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ELEMENTS 
 
 OF 
 
 NATURAL PHILOSOPHY. 
 
iLontJOn: CAMBRIDGE WAREHOUSE, 
 
 17, Paternoster Row. 
 
 CTamirnifie: DEIGHTON, BELL, AND CO. 
 
 ILcipjig: F. A. BROCKHAUS. 
 
it 
 
 -ELEMENTS 
 
 NATURAL PHILOSOPHY 
 
 SIR WILLIAM THOMSON, LL.D, D.C.L., F.R.S., 
 
 PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF GLASGOW, 
 
 AND 
 
 PETER GUTHRIE TAIT, M.A., 
 
 PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF EDINBURGH. 
 
 PART I. 
 
 _ SECOND EDITION, 
 .f^T% B A R iT 
 
 AT THE UNIVERSITY PRESS. 
 i8;9 
 
 \_All Rights reserved^ 
 
VI PREFACE. 
 
 of notes of a part of the Glasgow course, drawn up for 
 Sir W. Thomson by John Ferguson, Esq., and printed for 
 the use of his students. 
 
 "We have had considerable difficulty in compiling this 
 treatise from the larger work — arising from the necessity for 
 condensation to a degree almost incompatible with the design 
 to omit nothing of importance : and we feel that it would 
 have given us much less trouble and anxiety, and would 
 probably have ensured a better result, had we written the 
 volume anew without keeping the larger book constantly 
 before us. The sole justification of the course we have pur- 
 sued is that wherever, in the present volume, the student 
 may feel further information to be desirable, he will have 
 no difficulty in finding it in the corresponding pages of the 
 larger work. 
 
 " A great portion of the present volume has been in type 
 since the autumn of 1863, and has been printed for the use 
 of our classes each autumn since that date." 
 
 To this we would now only add that the whole has been 
 revised, and that we have endeavoured to simplify those 
 portions which we have found by experience to present 
 difficulties to our students. 
 
 The present edition has been carefully revised by Mr W. 
 BURNSIDE, of Pembroke College : and an Ltdex, of which we 
 have recognized the necessity, has been drawn up for us by 
 Mr Scott Lang. 
 
 W. THOMSON. 
 P. G. TAIT. 
 
 January^ 1879. 
 
CONTENTS. 
 
 DIVISION I. Preliminary. 
 
 PAGE 
 
 Chap. I. Kinematics i 
 
 „ II. Dynamical Laws and Principles . . 56 
 
 „ in. Experience in 
 
 „ IV. Measures and Instruments . . . 122 
 
 DIVISION II. Abstract Dynamics. 
 
 ., V. Introductory 136 
 
 ,, VI. Statics of a Particle. Attraction . . 140 
 
 ,, VII. Statics of Solids and Fluids . . . 199 
 
 Appendix 282 
 
DIVISION I. 
 
 PRELIMINARY. 
 
 CHAPTER L— KINEMATICS. 
 
 The word Dynamometer occurring in the Index on p. 288 should 
 have been Ergometer, that being the term which we shall in 
 future use to denote this class of instruments. 
 
 considered without reference to the bodies moved, or to the forces 
 producing the motion, or to the forces called into action by the 
 motion, constitute the subject of a branch of Pure Mathematics, 
 which is called Kinematics, or, in its more practical branches, 
 Mechanism. 
 
 5. Observation and experiment have afforded us the means of 
 translating, as it were, from Kinematics into Dynamics, and vice versd. 
 This is merely mentioned now in order to show the necessity for, 
 and the value of, the preliminary matter we are about to introduce. 
 
 6. Thus it appears that there are many properties of motion, 
 displacement, and deformation, which may be considered altogether 
 independently of force, mass, chemical constitution, elasticity, tempe- 
 rature, magnetism, electricity ; and that the preliminary consideration 
 of such properties in the abstract is of very great use for Natural 
 
 T. I 
 
OF THE 
 
 UNIVERSITY 
 
 DIVISION I. 
 
 PRELIMINARY. 
 
 CHAPTER L— KINEMATICS. 
 
 1. The science which investigates the action of Force is called, by 
 the most logical writers, Dynamics. It is commonly, but erroneously, 
 called Mechanics ; a term employed by Newton in its true sense, 
 the Science of Machines, and the art of making them. 
 
 2. Force is recognized as acting in two ways : 
 
 1° so as to compel rest or to prevent change of motion, and 
 
 2" so as to produce or to change motion. 
 Dynamics, therefore, is divided into two parts, which are conveniently 
 called Statics and Kinetics. 
 
 3. In Statics the action of force in maintaining rest, or preventing 
 change of motion, the 'balancing of forces,' or Equilibrium, is 
 investigated ; in Kinetics, the action of force in producing or in 
 changing motion. 
 
 4. In Kinetics it is not mere inotion which is investigated, but the 
 relation oi forces to motion. The circumstances of mere motion, 
 considered without reference to the bodies moved, or to the forces 
 producing the motion, or to the forces called into action by the 
 motion, constitute the subject of a branch of Pure Mathematics, 
 which is called Kinematics, or, in its more practical branches. 
 Mechanism. 
 
 5. Observation and experiment have afforded us the means of 
 translating, as it were, from Kinematics into Dynamics, and vice versd. 
 This is merely mentioned now in order to show the necessity for, 
 and the value of, the preliminary matter we are about to introduce. 
 
 6. Thus it appears that there are many properties of motion, 
 displacement, and deformation, which may be considered altogether 
 independendy of force, mass, chemical constitution, elasticity, tempe- 
 rature, magnetism, electricity ; and that the preliminary consideration 
 of such properties in the abstract is of very great use for Natural 
 
 T. I 
 
2 PRELIMINARY. 
 
 Philosophy. We devote to it, accordingly, the whole of this chapter ; 
 which will form, as it were, the Geometry of the subject, embracing 
 what can be observed or concluded with regard to actual motions, 
 as long as the cause is not sought. In this category we shall first take 
 up the free motion of a point, then the motion of a point attached 
 to an inextensible cord, then the motions and displacements of rigid 
 systems — and finally, the deformations of solid and fluid masses. 
 
 7. When a point moves from one position to another it must 
 evidently describe a continuous line, which may be curved or straight, 
 or even made up of portions of curved and straight lines meeting 
 each other at any angles. If the motion be that of a material particle, 
 however, there can be no abrupt change of velocity, nor of direction 
 unless where the velocity is zero, since (as we shall afterwards see) 
 such would imply the action of an infinite force. It is useful to con- 
 sider at the outset various theorems connected with the geometrical 
 notion of the path described by a moving point ; and these we shall 
 now take up, deferring the consideration of Velocity to a future 
 section, as being more closely connected with physical ideas. 
 
 8. The direction of motion of a moving point is at each instant 
 the tangent drawn to its path, if the path be a curve ; or the path 
 itself if a straight line. This is evident from the definition of the 
 tangent to a curve. 
 
 9. If the path be not straight the direction of motion changes 
 from point to point, and the rate of this change, per unit of length 
 of the curve, is called the Curvature. To exemplify this, suppose 
 
 two tangents, PT, QU, drawn to a circle, 
 and radii OP^ OQ, to the points of contact. 
 The angle between the tangents is the 
 qj change of direction between P and Q, 
 and the rate of change is to be measured 
 by the relation between this angle and the 
 length of the circular arc PQ. Now, if $ 
 be the angle, s the arc, and r the radius, we 
 see at once that (as the angle between the radii is equal to the 
 angle between the tangents, and as the measure of an angle is the 
 
 ratio of the arc to the radius, § 54) r9 = s, and therefore - =- is the 
 
 measure of the curvature. Hence the curvature of a circle is in- 
 versely as its radius, and is measured, in terms of the proper unit of 
 curvature, simply by the reciprocal of the radius. 
 
 10. Any small portion of a curve may be approximately taken 
 as a circular arc, the approximation being closer and closer to the 
 truth, as the assumed arc is smaller. The curvature at any point 
 is the reciprocal of the radius of this circle for a small arc on each 
 side of the point. 
 
 11. If all the points of the curve lie in one plane, it is called 2. plane 
 curve, and if it be made up of portions of straight or curved lines it 
 
KINEMATICS. 3 
 
 is called 2. plafie polygon. If the line; do not lie in one plane, we have 
 in one case what is called a curve of double curvature, in the other 
 a gauche polygon. The term ' curve of double curvature ' is a very 
 bad one, and, though in very general use, is, we hope, not inera- 
 dicable. The fact is, that there are not two curvatures, but only 
 a curvature (as above defined) of which the plane is continuously 
 changing, or twisting, round the tangent line. The course of such 
 a curve is, in common language, well called ' tortuous ; ' and the mea- 
 sure of the corresponding property is conveniently called Tortuosity. 
 
 12. The nature of this will be best understood by considering the 
 curve as a polygon whose sides are indefinitely small. Any two 
 consecutive sides, of course, lie in a plane — and in that plane the 
 curvature is measured as above; but in a curve which is not plane 
 the third side of the polygon will not be in the same plane with the 
 first two, and therefore the new plane in which the curvature is to 
 be measured is different from the old one. The plane of the curva- 
 ture on each side of any point of a tortuous curve is sometimes called 
 the Osculating Plajie of the curve at that point. As two successive 
 positions of it contain the second side of the polygon above men- 
 tioned, it is evident that the osculating plane passes from one position 
 to the next by revolving about the tangent to the curve. 
 
 13. Thus, as we proceed along such a curve, the curvature in 
 general varies ; and, at the same time, the plane in which the cur- 
 vature lies is turning about the tangent to the curve. The rate of 
 torsion, or the tortuosity, is therefore to be measured by the rate at 
 which the osculating plane turns about the tangent, per unit length 
 of the curve. The simplest illustration of a tortuous curve is the 
 thread of a screw. Compare § 41 {d). 
 
 14. The Integral Curvature, or whole change of direction, of an arc 
 of a plane curve, is the angle through which the tangent has turned 
 as we pass from one extremity to the other. The average curvature 
 of any portion is its whole curvature divided by its length. Suppose 
 a line, drawn through any fixed point, to turn so as always to be 
 parallel to the direction of motion of a point describing the curve : 
 the angle through which this turns during the motion of the point 
 exhibits what we have defined as the integral curvature. In esti- 
 mating this, we must of course take the enlarged modern meaning 
 of an angle, including angles greater than two right angles, and also 
 negative angles. Thus the integral curvature of any closed curve or 
 broken line, whether everywhere concave to the interior or not, is four 
 right angles, provided it does not cut itself. That of a Lemniscate, 
 
 g is zero. That of the Epicyloid @ is eight right angles ; and 
 
 so on. 
 
 15. The definition in last section may evidently be extended to 
 a plane polygon, and the integral change of direction, or the angle 
 between the first and last sides, is then the sum of its exterior angles, 
 all the sides being produced each in the direction in which the 
 
 I — 2 
 
PRELIMINARY, 
 
 moving point describes it while passing round the figure. This is 
 true whether the polygon be closed or not. If closed, then, as long 
 as it is not crossed, this sunri is four right angles, — an extension of 
 the result in Euclid, where all reentrant polygons are excluded. In 
 
 the star- shaped figure ^^ , it is ten right angles, wanting the suna of 
 
 the five acute angles of the figure j i. e. it is eight right angles. 
 
 16. A chain, cord, or fine wire, or a fine fibre, filament, or hair, 
 may suggest, what is not to be found among natural or artificial pro- 
 ductions, a perfectly fiexible and i?iextensible line. The elementary 
 kinematics of this subject require no investigation. The mathematical 
 condition to be expressed in any case of it is simply that the distance 
 measured along the line from any one point to any other, remains 
 constant, however the line be bent. 
 
 17. The use of a cord in mechanism presents us with many 
 practical applications of this theory, which are in general extremely 
 simple; although curious, and not always very easy, geometrical 
 problems occur in connexion with it. We shall say nothing here 
 about such cases as knots, knitting, weaving, etc., as being exces- 
 sively difficult in their general development, and too simple in the 
 ordinary cases to require explanation. 
 
 18. The simplest and most useful applications are to the Pulley 
 and its combinations. In theory a pulley is simply a smooth body 
 
 which changes the dh'ectmz of a flexible and inextensible 
 cord stretched across part of its surface ; in practice (to 
 escape as much as possible of the inevitable friction) 
 it is a wheel, on part of whose circumference the cord 
 is wrapped. 
 
 (i) Suppose we have a single pulley B^ about which 
 the flexible and inextensible cord ABP is wrapped, and 
 suppose its free portions to be parallel. 
 If {A being fixed) a point P of the cord i'P' 
 
 be moved to P' , it is evident that each 
 
 of the portions AB and PB will be • ^ 
 
 shortened by one-half of PP". Hence, i_j^'< 
 when P moves through any space in ■ 
 
 the direction of the cord, the pulley B 
 moves in the same direction, through 
 half the space. 
 (2) If there be two cords and two pulleys, the 
 ends AA' being fixed, and the other end of AB 
 being attached to the pulley ^'— then, if all free 
 parts of the cord are parallel, when P is moved to 
 P', B' moves in the same direction through half the 
 space, and carries with it one end of the cord AB. 
 Hence B moves through half the space B did, that 
 is, one fourth of PP', 
 
 rP' 
 
 <^ 
 
 B 
 
 K^' 
 
KINEMATICS, 5 
 
 (3) And so on for any number of pulleys, if they be arranged 
 in the above manner. Similar considerations enable us to deter- 
 mine the relative motions of all parts of other systems of pulleys and 
 cords as long as all the free parts of the cords are parallel. 
 
