their course. It is intended to impart such a general knowledge of the science as will enable them with advantage to continue the pursuit of the subject in the more advanced work of the Physical Labo('ratory. It will be the aim of the author to present the student with a concise statement of the leading principles of the science, and to explain the meth(ls of investigation by which they have been ascertaine(l. In the preparation of the work constant reference has been made to the or'iginal memoirs as well as to the standard text books upon Physics. In the chapters relating to mechanics, the Elements of llechanical Philosophl, by Prof. W. B. Rogers, long out of print, has been of the greatest service, and valuable hints have been derived froln the elenmentary works of Smith, IPeck and Mayer. The text-books of Thomson and Tait, Rankine, Todhunter and Price have furnished many suggestions. A list of standard text-books upon General Physics will be added hereafter, as well as a collection of physical tables. The references appended to various chapters are not intended to be exhaustive, but may serve as aids to the student who is desirous of' investigating the subject more thoroughly. The work will be put in a more permanent fbiorm, and properly illustrated, as soon it reaches a suitable stage of- progress. The present shellet. have been printed for the temporary use of the students. Boston, January, 1873.

ELEMENTARY PHYSICS. CHAPTER I. GENERAL METHODS OF PHYSICAL SCIENCE. 1. Physical Sciences. All those branches of human knowledge which have for their end the study of material objects and the phenomena of the external world, are included among the Physical Sciences. They may be divided into two general classes; first, those which aim simnply at a classification of objects, and, secondly, those which endeavor to ascertain the laws of the phenoimena of the material universe. The former class comprehends the various branches of Natural History. It includes Zoology, which describes and classifies the various forms of animal life; Systematic Botany, dealing with the classification of plants; Anatomy, both vegetable and aniintal, describing the structure of the organs of living things; and Mineralogy, which concerns itself with the constitution of the rock amasses of the globe. The second class comprises Mechanical Philosophy or Physics proper (of which Astronolny is a branch) and Chlemistry, which investigate the general laws of natural actions; Dynamical Geology, which studies the laws regulating the structure of the earth, and Physiology, whose province is the dcleteirniiation of the filoctions and mode of action of the orgtans of animals and11 pla.-nts. 2. Physics. In its orliginal and most extended sense, Physics, or NIatural Philosophy, as it is sometimes called, comprehended the entire range of physical as distinguished from mental science. The term is still occasionally used in a comprehensive manner to denote the second class of' sciences of which we have spoken. 1

2 GENERA L METHODS. In its limited sense, and the one in which it is usecd in this work, it is applied only to that branch of science which investigates the laws of physical actions in wbhich there is no permanent change in the general properties of the body acted upon. It is thus distinguished frol Chemistry, which deals with actions in which the properties of the body affected are perm:anently modified or entirely changed. A few illustrations will sulfice to make this plain. An unsupported body falls to the earth, Here we have a physical action in the motion produced, but the ge neral properties of the body'are not altered thereby, for it has- the same weight, color and other characteristics as, before. The Ii'wx of its downward motion is then a subject lying owithin the proviznce of physics. If the body is elastic it rebounds on reachirtg thle earth; there is still no change of its general properties, so that the laws of elasticity are physical and not chemlical. The motion ensuin.g when a piece of iron is brought near to a magnet, and the vibrations of a musical strihng, are additional examples. On the other band, if a spark be appliec to a mixtare of oxygen and hyChdiogen they unite with an explosion, forming an entirely new substance, water, possessing properties totally different from those of either of the gases composing it. This, then, is a chemical chalge. Again, iron when exposed to moist air rusts, because of the ulnion of oxygenx with the metal; a new body, oxide of iron, is, thus obtained, unlike either constituent. The laws of such changes are chemical laws. As would naturally be expected, there are many cases in which it is difficult to decide whether a given action is chemiell or physical, or a mixture of both. W~e also have occasion to examine the physical effects of chemical changes, so that there are lines of investigation in which it is difficult to decide where one ends and the other begins. 3. Deductive and Inductive Sciences. The various sciences may be divided into two general classes, according to the mode of reasoning employed in the investigation of the facts pertaining to them. These-methods are known as cect lctive and inductive recasoning, and the sciences characterized by their respective use as the Deductive and the Inductive Sciences. Deductive reasoning starts with certain general fundamental facts and definitions, and from these evolves truths which follow necessarily from the premises. Induction observes particular facts, and by analyzing and comparing them rises to more general truths. Deduction reasons from the general to the particular; induction from the particular to the general. Pure Mathematics furnishes us with an example of a deductive science. Certain definitions are laid down, anld froniom these in connection with the axioms, or fundamental truths relating to magnitude, the whole science is evolved without the necessity of going outside of the mind itself for any portion of the demonstration. The most abstruse theo

LOGIC OF INDUCTIVE SCIENCE. 3 rems follow directly from the fundamental simple ones. The Physical Sciences, on the contrary, are based on induction. Their end is to form a natural classification of objects, and to ascertain the general laws of the forces acting ulponl matter, and in no way can this be done except by the comparison of numerous particular cases. Hence we must reason from these to general laws. For example, a natural classification of animals can be attained only by observing the different' species, noting their fundamental points of agreement and dlifference, and thus arranlging them in groups connectedl with each other by certain structural peculiarities, which at the same time separate them firom all other anirals. Having thus briefly stated the difference between these two kinds of reasoning, let us consider more fully the process of investigation by which the general laws of the physical world are ascertained and logically arranged. 4. Physical Law, Theory, Hypothesis. We shall fiequently have occasion to use the terms law, theory and hypothesis. By a physical law is meant simply the constant method according to which a cause acts. "It is the eustom of philosophers whenever they can trace regularity of any kind to call the general proposition which expresses the nature of that regularity a law, as when in mathematics we speak of the law of decrease of the successive terms of a convereging series." I A theory is a tirue, complete and philosophicalf explanation of connected phenomena and their laws; as, for ex;lllple, the theory of gravitation or the theory of sound. A hypothesis is a supposition as to the nature of any law or cause macle in the absence of absolute proof; as, for example, the hypothesis of an electlic fluid. When a hypothesis is extended to include a number of phenomena, and is proved. to be true, it becomes a theory. The term theory is frequently applied more loosely than ill the above definition to denote any plausible and clearly defined hypothesis. 5. Observation and Experiment. The facts whichh form the oasis of physical science are ascertained in two ways, by observationm amd by experiment. Observation is the careful ekarnination of phcnllomena as they occur in the order of nature; experiment is the pro luction of phenomena by causes under our own control. It is evident that the latter method of investigation is a great extension of the former, and will often enable us to ascertain laws which could never be known without its aid. To illustrate by an example, suppose that we wish to know which of the components of the atmosphere, oxygen or nitroenl, enables it to support animal life. To ascertain this two processes are open. We may (1) notice which constituent of the altmnosphere disappears when an 1 This quotation, together with several others in the present chapter, is from Mill's rsductive Loic.

4 GENERAL METHODS. animal is immersed in a limited quantity of air, or (2) we may immerse animals in oxygen and in nitrogen and compare the effects. Both of these methods require us to perform an experiment, and as nature does not furnish either oxygen or nitrogen in a pure state, and also as the oxygen consumed by animals from the atmosphere is replaced by that given out by plants, we could never have ascertained the truth in question fiom pure obselrvation. Indeed, without the process of experimentatiol we could not even know of the existence of two ingredients in air. In some sciences, however, as Astronomy, the former method is the only one possible. 6. Induction. Having by these methods examined a large number of phenomena, we next approach the process by which we determine the causes of these phelnomeno a and their laws of action. The great logical principles upon whlich our procedclne is based are (1) that "every fct wVhich has a beginning has a cause," anlc (2) that "the course of nature is uniform," the first of these being an intuitive idea, the second a truth -ascertained from universal experience. Hence, when inder certain influencing circumstances a given event occurs, we are entitled to infer that it will recur under the samne circumlstances, an(1 it is thlle first object of science to ascertain exactly what these circumstances are, for they are the cauzses of the phenomena. In physical science we limit ourselves strictly to what are known as physical cqatses, that is the phenomena which are invariably and necessarily followed by some other phenomenon. The criterion by whicth we know that any combination of phelomena is the cause of another phenomenon is, that between the first and last there is inviariable and?u'cozclitio2cal sequence. It is not sufficient that the sequence be invalriable, for in this case both events might be consequents of a single cause. Thus day and night invariably follow each other, but do not stand in the relation of c1ause and effect. The antecedence of the suppose( cause must also be unconditioned, that is, its relation to the effect must be demonstra'bly such that without it the effect could not be produced. This -lemonstration can be given only by experiment ori by the process of deduction. The constant endeavor of science is to find the smallest number of causes that wrill account for all natural phenomena, or as Mill expresses it, to find 1" the fewest and simplest assumptions which, being taken for granted, the whole existing order of nature would result," These primary assumptions we call Laws of Nature. The causes of the phenomena studied by the processes of observation and experiment are to be ascertained only by a rigorous analysis and comparison of the circutlstances attending their production. By investigating a nurmber of isolated cases and ascertaining those circumstances which are present in all, we obtain a more general result, which we imay iinfer to be true in analogous instances, even though these may be beyond the range of our present observation.

LOGIC OF INDUCTIVE SCIENCE. 5 A single exanmple will illustrate this. Explerimental tests show thlat wbhen oxygoen ancd hydrogen unite to form water, the proportions are invariably 8 parts by weiglht of oxygven to 1 of hydrogen. So also when chlorine and hydrogen unite to form ebllorhydric acid, the invariable proportion of the constituents is 35.5 parts of the former to I of the latter. In all other cllemical comrnlounds which we have examined, a like constancy of proportion is olserved. HIence we lay it down as a general law, that the )proportions in which substaaces unite to form a clleical compound are fixed, definite and invariable;,and are justified in predicting tlat the law will hold for any unknown chermical compound that imay be discovered in the future. There are several processes too often confounded with induction which should be carefully (listinuuishedl fi om it. (1.) Any logical operation wvhich contains nothing in its conclusion which is not stated in the premises from which it is drawn, i. e., in -which the conclusion is not wider than the premises, is not properly induction. Thus were we simply to say that all chemlical compounds wshich we have excmied combine in definite proportionsl this would not be an induction. It would be merely a short way of expressing the several facts already known, that oxygen and hyclroen, etc., combine thus. We perform a process of inductive reasoning only when we infer fi'om t'he known instances to all instances of chemical colnbination. (2.) Ma'tlhemiatical proofs, thou~gh leading to general propositions, are not inductions, because there is no inference from known to unknown cases. Suppose, for example, that we have proved that the three angles of a trianllle are equal to two right angles. This is in reality only shown to be true of th0e particular triangle before us in our mind, or drawn on paper, but since we see that the same method of' proof will apply to any other triang-le we assume the theoreml to be a general truth. The reasoning is not,;'All the trianoles that I know have the sum of their angles equal to two right alngles, hence I infer that all triangles possess this property," but, " The method of reasonino, applied to this particular trianogle evidently applies to all possible trianiles, and hence the theorem is true of all." Mill suolgests that if this is to be allowved the title of induction at all, it should be called IndlUction by parity of reasonilg. (3.) Another process often confounded with induction is that which is called by Whewell the Colli/qarion of Facts, which'is a mere clescription of observed facts, and not an induction from them. To illustrate, when the astronomer Kepler announced as the result of his observations on the planet Maars that it moves in an ellipse, he simply expressed in that onle statement that the successive positions of the planet in the heavens lie in such a curve. There was no inference of unknowvn from known. Instead of saying' that on successive evenings Mars occupied certain definite positions, and stating thlese in order, lie said that these positions are such as would be occupied by a body revolving about the sun in an ellipse of known dimensions and position. 7. Deduction. The general laws of any series of phenomena once obtained by the inductive process, we are fiequently enabled to extend our knowledge very grleatly by the application of deductive reasoning. For assuming the tluth of the law in general, we

6 GENERAL METHODS. may calculate what effects should occur under given conditions of its action, and confirmn these calculations by direct experiment. Thus, to take a very simple instance, supposing the chemical law of definite proportions to be proved, on finding any new substance we may assume that if its constituents are chemic:ally united, their proportions are the same in all specimens, and if on analyzing different slinmples of it we find this is not the case, we may justly conclude that the union is not purely chemical, thus obtaining a new fact reogarding the substance. If the results of deduction coincide with those of observation and experiltent, the strongest confirmation of the theory in question is produced. In case of a wvant of coincidence, it does not necessarily follow that the theory is false, inasmuch as there may be certain circumstances which we have neolected to take into account in our calculations. Deduction often brings facts to our knowledge which it would be very difficult, if not impossible, to ascertain by inductive reasoning alone, since it directs us to the proper course of experimentation in any given case. In the "interrogation of nature" everything depends upon the question put, as the only answer given is, " Yes " or "No." Deduction infbrms us as to the question which we should ask. As an example in illustration, suppose that we wish to know the maximum amount of work which could be obtained with a perfect steam-engine burning a known quantity of fuel. We are incapable of answering this question by direct experiment until we can construct a perfect engine, and even in that case it would be impossible to prove it to be perfect by inductive melthods. If, however, we accept the truth of the law, that a given quantity of fuel when burnt can produce only a definite quantity of mechanical work, we have'an immediate answer, since a perfect engine would utilize all the heat produced by the combustion, and the equivalent of this heat expressed in units of mechanical work would evidently be the amount required. Our experimentation should therefore be directed to the obtaining of this equivalent. From what we have said it will be seen that the process of deduction to be complete must embrace three distinct operations; (1) Induction, (2) Ratiocination, (3) 5Terification. As the basis there must be some law or laws previously ascertained by induction as general facts from which the unknown particulars may be deduced. Hence to pursue such a deductive investigation we have first to learn the law of each separate cause which contributes toward the unknown effect.' Next comes the determination of the effects from a knowledge of the causes, which is an operation of reasoning often of the utmost complication, and which even in some of those cases apparently the most simple, is so baffling that we have as yet obtained only approximate results. Finally, every result of deductive reasoning should be verified by observation or experiment, that we may know whether in our calculations we have neglected to consider any of the attendant circumlstances. 1 The case of deduction from hypothesis is considered in ~ 9.

LOGIC OF INDUCTIVE SCIENCE. 7 8. Application of Mathematics. The methods of deduction assume an incalculable importance in Physics, because we are thus enabled to apply the processes of mathematics to determine what will be the result of the action of various concurring forces whose laws have l!ready been ascertained. The results thus obtained will be equally certain with the prior inductions on which the general laws are b::ised; provided, of course, that no elements are omitted from the calculation. It should, however, be observed that the remark in the preceding paragraph concerning apploximate solutions frequently applies to this use of mixed mathematics, since even the most subtle processes of the calculus often fail to furnish more than these. Those subjects to which mathematics is thus applied, are fiequently known as the Phbysico-mctthematiccal Sciences, and there are now few portions of Physics which have not a more or less developed mathematical theory. 9. Hypothesis. In the absence of certainty as to the causes or mode of production of phenomena, we frequently make use of hypotheses regarding thenm, and then by the application of deductive processes ascertain whether the result of these calculations coincides with the fitets obtained by observation. That is, in the course of reasoning explained in ~ 7, we substitute a supposition for the first process there mentioned, tha:t of a prior induction. If the supposition wit: which we st-:lrt is reasonable, this becomes a valuable addition to our methods for the discovery of truth. I;But it should be borne in mlind that the nmere fact of coincidence between observed phenomena and the results deduced firoml the hypothesis does not necessarily show that the assumed hypothesis is true. To prove this it must also be demonstrated that no other hypothesis will explain the kllown facts, and unless the supposition is of a nature to allow of such at demonstration, it is not perfectly satisfactory, since it is capable neither of being proved nor disproved. Any hypothesis, howeverl, if plausible, may be of great service as a guide to experimentation. 10. Explanation of Laws. In speaking of physical phenomena we constantly make use of the terms eoxplanation f phenomelna or explanation of lawcs. It is desirable that a clear conception should be formed of the meaning of these. Any fkct is said to be explained when the laws of causation on which its production depends are stated. In like manner, a law of nature is explained by showing it to be a particular case of a more general law. There are three methods of explanation of laws which apply in different cases. (1.) The first is that in which several causes unite to produce a joint effect equal to the sum of the effects which each would produce were it acting alone. In this case "the law of complex effect is explained by being resolved into the separate laws of the causes which contribute to it." Thus in the consideration of curvilinear motion we shall explain the law of the movement of a planet about the sun by showing the motion to be the result of the joint action of two forces, one of which is continually drawing the planet towards the sun, while the other continually

O8 GENEIRAL METIHODS, tends to cause it to fly off in a tangent to its orbit. The orbital motion is thus e:plaineel, as it is shown to result from forces with whose laws we are already fmlliliar. (2.) "A second case is when between what seemed to be the cause, and what was supposed to be its effect, further observation detects an intermlediate link; a faet caused by the antecedent, and in its turn causing the consequent, so that the cause at first assigned is but the remote cause, actino throungh the intermediate phenomena." The bleaching action of chlorine is explained in this manner. Between that gas and certain bases, specially hydroogen, there is an affinity so great, that chlorine will decompose most compounds of the metallic bases or hydrogen. These are constituent elements of almost all coloring matters, so that the latter are decomposed by chlorine, and their color caused to disappear. Here, the intermediate faclt is the affinity of chlorinle -for hydrogen. In like manner the efficacy of chlorine as a disinfectant is explained, since hydrogen is an essential component of the infectious matter. (3.) The thircd metlhod of explanation is what has been called tlle szubsumptioz of one law under another. It is "the gathering up of several laws into one more general law which includes theml all." As the best possible example of this we cite the generalization of' Newton, proving that the force acting to keep the planets in their orbits by tending to draw them towards the sun, and that causing a body to fall to the earth, are botl due to an attraction exercised by every particle of matter in the universe upon every other particle. If the student on reaching that point will carefilly read the section of this work treating of the history of the law of universal gravitation, in connection with the present chapter, he will find excellent examples of almost every process which we have described in the preceding pages, as the series of inductions which led to the establishment of that law is the most brilliant in the history of science. Our explanatioto s, as we call them, give us no insigllt into the inner causes of phenomena. "XWhat is called explaining one law of nature by another, is but substituting one llystery for another, and does nothing to render the general course of nature other than mysterious; we can no more assign a ~wh/ for the more extensive laws, than for the partial ones. The explanation may substitute a mystery which has becolie familiar, and has grown to seem not mysterious, for one which is still strange. Anld this is the meaning of explanation in common p'arlance. But the process with which we are here concerned often does the very contrary; it resolves a phenomenon with which we are fiimiliar, into one of which we previously knew little or nothing, as wlhen the common fi-ct of' the fall of heavy bodies is resolved into a tendency of all particles of matter toWVards one another. It must be kept constantly in view, thcrefore, that when philosophers speal of explaining any of the phienomlena of nature, they always mean (or should mean) pointing ourt not some more famnliar, but merely some more general phenomenon of' which it is a partial excmplification, or some laws of causation which produce it by their joint or successive action, and froiom which, therefore, its conditions may be determined deductively. EIvery such opteration brings s us a step nearer towards answering' the question, vwhich was stated some time ago as compreheniding the whole problenl of the investigation ot nature, viz.:'What are the fewest assumptions which being granted, the order of nature as it exists would be the result?'What are the fewest general propositions froim which all the uniformities existing in nature could be deduced? "The laws, thus explained or resolved, are sometimes said to be accounted

LOGIC OF INDUCTIVE SCIENCE. 9 for; but the expression is- incorrect if taken to mean anything more than what has been already stated. In minds not habituated to accurate thinkino, there is often a confused notion that the oeneral laws are the causes of the partial ones; that the law of' general gravitation, for example, causes the phenomenon of the fall of bodies to the earth. But to assert this would be a misuse of the word cause; terrestrial gravity is not an effect of general gravitation, but a cacse of it; that is, one of the particular instances in which that general law obtains.1 To account for a law of nature means, and can mean, no more than to assigrn other laws more general, together with collocations, which laws and collocations beinfg supposed, the partial law follows without any additional supposition." 2 The cause of these ultimate laws of action is to be found only in the design of' the ruling Mind of the universe. 11. Construction and Correction of Theories. In the nlathemeaticdal discussion of physical questions, we generally construct our theories by first considering only the simpler and more essential elelnentary forces acting to produce the phenomena in question. These discussed, other forces which merely modify the general result are taken into account in the form of corrections. To illustrate, suplpose we wish to deduce the curve followed by a body, as a clannon-ball, projected into the air, a problem of the greatest importance in the science of gunnery. The consideration of the two principal forces actingo upon the ball, viz., the original force of projectionl, tending to make it proceed in a straight line, and the attraction of the earth, iwhich constantly draws it towards the ground, shows that the path would be a certain mathematical curve known as the parabola, were: there no disturbing forces. Such disturbing forces exist, however, in the resisting air through which the projectile imoves, so that it is only in sa vacunum that the parabolic curve could be strictly followed. T'he effect of atmosphleric resistance in altering the path must then be taken into account as a correction to the original theory, which can be done if we know the relation between the retardation and the velocity of the projectile. Often, as in the case under consideration, these corrections are very difficult to apply, owNing' to our ignorance of the exact law of the disturbing action, or to complexities in its application whllen known. In like mianner, when considering the theory of machines, we filrst suppose their component parts to possess no weight, to be absolutely rigid or perfectly flexible, and to act upon each other without friiction, and we determine the law of the transmission of power under these conditions. We afterwards aseertain what loss of power there will be on accoulnt of fiiction, and also estimate the effect of the weight of the different parts, and their want of perfect rigidity or flexibility. Not until all these 1 It is customary, however, to speak of the fall of bodies as being caused by #gpavitation, meaning by this not the law, which is merely the statement of the general fact, but that force (of the nature of which we are profoundly ignorant) which causes the tendency of particles of matter to approach each other. 2 Mill, System of Logic, Vol. I., I3k. mIi., p. 529, 6th Ed. 2

10 GENERAL METHODS. corrections have bDeen applied to the original theory, does it meet the requirements of practical science. 12. Discs sion of Observatior.: Analyrical and Graphical Methods. After making a series of experiments or observations with the intention of determining the law of any phenomena, there are two methods of mathemnaticelI discussion that we may apply to them. These ale (1) the Analytical, and (2) the Gracphical Jethoc. An alytical Method, in applying this method, we make use of algebraic symbols, designating each one of the quantities considered by a letter, and discussing their relations by the analytical processes of pure algebra or the calculus. Graphical ~Method. This is geometrical, the quantities considered being represented by lines, fromn whose relations the relation of the quantities themselves is dedluced. The first of these methods is more accurate than the second, but is at the same time more difficult of application owingl to accidental errors in observation, and the complexity of manlny laws. The graphical construction appeals at once to the eye, and is frequently of the greatest service, because of the ease with which the relations among the lines of the figure, and hence among the quantities, are determined. As a simple illustration of the use of the graphical method let us take the case of a daily record of thernometrical observations extending throuoh a month. We draw two lines, OX and OY, Fig. 1, at right angles to each other, for purposes of reference. These are known as the axis of X and the axis of Y, respectively. Their point of intersection O is called the origin of coordinates. Lines drawn from OY parallel to OX are called abscissas, which are positive to the right, and negative to the left of OY, and lines from OX parallel to OY, ordinates, which are positive above, and negative below OX. We represent times by abscissas and temperatures by ordinates, laying off on OX spaces proportional to the time from the beginning of the observations, and parallel to OY spaces proportional to the temperatures at those times. Passing a curve throuoh the points thus determined, we have the desired thermogram, or temperature curve. The curve of Fig. 1, is constructed from the data given in the annexed table, and represents the mean daily temperature on the summit of Mt. Washington during the month of January, 1871. Date; Temperature. Date. Temperature. Date. Temperature. Jan. 1.' 3.0 Jan. 11. 21.5 Jan. 21. 10.5 ~' 2. 8.2 " 12. 32.5 " 22. -28.5 " 3. -1.6 " 13. 36.0 " 23. — 2.6 4. -- 14.8 I" 14. 33.5 " 24. - 2.5 5. 8.7 15. 30.2 " 25. -15.9 " 6. 12.5 " 16. 31.7 "26. 3.8 " 7. - 7.2 17. 2.7 "27. -4.5 8. -5 18. 1.5 " 28. -- 5.6 9. -7.7 "19. 8.0 "29. 6.5 "10. 0.7 20. 15.5 "30. 10.5 "31. 27.7

LOGIC OF INDUCTIVE SCTENCE. 11 It is convenient to record many continuous phenomena in this manner. As an application of the graphical method to the discussion of physical laws, we give the following illustration. Let it be required to find the relation between the volume of a quantity of gas and the pressure to which it is subjected. Suppose that experiments have given the results shown in the following table. Pressure. Volume. Pressure. Volumne. Pressure. Volume. Pressure. Volume. 30 100.00 - 50 60.00 70 42.86 100 30.00 32 93.75 55 54.34 80 37 50 110 27.27 36 83.33 60 1;.00 90 33.33 120 25.00 40 75.00 65 46.54 Denote pressures by abscissas and volumes by ordinates, and. construct the curve A C, Fig. 2. This will be found to be an equilateral hyp!1ersbola, of which OX, OF are asymptotes, a curve having the property that the product of the ordinate at any point by the corresponding abscissa, is a constant. Since the ordinates have been made proportional to the volumes of the gas, and the abscissas proportional to filte pressures, it follows that when a gas is compressed, the product of its volume by the pressure upon it is a constant, which is the law sought. A consideration of Fig. 2 will also show an easy manner of finding the volume of gas corresponding to any pressure not given in the table. Thus if we wish to ascertain the volume under a pressure of 45, we simply lay off Oa- 45, and draw ab perpendicular to OX till it meets the curve at b: the line ab- 66.7, represents the volume wished, since Oa X ab is a constant, and Oa has been made proportional to the given pressure. From this example it will be seen that the graphical method gives a simple way of determining intermediate values of a variable, or interpolating., as it is called. LiST OF WORKS OF REFERElNCE. For further information concerning the subjects treated in the preeeding paces, the student may consult, A Sysftem of Logic, by John Stuart Mlill; Book III., On Induction. Preliminary Discourse on the Study of Natural Philosophy, by Sir J. F. W. Herschel. Philosoph?1 of the Inductive Sciences, by WVm. Whewell. Hlistor!j of the Inductive Sciences, by Winm. Whewell. Cours dle Philosophie Positive, par Auguste Coote, Lecons 1-39. (Astronomie, Physique, Chimie.) More extended notices of the Analytical and Graphical Methods, and the general discussion of observations will be found in Lecture Notes on Physics, by Prof. A. Ma. Mayrer (Phila., 1868), pp. 28-49. Elements of Physical Manipulation, by Prof. E. C. Pickering (Boston, A. A. Kingrman, 1872), pp. 1-25. In these two works a number of valuable references are given to articles bearing upon the subject in question.

12 WEIGHTS AND MEASURES. CHAPTER 1I. WEIGHTS AND M:EASURES.-INSTRUMIENTS OF MIEASUREEMENT. 13. In physical investigations it is constantly necessary to determine various nma-nitlldes, so that it will now be advisable to consider the two systems of weights and measures in common use, which are known as the English -,and the French System. English System; Linear Measure. The unit'of English linear measure is the yacrd of 36 inches, a length which has come down to us from very early tilhes. The statutory ma.gnitude of the yard: is said to:have'been originally determined in 1120 by the lengtlh of' the arm of Henry I. There being a great discrepancy among the different yards in use in Great Britain, Graham, in 1742, after a comparison of all the yards and ells of best, authority, constructed a standard of, which Bird made two exact copies, one in 1758, and a second in 1760. An Act of Parliament in 1824, declared that the length of the yard constructed in 1760, when a.t a temperature of- 620 Fahrenheit, should be considered as the legal standard of linear measure for the kinCgdom, providing for its recoverv in case of loss by making it to consist of 36 inches of such length that 39.13929 of them should be equal to the invariable lenoth of the pendulum beating seconds at London, in a vacuum, and at the level of the sea. This Act therefore made the length of the seconds pendulum the basis of the English System. The length of such a pendulum had been carefully measured by Capt. Kater, who determined its value in standard inches of the scale of' 1760 as (iven above.'rhe standard scale, together with that of 1 758, was destroyed by fire on the burning of the House of Plarliament in 1834, and it became necessary to construct a new one. The scientific committee to whom the matter was referred, did not deem it expedient to adopt the mnethod of restoration by the use of the penldulum, as provided in the Act of 1824, because serious sources of error had been shown to exist in the processes usedl by Kater in his mealsurements, renderinog it impossible to verify these within -5.0 of an inch, while a copy of any existing scale could be constructed so as to vary far less than this amount from the original. They therefore advised that all the most authentic copies of the fobrmer standard should be compared, and a new one constructed from these, which was accordingly done. They also advised that the new standard should in no way be defined by reference to any natural basis, such as the pendulum, and in the Act of 1855 legalizinjg the new standard, the yard is defined as being the distance at 62~ Fahr., between two marks on a certain standard bar. The British yard, therefore, is no longer based on the lenoth of the seconds pendulum at London, but is simply a copy of a material scale kept in the Exchlequer Ofice at Westminster. The unit of linear measure adopted by the United States is essentially identical with that of Great Britain, The standard at the Office of Weights and Meaisures at W' ashington is a brass scale 82 inchles in length, constructed by Troigliton for the Coast Survey.

EN GLISH SYSTEM. 13 The distance from the 27th to the 63d division of this at 620 Fahr., is equal to one U. S. yard. Two copies of the new British standard, one of bronze and one of iron, presented to the United States by the English government, compared with the Troughohton scale, show that the U. S. standard exceeds the present British one by 0.00087 in. The units for the measure of surface and solidity-, as the square and cubic foot or yard, are obtained directly from the liecar unit. 14. Weight. The unit of weight at )present adopted'by Great Britain is the Avoircdzpois poutnd of 7000 grnains, the standard being a platinum weight kept in the Office of the Exchequer. The oldest English legal standard known was the Tower pournd, as it was called, which was used even before the Norman Conquest. and was sotmyewhat lilghter than the present Troy pound of 5760 grains. It was employed at the Mint as the standard for coinage up to the time of Henry VIII,, by whom it was abolished in 1527, though the Troy pound was more or less: in use, at least as early as the reign of Henry V. (1413-1422). There was also at this time another pound in general use, called the libra mercatoria, weifhino about 6750 grains. When the Avoirdupois pound was first introduced is not definitely known, but it appears in statutes of the time of Edward III. (1327-13 77), and like the Troy pound was probably introduced by the Lombard bankers of that day. It seems to have derived originally from a Greek weight, the mina, of 6945.3 grains, which the Romans subdividedl into 16 ounces, while the Troy pound came fitom the Roman weight of' 5759.2 grains, the Tx~ part of the Talet of Alexandria. which thliey divided into 12 ounces. During the reign of Elizabeth (in 1582 and 1588) Avoirdupois weights were deposited in the Exchequer, but the Troy pound continued to be the legal standard. All the early weights were carefully examined by Grahliam in 1742, and in 1758 a Parliamentary Committee recommended the construction of a, new standard Troy pound, by which all other weights should be regulated. Tbhe bill carrying these recommendations into full effect was never passed, but three one-pound weights were made, one of which was deposited in the Parliament House. In 1818. the question was revived, and a committee, among whose members were Youno:, Wollaston and Kater, was appointed to investigate the subject. They advised that the Troy pound of 1758 should be the standard for the kincldom, and that the Avoirdupois pound, which had never been lehgally defined, should be declared to contain 7000 grains Troy. They also defined the grain weight by adopting the results of Shuckburgh and Kater, according to which a cubic inch of distilled water, wei(ghed in air by brass weights at a temperature of 62~ Fahr., with the barometer at 30 inches, was equal to'252.458 grains, of which 5760 constituted a Ti'oy pound. In case of loss, the standard pound was to be restored by reference to the weight of a cubic inch of water. Their report became a law in 1824, but only ten years later the standard pound of 1758 was destroyed by fire. In 1838, Baily, Sir J. P. W. W. Herschel and others were requested to take into consideration some means for the restoration of the lost standards. In 1841, they reported adversely to restoring the pound in the manner designated in the Act of 1824, owing to the great discrepancy existing among various determinations of tlie weight of the cubic inch of water, which rendered

14 WEIGHTS AND MEASURES. the accuracy of Shuckburgh's results very questionable. In fact, the weights assigned to a cubic inch of distilled water by different experimenters exhibit a variation amounting to T%,- of the total amount, while a copy of any existing standard could be mad(ie which should not vary fromn the original by more than T50:Wx0 of its actual weight. The committee therefore advised that the new standard be made by a comparison of the best authentic copies of the old standard. They also recommended that the Avoirdupois pound be made the legal standard rather than the Troy pound, since the former was in much more reneral ulse. The construction of the new standard was undertaken in 1843 by Prof. WV. H. Miller, who collected thirteen authentic copies of the original Troy pound of 1758. Eleven of these were finally rejected from consideration, as they were made of brass and were considerably oxydized, and the two remaining, which were of platinum, were the only ones used in the determination of the exact weight of the standard of 1758. From measurements of these, a Troy pound of 5760 grains was constructed of platinum, which was regarded as identical with the former standard, and from this a platinum standard equal to IT 000 of the weight of the Troy pound was mnade, together with four copies. This was constituted the legal standard Avoirdupois pound of' Great Britain by the Weight and M-easure Act of 1855. Experiments made upon the actual standard when completed, showed it to really weigh 7000.00093 grains, of which the old standard contained 5760. The United States Standlard is the Troy pound of the U. S..Mint, copied by Capt. Kater from the Briitish Troy pou)dl of 1758. An Act of Congress of 1834, defines it as being equal in weight to 22.794422 cubic inches of distilled water at its ma-sximumn density, but the incorrectness of the weight of a cubic inch of water under those circumstances, which was assumed by that act, causes the definition of the actual pound weight to be in error. 15. Capacity Measures. The British unit for all measures of caplacity is the Imnpercial gallon, which cont:ains 10 Avoirdupois pounds of distilled water at a temperature of' 620 Fahr., with a barometric height of 30 inches. According' to the best determinations, it contains 277.123 cu. in. under those circumlst.lnces. The Act of 1824 stated its contents as 277.274 cubic' inclies, which has since been shown to be,an incorrect value. The U. S. Standard is the wine gallon for liquids, and the Winchester bushel for dry measure, derived fiom the English measures of the saine name. The former measure is defined by Act of Congress as containing 231 cubic inches, the latter 2150.42 cubic inches at 390 Fahr., 30 inches barometer. But unfortunately the capacity of the British Winchester bushel is 2150.4 cubic inches at 62~ Fahr., so that at our standard temperature it contains only 2148.9 cubic inches, and at no common temperature are the two bushels equal. The same is the case with the wine gallon. The British Imperial bushel, a measure in common use, but not a legal standard, contains 8 Imperial gallons or 2216.984 cubic inches at 62~ Fahr. The method of ascertaining the number of cubic inches contained in the standard gallon or bushel, is by finding the weight of the quantity of distilled water necessary to fill it under given circumstances of temperature

FRENCH SYSTEM. 15 and atmospheric pressure. This weight divided by the weight of a cubic inch of water, gives the capacity. This method evidently requires a precise knowledge of the weight of' a cubic inch of water, which, as we have stated, is so difficult of exact determination that somewhat different results have been obtained by the most skilfibl experimenters, causing a slight variation of the actual firon the legal contents of the U. S. standards. T'he greatest possible deviation of these from the statutory capacity is, however, so small as not to be of very great practical consequence, the gallon and bushel never being used in any but commercial measurements. There has been even more variation in the values of the various capacity measures of past times than almong weights. The capacity of the ordinary wine gallon has gradually risen froln 216 cubic inches, its contents in 1299, to its present value of 231 cubic inches. The Winchester bushel has had equally variable values, having risen from 2114.68 cubic inches in 1266, to 2150.4 cubic inches, its present capacity. The British Imperial standard gallon is a unique measure, and is derived directly from tlie Avoirdupois pound, while the other measures were originally obtained from the Tower and Troy pounds. 16. Circular or Angular IMeasure. The unit of angular measure is the degree, which is the, of a circumference. It is subdivided into 60 minutes, and each of these minutes into 60 seconds, smaller anlgles being indicated by decimals of a second. Until within two or three centuries it was customary to carry this sexagesimal subdivision still farther, each second consisting of 60 thirds, the third, of 60 fouwlhs, and so on, a cumbersome method, which has happily gone entirely out of use. 17. French or Metric System. Hlistory. This system was adopted in France in 1795, during the Revolution, and was intended to furnish a unifolrm system of weights and measures, all of which should be based upon a single linear unit, and the various subdivisions of which should be purely decimal. As early as 1790, Talleyrand proposed the establishment of a general system, in the construction of which all the civilized nations of the world should be invited to joil, and suggested the pendulum beating seconds as an invariable standard firom which to derive the linear unit. In accordance with the proposition, a party of five 1 from the Academy of Sciences was appointed to investigate the subject. This committee reported in 1791, advising that the project should be carried out, but preferred as a standard of reference the quadralnt of the earth's meridian included between the equator and the pole, which was then supposed to have the same value in all longitudes, while the seconds pendulum was known to vary in length with the locality. They also suggested the employment of the pendulum as a secondary standard, fiom which to recover the other in case of its loss. In accordance with their recommendation, which was immediately transmitted to th8 Assembly, Delambre and MEchain were deputed to ascertain the length of a quadrant of 1 Borda, Lagrange, Laplace, Monge and Condorcet.

16 WEIGHTS AND MEASURES. the meridian,:and for this purpose measured a meridional arc of about 9 1-2 defgrees, passing throug:h France from l)unkirk to Barcelona. This work occupied a number of' years, but the Assembly made use of a value of the quadrant given by some earlier measurements, and promulgated the system in 1793, not adopting the present nomenclature, however, until 1795. At length, in 1 799, an international convention was assembled at Paris, which finally decided the values of the various standards of weights and measures. 18. Linear Measure. The French arc, compared with arcs previously measured in Peru and Lapland, gave as the distance frokm the equator to the pole, 5,130,740 toises, the toise being an old French measure equal to about 2.132 English yards. It was decided that ~1oooo of this distance should be called a metre, and made the basis of the new system. The unit adopted in IDellambre and M6echain's:measurements was the'oise of Perc, the same that had been used in the earlier measurements of the Peruvian arc by Bouguer and La Condamine. The Act of 1799 decrees the metre to be _-4329- of the length of the standard Toise de P&rou, which is an iron bar, correct at 62.25 degrees Fahr., made in 1735 under the direction of Godin. A standard metre of platinum was made from this by L6noir. The value of the metre in English inches is 39.370432.1 More extensive geodetic measurements made during the present century show that the value of the quadrant obtained by the French astronomers is too small. Bessel gave as its true value 10,000,856 mletres. Moreover, owing to the recently lemonstrated fact that the earth is not an oblate spheroid, but more nearly an ellipsoid with three unequal axes, it follows that the value of the quadrant of a meridian varies with the longitude. Hence the basis of' the French system is in reality as much a local one as is the pendulum. It should also be borne in mind that from the terms of the law of 1799 the French metre, like the present English foot, is simply the length of a legalized nmaterial scale. The metric system is purely decimal. The larger divisions of linear mealsure are the myrictmetre 10,000 metres, the kilometre 1,000 metres, the hectonetre- = 100 mnetres, and the dekacmetre10 metres. Those less than 1 metre are the dcecimetre = -j metre, the cenztimetre — T metre, and the millimetre - = o6 metre. iMeasures of surface andc solidity are formned fronm those of linear dimensiols, precisely as in the English system. 19. Weight. Th'e unit of weight is the gramme, which was ordered by statute to be the weight of a cubic centimetre of pure water in vacuo, at 39.1 degrees Fahlr. The actual standard platinum weight (by Fortin) of 1000 grammes, in the French Alchives, is 1 According to the measurements of Capt. Clarke in 1866. Two other measurements of the metre have been-made; that of Kater (1821), which gave as a result 39.37079 in., anld that of Hassler (1832), giving 39.3810327 inl. The latter is the value which has been used by the U. S. Coast Survey. The results of' Capt. Clarke are those now in use by the British Ordnance Department.

FRENCH SYSTEM. 17 a little greater than it should be according to this law. The gramme is divided into decigrammes, centigrammes and milligramimes, equal to JlO, ri'- OWo grammnes respectively. The larger weights are the dekagrammne - 10 grammes, the hectogrcamme 100 grammes, and the kilogramme - 1000 grammes. The weight of the standard kilogramme above mentioned is 15432.34874 grains, or 2.204,621,250 lbs. Avoirdupois, which gives 15.43234874 grains as the weight of the gramme.1 20. Capacity 1Measure. The unit of capacity measure is the litre, which is the volume of a kilogr:alme of pure water at 39.1 degrees Fahr., and 30 inches barom:eter. It was originallyintended to be a cubic dlecimetre, but is somewhat larger, owing to the excess of weight of the standard kilogrammle mentioned above. It is equal to 61.02499 cubic inches.2 21. Circular Measure. The French also applied the decimal system to angular measure, and a number of the circles used in the observations for determining the lencth of the French meridional arc were thus divided. Each quadrant contained 100 grades or degrees, which were again subdivided into tenths and hundredths. Hence the centesimal degree was equal to 0.9 of a sexamgesimal degree. As the sexagesimal method of division was universally employed long before the metric system originated, the French division of the quadrant has never been extensively used, even by the nation which devised it. 22. Full tables of all French weights and measures employed in physical operations, together with their values in English units, will be found in the appendix to this volume. Owing to its great simplicity, the metric system is gradually displacing all others. At the present titme it is the most widely used of any, and is rapidly gaining in general favor, most of' the European nations having adopted it either wholly or in part. Its employment in all commercial transactions was legalized by Great Britain, in 1861, and by the United States, in 18663. Sir John F. XV. Herschel has proposed the polar axis of the earth as a more logical basis than any other for a system of weights and measures, since, unlike the pendulum andl the quadrant of latitude, it is not local in its nature. While this is very true, the wide acceptance of the metric system renders it undesirable to attempt a change which, even were it possible, would be chiefly of' theoretical advantare. WVhat is most desirable amonll different nations is somei system, at once uniform and simple, both of which requisites are already supplied; the derivation of the unit is a matter of but slight practical importance. Another advantage urged by Herschel, is, however, that if the present English inch were increased by W-LOu of its total value, the polar axis of the earth would contain just 500,500,000 such inches; and that 1728 such cubic inches of water would weigh 1000 ounces Avoirdupois if the present value of the ounce were increased by 1 of a grain. On the other hand, it is not probable that the most accurate Miller. 2 Clarke. 8 The student maytv be interested to know that the U.S. five-cent piece is 2 centimetres in diameter, and weighs 5 grammes. 3R

18 INSTRUMENTS OF MEASUREMENT. value that we possess of the earth's polar axis is sufficiently near to abso' lute correctness to warrant a change at present. 23. Unit of Tirae. The unit employed in physical observations extending over a considerable tine, is the mean solar day of 24 hours, that being the mean interval between two successive transits of the sun across the celestial meridian. The mean interval is taken because the time elapsing from day to day between the passages of the sun across the meridian is not constant, owing to the facts that the earth's motion about the sun is not uniform, and the sun's apparent path is not in the celestial equator, but inclined to it at an an(le of 23~ 281. Our ordinary clocks are adjusted to keep mean solar time. Astronomical clocks, on the contrary, are adjusted to sidereal or startime. A sidereal clay is the interval between two successive transits of the same star across the meridians, and is the exact period of the revolution of the earth on its axis. It is 3 m. 56.5 s. shorter than the mean solar day. The sidereal day, like the solar, is divided into hours, minutes and seconds. When astronomical clocks are used for the estimation of time in physical experiments, it is customary to reduce their indications to mean solar time by a simple proportion. For small intervals of time the unit in common use is the second. Until recently it was supposed that the period of the earth's rotation had not varied by 1 of its length since 720 B. C., but an error was found by Adams in the work of Laplace, on whose calculations the above supposition was based, and he has shown that the time of revolution is diminishing at the rate of 22 s. in a century. 24. Instruments of Measurement. Physical measurements are of five kinds, according, to the nature of the magnitude to be determined. Instruments for precise measurements may be classified according to their use, as follows:1. Instruments for the measurement of angles. 2. " " " length. 3. " c " volume. 4. 6 " " weight. 5. 4 " " tinme. It will be most convenient to consider the construction of the majority of these instruments as they present themselves in the sequel, in connection with the subjects of base lines, determination of the standards of capacity measures, balances, etc., but we-shall here describe a few which are of very general application. These are the graciucated circle for angular measurement, the vernier and the cathetometer, an instrument for the precise measurement of vertical distances. 1 More strictly, of the vernal equinox, but the difference is only one one-hundred and twentieth of a second in a day.

VERNIER. 19 25. Inatrument for measuring Angles. Angles are generally measured by circles divided at the circumference into degrees and fractions of a degree. The common Eagineer's Transit, Fig. 3, will give a general idea of an instrument for measuring angles. It is designed to measure either vertical or horizontal angles, and therefore carries two circles, one vertical, the other horizontal, which can be seen at C and D. At A2B is a telescope moving upon the same horizontal axis with _D, while the whole instrument turns horizontally about an axis attached to the centre of C. Screws, S (leveling screws), are attached, by which the circle Ccan be made exactly horizontal, also screws E and F called tangent screws, by which a slow motion can be given to C or D. At GC is seen a compass needle, which is sometimes of service, and at HKia level attached to AB, so -that the telescope may be made horizontal if necessary. The telescope AB, is furnished with two cross-hairs in its eye-piece, at right angles to each other (Fig. 4). Suppose it is necessary to measure the vertical angle between two points. The observer places his eye at the telescope, focuses it on one of the points, bringing the intersection of the cross-hairs to coincide with the image of that point, and observes the reading on the vertical circle given by the index 1 He then turns A-B about its axis until the second point comes into the field of view, and fixes the intersection of the cross-hairs upon it in the same manner as before. Tlhen. taking the new reading given by 1, the difference between it and the former reading is the desired angle. As the telescope mloves horizontally about the centre of C, it evidently makes no difference whether the two points are or are not in the same vertical. Horizontal angles are measured in a similar manner by means of the horizontal circle U. There is another index opposite 1; and in delicate rmeasurements the mean of the readings of both should be taken, which will eliminate any error from eccentricity in the mounting of the divided circles. 2,, gV2lraler. It is often necessary to measure lines and angles mInoe closely than it is convenient to divide the scale or circles used. In sach cases we m:ay estimate fractions of one division by the eye, especially if:ailed by a magnifying glass, with considerable:lccuracy. Various other metllodls h:ave been devised, one of the sitllplest an(i most &'enerally a.pplic:able of which is the Vernier, so called fiom its inventor, Pieri'e Vernier, of Brussels. (1631.) To understand its construction, let AB, Fig. 5, be an enlarged representation of a scale divided into maillimnetres, with which we wish to read to tentls of a millilletre. Sa tpiose that a scale DC is Tnmade by taking 9 divisions of AB (ill thle figure 110 to 119), and dividilng this length into 10 equ:li p:trts, numbered firom 0 to 10. Each division of DC will equal Ad of at millilnetre, and will be less thaIn one division of AB hby - mnll., so that if the 0 of D C, which is the vernier scale, be made to coincide with any division of AB,

20 INSTRUMENTS OF MEA.SUREIENT. as the 750th, division 1 of D C will fall short of 751 of AB by ~ mm., 2 will fall short of 752 by 920 min., 3 will fall short of 753 by Ta mm., and so on, until the 10th division of D C, which coincides with 759 of AB. To comprehend the method of using it, let it be requiredd o find the distance of some point S, Fig. 6, fiom any other point. The scale ASB is laid so that its 0 shall be opposite one of the points, and the vernier is slid alono A B until its 0 denoted by the arrow rests ag'ainst S. The distance from the other point as indicated by the scale, is evidently 751 and( a fraction millimetres, the whole number being given by AB. The fraction is obtained from CD, by counting its divisions upward friom 0, until one is reached which coincides with one of the divisions of AB. The number of this vernier-division gives the fractional reading, which in this case is the nulmber of tenths of a millimetre (-3) between 751 and the 0 line of thte vernier. For as each division of CD is less than one division of ABB by — Ir mm., if the coincidence takes place as in the figure at the 3d vernier-division above 0, the space between 0 and 751 must equal 3 scale-divisions minus 3 vernier-divisions, that is a- mmn. - _~ mm.-= -1 mm. Hence in the instance supposed, the total distance between the two points is 751.3 mm. From this it will be seen that in general, if we call n the number of divisions of the scale equal in length to n + 1 divisions of the vernier, in which case n 1 is the value of one division of the latter, and denote by x the distance between that scale-division which coincides with a vernier-division, and the scale-division next nx below the 0 of the vernier, the fiactional reading is x - n + i Verniers are applied to graduated scales and circles of all kinds, and furnish a ready means of increasing their delicacy of measurement. There are various modifications in use for particular purposes, but the principle of construction is the same in all.'I he smallness of' the fractions which are indicated by any vernier depends upon the number of its divisions. Thus if, as in Fig. 7, 9 scale-divisions are sub-divided into 10 parts, it reads to T1 }nmm. If 99 scale divisions were sub-divided into 100 parts, it would read to i~- mm., since each vernier-division would then = —-9 mm., and one scale-division minus one vernier-division -= 0 mm.- mm. --- m mmi Or, generally, if x in Eq. (1) is made equal to unity, - 1 will be the miniuum limit of reading in terms of the length of the divisions of the scale. Hence to find the limits of reading of any vernier, divide the length (of one scale division by the number of divisions of the vernier.

CATHETOMETER. 21 27. Cathetometer. In many physical investigations it is necessary to measure the difference between the vertical heights of two or more objects to which it is difficult to apply an ordinary scale. In such cases we make use of an instrument invented by Dulong and Petit, and known as the Gcathetometer (Fig. 7). Tile essential parts of this instrument are a heavy vertical pillar of metal AB, mounted upon a base Idif, which is furnished with leveling screws. The pillar AB carries a scale FG, divided into millinetres. A telescope CD is furnished with cross-hairs, and mounted horizontally upon a frame T1V; which can be raised or lowered by sliding it along AB. A screw S serves to give a very slow vertical motion to the telescope when desired. The frame T-carries a vernier E, and a level 1 is attached to C-D to detect any deviation fiom horizontality. To measure the vertical distance between any two points, after having carefully levelled the instrument, the fiame TN is moved until one of them lies in the axis of the telescope, which is focussed carefully, and its crosshairs are made to coincide exactly with the point by means of the screw S. The reading of the scale and vernier is then taken. This done, the telescope is raised or lowered until the other point coincides with the cross-hairs, when the scale and vernier are again read. The difference between these two readings is the vertical distance between the points. That the cathetometer may give accurate results it is evident that the axis AB must be exactly vertical, and that the line of sight of' the telescope must remain parallel to itself when raised or lowered. 28. Measure of Time. Measurements of considerable intervals of time are made with clocks or chroitometers. The mneasurement of very minute intervals is accomplished by various forms of instruments called chronographs. Many of these will be described in succeeding chapters. REFERENCE S. For additional information the student may consult Account of the Proportions of French Mleasures and Weights, fr'om the Standards of the same kept at the Royal Society, by G. Grahanm; Philosophical Transactions, Vol. XLII., p. 185. Account of Comparison of Standard I ards and several Weights with original Standard, by G. Graham; Phil. Trans., Vol. XLII., p. 541. Construction qf the new Imperial Standard Pound and its Copies of Platinunm, by W. R. Miller; Phil. Trans., Vol. CXLWVI., p). 753. (,Construction of the new Standard of Length, by G. B. Airy; Phil. Trans., Vol. cxvvLI., p. 621. (In this article will be fobund an extensive list of references to articles bearing on the subject of which it treats.) On the Correct Aojustment of Chemical Weights, by Win. Crookes; Chemical News, Apr. 19, 186-7.

22 WEIGHTS AND MEASURES. Comparison of Standards of Leng'h of England, Fracnce, Belgium, etc., by Capt. A. R. Clarke; (London, Ordnalnce Survey Department) pp. 105, 163. Tables, M1eteorological anld Physical, prepared for the Smithsotian Institution, by Arnold Guyot (Washington, 1858). The Yard, the Pendulum and the Metre, a Lecture by J. F. W. Herschel, published in Famzilar Lectures on Scientific Subjects (London, A. Strahan & Co., 1867), p. 419. The Metric Sbystem, by Charles Davies, (New York and Chicago, A. S. Barnes & Co., 1871) contains the lecture cited in the preceling reference, together with the Report on Weights and.lIeasures, by John Qulillcy Adams, (1821) and the Report of the University Convocation of the State of New York-, (1870). The Ml3etric System, with Appendix on Capacity 31easures, by F. A. P. Barnard. (New York, 1872.) Base du System 3Id'rique, Tome III, par Jean B. J. Delambre et Pierre F. A. Mlchain. Recueil Officiel des Ordonnances et Instructions sur la Fabrication et la Verification des Poids et AMesures. (Paris, 1839.) Atlas des'Poids et Allesures dresse' en Execution de l'Orldolnance Royale. (Paris, 1839.) Rapport sur les Poids et 1Mesures envoye's anz Cozverment (les Etats-Unis, par A. Vattemare. (Measures executedl by I. T. Sdlberliann.) Rapport sur la Revision des Etalons en 1867 et 1868; Annales du Conservatoire des Arts et ile'tiers, Tome ix., No. 33, ler Fascicul-, p. 1. Also consult references under the chapter of this work treating of the Pendulum. The following Parliamentary Sessional Papers may be consulted for records of the legislation of Great Britain relative to the subject of Weights and Measures: On Weiq. hts and M](easures; Paccl. Papers, Reports of Colzmmittees, 1813-14, Vol. III., No. 290. Also Rep. Connzoittees. Jaz. to,Julq, 1821, Vol. Iv., No. 571; and Report of Comzmissioners, by Clark, Gilbert, Wollaston, Young and Kater, No. 383 of same volume. lIinutes of Evidence on Wellqht and lieasure Bill; Rep. Colnmzitlees, 1824, Vol. Iv., No. 94. Also Rep. Comzittees, 1834, Vol. xiv., No. 464. Report on Weights and Mleasures, and Mi1inutes of Evidence; Rep. Committeel, 1835, Vol. xIv., No. 292. Report of Comnissioners appointed to consider the Steps i'o he talken for the tRestoration of the Standards of Weight andl 3easure, by Airy, B:lily, Bethune, Herschel, Lefevre, Lubbock, Peacock and Sheepshanks; R1,ep. Comnzissioners, 1834-5, Vol. I., No. 177. Report of Commissioners appointed to consider Steps for the Restorationz of Standards; Rep. Commnussioners, 1842, Vol. xI., No. 356. Report of Commznittee appointed to superintend the Construction of the Parliamentary Standards, by Airy, Rosse, Wrottesley, Lefevre, Lubbock, Peacock, Sheepshanks and Miller; Rep. Commzissioners, 18.54, Vol. XIx. Report of Committee recozmmending the Legalization of the Mletric System, together with 3Miinutes of Evidence; Rep. Committees, 1862, Vol. VII., No. 411. Bill authorizing the Use of the Aletric System; Parl. Bills, 1864, Vol. Iv., Nos. 24, 165. Report on the Exchequer Standards of WTeight and 3M1easure, by II. WV. Chisholm, with Notes by G. B. Airy; Accounts and Papers, 1864, Vol. LVIII., No. 115.