 Of course, if a pulley \>Q.jixed^ the motion of a point of one end of 
 the cord to or fro77i it involves an equal motion of the other end 
 from or to it. 
 
 If the strings be not parallel, the relations of a single pulley or 
 of a system of pulleys are a little complex, but present no difficulty. 
 
 19. In the mechanical tracing of curves, a flexible and inextensible 
 cord is often supposed. Thus, in drawing an ellipse, the focal pro- 
 perty of the curve shows us that if we fix the ends of such a cord 
 to the foci and keep it stretched by a pencil, the pencil will trace 
 the curve. 
 
 By a ruler moveable about one focus, and a string attached to a 
 point in the ruler and to the other focus, and kept tight by a pencil 
 sliding along the edge of the ruler, the hyperbola may be described 
 by the help of its analogous focal property ; and so on. 
 
 20. But the consideration of evolutes is of some importance in 
 Natural Philosophy, especially in certain mechanical and optical 
 questions, and we shall therefore devote a section or two to this 
 application of Kinematics. 
 
 Def. If a flexible and inextensible string be fixed at one point 
 of a plane curve, and stretched along the curve, and be then 
 unwound in the plane of the curve, every point of it will describe 
 an Involute of the curve. The original curve is called the Evolute of 
 any one of the others. 
 
 21. It will be observed that we speak of an involute, and of the 
 evolute, of a curve. In fact, as will be easily seen, a curve can have 
 but one evolute, but it has an infinite number of involutes. For all 
 that we have to do to vary an involute, is to change the point of 
 the curve from which the tracing-point starts, or consider the invo- 
 lutes described by different points of the string ; and these will, in 
 general, be different curves. But the following section shows that 
 there is but one evolute. 
 
 22. Let AB be any curve, PQ a portion of an involute, pP^ qQ 
 positions of the free part of the string. It will be seen at once 
 that these must be tangents to the arc 
 AB at / and q. Also the string at 
 any stage, as pP, ultimately revolves 
 about p. Hence pP is normal (or per- 
 pendicular to the tangent) to the curve 
 PQ. And thus the evolute of PQ is y£ 
 a definite curve, viz. the envelop of (or 
 line which is touched by) the normals drawn at every point of PQ, 
 or, which is the same thing, the locus of the centres of the circles 
 which have at each point the same tangent and curvature as the 
 curve PQ. And we may merely mention, as an obvious result of the 
 
6 PRELIMINARY. 
 
 mode of tracing, that the arc qp is equal to the difference oi qQ and 
 pP, or that the 2x0, p A is equal to pP. Compare § 104. 
 
 23. The rate of motion of a point, or its rate of change of position^ 
 is called its Velocity. It is greater or less as the space passed over 
 in a given time is greater or less : and it may be uniform^ i. e. the 
 same at every instant ; or it may be variable. 
 
 Uniform velocity is measured by the space passed over in unit of 
 time, and is, in general, expressed in feet or in metres per second ; 
 if very great, as in the case of light, it may be measured in miles per 
 second. It is to be observed that • Time is here used in the abstract 
 sense of a uniformly-increasing quantity — what in the differential cal- 
 culus is called an independent variable. Its physical definition is 
 given in the next chapter. 
 
 24. Thus a point, which moves uniformly with velocity v^ describes 
 a space of v feet each second, and therefore vt feet in / seconds, 
 t being any number whatever. Putting s for the space described 
 in / seconds, we have s = vt. 
 
 Thus with unit velocity a point describes unit of space in unit of 
 time. 
 
 25. It is well to observe here, that since, by our formula, we have 
 generally s 
 
 and since nothing has been said as to the magnitudes of s and /, we 
 may take these as small as we choose. Thus we get the same result 
 whether we derive v from the space described in a million seconds^ or 
 from that described in a millio?ith of a seco?id. This idea is very useful, 
 as it makes our results intelligible when a variable velocity has to be 
 measured, and we find ourselves obliged to approximate to its value 
 (as in § 28) by considering the space described in an interval so 
 short, that during its lapse the velocity does not sensibly alter in value. 
 
 26. When the point does not move uniformly, the velocity is 
 variable, or different at different successive instants : but we define 
 the average velocity during any time as the space described in that 
 time, divided by the time ; and, the less the interval is, the more 
 nearly does the average velocity coincide with the actual velocity at 
 any instant of the interval. Or again, we define the exact velocity at 
 any instant as the space which the point would have described in one 
 second, if for such a period it kept its velocity unchanged. 
 
 27. That there is at every instant a definite velocity for any moving 
 point, is evident to all, and is matter of everyday conversation. Thus, 
 a railway train, after starting, gradually increases its speed, and every 
 one understands what is meant by saying that at a particular instant it 
 moves at the rate of ten or of fifty miles an hour, — although, in the 
 course of an hour, it may not have moved a mile altogether. We 
 may suppose that, at any instant during the motion, the steam is so 
 adjusted as to keep the train running for some time at a uniform 
 velocity. This is the velocity which the train had at the instant in 
 
KINEMATICS. 7 
 
 question. Without supposing any such definite adjustment of the 
 driving-power to be made, we can evidently obtain an approximation 
 to the velocity at a particular instant, by considering (§ 25) the 
 motion for so short a time, that during that time the actual variation 
 of speed may be small enough to be neglected. 
 
 28. In fact, if v be the velocity at either beginning or end, or at 
 any. instant, of an interval /, and s the space actually described in 
 
 that interval; the equation v = - (which expresses the definition of 
 
 the average velocity, § 26) is more and more nearly true, as the 
 velocity is more nearly uniform during the interval /; so that if we 
 take the interval small enough the equation may be made as nearly 
 exact as we choose. Thus the set of values — 
 
 Space described in one second. 
 
 Ten times the space described in the first tenth of a second, 
 
 A hundred „ „ „ hundredth „ 
 
 and so on, give nearer and nearer approximations to the velocity at 
 the beginning of the first second. 
 
 The whole foundation of Newton's differential calculus is, in fact, 
 contained in the simple question, 'What is the rate at which the 
 space described by a moving point increases?' i.e. What is the 
 velocity of the moving point? Newton's notation for the velocity, 
 i. e. the rate at which s increases, or the Jluxion of J", is s. This 
 notation is very convenient, as it saves the introduction of a second 
 letter. 
 
 29. The preceding definition of velocity is equally applicable 
 whether the point move in a straight or a curved line; but, since, 
 in the latter case, the direction of motion continually changes, the 
 mere amount of the velocity is not sufficient completely to describe 
 the motion, and we must have in every such case additional data 
 to thoroughly specify the motion. 
 
 In such cases as this the method most commonly employed, 
 whether we deal with velocities, or (as we shall do farther on) with 
 accelerations and forces, consists in studying, not the velocity, accele- 
 ration, or force, directly, but its resolved parts parallel to any three 
 assumed directions at right angles to each other. Thus, for a train 
 moving up an incline in a N.E. direction, we may have the whole 
 velocity and the steepness of the incline given ; or we may express 
 the same ideas thus — the train is moving simultaneously northward, 
 eastward, and upward — and the motion as to amount and direction 
 will be completely known if we know separately the northward, east- 
 ward, and upward velocities — these being called the components of 
 the whole velocity in the three mutually perpendicular directions 
 N., E., and up. 
 
 30. A velocity in any direction may be resolved in, and perpen- 
 dicular to, any other direction. The first component is found by 
 multiplying the velocity by the cosine of the angle between the two 
 
8 PRELIMINARY. 
 
 directions ; the second by using as factor the sine of the same angle. 
 Thus a point moving with velocity V up an Inclined Plane^ making 
 an angle a with the horizon, has a vertical velocity Fsin^; and a 
 horizontal velocity Fcos a. 
 
 Or it may be resolved into components in any three rectangular 
 directions, each component being found by multiplying the whole 
 velocity by the cosine of the angle between its direction and that of 
 the component. The velocity resolved in any direction is the sum 
 of the resolved parts (in that direction) of the three rectangular com- 
 ponents of the whole velocity. And if we consider motion in one 
 plane, this is still true, only we have but two rectangular com- 
 ponents. 
 
 31. These propositions are virtually equivalent to the following 
 obvious geometrical construction : — 
 
 To compound any two velocities as OA^ OB in the figure ; where 
 ^ ^ OA, for instance, represents in magni- 
 
 /- "J^^ ^^^^ ^"^ direction the space which 
 
 / ^^■'</ would be described in one second by 
 
 / ^^^ y/ a point moving with the first of the 
 
 / ^^^ / given velocities — and similarly OB for 
 
 Z,.-^ / the second ; from A draw A C parallel 
 
 ^^_ / and equal to OB. Join OC-. then OC 
 
 ^ -A. is the resultant velocity in magnitude 
 
 and direction. 
 
 OC is evidently the diagonal of the parallelogram two of whose 
 sides are OA, OB. 
 
 Hence the resultant of any two velocities as OA, AC, in the 
 figure, is a velocity represented by the third side, OC, of the triangle 
 OAC. 
 
 Hence if a point have, at the same time, velocities represented by 
 OA, A C, and CO, the sides of a triangle taken ifi the same order, it 
 is at rest. 
 
 Hence the resultant of velocities represented by the sides of any 
 closed polygon whatever, whether in one plane or not, taken all in 
 the same order, is zero. 
 
 Hence also the resultant of velocities represented by all the sides 
 of a polygon but one, taken in order, is represented by that one 
 taken in the opposite direction. 
 
 When there are two velocities, or three velocities, in two or in 
 three rectangular directions, the resultant is the square root of the 
 sum of their squares ; and the cosines of its inclination to the given 
 directions are the ratios of the components to the resultant. 
 
 32. The velocity of a point is said to be accelerated or retarded 
 according as it increases or diminishes, but the word acceleration is 
 generally used in either sense, on the understanding that we may 
 regard its quantity as either positive or negative : and (§ 34) is 
 farther generalized so as to include change of direction as well as 
 change of speed. Acceleration of velocity may of course be either 
 
KINEMATICS. 9 
 
 uniform or variable. It is said to be uniform when the point receives 
 equal increments of velocity in equal times, and is then measured by 
 the actual increase of velocity per unit of time. If we choose as the 
 unit of acceleration that which adds a unit of velocity per unit of 
 time to the velocity of a point, an acceleration measured by a will 
 add a units of velocity in unit of time — and, therefore, a t units of 
 velocity in / units of time. Hence if v be the change in the velocity 
 during the interval /, 
 
 V = at, or a =^ J. 
 
 33. Acceleration is variable when the point's velocity does not 
 receive equal increments in successive equal periods of time. It is 
 then measured by the increment of velocity, which would have been 
 generated in a unit of time had the acceleration remained throughout 
 that unit the same as at its commencement. The average accelera- 
 tion during any time is the whole velocity gained- during that time, 
 divided by the time. In Newton's notation v is used to express the 
 acceleration in the direction of motion ; and, \i v = s as in § 28, we 
 have a = v = 's. 
 
 34. But there is another form in which acceleration may manifest 
 itself. Even if a point's velocity remain unchanged, yet if its direc- 
 tion of motion change, the resolved parts of its velocity in fixed 
 directions will, in general, be accelerated. 
 
 Since acceleration is merely a change of the component velocity 
 in a stated direction, it is evident that the laws of composition and 
 resolution of accelerations are the same as those of velocities. 
 
 We therefore expand the definition just given, thus : — Acceleration 
 is the rate of change of velocity ivhether that change take place in the 
 directiofi of motion or not, 
 
 35. What is meant by change of velocity is evident from § 31. 
 For if a velocity OA become OC^ its change is AC, or OB. 
 
 Hence, just as the direction of motion of a point is the tangent to 
 its path, so the direction of acceleration of a moving 
 point is to be found by the following construction : — 
 
 From any point O draw lines OP, OQ, etc., repre- 
 senting in magnitude and direction the velocity of the 
 moving point at every instant. (Compare § 49.) The 
 points, F, Q, etc., must form a continuous curve, for 
 (§ 7) OF cannot change abruptly in direction. Now 
 if ^ be a point near to F, OF and OQ represent two 
 successive values of the velocity. Hence FQ is the 
 whole change of velocity during the interval. As the O 
 interval becomes smaller, the direction FQ more and more nearly 
 becomes the tangent at F. Hence the direction of acceleration is 
 that of the tangent to the curve thus described. 
 
 The magnitude of the acceleration is the rate of change of velocity, 
 and is therefore measured by the velocity of F in the curve FQ. 
 
lo PRELIMINARY, 
 
 36. Let a point describe a circle, ABD^ radius R^ with uniform 
 velocity V, Then, to determine the direction of acceleration, we 
 must draw, as below, from a fixed point (9, lines OP, OQ, etc., 
 representing the velocity at A, B, etc., in direction and magnitude. 
 Since the velocity in ABD is constant, all the lines OP, OQ, etc., 
 
 will be equal (to F), and there- 
 fore PQS is a circle whose 
 centre is O. The direction of 
 acceleration at A is parallel to 
 the tangent at P, that is, is per- 
 pendicular to OP, i.e. to Aa, 
 and is therefore that of the 
 radius AC. 
 
 Now P describes the circle 
 PQS, while A describes ABD. 
 Hence the velocity of P is to 
 that of ^ as OP to CA, i.e. as Fto R; and is therefore equal to 
 
 r'^'^'r^ 
 
 and this (§.35) is the amount of the acceleration in the circular path 
 ABI?. 
 
 37. The whole acceleration in any direction is the sum of the 
 components (in that direction) of the accelerations parallel to any 
 three rectangular axes — each component acceleration being found 
 by the same rule as component velocities (§ 34), that is, by multiply- 
 ing by the cosine of the angle between the direction of the accelera- 
 tion and the line along which it is to be resolved. 
 