REFERENCES. 23 Bill to amend acts relative to Standard Weights and Measures; Public Bills, 1866, Vol. v., No. 166. First Report of W/arden of Standards, 1866-7; Rep]. Commissioners, 1867, Vol. xIx. Report of Commissioners appointed to inquire into the condition of the Exchequer Stancdards, and on the Abolition of 2Troy Weight, with Minutes of Evidence; Rep.' Commissione rs, 1870, Vol. xxvii. On the application of the Metric System to India; Re]p. Commissioners, 1870, Vol. LIII., No. 225. The legislation of the United States relative to Weights and Measures, and the reports to the Government on the construction of standards, may be found in the various Con#gressional Documents, as follows: Report of Committee on ]Fixing Standards of' WVeight and MlAeasure; Reports of Committees, 15th Congress, 2d Session (1818-19), Vol. vI., Doc. 109. Report on W7;eights and JMeasures, by J. Q. Adams, Secy. of State (Feb. 22, 1821); Executive Pacpers, 16th Cong., 2(1Sess., Doe. 109. Report of Committee to whom was referred the Report of the Secy. of State; Reports of Committees, 17th Co:ng., 1st Sess. (1821-2), Vol. II., Doc. 65. Comparison of 1t4eights and Mileasures of Length and Capacity, by F. R. HIassler; Exec. Doc. 22d Congr., Ist Sess., (1832.) Vol. vI., Doec. 299. Constructi(zn of Standards of Weights and lMeas-ures; Reports of Committees, Iou.se Rep., 23d Conlgr., 2d Sess., (1835) Vol. I., Doe. 132. Letter from F. R. Hassler, in Ruep., of Secy. Treas.; Exec. Papers., Htouse Rep., 24th Congyr., ist Sess., (1835) Vol. II., Doec. 32. Report onl Furnishing States with Standards; Rep. Comm., House Repr, 24th Congr., Ist Sess., (1836) Vol., I.,. Doe. 259. Ibid., Vol. II., Doc. 449. Report on Progress of Construction of Standards, by F. R. IHassler; Exec; Papers, 25th Congr., 2d Sess., (1837-8) Vol. xi., Doe. 14. Report on Construction and C7ompletion of Standards for all the States of the Union, by F. R. Hassler, (July 4th, 1838); Ibid., Doe. 454. Reports by F. R. HIassler; Senate Doc., 26th Congr., 1st Sess., (1839-40), Vol. II. Doe. 15; Vol. viII., Doc. 608. Also Sen. Doc., 26th Congr., 2d Sess., (1840-1) Vol. xi., Doe. 20. Report on Progress in Constructitzn of Standards of Liquid and Capacity M'easure, by F. R. Hassler; Sen. Doc., 27th Congr., 2d Sess., (1841 —2) VoL III., Doc. 225. Reports by F. R. Hassler; Sen. Doc., 27th Congr., 3d Sess., (1842-3) Vol. II., Doe. 11. Report 7elaive to Weights, lIeasures and Balances, by F. R. EIassler; Exec. Doc., 28th Congyr., 1st Sess., (1843-4) Vol. IV., Doc. 94. Reporls ona WVeights and Mleasures, by A. D. Bache; Exec. Doc., 28th Cogr., 2d Sess., (1844-5) Vol. iv., Doc. 159. Exec. Doc., 29th Congr., 1st Sess., (1845-6) Vol. vI., Doe. 225. Exec. Doc., 3., th Cgr., 1st Sess., (1847-8) Vol.. Ix., Doe. 84. Rteport on Construction and Distribution of Weiqhts, 3Measures and Balances, and on Co -mlparison of Foreign Standards, by A. D. Bache; Exec. Doc., 34th Congr., 3d1 Sess., (1856-7) Vol. VI., )oe. 27. Report on WVeights, Mieasures and Balances, by A. D. Bache; Exec. Doc., 35th Congr., 2d. Sess., (1858-9) Vol. vl., Doc. 6. (This report contains a resume of the work relative to the construction of standards previous to

24 PROPERTIES OF MATTER. 1858. Also the titles of all State Acts bearing on the subject from 1819 to 1804.) Report of Committee appointed for Purpose of investigating ihe.Metric System, and accompanyiny Bill; Acts and Resolutions oJ' 39th Congr., 1st Sess. (1865-6), p. 350. For a detailed account of the various forms of Vernier in practical use, see A Treatise on Land Surveying, by W. M. Gillespie. (New York, Appleton & Co., 1867.) For methods of graduating scales consult Gasonmetry, by Robert Bunsen, trans. by H. E. Roscoe (London, Walton & Maberly, 1857); p. 25. 7lrait e' iementaire de Physique, par P. A. Daguin; Tome I., p. 22. Cours de Physique de 1'Ecole Polytechnique, par M. J. Jamin; Tome I., p. 25. Edinburgh Encyclopedia, Article Graduation, by Troughton. CHAPTER III. PROPERTIES OF MATTER. 29. Matter. — General Properties. Whatever can be perceived by the ordinary operations of the senses is matter. It possesses two essential characteristics: (1) extension, or the property of occupying space, and (2) imrpenbetrability, by virtue of which no two bodies can occupy the same space at the same time. The second of these sometimes seems to be contradicted by experience. Thus when an inverted bottle is immersed in water the liquid rises to some distance inside; but in this case the air is not penetrated, it is merely compressed, so that a portion of the space which it formerly possessed is occupied by the water. A stone sunk in a vessel of water does not penetrate it, but displaces a quantity equal to its own bulk. And in all other cases in which at first sight one body seems to penetrate another, it will be found that there is really a displacement of matter. Both the above characteristics are essential to the existence of matter. A shadow occupies space, but is not matter as it is not impenetrable. 30. Three States of Matter. Matter exists in three different conditions, which are known as the solid, liquid and gaseous states. Solids are distinguished by possessing a definite form which resists any change with considerable force. Liquids take the form of the vessel containing them, their particles not being firmly united like those of solids, but moving upon each other with the greatest ease. Gases possess this mobility of particles in commlon with liquids, but in addition to this their particles have a constant tendency to separate from each other, so that some ex

PARTICULAR PROPERTIES. 25 terior effort, such as the resistance of the walls of the vessel containing them, is necessary to limit their volume. This expansive tendency may be shown by placing a bladder partly filled with air in a receiver in which we can proclduce a vacuum by means of an air-pump. When the external air is removed the bladder gradually swells out, owing to the expansion of the air which it contains on the removal of the external atmospheric pressure, which previously kept it under a more limited volume. As examples of the three states of matter, we may mention iron, wood and stone, among solids; water, alcohol and mercury, among liquids; and air, carbonic acid gas and common illuminating gas, among gases. Liquids and gases are often collectively calledfltcids. The same substance may exist in each of the three states of matter according to the external circumlstances. Thus ice when heated assumes the liquid form of water, and if this is heated still more it finally assumes a gaseous state by passing into steam. Sulphur also assumes the solid, liquid and gaseous forms, successively, when heated, and many substances, as carbonic acid, lauglhing gas, etc., which are gaseous at ordinary temperatures, can be made to assume a liquid, and finally a solid form under the influence of great pressure and extreme cold. 31. Particular Properties of Matter.-Divisibility. In addition to the preceding there are certain particular properties which are common to all kinds of matter, though not essential'to our idea of its existence. (1.) Matter is divisible. We know of nothing which can not be separated into parts, these again into smaller parts, and so on. The extent to which matter is divisible will be appreciated from the following examplnles. Gold leaf has been hammered into leaves but ni mm. in thickness. Silver wire gilded on the exterior has been drawn out to such a degree of fineness that the coating of gold was only 80-s oo min. thick. The silver core was then dissolved away by acid, leaving a gold tube of this excessive thinness. With a microscope it is possible to see a particle -,- mm. in diameter, so that we are able to divide gold into particles 40oo mim. in diameter, and so)t oo min. in thickness, each of which possesses all the qualities of the metal. Platinum wire has been drawn so fine that it took 200 metres of it to weigh 1 centigramme, its diameter being but T-T, man. This was done by enclosing a platinum wire in a closely-fitting silver tube, drawing out the compound wire thus made, and then removing the silver coating with acid. The thickness of a soap bubble just before bursting is but Looo minm. A gramme of carmine will tinge 60,000 gr. of water, or about 9,000,000 drops, so that in one of these drops there is but 9 o o o o o grammes of coloring material. By optical methods it is possible to detect even,, o o o-oo of a gramme of soda. It is said that a single grain of musk has perfumed a room 12 feet 4

f26 PROPERTIES OF MATTER. square for several years without sensible loss of weight; and an extract of spirit of musk has given a distinct odor to 2,000,000,000 times its weight of a certain liquid. These examples might be multiplied indefinitely, but we will mention only a single additional one, drawn from the organic world. Certain microscopic plants, the. Diatoms, are covered with a siliceous shell, weighing but 30 o o o00o o o o of a gramlme. On this shell we can see stride not over I mm. in width and thickness, thus perceiving a portion of siliceous matter, whose weight would be only about 7 o o o o o o o o of a gramme. 32. Atoms and IlMolecules. The question here arises whether this division could be carried on inldefinitely. Early in the history of science this was a disputed subject. Anaxagoras and Aristotle held matter to be infinitely divisible, while Leucippus taught that there were ultimate indivisible particles, to which Democritus gave the name of atoms. (B. C. 500.) The atomic doctrine was also taught by Epicurus. In modern times Descartes rejected the theory, while Gassendi maintained it. Up to a comparatively recent time these opinions were tmere philosophical speculations resting upon no solid basis, but in later years certain laws of chemical combination, crystallography and molecular physics, have shown it to be highly probable that matter is composed of ultimate particles of inconceivable, though finite minuteness, which are physically incapable of further subdivision by any known process. These atoms are grouped together to form larger masses, called molecules, which may again be arranged in still more complex aggregations. The molecules of a body are separated from each other by spaces called clpores, which are much greater in size than the atoms themselves. Most bodies are composed of compound molecules formed by the chemical union of atoms of unlike nature to form a new substance. Hence it is convenient to distinguish between integrant and constituent molecules, constituent molecules being aggregations of similar atoms, while integrant molecules are formed by the union of the constituent molecules of dissimilar substances. Thus marble is composed of integrant molecules of carbonate of lime, while each of these integrant particles is, in its turn, composed of constituent molecules of calcium, carbon and oxygen. The exceeding minuteness of molecules renders it impossible to obtain any direct measurement of their size, but Sir Win. Thomlson has shown 1 that the mean distance between the centres of contiguous molecules of matter is probably less than T'UOOOO-6oo and greater than 2o o o o o o of a centimetre. 33. Compressibility. All substances are compressible. On the application of external pressure to any body, its volume is 1 Article in Nature, Mar. 31. 1870, p. 651.

PARTICULAR PROPERTIES. 27 diminished. This can only arise from the closer approximation of its molecules, which renders it evident that these are not in absolute contact. The compression of solids is noticeable in any structure supporting a great weight. The stone pillars sustaining the dome of the Pantheon at Paris, were sensibly compressed when that structure was erected upon them. In the operation of coining, the metal check upon which the impression is struck is noticeably diminished in volume. The compressibility of liquids is less evident, and until within a century they were quite generally believed to be absolutely incompressible. In 1762, Canton first showed this idea to be erroneous, and found that water was compressed about TnU-oo-65o of its volumne by a pressure of I atmosphere (1033.6 grammes to a square centimetre). More accurate experiments since that time have confirmed this general result. Gases are by far the most easily compressed of all substances. This may be shown by means of an air-tight piston moving in a glass tube closed at one end. The piston can be forced into the cylinder, compressing the air contained in it. On withdrawing the pressure the gas expands, forcing the piston back until it has attained its original volume. This last effect is due to the perfect elasticity of the gas, a force tending to make it return to its primitive volume. The saime result occurs in the case of a liquid, and in a less degree with solids, the elasticity of the latter class of substances being in general very imperfect when compared with that of the two former. 34. Expansibility. On the application of stretching forces to bodies, or on heating them, their volume is found to increase. The expansion of solids by external forces may be, shown by stretching a wire or an India-rubber tube. Expansion by heat can be rendered evident by means of a ball of iron, made of such size as to just pass through a ringr of the same material when cold. If heated the ball will no longer do this, on account of its increased volume. If a quantity of liquid be contained in a tube with a bulb at one end, its level will be seen to rise in the tube when heated, and if such a bulb-tube contain air, or any other gas, a drop of liquid being placed in the tube to serve as an index, the expansion of the gas is so grealt that it will be apparent when the bulb is simply clasped by the ha:nd. 35. ]Porosity. The fact that the molecules of any substance are not in contact, which we have allready noticed, is shown to be true by the phenomena of compressibility and expansibility. The name pores is given to these intelrmolecular spaces. The porosity of substances can be proved directly in v: Lrious ways. An exlperiment celebrated in the history of science is that first performed by Lord Bacon, and afterwards by the Academy of Florence (1661). Wishinog to determine whether water was compressible, they filled a hollow silver sphere ifull of that liquid, and after closing it

28 PROPERTIES OF MATTER. tightly flattened it under a screw-press. This operation of course diminished the capacity of the sphere. At the close of the experiment the exterior was foun,l to be covered with a fine dew, showing that the water had been forced through the pores of the silver. The Academicians drew fioln this two conclusions: first, that the pores of silver were l;lrger than the molecules of water, which is true, and secondly, that water was absolutely incompressible, which is erroneous. Other examples occur in the arts. Thus the manufacture of steel fiom wrought iron depends upon the absorption of a portion of carbon by the metal. It has been shown that platinmn and cast iron, when heated to redness, are porous to gases. The porosity of liquids and gases is seen in the fact that in most chemical combinations the volume of the resulting compound is less than the sum of the volumes of the components. T'he same is true in the case of many solutions. Thus anhydrous alcohol and water mixed in the proportions of 116 to 100, sustains a diminution in volume equal to 3.7 per cent., and with other proportions there is still a diminution, though not as much as with those mentioned. The same thing occurs on mixing water and sulphuric acid. The porosity of liquids is farther shown by their ability to absorb gases. Thus water at 60~ Fahr. will. absorb about 720 times its bulk of ammonia gas, the resulting increase of bulk being only 50 per cent. In the case of gases, simnilar contractions are observed. Thus 4 volumes of nitrogen unite with 2 volumes of oxygen to form 4 volumes of protoxide of nitrogen, the resulting compound having a volume of only twothirds that of its constituents when in an uncombined state. In all these cases it will be seen that the particles of one kind of nmatter enter the pores of another, but do not penetrate the atoms, so that the statements already made relative to the impenetrability of matter are in no way invalidated. W~e should carefully distinuislh between the intermolecular pores, of which we have been speaking, and the larger interstices existing amnon(r the particles of solid bodies to wihi(ch the namie pores is also applied. TJhese last, often called organic or structural pores, according as they occur in organic or inorganic hodies, arne so large as readily to be seen with the aid of a microscope, or even with the unaided eye. Such pores may be seen in wood, and in many minerals. By slight pressure, mercury can be forceld through the organic pores of' a piece of' wood, and hydroplhane, a kind of agate, is opaque when dry, but on immersion in water absorbs that liquid, increasing in weight, and at the samne time becoming translucent. The disintegration of rocks by frost is an effect due to the structural pores. In the wet weather of autumn, water enters them which in winter is frozen, and by its expansion causes the rock to crumble. 36. Mobility. Constant experience shows us that matter has the property of mobility, or capability of motion from place to place.

PARTICULAR PROPERTIES. 29 37. Indestrnctibility Matteris incdestructible. However greatly the external appenarance of a body may be chalnged, not the smallest particle is actually destroyed. Thus, when a piece of wood is burnt, the only visible remainlder is a small quantity of ashes, yet were we to collect the carbonic acid and aqueous vapor formed durin: the combustion, and weigh them, the sunm of their weights, together with that of the ashes, would exactly equal the original weight of the wood. The s:ame is true in all other calses in which matter is seemingly destroyed, owing to its assuming an invisible state. 38- Ether. The phenomena of light, heat and electricity, have led to the supposition that in. addition to the three ordinary states of matter there is a fourth condition in which it may exist, and that such matter, known as the ether, is universdally distributed, penetrating the pores of all bodies, and diffused throughout all space, existing even in the lmost perfect vacuum. The m.ajority of scientists regard this ether as a subslance distinct from all others, thougch some maintalin with much pIlausibility that it is merely ordinary matter in a very tenuous state.' Ether has fiequently been called an impondcerable fluid, owing. to an idea once very generally entertained that it possessed absolutely no weighlt,. It is maore probable, however, that it is ponderable, like all other matter, the impossibility of our weighi ng it proceeding friom the fact that we canllot render any substance devoid of it, a necessary condition for ascertaining its weight. 39. Forces. The va ious changes which are constantly taking place in matter invariably aplpear to us as forms of motion, either of the body as a whole, or of its molecules. Any change of place is evidently- a motion as a whole, and we shall see hereafter that all other changes, as, for example, those of templ:erature, are forms of molecular motion. In general, whatever produces the effect we denominate a force, so that we may also define a force as anything that produces, or tends to produce, motion. That the actual production of motion does not necessarily followx upon the.action of a force, arises from the fact that the tendency to motion thus irnpressed is resisted by some other force opposed to the first. Thus, a body suspended upon a sprin-g-balance tends to fall to the earth because of its weight; but it does not fall, since this downwardl force stretches the spring and develops in it a certain amount of tension, which is exactly equal and opposite to the weight of' the body. The frces acting upon matter are of two kinds, cttractive and repzlsive forces, the first tending to make particles or masses of mrltter approach, the second tending to cause them to recede fiom See An Essay on the Correlation of Physical Forces, by W. R. Grove (New York, Appleton & Co., 1865), E. L. Younmans, Editor; p. 133, et seq.

30 PROPERTIES OF MATTER. each other. We may also classify forces according as they act at sensible distances and between mnasses, or only at insensible distances and between molecules. The latter are denominated moleculcr forces. The princip:ll forces acting at sensible distances, i. e., at distances exceeding T-0 of a millimetre, are the attraction of gravit~ation, and the attractive and repulsive forces of magnetism and electricity, while those acting at' insensible distances are cohesion, adhesion, capillctrity and chemnical canity, together with the repulsive force due to heat, and certain other molecular repulsions. Gravitation acts among all bodies, causing themn to tend towards one another; the forces of electricity and magnetism are manifested only under peculiar circumstances, and may be either attractive or repulsive. Cohesion acts to unite particles of the same kind of matter in a mechanical union; thus the molecules of a piece of iron are held together by this force. Adhesion causes a mechanical union of.particles of different kinds of matter, as the particles of two pieces of wood are united by glue. Capillarity is a manifestation of force developed by the united action of adhesion and cohesion; and chemical affinity causes the chemical union of unlike particles of matter to form an entirely new substance. These, together with the repulsive forces which we have mentioned, will be discussed in detail hereafter. The attractive forces acting among the molecules of a body would cause them to approach and come into absolute contact, were it not fbr the coincident action of a molecular repulsive force. This just balances the molecular attraction, so that the particles are kept at a certain distance apart. If they be forced nearer together the repulsive force is increased, and they tend to separate, while if the distance between them is increased by any external force, the molecular attractions cause them to tend to approach and assume their original position. 40. Polarity. Certain bodies have the property of exerting different forces at opposite ends, so that a body which is attracted by one end will be repelled by the other. The simplest illustration of this action is the case of two magnets. Suppose them to be balanced so as to move freely about a vertical axis; then, as is well known, one end of each will point towards the north. Denote by A the end of each, which assumes this position, and by -B the end of each pointing toward the south. Then it will be found that if the A end of one magnet be presented to the B end of the other, an attraction will be exerted between them, while if the A end of one be'approached to the A. end of the second, or if the two B ends be placed near together, they will repel each other. REFERENCEs. Size of Molecules, by Sir Wm. Thomson; Naature, Vol. I., p. 521. Letter on - the Size of Molecules, by Sir Win. Thomson, read before the

FORCE AND ITS MEASURE. 31 Manchester Lit. and Phil. Soc., Mllar. 22, 1870. Published in Nature, Vol. II., p. 56. Nature of Ether. See Essay on the Correlation of the Physical Forces, by W. R. Grove. (Youznman's C'ompilation.) CHAPTER IV. FORCE AND ITS MEASURE..- LAws OF MOTION. 41. M1otion. Motion is a progressive change of position in space. We can be acquainted with none but relative motions, for we have, no means of' ascertaining the fact that a bbdy really changes its position except by comparison with some other body not affected with the same movement. To know the absolute motions of any body this point of comparison must be absolutely at rest. But'we can find no such point. The earth and other planets have a double motion about their axes and around the sun, the sun moves on its own axis, and also probably in an orbit around one of' the fixed stars. Hence terrestrial bodies can only be at rest relatively to each otler and to the earth, and as we must refer their motions to bodies which are moving themselves, the absolute motion of any point can not be determined. Moreover, an apparent condition of rest of at body as a whole,. is compatible with movements of great energy among the particles of which it is composed. We shall see as we proceed that we have reason to believe that the molecules of all substances are in a constant state of vibration to and fio, these vibrations being so minute anda rapid that we can not perceive their existence except by their effects. There is no such thing, then, as absolute rest, and when the term rest is used it is to be understood as relative. 42. Velocity. The rapidity of motion is measured by the velocity, which is the linear space passed over in a unit of tille. Thus a body moving over 10 metres in one second is said to have a velocity of 10. 43. Force. - Pressures and Impulses. A force, as already stated, is any cause tending to produce or modify motion. Mechanics is the name given to that branch of physics which treats of the laws of force in general. Mechanical forces are commonly divided into pressures and impuloese, according as the time of their duration is sensible or insensible in magnitude. Thus the weight of' a body resting upon a surface is an example of a pressure, while the blow of a hammer in driving a nail is an example of an impulse. In the case of the supported weight the pressure tends to pro

32 FORCE AND ITS MEASURE. duce motion, but this tendency is resisted by the reaction of the sulface on which the body rests. If this be removed motion ensues, and the body falls. Evidently the tendency of' an impulse may be resisted in a similar manner. The difference between these forms of force, however, is only one of degree and not of kind, for impulses are, in reality, only pressures acting during a very short time. The impulse given to an arrow by a bow, for example, is due to the continued action of the bow-string for a fraction of a second. The impulse given to a cannon-ball is caused by the pressure of the gases fonrmed by the combustion of the gunpowder duriing the time that the ball occupies in passing through the whole length of the bore. So when motion is destroyed, as in the case of a cannon-shot fired into a wall, the body is not brouglht to rest immediate.ly, but penetrates a certain distance, in proportion to the magnitude of its moving force. Since an impulse is a pressure acting (luring an infinitely short timne, it follows that any pressure may be considered as caused by a succession of impulses repeated at infinitely small intervals. 44. Measure of Pressures. In order to estimate the magnitude of a pressure we make use of instruments called (cynamoemeters. The common spring-balance (Leroy's clynamometer), Fig. 8, is an example. It consists of a helical steel spring fixed in a friame, and connected with an index moving over a graduated scale. On the application of a force at B the spring is coiled more closely, and from the amount of this coiling, as shown by the index _;, the force is estimated. Another form of dynamnometer is shown in Fig. 9. The arcs A C, BD), are connected with the arms of a steel spring }, which is bent more closely together in proportion as the weight suspended firom C is greater. The magnitude of the force is read by the scale upon 2BD). Various other forms of dynamometers are used, varying in construction according to the nature and magnitude of the force to be measured. The graduation of the scales of all these instruments is performed by experinlent, known weights being applied, and the corresponding positions of the index make-d. If the plessure is so exerted as to cause motion, as in the case of a horse drawing a canal-boat, or a locomotive moving a train of cars, the resistance or pressure produced can be measured by interposing a dynanometer in the line of its action. 45. lieasure of Moving Forces. In many cases, especially where the moving forces are impulses, it is impossible to measume them in the manner described above, and hence it is important to have some other means of estimating their magnitude. A method by which we may compare all forces is by their effect in producing or destroying motion. If, for example, we see a man throw a ball with a velocity of 5, while another man throws the sam8 ball with a velocity of 10, the time of action of the arm upon the ball being the same in both cases, we infer that the second man exerts twice as much force as the first. This is merely reasoning from the

MOMENTUM. 353 effect to the cause, measuring the latter by the former, on the principle that the cause must be proportional to the effect. It will here be necessary to define two terms which we shall frequently employ. The mass of a body is the quantity of matter that it contains. Equal mnasses contain equal quantities of matter. Acceleration is the velocity generated in a body by the action of a force during a unit of time. The measurement of forces by the motion which they produce depends upon the following proposition, which is verified by universal experience. (I.) Constant forces are proportional to the products of the masses on, which they act, by the accelerations impressed upon those masses. Hence if F, F', be two forces, and 1 AlM', masses upon which they impress accelerations a, a', respectively,: _F':: la: AM'a'. (2) Thus if two forces acting for 1 second generate velocities 1 and 2 in bodies whose masses are 3 and 5, respectively, the forces are to each other as I X 3: 2 X 5, or as 3: 10, that is,: F':: 1 X 3: 2 X b::3:10. If in (2) we suppose the masses of the bodies to be the same, 1= f-AM', and F: _F':: Mal: Ala', or F.: a':: a:a' (3); that is, (II.) Constant forces are to each other as the accelerations which they impress on equal masses. Thus if two forces acting for the same time colmmunicate velocities 3 and 5 to a body, they are to each other as 3: 5, that is, F: Fl:: 3: 5. If in (2) the accelerations a, a' are equal, F: F':: Ma:M'a or, F: F':: AF: A' (4); that is, (III.) Constant forces are to each other as the masses on which they impress egqual accelerations. Hence if the forces F, Fl, acting during a unit of time, communicate equal velocities to masses 8 and 15, the forces are to each other as 8: 15, that is, F iF'::8:15. The preceding propositions are of fundamental importance, and should be thoroughly mastered by the student before proceeding farther. 46. Momentum. The product of the massa of a body and its velocity is known as its mnomentum or fJbrce of motion. The momentum of a body is therefore proportional to the quantity of matter which it contains, and also to the velocity with which it moves. If bodies whose masses are as 3: 5 have the same velocity, their momenta are as 3: 5; if the bodies are equal, and their velocities 2 and 3 5

34 FORCE ANI) ITS MEASURE. respectively, the momenta are as 2:3; and if the masses are 3 and 5, and their velocities 2 and 3 respectively, the momenta are as 3 X 2: 5 X 3, or as 6: 15. The fact that momentum depends on both velocity and mass, is susceptible of numerous illustrations. A small cannon-ball moving with a high velocity is capable of doing an immense amount of' damage, though its mass is comparatively small, since its rapid motion gives it a great momentum. A large ship, on the other hand, though progressing with an almost imperceptible velocity, will overcome X great resistance. Thus strong cables are sometimes snapped when the vessel scarcely seems to move, its great mass making up for its small velocity. So a slowly-moving iceberg coming in collision with a ship may destroy it. 47. Momentum a Measure of Force. Proportion (2) may be stated thus: -Forces are proportional to the momenta generated by their constant and untform action dluring a unit of time, or during equal times. The momentum impressed upon a body by any force acting for a unit of time is, then, a measure of that force. The momentum which can be generated by the action of any force during a given time is evidently a constant quantity, whether the body acted upon be large or small. Hence if a large body is put in motion by a given force it receives a proportionally less velocity than one of smaller size. In fact, if thea force F will cause accelerations a, a', in bodies of mass 1 31M', respectively, the momenta generated are Mlia and Jl a'. As these are equal, Mt = — y'a', whence a: a'::': M (5); that is the velocities 1 impressed upon two bodies by the action of the same or equal forces during equal times are inversely as their masses. If a llass is already in motion, and a force acts in opposition to it, the momentum destroyed by the force is equal to that which it would generate in the same time by its action on that mass when at rest. Forces are therefore proportional to the momenta destroyed by them in equal times. In all the preceding discussion two things have been assumed; first, that the action of the forces is constant and uniform; and(, secondly, that they act during equal times. Any variation in the intensity of the force acting would of course render our demonstrations invalid, because we should, in reality, be comparinl different forces at each moment. The forces compared must also act during equal times, because it is obvious that the longer a force acts the greater the velocity it will produce. 48. Measure of Mass. We have already defined the mass of a body as the quantity of matter contained in it. The question now arises, how shall we measure this? One method is furnished by the proportions just demonstrated. Since the momen1 We can evidently write for a, a', the velocities generated by the action of the forces for a unit of time, any velocities whatever, by making our unit of time larger or smaller, as the case may be.

MEASURE OF MASS. 35 turn which a given force can generate in a unit of time is a constant, and the force is therefore measured by this momentum, by choosing a suitable unit of mass,l we may write F = Ma; whence a (6). The mass of a body may then be expressed by the constant ratio between any force and the acceleration zwhich it vill produce when acting up2on that body. It will be shown in a succeedingo chapter that any body when allowed to fall freely under the influence of gravity acquires a velocity of 9.8 metres per second, a quantity which is usually designated by the letter q. The force causing it to descend is evidently its weight W. Here a force W generates an acceleration g, and as forces are proportional to their accelerations (3), F: TV:: a: g (7), whence = g- or M= - (8)' a g # The mass of any body is therefore its weight divided by the velocity which it would acquire by fallilng freely bfor one second. This quotient is a constant, for if W17varies firom any cause, g, tlhe acceleration produced by it, must also vary in the same ratio (3), so that the value of -will remain unchanged. This mode of indicating the mass of a body is the one which has usually- been follow-ed in modern treatises upon the subject, but some inconveniences, the nature of which we shall presently explain, have more lecently led to the use of another nmethod.2 The weight of a body may be considered under two aspects: First, as denoting the force with which it is drawn towards the earth; and secondly, as denoting the mass of the body as compared with the mass of an arbitrary- standard, such as the ranume or pound. Now if by the weight of a body stated in grammes or pounds, we understand the force with which it is attracted towards the earth, that is if we consider the standard unit of weight to be a unit of f)ree, we cannot express the nmass of a body in grammes or poullds, because, as we shall see hereafter, the weight of a given quantity of matter, and hence the quantity of matter contained in a: given weight is not the same at all places upon the eartlh's surface, so that a body of constant mass will possess different weights at the equator and poles. But if bv the weight thus expressed we understand merely the quantity of matter contained in the body, compared with the quantity of matter contained in the arbitrary standard of weight of the French or English systems; For the momentum llIa being pr1oportionsal to the force F, the latter must equal a constrlt multiple of the momentum, whatever be the uliit of masss chosen, and if this unit be taken so that when F = 1 and a = 1, M also equals 1, or F =- Ma. 2 See A Tr eittise onr Natural Philosophy, by Thomson and Tait, Vol. I., p. 166.

36 FORCE AND ITS MEASURE. that is, if by a gramme or a pound Awe understand a quaintity o matter equal in mass to the standard gramme or pound, we can express the mass of a body in such lllit, Which, though arbitrary, are yet defilite and invariable. Inl this sense it is perfectly correct to speak of the mass of a body as being so many griammes. Now in common life this use of wei'ht as an estimate of mass is far more general than its use as an estimate of force. In strict language, to be sure, weight is the downward tendency of a body, but we ordinarily employ weights "for the purpose of measuring out a definite quantity of' mlatter; not an amount of matter which shall be attracted to the earth with a given force." Hence our standards of weight are primarily intended as units for measuring mass, and it is a secondary application by which we use them to estimate forces. The latter method of measuring mass is therefore far more simple than the former. 49. Unit of Mass. According to the first method of measurement the unit of mass is the mass of g times the standard unit of weight. For if the unit of mass is so chosen that Mi- - it is evident that if 3M 1, TW must be numerically equal to g. Hence the unit of mass is the mass of g grammes, or g pounds of matter. The objection to this mode of estimating mass is now apparent. Since the quantity g is a variable, the unit of mass is also a variable quantity, and it is exceedingly desirable to have some unit which shall be illvariable. This is furnished in the second method of estimating mass, as in that case the standard graimme or pound is taken as the unit. This gives us an absolute unit of mass, which can be obtained in no other manner. This unit was first brought into general use by Gauss. 50. Unit of Force. The unit of force is the force which, by acting upon a unit of mass for a unit of time, generates a unit of velocity. The unit of force, according to the first system, is the gramme in French, the pound in English measures. For the unit of mass in that system is g times the unit of weight, and the force with which this is attracted towards the elarth is g grammes or y pounds, which in 1 second generates a velocity of g feet, or metres. Hence, as forces are proportional to the accelerations which they produce in equal masses, the force which would generate an acceleation of 1 unit is equal to of that which generates an acceleration of g units, that is, to 1 gramnme, or 1 pound. This is called the gravitation utnit of force, and the system of measurement of masses and forces derived from it is called the gravitation system of measurement.

UNIT OF FORCE. 37 According to the second of these methods of mlealsurement, the unit of mass being a standard gramme or pound, the unit of force is the force which, acting upon a national standard unit of mass for a unit of time, generates a unit of velocity. This unit of force is numerically equal to the unit of mass divided by the acceleration of gravity, that is, to - grammes, or pounclds, the value of g being the acceleration at Paris or at London, according to the system used. It is known as Gauss' ctbsolute unit of force, and the system of measureient of masses and forces based upon it as the absolute system of measurement. To obtain a clearer idea of the value of the absolute unit of force, we must know the numerical value of g. The value of g at London is 32.1889 ft.; and at Paris 9.8087 metres. Hence the British absolute unit of' force is eqlal to the weight of n-l.b-,g -ls. at London, and the French absolute unit is equal to the weight of s grarmmes at Paris. That is, in round numbers, the British absolute unit of force is equal to the weight of about half an ounce, and the French unit to about the weight of ~ of a gramme. We may evidently define the French unit of force as that force wohich, by acting upon a cmass of 1 gramme for 1 second, generates a velocity of 1 metre, and the British unit as that force which, by acting apon a?Zmass of 1'pouc for 1 second, generates a velocity of 1 foot. To transform forces expressed in gravitationl units to absolute units, we must evidently multiplly their numerical value by yg. Thuls a force of 10 grammes is equal to 10 X 9.8087 French absolute units of force, and a force of 10 lbs. is equal to 10 X 32.1889 British absolute units. The student will notice that the equation TV 1-MJ, which according' to the first system of mass nleasurement denotes the weight of a body of mass 1J/ in the second system denotes the number of absolute units of force in the downward tendency W,; caused by gravity. 51. Representation of Forces by Right Lines. In many problems it is' convenient to use a graphical method of representing forces. A force is defined when its macnitude, direction and point of application are given. Hence we may represent the relative lmagnitulde of forces by straight lines, whose lengths bear the same relation to each other as the numerical values of the forces themselves, while the directions of these lines may indicate the direction of the forces, afd the point firom which the lines are drawn the point of application. Thus two forces of 1 and 3 kgr., applied at a sin<gle point and inclined 20~ to each other, would be represented by lines AB, AC, Fig. 10, of lengths 1 and 3 units respectively, drawn fiom a point A, and making an angle BA C -_ 20Q.

38''THREE LAWS OF MOTION. 52. Three Laws of Motion. Law I. The elementary principles relalting to the phenomena of motion and force were arranged by Newton undler the laws often known as the _Newtonian _Laws of Motion. LAW I. A bo dy at rest continsues in, that state, an:td a body in, motion proceecds umn.iformny in a straigqht line, unless actedupon by some external force. This law is directly deducible firom our fundalnent: l ideas regarding clause and effect. Since every effect requires some cause to produce it, it follows that a bo(ly unaffected hy any force must remain in the state in which it already exists.. A body at rest has no power to put itself in motion; a body in motion h-s no power to bring itself to rest, or to deviate from its path, but must continue to move in a fixed direction. The law is also verified by all experience. The ilcapacity of bodies at rest to chainge their condition is so obvious as to need no illustra~tion. But the case when they are in invtion is not 1lwrays so clear. Thus a. body thrown horizontally dloes not move uniformly in the line of projection, but proceeds Nwithl.: gradually diminishing horizontal velocity, until it comes to rest onl the ground. This decay- of motion seems to contradict the secolld portion of the law. But a closer examination shows thalt there is here a force acting upon the body constantly tending to destroy its motion, viz., the resistance of the air; w-hile another force, the attraction of gravitation, continually draws it towards the earth. Ag'ain, a ball rolled upon a roald does not move on indefinitely. but soon slackens its velocity-, and is finally brou.ght to rest. This is no more an exception to the law, however, than the other c:ase, for it is the firiction of the ball upon the road that c;auses its loss of motion. If the ifriction is diminished the distance to which the ball will roll is proportionally increased: on a comllnton road it stops very soon, on a smoothl bowling-green it roll's much farther, while on a sheet of cletar ice it goes a very long (list.:llce before stopping. In cases where the friction and resistance of the air are reduced to a minimum, we obtain a very long continualee of motion. Thus a nicely-balanced wheel, moving- on fine, wvell-oiled be<arings, on being set in rotation will revolve for a very long while. A(ain, an ordinary clock-penduluml, if detached from the rest of the works and swung, will come to rest after a short interval, but some pendulumns of very delicate construction, moving with exceedingly little friction will swing in the air for nine or ten hours, and in a v:icuuml for twenty-four hoursl'. Since, thenl, whenever a diminution of motion occurs, we are able to find forces causing it, and since the condition of uniformity of motion is approached in proportion as we do away with these forces, we are justified in concluding that could we eliminate them altogether the loss of niotion would entirely disappear. So in all cases of motion in curved

LAW OF INE'l:TIA. 39 lines, as in those just cited of the wheel and pendulum, we find forces at iwork which deviate the body fiom the rectilinear path it would otherwise p-ursue. We therefble conclude that the law, as stated, is tlrue in all cases. Still filrther, the consequences deduced fiom other laws of mnattel, supposing this law to be true, agree with the results of observation, which could not be the case if the law twere false. Thus in Astronomy, the methods used in predicting eclil.pses assume that the motion of the ealth and moon is not altered except by the action of some external force, I:nd as the observed and computed timles agree, we are justified in asserting the truth of our proposition. The same remark applies to the relaining laws of motion. 53. Resisting Medium in Space. The question will be asked here, " Is there any example of' permanent motion in nature? " The revolution of the planets around the sun ofirs the nearest approach to perpetual motion with which we are acquainted, but even in this case there is evidence of the existence of a resisting medium actino to slacken their velocities. [lhis medium is so light that its effects are only perceptible in the retardation of a single comet (Eneke's), but if it exists it must nevertheless act in the same manner upon all bodies circulating about the sun. The reality of the existence of a resisting medium, though generally believed, is dliscredited by as high an authority as Sir J. F. W. Herschel, who explains the retardation of Encke's comet in a different manner. 1 54. Phenomena illustrating the First Law of Motion. The truth expresse(l in Law 1. is the principle of the InJertia of Matter. The term inerhiia in strict lanruage means simply the inability of matter to change its state except under the action of' some fobrce. It is also universally used in a somewhat different, though analogous sense, as denoting that property oj' matter because of which a definite force is necessary to produce a gyicen chanle in the existing state of a mzass, which is simply another mode of expressing the general idea of' momentum. By means of the first law of motion in connection with the principles of momentum, numerous familiar phenomena are readily explained. The apparent resistance experienced on setting a body in motion, or on stopping a moving body, is a consequence of its inertia. A person riding on horseback at a rapid rate is thrown forward if the horse suddenly stops, because the momentum of his own body carries him onward in the direction in which he was moving. The same thingr occurs to the passengers in a railway train stopped by a sudden application of the brakes. If, on the contrary, the car is started quickly they are thrown backward, the motion of the car not being immediately communicated to them. It is because of the inertia of matter that the effects of the collision of trains of' cars are so terrible. The locomotive is suddenly brought to rest, while the cars continue to move, thereby piling one upon another, and causing a general destruction. For a like reason when a vessel moving at full speed strikes a sunken reef, the spars upon her deck and even the sailors are sometimes shot violently forward over the bow. 1 See Herschel's Outlines of Astronomy, 11th Ed., 1871, ~ ~ 577, 570.

40 THREE LAWS OF MOTION.'Coursing owes all its interest to the intuitive consciousness of the nature of inertia which seems to govern the -measures of the hare. The greyhound is a comparatively heavy body moving at the same or greater speed in pursuit. The hare dolubles, that is, suddenly changes the direction of her course, and -turns back at an oblique angle with the direction in which she had been running. The greyhound, unable to resist the tendency of its body to persevere in the rapid motion it had acquired, is urged forward many yards before it is able to check its speed and return to the pursuit. Meanwhile the hare is gaining ground in the other direction, so that the animals are at a considerable distance asunder when the pursuit is recommenced. In this way a hare, though much less fleet than a greyhound, will often escape it." I We see a practical application of the first law of motion in the method often used of fixing an axe or hammer firmly on its handle. The tool being placed in a vertical position, with the head uppermost, is moved rapidly downward, so that the end of the handle strikes against some solid object. The motion of the handle is stopped, while the head moves on, thus fixing itself firmly. The principle under consideration may also be illustrated by a very simple experiment. Let a smooth card be balanced on the tip of one of the fingers. On this place a somewhat heavy coin. If, now, a quick, horizontal blow be given to the edge of the card with the fore-finger of the other hand, it flies from under the coin, leaving thlis poised upon the finger. The slight force of friction exerted by the moving card upon the coin, is not sufficient during the short time of its action to impress the motion upon the latter. If the card be pushed slowly the coin moves with it. 55. Time required to produce Change of State in a Body. From these examples, especially the last, we see that if' a force acts upon only a few points of a body, a sensible time is required to transmit its effects to every portion of the mass. Several curious phenomena are explained by this fact. A light stick supported by placing its ends upon the edges of two wine glasses will be broken by a quick blow, without affecting its supports, while a less sudden stroke will be liable to break them. In the former case the stick is snapped before sufficient time has elapsed for the blow to be communicated to the glasses. A rifle ball fired through a pane of glass cuts out a round hole, because it progresses too rapidly to allow the motion to be impressed upon any of the particles of the glass, except those immediately in its path, before it has passed entirely through. A slowlymoving bullet, or a stone thrown against the pane, cracks it in every direction. For the same reason the greatest damage in naval combats is caused by shot moving with a comparatively slow motion, as they cause far more splintering of the timbers than a swifter projectile. In like manner a cannon ball may be fired through a partly open door, scarcely moving it on its hinges. A candle fired from a musket will penetrate a board without being greatly crushed. A certain time is required to chlange the form of a body by crushinlg, and as this is greater than the time required to overcome the cohesion of the fibres of' wood composing the board, the candle goes through before ninch change in its form can take place. It is because of the time necessary to generate motion in a body that when a horse attempts to start a heavy load by a sudden pull, some part of the harness is liable to give way, while the load remains unmoved, though 1 Lardner, Treatise on Mechanics, p. 39.

LAW OF INDEPENDENCE OF MOTIONS. 41 a slow and steady pull would have put it in motion. In the case of long trains of' heavily loaded cars, a great gain ensues fiom the slight movement allowed by the couplings which unite them to one another. That the train may be set in motion it is ne, essary to overcome the friction of all the wheels resting upon the rails. The locomotive easily exerts force enough to put the first car in motion, as this can be fairly started before the second is moved at all, because of the slackness of the connecting coupling, and so on for each car in succession. The train is thus put in motion in parts, the inertia of the cars already moving aiding the engine in starting the remainder. Were the whole train rigidly connected, it would be impossible for an ordinary locomotive to cause it to move. When a person falls from a considerable height upon a rock he is severely hurt, but if upon a soft bed he is uninjured, because in the latter case the stopping of the motion is performed gradually, so that the violent shock otherwise produced is obviated. 56. Matter being purely passive, it follows that if any force, however small, act upon a body, it must produce a proportionate amount of motion, supposing the body to be absolutely fiee to move. In practical cases there is a certain resistance to motion offered by friction and other causes, which must be overcome before movement can take place. The least fbrce in excess of this will start the body.' Hence it is a general law that in the absence of resisting actions the smallest force is capable of moving the largest body. As an illustration of this it is stated1 that in calm weather and smooth water a very large ship can gradually be put in motion by the efforts of a child pulling at a rope attached to the bow. The effect of the force in producing momentum is of course exactly the same as if it were expended in giving a rapid motion to a small body. 57. Resistance of Media. The resistance offered by a fluid to the motion of' a body is due to the pushing aside of its particles by the moving mass, which thus loses a portion of its momentum. The denser the fluid, that is, the heavier its particles, the more resistance it offers to movement in it. Thus a delicate pendulum swung in the air will oscillate for some hours before being brought to rest, while if swung in water it will move for only a few minutes. 53. Law i]". If' several forces act upon a body simultaneously, each one of these produces the same efect in magnitude and direction as if it acted alone. This principle is known as the Law of the n;dependlence of 3lfotions. The truth of the second law of milotion may not appear at first sight. The effect of mechanical forces is to produce motion alonc their lines of action, and the qaestion will be askedl, "How c'n several motions in different directions coexist in a body?" The difficulty disappears, however, if it is recollected that all movement in a body is estimated by reference to some point not possessing the salne motion. The absolate motion of a mass can evidently have but a single (direction, but relatively to celrtain points a body may have several simultaneous motions. For examlple, a man travelling from southeast to northwest is at the same time moving toward the north and towards the west, so that after the lapse of an interval Leslie, Elements of Natural Philosophy; Vol. I., p. 30. 6

42 THREE LAWS OF MOTION. of time he will have passed over a certain number of miles in a northerly, and a certain other number of miles in a westerly direction. 59. Illustrations of Second Law of Motion. Various illustrations may be cited in proof of this law. The muscular exertion put fobrth in walking a given distance upon the deck of a steamer is the same, whether the boat is at rest or in motion. If the vessel moves at the rate of five miles an hour with reference to some point on the shore, and a person walks over the deck in the same direction and at the same rate, his velocity relative to that point will be ten miles an hour, and relative to the vessel five miles an hour. If, on the contrary, he walks in an opposite direction, his velocity relative to the vessel will be five miles, but relatively to the shore he will be at rest. Again, imagine a boy sliding upon a moving cake of ice in a direction at ri(rht-angles to its line of progression. The boy will be affected with both motions, moving down the stream as rapidly as if he were not sliding, and sliding as rapidly as if the cake of ice were at rest. So two persons standing on the deck of a steadily-moving vessel, can toss a ball to each other as readily as if the boat were at rest. To take the illustration first used by Galileo, a person on shipboard can write with ease, while in his pen coexist the motions caused by the hand, the swaying of the ship and its progressive motion, and the axial and orbital revolutions of the earth. But as the paper possesses all these movements except the first, the letters are traced exactly as if only this motion existed. A ball dropped from the mast-head of a moving ship will strike the deck at the foot of the mast, because the forward motion possessed by it in common with the vessel, is in no way diminished by the downward motion communicated by gravity. A heavy body falling from a balloon which is moving rapidly in a horizontal direction, "during its descent partakes of the balloon's motion, and until it reaches the earth is always seen perpendicularly under the car." A body dropped from the summit of a lofty tower, or allowed to fall down a deep mine-shaft, strikes the ground a little east of the vertical passing through the point from which it starts. This is because the earth's revolution on its axis gives to all objects a motion from west to east, the velocity being greater in proportion to their distance from the axis. Hence the easterly velocity at the summit of a tower is greater than at its base, and a body starting from that point, while falling downwards, must at the same time move toward the east more rapidly than the base of the tower, and hence strikes the ground a slight distance to the east of the vertical. The same explanation applies to the fall of a body down the shaft of a deep mine. A marked deviation of this kind has been shown by experiments made in some of the mines of Saxony. In the case of circus-riders who leap through hoops from the back of a horse running at full speed, and again alight on the animal, it is simply necessary to leap upward and not forward, because their motion in common with the horse carries them onward through the air with the same rapidity as if no leap had been made. Finally, a cannon-ball exerts the same mechanical effect, whether it is fired towards the east, west, north or south, though its velocity with regard to any fixed point in space is vastly different in these different cases, owing to the rotation of the earth. Thus suppose the ball to be propelled from the cannon with a velocity of 470 m. per second. At the equator the rotary motion of the surface of the earth is also 470 m.

LAW OF ACTION AND REACTION. 43 per second; hence, if at any station situated in that great circle the ball be fired towards the east, its velocity relatively to the centre of the earth is 470 - 470 m., or 0 m. That is, in the latter case the ball comes to rest relatively to the earth's centre, while the surface moves on, leaving it behind. To us travelling east with the surface, the ball appears to have a westerly velocity of 470 m. per second. Tile destructive effect of the ball would manifestly be the same in either case. 60. Law III. Action and reaction are equal and in opposite directions, or the actions of bodies on one another are equal ancl opposite. This law is established solely by induction. To exemplify its truth it will be sufficient to examine a number of phenomena of different classes, and note its applicability to all. 61. Illustrations. A magnet exerts a force of attraction upon a bit of iron, but the iron at the same time attracts the magnet with an equal force. If the magnet is fixed, and the iron free to move, the latter will approach the formner. If the iron is fixed and the magnet free, the magnet will move towards the iron. If both are free they will approach each oth r. And this action is equal as well as mutual and opposite. For it is fbund by experiment that in the last case the velocity imparted to the iron is as much greater than that imparted to the magnet as its mass is less. That is, if M, 3I' are the masses; and v, vf the velocities, we shall have l: Mt:: v': v. Hence the momenta are equal, for M -= M3v', and therefore the forces generating these momenta are also equal. The action of the magnet, then, is equal to the reaction of the iron. It is to be observed that the terms action and reaction do not denote two forces, but are simply convenient terms by which we express the mutual and opposite actions of the same force. The magnet and iron are drawn together by the single force of their mutual attraction. So when the elasticity of an uncoiling spring pushes apart two bodies connected with it, there is really but one force at work, which moves them in opposite directions, though it may sometimes be convenient to consider the motion as caused by two equal and opposite forces acting from the middle towards the end of the spring. If the masses of the bodies are in any given ratio, as 1: 2, for example, the velocity impressed on the smaller will be twice that of the latter, so that the momenta will be equal; thus again veriftinr the law undler consideration. In the case of a bullet fired from a gun, the force of the exploding powcler propels the ball forward at a higher velocity. At the same tilme, however, it forces the gun itself backward, causing the recoil or kickinq of the piece. If' the force of the recoil be measured, it will be found that the momentum of the gun equals that of the bullet. When a man-of-war fires a broadside, the whole vessel sways in the opposite direction. Many other examples may be mentioned. In the exercise of rowing, the water is pushed backward, while the boat moves forward. A person leaping fiom a small boat pushes it away from him, as he springs away from it. It is because of this principle that a person cannot lift himself into the air by pullinc at his boot-straps. The upward pull on the straps (action) is just balanced by the downward push (reaction) of the feet. A case is related of a gentleman who undertook to propel a boat by erecting a large bellows

44 THREE LAiS OV MOTION. at the stern, and blowing against the sails,1 the attempt resulting in a signal failure, because the reaction of' the current of air as it issued from the bellows neutralized its action upon the sail. it follows fi'on the law of the equality of action and reaction, that if a body in motion strikes upon one at rest, the shock is the same for both, because the loss of momnemtum which the moving body sustains affects it in the same manner as if being at rest it was acted upon by an equivalent force in an opposite direction. If two bodies moving in opposite directions impinge upon one another, the same force is exerted as if' one while at rest was struck by the other movino with a monmenturn equal to their united momenta. This explains why the shock is so violent when two vessels moving in opposite directions run foul of one another. In the case of bodies thus impinging upon each other, though action and reaction are equal, the weaker will evidently be the more injured by the impact. The fist of a pugilist sustains as great a shock as that part of the opponent's body which it strikes, but is not injured, because firon its structure it is fitted to endure the shock. But if by acci(lent fist meets fist, one person feels the blow as much as the other. 62. History of the Laws of Motion. Simple and fundamental as are the three laws of motion, it was not until the beginning of the 1 7th century that they were un(lerstoo(l. Kepler, with all his success in astronomy, was ignorant of the principles of inertia. He supposed that moving bodies, if left to themselves, woul(l coine to rest, and therefore imagined a constant force acting upon the planets to keep up their velocity. Galileo, during his earlier researches, thought that the only naturally uniform motion was that performed in a circle, but in his Dialoques on Mechanics, published in 1638, lie gives a correct statement of the first law, thlou4gh it is not made sufficiently comprehensive in its application. The law, in a general form, was announced by Galileo's pulpil, Borelli, in 1667., The truth of that portion,of the law relating to the tendency of bodies to move in right lines was recognized bv Bendetti, as early as 1585. The principle of the independence of motions was also announced in the Dialogues on 1Iiechanics, but its complete demonstration resulted friomn the establishmnent of the laws of the motion of the earth by the astronomers of the 17th century, foremost amiong whom was Sir Isaac Newton. The principles of momentum underlyingr the third law were known to Galileo, but the laws of impact of bodies, as related to changegs in momentumn (a knowledge of which was evidently necessary to a general statement 9f Law III.), were first correctly stated by Wren, Wallis and Huyghens, In papers communicated to the Royal Society about 1609. The terms in which it is now so frequently expressed, "Action and Reaction are equal and opposite," are those used by Newton. REFERENCES. For information upon the subject of Units of Force, see Treatise on Natural'Philosophy, by Sir Wnm. Thomson and Peter G. Tait; (Oxford, 1867) p. 166. Numerous examples of the application of the laws of motion will be found in Elements of Physics, by Neil Arnott. Arnott's Elements of Physics.

COMPOSITION OF MOTIONS. 45 A Treatise on ]Mechanics, by Henry Kater and Dionysius Lardner. Handbooks of Natural Philosophy and Astronomy, by Dionysius Lardner; First Course. For further information upon the subject of a resisting medium see Outlines of Astronomy, by Sir J. F. F. V. Herschel; 11th English Ed., ~~ 577, 570. Reports on Observations of En/cke's C'omet during its Return in 1871, by Asaph Hall and Win. Harkness (Washington, 1871); p. 33. CHAPTER V. COMPOSITION AND RESOLUTION' OF MOTIONS AND FORCES. 63. Statics and Dynamics. The science of mechanics is divided into Statics and Dynamcrics. Statics treats of b:llanced forces, or forces in equilibrium; dynamics of the action of forces in producing motion. The demonstration of the elementary principles of statics requires the preliminary consideration of the composition of motions; that is, of the laws determining the path described by a body in which several motions coexist. The fundamental theorem upon which they all rest is the law of the independence of motions. 64. Parallelogram of M'otions, If a particle be simultaneously impressed with two uniform motions tohich separately would cause it to describe the adjacent sides of a parallelogram in a given time, it will describe the diagonal of Sthat parallelogram in the same time, and uniformly. Suppose a particle at A, Fig. 11, to possess at the same time two uniform motions, one of which would carry it over the line AB, the other A-D in a given time. It will move uniformly over the diagonal A C in the same time. For by Law II. (~ 58, p. 41) each motion takes place as if the other did not exist, hence the motion along AlD cannot affect the movement of the particle in a direction parallel to AB, and it will therefore proceed as far in that direction as if it were not impelled along A D; that is, at the end of the given time it must be found somewhere on B C. Also the motion along AD produces its full effect, as if the particle were subject only to it, and hence causes the particle to proceed as far in a direction parallel to A-D as if it were not impelled along AB. The particle must therefore be on the line DC at the expiration of the given time; and since it is alsoson B C it will be found at their intersection C.