 38. When a point moves in a curve the whole acceleration may 
 be resolved into two parts, one in the direction of the motion and 
 equal to the acceleration of the velocity; the other towards the 
 centre of curvature (perpendicular therefore to the direction of mo- 
 tion), whose magnitude is proportional to the square of the velocity 
 and also to the curvature of the path. The former of these changes 
 the velocity, the other affects only the form of the path, or the 
 direction of motion. Hence if a moving point be subject to an 
 acceleration, constant or not, whose direction is continually perpen- 
 dicular to the direction of motion, the velocity will not be altered — 
 and the only effect of the acceleration will be to make the point 
 move in a curve whose curvature is proportional to the acceleration 
 at each instant, and inversely as the square of the velocity. 
 
 39. In other words, if a point move in a curve, whether with a 
 uniform or a varying velocity, its change of direction is to be re- 
 garded as constituting an acceleration towards the centre of curva- 
 ture, equal in amount to the square of the velocity divided by the 
 radius of curvature. The whole acceleration will, in every case, be 
 the resultant of the acceleration thus measuring change of direction 
 and the acceleration of actual velocity along the curve. 
 
KINEMATICS. ii 
 
 40. If for any case of motion of a point we have given the whole 
 velocity and its direction, or simply the components of the velocity 
 in three rectangular directions, at any tb7ie^ or, as is most commonly 
 the case, for any position ; the determination of the form of the path 
 described, and of other circumstances of the motion, is a question of 
 pure mathematics, and in all cases is capable (if not of an exact 
 solution, at all events) of a solution to any degree of approximation 
 that n^y be desired. 
 
 This is true also if the total acceleration and its direction at every 
 instant, or simply its rectangular components, be given, provided the 
 velocity and its direction, as well as the position of the point, at any 
 one instant be given. But these are, in general, questions requiring 
 for their solution a knowledge of the integral calculus. 
 
 41. From the principles already laid down, a great many interest- 
 ing results may be deduced, of which we enunciate a few of the 
 simpler and more important. 
 
 {a) If the velocity of a moving point be uniform, and if its direction 
 revolve uniformly in a plane, the path described is a circle. 
 
 (^) If a point moves in a plane, and its component velocity 
 parallel to each of two rectangular axes is proportional to its dis- 
 tance from that axis, the path is an ellipse or hyperbola whose 
 principal diameters coincide with those axes; and the acceleration 
 is directed to or from the centre of the curve at every instant 
 (§§ 66, 78). 
 
 {c) If the components of the velocity parallel to each axis be equi- 
 multiples of the distances from the other axis, the path is a straight 
 line passing through the origin. 
 
 {d) When the velocity is uniform, but in a direction revolving 
 uniformly in a right circular cone, the motion of the point is in a 
 circular helix whose axis is parallel to that of the cone. 
 
 42. When a point moves uniformly in a circle of radius R^ with 
 velocity F", the whole acceleration is directed towards the centre, and 
 
 has the constant value -^. See § 36. 
 
 43. With uniform acceleration in the direction of motion, a point 
 describes spaces proportional to the squares of the times elapsed 
 since the commencement of the motion. This is the case of a body 
 falling vertically in vacuo under the action of gravity. 
 
 In this case the space described in any interval is that which would 
 be described in the same time by a point moving uniformly with a 
 velocity equal to that at the middle of the interval. In other words, 
 the average velocity (when the acceleration is uniform) is, during any 
 interval, the arithmetical mean of the initial and final velocities. For, 
 since the velocity increases uniformly, its value at any time before the 
 middle of the interval is as much less than this mean as its value 
 at the same time after the middle of the interval is greater than the 
 
PRELIMINARY. 
 
 mean : and hence Its value at the middle of the Interval must be the 
 mean of its first and last values. 
 
 In symbols ; if at time / = o the velocity was F, then at time / it is 
 
 v= F+af. 
 Also the space (x) described is equal to the product of the time by the 
 average velocity. But we have just shown that the average velocity is 
 
 and therefore x = Vt + \at^. 
 
 Hence, by algebra, 
 
 V +2ax = V'+2 Vat + a'f - ( V-\- atf = v\ 
 or 1 z;- - 1 F^ = ax. 
 
 If there be no initial velocity our equations become 
 v-at^ x-\af, \v^ = ax. 
 
 Of course the preceding formulae apply to a constant retardation, as 
 in the case of a projectile moving vertically upwards, by simply giving 
 a a negative sign. 
 
 44. When there Is uniform acceleration in a constant direction, 
 the path described is a parabola, whose axis is parallel to that 
 direction. This is the case of a projectile moving in vacuo. 
 
 For the velocity ( V) in the original direction of motion remains 
 unchanged ; and therefore, in time /, a space Vt is described parallel 
 to this line. But in the same Interval, by the above reasoning, we see 
 that a space \af is described parallel to the direction of acceleration. 
 Hence, if AP be the direction of motion at A^ AB the direction 
 of acceleration, and Q the position of the point at time t\ 
 draw QP parallel to BA, meeting AP m 
 P\ then 
 
 AP=Vt, PQ^\at\ 
 Hence 
 
 AP' = ^^PQ. 
 
 This is a property of a parabola, of which 
 the axis is parallel to AB; AB being a 
 diameter, and AP a tangent. ' If (9 be 
 the focus of this curve, we know that 
 
 AP' = ^OA.PQ. 
 Hence 
 
 0A=—, 
 2a 
 
 B 
 
 Also OA Is known In direction, for AP 
 between the focal distance of a point and 
 
 and is therefore known 
 bisects the angle, OAC, 
 the diameter through it. 
 
 45. When the acceleration, whatever (and however varying) be 
 its magnitude, is directed to a fixed point, the path is in a plane 
 
KINEMATICS. 
 
 13 
 
 passing through that point ; and in this plane the areas traced out by 
 the radius-vector are proportional to the times employed. 
 
 Evidently there is no acceleration perpendicular to the plane con- 
 taining the fixed point and the line of motion of the moving point 
 at any instant; and there being no velocity perpendicular to this 
 plane at starting, there is therefore none throughout the motion; 
 thus the point moves in the plane. For the proof of the second 
 part of th^ proposition we must make a slight digression. 
 
 46. The Moment of a velocity or of a force about any point is 
 the product of its magnitude into the perpendicular from the point 
 upon its direction. The moment of the resultant velocity of a par- 
 ticle about any point in the plane of the components is equal to the 
 algebraic sum of the moments of the components, the proper sign of 
 each moment depending on the direction of motion about the point. 
 The same is true of moments of forces and of moments of momentum, 
 as defined in Chapter II. ^ 
 
 First, consider two component motions, AB and A C, and let AD 
 be their resultant (§ 31). Their half-moments round the point O 
 are respectively the areas OAB, OCA. Now OCA, together with 
 half the area of the parallelogram CABB, is equal to OBD. Hence 
 the sum of the two half-moments together ^ 
 
 with half the area of the parallelogram is .^^..^ 
 
 equal to A OB together with BOD, that is /jW 
 
 to say, to the area of the whole figure / j \ \. 
 
 OABD. But ABD, a part of this figure, 
 is equal to half the area of the parallelo- 
 gram; and therefore the remainder, OAD, 
 is equal to the sum of the two half-mo- 
 ments. But OAD is half the moment of the •4. B 
 resultant velocity round the point O. Hence the moment of the 
 resultant is equal to the sum of the moments of the two components. 
 By attending to the signs of the mom.ents, we see that the proposi- 
 tion holds when O is within the angle CAB. 
 
 If there be any number of component rectilineal motions, we may 
 compound them in order, any two taken together first, then a third, 
 and so on; and it follows that the sum of their moments is equal to 
 the moment of their resultant. It follows, of course, that the sum of 
 the moments of any number of component velocities, all in one plane, 
 into which the velocity of any point may be resolved, is equal to the 
 moment of their resultant, round any point in their plane. It follows 
 also, that if velocities, in different directions all in one plane, be suc- 
 cessively given to a moving point, so that at any time its velocity is 
 their resultant, the moment of its velocity at any time is the sum of the 
 moments of all the velocities which have been successively given to it. 
 
 47. Thus if one of the components always passes through the 
 point, its moment vanishes. This is the case of a motion in which 
 the acceleration is directed to a fixed point, and we thus prove the 
 second theorem of § 45, that in the case supposed the areas described 
 
14 PRELIMINARY. 
 
 by the radius-vector are proportional to the times; for, as we have 
 seen, the moment of the velocity is double the area traced out by the 
 radius-vector in unit of time. 
 
 48. Hence in this case the velocity at any point is inversely as 
 the perpendicular from the fixed point to the tangent to the path or 
 the momentary direction of motion. 
 
 For the product of this perpendicular and the velocity at any 
 instant gives double the area described in one second about the 
 fixed point, which has just been shown to be a constant quantity. 
 
 As the kinematical propositions with which we are dealing have 
 important bearings on Physical Astronomy, we enunciate here Kepler's 
 Laws of Planetary Motion. They were deduced originally from 
 observation alone, but Newton explained them on physical principles 
 and showed that they are applicable to comets as well as to planets. 
 
 I. Each planet describes an Ellipse [with comets this may be any 
 Conic Section] of which the Sun occupies one focus. 
 
 II. The radius-vector of each planet describes equal areas in 
 equal times. 
 
 III. The square of the periodic time [in an elliptic orbit] is pro- 
 portional to the cube of the major axis. 
 
 Sections 45 — 47, taken in connexion with the second of these 
 laws, show that the acceleration of the motion of a planet or comet 
 is along the radius-vector. 
 
 49. If, as in § 35, from any fixed point, lines be drawn at every 
 instant representing in magnitude and direction the velocity of a point 
 describing any path in any manner, the extremities of these lines 
 form a curve which is called the Hodograph. The fixed point from 
 which these lines are drawn is called the hodographic origin. The 
 invention of this construction is due to Sir W. R. Hamilton; and 
 one of the most beautiful of the many remarkable theorems to which 
 it leads is this : The Hodograph for the motio?i of a planet or comet is 
 always a circle^ whatever be the form and dimensions of the orbit. The 
 proof will be given immediately. 
 
 It was shown (§35) that an arc of the hodograph represents the 
 change of velocity of the moving point during the corresponding 
 time ; and also that the tangent to the hodograph is parallel to the 
 direction, and the velocity in the hodograph is equal to the amount 
 of the acceleration of the moving point. 
 
 When the hodograph and its origin, and the velocity along it, or 
 the time corresponding to each point of it, are given, the orbit may 
 easily be shown to be determinate. 
 
 [An important improvement in nautical charts has been suggested 
 by Archibald Smiths It consists in drawing a curve, which may 
 be called the tidal hodograph with reference to any point of a chart 
 for which the tidal currents are to be specified throughout the chief 
 tidal period (twelve lunar hours). Numbers from I. to XII. are placed 
 at marked points along the curve, corresponding to the lunar hours. 
 
 * Proc. R. S. 1865. 
 
KINEMATICS, 
 
 15 
 
 — B 
 
 Smith's curve is precisely the Hamiltonian hodograph for an imagi- 
 nary particle moving at each instant with the same velocity and the 
 same direction as the particle of fluid passing, at the same instant, 
 through the point referred to.] 
 
 50. In the case of a projectile (§ 44), the horizontal component 
 of the velocity is unchanged, and the vertical component increases 
 uniformly. Hence the hodograph is a vertical straight line, whose 
 distance from the origin is the horizontal velocity, and which is 
 described urtiformly. 
 
 51. To prove Hamilton's proposition (§ 49), let APB be a portion 
 of a conic section and S one focus. Let P move so that SP 
 describes equal areas in equal times, that is (§ 48), 
 let the velocity be inversely as the perpendicular 
 SY from S to the tangent to the orbit. \i ABP 
 be an ellipse or hyperbola, the intersection of the 
 perpendicular with the tangent lies in the circle 
 YAZ, whose diameter is the major axis. Produce 
 YS to cut the circle again in Z Then YS. SZ is 
 constant, and therefore SZ is inversely as SY, that 
 is, SZ is proportional to the velocity at P. Also 
 SZ is perpendicular to the direction of motion PY, 
 and thus the circular locus of Z is the hodograph turned through a 
 right angle about S in the plane of the orbit. If APB be a parabola, 
 ^ F is a straight line. But if another point UhQ taken in YS pro- 
 duced, so that YS. SU"\s constant, the locus of 6^is easily seen to be a 
 circle. Hence the proposition is generally true for all conic sections. 
 The hodograph surrounds its origin [as the figure shows] if the orbit 
 be an ellipse, passes through it when the orbit is a parabola, and the 
 origin is without the hodograph if the orbit is a hyperbola. 
 
 52. A reversal of the demonstration of § 5 1 shows that, if the 
 acceleration be towards a fixed point, and if the hodograph be a 
 circle, the orbit must be a conic section of which the fixed point 
 is a focus. 
 
 But we may also prove this important proposition as follows: 
 Let A be the centre of the circle, and O the hodographic origin. 
 Join OA and draw the perpendiculars 
 PM to OA and ON to PA. Then OP 
 is the velocity in the orbit : and ON, being 
 parallel to the tangent at P, is the direc- 
 tion of acceleration in the orbit; and is 
 therefore parallel to the radius-vector to 
 the fixed point about which there is equable 
 description of areas. The velocity parallel 
 to the radius-vector is therefore ON, and 
 the velocity perpendicular to the fixed line 
 
 OA is PM. 
 
 ^ , ON OA 
 ^'''-PM=AP 
 
 constant. 
 
i6 PRELIMINARY. 
 
 Hence, in the orbit, the velocity along the radius-vector is pro- 
 portional to that perpendicular to a fixed line : and therefore the 
 radius-vector of any point is proportional to the distance of that 
 point from a fixed line — a property belonging exclusively to the 
 conic sections referred to their focus and directrix. 
 
 53. The path which, in consequence of Aberration^ a fixed star 
 seems to describe, is the hodograph of the earth's orbit, and is 
 therefore a circle whose plane is parallel to the plane of the 
 ecliptic. 
 
 54. When a point moves in any manner, the line joining it with 
 a fixed point generally changes its direction. If, for simplicity, we 
 consider the motion to be confined to a plane passing through the 
 fixed point, the angle which the joining line makes with a fixed line 
 in the plane is continually altering, and its rate of alteration at any 
 instant is called the Angular Velocity of the first point about the 
 second. If uniform, it is of course measured by the angle described 
 in unit of time ; if variable, by the angle which would have been 
 described in unit of time if the angular velocity at the instant in 
 question were maintained constant for so long. In this respect the 
 process is precisely similar to that which we have already explained 
 for the measurement of velocity and acceleration. 
 
 We may also speak of the angular velocity of a moving plane 
 with respect to a fixed one, as the rate of increase of the angle 
 contained by them ; but unless their line of intersection remain fixed, 
 or at all events parallel to itself, a somewhat more laboured statement 
 is required to give a complete specification of the motion. 
 
 55. The unit angular velocity is that of a point which describes, 
 or would describe, unit angle about a fixed point in unit of time. The 
 usual unit angle is (as explained in treatises on plane trigonometry) that 
 which subtends at the centre of a circle an arc whose length is equal 
 
 to the radius; being an angle of = S7"*29578 ... = 57"i7'44"-8 
 
 TT 
 
 nearly. 
 
 56. The angular velocity of a point in a plane is evidently to be 
 found by dividing the velocity perpendicular to the radius-vector by 
 the length of the radius-vector. 
 
 57. When the angular velocity is variable its rate of increase or 
 diminution is called the Angular Acceleration, and is measured with 
 reference to the same unit angle. 
 
 58. When one point describes uniformly a circle about another, 
 the time of describing a complete circumference being JJ we have 
 the angle 27r described uniformly in T\ and, therefore, the angular 
 
 velocity is -= . Even when the angular velocity is not uniform, as in 
 
 a planet's motion, it is useful to introduce the quantity -=, , which is 
 then called the mean angular velocity. 
 
KINEMATICS. 17 
 
 59. When a point moves uniformly in a straight line its angular 
 velocity evidently diminishes as it recedes from the point about 
 which the angles are measured, and it may easily be shown that 
 it varies inversely as the square of the distance from this point. 
 The same proposition is true for miy path, when the acceleration is 
 towards the point about which the angles are measured : being 
 merely a different mode of stating the result of § 48. 
 
 60. The intensity of heat and light emanating from a point, or 
 from a uniformly radiating spherical surface, diminishes according to 
 the inverse square of the distance from the centre. Hence the rate 
 at which a planet receives heat and light from the sun varies in 
 simple proportion to the angular velocity of the radius-vector. Hence 
 the whole heat and light received by the planet in any time is pro- 
 portional to the whole angle turned through by its radius-vector in 
 the same time. 
 
 61. A further instance of this use of the idea of angular velocity 
 may now be given, to solve the problem of finding the hodograph 
 (§35) for any case of motion in which the acceleration is directed to 
 a fixed point, and varies inversely as the square of the distance from 
 that point. The velocity of P, in the hodograph PQ^ being the 
 acceleration in the orbit, varies inversely as the square of the 
 radius-vector; and therefore (§ 59) directly as the 
 angular velocity. Hence the arc of PQ^, described 
 in any time, is proportional to the corresponding 
 angle-vector in the orbit, i.e. to the angle through 
 which the tangent to PQ has turned. Hence (§ 9) 
 the curvature of PQ is constant, or PQ is a circle. 
 
 This demonstration, reversed, proves that if the 
 hodograph be a circle, and the acceleration be to- 
 wards a fixed point, the acceleratioti varies inversely 
 as the square of the distance of the moving point 
 from the fixed point. 
 
 62. From §§ 61, 52, it follows that when a particle moves with 
 acceleration towards a fixed point, varying inversely as the square 
 of the distance, its orbit is a conic section, with this point for one 
 focus. And conversely (§§ 47, 51, 52), if the orbit be a conic sec- 
 tion, the acceleration, if towards either focus, varies inversely as the 
 square of the distance : or, if a point moves in a conic section, 
 describing equal areas in equal times by a radius-vector through 
 a focus, the acceleration is always towards this focus, and varies 
 inversely as the square of the distance. Compare this with the first 
 and second of Kepler's Laws, § 48. 
 
 63. All motion that we are, or can be, acquainted with, is Relative 
 merely. We can calculate from astronomical data for any instant 
 the direction in which, and the velocity with which, we are moving 
 on account of the earth's diurnal rotation. We may compound this 
 with the (equally calculable) velocity of the earth in its orbit. This 
 resultant again we may compound with the (roughly-known) velocity 
 
 r 
 
 ^\ r? {^ A n y '%, 
 
 OF THE ^ 
 
 f UNIVERSITY I 
 
1 8 PRELIMINARY. 
 
 of the sun relatively to the so-called fixed stars ; but, even if all 
 these elements were accurately known, it could not be said that we 
 had attained any idea of an absolute velocity ; for it is only the sun's 
 relative motion among the stars that we can observe; and, in all 
 probability, sun and stars are moving on (it may be with incon- 
 ceivable rapidity) relatively to other bodies in space. We must there- 
 fore consider how, from the actual motions of a set of bodies, we 
 may find their relative motions with regard to any one of them ; and 
 how, having given the relative motions of all but one with regard to 
 the latter, and the actual motion of the latter, we may find the actual 
 motions of all. The question is very easily answered. Consider 
 for a moment a number of passengers walking on the deck of a 
 steamer. Their relative motions with regard to the deck are what we 
 immediately observe, but if we compound with these the velocity of 
 the steamer itself we get evidently their actual motion relatively to 
 the earth. Again, in order to get the relative motion of all with 
 regard to the deck, we eliminate the motion of the steamer alto- 
 gether ; that is, we alter the velocity of each relatively to the earth 
 by compounding with it the actual velocity of the vessel taken in 
 a reversed direction. 
 
 Hence to find the relative motions of any set of bodies with regard 
 to one of their number, imagine, impressed upon each in composition 
 with its own motion, a motion equal and opposite to the motion of 
 that one, which will thus be reduced to rest, while the motions of the 
 others will remain the same with regard to it as before. 
 
 Thus, to take a very simple example, two trains are running in 
 opposite directions, say north and south, one with a velocity of fifty, 
 the other of thirty, miles an hour. The relative velocity of the second 
 with regard to the first is to be found by imagining impressed on 
 both a southward velocity of fifty miles an hour ; the effect of this 
 being to bring the first to rest, and to give the second a southward 
 velocity of eighty miles an hour, which is the required relative 
 motion. 
 
 Or, given one train moving north at the rate of thirty miles an 
 hour, and another moving west at the rate of forty miles an hour. 
 The motion of the second relatively to the first is at the rate of fifty 
 miles an hour, in a south-westerly direction inclined to the due west 
 direction at an angle of tan~^|. It is needless to multiply such 
 examples, as they must occur to every one. 
 
 64. Exactly the same remarks apply to relative as compared with 
 absolute acceleration, as indeed we may see at once, since accelera- 
 tions are in all cases resolved and compounded by the same law as 
 
 velocities. 
 
 65. The following proposition in relative motion is of consider- 
 able importance : — 
 
 Any two moving points describe similar paths relatively to each 
 other and relatively to any point which divides in a constant ratio 
 the line joining them. 
 
KINEMATICS. 19 
 
 Let A and B be any simultaneous positions of the points. Take 
 
 G or G' in AB such that the ratio 7;^-^ or ^^^^ has a constant 
 
 value. Then, as the form of the relative 1 1 1 -+ 
 
 path depends only upon the length and G A. G B 
 
 direction of the line joining the two points at any instant, it is obvious 
 that these will be the same for A with regard to B^ as for B with 
 regard to A^ saving only the inversion of the direction of the joining 
 line. Hence ^'s path about A is A''s> about B turned through two 
 right angles. And with regard to G and G' it is evident that the 
 directions remain the same, while the lengths are altered in a given 
 ratio ; but this is the definition of similar curves. 
 
 66. An excellent example of the transformation of relative into 
 absolute motion is afforded by the family of Cycloids. We shall in 
 a future section consider their mechanical description, by the rolling 
 of a circle on a fixed straight line or circle. In the meantime, we 
 take a different form of statement, which however leads to precisely 
 the same result. 
 
 The actual path of a point which revolves uniformly in a circle 
 about another point — the latter moving uniformly in a straight line 
 or circle in the same plane — belongs to the family of Cycloids. 
 
 67. As an additional illustration of this part of our subject, we 
 may define as follows : 
 
 If one point A executes any motion whatever with reference to 
 a second point B ; li B executes any other motion with reference 
 to a third point C ; and so on — the first is said to execute, with 
 reference to the last, a movement which is the resultant of these 
 several movements. 
 
 The relative position, velocity, and acceleration are in such a case 
 the geometrical resultants of the various components combined 
 according to preceding rules. 
 
 68. The following practical methods of effecting such a com- 
 bination' in the simple case of the movements of two points are 
 useful in scientific illustrations and in certain mechanical arrange- 
 ments. Let two moving points be joined by a uniform elastic string; 
 the middle point of this string will evidently execute a movement 
 which is half the resultant of the motions of the two points. But for 
 drawing, or engraving, or for other mechanical applications, the 
 following method is preferable : — 
 
 CF and ED are rods of equal length moving freely round a pivot 
 at /*, which passes through the middle point 
 of each— C^, AB, EB, and BE are rods of 
 half the length of the two former, and so 
 pivoted to them as to form a pair of equal 
 rhombi CD, EE, whose angles can be altered 
 at will. Whatever motions, whether in a plane, 
 or in space of three dimensions, be given to 
 A and B, /'will evidently be subjected to half 
 their resultant. 
 
20 PRELIMINARY. 
 
 69. Amongst the most important classes of motions which we 
 have to consider in Natural Philosophy, there is one, namely, Har- 
 monic Motion^ which is of such immense use, not only in ordinary 
 kinetics, but in the theories of sound, light, heat, etc., that we make 
 no apology for entering here into some little detail regarding it. 
 
 70. Def. When a point Q moves uniformly in a circle, the per- 
 pendicular QP drawn from its position at any 
 instant to a fixed diameter AA' of the circle, 
 intersects the diameter in a point P^ whose 
 position changes by a simple harmo7iic motion. 
 
 Thus, if a planet or satellite, or one of the 
 constituents of a double star, be supposed to 
 move uniformly in a circular orbit about its 
 primary, and be viewed from a very distant 
 position in the plane of its. orbit, it will appear 
 to move backwards and forwards in a straight 
 line with a simple harmonic motion. This is nearly the case with 
 such bodies as the satellites of Jupiter when seen from the earth. 
 
 Physically, the interest of such motions consists in the fact of their 
 being approximately those of the simplest vibrations of sounding 
 bodies such as a tuning-fork or pianoforte-wire ; whence their name ; 
 and of the various media in which waves of sound, light, heat, etc., 
 are propagated. 
 
 71. The Amplitude of a simple harmonic motion is the range on 
 one side or the other of the middle point of the course, i. e. OA or 
 OA' in the figure. 
 
 An arc of the circle referred to, or any convenient angular reck- 
 oning of it, measured from any fixed point to the uniformly moving 
 point Qy is the Argu??te?it of the harmonic motion. 
 
 [The distance of a point, performing a simple harmonic motion, 
 from the middle of its course or range, is a sifjiple harmonic functio7i 
 of the time; that is to say 
 
 a cos {lit - e), 
 where a, «, e are constants, and / represents time. The argument of 
 this function is what we have defined as the argument of the motion. 
 In the formula above, the argument is nt — e.'\ 
 
 The Epoch in a simple harmonic motion is the interval of time 
 which elapses from the era of reckoning till the moving point first 
 comes to its greatest elongation in the direction reckoned as positive, 
 from its mean position or the middle of its range. [In the formula 
 above, put in the form 
 
 acosn 
 
 H} 
 
 e . 
 
 - is the epoch.] Epoch in angular measure is the angle described 
 n 
 
 on the circle of reference in the period of time defined as the epoch. 
 
 [In the formula, e is the epoch in angular measure.] 
 
 The Period of a simple harmonic motion is the time which elapses 
 
KINEMATICS. 21 
 
 from any Instant until the moving point again moves in the same 
 direction through the same position, and is evidently the time of 
 revolution in the auxiliary circle. [In the formula the period is 
 27r -, 
 n '-' 
 
 The Phase of a simple harmonic motion at any instant is an 
 expression used to designate the part of its whole period which it 
 has reached. It is borrowed from the popular expression * phases of 
 the moon.' Thus for Simple Harmonic Motion we nlay call the 
 first or zero-phase that of passing through the middle position in the 
 positive direction. Then follow the successive phases quarter-period, 
 half-period, three-quarters-period, and complete period or return to 
 zero-phase. Sometimes it is convenient to reckon phase by a number 
 or numerical expression, which may be either a reckoning of angle or 
 a reckoning of time, or a fraction or multiple of the period. Thus 
 the positive maximum phase may sometimes be called the 90*^ phase 
 
 or the phase -, or the three-hour phase, if the period be 1 2 hours, or 
 
 the quarter-period phase. Or, again, the phase of half way down 
 from positive maximum may be described as the 120° phase or the 
 
 — phase, or the \ period phase. This particular way of specifying 
 
 phase is simply a statement of the argument as defined above and 
 measured from the point corresponding to positive motion through 
 the middle position. 
 