46 COMPOSITION AND RESOLUTION OF MOTIONS. This motion is entirely performed in the diagonal A C. For let Ab, Ad, be the distances which would be passed over in any fi'action of the whole time if the motions took place separately. Then since, by supposition, the motions over A2B, AD are uniform, Time of passing over Ab: Time over AB:: Time over Ad: Time over AD; or, as the times are proportiollal to the spaces described, Ab: AB:: Ad: AD. Hence the parallelograms Abcd, AB C(D are similar, and the point c lies in the diagonal A C. But, by the first part of the demonstration, c will be the position of the body at the end of the given fraction of time; and as this is true whatever be the tilne chosen, the whole path must lie in A C. The motion in AC is uniform. For, by similarity of triangles, Ac: A C:: Ad: AD. But Ad: AD:: Time over Ad: Time over AD, And Time over Ad: Time over AD:: Time over Ac: Time over A C. Whence Time over Ac: Time over A:: Ac: A (; that is, the spaces described are proportional to the times occupied in describing them, in which case the motion must be uniform. Also the velocity in the diagonal is to the velocity in either side, as the length of the diagonal is to the length of that side, since each is traversed in the same time. 65. Triangle of Motions. It follows from the preceding proposition that if a particle possess simultaneously two uniform motions which, if taking place in succession, and each continuing for the same interval of time, would cause it to descr.be two sides of a triangle taken in order, it will describe the third side in the same time. For the motions which carry a particle over AD, AB, Fig. 12, if applied at A, would by acting successively, carry it over AD, DCU. Hence the effect of their simultaneous occurrence is the same in either case, producing a motion in AC, the diagonal of ADCUB, and third side of the triangle ADC. 668. Component and Resultant Motions. The motions which are thus combined are generally known as component or elementary motions, while the single motion due to their combination is called the resultant motion. 67. Illustrations of Composition of Motion. The composition of motions may be illustrated by placing a ball upon a level square table, and communicating to it two equal impulses alone the sides; it will be found to move in the diagonal. If a white ball suspended before a blackboard have a horizontal and a vertical motion communicated to it simultaneously by means of cords, it will be seen to move in an oblique direction. The apparatus shown in Fig. 12 illustrates these principles very clearly. Within a rectangular frame, ABCD, slides a second frame, EFGH. A white disc K slides with an easy motion upon a rod FN, at

COMPOSITION OF FORCES. 47 tached to the inner frame. A cord attached to the disc runs over a pulley P at the upper extremity of the rod, and is fastened to the outer frame at B. Now it is clear that if the frame EFGH be drawn in the direction of the arrow, the disc K is carried horizontally with the rod FN, while at the same time it moves vertically over FN, owing to the action of the string which is fastened at B. Under the combined effect of these two movements, the disc will be seen to traverse the diagonal GL. Practically, we notice the composition of motion in the case of a boat rowed across the river, while it is at the same time carried down the stream by the current. Thile boat moves obliquely, reaching the opposite shore in the same time as if there were no current, but at a point lower down the stream. If the boatman wishes to reach a point directly opposite he must evidently head up the stream, so that the resultant of the combined motions of the boat may lie in a straight line joining the two points. For example, suppose it is wished to cross a stream directly from A to C, Fig. 13, the current being sufficiently strong to carry the boat a distance A D, while the boatman can row it over A B. The boat must be rowed in the line AB, in which case the motions in AB and AD give a resultant motion in A C. The time of crossing is clearly that which it would take to pass from A to B in still water. 68. Polygon of Motions. If a particle possess simultaneously any number of unijbrtom motions, which occurring successitely, and each continuing for the sotme interval of time, would cause it to describe all the sides but one of a polygjon, it will describe the remaining side in the same time, and with a uniform motion. Let a body at A, Fig. 14, be impressed with uniform motions, which, if taking place in succession, would cause it to move over the sides AB, BC, CD of the polygon ABCD, the time of describing each side being the same. Then by the theorem of the triangle of motions, the combination of the movements parallel to AB, BC, gives a resultant motion over A C. Now combine this resultant with the remaining elementary motion parallel to CD. The resultant of these, which is also the resultant of all three motions, is over the line AD. But AD is the fourth side of a quadrilateral, of which AB, BC, CD, are the other three sides. If a fourth elementary motion were present it could be combined with AD, the resultant of the first three (in which case the resultant would be the fifth side of a pentagon), and so on, indefinitely. Hence the proposition is general. The unitbrmity of the resultant motion is proved precisely as in the case of two com}bined movements. 69. Kinematics. The science which treats of the relative motions of bodies considered independently of the causes producing them, is called Kinematics. 70. Composition of Forces. By the aid of the preceding propositions we shall be able to demonstrate the laws of the effect of several forces when acting simultaneously. Just fs elementary or component motions combine to produce a single resultant motion, component forces combine to produce a single resultant.force. The findamental theorem relative to the composition of forces is the following.

43 COMPOSITION AND RESOLUTION OF FORCES. 71. Composition of two Forces. - Parallelogram of Forces, If two forces are represented in magnitude and direction by the adjacent sides of a parallelogram, their resultant will be represented in magnitude and direction by the diagonal. Let F, F', be two forces represented in magnitude and direction by the sides AB, AD, of the parallelogram AB CD, Fig. 15. Then will their resultant R be represented in magnitude and direction by the diagonal A C. For suppose X, F' to act separately upon a particle at A to produce motion in it. Since forces are proportional to the accelerations which they produce in the same mnass, or equal masses (p. 33, Eq. 3), the particle will traverse AB under the sole action of F, in the same time that it will move over AD under the action of F'. The velocities generated by the forces would then be such as to cause the particle to describe two adjacent sides, AB, AD, of a parallelogram in equal times. Hence if both forces act simultaneously, the combination of these velocities will produce a resultant motion represented by A C, which measures the force producing it. But this is evidently R, the resultant force due to the combined action of F and F'. Hence F, F', and R nmust bear to each other the salie relations in magnitude as A, AD and A (C. A C also represents the direction of the resultlnt because motion can take place only in the line of the force producing it. 72.'Triangle of Forces. Since AB DC, and AD, D) C, A C, form three sides of a triangle, it is evident that the same forces which are represented in rimagnitude and direction by AlD, AB C, AC, two adjacent sides of a pallrallelogramn and its diagonal, are also represented by AD), -DC, A C, three sides of a triangle ALD C. Hence, if two forces are represented in magnitude and direction by two sides of a triangle, the third side will represent their resultant. 73. Particular Cases of Combination of Forces. It is evident that if the angle A- 180~, the forces F, F, act in direct oppositition, and the resultant will equal the difference of the components, that is, R - F - F' (9). On the other hand, if' A - 0, they act together, and the resultant is the sum of the components, that is, R = F + 1F (10).1 Hence if we combine these propositions, and distinguish all forces acting fronm left to right by the sign -+-, and those acting firom right to left by the sign -, the resultant of any rnumber of'forces acting in the sanme straight line equals the aclelbraic sum of the comJmonents. If R be this resultant, and Z F the sum of the components,2 R =' F (11). 1 Geometrically, this follows from the Parallelogram of Forces, because if the angle DAB (Fig. 15) = 180o, AC becomes equal to AB - AD; while if DAB = 00, AC becomes equal to AB -+- AD. 2 The Greek letter Z is frequently used to designate the sum of any finite number of quantities expressed by a symbol placed after it.

COMPOSITION OF FORCES. 49 74. Illustrations of the Composition of two Forces. The composition of two forces may be illustrated by the case of an arrow fired from a bow. The bowstring A CB, Fig. 16, is brought into a state of tension by the bent bow, hence each half of the string exerts a pull at C towards the extremity of the bow to which it is attached, so that the fbrces acting upon the arrow lie in the lines CE, CG. Representing their magnitude by CE, CG, the diagonal CF represents the magnitude and direction of the resultant. Hence the arrow, when the string is released, moves forward in the line CF under a force bearing the same relation to the tension'of the string in either direction that CF bears to CE or CG. 75. Solution of Problems. Problems relating to the composition of forces are readily solved by the principle of the Parallelogram or Triangle of Forces, either by graphical construction, or by the application of trigonometry. Let F and F' be the given components, a the angle made by their lines of action, and R the required resultant. (1.) Graphical Solution. - To solve the problem graphically, lay off AB - F units of length, and AD - FP units (Fig. 17), making BAD - a. Complete the parallelogram ABCD; and the diagonal A C will represent the resultant R; for it is the diagonal of a parallelogram whose sides represent the magnitude and direction of'the components. Or lay off AB = F units, BC = Ff units, making ABC 180~ - a, the supplement of the angle made by the directions of the forces. The third side A C of the triangle thus formed, ABC represents R, since AB, BC, represent the components F, F'. (2.) Trigonometrical Solution.-Construct the triangle ABC as before. In it are given two sides, AB, BC, and the included angle ABC 180~ - a to find the third side A C. 76. Illustrations. To illustrate these methods let us take a very simple problem. Suppose a canal boat (Fig. 18) to be drawn by two horses, one on each bank, pulling in the directions AB, A C, by means of ropes attached to the bow of the boat, and making equal angles with the line of its keel. Let the pull exerted by each horse be 50 kgrs., the angle of inclination of their lines of action being BA C = 90~. It is required to find the resultant effect in moving the boat directly forward. To solve the problem graphically, lay off AD - 50 units, AE = 50 units. The angle DAE = 90~. Complete the parallelogram ADFE, and measure the nunmber of units in the diagonal AF, which will be the number of pounds acting to move the boat alon(r A G. The triugonotnetrical solution requires the determination of the hypothenuse AF of a right-angled triangle AEF, of which two sides, AE and EF = AD are given. As each of these lines - 5, AF = /(502 + 502) - 70 ~ 71 units, whence R 70 - 71 kgrs., or a little more than seven-tenths the sum of the components. 77. Composition of any number of Forces. Polygon of Forces. If any number of forces are represented in magnitude and direction by all the sides but one of a polygon, the remaining side will represent a single equivalent force. This proposition follows from the Parallelogram of PForces in the same manner as the Polygon of Motions follows friom the Parallelogram of Motions. Let AB, AE, AF, Fig. 19, represent three forces acting at a point A. The resultant of two of them, AB, AE, found by completing the parallelogram AB BCE, is represented by the diagonal A C. Combining this resultant with the 7

50 COMPOSITION AND RESOLUTION OF FORCES. other force AAF, by completing the parallelogram A C-DF, we find AD to represent the magnitude and direction of the resultant of all three forces. Bult 2A7 B C, hence the components arce also represented by AB, B C, CD, and these lines form three sides of a quadrilateral, of which AD is the fourth side. As this process of combination can be pursued indefinitely for any number of additional forces, the proposition is general. 78. Solution of Problems relative to the Composition of more than two Forces. Hence if a number of forces act simultaneously upon a body, their resultant may be found graphically, as follows: Construct a polygon by drawing lines proportional to the forces taken in order and parallel to their directions, and join the extremity oC the line representing the last force with the starting point. The last side of the polygon thus formed will represent the resultant required. For example, suppose a material point A, Fig. 20, to be acted upon by four forces, AB - 5, AC - 10, AD - 15, AE _ 20, so situated that BAC- 25~0, CAD = 40~, DAE = 75~. To find their resultant draw ab - 5 units of length, and parallel to AB; through b draw bc = 10, and parallel to A C; next draw cd - 15, and parallel to AD, and then de _ 20, and parallel to AE. Finally join a and e; the length of line ae represents the maonitude of resultant, and the anole bae is the angle which its direction makes with tile direction of the force AB. The value of' ae can also be determined by trigonometry by calculating successively ac from atb, be and abc, ad from ac, ad and acd, and finally ae from ad, de and ade. A simpler method of solution is furnished by analysis, which will be explained in the chapter treating of the analytical method of studying forces. 79. Parallelopiped of Forces. If three forces not in the same plane are represented in magnitude and direction by three edges of a parallelopiped, the diagonal of that solid will represent a single equivalent force. At the point A suppose three forces applied, represented by the edges AE, AB, AD, of the parallelopiped A G, Fig. 21. Then will the diagonal A G represent their resultant. For by the Parallelogram of Forces the resultant of AB and AD is represented by A C. But A C is one side of a parallelogram AEG C, of which A-E, adjacent to A C, is also a side. Hence the resultant of A C and the remaining force AE, which is the resultant of all three forces, is represented by A G, the diagonal of the parallelopiped. The lines AB, B C= AD, CG = AE, form three sides of a gauche or twisted polygon (that is, a polygon whose sides are not all in the same plane), of which A G is the fourth side. Hence the present proposition is a particular case of the Polygon of Forces, and as the method of combination used could evidently be applied to additional forces lying in different planes, the Polygon of Forces is true, whether the lines of action of the forces lie in the same, or in different planes.

LAWS OF EQUILIBRIUM. 51 80. Equilibrium of Forces. Forces are said to be balanced, or in equilibrium, when their resultant is equal to zero, so that they counteract one another. 81. Equilibrium of two Forces. If two forces are applied at a single point, it is evident that they will be in equilibrium only when equal and directly opposed. 82. Equilibrium of three Forces. In the case of three forces in equilibrium, the resultant.of any two of them must be equal and opposite to the third. For if the resultant of two of the forces is not equal and opposite to the remaining one, it may be combined with the latter, producing an unbalanced resultant force (~ 70), which is contrary to the supposition. 83. If three forces acting upon a point can be represented in magnitude and direction by three sides of a triangle taken in order, they will be in equilibrium. At the point P, Fig. 22, let three forces, F, F', Br, be applied in the directions PA, PB, PC, these forces being represented in magnitude and direction by the sides PA, AD, -DP, of the triangle PAD taken in order. The resultant effect of PA and PB is a force represented by P.D, which being the third side of the triangle PAD is equal -and opposite PC. Hence the three forces are in equilibrium. It is to be noted that the forces are represented by the sides taken in order; that is, the. directions of the forces are those assumed by the sides in going round the triangle. The three forces at P act in the directions of lines drawn from P towards A, from A towards D), and from D towards P. 84. Conversely, if three forces are in equilibrium about a point, they can be represented in magnitude and direction by three sides of a triangle drawn? parallel to their lines of action. If the forces PA, PB, P C, are. in equilibrium, any one of them, as PC(7 must be equal and opposite to the resultant PD of the other two. But PA, ADL PB, and P1), are three sides of the triangle PAD drawn parallel to the directions of the forces, hence PA, PB and PCU, are represented by the three sides taken in order. 85. Since two triangles whose sides are perpendicular, each to each, are similar, it follows that three forces in equilibrium about a point can be represented by three sides of a triangle drawn at right-angles to their lines of action. 86. The theorem of the equilibrium of three forces may be put in still another form, which is sometimes usefill in practice. From the demonstration of ~. 82, putting the result in algebraic fbrmn, we have, F: R:: PA: PD:: sin PDA.sin BPD: sin PAD. (12.) F': R:: PB: PD:: sin PDB -sin APD: sin PBD. (13.) Also BP-D 180~ - BPC, APD- 1800 - APC, PAD - 180~ APB, PBD - 1800 APB.

52 COMPOSITION AND RESOLUTION OF FORCES. Substituting these values in the preceding equations, we have, F R:: sin BPC: sin APB. (14.) F': R:: sin APC: sin APB. (15.) Hence, when three forces are in equilibrium, any one of them is proportional to the sine of the angle included between the directions of the other two. 87. Equilibrium of any number of Forces. That any number of forces may be in equilibrium about a point, the resultant of all but one of them must evidently be equal and opposite to the remaining force. 88. If the forces acting at a point can be represented in magnitude and direction by the sides of a polygon taken in order, they will be in equilibrium. For by a course of reasoning similar to that used in ~ 82, it will be seen that the resultant of all but one of the forces will be represented by the last side of the polygon, and hence will be equal and opposite to the remaining force. 89. Conversely, if the forces acting at a point are in equilibrium, they can be represented in magnitude and direction by the sides of a polygon drawn parallel to their lines of action. The proof of this proposition is evidently precisely similar to that used in ~ 83. 90. Experimental Verification of Laws of Equilibrium. The laws of equilibrium of three forces may be shown experimentally by the apparatus represented in Fig. 23. Over two small pulleys, A, B, a cord is passed, to the ends of which weights are fastened. Weights, C, are also suspended from any point D on the loop, which is allowed to move freely, and assumes such a position that the three forces acting upon it are in equilibrium. If now we construct a triangle having its sides parallel to DA, DB, DC, respectively, we shall find that the lengths of those sides are proportional to the forces acting at D, parallel to their respective directions. The triangle may be laid off upon paper, or on a blackboard placed behind the apparatus, or better still, we can make use of proportions (14), (15), measuring the angles ADC, BDC, ADB. Each of the three forces will be found to be proportional to the sine of the angle included between the direction of the other two. The angles are most conveniently measured bv attaching a small bead to each of the threads (as shown in the figure), a decimetre distant from D. Then with a millimetre scale the distances of each bead from the others is measured. These distances will be the chords subtending the angles between the threads, which can then be obtained from a table of chords.1 If additional cords with weights attached to their extremities be fastened to D and passed over suitably-arranged pulleys, the apparatus of Fig. 25 becomes suitable for verifying the laws of the combination of a greater 1 For a description of several new pieces of apparatus for class experiments relating to this subject, see a paper entitled Apparatus Illustrating Mechanical Principles, by R. H. Thurston; published in the Journal of the Franklin Institute, 3d Series, Vol. LXII., No. 3, p. 192. Also consult Reference Table of works on General Physics.

LAWS OF EQUIIBRIUM. 53 number of forces. It will be sufficient to allow D to assume its position of equilibrium, and to construct a polygon, 11having its sides parallel to the directions of the various forces. These sides will be found to have the same relative magnitude as the forces themselves. Illn the case of a number of combined forces, however, the friction of the pulleys and resistance of the cords causes a considerable variation between the theoretical and experimental results. 91. Practical Illustrations. The Jib and Tie-Rod, Fig. 24, employed in the common hoisting crane, furnishes a practical illustration of the preceding propositions. In the construction of such a machine, the frame must be made sufficiently strong to sustain the heaviest weight that is to be lifted, and in order to ascertain the strength to be given to each part, the maximum force to which it will be subjected must be known. This can be done by a simple application of the laws of equilibrium. Let P be the maximum downward pressure in kilogrammes, exerted upon the axle of the pulley at S (which can be determined when the maximum weight to be raised, and the angle JWSD are known). This causes a certain pull (strain) upon the tie AB, in the direction BA, and a compression (stress) in the jib A C, in the direction AC. The rod AB will then be stretched, and the jib A C compressed, until the resistances to further chance (v(which act in the directions AO, AM)f produce a resultant equal and opposite to P. At this moment let the strain on AB be denoted by T, the stress in AC by T'. P', T and T' are in equilibrium about A, and hence may be represented by three sides of' a triangle drawn parallel to their respective directions. From A draw -A 1N, in the line of action of T', MMN in that of T, and AN in the direction of P, that is, vertically downward. The forces P, T, 7'' are proportional to the three sides NA, 11MN, A Al, of the triangle AMIN, drawn parallel to their lines of action. Tha.t is, P: T:: AN: MAN:: BC: BA. (16.) P: T':: AN: AMl:: BC: CA. (17.) BA. CA whence T = P -fC' (18.) P (19.) As P, BA, CA and BC are known quantities, the values of T, T' (expressed in kilogrammes) also become known, and these determined, the proper size of the beams AB, AC, can be calculated by means of the fbrmulva for the strength of beams. Another interesting application of the foregoing principles is the mechanical contrivance known as the Toggle-Joint, or Knee-Joint, Fig. 25. It consists of two bars, AB, A C, connected by a joint at A, the other extremities resting upon the firm base PQ, antl the movable plate ANN, upon which a powerful pressure is to be exerted. A force P is applied at A by pulling or pushing in the direction AP. This evidently tends to raise NIN, and thus presses upon any object confined between that plate and a fixed platform above it. The arms AB, A C, will evidently be compressed until the reactions thus developed are in equilibrium with P. To estimate their magnitude, upon AP lay off AE proportional to P. The reactions T, T', developed in AB, AC, will be represented by EF, FA, as AE, EF, FA, are three sides of a triangle drawn parallel to the directions of the forces in equilibrium. Hence P::: AE: EF::sin AFE: sin FAE. (20.) P T':: AE: FA:: sin AFE:sin FEA. (21.) whence T P - sin FAE T sin PEA ence T n APP (22) T P (23.) sin A FE' (23')

54 COMPOSITION AND RESOLUTION OF FORCES. An inspection of the figure will show that as INlV rises ullner the influence of the force P, the angle FAD becomes more ol)tllse. whence also FAE and DADE FEA also increase, while AFE (i inli:hlles. HE ence, under these circumstances the values of T, T', as given in (22), (23), also increase, becoming greater and greater as FAD approaches 180~, when T and T' equal infinity.' By the action of a comparatively smnll force we may thus produce an enormous pressure. The Toggle-Joint is frequently used in printing presses for brinoing the type and paper into close contact, in machines for cutting large thicknesses of paper, in the cotton-presses at lMobile, etc. Its great advantage is that when in operation the pressure exerted by it increases simultaneously with the increase of resistance caused by the compression of the substance acted upon; it l:l.s the, disadvantage, however, that to obtain a large range of vertical,novemlent. of the plate 1MAN, the dimensions of the press miust be very consid(erable. 92. The equilibrium of forces when the point of'appl;cation is at rest, is known as statical equilibrium. When, the forces acting upon a body in motion are balanced, we have dynamical equilibriuma, the treatment of which is reserved for a future portion of this work. 93. Resolution of Forces. In parnlarapls 70 -78 we have explained the manner in which several forces can be replaced by a, single resultant. We now take up the converse problem of finding, two or more component forces, which.are equivalent to a single force. This process is known as the resoluttion of forces. Suppose A(, Fig- 26, represent any force, and let it be required to find two other forces which acting together would produce the same effect as AC. It is merely necessary to construct any parallelogram AB C-D on A C as a diagonal. The sides AB, AD, will represent the required components. Or the triangle AD C may be constructed, in which case AD, D C, represent the cdinponents. Since any number of triangles, AB C, AG DC, AEC,' APC, Fig. 27, can be constructed on a single line taken as a lbase, it follows that if A C lepresent a force the lines AB andl _B C, AD and D) C, AE and EC, AF and FC, will equally represent two component forces. Hence any force can be resolved in an indefinite number of tways. To resolve a force into two components whose directions are given, we must knoQw the angles BA C, BUCA, between those directions and AC. We have then given two anole:.:end the includ(ed side of a triangle, to find the remaining sides, Nwhich represent the components sought. This may be done either by trigonometry or graphically. 94. Resolution of a Force into two rectangular Components. It is frequently necessary to resolve a force into two components at right angles to ench other. This is done by constructing a rectangle, oi0 a right-angled triangle, upon a line 1 Since in this case FAE and FEA each - 900, and A1FE = 00, and T= TI sin 900 sinl 00 P = -.

RESOLUTION OF FORCES. 55 representing the given c ree. Thus if A (1, Fig. 28, represent this force, AB and AD, or DC an(d AD, will represent th e components. C:,llinl R the original force, F the component represented by A'B or DC, F' that represented by AD, and denoting the angle BA C = A CD by a, in which case ]DA C = 90~ - a, we have AB - A Ccos a, AD = A C sin a, whence F = R cos a (24), F' = R sin;a (25), general equations for the resolution of a force into trwo( rectangcular components. 95. Equation of Relation between Components and Resultant. Denote the forces represented by AC, AB, BC, Fig. 27, by R, F, F', respectively, and call BA C = a. BCA = /. Then as AC AB cos BA!- + BC cos BCMJ, R - F cos a + F' cos P. (26.) Hence the resultant of any two forces is equal to the aelgebraic sum qf the producls of each'oonpoaenf, ito the cosine of the angle which it mwakes with the resultant. 98. Examples of Resolution of Forces. Let PQ, Fig. 29, be a body of weigllt W resting upon a horizontal table MITV. It is required to find the downward pressure upon the table when PQ is acted ul)on by a force R, representedl by C'A. Let 1t be resolved into two rectangular components, one F, represented by CD, at rio'ht angles to MN, the other, F, represented by RB, parallel to UIN. Of these forces, F alone exerts pressure upon the table, F' lmerely tending to make the body PQ slide along the surface on which it rests. H-lence the whole downward pressure = TV + F; or as F -- R sin a, down ward pressure - TV + R sin a. The tendency to move horizontally is evidently R cos a. Or, as another example, stl)pose that a body weighlling TV kilogrammes is to be raised by nmeans of ropes AE, AF, Fig.,30, and it is wished to determine the number of' kilogrammnes which must be exerted at the end of each rope, in ordler to start it. DIraw the vertical A C, having a length of TW units, andl complete the parallelograml ABCD by drawing BC, CD, parallel to iJF, AfE. Resolve WV into two components, one T,l parallel to AE, and a second 7" parallel to AF. Then T, 7'" will be the forces which must be applied at E and F to lift WT. From the triangle of forces ABC we lhave AB sin DA C W: 7':: AC: AB, whence T = — C S C lB (27.) BC sin BA C and T: T':: A C: BC, whence T' - WV A-t- W sin CBA' (28.) from which 7', 7' becomes known if the angles made by each rope with the vertical 1are determined. The exmnples given on p. 53 to explain the subject of statical equilibrium, also serve equally well to illustrate the resolution of forces. 97. Resolution of a Force into any number of Composnents. A force can also be directly resolved into any numnberl of' components by the principle of the Polygon of Forces, these colimponensts being in one or several planes. 1it is frequently sinmpler, however, to resolve the original force into two components, then cne of these components into two others, and so on. 98. Practical Examples. As a good example of'this, let us take the case of a vessel propelled by the wind. AB, Fig. 31, is a boat which is

56 REACTION OF SURFACES. loved forward by the action of the wind blowving in the direction indicated by the arrow WF. MIN is the sail. To show how the vessel proceeds under an oblique wind, let OP represent the magnitude of the whole pressure of the wind upon the sail MIN. Resolve it into two components, OD perpendicular to M1N, and OC i)arallel to it. The parallel component OC can have no effect to move the vessel because it acts upon the sail edge-ways; hence the whole imoving force is that due to the perpendicular component OD. Resolve OD into two other components, OE and OF, one of which, OE, is parallel to the keel of the vessel, and the other, OF, perpendicular to it. Then OE alone acts to push the vessel directly forwards, while OF tends to push it sideways. Hence the boat moves forward with a certain velocity due to the component OE, while at the same time it has a slight motion sideways (leeway), caused by the component OF. The reason why there is so little movement sideways in proportion to that forwards, is because of the greater resistance to motion in the former direction, caused by the shape of the vessel. The manner in which' two vessels can sail in different, and even opposite directions with the same wind, is easily demonstrated by the laws of the resolution of forces. In Fig. 32 the vessel AB is represented as proceedling in a direction opposite to that in which it moved in Fig. 31. The direction of the wind is the same as before, but the sail IMN is placed in a new position. Resolving OP into GI) and OC as before, OC is inoperative;* OD can be again resolved into OE parallel to the keel, and OF perpendicular to it. Hence the vessel moves on in the line BA under the action of the component OE. By varying the position of the sail, a vessel can be mlade to proceed in various directions, while the wind remains unchanged. In a boat with but one sail, the greatest advantage is gained when the wind is parallel to the keel, as its whole force is then exerted in causing a motion forward. The vessel is then saidl to be scudtlding or sailing before the wind. A ship sailing against the wind as closely as possible is said to be close-hauled. A large ship can sail so that her keel makes an angle of but six points (67~ 30') with the wind, and smaller vessels can sail very much closer. In a vessel with a lar4ge number of sails, a very favorable position of the wind is when it is at right angles to the keel (upon the beam), as the sails are then all acted upon with equal force, while if the wind is parallel to the keel the aft sails cut it off from those in'fmelt of them. CHAPTER VI. REACTION OF SURFACES. — COMPOSITION AND RESOLUTION OF FO0RCES ACTING AT DIFFERENT POINTS OF A BODY. 99. Equilibrium sustained by Reaction. When a body acted upon by a system of forces is at rest, equilibrium is often sustained by the reaction of one or more surfleces with which the body is in contact. The simplest example of this is when a bodly rests upon a horizontal plane, in which case the reaction

CONSTRAINED BODIES. 57 caused by the compression of the material of which the plane is composed, is equal to the weight of the body. Additional examples will be found in the case of the Jib and Tie-Rocd and the Toggle Joint already described. 100. Position of Resultant. If one surface be pressed against another in any manner, the resultant of the reactions at all the different points of its surface must be equal and opposite to the resultant of all the forees acting upon it. 101. Body pressed against a Curved Surface. When a body is pressed against a curved surface by the action of any number of forces, if there is equilibrium, the resultant of these must be normal to that surface. For if this were not the case, the resultant might be resolved into two other forces at right angles to each other, one normal to the curved surface, and the other tangential to it. The former would be opposed by the reaction of the surface, while the latter, being unopposed by any force, would produce motion over it, which is contrary to the supposition. Tlae resultant must also pass through the point of contact of the surface, as otherwise there would be a tendency to rotate about that point. Thus let B, Fig. 33, be a cube pressed against the sphere A. For equilibrium, the resultant of the forces acting on A must pass through the point of contact P, and be perpendicular to the curved surface of A. 102. Constrained Bodies. If a body is in such a condition that motion can take place only in certain directions, it is said to be constrainzed. Thus a body fastened by a pivot is constrained to turn about that pivot. A body fastened at two points is constrained to move about an axis joining these two points, and if fastened at three points not in the same straight line, it is capable of no motion whatever. 103. Action of Forces on Constrained Bodies. It will be profitable to examine a few cases of the action of forces on constrained bodies. The simplest case is that of a body resting upon an inclined plane. Let A, Fig. 34, be such a body, and let its weight, which acts vertically downward, be represented by the line TVW. Resolving this into two components, P perpendicular to the plane L, and F parallel to it, it is evident that the whole force of the component P is exerted in producing a pressure upon the plane at right angles to its surface, while the other component F tends to produce motion along L. A similar case of constrained motion will be noticed in the case of a body resting upon a horizontal surface, and acted upon by an oblique force.1 As another example, let P be a ring hung upon a curved wire AB, Fig. 35. If it be acted upon by a force F, norimal to the curve, it will remain at rest, being kept in equilibrium by the react:ion -F, equal and opposite to F. Itf; however, the force lie in any other direction, as in the line PR, it can be resolved into two components, one of which along PF is normal to the curve, and the other along PF I tangential to it. The former of these 1 See p. 55.

58 FORCES ACTING AT DIFFERENT POINTS. will be balance.d by the reaction of the wire, while the latter component will cause the ring to move along the wire. When a body is fastened upon a pivot, any force F, Fig. 36, whose direction is in tile line joining its point of application S with the pivot P, is directly opposed by the reaction of P. If the force h'as any other direction, as R, it will cause revolution about the pivot until the line PS11, becomes straight, the component F producing pressure on P, while F' causes the body to rotate. F' evidently diminishes as the angle PSI' increases, until it equals 0 when that angle assumes a value of 180~. 104. Forces applied at different Points of a Body. Hitherto we hlave treated of forces applied at a single point. We now proceed to consider their action when applied at different points of any connected system of' particles. The findamental principle upon which our reasoning is based is the Transferability of Force. The simplest case of transference is that of a force acting upon a rod in the direction of its length. The rod may be considered as composed of a line of particles, abed, etc., Fir. 37. Now if a pull be exerted at A, the particle a is moved from its normal position, so that its distance from b is increasedl This develops a molecular attractive force which acts upon b, causing it to move slightly towards a; thus in its turn c is acted upon, and so on through thd whole lenigth of the rod,, until the molecular tension exerted betwveen each particle is the same throughout the rod, and equal to the force, applied at A. The rod is then in a state of equilibrium, and the particle na exerts a pull upan any point, as B, to which it is attached,, equal to the force acting at A. Hence this power appears, to be transferred fi;om A to -B. If, on the other hand, the rod is pushed at A in the direction A]P,, the dist-ance between a and b is lessened, and the force thus developed is excited from particle to particle until the whole rodl is in a state of tension. The particle n then exerts a force upon -B equal to the pressure applied at A. It follows fromn what precedes, that the effect upon B will be the samie at whatever point of AB the power is applied. Hence equacl and opposite forces applied at the ends, of a 2rod or rope, are in equilibrium. For each imay be supposed to be applied to any single particle taken in their line of action. Also the resultant of two or more forces thus applied must equal their algebraic sum. 105. Principle of Transferability of Force, The efficiency of any force acting -upon a body is not altered' by transferring its point of' acplication to any point in its line of action. Thus let F be a force acting upon a body A-B, Fig. 38. Evidently the effect of F is. the same, whether it be applied at a, b, c,. d or e. This can be illustrated experirmentally by balancing AB on a pivot P. The force F being applied at a, let it b.e counterpoised by a weight WV, suspended from B. If then the point of' application of F be changed from a to b, c, d or e, the weight will still be found to balance it exactly.

EFFECT OF PIVOT. 59 196, Composilion and Resolution of' Forces applied at different iPon ts The principle of transference fiarnishes a reavdy method of determininng the resultant ef-ect of two or more forces acting at di- erent points of a body. Let _, F', be two forces applied to the blodly Al-B, Filg. 39, at the points aV,; and acting along the lines FiS3T F' Suppose these lines to be produced till they meet in some point 5, which may lie either within or without the bodri A-B. Then since S is in the continlation of F]211i, the effect of F will be the same as if applied at that point, and hence may be transferred to it. Also since S is in the continuation of' F'v' i may also be transferred in the same Ilmanner. The combined effect of the two forces is therefore the samle as if both were applied at 5, the intelsection of their lines of action. In this case the resultant would be found, as already explained, by means of the paralleloogram, so that if Sac, Sb represent Fi and F', SC will represent B, their resultant. Hence, to find the effect of two forces applied at diffelrent points of a body, prolongq their lines of action until they nleet in a point, and proceed to Jind the resultant as if both forces acted at that point. rThe resultant of any number of fbrces applied at different points of a bocly may be found by combining them two by two. 107. Equilibrium of Forces applied at diferent Points. The resultalnt of F and F' would be balanced by the application of an equal and opposite force anywhere in the line S/R. Hence,?f a body is kept (at rest by the action of three oblique forces appliecd at dcfferent points, these forces toould be in equilibritum if applied at a single point. 108. Effect of Pivot. If a pivot be placed anywhere in the line of action of the resultant SB, as at 0, the resultant may be considered as applied directly at that point, in which case it merely exerts a pressure upon the pivot without producing any tendency in the body to rotate around it. If the pivot does not lie in SBR there will evidently be a tendency to rotation.- Hence in the case of': body resting upon a pivot, if the resultant of all the capplied forces passes through the pivot, there will be equilibriunm. Conversely, if the applied forces are in equilibrium the resultant passes through the pivot.

60 STATICAL MOMENTS. CHAPTER VII. STATICAL MOMENTS. --- PARALLEL FORCES. - COUPLEs. AStatical llioments. 109. Moment of a Force. The tendency of a force to rotate a body about c fixed point is mecasured by the product of its intensity into the perpendicular distance from the point to the line of action of the force. This product is called the moment of the force. Thus the moment of the force F, Fig. 40, relatively to C is F X C'm. Let F, F',: Fig. 40, be two forces applied at,,i; N and tending to cause rotation in opposite directions about a pivot placed at the point C, and let R be their resultant. When R passes through C there will be equilibrium among the forces F, F', and the reaction of the pivot, hence the tendencies of F, F' to rotate A B will in that case be equal. From C draw Cmin, C/n, respectively, perpendicular to the directions of F, F', aind construct a parallelogram Sb (Ca, the adjacent sides of which, Sat, Sb, represent the magnitude and direction of iF F'. Then Cn Cm F: F' Sa: Sb:: sin b8SC: sin aSC:: S S -: C'n: (Cm, whence F X Cm F' X C(7, (27.) But F X Cm is the moment of F,~ and F' X Cn the moment of IF', relatively to C, and since we have shown that when their moments are equal the forces are balanced about C, these moments must represent the efficiency of the forces F, F', to cause rotation in either direction about that point. Hence, calling F any force, I the perpendicular distance from any point to its line of actions and Mits moment relatively to that point, we have -M= Fl. (28.) The perpendicular I is called the arm of the moment. The direction of the tendency to rotation is said to be righthandeed when it is in the direction of the moment of the hands of a watch, i. e., from left to right, and left-handed, when in an opposite direction. Thus in Fig. 40, F tends to cause a right-handed, and F' a left-handed rotation. The former are generally designated by the sign +, the latter by the sign -. 110. Experimental Verification. The laws of moments may be verified experimentally by means of the apparatus represented in Fig. 41. AB is a disc of wood balanced on a pivot passing through C, so as to remain at rest indifferently in any position. A weight D is then attached at B by a cord, while a second cord is fastened to any other point of AB as K, and passed over a pulley Al. Weights E are then attached to the latter

EQUILIBRIUM OF MOMENTS. 61 cord, and the body AB is allowed to assume its position of equilibrium. There is then equilibrium between the moments of E and D. The perpendiculars CB, CL, dropped from C upon the directions of the forces, are then measured, and it will be found that E X CL D X BC, in whatever position on the disc the point of attachment K be taken. 111. Resultant Moment of several Forces. If two or more forces tend to produce rotation in the same direction their united efficiency (resultant noment) is evidently equal to the sum of their nmomcnts.' If they tend to produce rotation in opposite directions, their united efficiency is equal to the excess of the sum of the moments in one direction over the sum of those in thle opposite direction. Hence calling right-handed rotations +, and lefthanded rotations -, the resultarmz morment of any number of forces relatively to a point is equcal to the al gebraic sum of the moments of the comnponenuts. Or, calling the resultant moment 2, 1 - = Fl. (29.) The moment of the resultant of two forces acting to produce similar rotations is equal to the sum of the moments of the components, since the resultant may be substituted for them without change of efficiency; and the moment of the resultant of two forces acting to produlce dissimilar rotations is equal to the difference of the mloments of the components, for a lklie reason. Hence as this course of reasoning may be extended to any number of forces, it follows that the moment of the resultant of any combination of forces equals the algebraic sum ofq the moments of the components. 01, calling -YFfl this sum, B the resultant of all the forces, and lo the length of its arm, RI0o = zI 1. (30.) As the preceding propositions hold, whatever may be the inclination of the forces to each other, they are true when the forces are parallel. 112. Equilibrium of Moments. When the sum of the moments in one direction equals the sum of the opposite moments, the resultant moment 0, which is expressed algebraically, _Rl0 - lZ = -0. (31.) To produce this, _R may become zero, or the arn l0 may assume that value. If the latter is the case, there is a simple tendency to a translatory movement of the body along the line of action of the resultant. And if the body be fastened at any point in that line, the force acting upon it will be balanced by the reaction of the pivot or other support on which it rests, as already shown. 113. Moment of a Porce relatively to an Axis. The moment of a force relatively to an axis is its tendency to produce rotation about that 1In this chapter we consider only the case in which all the forces lie in the same plane.

62 PARALLEL FORCES. axis. Let Pi be a force represented by PR, Fig. 42, and AB an axis. Draw Pill perpendicular to both PR and AB. If now the force PR be rf-'" ved into two components, one, F, represented by PF, perpendicular to AB, and the other F', represented by'F', parallel to A B, the fobrmer alone will have a tendency to produce rotation about that axis. The inomient of the whole force R relatively to AB is therefore the same as the moment of its component F, which is evidently the nmomnent of F relatively to the point M, that is to F X PM. Hence, the nmoment of a force relatively to an axis is equal to the pejel)enlicular distance between the axis and the line of action of the force, into that one of thle comnponents which is at right angles to the axis. Forces are in equilibriumn about an axis when the alegebraic sum of their moments relatively to it is zero. 114. Practical Application. An interesting application of thb principle of equilibriuml of' forces about an axis is the following. If three equal weioht.s, P P, P, Fig. 43, be applied at equal distances fi'om each other andl fiom an axis througoh 0, the allgebraic sumi of their mlomnents relatively to that axis will always be equal to zero, whatever mlay be the absolute position of the points of application, A, B, C. The algebraie sumi of the momients of P, P, P, is P X OlH X P + OL -P X OD- I (OH + O.L - OD), and it is to be proved that this product - 0. Join B, C, bisect BC in G and (lraw A G. As AB.C is an equilateral triangle A G is perpendiculhr to BC and passes through 0. Also AO- 20G, whence OD — 20K. Now OH -- OL- OH + OK + KL - 20OK, as I1L KL because of similarity of' the triangles BIKL, CKH. Ience OH -+ OLOD - OD - OD - 0, and P (OH+OL - OD) - 0 fr any position oi A, B, C. This is practically applied in punmping-machines in which three pumn: s are worked by a single shaft, their piston-rods bein(r attached to thrue cranks, making angles of 120~ with ea.ch other. The resistance of the three pumps is the same for all positions of the cranks, and perfect steadiness of motion is gained. Parallel Forces. 115. Resultant of two Parallel Forces. T]he resultact of two parallel forces acting in the same direction is parallel to them, and equal to their sum. Let F, F/', Fig. 44, be any two forces applied at the points A, B. Their resultant, 1R, is found by prolonging FA, Fj' B, till they meet in P, and constructing a parallelogram of forces, bPac. It is clear that R may be considered as applied at P, and that its direction will lie betweeni FA and iF/B'. Now imagine _F'B to be revolved about B in the direction indicated by the arrow. The point P continually recedes until the line F'1B assumes the position FB, parallel to FA, when P is at an infinite distance, and in this case, as all lines meeting at an infinite distance are parallel, P1R, the direction of the resultant, must then be parallel to'A and F2'B. Also, for all inclinations of the components to each other, the magnitude of the resultant B of the forces F, F' is given by the

RESULTANT OF PARALLEL FORCES. 63 equation -- _/F cos bPc +- F' cos aPc (26, p. 55). But when iF'B becomes parallel to FA, bPc - 0, aPc =- 0, whence - = - + F'. (32.) 116. Point of Application of Resultant. To find the point of aplication of the resultant on the line AB, suppose M, Flig. 46, to be that point. Imagine a force R' equal and opposite to the resultant B to be applied there. This force would balance the resultant, and must therefore be in equilibrium with the components 1?, F'. Hence the moments of -F, F' relatively to i; mast be equal, that is, -F X Ala- F' X ~ib, whence F: -F':: Mlfb: lIa. But by similarity of' triangles, Jlb': Ma:: 1IB': JA, whence F: F':: lIB: 1A. (33.) That is, the resultant divides the line joining the points of application of the component forces into parts whose lengths are inversely proportionctl to the adjacent components. 1 7. Two Parallel Forces in Opposite Direction s. If the forces are in opposite directions, the resultant is parallel to them, and equal to their difference. It lies onw the same side of both, next the greater comlponent, and has the same direction as that component. The three forces, F F', T', Fig. 45, are in equilibriunl, hence the resultant of F anld B' must be equal and opposite to l'"; that is, equal to B' - F (Eq. 32). It is also evident from the figure that the resultant is onl the same side of both F and B', next R', and in the same direction with it, The point of' application, _B, is so situated that the moments of F, B' are equal, relatively to it, in which case F:':: MIB: AB. (34.) 118. [Practical Illustration. To illustrate the application of the proceding propositions, suppose two weights of 10 and( 30 kgrs. respectively to be hung_' upon the extremities of a rod AB, Fig. 45, 2 Inetres in length. It is required to find the force R' whicl is requisite to support them, and the point at which it must be applied, neglecting the weilght of the ro(l. We hare r' R F + F', and as F - 10 kTrs,' = -30 kgrs., F' _ 40 kgrs. Also F: F':: ~MB: MA, or denoting AB by 1 and MB by x, F: F':: x 1 -x whene x 1F F' (35) Substituting for 1, F, F', their values as given above x - 2 X - - I., and I - x 1- m That is, the force R' must be applied at a distance of ~ m. from A1. 119. Resolution of a Force into two parallel Comiponents. Proportions (33) (34) furnish a nletlhod of resolving a single force into two parallel forces applied at given points. Thus, let it be required to resolve the force R into two parallel forces applied at points A, B. In the prop.ortion F: F':: MB: MA we know MB, MA, and in the equation -' -- = -'+- iF', we know R. From these the value of F, F' can readily be computed. If F, F' are to have opposite directions, propor

64 PARALLEL FORCES. tion (34), and the corresponding equation would be used in the same manner. It is also evident that if the magnitude of the components is given, their points of application can readily be found by the same proportions, To illustrate, if the force P'i 40 kgrs. is to be resolved into two parallel components F, Ft having the same direction, and applied at points A, B, distant from M/, 1-1 m. and ~ m. respectively, the magnitude of these is found as follows: R - F+ F' = 40 kgr.F: F':: IB: j1IA::: 1A whence F' - 3F, and R -F -j Ff = 4F =40 kgrs., whence F — 10. kgrs. F' =30 kgrs. 120. Centre of Parallel Forces. Since the point of application 1 of the resultant of the parallel forces, F,, F' divides the line AB joinining the points of application of the forces in a fixed ratio depending only upon the magnitude of F, F', it follows that whatever may be the direction of the forces. F, F, B; so. long as their relative intensities remain unchanged,, the position of Mll does not vary., This point is called the Centre of the iParallel Forces. 121. Resultant of any number of Parallel Forces. The position and magnitude of the resultant of more than two parallel forces may be found by combining them successively. Thus let -F,, F', F", Fig. 46, be three parallel forces applied to a body at the points A, B, C. To find their resultant we first combine F and F'. Joining A and B we have from (33) F: F': MB: MA, which determines the point M., Also, F= F + F'. We next combine this resultant R with the remaining component F". Joining MC, we have BR F -t F': F":: CX; M,, which determines. the point of application N of the resultant R' of all three forces. Also' = - + F" = F ~ F' + F". B' is evidently parallel to the components. The position of the centre, NV can be found either by a graphical constluction, or calculated by the rules of trigonometry when the positions of A,, B, C, are known. If any one. of the forces, as F", acted in an opposite direction, we should evidently have 2'- F+' -F' —' F", and its position would be found by (34). As this process of combination can be continued indefinitely, it follows that calling foreces acting in one direction +, those in an opposite direction _, tlhe resultant of any number of parallel forces equals their. algebraic sum, that is, - ='F. (36.), The preceding method of determining the position of the point of application of the resultant is simple when there are but few forces to be considered, but becomes very tedious with numerous forces. Another method is then adopted, which will be explained in the following chapter.

COUPLES. 65 122. Experimental Demonstration. The laws of parallel forces can be delnonstrated experimuentally by means of a very simple apparatus representedl in Fig. 47. A graduated bar, AB, is suspended friom the hooks of two delicate spring balances D, E. The weight of ARB causes a certain constant depression of' the indexes of the balances, which is read once for all. A weight 1V is then placed anywhere upon AB, as at C, and the balance readings taken anew, which being diminished by the preceding readin.s, show the pressures exerted by it at G and H, the points of' suspension. Calling these F, F, it will be found that in whatever position W may be placed, W - F +- F, and F''iT: C GH C. For verifying the laws with a greater number of parallel forces additional balances may be placed between D and E. Couples. 123, Definition. If two equal and opposite parallel forces not acting in the same straight line be applied to a body, their algebraic sum is 0, and hence they have no tendency to produce a motion of translation. But as their resultant moment with regard to any point cah never equal 0, their effect will be to rotate the body about an axis. Such a combination of forces is called a couple. The perpendicular distance between the lines of action of the forces is called the arn, of the couple, and the plane containing these lines of action the plane of the couple. Any line at right angles to this plane is an acxis of the cozuple. The terlms right-hancldedl and left-handed are applied to couples in the same manner as in the case of moments. 124. Effilenoy of a Couple. The rotary effect, or moment of a cosple is measured by the product of either force into the length of the arm. Let F, F', Fig. 48, constitute a couple whose arm is AB. Assume an axis at right angles to the plane of EF F', passing through any point P. The moment of F relatively to P F X P-JB, and the moment of F' relatively to the same point F- F' X PA - fi X PA,as F'` - _F The resultant moment of the two forces is therefore equal to F x P-B + F X PA = F(PB +- PA) - F X AB. If the axis be taken at a point not between the lines of' action of F;' and F', as P', we have nmoment of force F relatively to P' - F X PIB, mo1ment of force F relatively to P' - — F' X P'A - -F X P'A, as the moment is left-handed. Hence, res&ltant nmoment of F and F' - F X P'B - F X P'A F (P'B -- P'A) - F X A-B, as before. Hence calling Mthe moment of the couple, F the magnitude of the force acting at each end, and 1 the arm, 2W Fl. (37.) 9

66 COUPLES. 125. Transference of Couples. A couzple may be turned in its own plane, or moved parallel to itself without altering its eficiency. For the effect of the couple F, F', Fig. 49, to cause rotation a:bout P, is evidently not altered if the foices assume the positionsf, f', as the amn MiV equals AB. With any axis, as P', not situated between the lines of action of F, F', we have -Rotary-effect - f X mn — f X INV = F X AB. The moving of the couple to any other position in the same planle would simply be equivalent to nloving the assumed axis to some other point, as filoll P to P', which does -not alter the efficiency. A couple may also be trlansferred to any pllne parallel to its own plane without alteration of its efficiency, because its whole tendency being to produce rotation about an axis at right angles to its plane, the effect will be the same in whatever plane at right angles to the axis it may lie. 126. Reduction of Couples. From Eq. (37) it followvs that couples are equivalent when their moments are equal. That is, a couple of force 8 and arm- = 3, has the same efficiency as one of force - 2 and arm - 12, as in either case I - Fl - 24. Hence any couple mray be replaced by an equivalent one having a given arm. The value of the force _iti this case is foiund from the eqttution l - Fl, whence F - W (38). If, on the other hand, we wish the new couple to have a given force, we may find the corresponding length of the arm fiom the equation 1 - - 39). Thiis process is called the reduction of couples to the salle arm, or the samne forces. Fol example, let it be required to replace the couple F = 5, 1 - 10, by an equivalent one having an arllm 2. From 11 50 (38) we find the corresponding value of F to be -5 - - 25. An equivalent couple having an arm 2 must then have a force of 25. 127. Equilibrium of Couples. Since the sole effect of a couple is to produce rotation, no single force can be so applied as to hold it in equilibrium. It will evidently be balanced, however, by the application of an equal couple acting in an opposite direction. 128. Combination of Couples having the same: Axis.'The resultant moment of any niumber of couples lying in the same or parallel planes, is equal to the algebraic sum of their separate moments. For they maay all be reduced to equal arns, and so applied that the forces act in the same strlaight line. Calling Fl, F'l, F"l, — F"'l, the reduced couples, these wvill be equivalent to a single

COMBINATION OF COUPLES. 67 couple with a force - F + F' + F" - F"', and an arm 1, whose efficiency equals the algebraic sum of the component couples. Hence denoting the resultanlt moment of any number of couples by 1., we have Mr,- Fl. (40.) For equilibrium we must have YF1 - 0. (41.) 129. Representation of Couples by Lines. In many cases, especially where the planes of the couples are inclined to each other, it is convenient to represent the direction, magnitude and intensity of a couple in the followino manner. Let F, F', Fig. 48, be the couple which is to be represented. From any point 0 draw a line O3I at right angles to the plane in such a direction that to an observer looking from 0 towards Mll the couple shall seem right-handed, and let the length of OM be made proportional to the nunlber of units in the moment of the couple. 130. Combination of Couples having different Axes. Couples lying in planes inclined to each other can be combined by means of a theorem similar to the Parallelogram of Forces. Let F, F', f, f', Fig. 50, be two couples reduced to the same arm AB and actinc in planes FAlFlfaf', mnakingr with each other any angle FAf. Let the lines FA, fA, FIB, f'B represent the magnitudes of the forces of the couple acting at the extremities of' AB. The two forces F, f, give a resultant R represented in magnitude and direction by ARt; and F'f' give an equal and parallel resultant R', represented by BR,'. The four forces F, f, F'f' may therefore be'eplaceld by two parallel forces R, R', lying in the plane RAB. The resultant moment evidently equals R X AB. Hence couples may be combined andl resolved in the satme manner as ordinary forces. If instead of the forces F, F' f, f' we represent by A F, A F', Af; Af' the moments of the couples, A1R, AR', will evidently represent the moment of the resultant couple. Since the plane of a couple and its axis of rotation are at right angles the axes have the same inclinations to each other as the planes. Hence the moments of F, f may be laid off on the axes A F, Af, Fig. 51, instead of in the planes B C, BE. The line ARl will then represent the moment of the resultant couple, the plane of which is BD, at rilght angles with AlR. We have, therefore, the general proposition, if the momnelnts ani'd axes of two couples be represented by tto side.s of a pcarallelogram, the diagonal of that paralle/ogram represents thle moment and axis of the resultant couple. 131. Illustrations. The effect of two equal and opposite forces to cause rotation is seen in spinninr a common humming-top, the pull exerted on the string and applied at the circumference of the axle being one of the forces, and the reaction of the handle applied at the axis of symmetry of the axle the other. Another excellent example is the case of a light sphere of cork or wood kept rapidly rotating in the air by the action of a fountain-jet playing against the under side of it. The upward f'orce of the jet applied on the surface is equal and opposite to the fbrce of gravity (or weight of the ball) applied at the centre of the sphere, thus forming a couple.

68 ANALYTICAL STATICS. CHAPTER VIII. ANALYTICAL STATICS. 132. General Principles, Definitions, etc. When we have a large number of forces to consider, the prbcesscs of composition and resolution of forces are very much simplified by the use of analytical methods. It is customary in this case to refer tlhe directions of the forces, their points of application, etc., to coordinate axes at right angles to each other, as in Analytical Geometry. To the studlent unacquainted with that study the following explanation will give the main principles required for an intelligent study of thel present chapter. When the forces are all in the same plane we make use of two rectangular axes OX, OY, Fig-. 52, known as the axis oj X and( axis of T respectively, and collectively as the axes!f cordlioates. Lines drLawn fionm any i)oint parallel to the axis of Y and terminated by the axis of X are called ordinates, and are generally designated by the letter y, while lines drawn finom a point parallel to the axis of X alnd terminated by the axis of Y are called abscissas, and are designated by the letter x. The abscissas and ordinates of a point taken together are called the coer'diql:tes of that point. Thus Bill and BiV - MO are coordinates of B, B11 being the ordinate, BN the abscissa. The point 0 at the intersection of' the axes is the origqiz of coirdinates. A}bscissas of points lying to the right of the axis of Y-are generally affected by the sion +-; of -those lyino( to the left, -. Ordinates of points lying above the axis of X are +; of points below, -. Evidently the values of x and y for any point fix its position; thus the position of B is known when we have given 031 - x, AM3B?. To illustrate the application of this to mechanics; two forces of magnitude 5 and 4 applied at a singrle point, and making angles of 45~ andl 30~ with any given line are represented by taking that line as the axis of X (or Y if preferable), and the point of application as 0, and laying off' two lifies OA, OB of lengths - 5 and 4 units respectively, makinc angles of 450 and 30~ with OX. When the forces do not lie in a single plane they can be referred to three rectangular axes, X, Y, Z, Fig. 54, formed by the intersection of three planes at right anoles to each other. The position of a force in space is then represented by the angles u, 8, y, which its direction makes with the axes of X, Y and Z. The coor(linates of any point referred to a system of three coordinate axes are denoted by the letters x, y, z, indicating distances parallel to XY, Y or Z respectively. The projection of a line upon a plane is the distance between perpendiculars drawn to the plane from each of its extremities. The projection of a line upon another line is the distance between perpen(liculars drawn to the latter line from the extremities of the forlner. The projection of a line upon a plane or a second line is equal to the length of' the line into the cosine of angle which it makes with the plane or line upon which it is projected. 133. Analytical Statics in Two Dimensions. — Forces applied at a single Point. Let F, representled by OB, Fig. 52, be any force applied at a single point. It can be resolved into two rectangular components represented by lO1 and Bil. Denote the angle BOOM by a.