 72. Those common kinds of mechanism, for producing rectilineal 
 from circular motion, or vice versa, in which a, crank moving in 
 a circle works in a straight slot belonging to a body which can 
 only move in a straight line, fulfil strictly the definition of a simple 
 harmonic motion in the part of which the motion is rectilineal, if the 
 motion of the rotating part is uniform. 
 
 The motion of the treadle in a spinning-wheel approximates to 
 the same condition when the wheel moves uniformly ; the approxi- 
 mation being the closer, the smaller is the angular motion of the 
 treadle and of the connecting string. It is also approximated to 
 more or less closely in the motion of the piston of a steam-engine 
 connected, by any of the several methods in 
 use, with the crank, provided always the ro- 
 tatory motion of the crank be uniform. 
 
 73. The velocity of a point executing a 
 simple harmonic motion is a simple harmonic 
 function of the time, a quarter of a period 
 earlier in phase than the displacement, and 
 having its maximum value equal to the ve- 
 locity in the circular motion by which the given 
 function is defined. 
 
 For, in the fig., if F be the velocity in the circle, it may be 
 represented by OQ in a direction perpendicular to its own, and 
 
22 
 
 PRELIMINARY. 
 
 therefore by OF and FQ in directions perpendicular to those lines. 
 
 That is, the velocity of F in the simple harmonic motion is 
 
 FQ V 
 
 YY7\ V or -?Yr) ^Q > which, when F passes through O, becomes V. 
 
 74. The acceleration of a point executing a simple hannonic 
 motion is at any time simply proportional to the displacement from 
 the middle point, but in opposite direction, or always towards the 
 middle point. Its maximum value is that with which a velocity 
 equal to that of the circular motion would be acquired in the time 
 in which an arc equal to the radius is described. 
 
 For m the fig., the acceleration of (2 (by § 36) is -^ along QO. 
 
 Supposing, for a moment, QO to represent the magnitude of this 
 
 acceleration, we may resolve it into QF, FO. The acceleration of 
 
 F is therefore represented on the same scale by FO. Its magnitude 
 
 V^ FO F-" 
 is therefore ^^•^^= ^j^^ FO, which is proportional to FO, and 
 
 has at A its maximum value, yr-^, an acceleration under which the 
 
 velocity V would be acquired in the time ~r- as stated. Thus we 
 
 have in simple harmonic motion 
 
 Acceleration _ F^ _47r^ 
 
 Displacement ~ Q^^ T^ 
 where T is the time of describing the circle, or the period of the 
 harmonic motion. 
 
 75. Any two simple harmonic motions in one line, and of one 
 period, give, when compounded, a single simple harmonic motion ; 
 
 of the same period; of amplitude equal 
 to the diagonal of a parallelogram de- 
 scribed on lengths equal to their am- 
 plitudes measured on lines meeting at 
 an angle equal to their difference of 
 epochs; and of epoch differing from 
 their epochs by angles equal to those 
 which this diagonal makes with the 
 two sides of the parallelogram. Let F 
 and jP' be two points executing simple 
 harmonic motions of one period, and in 
 one line B'BCAA'. Let Q and Q be the 
 uniformly moving points in the relative 
 circles. On CQ and CQ describe a 
 parallelogram SQCQ'-, and through S draw SR perpendicular to 
 FA' produced. We have F'R= CP (being projections of the equal 
 and parallel lines QS, CQ, on CR). Hence CR^CF-hCF'; 
 and therefore the point R executes the resultant of the motions F 
 and F'. But CS, the diagonal of the parallelogram, is constant 
 
KINEMATICS. 23 
 
 (since the angular velocities of CQ and CQ are equal, and therefore 
 the angle QCQ' is constant), and revolves with the same angular 
 velocity as CQ or CQ' ; and therefore the resultant motion is simple 
 harmonic, of amplitude CS, and of epoch exceeding that of the 
 motion of F, and falling short of that of the motion of F'y by the 
 angles (2^15" and SCQ' respectively. 
 
 This geometrical construction has been usefully applied by the 
 tidal committee of the British Association for a mechanical tide- 
 indicator (compare § 77 below). An arm CQ turning round C 
 carries an arm QS turning round Q. Toothed wheels, one of them 
 fixed with its axis through C, and the others pivoted on a framework 
 carried by CQ, are so arranged that QS turns very approximately at 
 the rate of once round in 1 2 mean lunar hours, if CQ be turned uni- 
 formly at the rate of once round in 1 2 mean solar hours. Days and 
 half-days are marked by a counter geared to CQ. The distance of 
 S from a fixed line through C shows the deviation from mean sea- 
 level due to the sum of mean solar and mean lunar tides for the time 
 of day and year marked by CQ and the counter. 
 
 76. The construction described in the preceding section exhibits 
 the resultant of two simple harmonic motions, whether of the same 
 period or not. [If they are very nearly, but not exactly, of the same 
 period, the diagonal of the parallelogram will not be constant, but 
 will diminish from a maximum value, the sum of the component 
 amplitudes, which it has at the instant when the phases of the 
 component motions agree ; to a minimum, the difi"erence of those 
 amplitudes, which is its value when the phases differ by half a period. 
 Its direction, which always must be nearer to the greater than to the 
 less of the two radii constituting the sides of the parallelogram, will 
 oscillate on each side of the greater radius to a maximum deviation 
 amounting on either side to the angle whose sine is the less radius 
 divided by the greater, and reached when the less radius deviates by 
 this together with a quarter circumference, from the greater. The 
 full period of this oscillation is the time in which either radius gains 
 a full turn on the other. The resultant motion is therefore not 
 simple harmonic, but is, as it were, simple harmonic with periodi- 
 cally increasing and diminishing amplitude, and with periodical 
 acceleration and retardation of phase. This view is most appropriate 
 for the case in which the periods of the two component motions 
 are nearly equal, but the amplitude of one of them much greater than 
 that of the other. 
 
 To find the amount of the maximum advance and maximum back- 
 wardness of phase, and when they are experienced, let CA be equal 
 to the greater half-amplitude. From A as 
 centre, with AB the less half-amplitude as 
 radius, describe a circle. CB touching this 
 circle represents the most deviated resultant. 
 Hence CBA is a right angle ; and 
 
 smBCA=^. 
 
24 PRELIMINARY. 
 
 The angle BCA thus found is the amount by which the phase of 
 the resultant motion is advanced or retarded relatively to that of the 
 larger component; and the supplement of BCA is the difference of 
 phase of the two components at the time of maximum advance or 
 backwardness of the resultant.] 
 
 77. A most interesting application of this case of the composition 
 of harmonic motions is to the lunar and solar tides ; which, except 
 in tidal rivers, or long channels or bays, follow each very nearly the 
 simple harmonic law, and produce, as the actual result, a variation 
 of level equal to the sum of variations that would be produced by 
 the two causes separately. 
 
 The amount of the lunar tide in the equilibrium theory is about 
 2*1 times that of the solar. Hence the spring tides of this theory 
 are 3-1, and the neap tides only i-i, each reckoned in terms of the 
 solar tide; and at spring and neap tides the hour of high water is 
 that of the lunar tide alone. The greatest deviation of the actual 
 tide from the phases (high, low, or mean water) of the lunar tide 
 alone, is about -95 of a lunar hour, that is, -98 of a solar hour (being 
 the same part of 12 lunar hours that 28° 26', or the angle whose 
 
 sine is -7-, is of 360°). This maximum deviation will be in advance 
 
 or in arrear according as the crown of the solar tide precedes or 
 follows the crown of the lunar tide; and it will be exactly reached 
 when the interval of phase between the two component tides is 3*95 
 lunar hours. That is to say, there will be maximum advance of the 
 time of high water approximately 4I days after, and maximum retar- 
 dation the same number of days before, spring tides. 
 
 78. We may consider next the case of equal amplitudes in the 
 two given motions. If their periods are equal, their resultant is a 
 simple harmonic motion, whose phase is at every instant the mean 
 of their phases, and whose amplitude is equal to twice the amph- 
 tude of either multiplied by the cosine of half the difference of 
 their phase. The resultant is of course nothing when their phases 
 differ by half the period, and is a motion of double amplitude and 
 of phase the same as theirs when they are of the same phase. 
 
 When their periods are very nearly, but not quite, equal (their 
 amphtudes being still supposed equal), the motion passes very slowly 
 from the former (zero, or no motion at all) to the latter, and back, 
 in a time equal to that in which the faster has gone once oftener 
 through its period than the slower has. 
 
 In practice we meet with many excellent examples of this case, 
 which will, however, be more conveniently treated of when we come 
 to apply kinetic principles to various subjects in practical mechanics, 
 acoustics, and physical optics ; such as the marching of troops over a 
 suspension bridge, the sympathy of pendulums or tuning-forks, etc. 
 
 79. If any number of pulleys be so placed that a cord passing 
 from a fixed point half round each of them has its free parts all 
 in parallel lines, and if their centres be moved with simple harmonic 
 
KINEMATICS. 25 
 
 motions of any ranges and any periods in lines parallel to those 
 lines, the unattached end of the cord moves with a complex har- 
 monic motion equal to twice the sum of the given simple har- 
 monic motions. This is the principle of Sir W. Thomson's tide- 
 predicting machine, constructed by the British Association, and order- 
 ed to be placed in South Kensington Museum, availably for general 
 use in calculating beforehand for any port or other place on the sea 
 for which the simple harmonic constituents of the tide have been de- 
 termined by the 'harmonic analysis' applied to previous observa- 
 tions \ We may exhibit, graphically, the various preceding cases of 
 single or compound simple harmonic motions in one line by curves 
 in which the abscissae represent intervals of time, and the ordinates 
 the corresponding distances of the moving point from its mean 
 position. In the case of a single simple harmonic motion, the 
 corresponding curve would be that described by the point P in 
 § 70, if, Avhile Q maintained its uniform circular motion, the circle 
 were to move with uniform velocity in any direction perpendicular 
 to OA. This construction gives the harmonic curve, or curve of 
 sines, in which the ordinates are proportional to the sines of the 
 abscissae, the straight line in which O moves being the axis of 
 abscissae. It is the simplest possible form assumed by a vibrating 
 string ; and when it is assumed that at each instant the motion 
 of every particle of the string is simple harmonic. When the 
 harmonic motion is complex, but in one line, as is the case 
 for any point in a violin-, harp-, or pianoforte-string (differing, 
 as these do, from one another in their motions on account 
 of the different modes of excitation used), a similar construction 
 may be made. Investigation regarding complex harmonic functions 
 has led to results of the highest importance, having their most 
 general expression in Fourier's Theorem^ to be presently enunciated. 
 We give below a graphic representation of the composition of two 
 simple harmonic motions in one line, of equal amplitudes and of 
 periods which are as i : 2 and as 2 13, the epochs being each 
 a quarter circumference. The horizontal line is the axis of ab- 
 scissae of the curves ; the vertical line to the left of each being the 
 axis of ordinates. In the first case the slower motion goes through 
 1:2 2:3 
 
 (Octave) (Fifth) 
 
 ^ . 
 
 1 See British Association Tidal Committee's Reports, 1868, 1872, 1875 • ^"^ 
 Lecture on Tides, by Sir \V. Thomson (Collins, Glasgow, 1876). 
 
26 
 
 PRELIMINARY. 
 
 one complete period, in the second it goes through two periods. 
 These and similar cases when the periodic times are not commen- 
 surable, will be again treated of under Acoustics. 
 
 80. We have next to consider the composition of simple har- 
 monic motions in different directions. In the first place, we see 
 that any number of simple harmonic motions of one period, and 
 of the same phase, superimposed, produce a single simple harmonic 
 motion of the same phase. For, the displacement at any instant 
 being, according to the principle of the composition of motions, the 
 geometrical resultant of the displacements due to the component 
 motions separately, these component displacements in the case sup- 
 posed, all vary in simple proportion to one another, and are in 
 constant directions. Hence the resultant displacement will vary 
 in simple proportion to each of them, and will be in a constant 
 direction. 
 
 But if, while their periods are the same, the phases of the several 
 component motions do not agree, the resultant motion will generally 
 be elliptic, with equal areas described in equal times by the radius- 
 vector from the centre ; although in particular cases it may be uni- 
 form circular, or, on the other hand, rectilineal and simple harmonic. 
 
 81. To prove this, we may first consider the case, in which we 
 have two equal simple harmonic motions given, and these in per- 
 pendicular lines, and differing in phase by a quarter period. Their 
 resultant is a uniform circular motion. For, let BA^ B'A' be their 
 ranges ; and from 6>, their common middle point as centre, describe 
 
 a circle through AA' BB'. The given motion 
 of P in BA will be (§67) defined by the 
 motion of a point Q, round the circumference 
 of this circle ; and the same point, if moving 
 in the direction indicated by the arrow, will 
 give a simple harmonic motion of P', in 
 BA\ a quarter of a period behind that of 
 the motion oi P in BA. But, since A'OA, 
 QPO, and QP'O are right angles, the figure 
 QPOP is a parallelogram, and therefore Q is in the position of the 
 displacement compounded of OP and OP'. Hence two equal simple 
 harmonic motions in perpendicular lines, of phases differing by a 
 quarter period, are equivalent to a uniform circular motion of radius 
 equal to the maximum displacement of either singly, and in the direc- 
 tion from the positive end of the range of the component in advance 
 of the other towards the positive end of the range of this latter. 
 