FORCES IN SINGLE PLANE. 69 Then, as MAO _ BO cos a, BUl- BO sin a, it is clear that the component parallel to X F cos a, and that parallel to Y- F sin a. The direction of the components will be found by observing the algebraic signs of cos a and sin a. Thus if the force F lie in the direction indicated by the line OC, the angle a which it makes with X is MOC (counted on the same side of OX with OB), and both cos a and sin a are minus, MlIOC being greater than 180~. In this case the components would be - F cos a, parallel to X, - F sin a parallel to Y. 1 4. Resultant of any Nlumber of Forces. Suppose that any number of forces., F', F", F"', F,, are applied at a single point, and it is required to find their resultant. Take the point of application as the ori(,in of codrdinates, and denote by a, a', afl /, a -1, the angles made by the directions of the forces with the axis of X. Denote the resultant by R, and call 0 the angle which it makes with X. Resolving F, F', etc., into their components parallel to X and to Y, as explained in the preceding paragraph, and taking care to observe the algebraic signs of the functions sin a, cos a, etc., the results of the following tables are obtained. Components parallel to X. Components parallel to Y. F cos cc F sin a F' cos a' F' sin a' F" cos (/" F" sin a" "' cos a"' F"// sin a"' F" cos -" Fn sin an Hence, Total fo;rce along X - F cos a + F' cos a' + F" cos a" + F" cos a"' + Fn cos a" -- F cos a. Total force along Y_ F sin a- + F' sin a' + F" sin a'" + F"' sin a,"' + Fn sin a" IF sin a. Now conceive the resultant R to be resolved in the same manner. Its components are.R cos 0, along X, R sin 0, along Y. But the components of R along X must evidently be equal to the sum of the components of the elementary forces F, F', etc., along the same axis, and the components of R along Y must be equal to the sum:of' the components of F, F', etc.. along that axis. Hence R cos -8 F cos a. (42.) R sin 0 -- F sin a. (43.) Squaringi these equations and adding the results, R12 cose n -+ R2 sin2 a (2'F cos a)2 + (2-F sin )2, or as cos2 a - sin2 a=1, iR2 (2' F cos a)2 + (2 F sin.x)2; whence R - ( F cos )2 + (2 F sin -)2. (44.) Equation (44) gives the magnitude of the resultant. It remains only to determine its direction by means of the angle 0. R sina, Fsin a Dividing Eq. (43) by Eq. (42), Rcos a ZF cos.' whence c F sin tang 0 F- sF sa (45.) The algebraic sign of tang 0 evidently follows from the signs of 2'F sin a, i cos a. The value of R may also be obtained (having first determined the angle 0) from the equation R cos 0 - 2-F cos a, whence'-F cos a R s - (zF cos a) see &. (46.) cos 0 -

70 ANALYTICAL STATICS. That there may be equilibrium among a system of forces R must - 0, whence (zF cos,a).2 +- (EF sin.)2 _ 0, (47), which equation can only be satisfied when each of its terms - O. Hence the equations of equilibrium are (zF cos a) - 0, (48.) (EF sin a) — 0. (49.) 135. Resultant of Oblique Forces applied at several Points. In this case there is a tendency to rotation about an axis produced as well as a resultant translatory force. Let F. F', Fff"', F be forces applied at points whose coordinates relative to an assumed set of coordinate axes X, Y, Fio;. 53, are (x, y), (x',?y), (X", y"). (x', ey). It is required to find the magnitude and direction of the resultant, antl, also the resultant couples produced. Their directions are denoted by the angles a, an,,/, -X, as before. Let us first consider -the ease of a single force F applied at the point S (X,?). Resolving it into two components we have P - F cos a - comiponent?parallel to X. Q F- sin a = comiponenrt parallel to Y. Now let us imagine two opposite forces K, -- K, each equal and parallel to P, to be ap. lied at 0. Also two opposite forces T, - T, each equal and parallel to Q. The action of the system of forces F, I', etc., will not be altered by this addition. We lhave now actingo upon the body, (1) at the point S (x, y), the forces P - F cos., parallel to X, and Q F sin..parallel to Y; (2) at the point 0, the forces + K, -- K, along X, and + T, - T, along Y. The combined effect of these forces is as follows; - the forces K and T are together equal to a single force L - K2 + T2 =V (F cos.)2 + (F sin u)2 F, making an angle f with X of such value that T Q F sin a tang tang. ix whence ~ I InO K P F7 cos (, that is, L is equal and parallel to F and applied at 0. The equal and opposite forces P and - K form a right-handed couple, having an arm SAI y, and whose moment is conseFquently l yF cos a. Also the equal an(l opposite forces Q and -' form a left-handed couple with an arm SN x, and whose momlent is - QX -x F sin a. These couples being in the same plane, form a resultant couple 31 =- Py - Qx - y?/ cos a- xF sin a. Now suppose this process to be repeated for each of the forces F', F", Fn. We shall then have a similar result for each: that is, the force F' will be replaced by an equal and parallel force applied at 0, and a couple whose moment y'F' cos' - x'Ff' sin,/; the force Fn will be replaced by an equal and parallel force applied at 0, and a couple y?/f'" cos a" - x"F" sin u", and so for all the forces. Their combined efficiency will therefore be the resultant of F, F', F1", Fn applied at 0, and the resultant of' all the couples, which has a moment equal to the algebraic sum of the elementary couples. Using the same notation for the resultant as in the preceding cases, we have as the joint eflect of all the forces, R =,/ (22F cos.)2 +- (2-F sin.)2, applied at 0, (50.) taF cos. tang a = F Sin (51.) and the couple of moment -= (yF cos a - x F sin a). (52.)

FORCES IN SPACE. 71 If z (yF cos (- xF sin a) 0, the resultant passes through 0. If R - 0 and E (yF cos, -xF sin a) does not - 0, the resultant is simply a couple. When there is equilibrium amongc the forces applied at different points of'a bodlv,' —. 0, whence (ZF cos c() - 0, (ZF sin a) - 0, (53.) ald~ I (yF cos - xF sin,) -0. (54.) 136. Analytical Statics in three Dimensions. - Resolution of a Single Force into three Rectangular Components. Let F, Figz. 54, be any force in space which is required to be resolved into thlee components at righlt angles to each other. Take the point of application of F as the orisoin of co6rdinates, and let OF represent that force, its direction beinog indicated by the values of a, P, y, the angles which its line of action makes with the axes X, Y, Z respectively. Construct the rectangular l)arallelopipedl OBFE having OF as its.diagonal. The adjacent edges OA, OB, OC will represent the components required (~ 79, p. 50). It only remains to find analytical expressions for these in terms of F. Project OF upon the axes OX, OQY, OZ, successively. The figures ODFA, OEFB, OCFG, all being rectangles, 0. - OF cos u,, OB OF cos;, OC 1OF cos y. But 0Ai, OB, OC represent the rectangular components of' F, hence denoting these by P, Q2,, we have' P1' l' cos (5)3). ( - F cos J (54). S = F cos y (55.) 137. Composition of Three Forces at Right Angles. Conversely, the resultant of three rectangular components can be found analytically as followvs: Squaring equations (53), (54), (55), and adding the resultant we have 1-'2 + Q2 + S2.=.LC1(S2 Cos u +:12 COS2,1 + F2 Co y -- F2 (cos2 a + COS2, + COS2 y). But as Cos2 ac + Cos2 s + Ccos2 1 (by a proposition of analytical geometry), F2 - P + Q2 + R2, and F - / P2 + Q2 + l?. (56.) The same result may be reached geometrically, as OF- OA2 + 0OB2 + OC2. (57.) 138. Resultant of any Number of Forces in Space applied at a Single Point. Let F, f', F", Fn be any forces applied at a sipgle point. Take their point of application as the origin of' coordinates, and denote the angles made by the forces, and the axes of coordinates by the letters u,,,, Y r',i, /,' p", ycp, a", c, yn respectively. Denote their resultant by R, and let its direction be indicated by the angles c', fr, y'. which it makes with X, Y, Z. Now resolve F, F', etc., into rectangular components along X, Y, Z, as explained in ~ 136, taking care to observe the algebraic signs of the cosines of the direction-angles. These components are Along X. Along Y. Along Z. F cos a F cos t F cosy F' cos a' F' cos 9' F' cos y' 17" cos" " cos "" cos cos F"cos a" Fnm cos in Fn Cos y" From these it follows that Total jfrce along XF cos c + F' cos cc' + F" cos a" -t- F" cos - IF cos a. Total force alon Y F cos,s +- F' cos F" cos s" -+ F" cos -_ ZF cos,. Total force along Z F cos y +- F' cos y' -+ F" cos y" + k' cos yn - F cos r.

S72 iANALYTICAL STATICS. Now conceive the resultant R to be resolved into three components along the axes of' cordinates. These will be R cos ar long X. R cos {r along Y. R cos yr along Z. But the rectangular components of R along X, Y, Z respectively must evidently be equal to the sum of the components F, F', F F along the same axes. Hence Pt cos t_ -F cos s, (58.) Pcost'r YFcos p, (59) B- cos yr YF cos y. (60.) Squaring (58), (59), (60) and adding the results,'2 (os2 r+ coss r + — Cos y = (IFcos.)2 + (F0cos )2 -+ (XFcosy)2, or R - (E' cos a)" +- (ZF cos p)2 + (IF cos )2. (61.) This same value of R may also be found by considering R as a single force with components IF cos a, F1 cos f, EF cos y, and using the same geometrical reasoning as in the case of determining any force from its rectangular components as already explained( (~ 137). The direction of R is given by the equations F cos a IF cos _F cos'Y cos a —,1, (62.) cos,(63.) cos y' — (64) For equilibrium, -- V (IF1 cos a2 -H (+1 cos ) - + (lFF cos y)2 0, (65.) in which case (IF cos a)2 0, (IF cos )2 - 0,s (F cos )2= 0 (66.) Hence equations (66) are the equations of equilibrium. When forces in space are applied at different points they tend to produce a translatory motion, and also rotation about each of thle three axes Xi, Y, Z. This case being quite complex, the student is referred for its demonstration to more extended works on mechanics. 139. Resultant and Centre of any Number of Parallel Forces in Space. A much simpler method of obtaining the position of the centre of parallel forces than the construction explained in ~ 121, p. 64 is afforded by analytical methods based on the principle of' moments. Let R be a force applied at the point S whose coordinates are xe, yr, za (Fig. 55). If' B be parfillel to'OY its moment relatively to the axis OZ is Rx. If parallel to OZ its moment relatively to OX is Ry, and if parallel to OX its moment relatively to OY is Rz. Now suppose any number of parallel forces F, F', F" applied at points (x, y, z), (x, y', z'), (x, yZ, zn) to act upon a body, anld.suppose S, Fig. 56, to be the point of apiplication of their resultant R (centre of parallel forces). The magnitude of the resultant is equal to the sum of the components or, R EF. The coordinates of S are determined as follows: - Since the parallel forces composing the system may be turned in any direction without altering the position of the centre (~ 120, p. 64), we may consider them as acting parallel to each of the axes successively. First let us suppose them to be parallel to O Y. The moment of the force F relatively to OZ is then Ex; that of' F', F'z'; that of' Fn, Fxn. The sum of these moments is ZFx, and this must be equal to the moment of the resultant, which is xrZF. Hence xr;F - IEFx. Next suppose all the forces to be parallel to OZ. By the same course of reasoning we have yT EF zFy. Then

VIRTUAL VELOCITIES. 78 supposing all the forces to be parallel to OX, we have zr E; IF z From these three equations x' zF- zFx, (67.) yr ".IF zF.y, (68.) Zr YF ziFz (69.) we find the required coordinates to be Zx (70.) r F (71) z F (72.) For equilibrium there must be no tendency either to translation or ro:ation, hence R _- F -- 0, (73.) F x -, ZY y - 0, F z -- 0, (74.) are tile equations of eqluilibrlium. 140. Virtual Velocities.- Definition. ILet F, Fig. 56, be a force appliel at thll point A4, and suppose this point of application to be moved by some external force through an extremely small space, so that it assumes the position A' or Ai", the force i' still acting in the same direction as at first. From A' or A" draw a perpendlicular AlB or A"B'. The distance AB or AB' intercepted between the original point of' application A and the foot of this perpendictula r, is called the virtual velocity of' the force F, and is positive Nwhen it is laid off in. the (lirection in which the fboce acts, and ne-gative when opposite to that direction. Thus AB is positive, AB' neontive. 41. Principle of Virtual Velocities. If the forces acting upon a 6bo'l4 avre in e./il'iiri, oind a very, sin/al diplaCll cement be given to tihe body, the aelvfabic osUm (f the products of etach Jf,rce into its virtual velocity is zero. If' the forces are applied at a single point, represent this point by A, Fig. 57, andl the firces themrselves by It', F', F", Fn. Suppose the point of application to be rmoved over the very smlll distance AA'. Call a,' //, CZn the ail(les ma(le with X by the directions of the forces,/3, /', /3"," f the anglles lmade with Y, and denote the angle between X and the line AA' by 0. AB is the virtual velocity of F, which may be called v. Denote the virtual velocities of the other forces by v', v", vU respectively. It is to be proved tha-lt YFv 0. For any single force, as F, v - AB - AA' cos BAA' = AA' cos (a - 0) AA' (cos a cos 0 + sin a sin 6), whence Fv- F X AA' (cos a cos 0 + sill a sin 0). (75.) JIn like manlner for F', F', Fn we have F' 1' 1' X AA' (cos a' cos 0 + sin a' sin 0), (76.) F"'v" F "' X A A' (cos a" cos s 0 + sin a" sin 0), (77.) F _ v F" X AA' (cos a" cos 0 + sin an sin 0), (78.) Taking the sumn of these equations, noting that a, /, a', /', etc., are complemients, so thatt sin a cos /3, sin a' cos /3, etc., Fo +- F'v' + 1Fiv/" - F n -: ZFvo l A' [cos 0 (F cos a + F cos a + F" cos a" -+ F cos a ) + sin 0 (F cos 3 -- F' cos 13' -- F" cos /3" - F" cos /3)]. (79.) But, as -by supposition, there is equilibrium of the flrces. about A, it follows from equations (48), (49), p. 70,.that TF coos a F co s a' " n os a" + 1 cosa- 0 EF cos /3f F Cos/ 3 / cos 03 + Y oF" s t3/ + Fn cos pc 0, and as the terms of equation (79) enclosed between the brackets become - 0, the whole member - 0, and hence ElFv - 0. (80.) It follows fiom (80) that the resultant of any number of fobrces multiplied by its virtual velocity equals the algebraic sum of ealch of its 10

74 ANALYTICAL STATICS. components nmiltiplied into the correspondin virtlual velocity, that is, if we call V the virtual velocity of the resultant R., R Pi- V hi.v (81.) In case the forces are not ipplied at a single point, the tll eoem still holds. For siuppose the f,orees F F', ", Fn to be aplplied at difilferent points. It follows from the principl1 explained in ~ 106 (p. 59) that any t0wo of them, as F, F', produce the same effect as if they were both applied at the intersection of' their'lines of' action. Imaygine them thus applied, call II the resultant equal to their combined effe't, and denote by mo, rn, the virtual velocities of', F', wlhen transferred to the point supposed. According to the principles stated in the precedino paragraplh, R V - Fmy + -F. (82.) Now the virtual velocities of' F, F' are not affbe.cted by the transference of' their points of' application. For suppose A, D, Fig. 58, to ble these points, and let the body be slightly displaced by rotation, though a very small angle a abont an axis passing through any point IlI, and at ri(lht angles to the plane of the forces. The points A, D, mnove through very small arcs AC, DF, which may be considered as straight lines at rigoht angles to the radii iMA, ]iMD. Also AC -- MiAo, DF- MD=a. AB is the virtual velocity of F, DE of Ff. If F, F' were trfansfiil'red to G, their virtual velocities would be GK, GI. It is to be proved that G _K AB. We have v - AB - A C cos BA C - MlAa cos BA C- ilMAa cos A M1S IMAA -ail -Sa And m GK - GIH cos IHGK- l/K -Ga M cos CGM5 - iGa cos GMIS MGa l il- Sa; whence mz v. In like imanner it could be shown that GI - DE, or n - v'. Hence substituting these values in (82), RV F v+ F'v'. (83.) Now imagine R to be combined in the same manner with one of the remaining forces F", call P' the resultant of these, T/' its virtual velocity. By the same course of reasoning as that just employed, R' V' V + F"v"- F'v +- F'v' + Fv". (84.) In like manner, R"V' - R' V' - FE"v-" Fu + F'v' + F"v" + Fenn -FTv. (85.) But if there is equilibrium among the forces acting upon the body /P" 0, whence in that case ZFv = 0. (86.) Hence the theorem of virtual velocities is true whether the forces act upon the body at a single point, or have several points of application. The theorem also holds when the forces are in different planes, but the demonstration of this case is too complicated for the purpose of the present work. 142. Conversely, Zf Fev - 0, there will be equilibrium, among all the foices. For from (85) TFv R " 1"', whence if ZFv - 0, -" 0. In this case the forces imust either balance or form a couple. If they form a couple, the addition of an equal and opposite couple would produce equilibrium. Let P, P' it, u' be the forces and virtual velocities of such an equal and opposite couple. By the proposition zTFv- + Pu + P'' - 0,hence Pu __ P'' - 0. As P and P' are parallel forces having different points of' application, this equation can hold only when both fbrces - 0. The theoremls of' Virtual Velocities are of' great practical importance, as they coiimprehend( all the principles of the equilibrium of fbrces. A valuable application of them will be explained in treating of the mechanical powers (Chapter X.).

CENTRE OF GRAVITY OF LINES. 75 CHAPTER IX. CE3NTRE OF GRAVITY. 143. Definition. All bodies tend towards the earth by virtue of their weight. Since this tendency is impressed upon every particle, a body is really acted upon by a system of parallel forces having a vertical direction. The resultant of all these must equal their sum, which is evidently the weight of the whole mass. The centre of the system relnains at the same point, whatcver may be the direction of the forces relatively to the body (~ 120, p. 64), that is, in whatever position it may be placed. This ceitre is called the centre of grcavity of the body. Hence the cenztre of gravity of a body, or system of bodies, may be defined as the point through which the result-lant of the weights of the component particles always passes, whatever position the body or system may occupy. In considering masses as acted upon by gravitation, the whole weight may be supposed to be concentrated at the centre of gravity, so that if this point be supported the body will therefolre be balanced about it. Owing to these facts, a knowledge of the location of this point is often of great importance in mechanical problems, and we tlerefore proceed to a consideration of the methods of determining its position in various ca.ses.: C e'tt. r e of Gr avity of L'.Les The centre of gravity of a material straight line is at its middle poi~nt, for the resultant of the weights of the two extreme particles lies at a point midway between them. The same is true for the next two, and thus we may proceed with all the particles. The centre of gravity of two straight lines inclinzed at anr anf/le, lies upon the line joining their mziddle p2oints. Thus let AB/, A C, Fig. 59, be two lines united at A. Let C' be the middile poinit of A-, G" of _A I r The whole weight of AB may be considered as concentrateld at G', that of A (C at G". Joinlirng CG' G", the centre of gravity G of the vwhole system Nwill evidently lie uponl the line C' C". To find the positio)l of' G we have merely to determnine the point of al)plication of the resultant of the weights of A-2, AC. Sice the moiments of these.eights about G imltst be equal, G' ~: G" G t ~zi/ht 4 C o: ~ eiqhl AB. (87.) The centre of gravity of a crlledxtl line, -By, Fig. 60, lies somewhere within thle segmlent B(,i 1 In strict la'ttgnae, we slhotuld spelk of thie ccotte (cf ri-vitvy of a tltin rod instead of the centre of (grnvit-v of a litle, mtI of a tai1 soieta i!.i tel o)f.ilt ifce, Lit has le N-eights of portioll' of1 alo. c)e.l l)toti:::et 1 tpic (' ce!li(l:cdlticlI vol -:'c prlr, 3.tioanti I to their length, enid the Nwei.'lt.t of slteot:; of w!ifiorn'!itlcl: ts totl-i: lli.or:, we IlV for the satke of brevitv 11 tle tuse of,e forelll' tarm ot1 il.ie,rocde that their true maleitlg be kept in toiald.

76 CENTRE OF GRAVITY. 145. Centre of Gravity of Surfaces. The centre. of gravity of any plane sztrjtce which is symmetrical about a line lies ujpon that line. For let ABCD, Fig. 60, be any slchll,srftace symmetrical about BZD. As there is thle sa:me quantity of' 1mattel on either side of _BD, the centre of gravity of the surlflce must lie upon it. The line _BD is called:in axis of' symlnmetry. If any surface has two or more axes of symmetry, its centre of gravity lies at the intersection of those axes. Let AIB (I2 D be such a surface, symmetricall about ]BD) and A C. From lwh:lt we have already said, it is evident that thle centre of gravity lies on both B.D and A C. HIence it is at GC, their intersectioln. The centre of grlavity of a. para:llelolgram is therefore Iat the intersection of its dia;onals; that of a circle, ellipse, or any polygon, at its centie of figure. The ceJntre of' gravity of a ring is aIt its geometrical centle, andc tle samne is true of tle perimeters of all polygons. 146. Centre of Gravity of a Tria ngle. The centre of gravity of a triangle is on a line joining the vertex with the middle of the base, at a distance fr'om the vertex equal to two-thirds the length of that line. Let A-BCO Fig,. 61, be a trianlJe. From A draw AD) to the milddle p)oint D of B C, also fiomn _B draw BE7 to the middle, point E of A C. The lines A-D, BLE, each divi(le the triangle in(to two equnal.parits; hence tlhe centr'e of g.ravity must lie on each line, and is tlherefore at G, their point of intelrsection. To detelrmine the p-)osition of G, draw >'D. 5ED is parlllel to AB becaluse it divides AC, B C, pr(oportionally. Hence the triangIles A.B(CJ, ED( C, are similar, and _ED: AB::.D C B C Also as AB G, ED G 1are similar, EVD' AB:: GCD: GA, whence D C: B UC GD: CGA. But -DC -- BCby ConsIructiOln, helnce GD: A = - -AD, or AG -- zAD. 1:'7 Centre'of',avi'ty of' ny Polygo-a The centrle of g(ravrity of ainy polygon many be found by dividing it, into trianngles. Thius let ABCDE Fig. 62, be a polygon. Drnawing 3BE BD, it is divided into three tri-angles whose centres of gravity a tge'a -, g', g". Join g, g'. The centre of gravity, g"', of' the portion ABDE, muist lie on gg', and m:ly be fobtnd fiomn the proportion,c"'": g'g"' EBB ABE, ais the weights of EBD, ABE, rnily be supposed to be concentrateld at g, g'. Now join "'g''. The centre of gravity of the polygon ABJBCDE must lie on this line at a point G, so situated th tt g'" G: g"G:: ABE + EBD: RDBC. (88.) This process call evidently be applied to polygonls having any number of sides, and G may be fouaacl either graphically or by trigonometrical calculation.

CENbTRE OF GRAVITY OF SOLIDS. 77 148. Centre of Gravity of Solids. In the case of homogenous solids symmetrical about aly plane, the centre of gravity evidently lies in that plane. I-Ience the planes of symnietry determine the position of the centre of gravity. If a solid has two planes of symimetry its centre of gravity lies in their line of intersection, if three, at their point of intersection. Hence the centre of gravity of a cube, sphere or any regular polyedron is at its centre of figure.' The centre of gravity of lines, surfaces or solids, with the exception of the most simple ones, are best found by analytical methods requiring the use of the Calculus. 149. Centre of Gravity of Pyramid. As it is fiequently necessary in physical problems to know the position of the centre of gravity of a pyramid, we give a geometrical demonstration. The centre of gravity of a pyramid lies in the line joining the vertex wzith the centre of gravity of thle base, at a distabnce from the vertex equcal to threc-fourths the length of thcat line. a. Tricangulcr Base. Let AB CD, Fiy. 63, be a pyramid halving a triangular base A B C. Draw AE, BE from the verltices A, -B, of the triangles AD C, BD C to the mid(dle of their conmmnon base C-D. Let g be the centre of gravity of' BD C, g' that of AD C?. Draw Alg, Bg'. The centre of gravity of the pyranmid lies in Ag, for the 1)pyramid may be considered as built. up of a series of triangular prismns with bases parallel to B D C, and having a very snmill altitude; the centre of gravity of each of these lies in Ag, hence thecentre of gravity of the whole pyramid lies in. that line. It nlso lies in gBg', since -B may be taken as the vertex, and A C'D as the base of' the pyramid, which may then be considered a cornmposed of a series of prisms with bases parallel to AD C, and having infinitesinall altitudes. Hence as Ag, By' are in the same plane ABE, it must be at G, their point of intersection. To determine the position of G join gg'. As g Gg', A GCB are similar triang('les, gg': AB:: Gg: GA. Andl as gEg', BEA are similar, gg': A.B:: gE::BE. Combining these proportions, qE: BE:: Gq: GA But gE — ~ BE, therefore Gg = GA - - Ag. b. An'y Base. If tihe base of' t-he pryami(l is not a triang.le let it be any polygon wllhatever, BCDEF, Fig. 64. Divide the polygon into triangrles B2CD, BD.E, etc., and lclpass planes ABD, ABE thlrolghll the vertex A, andl the diagonals BD, BE. The whole pyramid is thus divided into a series of tria:ngular pyramrids. Let g be the centre of Igravity of the p)olTygonal bxse, and draw Ag. TI'e centre of gravity of the pyramid(1 is upon this line, since that solid may be supposed to be built up of a seies'of prisms

78 CENTRE ()F GRAVITY. with bases parlallel to B.CDEF,: and of very small altitude, each of wllose centres of gravity (and hellce the centre of gravity of the whole solid) would be onl Aq. It also lies in thle plane of the centres of gravity of the triangular pyramli(ds, into which the whole pyramidl llas been decolnlose(sd, tlhat is, in ai plane parallel to the base BCDEF, alnd cuttinll Ag.at aa distance of three-fourths its lenglth firom A. Helnce it must be at the intersection of that plane Nwith Ag, or at a point G so situalted tlhat A G- a Ag. 1 50. Centre of Gravity of Cone. Since acone mny be considere(d as a py-ramid with an infinite nuinler of faces, it follows th-at the centre of gravity of a cone is situatedl on the line joining its vertex withl the centre of its base, at a distance froim the vertex equal to - the length of tile line. 151, Centre of Gravity of System7 of Bodies. The centre of gravity of connected systems is readily found by the application of thle precedling princip)les. Let A, B, Fig. 6(5, be a system composedl of two bodies. Trhe nmass of A llay be considere(d as coneenltrated at its centlre of gravity g, that of' B ait g'. The cenltre of gravity of the systeim must evidently lie at somle point G on gg', so situated that the moments of A anlltl B relatively to it,re equal. Hence its position may be deteilined firomi the proplortion Gg: Gg' weight'B: weight A. (89.) In the case of a system containingo more than two bocdies the position of the cent.e of grav1ity may be deteirmined by a process similar to thl.t used in the case of an ir1iegular polygron (~ 146, p. 76). The apl:)lliationl otf these principles to extended systems of bodies is of 1gre:t imlnotalnee in lstronoimy. 152. C entre cf G-ravity of T, vw Bodies eonnected by a, o'acge-a:-;- oxns'od. To fid tlle celtre of gr:.-vity of two bodies connected by a ro(d x e first jiln(l thle celtre of g,'ralvity of the b)0olies themselves, Ilad theln finol tlhe centre of gravity of tihe rocl. By considerii n th!e veiloht of bocdies as concentrated at their centres of (gravitv, andl that of the rod as conce nttrated nt its middle point, we readily (etelmine the centre of gaviity of the whole system. Thus let A, ]B, Fig. 65, be the bo(lies, and.gg' tlhe loll()nrelouls ro()d connecting, thlet1. Tlhe centire o(t' gralvity of A an(-l B is at G, that: of' gg' at C. The centre of glaavity of the whole system is at x, so situated that Cx: Gx:: wceight of A + B: wueight of gg'. (90.) 153. DLtaiSn e of Cent re of.,i:ro,ity of two or more Bodies from a PIe. ne. The distance of the centre of gravity of any numSer of bodies.from a plctae is equal to tho te q'uolient arising fronz dzidzzn the product of' thze weight of each body iWo the disbt(zne of its cenztre of' graviy from that plane, by the sum of the weights of the bodies. This follows directly from equations (70), (71), (72), p. 73.

ANXALYTICeAL METHODS. 79 For let ZO', Fig. 56, be the pl'lne in question. The weights of the bodies mllay be represented by F, F', F", Fn, e:nd consideled as parallel forces applied at points (x, y, z), (x', /','), etc, The distances of the centres of gravity of the sepalrate bodies fiom ZO Y arae x,', n,) ar. Tllhe elentre of glavity of the system of blodies will therefore be tile centre of these parallel fboces, which lies at a distance xr firom Z O'; so situated that xr 2 h 154. General Analytical Methods. The equations of ~ 136 (70, 71, 72) furniisih general method of' determining the position of the centre of' gravlty of a system1i of bodies by reference to trilinear colrdinates. If we substitute for F, F', etc., in those eqla.tions the quantities 1l3, 11l', etc., the weieohts of the bodlies, and dienote by (x,?/, z), (X, y', z'), etc., thle coordinates of their cenltles of gravity, the codrdinates xr, y/, z, will give the position of the comlion centre of' gravity of' the system, this point being the centre of a combination of parallel forces IlI, 1Fl', ill", l'n applied at points (x,?j, x), (x'l,'?/, ), etc. Hence we obtain the f'ollowing equations, in which x, y,, yz, are the coordinates of the centre of gravity of the systein of bodies Mll, 11' 113", lin, X::,- tj (91.) /- 4 1j (92.), -- >11 (93.) 155. Application of the Calculus. The processes of integral calculus enable us to apIply the precedtlig Illetlod to tle determliniation of the position of, the centltre of gravity in the case of a bodly of any regular geolnetrical foirm, as all bodies mIlay be considered as systems of particles. Equations (91), (92), (93) are evidently true, whvlatever the nlaginitude of' the ma11sses co(ilposing thlem; hence, if we imagillne a body to be divided into a series of infinitesimall weight-elenments, they still hold. Suppose this division to take place in the case of any body, and let (l lbe the element of weight, x, y, z, the coordinates of' its centre of gravity. The Imomnients of any weight-eleinent relatively to the axes Z, X, Yrespectively are xdMll, yd'1, Z-dill, and the suml of the momlents represented in (91), (92), (93), by zlalx, 2I3'//, _Jlz, mlust be replaced by fxcdUI, fidIM, J"dli1, as dll is an infinitesimal. Also A1I nmust be replaced by /ill. I-Ience for any body whatever the coordinates of the centre of gravity are __ fxdh],l (94.) y qd3ill / l! (96. xr. (94.) (95.) (96.) fiH l JiMpu If the body under conslei, mtti p is hoino leneous di is equal to its volume, -,which is a volume element f1i V multiplied by the weiglit of a unit of' mass. As this latter factor is a constant, fll homlogene ous bodies -Xr./'dV (97.) yr- JV' (98.) z, fdV (99.) The imanner in which the body is supposed to be diviided to form weighteleiiients is diff'erent in dlif erent ca;es, as certain rmodes of imagina.ry division oftentimes simplify} the problem very greatly. The intem'altions must of course be taken between limits, as indicated by the condlitions of the problem to be solvedl. For the practical ap-plicationl of the preceding principles the student is referred to the standard treatises on analytical mechanics.

80 CENTRE OF GRAVITY. 156. Theoreirms of Pappus. An interesting application of' thle principles which we have delmonstrated in the precediig. pages is found in the centrobaryc mnethod of deterimining the volunles of solids of revolution, and the contents of surfaces of revolution. The principles of this method were first stated by Pappus of Alexandria (380 B. C.), and are hence called.after hinm. They are also known as the'Principles of Guldizus, from a Inatbhe maticianll who republished themn about 1640 in a worlk on centres of gravity of lines, surftaces and solidls. The first real deimonstration of them, however, appearled in 1647 in a work by Cavalieri.1 They are as follows:I. The volume qf the solid generated by the 9revolutioin of a surfaoce about an (axis in the same plane with it, is equal to the area of the surface multiplied by the circunmference described by its centre of gravity. It. The area of the surfjtce gernerated by the rotation of a line about anz axis lying ina the sqmue plane with it, is equal to the product of the line into the circumference described b its centre of gravity. Let AB C, Fig. 66, be a surtace generating tle solid, of which AB4 C - -D-E'/ is a part, by it;s revolution about the axis A. Any element of AB U, as, as,enerates a solid by its revolution, as indicated by the dotted lines, arld tle snln of the soli(ls thlls formed by the elements, P, P', etc., of which A-BC is comlposed, is evidently equal to the whole solid of re\volution. Supposing these elements to b)e very sniall, andl denoting by r, r', etc., their distances PiR, P'S fiolmn the axis YX, the plislatic solid described by P - 2,rr X P; tllat described by "P' -- 2-.r' X Pf, and sC on, f6r all the elements. Hence the volume of the whlole solid2,rr X P +H27r' x P' + 2-rr X PP.... + 2,rn X.pn - 2r(Pr +- Pt'r' P"r"... + Pnr"). Now if the distance of the centre of gravity of AB C fri'om be denoted by r0,, we have fronm the demonstration of ~ 152, p. 78, Prt + P'r' + P"r"... + Pn r'l ro(P + P' - P" +.. n). If' each of these equals be multiplied by 2-, we have 2r,(P- + P'q'- + P"r"... + Pnr") - 2, ()(P + P'+ P",.. P+ pn). But P +- P' +. pn P_ volumne of ABU, 2,r0O is the circumference described by its centre of glravity, and the first rnellber of the equation represents the suml of' the solids generated by thle elements of AB C, that is, the whole solid of revolution. Theolerem II. follows directly fiom Theolem I., since a line may be considerel as a surface of infinitesimnal widthl, in which case the solid of revolution grenerated becomes a surf ice of revolution. If the surface AB C revolves onlly through an anlle a instead of making a colnplete revolution, the volume of the solid generated - 2;r,0 X eaeac of szufce X 36X 1See The Theoremn of Pp.mls, by.1. B. Henck; Mlfathematicctl Monthly, Vol. I., p. 200.

EQUILIBRIUM OF BODIES. 81 As an illustration of the preceding let us take the case of the torus formed by the revolution of a circle about X. Let the'area of the circle be nR'2, and r0 thle distance of its centre from X. Then the volume of the torus generated - 2nr0 X 7R'2. Ag(ain let it be required to find the area of the circle generated by the revolution of its radius PR about X, by the centrobarye method. The length of the revolving line is R, the distance of its centre of gravity fiom R R X, r0 -. Hence the surface generated 2 2a- X R- 7r2. Eqfuilibrium of.Bodies as acffected by Position of Centre of Gravity. 157. Case of Suspended Bodies. Since a body is upheld when its centre of' gravity is supported, the nature of the equilibrium of a body is determined by the position of that point relatively to the points of support. We consider first the c.lse of suspended bodies. That any suspended body may be in equilibrium, its point of support and centre of gravity must lie in tile s-ame vertical, because the resultant weight is then applied directly at the point of suspension, and is balanced by its reaction. Three cases may arise: 1, when the centre of gravity is below the point of suspensioln; 2, when it is above. the point of suspension; 3, when the centre of gravity and point of suspension coincide. 1 58. Stable Equilibrium. If the centre of gravity is below the point of suspension, the body will return to its forlmer position of equilibrium on being removed from it. The equilibrium in this case is therefore stable. Thus let X1ia Fig. 67, be a body suspended at S. Let G' be the position of the centre of gravity when removed from its position of equilibrium, so that S and G' are not in the same vertical. The total weight of the body acts directly downward through G', and may be represented by G'A. This force may be resolved into two others represented by G'B, G'C, the former of which simply produces pressure on the pivot at 5, by the reaction of which it is balanced, while the latter produces al rotary motion about S as an axis. Hence the body can not come to rest until G' C becomes equal to zero, which occurs when G' takes the position G in the same vertical with S. 159. Unstable Equilibrium. Un~stable equilibrinum occurs when the centre of gravity is above the point of suslpelnsion. In this case, if the body be moved so that the two are no longer in the same vertical, it will not return to its former position, but will revolve about the point of suspension till the centre of gravity reaches the lowest possible position. This will be seen by ans inspection of Fig. 68, in which G'A represents the weight of the body, G'B, G'C, its components as before. 160. Neutral or Indifferent Equilibrium. A body suspended at its centre of gravity will remain at rest indifferently 11

82 EQUILIBRIUM OF BODIES. in any position, the total weight being always applied at the point of suspension, however the body may be moved. This is neutral equilibrium. The various conditions of equilibrium mlay be illustrated experilentally by suspen(ling a wheel successively on axes passing above, below and through its centre. 161. Experimnental lM~ethod of finding Position of Centre of Gravity. From what precedes it appears that the equilibrium of any sulspended body is stable only when the centre of gravity is at the lowest possible point, so that if any body be suspended freely its centre of gravity will assume this position. We are thus furnished with a method of finding the centre of gravity by experiment. For examllple, suppose it is required to find this point in the case of a thin sheet of metal 1MN, Fig. 69. If it be suspended fiomn a point S the centre of gravity will assume the lowest possible position. Hence if a plumb-line SB also be hung fiom S the centre of gravity of the surface of l N must be somewhere on the line ST. Marking this line on MN, let the body be suspenied fioln somle other point, as S. The centre of gravity of the surface will now be on SB, and as it is also on ST, it will be at G, the intersection of these two lines. The centre of gravity of the plate must therefore be on a perpendicular to the surface passing through G, at a depth equal to half the thickness of the plate. 162. Measure of Stability of Suspended Bodies. The stability of ally suspended body is measured by the force required to hold it in an inclined position, which is equal to the fobree with which it tends to reassume its position of stable equilibrium. The stability is proportional to the weight of the bocdy multiplied by the cdistance of the centre of gravity below the axis of suspension. Let MN, Fig. 70, be the body ulnder consideration. G its centre of gravity, and S the axis of suspension. The total force tendinig to'move it is the moment of its weight, or TW X SP. If now we suppose the centre of gravity to be lowered to G', and the weight to be changed to W, the new moment - W' X SP'. Hence calling _3land 3' the stabilities in the two cases, we have 1: 1M': X SP: W X SP', or as SP: SP':: SG;SG',. M: i':: WX XSG: W' X SG'. (100.) 163. Equilibrium of Bodies resting on a Horizontal Plane. A body resting upon a horizontal plane is in equilibrium whenever the vertical passing through the centre of gravity falls within the base, as in that clse the weight is directly opposed by the reaction of the supporting surface. Thus if G be the centre of gravity of AB, Fig. 71, the body will stand, as GMC falls within the base; but if it be raised to G' by the addition of the piece A C, the body will be overturned as G'31' falls without the base. The vertical through the centre of gravity is often called the line of' direction. The truth of this proposition may be verified by suspending a plumb-line f'orn G and G'. If the base of the body be curved, the line of direction must evidently pass

BODY WITH CURVED SURFAC:E. 83 through the point in which this base touches the plane on which it rests; otherwise the body -wkill roll until that position is assum led. The base of a body resting upon a point is evidently that point; if the body is supported by two points, the line joining thein may be considered as the base; if on three or nmore points, the base is the polygon included between the lines joining the points of support. 104. Illustrations and Applications. So long as the vertical through G lies within the base the body is secure. Thus the leaning towers of Pisa and Bolorna have stood fbor centuries, because althoughl tl:hey deviate fi-ont the perpendicular by a considerable anmount, the line of din('etion still fills so far within their bases thlat no ordinary shoc(k would be sIf: fcient to overthrow tllem. lle heigllt of' the tower at Pisa (Owhieh is 1he camcnparilze or bell-tover of the cathedral) is in round numbers 87 Il., and the diameter of its base is 17 im. It is inclined 4.7 m. firom the perj-endlicular, but the vertical through the centre of gravity fills 3 ml. within the base. The leaningo tower of Asinelli at Bologna. is 100 in. in heigrht, with an inclination of 1 im. The Garisenda tower in the sanme city is 63 m. high. with an inclination of 3 n..1 The pr.iciple explained in the first part of ~ 163 furnishes another experimental method of obtaiiining the centre of oravity of a body. This may be done by balancing it on its edge, in which case the centre of gravity nmust lie in the vertical plane passing through that edge. Balancnug a body in three differeat positions cleterlmines its centre of' gravity, which mulst be the point of intersection of three vertical planes passed throudgh the ed(ge in tihree successive positions of tihe supported body. The equilibriium of a body bounded by plane surfaces is stable when the body rests upon a side, and unstable wlen it rests'upon an angle. If a body be rolled it assumes positions of stable and unstable equilibrium alternately. The equilibriun of a body resting on a horizontal surface is evidently ismditerLent so far as mlotion in a horizontal plane is concerned. 165. Body with curved Surface. - Metacenter. The nature of the equiliblium of a body having a crtived surface is dletermilned by the relative position of the centre of glravity and a point kniownl as the metacenter. Let A UD, Fig. 72, be a solid bonnlded by a curved surface, and resting on a plane RQ. That it mlaly be im equilibrium the afxis AB must be vertical, as only in th:t position will the linle of direction ftill within the base. Sup-,ose ilowi that the body be mloved slightly to some new position in which A-B is not vertical. It will then be actecl pon by two forces, the resulta:lnt weight actino downward through G, and thle reaction of the surface IBQ acting ulpward along the line P.l The point i;, in which. AB and PM intersect, is called the mnetacenter of the body. The forces throuAgh G and 1c form a couple, as indicated by the arrows, the tendency of which is to il ake Al (D return to its orig1C)tlirlelble discrepllcv exists among these fiulies, is given bydifferent authori es. 1lhie (tdta above are reduced from iAppleton's Almes'Eicnl Encyclopedia.

84 EQUILIBRIUM OF BODTES. inal position, so long as thle centre of gravity is below JL. If it assumes the position G', above M, the tendency of the coulple is to overturn the body, and the equilibrium of the body ill its original positionl is unstable. If Mi and G coincide, the body remains at rest in any position. Hence the equilibrium is stable, unstable or incifferen.t, according as the centre of gravity is below, above or coincidlent with the metacenter, or, in genera-l, the eq2dlibriutm of a body with Ca curved surface is the same as if- it were suspended at its mnetacenter. 166. Equilibrium of a Body resting on Inclined Plane. Under these circumstan ces there is equilibrium when the line of direction falls within the bnse, precisely as in the case of a body resting on a horizontal suirface. 167. E General Criterion of Equilibrium. The dliferent conditions of equilibrium dlemonstrated in the preced.ing parag raphs may all be comprelhended in a single proposition, as follows: - -n, whatever mnanner a bocy is sutportecd, the equilibriumn is stable whien on vmoving it. the centre qf gravity ascen,cls, unstable when Whe centre qf gravity cdescedcs, acndc inclerent if it neither ascends nor cdescencds. This will be seen to be true of suspendeed bodies by a simple inspection of Figs. 67, 68. lil the case of a body restin- on a bise, ias,AB, Fi'o. 71, G will evidently rise if' AB is sligrhtly rotated about either ed(e oft its base. If AB rested on an angle, as B', the equilibrium woultl be unstable, and G would then occupy its highest possible position, so that on (listurllbino the body G would move in a descendingr curve. A considlration of' Ii(. 72 will show that when a body rests on a curved base, if the centre of' gravity be below 3li, as at G, it moves in an ascen(lin- curve on (listulrbinfg the body, while if it is at G' above ill, it moves in a descending curve, and if at ill the mnlotion is in a horizontal line. If the surface of the body possesses a dilferent curvature in different planes, the body may be in stable, unstable, or inlifferent equilibrium, according to the portion of its surface on which it rests. Thus an ellipsoid of revolution is in unstable equilibrium with regard to motion in any plane if placed on either extremity of its longer axis, while if placedl on its slhorter axis its equilibrium is stable with regard to motion in the vertical plane passing through its major axis, and indifferent with regard to any plane at right angles to this. 168. Experimental Illustrations. The prece(ling principles explain the (leportment of the toys so fiequently seen, which represent a grotesque human fiolure mounted on a curved base. The figure persistently resumies its upright position when laid upon its side. This is explained by the form of' the curved surface constitutinig the base, which is so constructed that the centre of' gravity shall be in its lowest possible position when the figure is upright. A method sometimes used to cause an egg to stand on one end is. also capable of a similar explanation. The yolk of an eggo is contained in a sac nearly concentric' with the white, and is somewhlat denser than the latter. By a vigorlous shaking the yolk-sac may be broken, and the heavy liquid then falls to the lower end of the egg when tbiz is

PRACTICAL ILLUSTRATIONS. 85 set uprilght, thus bringing the centre of gravity to a point so low that any motion of the egrc tends to raise it, Another experiment often shown is that of causing a wooden cylinder to roll up an inclined plane. This is done by weighltingr a portion of the cylinder with lead, so that the centre of gravity does not coincide with the centre of inmagitlide. The cylinder can easily be so placed that the rotation produced by the tendency of the centre of gravity to assume the lowest possible position at the same time causes the body to ascend the plane. The slope of the plane must of course be so small that the rise of the centre of gravity caused by the ascent of the plane shall be less than the fall due to the rotation of the cvlinder. A similar explanation applies to the case of two cones united at their bases, whlich can be made to roll up two inclined planes set at a suitable angle. In tlhis case, as in the last, the centre of' gravity really falls, since the rise caused by the rolling of the cones up the plane is less than the i:all produced by the simultaneous approach of the points of support to their vertices. 169. Practical Illustrations of IEquilibrium. All the attitudes and movements of' aninmals are regulated with regard to the position of the centre of gravitv of their bodies. In nan, when erect tlHe centre of gravity is about in the nmedian line of the body, but its place changes with every change of position. The whole art in feats of posturing consists in keeping the line of dlirection within the base formed by the feet. If' a limb be extended in any direction the centre of gravity mloves in the same direction, but if some other portion of the body be thrown toward the opposite sidie the equilibrium is preserved. When a porter carries a load upon his back lie must bend fborward, while a nurse carrying a child in her armls leans backward, otherwise the line of direction falls in one case behind the heels, in the other in front of the toes. For a like reason when climbin_, a steep hill we bend forward, and on descend(ling it we lean backward. The art of walking on stilts depends on the ability to keep the centre of gravity directly over the narrow base formed by them. The salne is true of' the art of walking or dancing on the tight rope. In this case the perfriller generally carries a heavy pole, by iaovingr which in any direction he can inore readlily bringr the centre of gravity to the (lesiredl position, thus miakinff his fe-ts far easier anrd inore (racefLti than if' units;isted by this aid. When a quadruped progresses it never raises both feet on one side simtultaneously, for in that case the cent~re of gravity wouldl be unsupporte(d, but the fore foot on one sidle anlI the hin l foot ow1 the other are lifted togrether in trotting, the fboe foot a littlet in adlvance in xvalkiilr, or else, as in galloping. the two fore le!s are lifted, andl the bo lN projected forward by the spring of the hind legrs. The p)riciples of e(quilibrium explain the different teats of balancing lotng poles, swords, etc., in an upright position upon the hand. Feats of this kind are miade much easier if a horizontal rotation is rgiven to the body which is b:iatlie(1, because the centre of gravity describes a small circle about a fixe,1 p:)int, so that even if' it is not directly over the base the diirection in whicch it tend(s to fall is constantly changing, causing a preservation of equilibriumi. It is thus that a top remains vertical when spinning rapidly, but falls when at rest. The same explanation applies to the coinIlon experiment of bldancing a plate upon the point of a sword. Titis is impossible it' the plate is at rest, but it' it be made to whirl rapidly no dificulty is experienced.

86 EQUILIBRTUM O)F BODIE.S. 170o Measure of the Stability of Bodies resting on a Plaane. - Satical Stabil.ity. We may consider tlhe stability of a body under two aspects: 1st, in relation to thle force necessary to mnove it fi'om its position of equilibriuln; 2d, in Ielation to the force'required to completely overturn it. The forces exerted in these two cases determine the statical and c?/ynamical stabclity qfo a body. Statical Stability. This is measured by the force necessary to begin rotation about ain angle of tlle body. Let 3f;A Fig. 73, be a mass resting on a horizontal plane, and suppose a force F to be applied along the line CF. If any obstacle be placed at IV, so that JIV Nicalnot slide along the phmne, F will tend to rotate YiNf about r1V. The moment of F relatively to an1 axis thr9ugh VT is P X D17:. The moment of the weigllt TV of the body tending< to lrevent motion is W X PT;. When nmotion begins, _ X.DN -- X PRN Hlence TV X P3X7 measures the statical stability of - NMAT and this is jointly proportional to the weight of the body and to the distance of the foot, the vertical through its centre of gravity fiom the axis about which rotation tends to take place. Hence the greater tlle weight andt the broader the base the greater tihe statical stability of a body, which is, however, independent of the heiglht of the centre of gravity. If the distance PlJ is different in different directions, the stability of iMN varies according to the direction of thle force F. The product W X PIVis called the moment of stabiity of the bodly liN. 171. Dynamical Stabili'ty, The dynamical stability of a body is me:asured by the moving force expended in overturnil'ig it. To accomplish this in the case of any mass MNV, Fig. 74, it must be revolved about one of its ang'les;, until the centre of gravity, which describes the curve GIl, reaches its highest point tI, after which the action of graivity will complete the overturning. Hence the Whole force is expended ill raising G thllough the vertic:ll heigllt lL. The power required to do this must be proportional to the weio'ht TWof the body, and also to thle hlei(rht ifL, through which this weight is raised; hence it is mIc:easured by their product W X flL. If thle clntre of' gravity ocenulies a more elevatedl position G', the dist:Ance Z1'J' L will be less tlhana ]Iii, Ilence less power will in that c:se be required to overtuirn the body. Theref;ore the dynamical stability of' a body increases with its weight, and dlininishes with the elevation of its centre of gravity. 172. Practical Illustrations of Stability. Many illustrations of the two kinds of' stability may be cited. A pyralmid owes its great firlness to tie breadth of' its base anlld the low position of the centre of gravity. W~alls are often strengthened with small expen(liture of' material by buildilnc offsets or by balter~in1 the side towlard vhlch the forces tending to

MECHANICAL POWfERS. 87 overturn them are directed. A hilgh carriage is upset much more readily than a low one, because the centre of gravity is higher, so that a smaller horizontal force overcomes its dynamical stability, and also if the road is sloping a less inclination is required to throw the line of direction outside of the base. For this reason wagons intended for carrying heavy loads are gen-:erally made so as to have the greater part of the load lower than the axles of the wheels. Quadrupeds have a inlch greater stability than man, because of the broader base formed by their feet. hence their young acquire the art of walking fGar more rapidly than a child, which must first learn the art of balancin(g itself on its legs. The human figrure is most stable wlhen the feet are so placed as to make the base as large as tpossible. If' the heels are in contact this occurs when the angle of' the feet is 90~. If the heels are separated by an amount equal to the length of' the foot, their angle should be 60~. CHAPTER X. M ECHANICAL POVWERS. 1 73. Di alqa tito -. The forces subject to our use can be applied only by means of machinery of greater or less complexity. Maelfines are either simlple or colmpouind. The elementary machines to which all systems may be reduced are called the mechanical powers. They are seven in nlumber, viz.: — the lever, wheel andc acxle, corTd, ptlleq/, inclinecl ]plaune, wedge, and screw. These maly agrlain, in principIe, be relduce(l to three, the lever, cord and inclined plane, the first of' these compllrehlening the lever andcl wheel and axle, the secolld the cord and p'tlley, the third the inclinec plane, the wzecdge andcl the screto. The force:applied to a mtlchine is ca llel the power; the olplposition which it is to overcome is the zoeiglht olr Jesistance. I. Lever. 174. D efiition. A lever is a rod either crooked or straight, moving about a fixed point called the fulcrumn. The theoretical lever, which we proceed to consider-, is supposed to be without weight, and absolutely inflexible. For convenienc:e of description, levers -are divided into tlhree classes, according to the relative positioni of the power, weight and fulcrum. When the fulcruml- is between the power ald the weight, the lever is saidcl to be of the Jirst order, Fig. 75; if the weight is between the power and fulcrum, the lever is of the second orcder, Fig. 76; and if the power is between the weight and fulcrum the lever is of the third ordcler, Fig. 77. In the two last

88 AllERI classes the powelr andl weight act in oplposite directions; in the first they act in the same direction. 175. Condfitons of Equilib rifum, There is equilibriulm with the lever when the moments of the powel and of the weight relatively to the fulcrum are equal. Thtis in the levrers represented inl Figs. 75, 76, 77, 78, e-quilibrium occuIrs when P X AF — WX BlF, orilwhen P: W:: BF: A_. (101.) From (101) it; fllows that TVr- PI?_ (102.), which shlows that there is, i general, a lechanicld adv-antage gained in employing levers of the filst ilrid second( orders, while in those of the third order the powerl -cts at: disadlvltvantae. When the level is straiollth, and the forces act at right angles to it, as in Figs. 75, 76, 77, the arms of the fobces evidently coincide with the portions of the levei' lyingA between them.and the fulclrum. If CnLy'/11n6mber ofo forces act ty poit a( lever the're will be euzilibriGon when, the calgebrai c sumn, of their;moientts relttive[l/ to the fulecrum is zero., 176. Pressure upon the Fulcrum. P ald 1W bhinrl balanccdl about the fulcrum, this must sustain a certain plresusiIe due to their coincident action. From ~ 108 (p. 59) it appears that whenl P ancld TV are in equilibrium their resultant must pass through F, and the maonitude and direction of this resultant, which is the pressure on F, ( an b-e determined, as shown in ~ 106 (p. 59). if thb. forces,act in parallel directions, the pressture on the fulcrum in thatcz di ection le juals their al'llebriic sumn. The pressure upon the fulcrum in any o her dir(:ction c(n be found by resolving the resultant of P and IW into two compoe: nts, one in the requlired direction, the other at righlt-angles to it. In cas( several fbrces act upon a lever, the pressure on the fulcrm u equals he rest lta llt of al1 of tIem. 177. Familiar Examples. As fa. ilial exlamples of the diflerent kinds of levers, we mention the fot!owinur: 1.It Order. A- crowbarr used to raise a weioht. a hammer applied to dlraw i nail. maund thle handle of a common pump. Scissors, pincers, etc., are composed of two levers of' the first order, the pivot beingo the fulcrum, and the substamnce to be (. ut the resistance. 2d Order. An oar is a lever of the second kindl, the wnatet-r servino for the fulcrum, the boat beino the weoight, an, d the muscular force, of the arm the power. The common vertical ha y-cutter, anml the machine eused by apothecaries for compressinc corks, are additional examples. So also is the common wheel-barrow, the wheel being the fllesrum, and the load the weight. 3d Order. In levers of this class the power is always greater than the weight, since its a;Lrm is less than the weioht-'m.l. The treadle of a turningy-lathe, or of a (rindstonle, is a gcool exsalple. Tong-s are also levers of the third order. We hi-ve another ill.lstration it the use of' a fishinorod, the fish beingq the wei(ght, tle c land ra.ising the rodl the power, and the othler hand the fulcrums. The nmechanical disadvantage of this kind of lever is well seeon in the case of a man raising a lonl, Iladder a~gainst the I In the psieselt chi:tes' thle foi'ces o', all sippose t, lie in the samte plane perpendicular to the txis of tihe furlcenm.

COMPOUND LEVER. 89 side of a building. The foot of the ladder is the fulcrum, and the weight miay be supposed to be concentrated at its centre of gravity; as the power nmust be applied near the base, it must exceed the weight of the ladder until the inclination of the latter becomes very slight. 178. Relative Velocity.of: Power and Weight. A considerition of Figs. 75, 76, 77, 78, will show that if motion ensues about the fulcrum, Velocity of P-: Velocity of TW:: AF: BF:: TW: P. (103.) Hellce the power moves througli a proportionally greater distance than the weight in the first two classes of levers, while in the third class a small motion of the power causes a proportionally large motion of the weight. The latter principle is beautifully applied in the human system. Thus it is necessary to be able to move the hand over a considerable distance with the elbow-joint as a centre of motion, and this must be done by a comparatively slight contraction of the flexor muscle of the fore-arm. One of the extremities of this muscle is attached to the upper part of the humerus, the other to one of the bones of the fore-arm, at a point very near the elbow-joint. Hence when the muscle contracts, the extremity of the arm is carried over a distance greater than the amount of the muscular contraction, in proportion as its length is greater than the distance from the joint to the point of attachment of the muscle. There are also several examples of levers of the first and second classes in the humanuskeleton. 179. Compound Lever. The cornpound lever is a system of simple levers, so arranged as to act upon one another successively, as shown in Fig. 79. The power, P, being applied at the extremity of the arm1 L of the first lever, the arm I acts upon _L' of the second lever, I' in its time acts upon L", and so on. The weight is applied at the extremity of 1". Supposing the forces to act at right angles to L, 1", the efficiency of the conlpound lever may be found as follows: The pressure acting on 25 - P, whence by (102.), p. 88, it follows that the pressure acting on 19' or TV19 19' 19 1' P -. In like manner, the pressure on: 1' - W" - TVf - =P - *'l,' In the same way we find the pressure exerted by F"to sustain the 19" L 19' 19" weight ITV", is WV" - W — P' l- l (104.), whence P X L' x L"-L W" X l X I' X'". (105.) If the forces do not all act at right-angles to the levers, 19, 9', L",, 1',,1" must be considered as the arms of the forces P, TW, W', WV", that is, the perpendiculars fiorm F, F', F", dropped upon their lines of action. It is evident that the above demonstration applies equially to all three orders of levers. Hence there is equilibrium with a compound lever when the power multiplied by the contzinued product of the power-arms is equal to the weight multiplied by the continued product of the weight-arms. 12

90 LEVER. 180. Effect of Weight of Lever. Hitherto we have neglected the weight of the lever altogether, which it is not generally admissible to do in practice. It remains, therefore, to explain the method of taking this into account. The weight of the lever may always be regarded as an additional force applied at its centre of gravity and acting vertically downward. For prismatic and cylindrical levers, the formulae may be deduced in the foliowinog manner. Call w2 the weight of the lever in all cases. As this acts through g, the middle point of the lever, the conditions of equilibrium are that the algebraic sum of the moments of P, W, and w, relatively to F, shall be zero. Calling the power-arm L, the weight-arm 1, and denoting right-handed moments by +, left-handed by -, the algebraic expression of these conditions for levers of the first order is, PL + w'(L 1) PL -w I(L - )'+ T-V1 O, whence W P- (106.) For levers of the 2d order, PL (WI + ww ~L) - 0, whence W = (107.), and for levers of the 3d order, PL - (WI + w l) O, whence W -. (108.) 181. Body resting on Props. An important application of the principle of the lever is the determination of the pressure sustained by each of two or more props supporting a loaded beam. Let AB, Fig. 80, be a beam resting on two props, A, B, and sustaining a weight W. Suppose B to be a fulcrum, about which W tends to turn the beam A B. The pressure exerted on A will be equal to the power which must be applied at A to hold W in equilibrium. Call this power P. Then from (101.), BC P: W:: BC: AB, whence Pressure on A = P = -WAB. (109.) In like manner, supposing A to be the fulcrum, the pressure on B, (P') is found from the equation, AC P': W:: AC: AB, whence Pressure on B = P' WAB. (110.) The total pressure found by adding these two together is ZBC AC\ P +P'= W x (B + B)= W' as might be inferred from the laws of parallel forces. In this demonstration we have not taken into account the weigoht of the beam AB. When this is necessary equation (107) may be used by considering a power (P) applied at A and B successively, to balance W + w, and solving relatively to P. 182. Balance with Equal Arms. Prominent among the applications of the lever are the various forms of instruments used for weighing, as the common balance, steelyard, platformscales, etc. We proceed to explain the most important of these. The common balance consists essentially of a lever of the first order, AB, Fig. 81, supported at its middle point F by means of 1 For W x gF = moment of w relatively to F. But gR B- AB = M(AF + FB) -=-(L + 1). Hence Bg -BF= gF = Y2(L ) - I= (L- ).

CHARACTERISTICS'OF A GOOD BALANCE. 91 a knife-edge, and having two scale-pans C, D, suspended from its extremities. In one of these pans is placed the substance to be weighed, in the other the standard weight necessary to balance it. The lever AB is called the beam of the balance, F the fulcrum, and A, B, the points of suspension of the pans. An index I is generally attached to show when the beam is horizontal. 183. Characteristics of a good Balance. A good balance must possess the following characteristics: 1, truth, that is, it must be so adjusted that the beam shall be horizontal when the weights in the two pans are equal; 2, stability, the beam should tend to return to its horizontal position when deflected from it; 3, sensibility, the beam should be deviated from its horizontal position by a very slight excess of weight in either pan, and the greater the deviation with a given weight, the greater the sensibility. To ensure truth (1), the two arms of the balance must be of exactly the same length and weight, otherwise the weights in the pans would be unequal when the beam is horizontal. To find whether this condition is fulfilled, a body is placed in one pan, and in the other the weight necessary to balance it. The contents of the pans are then transposed, and if the arms are equal the equilibrium is still maintained. (2.) The scale-pans should evidently be of the same weight, so that the beam may be horizontal when they are empty. (3.) All the parts of the balance should be symmetrical about two planes passing through the centre of gravity of the beam; one of these planes contlining the longer axis of the beam, the other being at right angles to it. To ensure stability, the fulcrum F, Fig. 82, and the centre of gravity of: the beam G, should not coincide, but should both lie in the same perpendicular FG, to the line AB joining the points of suspension, the centre of gravity being below the fulcrum. (1.) G should not coincide with F, for if this were the case the equilibrium of the beam would be indifferent, and hence when inclined it would remain in that position, with no tendency to return to horizont:tlity. (2.) G and F should be in the same perpendicular to A B, for otherwise the centre of gravity would not occupy its lowest iposition when the beam is horizontal. (3.) G should be below F, for if it were above, the equilibrium would be unstalble, and the beam would overturn on being deflected from horizont;ality. As the sensibility of a balance depends upon the weight necessary to deflect the beam through a given angle, it is evidently increased by everything which diminishes the fiiction at the fulcrum and points of suspension. Hence for fine balances it is customary to make the fulcrum F of hard polished steel, halvillg a sharp knife-edge, and supported upon a plane surfh. ce of'agate. The points of suspension are also made of wedles of steel. The three knife-edges should be placed so as to be horizontal, and per

92 LEVER. pendicular to the plane of the beam. They should also lie in the same straight line AB, for the M eights of the bodies placed in the scale-pans may be considered as applied at A and B, and if. the centre of these equal and parallel forces is at F, they will produce no tendency to motion about. that: point, whatever may.be the position of the beam, so that the sensibility will be independent of the weights in the pans. If, however, F is above the line AB, the centre of the forces will be below F, and hence will tend to keep the beam horizontal, thus diminishing the sensibility of the balance in proportion to their magnitude. If F were below AB, the equilibrium would be unstable when any considerable weight' was contained in the pans. The other conditions of sensibility may be deduced as follows: Suppose weights P, Q, to be placed in the scale-pans,.being greater than Q. The beam is deflected in the direction:of the greater weight, assuming an inclined position, as shown. in the figure. (Fig. 82.) There is now equilibrium between the moment of the deflecting. force P- Q (the excess of weight inl the left hand pan) applied at A, and the weight of the beam, to, applied at G, and tending to restore the beam to horizont a lity. Hence. when the beam comes to rest in an inclined position, the moments of these forces relatively to F must be equal; that'is,. (P - Q) X C F — w X mF.' Denote the angle CFA -m GF by a. Then.CF AF cos a, mF = GF sin a. Substituting these values in the above equation, (P - Q)X AF cos a -- w X GF sin a, whence Not etan (P — Q)X AF (110.) Now the angle a., through which the beam is inclined by a given excess of weight P - Q, measures the sensibility, which therefore increases as tang a increases; that is, (1.) as the arm lAF is greater, that is, as the beam is longer; (2.) as the weight of the beam w is less; (3.) as the distance of' the centre of' gravity of the be'aml below the fulcrum (FG) is less. Practically these conditions are true only within certain limits. If the beam is too light, or if the arms are very long, they bend under the action of the weights in the pans, thils bringingy the centre of:tlhese forces below F, and so diminishing the sensibility. Also if G is- too near F, a light weight deflects the beam by an inconveniently large amount. 184. Measure of the Sensibility of a Balance, The sensibility of balances may be compared by observing the angle through which -their beams are inclined by a weight placed in either pan. Thus if a weight of 1 mglr. incline one balance through half a degree, and another through 1 degree, the' second is twice as sensitive as the first. An easier method is by determining the smallest additional weight that will turn the balance per

CHEMICAL BALANCE. 93 ceptibly when this is loaded by a given, amount. For example, if a balance carries a load of 1 kgr. in each panp, and the smallest weigllt capable of deflecting its beamn is 1 mgr., the balance is said to be sensible to,)01ff0 of that load; thalt is, with such a balance the weight of 1 kgr. can be determined within I mgr. Theoretically, the sensibility of a balance constructed as described, is independent of the load, but pr:actically it diminishes as the load:increases, because (1.) a slight bending of the beam occurs; (2.) the, friction between the knifb-edges andc their supports is increased by the additional pressure.; and (3.) it is practically impossible to get the three knife-edges exactly in line. For these reasons the weight with which the balance is loaded must always be stated when the sensibility is estimated by this method. Still another way of cletermining sensibility is by observing the number of oscillations made by the beam in a oiven time. Whenever the beam is removed fi:oml its position of equilibrium, it perf;oms a series of oscillations about that position before coming to rest. The slower these oscillations the greater the sensibility of the balance. The.reason of this will be seen fiom Fig. 82. Suppose the pans of the balance to be empty, and let the beam be moved from its horizontal position:luntil it occupies the inclined position AB. The moment of the weightt;of the beam, which is zo X GF sin c, w\\ill now be exerted, to cmause it to return to horizontalitv, and it will vibrate about the line C)D in gradually dilinishing arcs, finally coming to rest with its axis coincident with that line. Now it is evident tlat the less the mnoment of tihe impelling force, that is, the less the value of w and Gi', and hence greater the sensibility, the slower the oscillations.. A good chemical balance should weigh to ~ n mgr. when each pan- is loaded with 60 grammles. The best balalnces detect a difference of load of 1 mgr. with a weig ht of 2 kgrs. in each pan, a sensibility of' a,-, under tihat load. A large balance constructed by Saxton, of Washington, and exhibited at the Paris Universal Exposition of 1867, tullfed on the addition of a weight of 1 mgr. when loaded with 20 kgrs. in each pan, a ratio of turnin'g-weight to load of,- o a id a large Alnerican balance in the Conservatoire (des Arts et.Metiers at Paris, detects a variation ofJ' I- mglr when loaded with 25 kgrs.' 185. Construction of delicate Balances. Fig. 83 represents a delicate chemical balance. Tlle beam AB is of brass, and of such a flrim as to combine strength with lightness. The fulcrum, F,.is a knife-ecldge of hardened, steel, resting, upon a horizontal plate of agate firmly fixed to the column GCD, sustaining the balance. A long index attached to A/B and moving before a 1 For a table giving the sensibility of the principal balances exhibited at the Exposition of 1867, see Report of U. S. Commissioners, Vol, III., Industrial Arts, by F. A. P. Barnard; p. 485.

94 LEVER. graduated scale G, serves to detect the slightest inclination of the beam. The pans P, P', are suspended fiomn delicate knife-edges, A, B, and so adjusted as to bring the centre of gravity of each pan directly beneath the point of support. The beam is also furnished-with a vertical adjusting screw, E, by means of which the centre of gravity can be brought as near to the fulcrum as is desired, thus regulating the sensibility according to the delicacy desired in any particular case. To prevent unnecessary wear of the knife-edge forming the fulcrum, the balance is arranged so that by turning the screw, 0, the beam is lifted, and the fulcrum no longer rests upon the agate plate, in which position it is allowed to remain when not in use. A similar device is sometimes adapted to the wedges supporting the scale-pans. To screen the instrument from currents of air it is customary to inclose it in a glass case which is furnished with leveling-screws V; V', in order that the fulcum upon which the beam rests may always be brought to a horizontal position. 186. Construction of Weights and Method of Weighing. The large weights used with delicate balances (1 gramme and upwards) are generally made of brass, those of lesser denominations of sheet platinum. The brass weights are carefully turned and adjusted by filing; the smaller ones are made by cutting out a rectangular sheet of platinum of a known weight, as 1 gramme, and dividing this into equal parts, 10 for decigramine weights, 100 for centigrammlles and so on. Occasionally platinum wire is used instead of sheet platinum. As very small weights are inconvenient to use, milligratumes and fractions of a milligramme are frequently estimated by a different method. One arm of the scale-beam is divided into a scale of equal parts, and a small U-shaped bit of wire of known weight, called a rider, is placed upon it. The rider balances a greater or less weight in the opposite pan according to its position on the beam. The beam is readily graduated as follows. The rider weighing 1 cgr. is placed upon it and moved until it just balances a half' centigramme weight, placed in the opposite pan. This point is marked, the weight removed from the scale pan, and replaced by a milligramme weight. The rider is now moved towards the fulcrum, until equilibrium is restored, and this point is also marked. Evidently when at tlie latter mark, the rider has the same effect as a weight of 1 mgr. in the adjacent pan, and when at the first made mark it corresponds to a weight of 5 Ingr. If now the space between these be divided into 40 equal parts, and the division continued towards the extremity of the beam, each division will correspond to -j- mgr. The Arc of Precision of Gallois has recently been substituted for the method of weighing by the rider, by several of' the best French and German makers. " This is an expedient for making delicate determinations of fractional weights, by deflecting more or less to the right or left an index needle attached to the beam of the balance beneath the centre of gravity. When this index points directly downward like the ordinary fixed needle of the balance, its effects upon the equilibrium of the balance is of course zero. When deflected so as to form an oblique angle with the horizontal axis of the beam, it contributes a portion of its weight, dependant on the amount of deflection, to the side toward which it is inclined. When, in the process

METHOD OF DOUBLE WEIGHINO. 95 of weighing, a position of the needle has been found which produces equilibrium, the fractional wei(ght contributed by the needle is read upon a circular arc, which is situated immediately behind it, and suitably divided. The division, of course, must be made experimentally at the time of' the construction the balance."l This method is more rapid than-the ordinary process. Which is the more accurate, remains to be determined by experience. 187. Weighing with a False Balance. If one of the arms of a balance is longer than the other, the weights required in each pan to produce equilibrium will be inversely proportional to the length of the adjacent arm (Eq. 101, p. 88). A given weight in one pan may therefore be made to counterpoise a greater weight in the other by simply lengthening the arm of the balance to which it is applied. The dishonest dealer is thus furnished with a ready method of defrauding his customers, by placing. the goods to be sold in the pan suspended from the longer arm. Such a deception may be detected by transferring the goods to the opposite pan, and replacing them by weights which previously balanced them, in which case the equilibrium will no longer be sustained. It is possible, however, to weigh truly with such a false balance. Let the article be wei-hed successively in each pan, the apparent weights thus found, being P, P', respectively. Callx the true weight, and L, L', the arms of the beam. Then L L' x:: P, and L: L': P': x; whence x: P P':x, or x- /P'. (112.) The true weight is therefore a mean proportional between the apparent weights. 188. Method of Double Weighing. Another means of accomplishinog the same end is the following. The body to be weighed is placed in either of the scale-pans, and accurately balanced by means of fine shot or sand. It is then removed, and weights substituted for it until equilibrium is made with the shot or sand used as a counterp)oise. The weights thus added evidently equal the weight of the body, as both have been applied at the extremity of the same arm to counterpoise a constant weight. As no balance can be made with absolutely equal arms, it is advisable to use the present method whenever great accuracy is required. The process which we have just explained is commonly known as Borda's Method of Double Weighing, after the physicist of that name who has generally received the credit of having invented it. That distinction, however, properly belongs to Pere Amiot, who made use of it some time previous to Borda. 189. Roberval's Balance. Fig. 84 represents a section of a form of balance often used where only a moderate degree of accuracy is required. The pans are attached to the vertical bars A C, BC' which are fastened by pivots to the extremities of the horizontal beamls AB, A' B', so that whether the pans riseor fall, A C and BC' remain vertical. An index I serves to show when equilibrium is attained. In this apparatus it is indifferent on what part of the pan the weights are placed. For let P, P' be the weights placed so as to act at different distances from the fulcrums F,F'. Imagine two equal and opposite forces, P", PI", each equal to P, applied at C, the centre of the pan on which P rests. Also two equal and opposite forces pv', Pv, each equal to P' applied at C'. The addition of these does not alter the equilibrium of P,. P'. The, balance is now acted upon by two equal forces P", P'p, applied at C, C','Report of U. S. Commrs. to Paris Universal Eposition of 1865; Vol. Il., p. 484.

96 LEVER. which balance each other, and also by the couples P - P/", P' - P', which tend to produce rotation of the whole system of levers. The efi'e(.ct of' these couples is balanced by the resistancee of the pivots, hence the final result is the same as if P, PI were transferred to C, C', the centres of the scale-pans. 190. Steelyard. Another instruiment adapted to the weighing of articles of' moderate size is the common steelyard, the use of which dates back to the time of' the ancient Ronians. It consists of an iron bar A C, Fig. 85, near one extremity of which aitee two hooks resting upon pivots at A, B. Tlhe instrument is suspendled by the hook H, and the article to be weighed, WT, is hung upon the other hook. Wis balanced by a small counterpoise P, which is movable along the beam. -The greater the distance BD, the greater will be the weight balanced by P.. The beam is furnished with a scale which is graduated in the following m-anner. If the hook B is placed at the centre of gravity of the beam, the graduation is effected by the process described in the explanation of the use of' the rider (~ 182 p. 94) More'commlonly, however, the centre of gravity is at some point as G, beyond B. In this case, let E be a point from which the counterpoise must be suspended to just balance tlle beam when no weight is suspended from A. E is the zero of the scale. For calling tw the weight of the beanm, w X GB - P X EB. Let a weight W, be applied at A, anl suppose P to balance this when at Ail. Then, W X AB-w X BG + P X BM- P X EB:-+ P X BM- P X EAL If now a weight, W,', be suspenddeid rom A, and D be the position of' P when T1' is counterpoised by it, WT X AB -P X EB + P X BD-P X ED. in these equations AB and P are constant, whence W1: TV'::EM: ED, (113.) Or the weight balanced by P is proportional to the distance of P from E. Hence if 1W be taken. as 1 kgr., and W' as 20 kgrs., and the space ED divided into 20 equal parts, each of' these will correspond to 1 kgr. When P is placed at the extremity of the arm A C, it denotes the maximum weight which the instrument is' capable of measuring, but the same beam may be used to determine greater weights by usinog a heavier counterpoise, or by suspending the weight T from a pivot nearer B, in which case a correspondingograduation is made on the other side of BC. In the Danish Balancre, Fio. 86, the counterpoise is fixed at one end of the beam, the article to be weighed suspended from the other end, and the beam is balanced upon a movable knife-edge fulcrum, from the position of which when equilibrium occurs, the suspended weight is determined. 191. Bent Lever Balance. This form of balance consists of a bent lever ABC, Fig. 87, moving upon a pivot at B. At one end is a hook A, to which a scale-pan is attached, and at the other a fixed counterpoise C. Let us denote the total weight of' the beam and pan by w, their common centre of gravity being at G. On placing a weight Win the pan, its effect is to raise the arm BC. But as BC rises, the leverage of C is increased, while the leverage. of f:is at the same time diminished. Hence. the arm BC ascends until w X BM -'W X BA. A circular scale PR is' o.ttached to the balance. It is best graduated experimentally by placing weights of 1, 2, 3 kgrs. in the pan successively, and marking the point denoted by the index with corxesponding numbers. 192. Platform Balance. The various forms of machines used for weighing bodies of considerable magnitude depend upon the principle of the compound lever. Fig. 88 shows the construction of one of the most corn

WHEEL AND AXLE. 97 mon forms. It consists of two V-shaped levers AIB, CID, resting upon fulcrums A, B, C, D, fastened at I to a fourth lever FE. A vertical rod EH connects FE with the balance-beam HG. MN is a platform on which the article to be weighed is placed. This platform is furnished with four steel legs, a' b' c' d', which when in place rest upon the levers AI, BI, Cl, DI, at the points a, b, c, d. The pressure upon the platform is communicated to EF by means of the four lower levers, and thence by the vertical connecting-rod to GH, where it is balanced by a movable weight P, as in the steelvard. The V-shaped levers are used instead of connecting the platform directly with I, in order to render the effect of the mass weighed the same on whatever part of the platform it may rest. To compute the ratio of the pressure W, on the platform to the counterpoise G, we may consider the whole pressure as applied to either one of the four arms AI, BI, CI or DI. Suppose it to be applied at a. Then by the principle of the compound lever (Eq. 105, p. 89), the power multiplied by the continued product of the power-arms is equal to the weight multiplied by the continued product of the weioht-arms. That is, P X GF' X EF X AI - W X HF' X IF X Aa. Thus if the length ofAI - 20, Aa- 4, EF = 20, IF - 5, GF' _ 10, HF'- 1, G X10 X 20 X 10=- W X 1 X 5 X 4; whence 2000G - 20VW, or W - 100G. A weight of 1 kilogramme at G will therefore sustain a mass of 100 kilogrammes upon the platform. An additional counterpoise P, is also furnished, the arm HG being graduated in the manner described in the case of the steelyard. II. Wheel and Axle. 193. Wheel and Axle. — Laws of Equilibrium. The wheel and axle consists of a cylinder or axle turning on gudgeons, to which is attached a larger cylinder, called the wheel, as shown in section in Fig. 89. The power is applied at the circumference of the wheel, the weight at the circumference of the axle, in such a manner that they tend to produce rotation in opposite directions. The power and weight are generally parallel, and applied along tangents to the sections of the wheel and of the axle respectively. To ascertain the law of equilibrium, it is to be recollected that the moment of the power and weight relatively to the axis through C, must be equal. Hence calling P the power, WV the weight, R the radius of the wheel, r the radius of the axle, P X AC - WX WB C, or PR,= Wr; whence P: V:: r: R (114.); that is, there is equilibrium with the wheel and axle when power: weight:: radius of axle: radius of wheel. From (114.) it follows that W= P R. (115.) If the power does not act tangentially, but has an oblique direction, as AP, Fig. 90, we have for equilibrium, P X CD - W X r, or calling a the angle made by AP with the tangent AD, P X R cos a =- W X r. When the...pQwer and weight. are applied by means of a rope the halfthickness of the rope miiust be added to the radii R and r to obtain a cor

98 WHEEL AND AXLE. rect theoretical result. In th's case, calling! the. llickness, we should have -)(R + II) w (1 + i t). (I, I.) Practicaly, however, tIis correction is un!ec(t ssary,, -as it is lcs ttlian the deviation tirom they prodtluced by friction, r'io'iiit of ('corda'e, ctc. The totall pressure on tlie pivots sustaiilling the ml1achine= P + IV, if' these iorces are parallel. 194.'1The chief advantag'e of' the wheel and axle lies in the power it gives us of continuing the inotion w1lich raises the weioht. so that %with a machine of modlerate size we call raise a heavy weihllt throuill alny re(lpit'ed (listance. Praetical applications are seen in the comimion windlass, cranle and coapstan. Thile wheel ant l axle may evidently be considered as a lever, of whlich i,?'1 are tilt' arllus. 195. Chinese Capstan. Since the weivht is greater than the power in the ratio of thll ralius of' the wheel to the le;~(liLs of the axle, the wveillt sustained by a givein torce mlay be increasedl, (ither,by inCreasincg the size of the wheel, or dliLnishing hIin the size of thel axle. Practically, however,l it is inconvenient to use a'very large wlleel, -while a small axle d(oes Tnot possess the necesslary strength. In llhe lmachine known as the Clhinese (',pstan, Figi. 91, this difficultv is overcome in the following nmaniner. The axle is mulvade in two parts, D amdl C, one of' which has a greater dliameter thLtn the odielr. The rope sustainin' the wei lit W1l is wound about tlihs axle, so that its two sectins. ED, DFC, tend to prodnce mnotion about the axis AlP in tdielrent directios. l'o fintd thel conditions of' eluilibriulll, call r the radiuls of the larC;er axle, / 1that of the slteluler one. The weig ht IV is sustaiinl by thle colrds ED, F C, each of wllch thelefore exerts a force ->-W I tan'entialit to the axis about whlicli it is wound. Hence that portion ot the wc(i'lht susta;ined by (C'F teiids to produce motion in the dlieetio, of thu arrlow. havini' a nioment _ -/s', and acting' in the samne ditection as the power P, w lile the plortion sustainecl by E1l) tends to produce nlotion ill an opposite (lireetion, htavint a nlloment - l'?. I-ence eq(uilibri:i ensues when PRl -- ~ Ir" - 4 H', or PR -j W(rQ- r') whencme P: TI:: 1( - r'): Pi. (I 17.) The efficiency of' this llachine theretboie depentls not upon the ablsolute size of' the axles, but unirely llipon the. difference between their radii, so that they nay be imade as largue as is necessary to bear the stl'ain put upon them without d..minislino the nmechani cal adv n ta,-e. 196. Combined VWheels and Axles. It is often necessary to colbl'ne whl:eli s;an(l axles, and when ilihe object is to o:in an increase of powetr, the method nlost comitnonly ad~ pt!l is that of toothed wheels.'I he teeth of' tile wheels whithl gear into ealch other being malde of snitalile fbrm, tlotion is (oluli;unicatred bh the le"ssiure of' the teethi of the first wlieel, or driver, ais it is calledl, upon thioe of the second wheel, or /follower.'IThe conditions of' equilibnium may be Ibind tlius. Let I Bi. A/i'B Fig.92, be two cogged-rwheels havingr their ceniites at C, C'. (Cl I,, e radii of' the wvlhe.ls, r' the rtadii of' theil axles, ulpon wch Pt and t act. It' P and 11 are supposed to be in equilibrium, the Ibioce at D comiinunicated fi'om P by nieans of the wheel ALB, and tending to prodlcee a ri hllthanded rotation in A'B'. must be balanced by an e(lual oth(ree acting in an opposlte direction, comminunicated firon TV b i means of the whlitel A'13', andt tending' to produce a riilit-handed( rotation in A B. Callingo this oiree P', theli in the nwheel A', B;PR - Pr, and in A'b, 1,'1- WV'. To estimate the efiect of' the cooped-whleels lcne,,tulppoe lte axles to thxeve equal radii, that is r- = r. Then fiom the preceding equations, --- -- -— ~E —-----

TRAINS OF WHEELS. 99 P: R:: F, andl T: Ft':: F:?r, whence P: TV R: i' (1 18.), that is, powCer is to eeifJt as the r1dli/.s of the cogqe(l-uwheel on whRich P octis, is to the mtadi. of t!he wvieel acled uponz lo, Iy. As the circunifertences of A B, A'B', are poport'onal to thueir ratIii, propo'tion (118) illay be written P T c::ircatur.fre/ce of A 1: c (irciofererce o/ A'B'; or since the teelth are eCi.al in size, the nuniber of thetim cont; iictd in the circiumferences (f A f. A'[, I a re proplr tionla to thos- circniuiereleces, anl we may lwrrite the equlition of' eq1liil)riumn as follows, d(enotin' by b, n'u, the nunlibel of teeth in the dlriver andt fbllower reslectively, P 1 I:: ~: s'. (119.) 197. System of any Number of Wheels. To find the conditions of' eClqilibrium in a systein of any nlumber of wheels in which power is trans-litted by Ineans of' cogs, as in Fi'. 93, in rhich Ihe teeth t'i eaeh wheel are put in motion by the te eetl of a pinion attached to the precedling wheel, we proceed.as follows: Let ]i, 1l' It', be thle radii of the wh]eels,'.', 1, tile radlii of' the correspondlin~ pinions. Denotin'r by F", h I, "', etc., the powers applied by the p)inions at tilhe cire ftulllfelec of tle wherlels AB, A'B', A"B", bwe have, by the course of reasoninz aplqpliel in the preceding parag(raph, PR Fr, FR-' = ]F'r', F'l"/ =' t Wr'. Ilence B P T -- F,, _ -- 7 l l - 1 — P - and P: T;:: X r'X r": RX R t x R'. (120.) That is, pover is to uweqht cas the co(,intlell /)'ro/tlt of the (radii of the 1])ilionlS is to the coom.ined ro'lhc! l of tle terodeii (If the whe els. 193. Spur, Crown and Bevel Wheels. A wheel in \ilich thle teetli;1 l:isel 1lo11i tilt e circnultlrice as ill F1;'. 93, is called a spur L-z heel. If ttle te.eth ite raised l)orn the side i,f thle wheel. andi li,r;llel to tlie axis as in Fi. 9 f, it is a c'ooa.-lwheel, an!i if' they:t,'t risc(l "1'p n a siilil('e illClinteI to tllt )lane of the wiheel (Fig,,. 9.), it is kniown ans a becelledI-liheel. Anl inslectioin of' t the fiilrles wvill show\l tllit spnur-e-,eels c-()11)111ii('iItt hllotionll in theti owvI p)ltlles, while a crown-whleel een-taing' itth a SluI p' n on (F g. 9 1), causes a olllti ln of the lattl' aboutan a axis at rim'llt and1 es to its own.i Bevelled wheels are torbnmed of firustrta of cones, and coiiiiiiuiiciate iiiot'on fi'omi, one;txis to anotlherl incilined itt n11y Iangle to it. Wlhieii a. pi'on works in a strai-lit bar, havinog teeth raised upon it, the combination is knoiwn as a r'ack: arnd pi2/ion. 199. Form of Teeth. In coinibiations of circular whcels, the teeth Slholiil( be so c(ut that a tper fetl ly uit'oibrin motion may' be trallsinitted from one wheeltl to tle other. lThius thli two co,-whlieels represented in Fig. 96, shcild inl e,ve with tile s;iInie relative velocites that Awoulld ensue it' tlhe two cilrces A.4, A'13, called the pitch-circl'es of' tlhe wheels wev'it allla/e to roll upon eatl1 otiicr. Tihat this ma.y be tlhe easi', it is necessary tlhat SAl, tllt lerpeidoicilullar to tlhe suLrfaces of' contact of the teCtlh at any' ioment of' the revoltiiton shall pass thll'ougl D, thle lp)oiit of conltact of tlhe,itell-circles. To /denllilostlrate this, sllq)oset the circ. e AB tu roll upoin At. T tl! tn-l linear velocity of' any point as D in thle cilrcnifiit'leces of' tile cii('les, iiust be the samie. in e;tch, by the colnitions ot' tle mlltiOln. 1enote 1)y -, the aide tlI'oul'h wh l'i A lb revolves in aim elementllt of' tinle anll b1,,' hat l)alssetl tlhrongoh lby A'B; a antl a' aile cal.led the (t,,n/.ll/r' re'oi ies ot'e, B, A'/'.'11 leun'tlms of' t!e arcs Dfn, DC' sul)teiiuliif tlote Is'i'dles, must be e/ ll. Now as cre t- arg!e X'(radi',, DK _ ('D X a, DK' — C']) X a, Yhence CD X a = CiD X a' or a: a': C'"D; CD. (121.) 1 Wlien teeth are cut upon an adxle it is called'a pillion.

100 CORD. Suppose now that the wheels are furnished with teeth. To fulfil the required condition,,proportion (121) must still hold. Consider the wheels as revolving, and let R be the point of contact of two teeth. The pressure of the driving tooth upon the following one causes motion, the teeth being in what is known as sliding-contact. By the conditions of the motion, the two teeth must at the moment considered, have equal velocities in the direction MS. The space passed over by the tooth K in an element of time- CM X a, that passed over by the tooth IC' - C'N X a1, and as these spaces are equal a: a':: a C'N: CM (122). If the values of the angular velocities in the case of motion communicated by teeth are supposed equal to those obtained by the rolling of the pitch-circles upon each other, a comparison of proportions (121) (122), gives C'D: CD:: C'N: CM, (123), a condition which can hold only when the triangles CDM, C'DN are similar, in which case MN must pass through D, the point of contact of the pitchcircles. 200. In all well-constructed gearing the teeth of the wheels are so cut as to fulfil this condition. Great ingenuity has been spent in devising the forms of teeth suitable in different cases. For the details of such investigations, the student is referred to any of the standard treatises on Mechanism. It may be stated in general terms that it follows from the preceding demonstration, that the teeth should have the form of epicycloids, generated by the rolling of a circle of diameter determined by the size of the teeth, upon the pitch-circles of the wheels. III. Cord. 201. Case of Forces applied at different points of a Cord. The forces acting at various points of a cord are in equilibrium when they bear the same relations to each other in magnitude and direction, as if they were in equilibrium about a single point. Let us first take the case of three forces. AB, Fig. 97, is a rope, at one extremity of which are applied forces, P, P', and at the other a force P". To find their relations when in equilibrium, lay off AP, AP', representing P, P' respectively, and complete the parallelogram APRP'. R, represented by AR, is their resultant. Now the rope being flexible it must assume such a direction that the directions of X and the remaining force P" shall be in the same straight line, RP". Hence the force P" required to balance R, must be equal to R, and the forces P, P", are represented in magnitude and direction by the three sides, AP, PR, RA, of the triangle APR, in which case they would also be in equilibrium if applied at a single point. The same theorem holds for any number of forces. Let AB CD, Fig. 98, be a cord with forces P1, P2, P,, P4, P5, P6, P7, P8, applied as there shown, and represented in magnitude and direction by the lines AP,, AP2, BAP., 2BP4, etc. Thle resultant of P1, P2 (-1), takes the direction ARl, which must lie in the prolongation of BA, and hence may be transferred to B without alteration of its effi

FUNICULAR POLYGON. 101 ciency (~ 105, p. 58). In like manner, the resultant R2 of R1, P3, P4, must lie in the prolongation of C(B, and hence may be transferred to C. The resultant R3 of BR, P8, P6, must lie in the prolongation of D C, and may be transferred to D without alteration of the equilibrium. If, then, there is equilibrium, the resultant R, arising from the combination of P,, P, P3, P4, Ps, P6, must be equal and opposite the resultant, 8, of P,, P8, which would also be the case if all the forces under consideration were applied at a single point. In this demonstration the cords are supposed to be perfectly flexible and destitute of weight. 202. Illustrations. The cord is extremely useful in changing the direction of the motion, but as the tension is the same throughout its whole extent, there is no mechanical advantage gained. Most interesting examples of the employment of the cord for this purpose, occur among the muscles of the human body. The tendon of the muscle raising the eyelid (Fig. 99,) runs through a loop A, so that the contraction of the muscle in the direction BA, moves the lid in the direction CA. Similar examples may be seen in the tendons of the fingers and toes. 203. Weight Supported by Segments of a Cord. If a weight W be attached to a cord in the manner represented in Fig. 100, the forces TT" which must be exerted at EF, to sustain it, may be found from equa5ions (27), (28), Tz sin DAC si tions (27), (28) T Wsin DA C T'= Wsin CBA If we suppose BAC- - DAC, T' -T. In this case, denoting BAD by 2a, T= T' sin a sin a W Wsin 2a - 2 sin a =os a 2 cos.- (124.) The magnitude of the forces T, T', which measure the tension of the rope, is evidently inversely proportional to the cosine of half the angle made by the segments of the rope (cos a). If these segments are parallel, in which case a = 0, T - T'= W. That the segments of the cord may be in the same straight line, that is, that the rope may become horizontal, the angle BAD must = 1800. That W W is, a = 90~, in which case T- T' - - = o-' That is, it requires an infinite power to cause loaded cord to become horizontal. Still further, as the whole weight of any material cord may be considered to be concentrated in its centre of gravity, and to act vertically downward through that point, it is impossible to stretch any material cord into a horizontal position.l It is also evident from the preceding demonstration, that the smallest power can he so applied as to produce an unlimited amount of force. 204. Funicular Polygon. The principles of the transmission of pressure by a cord, determine the position taken by a rope when acted upon by a number of forces applied at different points, as represented in Fig. 101. The figure ABCDEF, formed by the segments of the rope, is known as the funicular polygon. A similar figure is assumed by a series of beams united at their extremities by pivots and suspended as shown in Fig. 102. The forces acting upon the polygon in this case, are evidently the weights of 2 Compare case of Toggle Joint, ~ 91 p. 63.

102 PULLEY. the sepm.rate pieces tA B, BC; CD, DE. which may be considered as concvnu tlate(l at telircil re sp'ctie (cen'lties of' a1lv itv. It is fbunl(i tllhat if' the fiiincill'1r polygon be invert(ed ( as s:.own by the (lotted lilies of' the figrel). it is the st;')lng-est firill ill n'lic]l a i' VCt nuibert of bef's c111 be Uziited to resist pressurel, a t:eict of illpolrtace n tIlie art of (onvstrilctiol. 203. Catenary. If' tIle llnit,,) of the pi(,ces- AB, BC, etc., be incre;sed;l il lefilltelv tlii f'lin il )'lalr oly',)n hIs'an ina iite ilu!nl)eir of' rilcs, antd becmles a curve (Fi'. 103). Vli;c'l is ktilow asw tile cterlli,. Au i1loaltedl cordl. or a chlin coinpoted(l of a (reat riivl-v lin's, wlihen silspe ile (l tiv both enils, is all examline of this curve. A knowxlte de' of the! roperties of the catenanrv is of' the (reatest value in the construction of suspension bridglIes, which are simply bri'ides suspendedil ifol two pairallel ch:iins, (catenuiries) stretcheLd across a river. If the position of the extrelitics A.1 B, the total length of' the catenary A CB. alnd its weilght aire known, it is possible from analytical constrnction, to dete'rmine tle tension at aiy pll)itut of the curve, bothi when unloadedl anl vlien loadedl wihli a giveii weidlt, so that the strength necessary to be given to the chain-cables of' such a bri(lot' to enable themn to resiszt the strain brought to bear upon them, cai ea-ily be caicIla:teld. Thet inverted catenary is used in the construction of domnes and arches of masonry. IV. Pulley. 206. Pulley. The pulley consists of a ciculliar disc, AB, Fig. 104, mnoving uipon an axis:At (, a:id hLivinig a cord or ba-ld to whlicihe the power f:ud weiollt are applied, l:iid ill a grlooe runlling overl its circumtfetrenCe. The:xle stupl)potillg the mnachine mn:aVy he fixed( or othlerwlise. In thle first cse, the pulley is solid to becfixzed, PFir. 104, in thle second m2oucvabe, Fig. 105. 207. Conditions of Equilibrium in Fixed Fulley. The power beinig applied at P, aInd tle w\eight at't V, these tenll to ritate the pullley in opposite directions; lhen(ce, fbr eq(nilibriimnl, tlheir imoments must be eq-ual; that is, TVW X (B =- P X A (7, or as A C - CB, both b)eilg radii ()f the salte circle, -W= P. (125.) Hellce, with a fixed pi&u.ley there is ezquilibrium when the poweir cappi:ied is equal to the wei.qht. 208. Coditl. onis of Equilibrium in V Movable EPulI y. Case 1. Whlen the Segy/ments qof the Corid.are plarle el. Fig. 105 represetnt s siiucli: lulley, with tle p)ower P allpplied:at A, andi thle weighctt IV suspendedl firoiii thle axis C. W:l:ndl P c:cll ten:l to r(ot'ite the pltilley about B, but in opl)osite dlirectionls, aiid for equilibrtium their lltoliients must be equtyal; that is, T X BCC -P X AB. _But AB'2BC, ts AB is the dia;inetel of the pulley, hence IV X B C P X 2B(, or W — = 2P. (12'6.) Thli:t is, w\ithl a sinogle iinov:bl)e pulley, it' the seglllents of the cord ane para'llel, there is equcilibritm when the weight equals twice the powero. Ctase II; WM7en the Seqmernts of the Cored are not parC'allel. Fig. 106 represents a pulley in which the segments of the cord are

COMBINATIONS OF PULLEYS. 103 ob1liqe. Suppose the powe oer P to be a~pplied ".] ii, )tl.!e Hlie PA/,:TV.ld } to)'4ustpell'l ti'Oll (:.!!ioll C-, oellotfe lh 1:',.:e /),'2 lllrl ce l)y th]e se:ielcntt s of ti'e cord 1by 2;. t:ldl P eled to toatete 1 le ltdley ill olplpsite dlile(t;iolls;:bol.;t B, tile first po)ilt, ft colt.:ct of tile cord Nw-iihl thle pultlle3. Thle moellne!t of TV - TV X Bjil, tliat of P- P X B.D), _.D beillhn drawml frlolrl _B pel1'-'t:llendicll ll to PA plroduced. Henct, for equilibnmillun, TV X BBJl - P X BD. IBut Bl — BAB sil AV'J -i / iTN sin,, o:dl dB.D = flsinl 1BN7D - BN siln AIB JY si A s'2,. I-Ie!ce TV X BN sin. P X.IlBNsin 2,/, or TV sinl- _ P sill 2, vlwhence -Wsinl.. - P'2 sin a. cos,, ";nd IFV - 2 P cos 1. (127.) Th:l t is, if t-le seolellts of' thle corll:1ire oblique, there is egaiiibriumn when the 9weight eqtuals twoice the power into the cosine of' hcw' the ac cle mntcde by:the se9gmen.ts. If the sergments become paral!el, in which ease 2 a- 00 awl cos a - 1, = 2 1', as (tlllonstrate(l undler CGoe I.'I hat the segnments lmay) lie in the satle straight line, 2 a iust become 180~, in whlich case, cos a = 0 and P - V c, as in the case cf the cordl unlder silmiar conditions. 209. Pressure on Support. With a fixe(l pulley in which the seOnenlts of tie cord are lparallel, the 1)ressure oil the axis of tihe pulley evdlently P -o+. With a tovable llley;and p.arallel scti}ceittts o1 colrl, the.!'essT'e onl tlse hook l -1 - I 1', as the weight is slstaiue(l by the two seginlents AIP, 1-1H. AWhenl the segtmlents mlake an oblique anrile, llhe presslure exerted ilI)o0l the hook in any given direction imay lite calculated by mleans oft the tri:ngle of folres. 310. Combinations of Putleys.-Block and Tackle. ComlbinatiOls of t uilet1s are eimp oyed in the conimon blick atd lac/cle, and in various other ftors of hoistinr lmlachines. There are several ways of arrangig'il tlhe pulleys conll)osSing tlhe systellm, tile itiost imll)rtant of wliich we priocee(l to consi.tle. Two comlllonll forms of hoisting Itllleys, known as the block and tackle, are shown ill irgs. 107, 18, il wlich systemsl tle santre roe ruins over all the t )llleys. In both tigures A, B, are fixed, andl C, D, miovalble pulleys. Since tile weighlit }IV is su)pp:)orted by the united action of the senlllents of the cords, A C, CL, BD, DE, each of these must stustail an equal strain. If there are tn segments, each will bear -tth of the whole weight, and as the tension of' the cord is lle sanme tlhroughout its whole extent, thle power P'required( to balanceJ TJ is. HIence TV — n P (128.) Thalt is, i) Cany colinCalion of pulleys in which one r,, pe runs lorneugh lle who.'e sqsi'emn, i/e weight is equtal to the p)ower multip)lied b/ thle 1tnumber oJ' segm';nms (oJ' the coi-(l. Th''e downward pressure at 1 = T - P = n P -- P P (n -- 1,) (129). 211. Geometrical Pulley. The system represented in Fig. 109, is known as the (Geometrical _Ptitey. In it there are as llany' sep;arate ropes as there are mlovable pulleys. The relation of weight to powver mally be 1 P is supposed to be so applied that HINC, CNP are equa;l.

104 PULLEY. found as follows. If a power P be applied to the pulley A by means of a rope running over a fixed pulley S, we find from the law of the movable pulley, that the upward force exerted by A upon B 2 P _ P X 2. Since this force 2 P becomes a new power applied to the movable pulley C, we have Upward TForce exerted by B upon C - 4 P = P X 22. In like manner. Upward Force exerted by C upon D - 8 P = P X 23. Hence the weight which D can uphold = 16 P - P X 24. In the same manner for any number of pulleys as n, we should find that W - P X 2n, (130), so that with this arrangemnent the weight is equal to the power exerted, multiplied by 2 raised to that power whose exponent is the number of movable pulleys. The pressures upon the hooks are as follows: — Pressure on N W — P X 2 —- P X 2u-1 Pressure on M -- Pressure exerted by D on C = i Pressure on N - P X 2n-' P + 2nPressure on L Pressure exerted by C on B = i Pressure on M-= P X 2-2- P X 2'-3 Pressure on H_- a Pressure exerted by B on A = ~ Pressure on L-i X 2n-3- p X 2n-P 212. Another system of pulleys is shown in Fig. 110. The power applied at P produces by means of the rope PA H an upward pull at H = P. The rope ABG exerts an upward pull at G = 2 P, the rope BCF exerts an upward pull at F = 4 P, and the rope CDE exerts an upward pull at E - 8 P. Hence, the total force exerted on W,- P + 2 P + 4 P -+ 8 P 15 P or W_ - P X 2`-1 (131), if n number of movable pulleys. The pressure upon the hook K = P + W -P + P (2- ) = 2n P. 213. The relation of weight to power in the various combinations of these systems frequently occurring in practice may easily be calculated by a process similar to that by which the preceding formulae were obtained. For example, let us take the arrangement of' pulleys shown in Fig. 111. The power P is applied to a cord running over the pulley A, which alone would cause an upward force on B - 2 P. The tension of the cord running from A over the fixed pulley D- 2 P. This force is also communicated to B, as the cord is fastened to the axis of that pulley, causing the total upward pull exerted by B to be 4 P. Hence in this case W 4 P. In a similar manner it may be shown that with the combination represented in Fig. 112, W 5 P. 214. Differential Pulley. The liability of a system consisting of a large number of pulleys to get out of order, and the great length of rope required for such combinations, together with a number of other practical difficulties, render them undesirable for ordinary use. The device known as the Differential Pulley (Fig. 113), remedies some of these difficulties. Its principle is similar to that of the Chinese Capstan described on p. 98. The upper pulley, AE is fixed, thelower one FG, is movable. AE is composed of two pieces virtually forming two pulleys EB, AD, firmly connected, and moving about a common centre C. An endless chain BPDAGE runs over both pulleys as indicated by the arrows in the figure. The weight is hung from the axis of the movable pulley. To find the conditions of equilibrium in this machine, call P the power, W the weight, r the radius of EB, r' the radius of A C. Then the fixed pulley is acted upon by three forces as follows. P, acting with an arm CB (r), and i W with an arm A C (r'), both of which tend to produce a right-handed rotation; and 1 W with an arm EC (R), tending to produce a left-handed rotation. For equilibrium the opposite moments of these forces must be equal, that is,

INCLINED PLANE. 100 P r+ 1 r'" — WV r, or P.- I 1V (r-r'), whlence P: W (r-r')::. (132). The efficiency of' this fborm of' pulley therefbre depends simply upon the difference between the radii of the two portions of the fixed pulley. The friction of the differential pulley is enormous, but a practical advantage is gained thereby, from the f:ct that it will not run down when the power P is removed, the weight tw not being sufficient to overcome this resistance. 215. Effect of Weight of Pulleys. In the foregoing discussions we have iiot taken the weight of' tlle pulleys into ac(ouint, which is necessary in practice, as they are fiequently very heavy. With systems of movable pulleys this weight generally acts in opposition to the power, rendering a considetrable portion of the applied force practically useless. A still greater loss of power occurs because of' the fiiction and rigidity of the cordagRe employed, so that a very large percentage of the power is lost. The different velocities with which the various pulleys composing a system move, is also a source of' practical difficulty. For these reasons complicated systems of movable pulleys are rarely used. The greatest value of' the pulley is to change the direction in which the power acts. V. Inclinecd Plane. 216. Definition; Conditions of Equilibrium. The inclined plane is considered as a perfectly hard and inflexible plane, inclined at an angle to the horizon. ro find the conditions of equilibrium, suppose W to be the weight of a bodly resting upon the inclined plane of which M2YN is the vertical projection. Let the power balancing it be denoted by F, acting in the line AF,1 and making an angle p with the inclined plane. Denote the inclination of JlX by a. The weight, W, of the body acts vertically downward along the line A W: Let A BW represent the magnitude of' this force, and resolve it into two components, P and P2, one of which is perpendicular to the plane 1X1VX and the other parallel and opposite to the force F. These components are represented by AP, P W, respectively. The total force of the component P is exerted in producing a pressure upon the inclined plane MIV, to which it is perpendiculalr, and is wholly balanced by the reaction produced; hence it is merely necessary that the remaining component BR shall be balanced by the power F, in order that the weight may be supported; that is, for equilibriuma, the force -F must be equal to thle conmponent R. If this is the case, the three sides, A W;, WP, PA, of the triangle A WP, taken in order, Will represent the three forces, W,; F, P, under whose united action the weight is sustained. Hence, for equilib)rium, F: W:: TWP: VA::sin WAP: sin APW. But WAP = a., AP W = 90~ + -3, hence lWe confine our attention to the c'ase in which the force F lies in the plane MN.O, at right-angles both to the inclined plane and to the horizon. 14

106 INCLINED PLANE. F::: sil a: sin 90~ + C, or F: Y:: sina: cos j. (133.) That is, with the inclined plane there is equilibrivzn when poewer is to weight as the sine of the angle of inclination of the plane is to the cosine of the acngle made by the power with the plane. 217. Pre:ssure on Plane, The pressure on the plane, P, is represented by AP, land may be determined from the proportion P: W-:: AP: A W:: sin A WP: sin AP W: sin (90 - a - p): sin (90~ + (), or P: W:: cos ( ) +: cos ( (134.), a proportion from which the pressure on the plane may be computed in terms of the weight of the body considered. By combining proportions (133), (134), we obtain the fobllowing: P: F:.: cos (a -- jS): sin a. (135.) 218. Special Cases. Proportion (133) is much simplified, when the power F_ acts either parallel to the plane, or horizontally. In the former case, FA and WIP become plarallel to AIN, consequently = 0~. Thereibre, (133), (134), (135), assume the forbn F: W:: sin a: cos 0: sina: 1. (136.) P: W:: Cos a: cos 0:: Cos a: 1. (137.) P': F:: cos a: sin a:: 1: tang a. (138.) Denote the length of the inclined plane, iXl,, by L, the height, 2VO, by H, and the base, MiO, by B. Then H~ B H sin - cos a L tang a B. By substituting these values for sin a, cos a, tang a,in (136), (137), (138), the following proportions are obtained. H F: W:: sin ca: 1::: 1, whence F: WT::: L. (139.) P: W:: cosa::: 1, whence P: Y:: B: L. (140.) P:F::1:tanga l:1,whence P: F: B:. (141.) That is, when the direction of' the power is parallel to the plane, I. Power is to weight as the height of the plane is to its length. II. Pressure on plane is to weight as base of plane is to its length. III. Pressure on plane is to power as base of plane is to its height. If _F is parallel to the base of the l)l;ie, p - a(9, in which case proportions (133), (134), (135), assume the following forms: F: W:: sin a: cos (- a):: sin a: cos a::: B, whence F: W T::: B. (142.) P: W:: co o 0~: cos a:: cos a-: I:B twhence P: W::L:B. (143.)

PRACTICAL EXAMPLES. 107 P: F:: cos 0~: sin a: 1: sin a: 1' whence P: F:: L: H. (144.) That is, when the direction of the power is parallel to the base of the plane, I. Power is to weight as height of plane is to base. II. Pressure onplane is to weight as (ength of plane is to base. III. Pressure on plane is to power as lenygth of plane is to height. 219. Direction of Power for Maximum Weight and Pressure. Solving (133) relatively to IV, we have TV F- cos, a quantity which for anyl given inclined plane is proportional to cos /3, and therefore has its maximum value when / - 00 and cos /3 - 1. Hence a given power will sustain the greatest weight whe~n it acts patrallel to the plane. The greatest pressure upon the plane produced wshen a given weight is resting in equilibrium uponz it occurs when the pow;or is parallel to the base of the plane. For from (135) P - F sin (, a quantity proportional to cos (a + /3), and wllich is greatest when cos (a + /3) 1, in which case B - a- a, and F is parallel to the base of' the plane. The maximum pressure is therefore P - si-n In the foregoing demonstrations, no account whatever has been taken of fiiction, an element which causes a great variation froml the preceding results. Its effect will be discussed in a succeeding chapter. The mechanical advantage of the inclined plane lies in the fact that a portion of the weight is supported by the plane itself. 220. Conditions of Equilibrium of Bodies Resting on Contiguous Planes. Suppose two bodies, A, B, of weights TV, 1T' respectively, to rest on two contiguous planes of the same height as shown in Fig. 115, B is connectel with A by neaiis of a cord passing over a pulley at N. It is requiredl to findl the relation of the weights JW, TV' that the systenm may be in e(juilibriulm. Call P the power parallel to NO required to sustain W, P' that parallel to lIN required to sustain W,' and denote NR by H, ArN by L, NO by L'. As thle power in both cases is parallel to the plane, H P: TW:: H: L, whenceP W T~L P': W'::H: L', whence P' - TV' But that W and W' may be in equilibrium P must be equal to P', in which case W - W''L, or W': TW':L: L', (145) that is, the weights must be to eRch other as the lengths of the planes in which they rest. 221. Practical Example. The best examples of inclined planes are seen on roads leading over hills, the power required to draw a vehicle over them increasing with their steepness. In passing over a level road, the horse has merely to overcome the fiiction of' the wheels, but on an inclined plane he has in addition to lift a fraction of the load, the magnitude of which

108 INCLINED PLANE. depends upon the steepness. Thus if the rise is 1 m. in 20 m., 1 of the load must be lifted, for 1 m. in 40 m., l- of the load, and so on. Interesting illustrations also occur in the various mountain railways which have recently been constructed both in this country and in Europe. The first of' these in date of construction is the Mt. Washington Railroad, which was begun in May, 186G, and finished in July, 1869. The road starts front a point on the west side of the mountain, 813 in. above the level of the sea, and ascends with an average grade of 1 in. in 4 in. to the summit, which lies at an elevation, 1918 in., making the vertical heiglht of the inclined plane equal to 1105 in. The grade being far too steep to be ascended by an ordinary locomotive, which acts solely by the friction of the wheels on the rails, a third rail is introduced, lying midway between the other two, and furnished with cogs. The middle rail is in fact a rack. A strong cogged-wheel driven by the locomotive works in this rack, and thus causes the locomotive anti attached car to ascend. The steepest grade upon the road is 375 man. per metre., or a little over 1 nm. in 3 m. There are nine curves with radii varying from 151 in. to 288 in., and the road passes at one point over a trestle-work bridge 9.2 in. in height, and with an inclination of' over 1 m. in 3 m. The locomotive at present used weighs 5900 kgr. The carriages for passengers resenible a horse-ear, and seat 48 persons. The locomotive is always below the car whether ascending or descending. The actual time of ascent is 90 minutes, a rate of over 3 km. per hour; the time of descent is somewhat shorter, and the descent is accomplished by mleans of gravity alone. The arrangements for controlling the motion are excellent, consisting of (1) a form of firiction-brake attached to the driving-wheels, (2) the power of reversing the driving-wheels, and (3) an atmospheric brake attached to the cars. The road is kept open firom the middle of June to the first of October, andl is well patronized during the season of summer travel. The inventor of the peculiar form of' locomotive and rails used in its construction is Mr. Sylvester Marsh, to whose energy the road owes its existence. Another very remarkable mountain railway is that of Mt. Rhigi, in Switzerland, which was constructed by Riggenbach, upon the same general plan as the one at Mt. Washington. It was begun in Nov., 1869. and finished late in the sunimmer of 1870. The vertical height of Mt. Rlhigi above the sea is 1800 m., and above Lake Lucerne 1360 m. The road is 5340 in. in length, with a total ascent of a little over 1200 m., an average slope of over 220 mmnn. per metre. There are several curves, all of which have a radius of 180 in. At one part of its course the railroad passes through a tunnel 80 m. long, and shortly after emnerging froml this crosses a ravine 23 in. deep, upon a bridge 75 m. in length, and havino an inclination of 250 ninn. per nletre. The weig'ht of the loconlotive iwhen ready for use is about 12,500 kgr., bein(r muclh larger than those used in this country. The velocity attained is 6.4 km. per hour, so that the ascent occupies only about that time. The passenlger cars seat 54 persons, and when empty weigh 3970 kgrs. It is customary to have a man walk before the engine to clear the road of any obstructions that may occur. The mountain railroad leading from Ostermundingen to connect with the railroad to Berne, is also of importance. It was constructed by Riggenbach. Unlike the two roads already mentioned, it was built, not for the purpose of carrying tourists to the inountain-siimmnits, but to tran sport the buildling stones quarried at Osternmundingen to a place from which they could be taken by the ordinary railroads. It is composed of two nearly

PRACTICAL EXAMPLES. 109 level portions united by a grade of 100 mm. per metre, which is half a kilometre in length. The total length of the road is a little over 2 klm. The existence of both levels and inclines upon the road, rendered necessary a modification of the Rhigi locomotive, as the cogged-wheel and rail are unnecessary upon the levels. Hence the middlle rail is made several centiinetres higher than the exterior ones, and is omitted upon thelevel portions of the road, so that the cogged-wheel engaging in it has such an elevation above the ground, that it does not at all interfere with the working of the locomotive when upon the level. The railroad up Mt. Cenis is perhaps more widely known than any of the preceding. The break in railroad connections (77 km.) from St. Michel to Susa led the government of Sardinia to undertake the construction of a tunnel through the mountain, in order to secure a continuous line of rail from France to Italy. The extreme length of the tunnel (12,220 in.) caused its progress to be very slow, and while the excavation was going on, Mr. J. B. Fell, an English engineer, proposed to construct a line over the mountain, using a peculiar system of rails and locomotive. Preliminary trials of the system were made in England, and a line was afterwards built up the side of Mt. Cenis. The length of this railroad is 1960 m., and its vertical ascent is 151 m. The line begins at a height of' 1622 in., and terminates at a height of 1773 m. above the sea-level. Its mean grade is 1 rn. in 13 nm. (77 niun. per metre), and the maximum is 1 m. in 12 inm. (83 llm. per mietre). There are a number of curves with radii of firom 40 m. to 300 m. The device for enabling the locomotive to climnb the steep grade is somewhat different from those in use on the railroads at Mt. Washington and Mt. Rhigi. It consists of a middle rail, which is firmly embraced by two horizonltal drivers, which furnish the necessary fiiction. There are also four vertical drivers of the ordinary form, which are used alone when the grade is less than 40 mm. per metre. Another railway upon the Fell system is now in process of construction in Brazil. It is designed to convey the coffee raised in the elevated district of Cantagrallo over the Organ Mountains to the lowlands. The lencth of the railway is about 12,500 m., and its gauge is 1.1 mn., which is identical with that of the Mt. Cenis road. The average grade is about 77 rnm. per metre, and the curves have a radius of 40 m. The centre rail will be used only on the steep inclines, which is also the case on the Mt. Cenis road. Still another celebrated example of the mechanical power under consideration is the Timber Slide of Alpnach. This is a gigantic inclined plane about 14,000 m. in length, leading firom the forests on the sunimmit of Mt. Pilatus, one of the Swiss Alps, to the borders of the Lake of Lucerne. It was constructed in 1812, to convey the fir-trees from the mountain-forests to the lake, fiom which they could easily be carried to the Rhine. The mean inclination is 30 14' 20", the greatest slope being 22~ 30'. It is necessarily somewhat circuitous, owing to the unevenness of the ground. The trees, with their branches lopped off and divested of their bark, are placed in the trough of the slide, and descending by their own gravity, acquire a velocity so great that they reach the lower end in a very few minutes. VI. Wedgqe. 222, Explanation of Action. If we suppose the power P, to be applied to 0NO, as represented inl Fig. 116, pushing the inclined plane in the direction OM1 and thus raising W, we have

110 WEDGE. the mechanical power, known as the wedge. The relation between P and W may be estilmated as follows:- Since the effect of a horizontal fbrce in moving the plane is the same at whatever point of 1vNO it may be applied, suppose it to act along PB, a horizontal line passing through A, the point of contact of JfrXwith the vertical throullh the centre of gravity of the body resting upon it. Draw AB, B C, parallel to the directions of P and W, respectively, and complete the tri:angle ABC by drawing A C perpendicular to 21VZ as the reaction of the plane (B) assumes that direction. Then P, Wand B2, are represented in magnitude and direction by BC, AB, A C(, respectively. Hence P: V:: BC: AB:: NO: O, or P: V::: B. (146.) Also P': BR':: BC: AC:: NVO:'MN, or P::: IT:. L. (147.) That is, power is to weight as the back of the wedge is to its base, a-nd power is to reaction on face as thze back of the wedge is to the length of its face. 223. Common form of Wedge. The most common form of wedge consists of two inclined planes mnountedl base to base, as shown in Fig. 117. Suppose the power 2P to be applied at the middle of the back, and to act along the line 1DA, which bisects the angle BA C. The advance of the wedge is opposed by pressures B, B2', at right-angles to the faces AB, AC, and if' there is equilibrium the directions of 2P, Bi, R', must meet at some point, as E. Let the forces BR, iB', be resolved into two rectanoular components, respectively parallel and perpendicular to LAD. The components perpendicular to AD are R cos a, R' cos a, which are equal to each other if the wedge is isosceles, and hence balance. The sum of those components of the resistances which are parallel to DA, must be equal to 2_P; that is, for equilibrium, 2 P - 2B sin a, or P- R sin a (148); whence P: R::sin a:: 1 B': B AB. (149.) Hence there is equilibrium with the wedge when power is to weight as half the back is to the face of the wedge. 224. Practical Applications. The wedge is generally usedl for purposes of cleavage, as in splitting timber. A cleft is made, into which the edge of the iwedge is introduced, and the blows of a mallet are applied to drive it forward. As the theorems which have just been given apply to the case of equilibrium under pressures, while the wedge is usually driven by percussion, they are of very limited use, and hence of little practical use. Were it not for friction, which is not taken into account in their deduction, the wedge would lose most of' its value. In fact, all that can be said is that the cleaving power of the wedge increases as its angle dimninishes. Most of our cutting instruments are applications of the wedge, as the chisel, saw, knife, razor, etc. The angle of the instrument depends upon the tenacity of the substance which is to be cut, as the diminution of' the angle must practically be limited by the corresponding decrease. of strength.