 82. Now, orthogonal projections of simple harmonic motions are 
 clearly simple harmonic with unchanged phase. Hence, if we pro- 
 ject the case of § 81 on any plane, we get motion in an ellipse, of 
 which the projections of the two component ranges are conjugate 
 diameters, and in which the radius-vector from the centre describes 
 equal areas (being the projections of the areas described by the 
 radius of the circle) in equal times. But the plane and position of 
 
- KINEMATICS. 27 
 
 the circle of which this projection is taken may clearly be found so 
 as to fulfil the condition of having the projections of the ranges 
 coincident with any two given mutually bisecting lines. Hence any 
 two given simple harmonic motions, equal or unequal in range, and 
 oblique or at right angles to one another in direction, provided only 
 they differ by a quarter period in phase, produce elliptic motion, 
 having their ranges for conjugate axes, and describing, by the 
 radius-vector from the centre, equal areas in equal times. 
 
 83. Returning to the composition of any number of equal simple 
 harmonic motions in lines in all directions and of all phases : each 
 component simple harmonic motion may be determinately resolved 
 into two in the same line, differing in phase by a quarter period, 
 and one of them having any given epoch. We may therefore reduce 
 the given motions to two sets, differing in phase by a quarter period, 
 those of one set agreeing in phase with any one of the given, or 
 with any other simple harmonic motion we please to choose (i.e. 
 having their epoch anything we please). 
 
 All of each set may (§ 75) be compounded into one simple har- 
 monic motion of the same phase, of determinate amplitude, in a de- 
 terminate line ; and thus the whole system is reduced to two simple 
 fully-determined harmonic motions differing from one another in 
 phase by a quarter period. 
 
 Now the resultant of two simple harmonic motions, one a quarter 
 of a period in advance of the other, in different lines, has been 
 proved (§ 82) to be motion in an ellipse of which the ranges of the 
 component motions are conjugate axes, and in which equal areas 
 are described by the radius-vector from the centre in equal times. 
 Hence the proposition of § 80. 
 
 84. We must next take the case of the composition of simple 
 harmonic motions of different periods and in different lines. In 
 general, whether these lines be in one plane or not, the line of 
 motion returns into itself if the periods are commensurable ; and if 
 not, not. This is evident without proof 
 
 Also we see generally that the composition of any number of 
 simple harmonic motions in any directions and of any periods, may 
 be effected by adding their components in each of any three rect- 
 angular directions. The final resultant motion is thus fully expressed 
 by formulae giving the rectangular co-ordinates as 'complex harmonic 
 functions ' of the time. 
 
 85. By far the most interesting case, and by far the simplest, 
 is that of two simple harmonic motions of any periods, whose 
 directions must of course be in one plane. 
 
 Mechanical methods of obtaining such combinations will be after- 
 wards described, as well as cases of their occurrence in Optics and 
 Acoustics. 
 
 We may suppose, for simplicity, the two component motions to take 
 place in perpendicular directions. Also, it is easy to see that we can 
 only have a reentering curve when their periods are commensurable. 
 
28 
 
 PRELIMINARY, 
 
 The following figures represent the paths produced by the com- 
 bination of simple harmonic motions of equal amplitude in two rect- 
 angular directions, the periods of the components being as i : 2, 
 
 and the epochs differing successively by o, 
 a circumference. 
 
 I, 2, etc., sixteenths of 
 
 In the case of epochs equal, or differing by a multiple of tt, the 
 curve is a portion of a parabola, and is gone over twice in opposite 
 directions by the moving point in each complete period. 
 
 If the periods be not exactly as i : 2 the form of the path pro- 
 duced by the combination changes gradually from one to another 
 of the series above figured ; and goes through all its changes in the 
 time in which one of the components gains a complete vibration on 
 the other. 
 
 86. Another very important case is that of two pairs of simple 
 harmonic motions in one plane, such that the resultant of each pair 
 is uniform circular motion. 
 
 If their periods are equal, we have a case belonging to those 
 already treated (§ 80), and conclude that the resultant is, in general, 
 motion in an ellipse, equal areas being described in equal times 
 about the centre. As particular cases we may have simple har- 
 monic, or uniform circular, motion. 
 
 If the circular motions are in the sajtie direction, the resultant is 
 evidently circular motion in the same direction. This is the case 
 of the motion of ^ in § 75, and requires no further comment, as 
 its amplitude, epoch, etc., are seen at once from the figure. 
 
KINEMATICS. 29 
 
 87. If the radii of the component motions are equal, and the 
 periods very nearly equal, but the motions in opposite directions, 
 we have cases of great importance in modern physics, one of which 
 is figured below (in general, a non-reentrant curve). 
 
 This is intimately connected with the explanation of two sets of 
 important phenomena, — the rotation of the plane of polarization of 
 light, by quartz and certain fluids on the one hand, and by trans- 
 parent bodies under magnetic forces on the other. It is a case of 
 the hypotrochoid, and its corresponding mode of description will be 
 described in § 104. It may be exhibited experimentally as the path 
 of a pendulum, hung so as to be free to move in any vertical plane 
 tlirough its point of suspension, and containing in its bob a fly-wheel 
 in rapid rotation. 
 
 88. [Before leaving for a time the subject of the composition of 
 harmonic motions, we must enunciate Fourier's Theorem, which is 
 not only one of the most beautiful results of modern analysis, but 
 may be said to furnish an indispensable instrument in the treatment 
 of nearly every recondite question in modern physics. To mention 
 only sonorous vibrations, the propagation of electric signals along 
 a telegraph wire, and the conduction of heat by the earth's crust, 
 as subjects in their generality intractable without it, is to give but 
 a feeble idea of its importance. Unfortunately it is impossible to 
 give a satisfactory proof of it without introducing some rather trouble- 
 some analysis, which is foreign to the purpose of so elementary a 
 treatise as the present. 
 
 The following seems to be the most intelligible form in which it 
 can be presented to the general reader r — 
 
 Theorem. — A complex harmonic function^ with a constant term 
 added, is the proper expression, in mathematical langiiage, for any 
 arbitrary periodic function ; and consequently can express any function 
 whatever between definite values of the variable. 
 
30 PRELIMINARY. 
 
 89. Any arbitrary periodic function whatever being given, the 
 amplitudes and epochs of the terms of a complex harmonic function, 
 which shall be equal to it for every value of the independent variable, 
 may be investigated by the ' method of indeterminate coefficients.' 
 Such an investigation is sufficient as a solution of the problem, — to 
 find a complex harmonic function expressing a given arbitrary 
 periodic function, — when once we are assured that the problem is 
 possible ; and when we have this assurance, it proves that the reso- 
 lution is determinate ; that is to say, that no other complex harmonic 
 function than the one we have found can satisfy the conditions.] 
 
 90. We now pass to the consideration of the displacement of a 
 rigid body or group of points whose relative positions are unalterable. 
 The simplest case we can consider is that of the motion of a plane 
 figure in its own plane, and this, as far as kinematics is concerned, 
 is entirely summed up in the result of the next section. 
 
 91. If a plane figure be displaced in any way in its own plane, 
 there is always (with an exception treated in § 93) one point of it 
 common to any two positions ; that is, it may be moved from any 
 one position to any other by rotation in its own plane about one 
 point held fixed. 
 
 To prove this, let A^ B be any two points of the plane figure in a 
 first position, A\ B' the position of the same two after a displacement. 
 The Hnes AA\ BB' will not be parallel, except in one case to be 
 presently considered. Hence the Hne equidistant from A and A' 
 will meet that equidistant from B and B' in some point O. Join 
 OA, OB, OA', OB'. Then, evidently, because OA' = OA, OB' = OB, 
 and A'B' = AB, the triangles OA'B' and OAB are equal and similar. 
 Hence O is similarly situated with regard to A'B' and AB, and 
 is therefore one and the same point of the plane figure in its 
 two positions. If, for the sake of illustration, 
 ^B we actually trace the angle OAB upon the 
 plane, it becomes OA'B' in the second posi- 
 tion of the figure. 
 
 92. If from the equal angles A' OB', A OB 
 of these similar triangles we take the com- 
 mon part A' OB, we have the remaining 
 angles AOA', BOB' equal, and each of them 
 is clearly equal to the angle through which 
 the figure must have turned round the point O 
 to bring it from the first to the second position. 
 The preceding simple construction therefore enables us not only 
 to demonstrate the general proposition (§91), but also to determine 
 from the two positions of one line AB, A'B' of the figure the 
 common centre and the amount of the angle of rotation. 
 
 93. The lines equidistant from A and A', and from B and B', 
 are parallel if ^^ is parallel to A'B' ) and therefore the construction 
 
KINEMATICS. 
 
 31 
 
 A 
 
 fails, the point O being infinitely- 
 distant, and the theorem becomes 
 nugatory. In this case the motion is 
 in fact a simple translation of the 
 figure in its own plane without rota- 
 tion — since as AB is parallel and equal 
 to A'B\ we have A A' parallel and equal to BB' \ and instead of 
 there being one point of the figure common to both positions, the 
 lines joining the successive positions of every point in the figure are 
 equal and parallel. 
 
 94. It is not necessary to suppose the figure to be a mere flat 
 disc or plane — for the preceding statements apply to any one of a 
 set of parallel planes in a rigid body, moving in any way subject to 
 the condition that the points of any one plane in it remain always in 
 a fixed plane in space. 
 
 95. There is yet a case in which the construction in § 
 nugatory — that is when A A' is parallel 
 to BB', but AB intersects A B'. In 
 this case, however, it is easy to see at 
 once that this point of intersection is the 
 point O required, although the former 
 method would not have enabled us to 
 find it. 
 
 96. Very many interesting applications of this principle may be 
 made, of which, however, few belong strictly to our subject, and we 
 shall therefore give only an example or two. Thus we know that 
 if a line of given length AB move with its extremities always in two 
 fixed lines OA^ OB, any point in it as P describes an ellipse. (This 
 is proved in § 101 below.) It is required to find the direction of 
 motion of P at any instant, i. e. to draw a tangent to the ellipse. 
 BA will pass to its next position by rotating about the point Q ; found 
 by the method of § 91 by drawing per- 
 pendiculars to OA and OB at A and 
 B. Hence P for the instant revolves 
 about Qj and thus its direction of 
 motion, or the tangent to the ellipse, 
 is perpendicular to QP. Also AB in 
 its motion always touches a curve 
 (called in geometry its envelop) ; and 
 the same principle enables us to find 
 the point of the envelop which lies in 
 AB, for the motion of that point must 
 evidently be ultimately (that is for a very small displacement) along 
 AB, and the only point which so moves is the intersection of AB, 
 with the perpendicular to it from Q. Thus our construction would 
 enable us to trace the envelop by points. 
 
3 2 PRELIMINAR K 
 
 97. Again, suppose ABDC to be a jointed frame, AB having 
 a reciprocating motion about A^ and by a link ^Z> turning CD in 
 
 the same plane about C. Deter- 
 
 ■ 4 y ^ mine the relation between the 
 
 angular velocities of AB and 
 CD in any position. Evidently 
 the instantaneous direction of 
 motion of B is transverse to 
 AB^ and oi D transverse to CD — 
 hence if AB, CD produced meet 
 in O, the motion of BD is for an instant as if it turned about O. 
 From this it may easily be seen that if the angular velocity of AB 
 
 be <o, that of CD is -^^-^ -j^^=- w. A similar process is of course 
 UJd CD 
 
 applicable to any combination of machinery, and we shall find it very 
 
 convenient when we come to apply the principle of work in various 
 
 problems of Mechanics. 
 
 Thus in any Lever, turning in the plane of its arms — the rate of 
 
 motion of any point is proportional to its distance from the fulcrum, 
 
 and its direction of motion at any instant perpendicular to the line 
 
 joining it with the fulcrum. This is of course true of the particular 
 
 form of lever called the Wheel and Axle. 
 
 98. Since, in general, any movement of a plane figure in its plane 
 may be considered as a rotation about one point, it is evident that 
 two such rotations may, in general, be compounded into one ; and 
 therefore, of course, the same may be done with any number of 
 rotations. Thus let A and B be the points of the figure about which 
 in succession the rotations are to take place. By rotation about 
 A, B is brought say to B, and by a rotation about B' , A is brought 
 to A'. The construction of § 91 gives us at once the point O and 
 the amount of rotation about it which singly gives the same effect 
 as those about A and B in succession. But there is one case of 
 exception, viz. when the rotations about A and B are of equal 
 
 amount and in opposite directions. In 
 this case A' B' is evidently parallel and 
 equal to AB, and therefore the com- 
 pound result is a tra?islation only. Thus 
 we see that if a body revolve in succes- 
 sion through equal angles, but in oppo- 
 site directions, about two parallel axes, it finally takes a position 
 to which it could have been brought by a simple translation per- 
 pendicular to the lines of the body in its initial or final position, 
 which were successively made axes of rotation ; and inclined to their 
 plane at an angle equal to half the supplement of the common angle 
 of rotation. 
 