SCREW. 111 in the tool itself. IHence the harder the substance on which the tool is to be usecd, tle less aculte is the angle. Thus chisels fur cuttiilg wood have an angle of' about 3Q0, for iron an angle of from 50~ to 600, anl fior brass from 800 to 90~. VII. Screw. 225. Construction of the Screw, The screw is formed by winding a narrow inclined plane about a cylinder, as shown in Fig. 118. The spiral iB C)DEF fobried by the plaiie is called the trecad of the screw, and works in a sinilTl threadl cut on the interior of a block called a nzut, PQ, Fig. 119. On turning the screw aroun(d, it is forced to advance oi 1ececde in the nut, according to the direction of the motion. The power is generally applied at the extremity of a lever AB, Fig. 119 x which is attached to the head of the screw. To find the relation between the power and the plressure exerted by the screw, we have simply to considler the e:lse of a body L, Fig. 118, resting on an inclined plane. On turnillg the screw, i_ will be pushed up the plane with a certain foice Which is evidently equal to the force with which the screw will descend if L is-fixed. This last supposition is evidently precisely the case of a screw with a fixed nut, for L may be supposed to embrace a considerable portion of the thread. The power P turning the screw is applied parallel to the base of the plane, hence from (142) P: W:: _f: B. Now, considering only a single tuln of the thread, this formns an inclined plane of which the base is the circumlference of the cylinder on which the thread is fornmed, -mid thle height is the distance between the threads. Hence, ca-lling, the power applied, W the pressure exerted by the screw, d the dlistance between the threadls, and r the radius of the cylihndeur, P: W:: d: f2,r, (149). Proportion (149) supposes the power to be applied at sulrlice ot the cylinder. If the lever C V(Fig. 120) be attached, a force P acting at AT will produce a pressure P' at Ai, which may be determined fiomn the law of momlents. If BR 1c the radius CX, r the radius CAW of the cylinder, we have PR= P'r, whence P'_ P- Substituting this value of P' in (149) we 1ave.P:W:: d: 2,rr or P: IW:: d: 2B,.R (150). That is,'with the screw there is equilibrium when the power c2pplied to t]he lever is to th/e pressure producteed by the screw as the distance between the tghreads of the screw is to the circumference described by that portion of the lever arm to which the power is applied. Froim (150) we obtain the equation W — P -- - (151). The effect would evidently be the same if the screw were fixed and the nut movable, so that the proposition is general. The preceding theorem is incorrect in practice, owing to the enormous friction produced by the threads.

112 scREW. 226. Definitions. It will be seen that at each turn of the screw it moves over a space equal to the distance between the threads. This distan:ce is known as the pitch of the screw. It is generally determined by noticingo the number of ridges occurring in an inch of the lengtll of the screw. The angle made with a horizontal line by a tangent to the thread of tihe screw is the angle of the screw. A screw generated by the revolution of a single band around a cylinder, as already described, is said to be singlethreaded. To increase the strength of the instrument, it is not uncommon to form the screw by cutting two parallel threads side by side, making a double-threaded screw. This increase in the number of threads does not alter the pitch, which is dependent only on the angle of the screw and the size of' the cylinder upon which it is cut. Two forms of screw-thread are used in practice, the square thread, Eig. 121, and the V threcd, Fig. 122. The square thread is the most powerful because the reaction of its surface R1B, Fig. 121, is parallel to the elements of the cylinder of' the screw, so that the total furce applied acts to overcome the resistance, while with the V thread these reactions are oblique to the elements of the cylinder, and hence a portion of the applied force is uselessly exerted in producing a pressure tending to burst the nut. The V thread is, however, stronger, because there is less material cut away in forming it. A deep screw is evidently less strong than a shallow one, but it is more durable, as the greater amount of bearings surface prevents it froml wearing away so easily. Screws are used in practice whenever it is necessary to exert a great pressure though a small space, as in pressing books, extracting oil fiom seeds, raising buildings, and the like. Boring instruments as augers, gimlets, and corkscrews, are a(lditiona.l examples. 227. Differential Screw. Equation (151) shows that the mechanical advantage of' a scrlew may be increased in two ways; (1) by increasing the lenoth of the lever arm, and (2) by diminishing the distance between the thlreads. The first of these methods greatlv increases the size of the ilachine, while the second is practically linmited, owing to the weakness of extremely fine threads. The difficulties are avoided in the machine known as the differertial screw, or Heanter's screw, one of the forms of which is shown in Fig. 123.1 Two screws AB, CD, are cut upon a single cylinder, the pitch of AB being greater than that of CD. AB works in a nut EF, and CP enters a second nut, which is capable of a vertical motion between the guides DEFG, and is prevented from turning around with the screw. To understand its working, suppose the pitch of A B to be X of an inch, and that if CD to be 5 of an inch. On turning AB once around, it advances through a spa.,e of 8 of an inch. The smaller screw CD at the same time turns in the nut GH, and by its motion as it enters the nut draws this towards EF by an amount of' - of an inch. Hence by a single turn GH is moved away from EF i of an inch by the motion of A B, while it is moved towards EF 2- of an inch at each turn, and hence the same effect is produced as if there were a single screw with a pitch of 1 of an inch, the difference between the pitch of AB and that of CD. The differential screw is therefore in equilibrium when power is to pressure produced as the differences of the pitches of the two parts of the screw is to the circumference described by the power. Since the effect is dependent merely on the difference of the pitch and not on its absolute value, the ratio of pressure to power may evidently be in1From the inventor, John Hunter, the celebrated surgeon.

MICROMETER SCREW. 113 creased to any desired extent without diminishing the strength of the apparatus. Suppose, for example, the pitch of AB to be 1- in., and that of CD 30 in., and let the end of the lever-arm describe a circumference of 10 inches. A power of 10 lbs. applied at the extremity of the lever will cause a pressure of' (10 X 50)'( (L - O) 30,000 lbs. 228. Endless Screw. Fig. 123 represents what is known as an endless screw. It consists of a wheel AB, called a worm-wheel, furnished with oblique teeth which engage in the thread of the screw CD, which is known as the worm. If the screw is single-threaded, the wheel advances by a single tooth fobr each revolution of CD. If there are two, three or more threads, a corresponding number of teeth pass at each revolution. The reduction of velocity taking place in the transfer of motion from the screw to the wheel, renders this contrivance useful for registering the number of revolutions of an axis. Thus if the wheel AB'has 50 teeth, it will make but 1 revolution, while the screw makes 50. If, now, the axis AB carry a pinion acting in its turn on a cogged-wheel, the velocity of the revolution may be reduced to any desired extent. For instance, if the pinion has 5 teeth and the wheel 100, the screw will revolve 1000 times while the coggedwheel revolves once. The number of revolutions of the screw-axis is read by means of an index attached to the axis of the last wheel. When any considerable force is to be transmitted by the endless screw, the screw is made to drive the wheel, because the friction would prevent the wheel from driving the screw, in addition to the fact that such a combination would involve a mechanical (lisadvantage. In liaht machinery, however, where there is little friction, the wheel may be made the driver. This combination is frequently used to regulate the velocity of a train of wheels by connecting the axis of the screw with a fan-wheel, the motion of whose wings is so much resisted by the air as to keep the velocity within suitable limits. 229. Micrometer Screw. By means of the screw we are enabled to measure small linear distances with great accuracy. When the screw is turned, the space passed over by its extremlity is known immediately from the number of revolutions, provided that we know the pitch, and if a graduated circle is attached to the screw-head a minute fraction of a turn can easily be measured. Suppose, for example, that a circle divided at its circunmference into 100 parts be thus attached to a nicely-made screw having a pitch of 1 millimetre. If the screw makes one complete revolution, it will advance 1 millimetre, hence for TI- of a revolution, the length of one of the divisions of the graduated circle, it will advance m —6 mml., which distance therefore corresponds to one of the scaledivisions. With a first-class micrometer screw we can measure a length within x.s~ of a millimetre. 230. Sheet Metal Gauge. A practical application of this principle is shown in the gauge used for measuring the thickness of sheets of metals, paper, etc., Fig. 124. " The piece in the fobrm of the letter U has a projectinc hub. a, on one end. Through the two ends are tapped holes, in one of which is an adjusting screw, B, and in the other the gauge-screw, C. Attached to the screw C is a thimble, D, which fits over the exterior of the hub, a. The end of this thimble is beveled, and the beveled edge graduated 15

114 SCREW. into 25 parts, and figured 0, 5, 10, 15, 20. A line of graduations 40 to the inch is also made upon the outside of' the hub a, the line of these divisions running parallel with the centre of the screw C, while the graduations on the thimble are circular. The pitch of the screw C being 40 to the inch, one revolution of the thimble opens the gauge - or 5 of an inch. The divisions on the thimble are then read off for any additional part of a revolution of the thimble, and the number of such divisions is added to the turn or turns already made by the thimble, allowing 25 for each graduation on the hub a. For example, suppose the thimble to have made 4 revolutions and one-fifth. It will then be noticed that the beveled edge has passed four of the graduations on the hub a, and opposite the line of graduation will be found on the thimble the line marked 5. Add this number to the amount of the four graduations which is W1~~Q-, and equals 105o which is the measurement shown by the gauge.l" To show the method of using the instrument, suppose we have to determine the thickness of a sheet of metal EF. We first turn the screw till the ends B and C are in contact. The reading on the scale should then be 0. Then turning the screw till the sheet will pass between B and C, we place it as shown in the figure, and again revolve the screw until the end just touches the metal, taking the reading of the scale as before. The second reading evidently shows the number of revolutions made by the screw in passing over a linear distance equal to the thickness of' the sheet of metal, and this multiplied by the pitch of the screw is the thickness sought. If the material of the sheet is very soft, it is placed between two plates of glass, the reading being taken when the metal is in place and after it has been removed. The difference of the readings gives the thickness sought. 231. Dividing Engine. The micrometer screw is also applied in the construction of the engines used for dividing linear scales. The principle of the instrument is as follows. A diamond, or other graver, is attached to an arm, which is connected with the nut of Ia carefully-made screw, so as to be carried forward when this is revolved. Directly under the graver, and parallel to the line of its motion, is placed the bar to be divided. The' graduated head of the screw is tnurned through a certain number of divisions corresponding to the spaces to be marked off upon the bar. A movement is then given to the graver, cutting a line upon the bar, the screw is again revolved, and thus the process is continued. The best dividing engines have devices for rendering the whole action automatic, and also for making every fifth and tenth mark on the scale larger than the intermediate ones. Oftentimes the graver is stationary, and the bar to be graduated is moved by the screw. A similar device is used in the simplest method of dividing circles. The plate to be graduated is attached concentrically to a heavy circular wheel-, to which a tangential microimeter screw of known pitch (tcangent-tscrew) is fitted. The rotation of the tangent-screw through a single turn corresponds to the rotation of the wheel and the attached plate through a certain fraction of a degree. At each turn of the screw a mark is made 1 Catalogue of Darling, Brown and Sharpe.

APPLICATION OF VIRTUAL VELOCITIES. 115 upon the plate by the gravelr, and the process is continued until the whole circumference has been traversed and completely oraduated. Additional examples of the use of the micrometer screw occur in reacding-microscopes, the spicler-line micrometer, etc. 232. Combinations of the Mechanical Powers. The mechanical powers may be combined in various ways. The mechanical advantage gained by any particular combination may be determined by estimating the effect of its component parts separately. For example, Fig. 125 represents a form of machine sometimes used for raising ships fiom the water. The vessel is drawn up an inclined plane by means of the wheel and axle (capstan) HG, acting upon the system of movable pulleys EF. If we suppose'the radii GH, GI, to be 2 metres and one-fourth metre respectively, and the slope of AC to be 1 metre in 10, the power exerted at K, on applying at H a force of 1000 kgrs., will be 32,000 kars., which would be sufficient to draw a vessel weighing 320,000 kgrs. up the plane AC, supposing no power to be lost by friction. VIII. Application of the Principle of Virtual Velocities to the _Mechanical Powers. 233. General Principles. In the use of the various mechanical powers there is no absolute gain of force. For it will be recollected that the force of motion of any body is equal to its mass multiplied by its velocity, and it is found that if the power and weight are in equilibrium, and a motion is impressed upon the machine in the direction of either of those folrces, the power multiplied by its velocity equals the weight multiplied by its velocity; or, calling P the power, W the weight, V, Vl their respective velocities. PV- TI'V' (152), whence P: W V': V. (153.) Hence, the velocity of the weight when compared with that of the power, is diminished in the same ratio as the weight balanced by that power is increased by the use of the machine, or as it is generally expressed, what is gained in power is lost in time. The demonstration of this principle is simply an application of the general theorein of Virtual Velocities. (~ 141, p. 73.) We proceed to consider the various mlechanical powers in order. Our method of proof will be to show that assumin7g the principle of virtual velocities, the law of equilibrinlu a (ldeduced for each power follows. 234. Application to Lever. Let us take the case of a bent lever, A4PtBR, Fig. 126. P bfing the power, and TV the weight balancing it. Suppose' now that a slight displacement be given to the machine, so that A takes thme position Al, and B moves to B'. The power now acts along A'1J', the weight alon B'W'. Draw A'm perpendicular to A P, and B'n perpendliular to the prolongation of TVB. Am and -Bn are the virtual velocities of P and W, Am being positive, as it is laid off' in the direction of the action of P, and Bn negative, because it is laid off opposite to the direction of the action of TV. The ares A A', BR', being very small, may be considered as strai.ght lines perpendicular to A F and BF, respectively. By the principle of' virtual velocities, P X Am-WT X Bn-0. (154.) Now Am - AA' cos mAA', also are AA' AF X angle AFA', and cos mAA' - cos (PAF- 900~) sin PAF. Hence Am - AF X angle AFA' X sin PAF.

116 APPLICATION OF VIRTUAL VELOCITIES. In like manner, BRn - BB' cos nBB'. But BB' - BF X angle BFB', and cos nBB' cos (TBF - 900) sin WBF. Hence Bn _ BF X angle BFB' X sin TYBF. Substituting these values in equation (154), P X AF X angle AFA' X sin PAF - W X BF X angle BFB' X sin WBF 0. (155.) But AFA' - BFB', from the conditions of motion, whence P X AF X sin PAF WTV X BF X sin WBF 0 (156), or P: W:: BF sin WBF: AF sin PAF (157), which is the condition of equilibrium, as deduced for the lever in proportion (101), p. 88, as AF sin PAF, BF sin WBF, are the arms of the forces P and W. 235. Application to Wheel and Axle. Let ACB, Fig. 127, represent a wheel and axle. The forces applied are P and W, acting vertically downwards, and the upward reaction R _ P + W, at C. The machine being rigid, the forces may be considered as applied in the same plane. The virtual velocity of R 0, as the wheel and axle revolve about C. Suppose a motion impressed upon the machine so that P moves to P', thus raising W to W'. The original point of contact of the rope P is A, and the wheel moves from A to A', and the original point of contact of the rope Wis B, and the axle moves from B to B'. PP' -=AA' is the virtual velocity of P, and — WT' -- BB', that of TV. For equilibrium, P X AA'- W X BB'- 0. (158.) But AA' = AC X angle A CA', and BB' - BC X angle BCB', or, calling R, r the radii of the wheel and axle, AA' = R X angle ACA', BB' r X angle BCB'. Substituting these values in (158), P X R X angle ACA' - W X r X angle BCB' = O, or, as the angles ACA'BCB',areequal, P X R T X r O,orP: W'::r:R (159), the condition of equilibrium demonstrated in ~ 193, p. 97. 236. Application to Pulley. With a fixed pulley and parallel cords, the truth of the principle is evident, as the power and weight move over equal distances. The most general case of the movable pulley is that in which the cords are inclined, as represented in Fig. 128. Let C, P, W, be the original positions of the pulley, power and weight. On impressing a slight motion upon P, so that it moves to P', C rises to C'. and W to W', so that WIW' CC'. PP' is the virtual velocity of P, -WW' that of W. Draw Cm, Cn, circular arcs having their centres at M and H. When the displacement is very small these become straight lines perpendicular to Hn, UHn. In that case Cm - Cn - CC' cos H C'CC' CC' cos ICM - CC' cos a, using the notation of ~ 208, p. 102. Since the shortening of the whole of' the cord MICH, is equal to the lengthening of MP, PP' m C + nC - 2CC' cos a. For equilibrium, P X PP' - W X CC' -O. (1 0.) Substitutincg the above value of PP', P X 2CC' cos a = W X CC', whence W- 2P cos a (161), which is identical with equation (i27), p. 103. 237. Application to Inclined Plane. Let A, Fig. 129, be the ori(inal position of the point of application of F and TW, and suppose it to be displaced alone the line AB to the point B. Then Am is the virtual velocity of F, and -Bn that of WT. For equilibrium, F X A Al - W X Bn 0. (162.) But Am --- AB cos B, and Bn - AB sin a; hence, substituting these values in (162), F X AB cos B - T X AB sin a 0, or F: W: sin a: cos B (163), as demonstrated in ~ 216, p. 106.

VELOCITY-RATIO. 117 238. Application to Wedge. Calling 2P the power, and R, R', the equal reactions, we have in the case of a small displacement of the wedge from the position ABC to A'B'C', Fig. 130, virtual velocity of P mm' _ AA', virtual velocity of R — nn'- -Ap. If there is equilibrium, P X AA'-R X AP _ 0 (164), or P: R:: Ap: AA'. But Ap: AA' BN: BA, whence P: R:: BN: BA (165), which is identical with the conditions of equilibrium demonstrated in ~ 223, p. 110. 239. Application to Screw. The application of the theorem to the screw is obvious, since the power must move through the circumference whose radius is the lever-arm, while the weight moves over a distance equal to the pitch of the screw. For any small displacement the virtual velocities would be in this ratio, whence P: W d 2rR (166), as already demonstrated. 240. Velocity-Ratio. The ratio of the virtual velocities of the V power and weight, -V, is called the velocity-ratio of those forces. From what has already been demonstrated, will be seen that in all cases the mechanical advantage gained by the use of any machine is expressed by the velocitey-ratio of the power and weight. This theorem gives a simple method of determining the theoretical efficiency of any combination of the mechanical powers. REFERENCES. Treatise on Mechanics, by Capt. Henry Kater and the Rev. Dionysius Lardner; Chap. xxi., On Balances and Pendulums. Reports of U. S. Commissioners to the Paris Universal Exposition, 1867. Vol. Iri. Machinery and Processes of the Industrial Arts, and Apparatus of the Exact Sciences, by F. A. P. Barnard; p. 484. Balances. Mount Washington in Winter, Chap. iv.; (Boston, 1871.) Description of M-t. Washington Railroad. Die Righi-Eisenbahn mit Zahnradbetrieb, beschrieben von Prof. J. H. Kronauer. Etude sur les Chemins de Fer de Montagnes avec Rail a' Cremaillkre, par M. A. Mallet (Paris, 1872). Article on Mt. Cenis Railroad, Journal of the Franklin Institute, Vol. L., p. 289. Mountain Railways; Van Nostrand's Eclectic Engineering Journal, Vol. viI, p. 407. (From Engineering.) Centre Rail System, Do., Vol. Vii, p. 394. Account of Cantagallo Railroad. (From Engineering.) Account of Slide of Alpnach; Works of Professor John Playfair, Vol. I. (Edinburgh, 1822.) For various forms of Dividing Engines consult references given on p. 24. Also consult Reports of U. S. Commrs. to Paris Universal Exposition, Vol. iII, Chap. xviii., On Afetrology and Mechanical Calculation, for various forms of comparators, spherometers, etc. See also the same volume, p. 11, for a description of Whitworth's Micrometric Apparatus.

118'DYAMICS. CHAPTER XI. UNIFORAm AND UNIFOFRMLY VARIABLE MOTION. 241. Definition. Dynamnics treats "of the relations between the motions of bodies and the forces acting amongst them." 1 It differs from Kinematics in that the latter subject considers only the relations of the motions to each other, independently of their causes. Motion is of two kinds: (1) motion of translation, in which the moving body simply passes from point to point, the paths of all the particles composing it lying in parallel lines, and (2) motion of rotation, in which the body under consideration revolves about an axis. These two varieties of motion frequently exist in com-, bination. A motion of translation may be rectilinear, in which case the path of the moving body is a straight line; or curvilinear, when the path is any curve. All motion is either uniform or variable. In uniform motion equal spaces are described in equal times; in variable motion unequal spaces are described in equal times. 242. Uniform Motion: Formulae. Since a body having a uniform motion describes equal spaces in equal times, if in. a unit of time it describes a space V; in 2 units it describes a space 2 F; and in t units a space t V; Hence, denoting by S the whole space passed over by the body in t units of time, with a constant velocity V, we have, S - t (167), whence t V (168), and V=- (169), the fornullm for uniform motion. 243. Causes producing Uniform Motion. There are two cases in which uniform motion may occur. (1.) If a body were projected with a definite velocity, in the absence of all disturbing f-rces, it would proceed indefinitely with a uniform motion, by virtue of its inertia. Practically, these conditions can never be exactly fulfilled, but as they are approximately realized, uniformity of motion is also approached. (2.) When the resultant of all the forces acting upon the moving body is zero, that, is, when'the magnitude of the impelling forces equals the magnitude of the resisting forces, the motion is also uniform. For an example, let us take the case of a train of cars. Here the train is moved onward by the impelling force of the locomotive, while the friiction of the -wheels is a constant retardclin force. Suppose the train to be in motion. Were there no retarding force, its inertia 1 Rankine, Applied Mechanics, p. 475.

UNIFORMLY ACCELERATED MOTION. 119 would carry it onward with undiminished velocity. But the friction on the rails, and other resistances, destroy at plortion of the motion in each unit of time, and if acting alone wouldi filn:lly (.onsume the whole o111tion, when the t-rain would come to rest. The locomotive, however, gives a constant impelling force, and if this is made just equal to the resistance, the loss of motionll by the latter is compensated by the gain from the former, andl the velocity of the train is unaltered. A body thus moving unifbrmly under the actionl of b)alanced forces is said to be in ciynamnical equilibrium. 244. Uniformly Accelerated and Uniformly Retarded Motion, and their Causes. When the spaces traversed in successive equal times increase, the motion is said to be accelerated; when they decrease it is said to be retared. If' the increments or decrements of motion are equal in equal times, we have uniformly accelerated or uniformly retardced msotion. ~aWhen a moving body is acted upon by any force exterior to itself, its motion is caused to vary. If this force acts in the direction in which the body is moving, the velocity will be increased, thus producing accelerated motion; if the force is opposed to the mlotion of thle body, the velocity will be decreased, producing retarded motion. If the force acting upon the body is constant, equall increments or decrements of motion will be generated or destroyed in equal times, causing uniformly accelerated or uniformly retarded motion. 245. Definition of Velocity. In uniformly accelerated or retarded motion the velocity of the moving body is constantly varying. The velocity at any instant may therefore be defined as the space over which the body would move in a unit of time were the accelerating or retarding force to cease its action. In uniformly accelerated motion the increment (t' velocity in a unit of time is called the acceleration. In uniformly retarded motion the decrement of velocity in a unit of time is called the retardation. The unit of time generally adopted is the second. 246. Laws of Uniformly Accelerated Motion. Formulae. I. The velocity acquired by a body moving with a uniformly accelerated motion is proportional to the time. For by the definition of this kind of motion, the increments of motion are equal in equal times. Therefore if a body u1nder the action of any fiorce acquires a velocity, a, in 1 second, in 2 seconds it will acquire twice that velocity, or 2a, and in t secolld(s t times a. Hence calling a the acceleration, t the time, and v tlhe velocity at the end of that time, we have v - at (170), a formula giving the relation between the acceleration, time and velocity, a.lll showing that the velocity varies directly as the time. II. The szccessive spaces traversed by a body movigg ntithl a uniformly accelerated motion are proportional to the squares of the time occupied in passing over them. The space traversed by a body dur

120 UNIFORMLY VARIABLE MOTION. ing an extremely small interval of time may be regarded as moved over with a uniform motion. Let t be the time which the body occupies in passing over a space s, and let that time be divided into nz equal parts, n being a very large nulmber. Each one of these parts will equal 9. Calling a the acceleration, the velocities after successive intervals of time,,t 32t. - will be, act t I t cording to equation (170), aC, 2a t, 3a, na - and supposing the velocity to be uniform during each of these intervals of time, the corresponding spaces passed over will be t2 t2 t2 t2 a -, 2a, 3a, napd, by equation (167). The space s traversed in time t will evidently be the sum of these small int2 12 12 t2 tervals, therefore s - a? + 2aw- + 3a, +-.*..... na -, or t2 s a 2 (1 - 2 3 +. n). The quantity within the parentheses is the sum of the terms of an arithmetical progression, (n + 1)n t2 [(n + 1)n] at2 at2 and, hence s= a (L - ) or s 2 + 2' - 2 2n' A consideration of the cause of uniformly accelerated motion will make it evident that only when the interval of time considered is infinitely small, that is, when n is infinitely great, does the supposition which has been made, that the spaces traversed during the interval A are uniform, apply to the kind of motion with which we are now concerned. Hence, that the equation just deduced may be applicable'to uniformly accelerated motion, n must be made - o. Inserting this value therein, the formula becomes, at2 at2 at2 s = - + -- 2 +- 0, whence s -1 at2 (171), a formula showing the relation between the sp:rce traversed, the time occupied in passing over it, and the acceleration, and from which it follows that s varies as t2, or the spaces are proportional to the squares of the times. III. In uniformly accelerated motion the velocities are proportional to the square roots of the spaces passed over in generating them. For from (170) v = at, and t = a' and from (171) s = 2at2, and t /2sc /2s V. Equating these two values of t, we have v - or solving relatively to v, v - /2as (172.), in which v varies as the square root of s.

UNIFORMLY RETARDED MOTION. 121 IV. In uenbformy accelerated motion the s:aces passed over in successive equal intervtls of timze ore propoi tional to the odd numbers, 1, 3, 5, 7, 9, etc. Thlis foliows froim formula (171), s - lact2. For in intervals of time 1, 2,.3, 4, 5, the spaces traversed are -a, -4a, — a, -26-a,'2-c.-a The spaces travelled over in each successive interval will be found by subtracting these spaces, l-a from -a, 4ca from 9a, 9-a fiom I-6a, etc., and are respectively equal to la, 3aa, a, la 9-a quantities which are to each other as 1, 3, 5, 7, 9, and the same reasoning cn evidently be applied to a greater number of successive intervals. The same fact nmay also be shown to follow fiom a consideration of the first and second lawNs of motion in connection with forlmula (171). Tlhe space passed over in a unit of time is l-a, and the velocity at the end of that time is a. By virtue of its inertia alone, in a second unit of time the body would pass over a space a, while the additional space traversed, owing, to the continned action of the accelerating fiorce is -ca. The total distance tra.velled over during' the second unit of time is therefore Ma. The velocity at the end of this secolnd interval is 2a. During a third interval the body will pass over a space 2a, because of its inertia, annl -a, because of the continued action of the folce, in all -a. The same reasoning could be applied to a greater number of equal intervals, showing that in the fourth the space is l-a, in the fifth 9-a, and so on. In intervals 1, 2, 3, etc., the spaces traversed therefore vary as the numbers 1, 3, 5, 7, 9. 247. Body projected in direction of Accelerating Force. If the body has an initial velocity, V; the velocity at the end of time t, will be v - V + at. (173.) For by the second law of' motion the accelerating force will produce the same effect as if the body were at rest, hence the total velocity will be that due to the inertia of the body plus the velocity generated by the accelerating force. The space described in time t, will be s - Vt + Jat2. (174.) For the space traversed by virtue of the initial velocity will be Vt, while that described because of the acceleration is ~at2. The total space is therefore equal to the sum of these two. 218. Formule for Uniformly Retarded Motion. In uniformly retarded motion the constant force acts in opposition to the original motion of the body, thereby tending to bring it to a state of rest. The amount of motion' destroyed in a unit of tile by a given force acting in opposition to a pretixisting'motion, is equal to that which the same force would generate in a body starting from a state of rest under its influence. From this it follo\ws that at the end of a time t, the velocity of a: body moving Nwith a uniformly retlarded motion is V - at (175), a being the retardation, and V the initial velocity. For as a is the quan16

122 UNIFORMLY VARIABLE MOTION. tity of lmotioln destroyed in 1 unit of time, the velocity, at, must be destroyed in t units. The space traversed in time t, will be s - Vt - MaP (176), which is the distance Vt, over wlich the body would move by virtue of its inertial velocity J7 minus the spa ce -}at2, throulh which it would move in an opposite direction, were it solicited from a state of rest by the retarding force. This is a direct consequence of the second law of motion. 249. Laws of Uniformly Retarded Motion. I. From these formulne it follows that the space described in extinguishing a given velocity is equal to that described in the generation of the same velocity under the action of the sameforce. For the time required to destroy a given inotion equals the time required to generate that motion. Let V be the given initial velocity, which is supposed to be extinguished in time t. Then 1V at. By the formula (176) the space s Vt _ -cett2. Sllstitutinug in this the above value of j; s - at2 - -cat - cat2, which is also the space described in time t, by a body starting fiom rest, and moving with a uniformly accelerated motion. I1. YThe velocity at any particular point of the space described is the same awhether the body startsfrom rest and acqui'res a velocity v, or starts with a velocity v and is gradually brought to rest. Let AB, Fig. 131, be the path described with a uniformly accelerated motion, and _BA that described with a uniformly retarded motion. When moving fiom B to A the velocity possessed by the body on arriving at any point G, is just sufficient to carry it up to A. From the preceding proposition it follows that this velocity is exactly equal to thalt which it would acquire by passing over A C, under the action of the force causing the acceleration. 250. Case of Falling Bodies. A body filling vertically through the air moves with a uniformly accelerated motion, as it is acted upon by a constant accelerating force, its weight, or gravity, pulling it towards the earth. A body thrown vertically upward moves with a uniformly retarded motion, the weight in this case acting in opposition to the projecting force. The acceleration produced by gravity is 9.8 n., and is usually denoted by the letter g. Substituting this letter in formulk (170), (171), (172), (173), (174), (175), (176),ancl for s writing h the heigcht, we obtain formulhe for the solution of all numerical questions relative to boedies falling fireely under the influence of gravity, as followsv gt (177), h -gt2 (178), v V12gh. (179.) Formule for uni=2TI I0f, -TF t 1+ 2 I-lt2 \ (18formly accelerated V V+- gt (180), i- t + gt2. (181.) m motion. " Formulae for univ = - gt (182), h Vt- 2gt2. (183.) formly retarded motioni. 251. Application of Formule. The various experimental verifications of the laws of motion deduced in the present

APPLICATION OF GRAPHICAL METHOD, 123 chapter will be found in the discussion of fallling bodies, the motion caused by gravity being the variety of.ccelerated motion most easy to -deale with. The a-pplicaltion of the precedingr formula will be mrade clear by the consideration of the following numerical exalilles. 1. Suppose that a body fllls fireely for 5 seconds, and we wishl to determine the velocity at the end of that time, and falso thle total space traversed. Substituting in (177), g -9.8, t - 5, we have, v -9.8 X 5 - 49.0 m. Also from (178), h - I X 9.8 X (5)2 _ 122.5 m. Or h could be found directly firom v, as v -= /2gh, whence 49 - / 2 X 9.8 X A. Solving this relatively to ],, the same value as before, h = 122.5 m. is obtained. 2. Suppose that the body were projected vertically downward with a velocity of 10 m., and that we wish to know its velocity after 5 seconds, and the space traversed in that time. From (180) v = V+- t 10 +- 9.8 X 5 - 59m.; and from (181.), s - Vt + - gt2 =10 X 5 -+ X 9.8 X (5)2 - 172.5 m. 3. A body is projected vertically upward with a velocity of 10 m., and it is required to find the time it will be in the air, and the height to which it will ascend. The time is found fiom (177), since the time occupied in destroying a velocity v equals that necessaly to generate it. Substituting for v ald 9 their values, we h]ave 100 -- 9.8 t, or t = ----- - 10.2 sec., which is the time of nscelit, and as the times of ascent and descent are equal, twice this time, or 20.4 is the total time that the body remains in the air. Tile height is found eitherfirom (179) or (183). For v=,V/2gh, or 100 -'~/2 X 9.8 X /h, whence A = 510.2 m. Or fiom (183), h -- Vt - gt2, h - 100 X 10.2 - - X 9.8 X (10.2)2 = 510.2 m. 252. Graphical Representation of Motion. Curve of Spaces. It is often convenient to study the laws of motion by the use of the Graphical Method. Any rectilinear movement of a particle is defined when we know its (ist.lll( fi'n somle fixed point of reference, at each successive unit of tiue,. If nace we draw two rectangullar axes, OX, OY, Fig. 132, representing successive units of time, by equal dislances measured on OX (abscissas), a1(l ducote, by vertical lines (ordinates) the distance of the moving body fi'tlll tlle point of reference 0 at the end of' each instant. By joining the cxtIetItities of these ordinates we may construct a curve representinr the im)tion of the body. For example, let us suppose that we wish to exhibit grapliically the law of motion of a particle vibrating- to and fro about a point D, Fig. 153. Let the particle occupy the positions B"', B", B', B, D, C, C1, C", C"', at the expiration of smuc.essive units of time, then returning throunh the points C"', C", C', C, D1, and so on. We assume thle niddle point, D, as the point to which the motion is to be referred, and lay off equal lengths, OB, BB', B'B", B"B"', etc., along the axis OX (Fig. 132), representing successive units of time. Those spaces at the left of' O are negative, and denote intervals of' time before the particle reaches D; those on the right are Fositive, and denote times after the particle has passed D.

124 UNIFORMLY VARIABLE MOTION. Now from each of these points, B"', B", B', etc., we erect ordinates of length proportional to the distances DB"', DB", DB', etc. (Fig. 133), representing positions at the left of D by negative ordinates, and those at the right by positive ordinates. Drawing a curve throuoh the extrenlities of these ordinates, we obtain a graphical representation of the mlotion of the particle. From this curve it will be seen that, for example, 3 units of time before reaching D the particle is distant from it by a space represented by B"S. The distance from D, two -units after passing it, is represented by C'AI, 5 units by C'iv, and so on, the ordinates first increasing as the particle recedes fiom D, then decreasing as it reapproaches D. and again becoming negrative when D is passed, andl the particle moves from ID towards f3". Having constructed the curve we can readily obtain intermediate values. Thus if we wish to find the distance of the particle fi-om D at the end of 31 units of time, we have simply to lay off OK - 3, and ascertain the length of the corresponding ordinate KL, which represents the space required. The curve constructed as explained, is known as the Curve of Spaces. 253. Determination of Velocity from Curve. From the curve of spaces it is easy to find the velocity possessed by the particle at any given distance fi'om D in the following manner. In the case of variable motion the velocity at any time may be considered as unifbrm, while the particle is passing over a very small distance. Let VW, Fig. 134, be a portion of the curve of spaces. Suppose B', Ml to be periods of time taken so near tofgether that the motion of the particle is sensibly uniform during the interval. The space traversed in this time is represented by OK, the time occupied in describing it by B'MI - NK. Hence the velocity space _ OK TL souglt, v time NK L' But NL = BB' represents a unit of time. Hence v = TL, that is, TL is the space which the particle would describe in a unit of time Were it to continue moving with the velocity possessed by it at the given instant. Since r = tang TNL, the velocity at any particular time is the tangent oJ' the aTngle mcadle with OX by the geometrical tangent to the curve at the point corresponding to that time. 254. Curve of Velocities. If we find the velocity corresponding to each successive unit of time, we may construct a cn1-rre of velocilies, representing tinmes by abscissas, and the correspon(ling velocities by ordinates, as in Figr. 135. Motions firom left to right are positive, those from rioht to left negative. If we have the curve of velocities given to construct the curve of spaces, this is easily (lone, for the chanre of distance of the particle fiom the point of reference (lduring an extremely small portion of tilme is CMI X llNVArea CSMI1N. The same mllethodwill give the space traversed in any other element of time. Hence the whole space traversed by the particle in a gire time is equal to the area of the corresponding portion of the curve of velocities. From the, curve of velocities we can ascertain the acceleration exactly as we have already ascertained the velocity fi-om the curve of spaces, since the acceleration is simply the increase of velocity in a unit of time. 255. Application to Uniformly Accelerated Motion. As a simple application of the preceding principles, we procee(l to apply theln to the demonstration of the laws of' uniformly accelerated motion, which we

MOTION ON INCLINED PLANE. 125 have already proved by analytical methods. As the velocity is proportional to the time we construct the curve of velocities by laying off successive equal distances, OA, AB, BC, etc., on OX. These represent times. From each of the points A, B, C, etc., we raise perpendiculars AA', BB', CC', etc., of lenoths proportional to the corresponding times, that is, we make AA' BB' OA: OB, and so with all the rest. The line 0E' is the curve of velocities, which, in the case of uniformly accelerated motion, is reduced to a straight line. From what we have already said the space traversed in any given time equals the corresponding area of the curve of velocities. Hence the space described in an interval of time denoted by OE is equal to the area of the triangle OEE' _ OE X ~EE'. If OA 1 second, in which case AA', the velocity acquired in that time, becomes the acceleration, and OE - t, we have OE: OA:: EE': AA'. or t 1:: EE': a, whence EE' at. Hence area OEE' = OE X -EE' =t X lat Iat2. 256. Comparison of Forces with Gravity. The acceleration which will be produced when any force acts upon a body at liberty to move is readily found from proportion (7), (p. 35.) Calling Wthe weight of the body, and _F the force, expressed in kilogrammes, a the acceleration produced by F, and g the acceleraF tion produced by gravity, F: TW:: a: g, whence a - gW (184), a formula giving the acceleration when the force is known. For example, suppose a body weighing 20 kgrs. to be acted upon by a force of 10 kgrs., and it is required to find the acceleration. From (184) we have a - 9.8 X 0- 4.9 m. 257. Motion down an Inclined Plane. We now proceed to investigate the subject of the motion of a body rolling down an inclined plane. Let A, Fig. 137, represent any body resting upon an inclined plane of which NO H fis the height, and 1NT- L the length. The body presses in a vertical direction with a force equal to Tr its weight. Let this force be resolved into two components P and F;, one of which, P, is perpendicular to the plane, and can have no tendency to cause the body to descend, while the component F, on the other hand, is parallel to the plane, and therefore exerts its whole influence to generate a motion down JNX. By similarity of triangles F: TV:: NO: MNX:: AI: L whence F — WA1 (185), a constant force. The body must therefore descend the plane with a uniformly accelerated motion. We wish to find the velocity and space in terms of the time and acceleration. romF (184) we have a= g d s From (184) we have a- -., and as FT - IVL, a -

126 UNIFORMIlY VARIABLE MOTION. Substituting this value of a in the general formulra (170), (171), (172), we have v gt (186), s -:g-t2 (187), v- 2gs (188), as formule for motion down inclined planes. Since = sin a, the above equations may be written, v gt sin a (189), s =- ~gt2 sin a (190), v - V/2g sin a (191). 258. Velocity Acquired in Rolling down Inclinaed Plane. If we make S =- L, equation (188) becomes v V/2gH, (192), which is the velocity acquired by a body rolling down the whole length E, of the plane, and which is independent of L, as that quantity does not enter into the value of v. But this is also the velocity acquired by a body filling vertically through the height Hff of the plane. Therefore the velocity acquired in descending any inclined plane is indepentdent of its length, and equal to the velocity acquired by falling through its vertical height. This proposition is also true for any portion of the plane, as C, Fig. 138. For the velocity acquired in rolling over AB, is equal to that gained in falling through Am, and that acquired in rolling over A Cto that gained in falling through An. Hence the velocity gained in passingr over A — AB = B C, is equal to that gained in fallinog through An-Am - mn. 259. Velocity acquired in descending a Series of Inclined Planes. The velocity acquired by descendinig a series of inclined planes is equal to that acquired by falling through their perpencicular height. Let AB, B C, (CD, Fig. 139, be a series of inclined planes over which the body rolls from A to D. From what has already been shown, it is clear that the velocity at B is equal to that which would be acquired in falling through,A3; in like lmanner the velocity acquired in rolling over BBC is equal to that generated in falling through mn, and that acquired in rolling over CD1 to that gained in falling through nE. IHence, supposing that no velocity is lost by the change in direction of the motion at B and C, the whole velocity generated in passing over A4B - B C + C-D is equal to that generated inll falling through the vertical distance Am + — mn -+- nE AE, which is the total hleight of the series of planes. 260. Body descending a Curve. If the number of planes becomes infinite, the broken line ABC'D becomes a curve, fiom which. it follows that the velocity acquired in descending any portion of a curve is equal to that acquired in falling. through its vertical height. 261. General Proposition.'Therefore in general, a body by descending from a given height to a given horizontal line will acquire the same velocity whether the descent is made perpendicularly or

MOTION DOWN CHORD OF CIRCLE. 127 obliquely, over an inclined plane, a series of inclined planes, or a curved surface. 262. Investigation of Loss of Velocity. Tle preceding demonstrations suppose that there is no change of' velocity produced when the body passes fi'om one of the planes to the following one. It is important to ascertain the conditions under which this supposition is correct. Suppose the body to pass fionm AB to BC, Fi(. 140, and denote the angle ABAI by 0, and the velocity on reaching B by v. Let this velocity v be resolved into two components, one parallel to B(C, the other perlpendicular to it, and equal to v cos 0 and v sin 0, respectively. The perpendicular component v sin 0 will totally be consunled in producing a pressure upon BC, while the motion over BC will be wholly due to the parallel component v cos 0. Hence the loss of' velocity v - v cos 0 v(1 - cos 0). That this quantity may equal 0, cos 0 must - 1, in which case 0 - 00.'This can only occur when tile broken line ABC becomes a curve, of which AB and BC are elements. There is always, therefore, some loss of velocity excel)t in the case of a body rolling down a curved surface. 263. Time of Descending an Inclined Plane. The time occupied zn p1assing over cnay inclined plane is to the time occupied in falling through its vertical heightj as the length is to the height. Let t1, th, be the timles occupied in falling over the length L and the height H, respectively, and v the velocity acquired thereby. Then from H (177) v -gth, and from (186) v g Lt,. Equating these two quantities, gth = g -ft from which it follows that t,: th: L: H. (193.) 264. Time of Descent down Chords of Circle. If a chord be drawn from either extremity of the vertical diameter of a, circle, the velocity acquired in falling over it is proportional to the length of the chord, and the time of falling over any sutch chord is independent of its length and equal to the time of falling through the vertical diameter. Let D) C, Fig. 141, be any chord drlawn from the extremity of the vertical diameter A C. Draw )]DB perpendicular to A;, and join AD. Denote A C by D, B C by 1, and D) C by I. The velocity acquired in passing over D)C is v- a/2gl. But as AD C is a _D C2 __2 right-algled triangle, B C - or Theefoe v 2g~-, which is proportional to L, the length of the chord. Again equating (192) and (186), two expressions for the final velocity v, we have V2gH = g t, whence t- g. But 12 -- HX D, whence t-, / gH- D a quantity independent of L and H. This value of t is also the time of filling through

128 CURVILINEAR MOTION. the vertical dilnmeter D. For, equating (177) and (179), gtV/~gh, whence t --. If for h we substitute D, the diame2-D ter, we have t g the time of falling through that diameter which is equal to the value already obtained for the time of descent down the chord D C.'The same method of proof can evidently be applied to any chord AD drawn fiom the upper extremity of the diameter A C, hence tlhe proposition is general. 265. Properties of Cycloid. If a body be supposed to descend through a given vertical distance AB, Fig. 142, by rolling over the paths AC, AD C', A EC, AlFC successively, it is evident that while the velocity acquired on reaching C would be the same in all cases, the time occupied in making the descent would vary. Hence the question rises, over what path would the body descend most rapidly? or, in other words, what is the curve of swiftest descent? It was first demonstrated by J. Bernouilli, that the curve known as the cycloid,1 possesses this property, so that if AD C' be a semi-cycloid having a horizontal base, the body will reach C sooner than by any other path. Hence the cycloid is often called the brachistochrone. Another curious property of the cycloid is that the time required to descend to C from all points on the curve AD )C is the same. Thus the time of descending D C' will be the same as that occupied in describing AD DC, as in the latter case, the greater steepness of the arc about A compensates for the increase in the distance traversed. This important truth was discovered by Huyghens. CHAPTER XII. CURVILINEAR MOTION. 266. Origin of Curvilinear Motion. Any moving body, if left to itself; continues its motion in a straight line because of its inertia. If any change occurs in the direction of the motion, it must evidently be due to some new fblorce, whose line of action is inclined to the path of the body; and for each succes1 The cycloid is the curve described by a point in the circumference of a circle rolling upon a straight line. Thus let DE, Fig. 148, be a circle, and P a point in its circumrnference. If the circle rolls over the straight line AB, the point P will describe the cycloid APB. AB is called the base of the cycloid, and from the mode of generation of the curve is evidently equal to the circumference of the generating circle.

CENTRIPETAL AND CENTRIFUGAL FORCES. 129 sive change of direction a separa-:te impulse of the deflecting force is necessary. A ciurve changes its direction at every point, therefore, in curvilinear motion, tlhe number of impulses of the deflecting force in a unit of time must be infinite; that is, the. deflecting force must act constantly. For exanmple, a body moving in the line AB, Fig. 143, if uninfluenced, by any additional force, will move onward in the same straight line towards G. If, however, when it arrives at B, it is acted upon by a force which alone would cause it to move over the line BD in a unit of time, accordling to the law of the composition of mnotions it will move in a new line BM31l found by laying off -BI;, the distance it would traverse in a uIlit of time by virtue of its original motion, and driawing the diagonal BJIf of the completed parallelogram BI1tCMID. The body will move in this li-ne 2BM, until the deflecting force, again acts at C, when it takes the direction (,Y. At each suiccessive impulse of this torce a chlange occurs in the path of the body, and if the number of these'irnpulses in a unit of time is infinitely great, as is the case with a1 constant force, the broken line AB GCF becomes a curve, the direction of which is constantly changing. 266. Definitions and Explanations. The deflectin force may be constant or varying in intensity. Its lines of actioin at different points of the curve may be parallel, or inclined to each other. If the lines of action at all points of the curve meet in a single point, this is called the centre of force, and the deflecting force is klnown as the centripetal force. The momentunl of the body, which tends to carry it on in a straight line tangent to the curve, is called the projectile or tangentialfborce. A stone projected obliquely into the air furnishes an excellent example of centripetal olrce. The projectile force tends to carry it in a straight line, fiomn which it is continually deflected by a force passing through the centre of' the earth, and is thus caused to move in a curve. 267. Relation between Centripetal and Centrifugal Forces. The centripetal force at any point of the curve is equal to that component of the projectile force which is directly opposed to it, and which tends to carry the body away from the centre of fobrce. Let Pit, Fig. 144, be an element of the curve. As the are PK coincides with the chord PK for infinitesimal lelgths, the body at a given instanllt may be considered as moving along that chord, which will therefore represent in magnitude and direction the resultant of the projectile and deflecting forces acting upon the body in question. Decompose th}is force into two others, representede by PRA, tangent to the are at P, and PQ, lying in the direction of the centre of fbrce, both of which lines are sides of a parallelo. graln, of which PKl is the dcligonal. PR will represent the pro. jectile, PQ the deflecting or centripetal fore. Again, resolve PR 17

130 CURVILINEAR MOTION. into two components, P1; directly opposite to PQ, and P-lTin the line of motion of the body, by constructing tile parallelogrnlm PLRK.. The component PL tends to carry thle body awVay from the centre, andl is opposed to PQ. But as PXJfQ is a parallelogram, PQ anfd RKare equal; and as _PILRK is a parallelogrlm, PIL is also equal to 1-KJ. Hence PL is equal to PQ, Mlich represents the centripetal force. That comnponent of the projectile force, which is directly opposed to the centripetal force, is called the centrifugalforce. 268. Additional Explanations. Curvilinear motion therefore larises fiom the simultaneous action of two forces, the projectile force, tending to carry the body onwards in a straight line, alnd the centripeotal force, continually deilecting it from that line. The celtriftugal flree is not a power separate fi'om tlhe projectile force, but is only that component of it which is always opposite the centrifugal force. The projectile force is evidently the momentum of the body.'When the revolving body is nieclanic-ally connected with the centre of force by a rod or string, the centripetal force is the tension of the particles of that rod-or cord, caused by the centripetal force. In the case of fiee revolvincg bodies, as the planets, the centripetal force is generally the attrletion of gravitt;ion. When a body revolves about a celtre of force, the line joining that centre to any point of the path of the body is called a rctdius vector. The curve pursued by the body is known as its orbit or trcajectorjy.'Z69. Law of Equal Areas. If a particle revolves about a centre of force, the radius vector describes equal areas in equal times. Let us first suppose that the central force -acts intermittently at equal intervals of time. Let the particle be at the point A, Fig. 145, imoving with such velocity as would alone cairry it to the point B in a unit of time, duing wllich time the deflecting force alone would carry it to G. Under the united action. of these olrces the particle will move in the diagon;l A C of the parallelograln A G C(B. On arriving at C the particle tends to move over the line CD - AC inr a illit of tinme, because of its inertia. The deflecting force, however, now gives it such an impulse as alone would cause it to move over C~G' (which may be gireater or less than.A G) in the same time. Impelled simultaneously by these two folrces, the particle will move over the diagonal CE of the parallelogram CG'ED. Drawln SD. The triangles SCE, SCD, are equal because SC and D)_E are parallel. The triangles SA C, SC.D, are a:lso equal, since AC=- CD, and the vertex S is comnlon to both. Therefore SCE _ SA C. In like manne r, SEF can be proved equal to S0C-1, and so on, for all parts of thle trajectory. But the ]preceding reas,)ning holds, however short may be the interval between the successive ilpulses of the deflecting

VETLOCITY IN ORBIT. 131 force; it is therefore true when that interval becomes infinitesimal, thalt is, when the force is constant. The lines A (,, CE, _E, then become elements of the curve in which the body moves. Since, therefore, the equal elementary triangles SAGC SO, S SEF, are described in equal times, the same is true for any equal finite areas, as each of these contains the same number of elements. 270. Convertse of preceding Proposition. ]f there is a point within any orbit so situated that the line connecting it with the revolving particle describes equal areas in equal times, that point is the centre of force. Let A CEF, Fig. 145, be the orbit, and S a point so situated that the elementary areas SA C, SCE, SEE, described in successive elements of time are equal to each other. Then will S be the centre of the centripetal fbrce acting upon the revolving body. Prolonmg A C, making CD - A (7, and sull.ppose the deflecting force to act intermittently, as in the precedinc proposition. CD is evidently the space which would be traversed in the second element of timle, were there no deviating force in action. But CE is the space actually traversed under the united action of both the projectile and deviating forces. Join DE, and complete the parallelogramlll G'IDE. CG', which is equal and parallel to DEY, evidently represents the direction and nmagnitude of the deflecting force. To demonstrate the present proposition it is merely necessary to prove thalt the direction of this deflecting.fohree is toward S, that is, that CG' lies in CS. Now the triangles SAGC, SCE, are equal by hypothesis; the triangles SA, SGCD, are also equal, since they have equal bases, AC, GD, C, nd a comlnon vertex, S. Hence SCD = SCE, ian(l as these trianlgles have the same base, SG, their vertices nmust lie in a line parallel to SGC, that is, D-Eis parallel to SC, and llence CG' lies in SC. In the samne n-malnner it can be proved the direction of the deflecting force at all other points of the orbit, as E, E, etc., is directed toward S, hence S is the centre of force. The two preceding propositions, together with the three following are extremnely important fiom an astronomtical point of view, as will be lnore filly seen in the chapter upon Gracvitation. 271. Velocity at different Points of Orbit. The denlonstration given in ~ 269 fiurnishes a silmple method of comparing the relative velocity of a body at dif-ferent points of its orbit. Let v, vt, be the velocities possessed by a particle at two points of its orbit, P, P', Fiog. 146, and let SPQ, SPfQ', be equal elementary areas describedl by the radius vector in an infinitely short tille. The arcs PQ, P'Q' being elements of thle ctrve, may be considered as beingl traversed wTith uniform velocities, v, v', in time, t, -a(nd P Q vt, P'Q' vt. Draw P T, P']", tanlgents to the curve at the points P, P', and S', ST', perpendiculars let

132 cURVILINEAR MOTION. fall upon those tangents from S. Denote ST by A, ST' by A'. The triangles SPQ, SP'Q',.are equalf to vt X A, v't X A', respectively. Hlence as SP =- SP' Q', v X A =v' X A', whence v: v': A': A. (194.) Hence the velocities at different points of an orbit are inversely proportional to the perpendiculars let fall from the centre of force upon the tangents to the orbit drawn fror those points. 272. Law of Force in Orbit. The centripetal force in different orbits, and at different points of the same orbit, is proportional to the deflection of tile curve corresponding to arcs described in equal infinitely short times, since the deflection is wholly caused by that force, which may be considered as constant during the time of describing elementary ares. We are thus furnished with a method of comparing central forces, for if we know the form of an orbit, and the position of the centre of force, we can readily compute the deflections at any point, and thence the relative values of the centripetal force. 273. Elliptical Orbit. The most interesting application of this principle is the case of a particle revolving in an elliptical orbit about a centre of force situated at one of the foci. Let A, Fig. 147, be that focus of the ellipse in which the centre of force is located, and SPQ, SP'Q', equal areas described in- an infinitely short time. The deflections RQ, R'Q', parallel to the radii-vectores, SP, SP', measure the intensity of the centripetal force at tile points P, P'. But by a property of the ellipse Q: R'Q':: SP'2: SP2 (195), that is, when the orbit is an ellipse, with the centre of force at one of thejbci, the centripetalforce at differentpoints of the orbit is inversely proportional to the square of the distance of the revolving particle from the focus. The same law of force csal be shown to hold when the particle describes either a parabola or a hyperbola. 274. Centripetal Force in Circular Orbit. The curvature of the circle being the samle for every portion of the curve, the centripetal force in such an orbit is of constant intensity. To determine its value, let C, Fig. 148, be the centre of the circle in whose circumference the particle revolves, PQ an arc described in an infinitesimal time, PR the space over which the body would pass in that time in virtue of the projectile force, PM-h = -Q, the space over which it would pass in virtue of the centripetal fo)rce. Call v the velocity of the particle, t the time of describing the arce PQ, and R the radius of the circle. The arc PQ and the chord PQ will coincide, as the arc is an element of the circumference, and PQ vt. By a property of the circle, pQ2 = PM X PD, or v2t2 P- P X 2R, whence v2t2 P - 2R Call f the acceleration which the centripetal force

CIRCULAR ORBIT; 133 is capable of generating in the particle in a unit of time. Then, by the laws of uniformly accelerated motion, PF- = ft2. Equlatv2t2 ing this with the value of PJlf previously found, jft2'2N whence f -f (196.) That is, the acceleration which the centripetal force is capable of producing by its uniform action during a unit of time is equal to the square of the velocity of revolution divided by the radius of the orbit. Krowing this acceleration we can easily deduce the value of the centripetal force in units of pressure, or kilogrammes. Let F be the centripetal force thus estinlmated, and WV the weight of the body. Then, since forces are proportional to their accelerations, F: W::f: g, whence Fi- IV-'= Substituting the value off, given in (196), we have, F (197), or for substituting its value, 2, as given in equation (8), we have F-= M,.(198.) The centripetal force expressed in units of prlessure is therefore equal to the mass of the particle into the square of its velocity, divided by the radius of the orbit. In formula (198) if M is made equal to -, the force /F is given in gravitation units. If M be made equal to the mass of the body in absolute units of mass, i.e., in grammes or pounds, the result is expressed in absolute units of force. For g being Wv2 V2 the number of gravitation units, - v X g = Wi, is the number of absolute units of force in F (~ 50, p. 37), in which expression we may write M, the mass in absolute units, instead of W. The centripetal force of anly extended body, or system of bodies, can be shown to be the same as if the total mass of that body or system were concentrated at its centre of gravity. Hence the propositions which have been demonstrated as true for particles, call be applied to extended masses by simply considering the whole mass thus concentrated. 275. Different Expression for Preceding Formulas. Formula (198) can be put in another fornm, which is often of convenience. Let T be the time of revolution in the orbit. 2=rR Then vT = 2-r, and v =- T, Substituting this value of v in (198), we have F M,' (199.) Therefore for bodies of the

134 CURVIIIN EAR MOTION. same mass revolving in circular orbits of cbferenft radii, the centripetal force is proportional to the radius of the orhit, anjid inversely as the square of tthe time of revolution. 276. Velocity in Orbit. The uniform velocity tioth whtich a body revolves in a circular orbit is equal to that which the centripetal force would generate by its uniform action upon a body falling through half the radius of the orbit. For by (196) f — =.h, and v = -/fR. Suppose the centripetal force to impel the body from a state of rest until it attains a velocity v. Call S the space described. Then from (172), v.- /s 2s, We have, therefore, v7f o _ ving, whence s = -y 277. Body revolving in Vertical Orbit. Tf a body heldl by a cord be made to revolve in a vertical circular orbit, it is evident that the tension of the cord will vary firom F - 1T to F + IV, (F beillg the centrifugal force, and W the weight of the body) accorlding as the body is at its uppermost or lowest position. In order that a body may revolve in such an orbit, it is necessary that the centrifugal force shall be at least equal to the weight of the body, as otherwise the uppermost point of the curve could not be passed. It is easy to calculate from this the dlinimum velocity which must be given to the body to enable it to perform its revolution. For, with this minimum velocity, F= W. Hence TfW - an(l v -Vy.(200.) g r 278. Experimental Illustrations. The action of the centripetal and centrifugal forces may be illustrated by the Whirlig 7 able, one form of which is shown in Fig. 149. It consists essentially of a wheel A, to which a rapid rotation can be imparted by means of the larger wl:eel B connected with it by a cord or train of wheel work. If a frame EF (Fig. 150) be attached to the small wheel so as to revolve about the vertical axis A, a body IV moving freely u on a horizontal wire, will show the action of the centrifugal force by moving towards D. The experiment mnay be made quantitative, and the laws of central forces verified, by affixing to W a cord passinsr over the pulleys B, C, as represented in the figure, and to which a weight P is attached which can be varied at will. The table is whirled until P just begins to rise because of the centrifilgal force exerted from W by means of the cor(l. The velocity of rotation is noted, as well as the magnitudes of P and TX, which data furnish a means of verifying formula (198.) Many interesting experiments may be performed with the whirling-table, Fig. 151 represents a frame CD, containing two inclined tubes proceedin(g from a reservoir filled with water. On rot.ating this, the centrifilgal force generated, causes the water to rise in the tubes. If a ball of iron, S, is placed in one tube, and a ball of cork, T, in the other, the ball S, being heavier than the water, will be thrown to the farther extremity of the tube on rotating the machine, while the lighter ball of cork 7, will remain at the bottom, as the greater mass of the water causes it to possess a greater amount of centrifhgal force. If a glass of water be rotated rapidly in a horizontal plane, thle liquid is elevated about the borders and depressed at the centre, the depression having a paraboloidal form. A flexible hoop of metal fixed at the point A

PRACTICAL APPLICATIONS. 135 upon a vertical axis AB, and free at C, bulges out at its equator on being rotated, and assulnes the forln represented by the (dottedl lines. All the particles of the hoop tend to move as far as possible from tlhe axis, in ivirtue of their celtrifugal olrce, and ttlis being greatest ata those points having the greatest ra(lii of' rotation, the particles at the equator press so strongly outward as to cause the hoop to become elliptical. The action of the centrifugral force can also be shown, and its laws verified, by suspending a heavy ba.ll from a delicate spring balance, as in the figure (Fig. 153), and allowing it to swing as a pendulunm. The excess of the index readinDg when the ball is vibrating and at its lowest point, over the reading when it is at rest, gives the centrifibgal force directly. By varying the weight of the ball and the length of the sustaining cord, formlula (198) may be illustrated experimentally. 279. Practical Applications and Illustrations. A practical application of the foregoingl principles is illustrated by the machine often used in laundries for drying cloths, and known as the hydro-extractor. It consists of a large annular trough having its exterior surface perforated with holes, or made of coarse wire cloth. The wet cloths are wrunll partially dry, and then placed in the trough, to whic(h a rotation of firom 1200 to 1 500 turns per minute is then imparted. The water flies away from the cloth in virtue of' the centrifugal force generated by the rotation, rendering it almost dry in a few mlnutes. A similar machine is used in England in the operation of sugar-refining. The syrup being boiled in vacuo, crystals of' sugar fborn thlroughout its mlass. To separate the molasses from these, the whole mass is poured into the trough of a machine resembling that already described. When this is revolved rapidly, the molasses flies off through the apertures, leaving a clear mnass of soluble sugar behind. A small jet of steaml directed against the outside of the trough prevents the expelled syrup fioln collecting there in a coating, and obstructing the exit of the remainder. The same process has also been applied quite recently, to separate the liquid colors used in printing fiom the mlore solid matters with which thev have been mixed. These colors are usually very difficult to strain, but the present process seems to have rendlered the operation quite easy. An interesting illustration of centrifugal force occurs in the process of makinlg crown-glass. The workman first obtains a quantity of melted glass upon the extremity of a long iron tube, which he shapes into a bell-like forml. Then heating this again until the glass is softened, he places the tube horizontially upon an iron bar, and rotates it with great rapidity. The centrifugal force causes the bell to bulgre out, and finally to become almost flat and of' uniftborm thickness except at the very centre, where the instrument is attached, and ftolo it small sheets of' extremely brilliant and clear glass may be cut. The fznnilg-mnill, or blower, used for creatingr a blast of air, and applied to cleaning grain, the ventilation of buildings, etc., is an additional example of the application of the laws of centrifhugal force. It consists of a drum MM.1A, Flg. 154, within which is an axis carrying several paddles or vanes, N, N, is made to revolve rapidly. The centrifugal force produced in the air, carrie[l round by the fans, causes it to rush through the tangential exit tube S, in a constant stream. Fresh air is drawn in throughlll a large aperture near the axis of rotation. When the rotation of a body becomes exceedingly rapid, the centrifiogal force generated may be sufficient to overcome the cohesion of the particles composing it. Large millstones and fly-wheels, running at a high rate, of

136 CURVILINEAR MOTION. speed, have been known to burst, the fragments scattering in all directions, antd causing great destruction.t 280. Vehicle on Curved Road. When a carriage moves rapidly upon a curved road, the tendency of' the centrifilwal force is to overturn it. This will be understood by a reftirence to Fig. 155. The vehicle is acted upon by two forces, its weight, along the line G W, and the centrifugal force along GF. Suppose GWV, GF, to be made proportional to these forces then GR will represent their resultant, and if' that line fdlls without the base formed by the wheels, it is clear that the vehicle will be overthrown. As the centrifugal force is proportional to S, the greater the velocity and the less the radius of the curve, the greater the tendency to overturn. 281. Depression of Inner Rail on Curve. In the case of railroad trains, the velocity is so great as to render it necessary to counteract this effect of the centrifugal force. This is done by making the inner rail lower than the outer one. That the stability may be unaltered by the motion in the curve, the resultant GR, Fig. 156, of the weight of the vehicle and the centrifugal force must fall' midway between the wheels. To calculate the inclination requisite for this, let G W represent the weight of the car, and GF the centrifugal force. The slope of the road AB must be such that it R W is perpendicular to GR. But calling 0 the inclination, tang 0 -G F W v2 V2 or as F -g R tang 0= - (201.) g RI' n gK To find the linear elevation necessary to make tang 0 equal to this quantity, call h - BC, the elevation required, d the distance AB between the rails. As 0 is very small, AB may be considered as equal to A C, in h h v2 dr'2( which case tang 0 Hence I= - and h -- - (202.) In the formula v is taken as the ordinary running velocity of the trains. 282. Centrifugal Force at Equator of Earth. The rotation of the earth on its axis, carries all bodies resting upon its surface in parallel circles, thus generating a certain amount of centrifugal force. The velocity being greatest at the equator, the centrifugal force is greatest there. To calculate its amount, we must first find the velocity of a body situated at the equator, and substitute this together with the proper values of' g and R in equation (197). We have F 9- X ( t), in which t is the time of a single revolution, or 86,164 seconds. Assumingr R - 6,377,300 m., the mean equatorial radius, and g 9.8087 m., t 9.8087 X 6,377300X 9.808 X 6,3 7 2 X 3.14159 X 6,377,300 2 1 86,164 289 The centrifugal force acting upon a body at the equator, is therefore about. ag the weight of the body, and as on the equator, this weight an(l the centrifugal force are directly opposed, the weight of a mass is diminished in that ratio. If' the earth's velocity of rotation were to increase, the centrifugal force would grow greater and greater, till a point would be reached, when it I For an account of a very destructive accident of this kind, see Journal of thle Franklin bInstitute, Vol. axv., p. 86.

EFF:ECT ON FIGURE OF EARTH. would exactly counterbalance the force of gravity, and hence a body at the equator would lose all its weight. Tile velocity required can readily be fbund. That this may occur, we must have g or vu,2gyl, whence v 79.09 nm., or about 17 times their present velocity. 283. Variation of Centrifugal Force with Latitude. Let F denote the centrifhgal fbrce of a body situated at the equator, and F/ that of a body- at E (Fig. 157), in latitude 0. Since tlle revoltltion of the bodies about CD is accomplished in equal times, the centrifigal forces are proportional to the radii (199), that is, F: F:: C: DE. But DE CE cos 0 = CR1 cos 0. HenceF': FI:: CR: CR cos 0:: cos0 ('203), that is, the centrifugal force at any place is proportional to the cosine of the latitude of that place. 284. Effect of Centrifugal Force on Figure of Earth. The force F', represented by EG, Fi,,. 157, can be resolved into two comiponeilts, represente(l by I, E K,EK, the former of which is perpendicular to the surfiace, the other tangential to it. The tendency of the tangential colnponent upon any particle at E, is to force it towards the equator. and if the surface is suf)posed to be covered(1 with a liquid whose particles arei fiee to Inove, they will be (iriven onwards, aec.cumulatimn- about the e(uator, so that the mass will no longer be spherical, but will assume the formi of' an oblate splheroid. This accumulation will go on until the resultant AR (Fig. 158), of' A7' the centrifugal folcc, and Alll, the force of gravity, is normal to the surthce of the figure, when the whole force acting upon the particle is exerte(l to push it ag;inst that surfhce, and the particle will evidently remain at rest. The form thus assumed is called the spheroid of equilibrnzi. Geology shows that the earth was once in a fllid state, in whic:h case tlle spheroidal form would necessarily be assumeid. Knowing the velocity of mrotation, and the (limuensions and weight of the earth, the amount of flattenino' can bz calculated upon mIathemlatical plinciples. Tilis subject lhas been investigated by liiyglhens, Newton and Maclaurin. whose results agree, in rgeneral,: with those given by actual mneasuremlent, although fi'-omn another cause the figure of' the earttll deviates slightly firolm that of a spheroid. The oblate splheroid is not the only form of' equilibrium which can exist in the case of' a rotating liqlid mnass. The general problem of determining the possible forms of equilibrium in such a lmass is extremely difficult, an(l its solution has scarcely been a tte!mpted.,Jacobi has shown, howeveri, that an ellipsoid of three unequal axes. the least of whichis the rotat;on-axis, is also a form compatible with equilibrium. 285. Plateau's:Experiment. The efflect of the centrifugal force in proluteCin(r the spheroidal torlmn of the earth can be illustrated experimentally by a method d(evised by Plateau. A smiall quantity of' oil is pourled into a mlixture of alcohol and water, these liquids beinr combined in suLch proportions that the density of the mixture is just eqtual to that of the oil, which is therefbre relievetl frol the action of gravitv, -anl has no tendency either to sink or to rise to the sufa'cet. It can easily be collecterl ill a single niass, whlich assullles the siphel'rical form because of the internal molecular attraction actin( amongl its particles. If' now a vertical wire bearirn a slla.ll disc be passel downward thlroug' tlIe oil till the disc is at the centre of the sphlere, and tllei made to revolve rapidly, thte oil b1egins tI, revolve, with it. I See Thompson nnd Tait's Treatise o NVatutral Phloksophy, Vol. i, p. 617. 18

138 WORK AND ITS MEASURE. As the velocity is increased the spheroidal form is assumed, the bulging out at the equator becominr nomre and more prolllinent. Finally, when the velocity becomes sufic.iently great, a ringl of oil is detalhced froml the spheroid, and though entirely separated from the former, contitnues to revolve around it. CHAPTER XIII. WORK AND ITS MEASURE.- DYNArMICS OF RIGID BODIES. 286. Nature of Work. Work is performed whenever a foree produces motion in opposition to a resist:nce; as wh(-n a weight is raised in opposition to the action of gravity, when a force complresses a spring, or moves a body against the resistance of friction, or the resistance of the ail. It is ploportional (1) to the amount of the resistance, and (2) to the distance throu'fh which that resistance is overcome; that is, to the distance through which its point of application is mnoved. This if a body be lifted, there is work done, which is greater in proportion as the mass is greater, and as the distance through which it is raised is greater. Since the work donle varies directly as each of these quantities, it varies directly as their product, and by choosing a suitable uieit, the work performed may be said to be equal to the product of the resisting force into the distance tlhrough which it is moved. That is, if we represent the resistance by P, an(d the distance through which it is mloved by A, the work X will be EP- _PS.. (204.) For example, if a weight of 100 kilogrammes be raised through 10 letres, and a weight of 10' kgrs. through 5 metres, the quantities of work performed in the two cases are to each other as 1000 to 50. Both, these elements, motion and pressure, are necessary to the performance of work. There is no work done when a force merely exerts pressure against a surface withlout causing motion, as when a mass is supported upon a prop. Neither is there any work done when the force has no resistance to overcome. Thus a body moving in fiee space performs no work. It. is only when a resistance is overcome through a certain space that work enslles. 287. Case of Oblique Force. If the point of application of the resistance does not move along the line of action of the force P represented by AC, Fig. 159, but is constrained to follow some other path as A B, the work done is still equal to the resistance overcome into the distance through whi6ch it is overconle, or if the force AC, the resistance AE, and the

UNIT AND RATE OF WORK. 139 reaction AG are balanceld, the work is equal to that component of the force which is directly opposed to the resistance, multiplied by the distance through which the point of application is moved. That is, the work done in the case of Fig. 159 is represented by AB X A C cos 0, or P cos 0 X AB, which is equal to P X A llM, as AB cos 0 -- AM. Hence the work done by an, oblique jorce is equal to the magnitude of the force multiplied by the projection of' the pat/h of the point of application upli its line qf action. 288. Application to Virtual Velocities. It is evident that if the distance.lB is supposed to be very small, AMll is the virtual velocily of the force P, and P X A iM, which is called its virtual tmomrent, must represent the elementary quantity of work done during the time corresponding to that displacement. It follows from equation (85) p. 74, that the algebraic sum of the virtual moments of any numlber of component forces is equal to the virtual moment of' their resultant. Hence when lmoving against a resistance, the work (lone by- any number of component forces is equal to the work done by their resultant. There is therefore no gain of' work by the use of alny mechanical power, or the quantity oJ work twhich ca'n be peiformed by any force i.s a constant in whatever man.ner theJforce ma be applied. 289. Case of Variable Force. If the resisting force is variable, the path of its point of application may be supposed to be divided into a a number of equal parts so small in magnitude that the resistance mnay be considered as constant durinn the time occupied in traversing one of them. The elementary quantity of' work done in passing over one of these spaces, will be equal to the resistance into that space, and the total Iquantity of work will be found by a summation of the elementary quantities by the processes of algebra or the calculus. 290. Unit of Work. The unit of work in practical use by engineers is the kilogrcamme-metre in French, and the footpound ill English measures. The kilogramme-metre is the work done in movilg a tforce equal to the weight of 1 kilogramme through the space of 1 metre. The foot-pound is the Nwork done in moving a force equal to the weight of 1 plound through the space of 1 foot; neglecting the difference of g between London and Paris, the kilogramme-lnetre is equal to 7.2331 foot-pounds. In purely scientific investigations, however, these units are not employed. In such cases the unit of work is the work done in moving an absolute unit of force througLh a unit of space. Hence the absolute unit of work in French measure, is the work done in moving a force equal to the weight of. grannmmes at Paris through 1 mnetre, and in English measure the work done in moving a. force equal to the weight of p3-.~ ounds at London over 1 foot. 291. Rate of Work. The amount of work performed depends simply upon-the force overcome, and the distance through which it is overcome, andl is totally independent of,the time: occupied in its perfornmance. In other xwords, the work (lone is the same, wMhether it be performed mnore or less rapidly. In practical calculation, however, the element of tihne enters, lnd it is neces

140 WORK AND ITS MEASURE. sary to determine the quantity of work done in a given timne. The rate of work is the work dlone in a uhit of time, and is found by dividing the total work performed by the tilme occupied in its performance. That is, if PS be the work done in T seconds, PS R -- (205), is the rate of work. Thus, if a force move a resistance of 10 units over 100 metres in 5 seconds, the rate of,P2S the work, that is, the quantity performed in 1 second is S 10 X 100 _______- 200 units. Thile standard in ordinary practical use for comparing the rates of work of different motors, as steam-engine s, w:lter-wheels and the like, is tile horse-power, which is a force capable Of performing a work of 550 foot-pounds, or 76.0375 kilogranmme-metres per second. A horse-power is therefore a force capable of raising 76.0375 kgrs., 1 mn. per second, or 550 lbs., 1 ft. in that time. To illustrate the application of this standard, suppose that a steamer mnoves against a resistance of 10.000 kgr. at a rate of 10 km. per hour. It is reqluire(l to find the horse-power of an engine:-that will propel it with this velocity. The work done per second is 27778 kilogramme-metres, hence the required horse-power is 27o-7sw = 365.33 horse-power. 292. Accumulated Work. Equation (204) gives the work done by a force'when the resistance overcome, and the sl>ace through which it is overcome are known. But it frequently happens tlhat this distance, X, is not given, but instead of' it the velocity, V, with which the body is movinrg, is known. It is therefore important to fincl an expression for thle work which can be done by the body in question, in terlns of its mass and velocitv. This can readily be found by ascertaining the space over which the body would move with a uniformly retalrded motion before its velocity would be reduced to zero, and substituting this space as expressed in terms of Vin equation (204). If the body is moving with a velocity V, against a resistance P, which is capable of producing a retardation a, the distance S., V-2 over which the body will move, is S- c2a Substituting this value V2 p TV of Sin (204), we have, E-PS= P. But a 2a' a g whence E = M'V2 (206). The work which a moving body is capable of performingg is therejbre equal to half the mass into the sqaeare of its velocity. If iM is made numerically equal to -, as in the first system of' the measurement of forces, the resulting value of E is expressed in foot-pounds

KINETIC ENERGY. 141 or kilogramme-metres. If lk is made equal to the weight of the body in graintnes or pounds, as in the second system, the work E is expressed in absolute units. The quantity ~ MV2 is known as the vis-viva, or living force of the body.' Work thus stored up in a body is often called accumulated work. To illustrate this by an example, let it be required to find the amount of work in kilogramme-metres which must be exerted to bring to rest a cannon-ball weighing 15 kgrs., and moving with a velocity of 400 in. per second. The work required must equal the vis-viva of the ball, that is, E - ML 2 --- X V~So- Xa (400) 122,340 kilogramme-metres. 293. Energy.- - inetic Energy. Energy is the power of doing wolk, and is either Kinetic or Potentictl. Kinetic energy is the power possessed by a body of doing work in virtue of its force of motion, and is represented by the equation (206), wE -i IV2, in which E is the kinetic energy. The term kinetic energy was first introduced by Thomson and Tait. The term actual energy has also been applied to this kind of enelrgy by Rankine. If an unbalanced force acts upon a mass in motion, it is consumedcl in adding to the kinetic energy of that mass by continually increasing its velocity. The increment of energy during a given timte, that is, the work stored up, is _K = 1i( FV2 - V2) (207), in which V, VJ, iare the velocities at the beginning and end of the time considered. If the mass in motion moves against a resistance it evidently does work, which work is measured by the decreaqse of kinetic energy, that is, by K= _Lf ( V2 _ T12) (208). The storing up of energy may be illustrated by reference to the case of a cannon ball. By the continued action of the confined gases oenerated by the burning powder, the ball is moved along the bore of the gun with a continually accelerated motion, thus storing up a large amount of work, or, in other langu:ige, becolling possessed of a great amount of kinetic energy. This is grtadually diminlished by the resistance of the air, and finally, on striking any obstacle, as a wall, or an iron armor-plate, the ball penetrates until it has expended all its energy, when it comes to rest. The tota.l work done is expressed by equation (206). Or,:again, a stream of water, by running (lown its bed, acquires a considerable velocity, thus generating kinetic energy, which may be emniloyel in iruning a water-wheel against which it impinges, and so carry in;acllinery. The amount of worzk which this machinery can do in,grinding corn, or rolling iron, or in any other species of work, is (after deducting the loss by fiiction, and in other ways) exactly equal to the kinetic energy of the moving water. The same is true of a windmlill moved by air, and of any other forim of motor. 1 Vis-viva has generally been defined as equal to MVa2, but it is more simple to consider it as idea.tical with the work which the imass is capable of performing.

142 WORK AND ITS MEASURE. 294. Pctential Energy. This kind of energy is the power of doing work possessed by a mass in virtue of its position. if a body is so situateAd that it is acted upon by a force which will produc e motion inl it, as soon as some restraining force is removed, and thence generate kinetic energy, it is said to have potential energy. Thus, a weight suspended at an elevation will fall as soon as the cord sustaining it is cut. The potential energy of a mass is expressed by the amount of kinetic energy which would be developed in it under the unimpeded action of the forces soliciting it. For example, the potential energy with regard to the surf:ce of the earth, of such a body as we have supposed sustained at aul elevation, is E- PS, if P is the weight, and S the elevation, or.E=- 2MV2, if Vis the velocity which would be developed by the motion of the body until stopped by coming in contact mwith the earth. As other illustrations of potential energy, we may Inentior the case of a stretched spring, which is capable of doing work in returning to its normal state, and that of a head of water, which is capable of doing work by the kinetic energy acquired when it is allowed to flow freely under the action of gravity. The term potential energy was first used by Rankine. Helmholtz hadlcl previously applied to it the title sum o' the the isions, and Thonlson had called it statical energy. 295. Conservation of Energy. An examination of the examples adduced will show that there is a close relation between these two kinds of energy. We have seen how potential is converted into kinetic energy, Let us now consider the converse case, in which kinetic is converted into potential energy. The most fianiliar instance which can be adduced is the case of a body thrown into the air with a certain velocity. It possesses a definite amount of kinetic energy, which will be expended in raising the mass against the action of gravity, and will carry it to a certain height. Suppose the body to be stopped while at this uppeimost point of its course. The amount of kinetic cnergy then possessed is evidently zero, while the anmount of potential energy is exactly equal to the kinetic energy which it had on leaving the earth, as the velocity which it would acquire by falling freely froml the point now occupied is equal to that with which it was originally projected upward. It is clear that the amoutnt of potential energy developed here is equal to the amount of' kinetic energy consumed in developing it. Next let us consider the case of the body after it has begun to descend. As it falls its kinetic energy becomes greater with the increase in its velocity, while its potential energy becomes less, since it is continually approaching the ground. It will also be seen that for all points of its downward path the sum of its kinetic and potential energy must be a constant. For the potential energy at any point supposed will be that due to its distance fiom

CONSERVATION OF ENERGY. 143 the ground, and the kinetic energy that due to its velocity acquired by falling fiom the highest point of its path. Hence, calling S the total height reached by the body, S' the height considered, S — S' will be the distance of the body fiom the most elevated point reached by it. Now the kinetic energy is measured by the quantity E' = P(S -S'), and the potential energy by the quantity E PS'; and the sum of these two expressions will be seen to be a constant, equal to PS. Or expressing the same fact by means of the equation for accumul:ated work, E' -- I V'2 and E =.f( V2 - V12,) whence the sum of these is a constant equal to 13IV2. The same equations are evidently true for the ascending motion of the body. A similar course of reasoning can evidently be applied to the case of a stretched spring, or of a head of water. Hence in all of these cases we see that the sum of the potential and kinetic energies possessed by a mass is a constant. The preceding propositions are capable of being extended to a far more general form. It can be shown that in any system not acted upon by externalforces, and the masses composing which act upon each other along the line joining their centres of mass, with forces dependent only upon their mutual distance, and in no way upon their relative velocities, the sum of the potential and kinetic energies is a constant. This is a general statement of the fundamental principle of the Conservation of Energy. We shall see hereafter that there are various other forms of energy besides mechanical motion, into any of' which such motion may be transformed. The student will ask what becomes of the mechanical energy of a body stopped by fiiction, or the resistance of the air, or by being brought to rest on striking the ground. In all these cases kinetic energy disappears, and the body comes to rest, and yet there is apparently no potential energy developed in it. There is here a loss of mechanical motion, bult it is transformed into another form of energy: viz., heat, which, as we shall see hereafter, is itself capable of developing mechanical energy under proper conditions, and the heat produced is an exact mechanical equivalent of the kinetic energy disappearing in the mass. 296. Energy of Rotating Bodies. Hitherto we have considered only the question of the accumulation of' work in bodies having a translatory motion. In practice, however, it is fiequently necessary to make use of the energy stored in a rotating body, as a fly-wheel. We proceed to consider the question of the determination of the kinetic energy of such a body. 297. Angular Velocity. -Angular Acceleration. In this investigation we shall make use of the angulCar velocity of the body, which is measured by the angle throughl which it rotates in a unit of time, and is the linear velocity of any point situated at a distance fronm the axis of rotation equal to unity.

144 WORK AND ITS MEASURE. Angula~tr acce7eration is the acceleration impressed in a unit of time upon such a point. The linear velocity of any point in a rotating body is equal to the angular velocity multiplied by its distance fiom the axis of rotation. 298. Determination of Energy. Let ABD, Fig. 160, be a body rotating about an axis through C, and at right angles to the plane of the paper. Suppose m to be the mass of a parlticle of the body situlted at a distance r, from the axis of' rotation. Let wo be the angular velocity of the mass. The kinetic energy, k, of the particle is rMnv2, or as v - rw, k -= mr2A2. The energy of any other particle of mass m', and situated at a distance r' fiom the axis, is i-' -= m'r'2w2. Fol, other particles of masses, mn", mnT", situated with radii of' rotation r", r",''mr 1111"2w")2, Jkn nr rn2(a2, and similar expressions may be obtained for all the particles of the body. The total kinetic energy of the body is E, which is equ:al to the stum of the- eneroies of its particles, that is, E- Imr2,2wJ2 -+- Im'r'2wo2 + -m"/r"%2w2 +j Lq2 nrnw2,2 or denoting by z mrnr2 the sums of the products of the mass of each particle into the square of its radius of rotation, E = - oJ2.1mr2 (209), which expresses the amount of work which the body can perform before being brought to rest. 299. Moment of Inertia. The expression,mr2 is of very fiequent occurrence in treatises on dynamics, and is known as the moment of inertia of the body considered. We shall, in general, denote it by I1 Hence Vrnr2 - 1 (210). Equation (209) may evidently be written E = 1)2I (211). The kinetic energy qf a rotating body is therefore equal to its moment of inertia multi2plied by half the square of its angular velocity. The moment of inertia evidently varies with the form of the body, its mass and the position of the axis of rotation. For continuous bodies equation (110) must be changed so as to consider the masses m, m', etc., as infinitesimal mass-elements, in which case we should have I-.fr2dcm (212). 300. Determination of Angular Velocity. Equation (211) furnishes a means of determining the angular velocity generated in a body by the expenditure of a given amount of energy. Thus suppose a cord, with a weight at one end, to be wound round a wheel, and to put this wheel in rotation by unwinding under the action of gravity. The angular velocity produced by the descent of the weight through a vertical height S, is to be determined. Let P be the magnitude of the weight, and I the moment of' inertia of the wheel. The energy consumedt in accelerating the wheel is PS, and the kinetic energy generated 2PS in the wheel is iw2I Hence PS - ow2J, and w = — which is the angular velocity required.

RADIUS OF GYRATION. 145 301. Moment of Inertia about any Axis. If the moment of inertia of any body is known with regard to an axis passing through the centre of gravity, the moment of inertia relatively to any other axis can be found by means of the following dem onstration. The moment of inertia with respect to any axis is equal to its moment of.inertia with respect to an axis passing through its centre of gravity plus the mass of the body into thle square of the distance betuween the two axes. Let AB, Fig. 161, be any body, G its centre of gravity, and C the point in which the axis of rotation cuts the plane of the paper, and let m be any element of mass. Call the distance CG between the two axes d, and denote the distances Gnm, Cmn, by 1, r, respectively. Then by trigonometry, ()m2: Gnm2 + UG2 + 2 G C X Gm cos CGm; or, substituting for,(n, Gnm, CG, etc., their values, and noticing that cos CGqm - cos P Gm, r2: 12 +- d2 + 2cl cos P Gm. The moment of inertia of this particle relatively to C is mr2 - m12 +md2 + 2dml cos PGnC. As similar equations hold for each elemnent of mass, we have I --'mr2 -- m12 + 2 md2 +- 2 2dml cos P G-n, or as d is a constant, I:2 mr2 = m12 -+ d2m -n 2d. ml cos P Gm. But.2 m A — 1, the mass of the body, and ml cos P Gm is the statical moment of a particle relative to the centre of gravity G, since I cos PrGm - GP, hence nml cos GPm - O. Also Zml2 is the moment of inertia relative to G, which we will call Io. Substituting these values in the preceding equation, we have I - 10 + Md2. (213.) 302. Radius of Gyration. The radius of gyration is a radius at whose extrelnity the whole mass of a rotating body may be supposed to be concentrated without altering its moment of inertia. That is, if' p be that radius, and Mthe mass of the body, Mo2 = 1; whence (p2 M and p --. (214). That is, the radius of gyration of a body is equal to the square root of the quotient arisingfrom cdividing its moment of inertia by its mass. The following table, abridged from Rankine's Applied Mechanics,l gives the moment of inertia and rad(ius of gyration of a number of bodies relative to an axis passing through the centre of gravity, 1Page 518,

146 DYNAMICS OF RIGID BODIES. TABLE. Body. Axis. Wi'eyght. o'. iShere of 1rtadliu, Diaeere. 4ite,.:tWr iwra W-32 SpheroMid of revolltion?1.- olar-se miaxis, a, equaltor ial radius,?r. Polar Axis 4 8-; waUr2 8- a4 2tar4 Circular cylitder- L ogilatlenglh 2a, radlius r. dinal Axis. 2awalr2 n warC2 I r2 Doa IHO.V.Cverc r aDo.Tri$e " D)icmeter. 21 wa.r2 I wva.'2(3r,2- + 4 a2) 4 + 3 Rectanqu!lar Prism - dliilaensions 2a, 21l, 2c. ARxis 2a., 8tcvbc w8 bc(b2 - c2) 8;2 + C") 303. Fly-Whcels. A knowledge of llhe subject of moments of inertia. is of a pl'riL at tical inllpottan nee in tlhe constoruction of oll kiinds of m::linuep- wh ovl pici possess 1rotary motions. As a single example of' the ptee ii:llJ princil)lt s, let us consider their ti)plicatioIc in tlhe ease of' flyxwlee.s. In a11 inaethinres inl wVihl tilher is any variation in the magnit(lde. either of the inpil)eihnar. ower, ori of tle iresistaice opposed, there i:ust evidetntly be a. lilttl ati i,l i:, al it. is'nereally the case in aic. ihincs that (dring sueesO;'ixve ilitievals ofint, tie lle va rliatioll in Iowe; applied ln( resistlncee off0treh l is suchl tl at tlle eliergy irceive(l is alternatcly (,reater iand less thall the work to bel Irel 0nllllted. Nrow it is evildent that when there is an excess of' eilerly Ireciveed over resistanoce overconle, this will be co0iiume id in ad(i.ing io the 1ki1etic energy of' the nachine l)y inereasinlr its velocity, and that when the enerugy al)llied is less thba.n the resistance to be overcome, t!he kinietic etergy of tile mllhine di minishes, beiu, consumed in overcolingir the xc ess of' the resistaniec offerel over the inipelling' ftirce ap:)lied at thlat ioment, t ihus slalckening tile stlC1. 1 ihis alterinate ilclrtse alndl dilinu4lon of spctl evi(olently varies according t to tlhe ail.e of the mlotive poxwver, and the work pert lrined. Ict us call V, V', the maximuim and( minimium velocities of the driving point, or thait part of the machilne ap} 1 ed to overcomie tile resistance, in whiiich case the fi ttli uaio of spioetl wvill be represe, ted by the quantity V - V'; and let VO be the mean velocity, eqlga.l to V'Ihlle quotient -- C (215), will express the ratio of thle lluctultion of velocity to.the nllean xelnoity allnd is ca;llel l the'oPe~',ieLi, o. f/ fl(tclu 1i0, of spee(/. Now if' we denote y TV'I a mass xihch, ii' itut:ted;at t l: irix ilg-point of tlhe enline. oul(d haiive the saitie actual e:l.erly as lie tii ie il:-chile, thell XllluXinii kinetiic ie-igy of IV *V2 IV V' the mass will Le - nd tile miinilnuu m so that the variltion 2' g 2z;w x weight of unit of volume.

PRESSURE UPON AXIS. 147 in kinetic energy will be Q - (216.) The mean kinetic energy is I - Vi (217), or as V~- 2V+'J g2 2 M.v(+ V'.)2 (218.) Q V- 2.... Hence, dividing (216) by (217), Q C, or substituting the 2M — V' I value of M, given in (217), C — (219) Hence the coefficient of fluctuation can evidently be ascertained by experimentally determining Q and V0, if WV is known. Now in most machinery, if there were no regulating apparatus, this coefficient woul(t be extremely great, as the variation of thle relation between the effective force and resistance would render the difference between V and V' very large; the excess of the resistance over the energy applied might be sufficient to stop the machine, anl would at any rate cause se10ous jarrcing. To obviate this, a.fqy-wlAeel can be attached to the rotating shaft of the en!ine. This is at wheel with a heavy rim, which possesses a great moment of inertia, and stores up energy when theme is an excess of driving power, and gfiveq it out to overcome the resistance when the (drivincr power diminishes. It is clear that the coefficient of fluctuationl ill in this way be creatly diminished, as V, Vr. will become more nearly cqual. In practice the fiv-wheel is made of such dimensions as to reduce this coefficient to about - fbr ordinary machinery, and 1 to ~- for delicate machinery. The dimensions of the wheel in any given case can be ascertained as follows. Let 1 be the coefficient of fluctuation which is to be obtained, n andi let the angular velocity with which the wheel is to revolve, as determined by the kind of work to be done, be denoted by,). Then equation (219) becomes - -= Q If I be the moment of inertia of' the fly-wheel necessary, and,,, the proper angullar velocity,, 2 will be its kineti:. energy, which may be taken as equal to the energy of the mass TWT, supposed to be concentrated at the driving-point. Hence,-2J'V-;V02, I qQ and 1T'Vo-o — _,li. Substitutingr this value in (219), —'-TQ (220), whence I,Q (221), from which equation I can be determined. Capt. Von Schuberszky, of the Russian engoineers, has proposed to furnish heavy fieiiht trains with lanroe fly-wheels, placed on a truck imme(diatel behlind the locomotive, which will store iup energy whiile d(escending inclines, and give it out as useful driving power when a level or ascent is reached, instead of wasting it by the use of brakes.1 30-. Pressure Upon A:Tis of Dlevolving Body. When a bodty revolves about an axis there is in genernl a pressure produced uplon that axis, due to the centitfiLgll force of the particles composing the mass. The axis may, however, be so situated See Rep. Conzmrs. to Paris Exposition., Vol. III., p. 1583.

148 DYNAMICS OF RIGID BODIES. that the centrifugal force due to any particle is just balanced by that exerted in the opposite direction by another particle, in which case there will evidently be no pressure upon the.axis. The effect of an unbalanced centrifugal force is twofold, tending (1) to shift the axis of rotation as a whole, i. e., to produce a translatory motion of the axis; and (2) to change the angular position of the axis. To undelstand this let us consider Fig. 162, which represents a body ADBB C, revolving about an axis AB. The portion ADSB, lying upon the right of the axis, evidently tends to move it in the direction _ED, anld the portion ACB, lying to the left of the axis, tends to move it in the direction ECU. But as the centrifugal force of AiDB is greater than that of A CB, owing to its greater mass, and the greater distance of most of its particles fiom AB, there will be a resultant force, tending to shift the axis as a whole to some new position in space; in the case represented in Fig. 162, to a line at the right of AB, and parallel with it. To see how the tendency to angular deviation is produced, let us notice the body A GBJ,, Fi'g. 163, which revolves about AB as an axis.:Now the centrifiugal forces of A /GDB, BHCA, act in opposite directions, and as A GDB is greater than RBHGA, there will be an unbalanced centrifuigal force in the direction E.D, as shown in the preceding case. Also since the mass of the body is not symmetrically distributed about CD, the centrifugal force of the portion A GDE minus that of A CE, can be represented by a single force F, applied at some point, as S, within A GDE, while the centrifugal force of (HB-E minus that of D)EB, may be represented by a single force F' applied at T. These two forces evidently act as a couple (centrifzygal couple) to produce rotation about an axis at iight-angles to the plane of AB, and hence tend to produce aIn aangular deviation of that axis. 305, Free Axes and Principal Axes. A consideration of Figs. 162, 163, will show th;at if the axis of revolution passes through the centre of gravity of the body AIDBCU, the tendency to shifting it as a whole disappears, since the mass, and consequently the centrifugal force, is equally distributed on either side of the axis. That there may be no tendency to deviation of the axis, in which case the centrifugal couple must reduce to zero, the axis AB must also be an axis of symmetry of the body, since in this case the forces F, ~F', will be directly opposed, as shown in Fig. 164. It is evident that if the axis of revolution, AB4/, is parallel to an axis of symmetry, without passing through the centre of gravity, as shown in Fig. 162, there will be no tendency to angulfar deviation, though there is a tendency to translation of the axis. Any axis about which a body may revolve without causing any lendency either to shift the axis as a whole, or to produce angular

AXIS OF STABLE ROTATION. 149 deviation, is called a free axis or permanent axis, because if the bodly is perfectly free to move, the axis will not change its position. It is proved by analysis that every body, or system of bodies, has at least three free axes, which are at right-alngles to each other, and intersect at the centre of gravity of the body or system. Any axis about which a body may revolve without causing any tendency to an angular deviation of the axis is called a principal axis. For each point of any body, or system of bodies, there are at least three principal axes at riglht-angles to each other. For points on either of the free axes, the principal axes are parallel to the fiee axes. The latter are evidently the principal axes passing through the centre of gravity. It can be shown that the moment of inertia of the body or system, relatively to one of its principal axes is greater, and relatively to another is less, than relatively to any other axis through the point considered. In an ellipsoid having three unequal axes, the fiee axes are the three axes of that solid; in a right elliptical cylinder the axis of the cylinder, and the major and minor axes of the elliptical section of the cylinder at its middle point, are free axes; any diameter of a sphere is a free axis, also any diameter of the equator of an oblate or.prolate speroid, together with its polar axis. 306. Practical Applications. In any rotating piece of machinery it is desirable that the resultant centrifugal force be reduced as much as possible, since the friction and strain of the machine are increased thereby, while the continual chaunge in the direction of the centrifugal force as the moving body rotates, may give rise to dangerous jarring of the machine. Hence every rapid rotatory motion should, if possible, have its axis coincident with the free axis of' the rotating mass. The practical application of this is illustrated in the driving-wheels of locomotives. As the heavy cranks to which the connecting-rod is attache(d, would, if unbalanced, produce serious jolting of the locomotive as they revolve, the space between the opposite spokes of the wheel is filled up solid, that the axis of revolution may be as nearly as possible a free axis. 307. Axis of Stable Rotation. Although a body revolving accurately about any one of its free axes has no tendency to deviate from it, yet it is only in a condition of stable rotation when that axis is the one relatively to which the moment of inertia of the body is the greaitest, which is, in general, the slhortest axis. It is only in this case that it will return to its original axis if slightly displaced fiom it. To show this, suppose CllDE, Fig 165, to be a body rotating about (JD, the longcest of its free axes. So long as the position of the body is undisturbed, the revolution will continue unchanged, but if it is slig-htly displaced in any way, so as to assume the position C'I'D)'E', Fig. 166, the rotation continuing about AB will generate a centrifugal couple P - FI', which tends to deviate the body still more from its original position, and

15.0 DYNAMICS OF RIGID BODIES. which Nwill (1isappear only when the shorter a:xis, E'fT', coinlcides with tile <xis of rot atioln. Hence the body is in stable equilibrillm only when it is rotating aIbout its shortest diameter, i.e., about that one of its fiee axes witlh iegalrd to which the moment of ineltia is the greatest possible. Tllus a rillcr rotates in stable equilibliun only when revolving about an -axis perlpendcicilar to its plane; an ellipsoid of three unequal axes only when revolving about its shortest ditameter, and tlan oblate sp)leroil when revolvinig about its polar axis. 308. Case of the Planets. It is worthyl of notice that the oblate spheroil having its polar axis for an axis of rotation, the form naturally assumed by the planets, is one of the very few cases in which the axis of rotation is permanetl nt. For in case of any external disturbance, producing.a temlpolrary swervingr of the body, the ten(lency of the centrifigal force is to cause it to return to its original axis. Such perturbations,are const;antlt acting upon every planet, and were not the variations caused by thell colnfinled within very narrow lilits, they would lead to a continual change in the axis, and hence to a continual variation in the seasons, and in the distribution of' land and water. Were the eartlh a prolate spheroid, for example, there would be no power to prevent the passingr of the rotation-axis firomn on-.position to another, on the occurrence of any perturbation, and anll- o-,;clilhana/e, if once introduced, would go on forever, 309. Elxperimental Illustrations. The stable and unsta:ble equilibrium of rotattiln bodies can be illustrated experimentally witlh the apparatus represented in Fig,. 167. AB is a body suspended by a corld, by means of which it can be made to revolve very rapidly. If' it be so lulng as to rotate labout any axis but its shortest one, the sliglht disturbance produced by the swavying, of the string will displace it a little, and the bodly will then move until it assumes the position A'B'. A prolate spheroid, rotating about its longest axis, or a rinr rotating about one of its diamletens, shows this very clearly. An oblate spheroid made to revolve about one of its equatorial axes, will rise until it revolves about its polar axis, but if made to revolve originally about'its polar axis, the equilibrium is seen to be stable. 310. Composition of Rotations. When a body is affected with two or Inore rotations about inclined axes, these rotations may be coinbined precisely as we comnbine couples. Thlus suppose a bo(ly to be acted upon by two forces which sepa-rately would ca, use it to rotate about the axes AB, AC, Fig. 168, with angular-velocities a, a', respectively. To find the resultant axis of rotation, we ma'r consider the rotatin~ forces as f'rmllgr two couples with incliined axes, and determine the mianitude and direction of the resultant couple, which will represent the resultant iotatilon iiin nagnitu(le and direciionl. Hence we simply lay off on AB a distance A b, proportional to a, and on A C a distance Ac lroportionall to a', and complelte the pLarallelogram AbdIc, the diagonal of' whichl, Ad, repreellents the magnitude of the resultant rotation, and( is also its axis. It is clear that in the sanme manner we nimar combine three or more rotations about axes lying in the same, or in- dilfl'enret planes. 311. Gyroscope. Thle phenomnena of the composition of rotations are excellently illustrated by mleans of the gqroscope. This instrument consists of a heavy, well balanced wheel, AB, Fig. 169, revolving with as

GYROSCOPE. 151 litt!e friction as possible about an axis DC. This axis is liheld in a ring, KH, to wrhi(h is attached a projecting rod 1EF, with a pivot E, Awhich llmay be placed in a socket G. If a rapid rotation be now communicated to the wheel AB by unwin(lino a cord fiiom the axis CD, and the pivot E be rested in the socket G, with the axis horizontal, the instrunment awill not fall fi'om the support as inight be expected, but Aill remain niearly horizontal, and rotate slowly about E, in the (lireotion indicated by the arrow. To exlplain this action, let us consider the motions with which the wheel is affected. Suppose tile axis of revolution to be so placed as to coincide with OX, Fi(r. 170. The wheel has a rotation about CD, Fio. 169, dule to the original impulse. But as soon as E is placed in G. the weioht of the apparatus, whose centre of gravity is then unsupported, oives it a tendency to fall, thus rotatino about the point E, and in the plane of CD; that is, about an axis O Y, Fig. 1 70, at rirght-angles to OX. Let the angular velocity of the rotation about OX be represented by Oa, that about OY by Oh.'rhe direction ntl mtionitude of the resultant rotation will be represented by Oc; hence this resultant motion must take place about the axis OX', and the metallic axis of the gyroscope (CD, Fi,. 169) will move so as to approach this position, thus assumingr a retroorade motion. But as the axis about which the rotation due to gravity takes place is always at riglhtangles to thle axis of revolution of the wheel, OY will recede as the wheel IIIoves towards it, and hence a continuous retroorade revolution will take plxce about OZ. But the preceding construction is complete only for the first instant of the motion. For as soon as the movement about OZ has beguin, there are evidently thtle simultaneous rotations to be compounded instead of two. To ascertain the effect of these, let us consider Fig. 1 7 1, in whaich Oc repre~ sents the rotation about the axis OX', Ob that about 0Y, and Oe that about OZ. The resultant rotation will be represented by Of, the diagonal of the parallelopiped Ofce, of which Oc, O/, Oe, are adjlacent edges, Hence the instrument will move so that its axis may assume this position, But as this movement takes place OY1 evidently recedes, and so the orbital, rotation is continuous. Also since Qf is inclined above the horizontal plane X'OY, the axis of the gyroscope tends to elevate itself: As Of is greater than either Oc or Ob, the angular orbital velocity would continually increase, were it not counterbalanced by fiiction and the resistance of the air. The effect of the orbital motion is to elevate the axis of the gyroscope, that of gravity to (lepress it; hence the axis rises or falls, as one or the other of these predomlinates, which latter circumstance is determined by the angular velocity of the wheel, andl the position of the centre of gravity of the total mass of the gyroscope. If' the instrument is balanced about E. Fig. 169, by suspending a weight froin F, so that the centre of gravity is over the pivot E, there is no tendenev to rotation about O Y, and hence no orbital revolution. If the weight at /l is so great as to bring the centre of gravity of the instrument to the left of' E the gyroscope tends to rise, and the orbital motion becomes dth-ect. The same effect occurs if the rotation of' the whileel be reversed. If the gyroscope be held in the hands and made to rotate rapidly, on attempting to incline it in one or another direction, a strolig resistance to such change will be experienced by the hand. The gyroscope also offers an experimental illustration of the astronomical phenomena of the constant parallelism of the earth's axis to itself; the precession of the equinoxes and nutation.

152 DYNAMICS OF RIGID BODIES. 312. D'Alemlbert's Principle. When a rigid body is caused to move under the action of any external force, it generally occurs that some of its particles move slower, and some faster tlhan they would move if there were no rigid connection between them. Suppose, for example, that we have two balls, one of gold 1lnd the other of' some light substance, as pith. We shall see hereafter that these bodies would fall through a vacuum with equal velocities, but that the resistance of the atmosphere retards the pith more than the gold ball, so that in the air the descent of the latter will be more rapid than that of the former. Sutppose now that the two are firmly connected. It is clear that they must now both fall with the same velocity, and that the gold ball moves less rapidly than if fiee. A portion of the impelling force impressed on the gold ball is not expressed in its momentum, while the momentum expressed in the pith is in excess of that impressed upon it, this difference arising from the connection of the bodies. Now the total expressed force must be equal to the total impressed force, as no energy can be lost; hence the pith ball muLst gain in momentum exactly as much as the gold ball loses. The same law evidently holds for all cases of motion in rigid bodies, whose particles tend to move with different velocities. Hence, in general, the resultant of the impressed forces must be equal to the resultant of the expressed forces. This theorem is known as _D'Alembert's Principle, from the mathematician who first enunciated it before the Academy of Sciences of Paris, in 1742. 313. Angular Acceleration of Body. LetAD), Fig. 172, be a body capable of moving about an axis through C, at right-angles to the plane of the paper, and suppose it to be acted upon by any force T7, applied at a distance B C - R fiom U. It is required to find the angular acceleration which the body assumes. Denote the angular acceleration, that is, the angular velocity generated in 1 second by w, and consider first the case of a single particle P', of mass m, at a distance r fiom C(. The velocity of P is rw, and its momentum mnrw, The moment of this momentum relatively to C is mrw X r = mr2%. For other particles, in', mn", mn at distances r', r"', r, fiom the axis, similar expressions, m'r'2%, m"r"20), mn rn2, would be obtained, and the sunm of these, or ynmr2w is evidently the moment of the mornentum of the whole body relatively to C. Now the force generating this momentum is the moment of' F relatively to C, that is, F X N, and as the total impressed force acting upon AD must equal1 the total force expressed in its motion, according to D)'Alembert's Principle, we have PFR Z= mr2w, or _FIBR woinmr2 (222), as w is a constant, whence aw - 2 mr- 2 (223), or denoting 27mr2, which is the moment of inertia of' AD relatively to U, by I - -. (224.)

CENTRE OF OSCILLATION. 153 If the force generating the motion is the weight of the body, acting through G, its centre of gravity, the impressed force will be TV X G C WRO0, whence in the case suplposed, - = l (225). 314. Centre of Oscillation. The angular velocity which would, in the latter case, be assured by any one of the particles of which the body AD) is composed, would be found from the equation w P-, in which o)r is the angular acceleration " which the particle tends to assume, w, the weight of the particle, rr its distance fiom C, and i its momnent of inertia relatively to the axis of rotation. As P - mr2p, p r- Y (226), mr r (226), whence it appears that the particles mnost distant fiom n tend to rotate more slowly than those nearer that point. The angular acceleration, w, of the whole body, will thereftre be greater than that which would be assumed by the partieles farthest firom C, were there no rigid connection, and less than that which would be assumed by the nearer particles. Hence there iwill be a point somewhere upon the line B C, which will possess the same angular acceleration as if it were not rigidly connected with the rest of the body. Let 0 be the position of that point. We wish to determine its distance, CO - 1, fiom the axis through C, upon which it is suspended. The angular acceleration of' 0, were it free, would be + (Eq. 226). This must equal %w, the angular acceleration of the body AD. That is, - or as ]Wg,' -- g, whence 1 - I (227). The point 0 is therefore at a ciistance from C ejqual to the moment of inertict of the body relatively to the axcis throtugh that point, dividced by the stcatical 2moment of' the weiyght relatively to the scame axis. The point 0 is catlled the centre of oscillation of the suspended body AD. In case the body vibrates under the influence of its weight, it perfbrms its oscillations in the sanme time as if th3e whole mass were concentrated at 0, since the angular accelerations in the two cases would always be the same. 315. Convertibility of Cen'tres of Suspension and Oscillation. A proposition of great importance is the followingg. The centres of suspension and oscillation, are mutually convertible. If 0 is the centre of oscillation of a body suspended at C6, and 20

154 DYNAMICS OF RIGID RODIES.'vibrating under the influence of its weight, then if the body be suspended from O its new centre of oscillation will be at C. To demonstrate this, let 10 be the nloment of inertia of AD, Fig. 172, relatively to its centre of gr:vity, and I' relatively to the axis of suspension through C. Then denoting the distance from C to G by d, that from G to O by d', we have from Eq. (213), p. 145, I'-I- + Md2. But CO 1 - I 10+Md2'0 ~d ~ -d d + /e- (228.) Suppose now that the body were suspended at 0. Call I' the distance from the new axis of suspension 0, to the new centre of oscillation, and lr" the moment of inertia of the body relatively to the axis through 0. Then I' = I''+ Md' I( llld i~' dl' + -1 — 4 (229). Now from (228), as I=- d d',d+d' d+ whence dd' - and d Substituting this value in (229) we have'- d' + d = 1 (230), whence it follows that the new centre of oscillation is at C, the former point of suspension. 31 6. Centre of Percussion.. The centre of oscillation can also be shown to be the centre of percussion, which is the point at which a body suspended from an axis may be struck by a blow in its plane of rotation without producing any pressure upon the axis; and, conversely, the point at which a body moving about an axis must strike an obstacle, that there may be no blow comIunicated to the axis. A knowledge of the position of this point is practically applied by experienced batters, who learn by practice to strike the ball with that part of the bat in which the centre of' oscillation of the system formed by the movina arm and bat, is situated. In this case, the whole force of the blow is spent in overcoming the motion of the ball. and no jar is sustained by the hand, otherwise an unpleasant stinging sensation is produced. The centres of oscillation and percussion of a straighlt rod suspended at one extremity lie in the axis of the rod at a distance from the point of suspension equal to two-thirds the length. Hence if a stick held in the hand, and swun(r about the wrist, strikes an obstacle at that point, no blow will be felt by the hand, and at no other point will this be the case. 317. Axis of Spontaneous Rotation. If a body be impelled by a blow acting at its centre of gravity, it can be shown that the motion which it receives will be one of translation only; hut. if it be struck at any other point, the body will become affected with a double motion of translalation and rotation, receiving the same motion of translation as if the blow had been delivered at its centre of gravity, and the same rotation as if' the body had been suspended on an axis through the centre of gravity. From this it follows that if an impulse be communicated to a free body, the combined translation and rotation will cause its motion to be the same as if it revolved about some fixed axis, this axis changing, however, from moment

AXIS OF SPONTANEOUS ROTATION. 155 to moment. The reason of this will be seen if it is rememnbered that the translatory motion is equally partaken of by all the particles, while the rotation causes some particles to move backwards, and others to move forwards, relatively to the direction of the translation, the rotatory velocity of any one particle being greater in proportion to its d(istance from the centre of gravity. If the backward motion of certain particles due to the rotation exceeds their forward motion due to the translation, they will evidently move backwards in space, and as the remaining particles move forward, there will be same point at which the backward motion ceases, and the forward begins. The resultant motion of' the body is therefore the same as if it revolved about an axis passing through this point. The position of this axis changes as the body moves, because the position of tile centre of gravity chan(es. Thtat axis about which the body tends to revolve at the first instant of its motion, is called the axis of spontanenzs rotation, and the successive axes about which such a body seems to rotate are called instantaneous axres. The position of the axis of spontaneous rotation cvidlently depends upon the point at which the body is struck, and it can be demlonstrated that the axis of suspension corresponding to a given centre of oscillation is the axis of spontaneous rotation for a body struck at the latter point. Hence if a fiee body be struck at any point, it begins to move as if rotating about an axis so situated that if the body were suspended from that point, the point struck would be the corresponding centre of oscillation. It will be seen that this fact affords an explanation of the principle already explained, that the centres of percussion andl oscillation are identical, for if a blow be struck at the centre of' oscillation, the axis of suspension will be the, axis of spontaneous rotation, and hence no shock will be sustained by it. REFERENCES. Treatise on Natural Philosphy, by Thoimson and Tait. Chapter on Work. Treatise on Infinitesimal Calculus, by Bartholomew Price; Vol. Iv,. Dynamics of Rigid Bodies. (Oxford, 1872. ) Mlanual of Applied Mechanics, by W. J. M. Rankine; (4th Ed., London, 1868) Part v., Dynamics. Marcnual of Machinery and Mill-work, by W. J. M. Rankine. (London, 1869.) Part Ir., Dynamics of MIachinery. Manual qf the Mechanics of Engineering and of the Construction of Machinzes, by Julius Weisbach. Translated by E. B. Coxe. Vol. i., Theoretical Mechanics, Section v., Dynamics of Rigid Bodies, Chap. I., Ir. (New York, 1870.) Correlation and Conservation of Forces. Essays by Grove, Helmhroltx, Mayer, Liebig and Carpenter. Edited by E. L. Youmans. (New York, 1865.)