 99. Hence to compound into an equivalent rotation a rotation 
 and a translation, the latter being effected parallel to the plane of 
 the former, we may decompose the translation into two rotations 
 
KINEMATICS. 33 
 
 of equal amounts and opposite directions, compound one of them 
 
 with the given rotation by § 98, and then compound the other with 
 
 the resultant rotation by the same process. Or we may adopt the 
 
 following far simpler method : — Let 
 
 OA be the translation common to 
 
 all points in the plane, and let 
 
 BOC be the angle of rotation k^, ' 
 
 about O, BO being drawn so that JB 
 
 OA bisects the exterior angle COB'. Evidently there is a point 
 
 B' m. BO produced, such that B'C\ the space through which the 
 
 rotation carries it, is equal and opposite to OA, This point retains 
 
 its former position after the performance of the compound operation ; 
 
 so that a rotation and a translation in one plane can be compounded 
 
 into an equal rotation about a different axis. 
 
 100. Any motion whatever of a plane figure in its own plane 
 might be produced by the rolling of a curve fixed to the figure 
 upon a curve fixed in the plane. 
 
 For we may consider the whole motion as made up of successive 
 elementary displacements, each of which corresponds, as we have 
 seen, to an elementary rotation about some 
 point in the plane. Let 6>,, O^, O^, etc., 
 be the successive points of the figure about 
 which the rotations take place, 0^, o^, 0.^, 
 etc., the positions of these points on the 
 plajie when each is the instantaneous centre 
 of rotation. Then the figure rotates about 
 (9j (or ^j, which coincides with it) till O^ 
 coincides with 0^^ then about the latter till 
 O^ coincides with ^3, and so on. Hence, if 
 
 we join (9^, O^, O3, etc., in the plane of the figure, and ^,, 0^, 0^, etc., 
 in the fixed plane, the motion will be the same as if the polygon 
 O^O^O^, etc., rolled upon the fixed polygon o^o^o^, etc. By 
 supposing the successive displacements small enough, the sides 
 of these polygons gradually diminish, and the polygons finally 
 become continuous curves. Hence the theorem. 
 
 From this it immediately follows, that any displacement of a rigid 
 solid, which is in directions wholly perpendicular to a fixed hne, 
 may be produced by the rolling of a cylinder fixed in the solid on 
 another cylinder fixed in space, the axes of the cylinders being 
 parallel to the fixed line. 
 
 101. As an interesting example of this theorem, let us recur to the 
 case of § 96 : — A circle may evidently be circumscribed about OBQA; 
 and it must be of invariable magnitude, since in it a chord of given 
 length AB subtends a given angle O at the circumference. Also OQ is 
 a diameter of this circle, and is therefore constant. Hence, as Q is 
 momentarily at rest, the motion of the circle circumscribing OBQA 
 is one of internal rolling on a circle of double its diameter. Hence 
 if a circle roll internally on another of twice its diameter any point in 
 
 T. X 
 
34 
 
 PRELIMINARY. 
 
 its circumference describes a dianieter of the fixed circle, any other 
 point in its plane an ellipse. This is precisely the same proposition 
 as that of § 86, although the ways of arriving at it are very different. 
 
 102. We may easily employ this result, to give the proof, promised 
 in § 96, that the point P oi AB describes an ellipse. Thus let 
 OAy OB be the fixed lines, in which the extremities of AB move. 
 Draw the circle A OBD, circumscribing A OB, and let CD be the 
 diameter of this circle which passes through P. While the two 
 points A and B of this circle move along OA and OB, the points 
 C and D must, because of the invariability of the angles BOD, 
 
 AOC, move along straight lines OC, 
 OD, and these are evidently at right 
 angles. Hence the path of P may 
 be considered as that of a point in 
 a line whose ends move on two 
 mutually perpendicular lines. Let E 
 be the centre of the circle ; join OE, 
 and produce it to meet, in F, the 
 line FPG drawn through P parallel 
 to DO. Then evidently EF^EP, 
 hence F describes a circle about O. 
 Also FP : EG :: 2FE : FO, or PG 
 is a constant submultiple oi EG; and 
 therefore the locus of P is an ellipse 
 whose major axis is a diameter of the 
 circular path oi F. Its semi-axes are DP dXong OC, and i^C along OD. 
 
 103. When a circle rolls upon a straight line, a point in its 
 circumference describes a Cycloid, an internal point describes a 
 Prolate Cycloid, an external point a Curtate Cycloid. The two 
 latter varieties are sometimes called Trochoids. 
 
 The general form of these curves will be seen in the succeeding 
 figures; and in what follows we shall confine our remarks to the 
 cycloid itself, as it is of greater consequence than the others. The 
 next section contains a simple investigation of those properties of 
 the cycloid which are most useful in our subject. 
 
 i 
 
104. Let AB be a diameter of the generating (or rolling) circle, 
 BC the line on which it rolls. The points A and B describe similar 
 and equal cycloids, of which AQC and ^^S* are portions, li FQR 
 be any subsequent position of the generating circle, Q and S the 
 new positions of A and B, QFS is of course a right angle. If, 
 therefore, QR be drawn parallel to 
 FS, FR is a diameter of the rolling 
 circle, and R lies in a straight line 
 AH drawn parallel to BC. Thus 
 AR = BF. Produce QR to 7] 
 making FT= QR^FS. Evidently 
 the curve AT, which is the locus 
 of T, is similar and equal to BS, 
 and is therefore a cycloid similar and ^ 
 equal to AC. But QR is perpen- 
 dicular to FQ, and is therefore the 
 instantaneous direction of motion of 
 Q, or is the tangent to the cycloid 
 AQC. Similarly, FS is perpendicular to the cycloid ^^S* at S, and 
 therefore TQ is perpendicular to ^7" at T. Hence ($22) AQC is 
 the evolute oi AT, and 2.rc AQ= QT=2QR, 
 
 105. When a circle rolls upon an- 
 other circle, the curve described by a 
 point in its circumference is called an 
 Epicycloid, or a Hypocycloid, as the 
 rolling circle is without or within the 
 fixed circle; and when the tracing-point 
 is not in the circumference, we have 
 Epitrochoids and Hypotrochoids. Of 
 the latter classes we have already met 
 with examples (§§ 87, loi), and others 
 will be presently mentioned. Of the 
 former we have, in the first of the 
 appended figures, the case of a circle 
 rolling externally on another of equal 
 size. The curve in this case is called 
 the Cardioid. 
 
 In the second figure a circle rolls ex- 
 
36 
 
 PRELIMINARY. 
 
 ternally on another of twice its radius. The epicycloid so described 
 
 is of importance in optics, and will, with others, be referred to when 
 
 we consider the subject of Caustics by reflexion. 
 
 In the third figure we have a hypo- 
 cycloid traced by the rolling of one 
 circle internally on another of four 
 times its radius. 
 
 The curve of § 87 is a hypotrochoid 
 described by a point in the plane of a 
 circle which rolls internally on another 
 of rather more than twice its diameter, 
 the tracing-point passing through the 
 centre of the fixed circle. Had the 
 diameters of the circles been exactly as 
 I : 2, § loi shows us that this curve 
 
 would have been reduced to a single straight line. 
 
 106. If a rigid body move in any way whatever, subject only to 
 the condition that one of its points remains fixed, there is always 
 (without exception) one line of it through this point common to the 
 body in any two positions. 
 
 Consider a spherical surface within the body, with its centre at the 
 fixed point C. All points of this sphere attached to the body will 
 move on a sphere fixed in space. Hence the construction of § 91 
 may be made, only with great circles instead of straight lines ; 
 and the same reasoning will apply to prove that the point O thus 
 obtained is common to the body in its two positions. Hence every 
 point of the body in the line (9C, joining O with the fixed point, 
 must be common to it in the two positions. Hence the body may 
 pass from any one position to any other by a definite amount of 
 rotation about a definite axis. And hence, ako, successive or simul- 
 taneous rotations about any number of axes through the fixed point 
 may be compounded into one such rotation. 
 
 107. Let OA^ OB be two axes about which a body revolves with 
 angular velocities o>, w^ respectively. 
 
 With radius unity describe the arc AB^ and in it take any point /. 
 Draw /a, //? perpendicular to OA, OB respectively. Let the rota- 
 tions about the two axes be such that that about 
 OB tends to raise I above the plane of the 
 paper, and that about OA to depress it. In an 
 infinitely short interval of time r, the amounts of 
 these displacements will be Wj/^ . t and — oo/a . t. 
 The point /, and therefore every point in the 
 line 01, will be at rest during the interval t if 
 the sum of these displacements is zero — i.e. if 
 
 o)j . //3 = (0 . 7a. 
 
 Hence the line 01 is instantaneously at rest, or the two rotations 
 about OA and OB may be compounded into one about 01. Draw 
 
KINEMA TICS. 3 7 
 
 Ip, Iq, parallel to OB, OA respectively. Then, expressing in two 
 ways the area of the parallelogram IpOq, we have 
 
 Oq,ip = Op.Ia. 
 
 Hence Oq -. Op :: m^ -. w. 
 
 In words, if on the axes OA, OB, we measure off from O lines 
 Op, Oq, proportional respectively to the angular velocities about 
 these axes — the diagonal of the parallelogram of which these are 
 contiguous sides is the resultant axis. 
 
 Again, if Bb be drawn perpendicular to OA, and if O be the 
 angular velocity about 01, the whole displacement oi B may evidently 
 be represented either by 
 
 oi.Bbora. 7/3. 
 Hence n : oi :: Bb : I^ 
 
 :: 0/ : Op. 
 
 And thus on the scale on which Op, Oq represent the component 
 angular velocities, the diagonal 01 represents their resultant. 
 
 108. Hence rotations are to be compounded according to the 
 same law as velocities, and therefore the single angular velocity, 
 equivalent to three co-existent angular velocities about three mutually 
 perpendicular axes, is determined in magnitude, and the direction 
 of its axis is found, as follow* : — The square of the resultant angular 
 velocity is the sum of the squares of its components, and the ratios 
 of the three components to the resultant are the direction-cosines 
 of the axis. 
 
 Hence also, an angular velocity about any line may be resolved 
 into three about any set of rectangular lines, the resolution in each 
 case being (like that of simple velocities) effected by multiplying by 
 the cosine of the angle between the directions. 
 
 Hence, just as in § 38 a uniform acceleration, acting perpendi- 
 cularly to the direction of motion of a point, produces a change in 
 the direction of motion, but does not influence the velocity; so, if 
 a body be rotating about an axis, and be subjected to an action 
 tending to produce rotation about a perpendicular axis, the result 
 will be a change of direction of the axis about which the body 
 revolves, but no change in the angular velocity. On this kinematical 
 principle is founded the dynamical explanation of the precession of 
 the equinoxes, and some of the seemingly marvellous performances 
 of gyroscopes and gyrostats. 
 
 109. If a pyramid or cone of any form roll on a similar pyramid 
 (the image in a plane mirror of the first position of the first) all 
 round, it clearly comes back to its primitive position. This (as all 
 rolling of cones) is exhibited best by taking the intersection of each 
 with a spherical surface. Thus we see that if a spherical polygon 
 turns about its angular points in succession, always keeping on the 
 spherical surface, and if the angle through which it turns about each 
 point is twice the supplement of the angle of the polygon, or, which 
 
38 PRELIMINARY. 
 
 will come to the same thing, if it be in the other direction, but 
 equal to twice the angle itself of the polygon, it will be brought to 
 its original position, 
 
 110. The method of § loo also applies to the case of § io6; and 
 it is thus easy to show that the most general motion of a spherical 
 figure on a fixed spherical surface is obtained by the rolling of a 
 curve fixed in the figure on a curve fixed on the sphere. Hence as 
 at each instant the line joining C and O contains a set of points of 
 the body which are momentarily at rest, the most general motion of 
 a rigid body of which one point is fixed consists in the rolling of a 
 cone fixed in the body upon a cone fixed in space — the vertices of 
 both being at the fixed point. 
 
 111. To complete our kinematical investigation of the motion of 
 a body of which one point is fixed, we require a solution of the fol- 
 lowing problem: — From the given angular velocities of the body 
 at each instant about three rectangular axes attached to it to de- 
 termine the position of the body in space after a given time. But 
 the general solution of this problem demands higher analysis than 
 can be admitted into the present treatise. 
 
 112. We shall next consider the most general possible motion of 
 a rigid body of which no point is fixed — and first we must prove 
 the following theorem. There is on^ set of parallel planes in a 
 rigid body which are parallel to each other in any two positions of 
 the body. The parallel lines of the body perpendicular to these 
 planes are of course parallel to each other in the two positions. 
 
 Let C and C be any point of the body in its first and second 
 positions. Move the body without rotation from its second position 
 to a third in which the point at C in the second position shall 
 occupy its original position C. The preceding demonstration shows 
 that there is a line CO common to the body in its first and third 
 positions. Hence a line CO' of the body in its second position is 
 parallel to the same line CO in the first position. This of course 
 clearly applies to every line of the body parallel to CO, and the 
 planes perpendicular to these lines also remain parallel. 
 
 113. Let S denote a plane of the body, the two positions of which 
 are parallel. Move the body from its first position, without rotation, 
 in a direction perpendicular to *S, till S comes into the plane of its 
 second position. Then to get the body into its actual position, such 
 a motion as is treated in § 91 is farther required. But by § 91 this 
 may be effected by rotation about a certain axis perpendicular to the 
 plane *S, unless the motion required belongs to the exceptional case 
 of pure translation. Hence (this case excepted), the body may 
 be brought from the first position to the second by translation 
 through a determinate distance perpendicular to a given plane, and 
 rotation through a determinate angle about a determinate axis per- 
 pendicular to that plane. This is precisely the motion of a scrc^v 
 in its nut. 
 