156 GRAVITATION. CHAPTER XIV. GRAVITATION. 318. Law of Universal Gravitation. The force of gravity or weight, to which we have frequently had occasion to refer in preceding chapters, is only a particular case of the general law of universal gravitation, which we now proceed to consid(er It is firequently known as NArewton's Law, firom its discoverer, and may be stated as follows: Every /particle qf nmatter'in the universe attracts every other particle with a.force varying directly in the compound ratio of t/heir 9mnasses, ancd inversely as the square of their distance. It follo ws from this that if'we represent by V the attraction of a unit of mass upon a unit of mass at a unit of distance, the mutual attraction, G, of two bodies of masses m, m', at a distance d is expressed by the formula = 4G -/- (231). For if ~, be the attraction of one unit of mass upon another at a distance unity, the attraction of a unit of mass upon a mass m is,rm,. and the attraction of a mass m' 1upon m is V'mm'. If the distance instead of being unity is equal to cd, since gravitation varies inversely as the square of the distance, the attraction becomes - cl319 l. Method of Proof of Law. This proposition was first enunciated by Newton, who showed it to be a direct consequence of the principles known as the Three Lazos of Kiepler, which that astronomer had established by astronomical observation, These are, I. All the planets move in elliptical orbits, of which the sun is in one focus. II. The radius vector of any planet describes equal areas in equal times. III. The squares of the times of revolution of the planets are proportional to the cubes of their mean distances from the sun. The following is an outline of' the course of reasoning followed by Newton. In the filst place the proposition demonstrated in ~ 270, p. 131, shows thlat since the radius -vector of a )planet describes equal areas in equal times, there is a deflecting force, which lmust pass througrh the centre of the sun, which last fact is also a conseqolence of the law of gr:avitation. For analysis proves that the law being true, the srum of the attranctions exerted upon any external body by the particles of matter of a sphere (and approximately of a spheroid of small eccentricity), composed of concentric holuogeneous layers, is the same as if the whole mass of the solid were

PROOF OF LAW. 157 concentrated at its centre, to which centre, therefore, any body subject to the attractive influence of the sphere would be drawn. The law of the variation of the force in the inverse ratio of the square of the distance, follows fionom the ellipticity of the orbits, as shown in thle proposition demonstrated in ~, 273, p. 132. That peortion of the law relatinfg to the mass of the body also follows fiom the law of centripetal force in an ellilptical orbit. To show more clearly the miathemaitical reasoning employed, let us demonstrate the law in the simnlest case, thalt o)f a planet mnoving inl a circular orbit. As tlme celitlipetal or deflecting fnr(ce in the cnase of the planet is the attl-raction of the su;n, if we denote by G,G', the attractions exerted by the sun upon two planets at 4~2d distances d, d', from its centre, are by ~ 275,'p. 133, G = 31-, Gz' =f ]r2 --- from which it follows that CG: GC':: I2 _'2 / ~ But by IKepler's 3d Law,:T2: T'2:: ci3: d'3. Hence 3 _jF': -' d-'T2 (232), a proportion expressing Newton's Law. A similar dernonstration can also be given in the case of the ellipse, parabola or hyperbola, so that under the influence of gravitation a body may mnove n eitlier of these curves. Newton next verified the law of inverse squares in the case of the earth and moon, showing also by his methodl of proof thaClt the deflecting force acting on the planets is the same in nature as that which causes a body to fill to the ground. Let Ai, Fig. 172, be a portion of the mloons path, which it describes in 1 minute. Were it not for the deflecting action of its gravitation to the earth, it would move over the line AB in the same time. BEF, which is sensibly equal to -DR for smanll arcs, is therefore the amount of its (deflection while traversing A-R, that is, thle amount by which it falls towards the earth in one.minute. Knowing the diniensions of the rmoon's orbit, and( the velocity of describing it, this distance c.an easily be computed, and compared with the space through vwhich a body, at a distance fiom the centre of the earth equal to the rnoon's distance, would fill, assulling the l:aw to be true. Thle deflection of the arc A-R, fiom the tangrent AB, is found to be 4.9 m. The rmoon therefore fills to the earth 4.9 n. in 1 miniute. At the surface of the earth the space traversed by a falling body in I minute is 4.9 X 3600 in. Siulce forces are ploportional to their accelerations, if we call G, G', the force of terrestrial gravitation at the earth's surface, and at

158 GRAVITATION. the moon, and d, d', the radii of the earth and of the moon's orbit, G: G':: 4.9 X 3600: 4.9:: 3600 1. But d' d:: 60: 1. 11 Hence: GC'::d'2:d2, or G: G':: 2: -, 2 (233), that is, the force of gravity at the two points is inversely as the square of the corresponding distance from the earth's centre. Froin the law of gravitation, in connection with the three laws of mnotion, we can deduce the laws of Kepler, and also compute nmathematically the amount of the perturbations of the various bodies composing the planetary system. It should be carefully borne in mind that the attraction of the earth is not a single force, but the resultant of the separate attractions of each of the particles of which it is composed. That attraction is exerted by every particle, is shown, among other ways, by the deviation of a plumb-line from the vertical when near the side of a mountain. The irregularities on the earth's surface are, however, so small in proportion to the magnitude of the whole globe, that they may generally be neglected in astronomical calculations. 320. Attraction upon a Body situated on the Sur. face of a Sphere. The principles stated in the preceding paragraphs, furnish a method of ascertaining the relative attraction of two spheres upon a body of given mass situated at their surface. Let A, B, be two spheres of equal density, with radii 2B, BR', respectively. Since the total mass of each may be imagined to be concentrated at its centre, calling G, G', their attractions, M, M', their masses, we have, G: G' 2: ~.. But since the masses of the spheres are proportional to the cubes of their radii, 31': MI:: _:I B'3. Hence T Gh i's, t12 aTt or G: G':: R: _R' (234). That is, the attractions of spheres of equal density on a body situated upon their surface are proportional to the radii of the spheres. If the densities vary, the masses of the spheres are proportional to the volumne multiplied by the density, that is, M: 31':: Vd: V'd': d3cl: Ru3d'. Whence Bsd BR'3d' G: G': 2 R'2:: Rd: R'd'. (235). To show the application of this proposition, suppose that it is required to find the relative weight of a body at the surface of the earth, and at the surface of the sun. The mean radius of the earth is, in round numbers, 6377 km., that of the sun 709,700 km. The density of the sun is but one-fourth that of the earth. Hence from (235) denoting by G, the weight of a body at the earth's surface,

GRAVITY BELOW SURFACE OF SPHERE. 159 and by G', its weicht at the surface of the sun, G: G':: 6377 X 1: 709,700 X:: 1: 27.8, whllence G' = 27.8 G. A body weih'hing one kilogramme on the earth's surface would therefore weigh 27.8 kgrs. if transported to the sun. As forces are proportional to their accelerations, a body falling to the sun, when near its surface, would acquire an acceleration of 9.8087 X 27.8 m., or 272.68 m. per second. 321. Diminution of Gravity above Surface of Sphere. Since the resultant effect of the attraction of all the particles of matter composing a sphere upon a body outside of it, is the same as if the whole mass were concentrated at the centre of the sphere, it tollows that the attraction exerted upon any body varies inversely as the square of its distance fiom that centre. Hence if we call G, G', the attraction upon the body at the surface of the sphere, and at a distance h from the surface, G: G':: i 1: (R -+ h)2 (235), whence G' = G ( + h)2 (236), or performing the division, we have approximately G' G(1 - )(237). This formula can evidently be used to determine the weight of a body when elevated above the surface of the earth. Thus a body weighing 1 kgr. at the surface, if carried in a balloon to the height of 5 kin. above the surface would have for its weight G' =1(1 -- r37).998 kgrs. The acceleration g', which a falling body would acquire at any given elevation, may also be found from proportion (235). For 1 1 12 G C! g: g' 2 (R2+ t-h)2' whence g' =-g(R h)2 (238); approximately, g'=g(1 - )(239). Also g g' (i + i (240) approximately. 322. Gravity within Hollow Sphere. A body placed anywhere in the interior of a spherical shell of uniform thickness and density, will be }qually attracted in all directions. Let P be a body placed at any point within the spherical shell AB -(ab, and subject only to the attraction of the matter composing that shell, and let ab be an element of the surface. The lines of attraction of the particles composingr ab will be comprised within a cone aPb, whose base is the element ab, and whose vertex is P and their resultant will be in the axis of the cone. The elements of Pab, if prolonged throllgh P, will form another cone, PAB, having a vertex P and base AB. The lines of attraction of AB will evidently lie within the cone PA-B, and their resultant will coincide with its axis, being therefore directly opposed to the resultant of the attraction of ab. Hence ab AoB Attraction'of ab: Attraction of AB'.pZ:.pAB., Pa 2

160 GRAVITATION. )But ab: AB:: Pa'2:PA2. Hence Pa 2 PA2 Attraction of ab': Attraction of AB.:: Pa2 1:l 1. (241.) The body P is therefore equally attracted towards ab and towards AB. A like demonstration can be applied to every other element, and to any number of concentric shells; hence the body is in equilibrium. 323, Gravity below Surface of Sphere. Suppose a body to be situated at any point B, below the surface of a sphere AiFE, Fig. 174. Denote AC by B, B C by B'. The resultant effect of that portion of the sphere outside of BfHD is zero, by the preceding proposition. The whole resultant attraction at B is therefore that of the sphere BHD of radius R'. But the attraction when at A is that of the sphere AFE, hence calling G, G', the attractions at A and B, respectively, we have fiom (235), G G::: d B'd'c', or as the densities, d, d', are the same, G ~C':: B -: B'. (242.) Hence the gravity of a body sittated within a solid sphere is directly proportional to its cdistance from the centre. This proposition can not be applied to determining the weight of a body below the surface of the earth, as the density varies with the depth. The law in this case will be explained in treating of the mass of the earth. 324. Eifect of Spheroidal Form of Earth on Gravity. Since the earth is not perfectly spherical, but flattened at thle poles, it is evident that its attraction upon a body will not be the same at every portion of the surface. Analysis shows that gravity increases from the equator to the poles. The loss of weight, due to the want of sphericity of the earth, in the case of a blody carried from either pole to the equator is -1 part of the total amount. 325. Effnect of Centrifugal Force. The actual change of weight, as computed from the results of experiments, is much greater, beino T- part of the whole weight. This is owing to the flct that besides the diminution explained in the preceding paragraph, there is another and greater cause of decrease, owing to the centrifugal force due to the earth's rotation, one component of which is always opposed to the weight of terrestrial objects, and which increases as the latitude decreases. An approximate formula, showing the effect of the centrifugal force on the weight can be demonstrated friom Fig. 175. Let the centrifugal force at R., the equator, be denoted by F, that at E, in latitude 0, by F'. Then F' - F cos 0, as shown in ~ 283, p. 137. Let EG represeut the centrifugal force F' at E. Resolving this into two components, one EH, perpendicular to the surface of the sphere, the other EK, tangential to it, EH

HISTORICAL SKETCH. 161 EG cos GEH --- EG cos 0, whence if the component represented by Eli be called F", F" - F' cos 0 F cos2 0 (243), which represents the loss of weight due to the centrifugal force when F is the loss due to that force at the equator. This demonstration evidently supposes the earth to be spherical. 326. Corrected value of g. It can be shown that if g be the acceleration due to gravity in latitude 45~0, and at the level of' the sea, the acceleration, y', in any other latitude, L, at the level of the sea, will be g' g (1 - 0.002552 cos 2L). (244.) For any elevation, h, above the sealevel, the formula becoles g' _ g (1- 0.0025. 2 cos 2L) (1-P ), (245.) This value of g' is that which would occur in mid-air, as in the case of a bodlv let fall from a balloon. In the case of' pnountain summits, or tablelands, a somewhat different formula is generally used.l The existence of local disturbances renders it imn ossible to derive any general formula which will give the exact value of g for any particular'place. Hence this can only be determined by direct observation by methods to be detailed shortly. 327. Historical Sketch. The way to the discovery of the law of gravitation was opened by the establishmlent of thbe fact that the sun is the centre of the planetary system, as tauglht by Copernicus,2 whose work, De Revolutionibtts Orbirsn C(elestiume, was publisled in the year of his death, 1543, although he had promulgated his doctrines more or less extensively some years before. Dturing the latter part of the 16th, and early part of the 17th centuries, Kepler3 discoverel the three laws bearing his name. The first and second laws were made public in his work, Onl the AIotions of Mlars, published in 1609, the third in the Harmoonice Mundi, published in 1619, though the third law was surmised long before either of the others. Shortly after, Galileo 4 discovered the laws of monloentum, and by observations with the newly invented telescope, added fresh1 confirmation to the truth of the Copernican System. Fromn the three laws of Kepler, ip connection wvith tihe laws of momentumrn; Newton 5 established the law of' universal gravitation, as already described, during the years between 1666 and 1687. The various facts established by Newton in completing his theory are ap follows:6 - 1. The force by which the diftlrent planets are attracte4 to the sun is in the inverse proportion to the squares of their distances. This was shown to result from the third Law of h}epler. 2. The force by which the sampe planet is attractcd to th.e sun in different parts of its orbit is also in the inverse proportion to the square of the distances. This proposition follows from the first and second laws of Kepler, as shown in ~ 273. 3. The earth also exerts such a force on the nloon, aid this force is identical with the force of gravity. (See ~ B19.) 1 See Chapter on Pendulum. 2 Born at Thorn, Prussia, in 1473; died., 1543. 3 Born at Magstatt, Wiirtemberg, 1571; died, 1630, 4 Born at Pisa, in 1564; died, 1642. 5 Born at Woolsthorpe, 1642; died, 1727. 6 Whewell's Ilistory oJ the Inductive Sciences, Vol. I, p. 399. (Appletonus Edition, New York, 1865.) 21

162 LAWS OF FALLING BODIES. 4. Bodies act thus on other bodies, besides those which revolve around them; thus, the sun exerts such a force on the moon and satellites, and the planets exert such forces on one another. 5. The force thus exerted by the general masses of the sun, earth and planets, arises from the attraction of each particle of these masses; which attraction fbllows the above law, and belolgs to all matter alike. The last two propositions were demonstrated fiom various astronomical phlenomena by the most acute mathematica.l analysis, andl Newton frequently found it necessary to invent new methods for the solution of the particular problems whlich constantly arose in the course of his research. Tile theory of gravitation lhas been greatly extended since the time of its originator, so as to explain quantitatively the various planetary perturbations, and other problems of astronomy, by the labors of Laplace, Lagrange, D'Alembert, Clairaut, and nmany other scarcely less celebrated scientists. REFERENCES. Philosophice Naturalis Principia _M3athematica, by Sir Isaac Newton. Treatise on Infinitesimal Calculus, by Bartholomew Price, Vol. III. Treatise on Analytical Statics, by Isaac Todhunter. (London: Macmillan & Co.) Mlathematical Prionciples of M[Ueclhanical Philosophy, by John Henry Pratt. (Cambridge, 1836.) FIistory of the Inductive Sciences, by Wnm. Whewell, Books v., vi., vII. History of Physical Astronomy, by Robert Grant; (London: Henry G. Bohn.) Chapters I - xiii. Iawcs of Fcaling Bodies. 328. Laws. We now pass to the consideration of the laws governing the fall of bodies thrlllough the air. We shall, for the present, consider the distance flllen throngh to be so small that the force of gravity throughout that distance may be considered as uniform. 1. The velocity of a falling body is independent of its mass, whence all bodies fall with the same velocity. Since gravity is directly proportional to the mass, if G, G', be the forces acting on two bodies whose masses are M, M'1, respectively, we have, G: G' M l: J2'. Also, the momenta generated by these fbrces in equal times, are prolpotional to the forces themselves; helce, calling V, V', the velocities acquired by M, l11', in equal tinles, G: G':: MV: Ifl' V', whence 31: 1':M: if V', and V V'. This law was unknown until the time of Galileo, earlier philosophers havAing supposed that bodies fell with velocities proportional to their masses. Galileo, however, reasoned that the aggregate force acting upon a number of equal separate fa]ling bodies, a collection of balls, for instance, would neither be increased nor diminished by uniting themn in a single mass. so that each particle of' such a body would fall with the same rapidity as if it were free. He verified his reasoning in an experiment performed about

INCLINED PLANE. 163 1590, in which he dropped simultaneously balls of different material and -weigtht from the summit of the Leanincg Tower of Pisa, the balls being found to strike the ground at the same instant. 2. The velocity is independent of the material of which the falling body is composed. This follows directly from the flact that gravitation depends simply upon the mass and distance of bodies, and in no way on their composition. Experience seems in a measure to contradict these two laws. Thus, a ba.ll of mletal falls more rapidlly than a sheet of' paper. This arises, however, firomn the resistance of the air, which buoys up the extended surfitce of the paper, while it offers comparatively little opposition to the passa.ge of the, ball. If this disturbing force be r emovedl, the two fiall with equal velocities. This is illustrated by an experinment originated by Newton. A long tube' firon whlich the air can be exhausted, contains a coin and a fbeather. On hol(ling the tube vertically these are seen to fall, the coin very rapidly, the feather very slowly. If the air is now removed from the tube by means of an air-pumlp, and the experiment repeated, the two will fall equally fast. 3. The' velocity is proportional to the time of descent. 4. T]he velocity is proportional t the sqzare root qof the distance fallen:through. 5. Thie spaces traversed by a fallling bocld are prooportional to the squares of the times occupied inz cdescribing them. The truth of the last three laws may be inferred fronm the fact that falling bodies, being acted on by gravity, a constant force, their motion must be uniformly accelerated. We have next to explain the means by which these laws are verified. 329. Inclined Plane. The simplest method of demonstration is the one originally used by Galileo, and explained by him in his Dialogues on Motion. Since bodies rolling ldown an inclined plane are acted upon by a constant accelerating force, ~their motion is uniformly accelerated, and as a.- q 1 (p. 125), the less the height in proportion to the length, the less will be the acceleration, so that by giving a very small inclination to the plan:e, the body may be made to roll with sufficient slowness to measure the spaces traversed in successive seconds, which is dificuilt in thle case of a body filfling freely. It is merely necessary to fix a scale of equal parts upon the plane (Fig. 176), nln d letting a ball start from its sumimit, to note the number of divisions thait it passes over in 1, 2, 3, etc., seconds. They are founld to be;lr to eanch other the relation of' 1, 4, 9, etc., the squares of the times, which involves all the laws of uniformly accelerated motion. 2It As s- 2-gt j- (p. 126), we can substitute for s, t, 11,, their

164 LAWS OF FALLING BODIES. values, as thus measuredl, and by solving the equation relatively to g, obtain the value of the acceleraetioll of gravity. The firiction of the ball, however, renders the value as thus determinecld, somewhat too small. 330. Attwood9s Machine. Another method of verifying these laws is by the spparatus known as Attwood's /Xcchine, from the name of thie invertoi, Its principle is very simple. Suppose two equal weights, P, P, to be suspended from the opposite ends of a thread passinti over a pulley A, Fig. 177. These will relma.in in equilibrium so long as undisturbed, but if a small additional weight, p, be plaeed tpon otie of them, it will descend with an acceleraf;iotn dlependent upon the magnitude of the weights P P, p, To find this aceelelatiorn we must observe that were p to fall freely, the velocity generated in one second would be g. But in the apparatus the f1ll of p puts the weights P, P, and the pulley A in motion, as well as the imasls of' p itself. Hence as the the momnenhta generated in one second must be equMal in the two cases, if we neglect the effect of the weight of the pulley A, we have fiom proportion (5), p. 34, a::: 2P + p, whence a - g2pq_ +p (246), If, Ithelrefore, w-^e malke p very small, as conpared with 2P, the motion will be so slow that the spaces passed overl in successive' seconds canh be read on a scale placed behind the fallifng weight, In the preceding demonstration we have, foir simplicity, neglected the effect of the mass of the pulley A, and of the cord connecting P, P. This is not generally allowable'ii praectice, as the pulley is usually quite l;arge, We must therefore modify equation (246) so as to take this into account. The correction to be alpplied is. due to the inertia of the pulley alnd cocld, which must be overcome in older that A may revolve with the same velocity as that possessed by the weight, ThiN resistaice is a constalt, and prodlices the same effect as if there were ain additional weirght P', to be moved by the gravity of p. lModlifying (246) in accordance with thi, fiact we have a 2 (247.) The constant P', is best found by direct experimenot; A known wtig'ht p, is placed on P, and the acceleration a is measlured, The values thus obtained are substituted in equation (247), in which P' remains the only unknown quantityg which is therefore immediately determined once for all, The most approved form of Attwood's machine is represented in Fig. 178. The pulley A is mounted on fiiction wheels B, B, in order to reduce the resistance caused by friction. The time of' descent is measured by a second's pendulum C, which is so ar. anged as to nmake and break an electric circuit at every vibration, thereby-causing an electro-inagnet E to act upon a bell-hammer, so that the bell sounds once every second. The correspond

BARBOUZE S METHOD. 165 ing spaces are read upon the graduated scale LM. T is a table capable of sliding along this scale, and upon which the descending weiglht is received. To verify the law of the proportionality of the spaces to the square of the times, the veigcht P, together with a small additional weight p is I-laced upon a little table i/:A. This table is connected with the armature of the electro-lnagnet E in such a manner that when the circuit is completed by the pendulum, it is dletached, thus allowing the weight to fall with a uniforrmly accelerated motion. The weioght being in place, the pendulum is made to vibrate, and simultaneously wi th the stroke of the hell, the table is detached, and the weight beogins to fill. By repeated trial the table T is then adjusted so that the click of the descending weight as it strikes, and the sound of the bell striking the second second, shall coincidle. The correspondinc length of the scale is noted, and the experiment is then repeated, using intervals of 1, 2, 3, etc., seconds. The lengths of the corresponding spaces on LM are then found to be proportional to 1, 4, 9, etc., the squares of the times of descent. The law of the proportionality of the velocity to the time can also be proved directly with this apparatus. For this purpose there is a second table, M, consisting of a ring of metal. The weight p is made in the form of a bar, so that it shall be lifted from P by this rino, leaving the larger weights P,P, to move onwards uiniformlly, with the velocity possessed by them at the moment of the withdrawal of the accelerating force. If the table B is placed so that the weight is removed at the end of 1, 2, 3, etc., seconds, and the space passed over during the succeeding second is measured, these will be the velocities generated in 1, 2, 3, etc., seconds, which will be found to bear to each other the relation of the numbers 1, 2, 3, etc., that is, the velocities are proportional to the times occupied generating them. By determining a by experiment, and solving equation (247) relatively to g, the value of the acceleration due to gravity may be found. 331. Barbouze's M aclhine.l A modification of Attwood's Machine has been constructed by Barbouze of Paris, which is designed to give more accurate results than it is possible to obtain with the former apparatusd The axis of the pulley of an Attwood's lMachine is attached to a light, cylindrical drum, covered with lamnpblacked paper, Against the paper rests a style attached to a tuning-fork. A small additional weight is added at one end of the cord, as already described, and after having set the fork in vibration, the weight is allowed to fall. The druml, moving with the pulley, assumes a uniformly accelerated motion, and the vibrating style removes the Lalmpblack whenever it touches, leaving a sinuous line, the undulations of which are described in equal times, as the vibrations of' the fork are isoehronous. Owing to the nature of the imotion of the drum, the distance between corresponding points of the sinuosities increases, the distance of each ftiom the beginning of the curve measuring the space described by the 1 See Elementary Treatise on Natural Philosophy, by A. Privat Deschanel; translated by J.. D. Everett; Part I., p. 45. Also Report (f U. S. Conmeos. to Paris Uliversal.x position, Vol. Iri, p. 489.

166;LAWS OF FALLING BODIES. revolving cylindler. These distances are found to bear to each other the relation 1, 4, 9, 16, etc., which are proportional to the squares of the times fiom the beginning of the motion. In the machine, as actually constructed, the vibration of the tuning-fork is fiequently kept up by electricity, and the weight is allowed to falll by breaking an electric circuit. The best way of marking the intervals of time is first to allow the instrument to revolve while the fork is not in vibration. The style then traces -a line without undulations, ab, Fig. 179. If the fork is now sounded, and the body allowed to fill, a sinuous line, acdb, is described, the portions acc, cl, cdi, lying between its intersections with ab, being drawn in equal intervals of tinie. The other laws of acecelerated motion canll evidently be verified by removing the weight when part of the descent has been accomplished, as described in the case of Attwoocl's Machine, after which the spaces ac, cc, cib, will be equal to each other. A similar method has been used in the laboratory of tile Institute, in which a freely-falling glass plate covered with lamllpblack, is pressed against a style attached to a vibrating fork. 332. Horizontal Accelerating Machine. This apparatus is similar in prilnciple to Attwood's Machine. A wnagon is arranged so as to move over a horizontal table, ALB, Fig. 180, thle motive force being the small weight, p, attached to it by a cord running over a pulley. A scalle is placed on the talble so that the position of the wagon can be noted at any moment by means of an index 1. To demonstrate the law of the spaces, it is merely necessary to observe the number indicated by the index, at the expiration of 1, 2, 3, etc., seconds from the beginning of the motion. If P be the weight of the wagon, cord, etc., p the added weight, the accelemtion a = g p. (248.) 333. Mllorin's Machine. In this apparatus, which is represented in Fig. 181, the falling body P is allowed to descend fieely under the influence of gravity, being guided in its descent by two vertical wires, EL, IS. A/TXis a cylindrical drum (in the original,pparatus about 2 metres high), which is covered with paper, and is made to revolve uniformly about a vertical axis by means of clock-work. A pencil attached to P presses against this paper, so as to make a mark when the body fllls. Evidently if the cylinder were at rest the line would be vertical; while if the body remained at rest, and the cylinder revolved, the line traced would be a horizontal circle, OX. But if both 1XNVand P are in motion simultaineoulsly, a curve will be traced, the nature of which depends on the law of motion of the falling body. If the motion of the bodly is uniformly accelerated, it is plain thllt calling E, ], two points on the curve, described in t and t' seconds, respectively, the vertical lines A-L, _BE,7 which are the corresponding spaces

PROJECTILES. 167 described by the falling body, will be proportional to the squnres of tile horizontal lines OA4, O-B, which measure the tinmes, since the motion of the cylinder is uniform. That is, if the paper be taken firom the cylinder and spread upon a plane surface, the clurve will be similar to that shown in Fig. 182, in which, for tTwo points, E, F, whose coordinates are (xy), (x'y'), we have, a-2?. y. The curve possessing this characteristic is a pccrabola. Now the curve actually traced by the f:lling body is found to be a parabola, hence its motion must be uniformly accelerated. 334. Addl'itional Methods. By far the best method of verifying the laws of falling bodies is by the use of some of tlhe forms of chronogrcaph, in which the registering of the time of descent is accomplished by electricity. The delicacy of such instruments far transcends any of those which we have described, but as the understanding of the apparatus involves a knowledge of the principles of electricity, they will be treated under the practical applications of that agent. 395. Case of Body falling from great Height. In treatingo of the laws of falling bodies we have'supposed them to be acted upon by a uniformly accelerating force. T'his is not strictly trine, however, for gcavity varies inversely as the square of the distance, so that for a'body falling fiom a very great elevaltion, the acceleration would increase as the body approached the earth. But in most problems in Physics, the height fallen through is comparatively small, so that the diminution of gravity above the su'face may be neglected. Indeed, at the heigoht of a kilometre it. would be but s of the total weight of the body. As the force acting upon a fllling body varies inversely as the square of the distance fr'om tile centre of the earth, it can be shlown by the calculus that there is a linit to the velocity which can be possessed by a body fialling to the earth. It can be proNved th'at a body filling' fiom an illnfinite distance would acquire a velocity of about 11 kilometres per second. Projectiles. 336. S tatement of Problem. In treatingo of the theory of projectiles we assume that the moving body is actedl upon by but two forces; (1) the orliginal impelling force, and (2) the force of gravitation. The disturbing effects caused by the resistance of the air are computed separately, and applied as corrections. Since the greatest horizontal distance to which a projectile can be thrown is comparatively very small, amounting to but a few minutes of arc, the directions of the force of' gravity at:all points of the path may be supposed to be parallel. It may also be regarded as invariable in intensity, since the greatest vertical height ever reached is less than a mile.

168 PROJECTILES. 337n Pat;h of Projec1tie. To ascertain the path pursued by a body thlrown into the air, let AB1, Fig. 183, represent thle direction in which i it is projected. Were there no deflecting or resisting force, the body would move uniformly in this line, bout since gravity is constantly drawing it towards the earth, it will, at the same time, have impressed upon it a uniformly accelerated downward motion. The resultant of these movements will be the path sought. Hence the point occupied by the projectile at the end of any interval of time, can be found in the following mannero Let v be the velocity of projection. Then at the end of 1 second the body will have moved over a distane AG C v, by virtue of the projectile force, while under the action of gravity it has fallen through a vertical distance GC = 2-y. The body will therefore be found at 0C at the expiration of the first second. In like manner, at the end of 2 seconds the position of the body will be found by laying off AiL = 2v, and from this point;, letting fall a vertical -JTD- 2g. At the end of 3 seconds it will be at _, AI being equal to 3v, and I-E to 9-g. So after 4 seconds it will be at F, and. after t seconds at X. The path of the projectile is therefore a curve passing through these points. As AG - v, AB vt, GC = g, B3Jf - gt2, G C: AB:: g': gt2:: 1: t2. Also AG:::v: vt;;: t, and AG: A2:: 1: tI, whence G C: B:-: A G: A B2 A2. (249,) The curve is therefore of such a nature that GC varies as AG2, or the deflection is pro.portional to the square of the corresponding tangent, a property characteristic of the par~zbola. The time occupied in describing the curve AE_/7I, is called the time of flight; the horizontal distance A1Vfrom the point of starting to the point at which the projectile reaches the earth, is the horizontal rangye; and the greatest elevation attained by the body is the vertical range, or flight. These quantities are dependent upon the initial velocity of the projectile and the angle of projection, and when the latter are given, can easily be found. We confine our invbestigations to the case in which the line Aliis horizontal. 833. General Equations. Let AG, Fig. 184, be the line of projection, making an angle, 0, with a horizontal plane, AN1: Call Y the velocity of projection, which may be resolved into two components, one of which, T/, V cos 0 (250), is horizontal, the other, V, - V sin 0 (251), vertical, During the movement of the projectile the horizontal component, VI,, is constant, hence the horizontal distance passed over in time t will be S1, = Jht. The vertical component, V7, is, however, uniformly diminished by gravity, according to the laws of uniformly retarded motion; hence, after t seconds, the vertical component will have become v -= 1 - gt, and the space passed over in that time will be,

RANGE 169 ~ =- Vt - ~gt2. We have, therefore, the following fundamental equ ati ons vi- -,c &0 0 (259) -'- cos Ot (25s), v V qt Vesiin C at (2~4), v 2 V g — g 2 il Ct E o....s (.)., (22 ) (1514), -i the t vertical and horizont-l compl-einu ts t7e -lin i::t, vel,:t y -', te n, le of -sei;tion 0 0:ndc the ti-me i'-nloor.ixne;e -;oillug i i C:olo n is knoiwn equations (200),S (gS95), t::e cc.O, t:dtn::i t- o0f:!e tolCr<:- e 0:-ibm tle Ict::pe of a:lyiFi-uven 1i, t::be' Go 6eCu;;@ai'.:,9 1i:::-',',,?:. n a.,3ce.,id-;og fi-omn, to 1 j; ee b8xs: p:;.-h,:'t r eached by tu e uproectile, t!:::t is 1 in ha etim le o f flgh t (~ 2', p. ~22) the toit,- vertcai coral)ent ofC holi ln is aestre'oet bl y -'iramvty. o lcee c:.: llg I 7 theo titne ot fig1lht we lie v i:Vv or y T z 2 V shin ), wh7ene T= formula givingo the'tihme of filht in terms of V and 0. &9'4c',;, ince the time of descent equals half the time of fiht, or 1 the vertical range,.5v, mnst be -,r substitutino t1he v,!ue of the time of -flight as given in F12 sin=, 2 (256), v~ zg (2057) This value may also be obtained!z,. by uLbstiueting for t in (255), 2' anLd inserting the value of S given in (256)~ 38'?: "l- iti:~::?,-!-~a- Fr o m thon (253), $-:V cos gt, in which if t S, w e h-ave thoe 7rizoontal trange, /- = V C os 0.7 (2figS.) 2 iF -n1 9 2 S s n OS 0 V sO (in) 20 Also since T 0 -s - V h (259), a formula gavinog thCe'horizontal range in termls of Reand 0. $4-2:,, L geg to AR, g,:. n equation (259), v- varies as sin 20, and is'ilelef re a maxiuman when sin 20 =- 1, in which case 0 - 45~. Heence mwith a given veloeity, the greatest hod lzonllta.l ranege is obtainedl when the angle of elevation is 45~. Also since sin 2(45~ + C4) -- si 2(45~,) O being any angle, the horizontal ranae is the saute for anglles equally above or below 45~. The maximum vertuical range evidently ocp-2'FV sin 20 curs when 0 - 90~, and equals 2i' (257). Since' V —'72 if 0 - 5~, h -"h - Bu tu - is the maximum vertical rangoe, hence with a given velocity the glreatest lhorizontal range is twice the greatest vertical range. 22

1 70 PROJECTILES. TI f!= 0~, andl the projectile is fired from an elevation, the time of fli ht i is cq t:ll to the time in which it would fill vertically from t,lt;,:evatt, i1 t: th(- c:tlth. This follows fi'omi (255), in which, by s; )sst i-utiilng 7' the tin,,e of fligllt fir t, Sv becomes equal to Pv, a (1 14 - -P= g ~'12, tlhe spiace iwhich would be traversed in T' seconds by a body falling freely. 343. Equation of Trajectory. To those fhmiliar with the processes of anally tical geometry, the following demonstration may be more satistilet!y tlhta the plrncciig. The equation of the curve OB111, Fig. 185, in which the projectile moves, is y = x tang 0 -- 2-V2 2 (260), which is the e(luation of a paa.labola with a vertical axis. The equation is derivedc in tile f/blowinl liinneri. Let B be any point on the parabola, with coordinates x, /. reachlcd by the projectile in t seconds, after the beginnlin( of its motion. Then y CA - A x tang 0 - AB. But AB = gt2, x.qx2 and as x V cos Ot, t cos, whence AB- 2V'os2b' whence y — = x tang( 0 2,V'q J x2. To find the horizontal range, in (260), make' 2 V cos2 0I 2y 0, and( solve relatively to x, which will be found to be equal to 2 VIz ecos2,9 tanrg 0 V2 sin 20 — alue -~ -, beas before. To find the vertical range, the maximuln alue of'?/ should be fbund Iiom equation (260), regarding x and y as variables. 344. Parabola of Safety. The curve tangent to all the parabolas, A CF, A D)I, ANE, etc. Fig. 186, in which a projectile moves when the anfgle of elevation varies from 00 to 90~, is a parabola, and is known as the prraboia of sajfety, because no point beyond it can be reached by a projectile fired with a given velocity. The range of a projectile is therefore included within a paraboloid of revolution, found by rotating the parabola of safety about the vertical axis A,4. In Fig. 186, A CDE is the parabola of safety in the case of a projectile whose maximum horizontal range is AE. 345. Effect of Resistance of Air. The theoretical curve of motion of:ny projectile being ascertained, it remains to see how this is modified by the resistance of the air. A moving body in passing tllrough the -atmosphere pushes the gaseous particles asicde, thus losing a certuin amount of energy, iwhich is evidently a measutre of the resistance offered. The resistance is evidently proportional to the density of the medium, and to tlhe surfllce of the projectile, since thle muss of the ail displaced vwaries directly as these quantities. Supposing no farthelr action to be exerted by the displaced particles, the re(sistince also varies as the square of' the velocity of the projectile. For suppose two balls A1 and B, of equal masses, A moving with times the velocity of B. A communicatess z times the velocity iven by B, to ch:li particle it displa:lces, a-nd displ:alces x tilmes as lany particles in a given time, thus experiencing xa2 times the reistance.

RESISTANCE OF AIR. 171 But there is another consideration. Only in the case of very small velocities is the supposition;llowavble that the:lir this to farther effect aifter co'ning in contact with the projectile. When the velocity is considerable, the air int fri )t of the bo(dy becomes compressed, that in front rarefied, thus grc:ltly increasing tile resistan ce,.l Recent experiments of Professor B:lshforth have shown that for velocities from 1100 to 1400 feet per second, the total resistance to motion varies approximately as the cube of the velocity, which is a sufficiently accurate ratio for practical purposes. At velocities of fiom 1400 to 1700 ft. the resistance is nearly as the square of the velocity. The actual resistance in pounds upon a spherical shot,:led upon an elongate(l projectile with an ogival headl is shown in the following table, abridgred from the results of' Bashforth.2 Resistance in Pounds. Velocity Spherical Shot. ~Elongated Shot. i7 feet. Diameter. Diameter. |1 ini. i 10 i. 15 in. I in. 5 in. 10 in. 15 in. 1000 4.4 110 -.38 986 2.3 58 233 524 1250 9.2 229' 17 2063 6.6 165 660 1484 1500 14.1 3.51 1 1406 3163 10.2 255 1019 2-293 1750 19.5 439 1954 3 9 7 2000 25.8 6-.5 2582 5,09 _ The efiect of this resistance is of course to cause an enormous diminution of[ asnge, as shown in the following table, which gives the range in yards of a 32 lb. shot, fired with an initial velocity of 1600 ft., in vacuo and in air. Elevation. 1T. 2~. 3~. 4~. Riange iln Vacuo.I 930 1840 2.. 3709 R nge in JAir. 780 1160' K 0 1690 Raeio. 1.19 1.58 1.90 2.19 Elon:cated shot are retardled less than balls, because they are better fitted for cleaving the air.3 The lonll ran(ges of rifled guns are h1-rgely due to the use of elongated, shot, the initial velocity beingr mluch less th'n in the case of smooth-bore armns.4 An ogival head, Fio. 187, offiers the least resistance. LSee Hutton's Tracts, iathematical and Philosophical, No. 37. 2 See 0.vel's Moden Artillery, p. 434. See precedinc table. 4 Owen's Modern Artillery, p. 12.

172 PROJECTILES.,-:.. t,::..., on Ti- r.:: The -raL-i.ett auton o0' mru inr' foDce by th,. resistmnee 01 the ru r%:se the' t nna' i.ea: o' pet do to vary greaty frem a arnb01..,;r: o vel r 0 to pet s e o i t.. ou f t". -e Sn.:ln t latter —:o':01 )I, 111 v e 1 -'.t(tio of)- th1 16.h oe a' m h m u moi e 1.cllned to tihe horizon t]h:n tlheel'? naiffl ini te tO. s%-for te srame reason thle I)t ricel,anllgle of mnoxo-lum r n ng A3 1 aot 45~ except for very low velocities. For inpid, jM.eetiles this ano'-le' 385~ "4:'7. EfSeot of ostation of Proje-tile. If the ball moves onward wsit". no rot ary M LLoon it is clear t11at the resistance of the air wvill rettrd all )pirts equally, and hence there, will be no tendency to deviate in any dir etion. The salne is truek if the ball rotates about the line of flilh as an axis, Ps in Fii.o 1e88 owever tn"re is a irotation'tboe t an axis at risht a'es'lin e of' f'uit-i11, Fig. 189, it is evident that there wvill'e a orenate:'eoesistance at A than atB, because at the former pollit the poojectili and rota. ry nmotions are in t-he same direl tion, while in thle latter -lhey ire opposite. Thle resistance at A is therefbre due to the sut or these' velocities, thiat t 7 t o th i ) iheir dilierence. s3u it c t vi8 e 1 L, -lai <bjj;r(ti[ p'3.iAAent4 is.v by a' aL.gnu's I show.- tial, in suel a cases" thlere would be a greaitLeo. —b. umospriC ~)ressure at A than1 a t B, hen e Iim pro Je"dle must de-ie.-toad,o Wee d-l; the ro.' itin in tie opposite cirection,1 the dirtetion of t,vi. in wl' e r v e e. Tce ay, Ii thuis mlovinc llX be i 1-. ed, l to the rh L t or the lefL, or its ra'9Oe wvill he varied, accordin, toe' di'ecti o Of its.3tation. Sich a moti:n is invar',.bly acquired by a ball firedl from a smootl —bore f l- ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ _ gun, becanuse',h b sin n e-ssaiily i(,, lari uer th- -, e diniatr of tlil r-oajcyt'ee, ths rebodsiu ~rohm, o ni plnn l point to anothe r i. its passage to the muinzln3e, i I'.[-!ne o oa y moition is i'nputte-, the Cireetioln of' whlh.'epenils chiefly oen eac Ilst point, of impactio Thus a ball strikin at B J, ust before!eavin tile l f run,1 roeceds witlh a ril; nht-humdnd rotation. T~as source of deviatinon cano be avoided by rfiuny thw onun. By m-eans 0o. se: g' rOvtits Cut in th bo'e, the shot is 1 a ie to assiuine a rotary io0tioil u tba ~ of 0. e pOieLe, o0 In)that on startin thfis rotation is at rinoltan:' e to a i ta, to tee tr"cti.'e 1itch of0 the screw vtaites in di'i. n;., s it ito' prejeetiL e nenenirally makes less than one complete r -o,1 ia il0.n b1 e I- e i ea' - " the p i ee he rolY.Eon eused bo ri-"' also neutralizes the eflect- of any iirecularitues o su rie, or'vait of' hi)-nigenwity of thel prio...io v which wouveld otlh-'o,-"evisep lro lce a n e_ l'o to ei t Aso an eAoni'-i...' irectile can be usci, thns seurnI a a much grneater rn-e. Even wivh rinled t uns, however, t!iere is s:i1 a io ~ ta 0 o0L th 1 e. theIoeti al path, ovina to (1) want of equa-ditx in the p:ojecetsla o tihe i poa dere(it (2) the oet wind, (o) tlhe rotaLtiat) On the I ari,tl anil (4) a 1oemenit In the C'retioni 0of te' iotation, c.aile(, de:icai'ionto or 4i, e.lsed -iy la coml nation of-l the brces of aa'aln resist-ance antid the rotation of' the ir jectile, anlCto-ous to the orbital revolution f01' the gyroscope,. 1 Owe'ls Mod'lern Artiler/, pp. 202, 2321. 2 This excess of priessire is due to thle fect that a laver of air in conltact with the ball i; set in rotationl with it by frictioil, tius condensing the air at A antd raretying it at B.

MEASUREMIENT OFP VELOCITY. 173 $4,o,Cmseuw-erCaet of Voelocty ofrojctes It reainsi to ri,.u1 ~ ~ ~-s.'~ he I ate'be. ne ise'b y'i: pC.~.-C AU,, ""'"" %1'C~i"7 -.l' c, C It c ists of ma i';A igo 1 cv. ws t i;v i rietl&i rapioly I iai bau Ia vert~ i: axis, T s-ot is 5_': s a'i i:,e a da'arieter o[tt;' theqo dcerum'Ln~.,;, lo: i th dr, at rest tt H tw:!;,:re.ie d woM.3111(I' di e a ica e y o 1_te, but o-1 i - th c evo u, a ce.t:.n are, 2 T3, will ojI li''-1'sei, bnow~c~.ye s et- n- _ on' s a rc *h ciC; psirnt oC a;.xt ot-he shot v-Il 1 -at, t iea,:' vcf Is t: e vOo-C c I y of S ~ tb e thns n whi tle tile s.i s h sinlt i ire'i: 1 Utn loi9 D tie diameterof 0 teruzn cd he t anse ib tene; by.:,1 2Cv v = (25 G ) A l.aa t m o thit, 5 ppms ivent: byc Gi'Lit, -n 16, c onsis t two p,aralil dis, 7, i — S 192 nivided by ad i to e t0~9 IonaeCted l Y a or "izon-tal shaft a-,nd ro in11 raiy. Te s. s firea l co th -' axis, and i ti i ore dc - - i eecrmi f te anode Obet "'en the r i-i pa 3si':: t inolh's th' hIeIis iI eah C L:. The objet'ti-on to t- Oi e -1 es of apts u t tine i not measu -red With suflOnIeit iaccurnacVy. A& methojl?!ied by Count Ru'for,4 to obtPnin111" tee veloucity or p-%e - ~ti 1s Slio s 11 a 23W iy3s oy eans' l 0 1 b i' th C 0 i:o ~-'"i powlz ae ts s in o e ireortien ateo'L;-toi t h e ij -- to iiV be en -,'. e i- f w e e -' n 1. v i wt' h!, r eA. 1' 0Si l, ISpvno v i 1 a t wei: 1 V t0Z h b a 1 i v OiS c9 (' o -3. ti, cn.i 1 -1' 51 1' % ii t 5' 1'1 o\ Ac. t "'rct in"-"sl ""'U. C 31 de ) of te el 1 to2 1, v -;.S3:-. r ii.d 1V Ico., scus in ended, i l ri:i: bf; m Lovi atout oa -ol (Pec V!es ite v bf nflevi (/ V;'li11 q e i hb+S n ~ 4 theAn ntmul-t -re extni " —,- swei"l Pi an Of th -'eU sn;', I BK tol.<;il ~~eou'oi-f' Y'l'' S wn'y -o'e o-, oIver y h.rs o,,-,,:t ar3e _o olIn velaOis.y of t-he rtioi to o 1 tt L f. hs 1r. -tiII, L m0valz> C1 Urin] 0 have, GV bei:n equnl to l,. T." iSf - be a t a kean as the verse -ds -'l l v Sat ar. An instru -nt' -sore extensiv e l y ec b-iused thia n sl an o lt s'e; -',"-. liew d'u' o eRCas, im i ony- y a1uton, Grby ea oy and o er% AIt oinsist s ofn a h -iosusy tblock 5f w iod, A, Pisc ons. 194-f, e by a r i m:hvMin? r;, kn;",, e'S- o_'1 bil 1on) k-n; ceJ:.:s at,_., e'1 iaint' - -j ii.i, eni of Qs,. i bl - ii biS it'c into i'......... we.ooCt cms( tol-eis i, oret qeC' idoeer, c aric' eo t e by'ci'st o nft t U oiut the front,n otion o f t le pendulum. fTllisg iet wii ls clay so t.at the ball nhtaay sink into thip -ws ithout ausing. acceny splinte ion. r pact o moe osn t% ~;ether ainxe e n q u calffirig iv the -wei,, ht o[' the:, b8ai v i';s 8 veloc: veloc ityhe of- the bl ock; andl V its vel c ity, m t —_ (P'F-1- t) F, wheneo v _ w) (263). It is custo'' t o hollow out the ~ront portion of the penduluni. fiiling it with clay, so that the ball iay sink into this without causing any splintering.

174 rPROJECTILES. The methods which we ha\ e (described have been used with more oir less success, but more recently the invention of electric imethods h11s given u, fi'r mnore accurate ncetho(ds of measurelent. The Bashforth, Navez-Leurs andt Noble chro;noscopes, in which electricity is used(, hlave afforde(1 most vahlaible results, in (leterininin( the velocity of the ball at different points of its course, both in the bore of the piece, and in its path through the air. We shall describe some of' these formls when treating of' electro-chronographs. These instrtunients are appliedl to ascertain the resistance of tile air by observing the decrease of velocity of the projectile as its distance froml tlhe gun increases. It is,-imply necessary to measure the velocity at points more and more distant fromn the gun. From this loss of' velocity we may (calculate the mlagnitudle of the resistance by comparing it wit-h gravity. Thus if' d be the loss of velocity in t seconds, TV the weight of' the projectile, F the resistance in kilogrammnes, observing that forces are proportional to the momenta which they destroy in equal times, we have, F: T: d: gi, whence F - -. (264.) 349. Historical Sketch. The earliest treatise on gunnery was published in 1561, by Santhach, who supposed that every projectile procee(led in a st'ai(rht line till its whole velocity was spent, and then ftll directly downwardIs. Nicolo TIartalea (1550-1554), a Venectian, rep)resented( the tra{jectory as first a strain ht line in the direction of the original projection, next an are of a circle, pursuedl till its d(irection became vertical, and finally terminatinr in a vertical line reaching to the,troundl. IHe also remllarkled that the angle of imaximnum range was 450. Rivius (1582) knew theat the body continually mnoved in a curve, but considered the devilation at first so slight that it mnighlit be consi(leed as a straight line. Galileo showed tihat the body it' unresisted woull move in a parabola. He also deterllined the law of aerial resistance for slow motions, as did Nexwton also. But lie greatly underrated the effect of this resistance, an(i supposed tlhat the ac(tual path of a pro(jectile was, in general, a near approximantion to tile curlve given by theory, while it is, in reality, completely altered. Galileo's opinion, however, was accepted as conclusive, and in thle works of Andlerson (1674) and Blondlel (1683), tLtis theory is strenuously upheld, and tables calculated upon it as a'bl.sis. Little progress was niade till, in 1742, Benjamin Robins published his NAew Principles of Gun?,aery. This contained the (lescription of' his newly-invented ballistic pendululn, and of his investigations on the explosive force of' powder, resistance of' the air and its efifect in retalrdingt projectiles. Robin's methods were still further im1proved by Huttrn, towards the end of the last century. The general theory has not changred grealtlr since, but in late years the application of the electlic nlet.hod of reristration Ii as been of incalculable advantagge in giving far more precise results than were before obtainable. RtEFERENCESS. Priunciples and Practiee of Modlern Artillery, by Lieut. Col. C. H. Owven. (London, John Murray, 1871.) Irealtise on Ar'illery, by) Capt. Boxer. United Stales Ordnance Manmal. (Philadelphia, J. B. Lippincott & Co., 1862.)

REFERENCES A ND ERR ATA. 175 Tile preceding works may be consulted for general information relative to ordnance. For the mathematical investigation of. the motions of projectiles, see'reatise on the Dynamics of a Particle, by P. G. Tait and W. J. Ste(dle. (Mlacmillan & Co., 1865.) p. 76, p. 228. T7reatise on) Ift nitesimal Calculus, by B. Price; Vol. Iv., Dynamics of Riqidl Bodies. Mathematical 7Treatise on the 3Motions of Projecliles, by F. Bashforth. (London, 1872.) Traitel de balistique exterieure, par N. Mlyevski. (Paris, 1872.) T'racts, IMathematical and Philosophical, by Chas. 1H-utton. (London, 1786.) See, also references in Owen's Modern Artillery. For mlethods of deter nining velocity, see New Principles of Gunnery, by Benj. Robins. (1742.) [Hutton's Edition, 1805.] New Expjeriments upon Gunpowder, with an Account of a new Method for determining the Velocity of all kinds of MIilitary Projectiles, by.Sir Benjamin Thoml)son (Count Ruimford); Philosophical 2ransactions, 1781. Mlechanics of Engineering, by Julius Weisbach; Coxe's Translation, Vol. i., p. 693. (Ballistic Pendulum). Mcldern Alrtillery, by Lient. Col. Owen; Appendix IV. Description of Bashfiorth', Navez-Leurs and Noble Chronoscopes, and other methods. Report of Co)m)missioners to Paris Universal Exposition, Vol. III., p. 563. For a fhll description of the Noble Chronoscope for determining the velocity of the ball within the bore of the gun, see Annales dlu Conservatoire des Arts et'Meftiers. Tome Ix., No. 35. (Tralnslated fioom the Preliminary Report of the Committee on Explosives, London.) For historical information consult Whewell's History of the Inductive Sciences. Bk. vI. Owen's Ilodern Artillery, p. 145. ERRATA. P. 29, ~ 37, line 8. After' weight of the wood" insert plus that of the oxygen added in the act of burning." P. 37, ~ 50, line 17. For'1 " read' ~6. P. 43, ~ 59, line 2. After "towalrds the east," insert " its velocity relatively to the centre of the earth is 470 m. + 470 m. - 940 m. If on the )ther hand it be fired towards the west." P. 46, ~ 65, line 6. For "' Fig. 12" read " Fig. 11." P. 49, ~ 76, line 10. For "pounds," read " kilogrammes." P. 53, ~ 91, line 39. For "PQ "read'LQ." P. 55, ~ 94, line 4. For " A'B" read "AB." P. 60, ~ 109, line 28. For "' movement" read "rotation." P. 62, ~ 114. line 7. For' "P X OH X P + OL " read " P X OH + P X OL. P. 62, ~ 114, line 11. F or 11 L - KCL " read " HKI KL." P. 62, ~ 114, line 12. For " similarity" read" equality." For "BKL, CKH'" CG1, BGM1.'"

176 ERRATA. Po 62, ~ 115, line 11. For "2 B 99 read l "F2'B." P. 63, ~ 113,:e 30 For; Fi]p. o 45 99 read F'i''. 46:' P.o u3 3 9 Ine 6o or'Y - rea d 0. Fo 6, ~ 121, l1e 8. Lie r S F r - PP -" readd' R F +L YF." P. 72, ~ 33 In equations (66), for; 09" read" 0 O." P. 76 ~ 147, 1neI. II For.y"'.: f' G " read 6G G: 9111 G ~P 77, ~ 149,' iie 90 For "'6 BF" read "66 BE" P0 836 6 170, ilne 18. After' foot " insert 6 of.99 P. 9/ S5 193, line 21. For" AD" reiCd 6 A 97." P. 95~ ~ 195, line 11.Z J Aor "AP"99 rad 6"A." P. 11I, ~ 224. The statement relativ e to the angles of chisels for brass and iron, given on the authority of Lardner, is erroneous.