KINEMATICS. 39 
 
 114. To understand the nature of this motion we may com- 
 mence with the shding of one straight-edged board on another. 
 
 Thus let GDEF be a plane board whose edge, DE, sUdes on 
 the edge, AB, of another board, ABC, of which for convenience 
 we suppose the edge, AC, to be hori- 
 zontal. By § 30, if the upper board 
 move horizontally to the right, the 
 constraint will give it, in addition, a 
 vertically upward motion, and the rates 
 of these motions are in the constant 
 ratio of ^C to CB. Now, if both 
 planes be bent so as to form portions 
 of the surface of a vertical right cylinder, the motion of DF parallel 
 to ^C will become a rotation about the axis of the cylinder, and 
 the necessary accompaniment of vertical motion will remain un- 
 changed. As it is evident that all portions of AB will be equally 
 inclined to the axis of the cylinder, it is obvious that the thread 
 of the screw, which corresponds to the edge, £>E, of the upper 
 board, must be traced on the cylinder so as always to make a con- 
 stant angle with its generating lines (§ 128). A hollow mould 
 taken from the screw itself forms what is called the nut — the re- 
 presentative of the board, ABC — and it is obvious that the screw 
 cannot move without rotating about its axis, if the nut be fixed. 
 If a be the radius of the cylinder, w the angular velocity, a the 
 inclination of the screw thread to a generating line, u the linear 
 velocity of the axis of the screw, we see at once from the above con- 
 struction that 
 
 aia : u \\ AC ; CB :: sin a : cos a, 
 
 which gives the requisite relation between o> and u. 
 
 115. In the excepted case of § 113, the whole motion consists 
 of two translations, which can of course be compounded into a 
 single one : and thus, in this case, there is no rotation at all, or 
 every plane of it fulfils the specified condition for *S of § 113. 
 
 116. We may now briefly consider the case in which the guiding 
 cones (§ 110) are both circular, as it has important applications to 
 the motion of the earth, the evolutions of long or flattened projec- 
 tiles, the spinning of tops and gyroscopes, etc. The motion in this 
 case may be called Frecessioiial Rotation. The plane through the 
 instantaneous axis and the axis of the fixed cone passes through the 
 axis of the rolling cone. This plane turns round the axis of the 
 fixed cone with an angular velocity O, which must clearly bear a 
 constant ratio to the angular velocity w of the rigid body about its 
 instantaneous axis. 
 
 117. The motion of the plane containing these axes is called the 
 precession in any such case. What we have denoted by O is the 
 angular velocity of the precession, or, as it is sometimes called, the 
 rate of precession. 
 
40 PRELIMINARY. 
 
 The angular motions w, O are to one another inversely as the 
 distances of a point in the axis of the rolling cone from the in- 
 stantaneous axis and from the axis of the fixed cone. 
 
 For, let OA be the axis of the fixed cone, OB that of the rolling 
 cone, and 01 the instantaneous axis. From any point P in OB 
 draw PN perpendicular to 01, and PQ perpendicular to OA. 
 Then we perceive that P moves always in the circle whose centre 
 is (2, radius PQ, and plane perpendicular 
 to OA. Hence the actual velocity of 
 the point /^ is 12 . QP. But, by the prin- 
 ciples explained above (§ no) the velocity 
 , of jP is the same as that of a point moving 
 in a circle whose centre is N, plane per- 
 pendicular to ON, and radius NP, which, 
 as this radius revolves with angular velo- 
 city CO, is 0) . NP. Hence 
 n.QP=i^.NP, 
 
 • or id '. Q, V. QP : NP. 
 
 118. Suppose a rigid body bounded by any curved surface to be 
 touched at any point by another such body. Any motion of one 
 on the other must be of one or more of the forms sliding, rollings 
 or spin7iing. The consideration of the first is so simple as to require 
 no comment. 
 
 Any motion in which the bodies have no relative velocity at the 
 point of contact, must be rolling or spinning, separately or combined. 
 
 Let one of the bodies rotate about successive instantaneous axes, 
 all lying in the common tangent plane at the point of instantaneous 
 contact, and each passing through this point — the other body being 
 fixed. This motion is what we call rolling, or simple rolling, of the 
 movable body on the fixed. 
 
 On the other hand, let the instantaneous axis of the moving body 
 be the common normal at the point of contact. This is pure spin- 
 ning, and does not change the point of contact. 
 
 Let it move, so that the instantaneous axis, still passing through 
 the point of contact, is neither in, nor perpendicular to, the tangent 
 plane. This motion is combined rolling and spinning. 
 
 119. As an example of pure rolling, we may take that of one 
 cylinder on another, the axes being parallel. 
 
 Let p be the radius of curvature of the rolling, o- of the fixed, 
 cylinder ; w the angular velocity of the former, V the linear velocity 
 of the point of contact. We have 
 
 C*3 
 
 For, in the figure, suppose P to be at any time the point of 
 contact, and Q and p the points which are to be in contact after 
 a very small interval t ; O, O the centres of curvature ; POp = 6, 
 PO'Q = cf>. 
 
KINEMATICS. 
 
 41 
 
 Then ^(2 = i^ = space described by point of con- 
 tact. In symbols 
 
 p<^ = o-6»= Vr. 
 
 Also, before 0'Q.2.Xi^ OP can coincide in direc- 
 tion, the former must evidently turn through an angle 
 
 Therefore wt = ^ + <^ ; 
 
 and by eliminating B and ^, and dividing by t, we 
 get the above result. 
 
 It is to be understood here, that as the radii of 
 curvature have been considered positive when both 
 surfaces are convex, the negative sign must be intro- 
 duced for either radius when the corresponding sur- 
 face is concave. 
 
 Hence the angular velocity of the rolling curve is in this case 
 equal to the product of the linear velocity of the point of contact into 
 the sum or difference of the curvatures, according as the curves are 
 both convex, or one concave and the other convex. 
 
 120. We may now take up a few points connected with the curva- 
 ture of surfaces, which are useful in various parts of our subject. 
 
 The tangent plane at any point of a surface may or may not cut 
 it at that point. In the former case, the surface bends away from 
 the tangent plane partly towards one side of it, and partly towards 
 the other, and has thus, in some of its normal sections, curvatures 
 oppositely directed to those in others. In the latter case, the sur- 
 face on every side of the point bends away from the same side of 
 its tangent plane, and the curvatures of all normal sections are 
 similarly directed. Thus we may divide curved surfaces into Anti- 
 clastic and Synclastic. A saddle gives a good example of the 
 former class ; a ball of the latter. Curvatures in opposite directions, 
 with reference to the tangent plane, have of course different signs. 
 The outer portion of the surface of an anchor-ring is synclastic, the 
 inner anticlastic. 
 
 121. Meimier's Theorem. — The curvature of an oblique section 
 of a surface is equal to that of the normal section through the same 
 tangent line multiplied by the secant of the inclination of the planes 
 of the sections. This is evident from the most elementary con- 
 siderations regarding projections. 
 
 122. Elder's Theorem. — There are at every point of a synclastic 
 surface two normal sections, in one of which the curvature is a 
 maximum, in the other a minimum; and these are at right angles 
 to each other. 
 
 In an anticlastic surface there is maximum curvature (but in 
 opposite directions) in the two normal sections whose planes bisect 
 the angles between the lines in which the surface cuts its tangent 
 
42 PRELIMINAR V. 
 
 plane. On account of the difference of sign, these may be con- 
 sidered as a maximum and a minimum. 
 
 Generally the sum of the curvatures at a point, in any two normal 
 planes at right angles to each other, is independent of the position 
 of these planes. 
 
 If - and - be the maximum and minimum curvatures at any 
 p a- 
 
 point, the curvature of a normal section making an angle 9 with the 
 
 normal section of maximum curvature is 
 
 - cos^ 6 + - sin^ 0, 
 P o- 
 
 which includes the above statements as particular cases. 
 
 123. Let F, p be two points of a surface indefinitely near to each 
 other, and let r be the radius of curvature of a normal section passing 
 through them. Then the radius of curvature of an oblique section 
 through the same points, inclined to the former at an angle a, 
 is r cos a (§ i2i). Also the length along the normal section, from 
 P to /, is less than that along the oblique section — since a given 
 chord cuts off an arc from a circle, longer the less is the radius 
 of that circle. 
 
 124. Hence, if the shortest possible line be drawn from one point 
 of a surface to another, its osculating plane, or plane of curvature, 
 is everywhere perpendicular to the surface. 
 
 Such a curve is called a Geodetic line. And it is easy to see that 
 it is the line in which a flexible and inextensible string would touch 
 the surface if stretched between those points, the surface being sup- 
 posed smooth. 
 
 125. A perfectly flexible but inextensible surface is suggested, 
 although not realized, by paper, thin sheet-metal, or cloth, when the 
 surface is plane ; and by sheaths of pods, seed-vessels, or the like, 
 when not capable of being stretched flat without tearing. The process 
 of changing the form of a surface by bending is called * developing.^ 
 But the term ^ JDevelopable Surface' is commonly restricted to such 
 inextensible surfaces as can be developed into a plane, or, in com- 
 mon language, ' smoothed flat.' 
 
 126. The geometry or kinematics of this subject is a great contrast 
 to that of the flexible line (§ i6), and, in its merest elements, presents 
 ideas not very easily apprehended, and subjects of investigation that 
 have exercised, and perhaps even overtasked, the powers of some 
 of the greatest mathematicians. 
 
 127. Some care is required to form a correct conception of what 
 is a perfectly flexible inextensible surface. First let us consider a 
 plane sheet of paper. It is very flexible, and we can easily form 
 the conception from it of a sheet of ideal matter perfectly flexible. 
 
KINEMATICS, 
 
 43 
 
 It is very inextensible ; that is to say, it yields very little to any 
 application of force tending to pull or stretch it in any direction, 
 up to the strongest it can bear without tearing. It does, of course, 
 stretch a little. It is easy to test that it stretches when under the 
 influence of force, and that it contracts again when the force is 
 removed, although not always to its original dimensions, as it 
 may and generally does remain to some sensible extent permanently 
 stretched. Also, flexure stretches one side and condenses the other 
 temporarily; and, to a less extent, permanently. Under elasticity we 
 may return to this. In the meantime, in considering illustrations of 
 our kinematical propositions, it is necessary to anticipate such phy- 
 sical circumstances. 
 
 128. The flexure of an inextensible surface which can be plane, 
 is a subject which has been well worked by geometrical investigators 
 and writers, and, in its elements at least, presents little difficulty. The 
 first elementary conception to be formed is, that such a surface (if 
 perfectly flexible), taken plane in the first place, may be bent about 
 any straight line ruled on it, so that the two plane parts may make 
 any angle with one another. 
 
 Such a line is called a 'generating line' of the surface to be 
 formed. 
 
 Next, we may bend one of these plane parts about any other line 
 which does not (within the limits of the sheet) intersect the former ; 
 and so on. If these lines are infinite in number, and the angles of 
 bending infinitely small, but such that their sum may be finite, we 
 have our plane surface bent into a curved surface, which is of course 
 'developable' (§ 125). 
 
 129. Lift a square of paper, free from folds, creases, or ragged 
 edges, gently by one corner, or otherwise, without crushing or forcing 
 it, or very gently by two points. It will hang in a form which is very 
 rigorously a developable surface; for although it is not absolutely 
 inextensible, yet the forces which tend to stretch or tear it, when it 
 is treated as above described, are small enough to produce absolutely 
 no sensible stretching. Indeed the greatest stretching it can expe- 
 rience without tearing, in any direction, is not such as can affect the 
 form of the surface much when sharp flexures, singular points, etc., 
 are kept clear off". 
 
 130. Prisms and cylinders (when the lines of bending, § 128, 
 are parallel, and finite in number with finite angles, or infinite 
 in number with infinitely small angles), and pyramids and cones 
 (the lines of bending meeting in a point if produced), are clearly 
 included. 
 
 131. If the generating lines, or line-edges of the angles of bending, 
 are not parallel, they must meet, since they are in a plane when the 
 surface is plane. If they do not meet all in one point, they must 
 meet in several points : in general, each one meets its predecessor 
 and its successor in diff'erent points. 
 
44 
 
 PRELIMINARY. 
 
 132. There Is still no difficulty in understanding the form of, say a 
 square, or circle, of the plane surface when bent as explained above, 
 
 provided it does not include any of these points 
 \ of intersection. When the number is infinite, 
 
 and the surface finitely curved, the developable 
 lines will, in general, be tangents to a curve (the 
 locus of the points of intersection when the 
 number is infinite). This curve is called the 
 edge of regression. The surface must clearly, 
 when complete (according to mathematical ideas), 
 consist of two sheets meeting in this edge of 
 regression (just as a cone consists of two 
 sheets meeting in the vertex), because each 
 tangent may be produced beyond the point 
 of contact, instead of stopping at it, as in the preceding diagram. 
 
 133. To construct a complete developable surface in two sheets 
 from its edge of regression — ■ 
 
 Lay one piece of perfectly flat, un- 
 wrin'Kled, smooth-cut paper on the top 
 of another. Trace any curve on the 
 other, and let it have no point of in- 
 flection, but everywhere finite curvature. 
 Cut the paper quite away on the con- 
 cave side. If the curve traced is closed, 
 it must be cut open (see second diagram). 
 The limits to the extent that may be left uncut away, are the 
 tangents drawn outwards from the two ends, so that, in short, no 
 portion of the paper through which a real tangent does not pass 
 is to be left. 
 
 Attach the two sheets together by very slight paper or muslin 
 clamps gummed to them along the common curved edge. These 
 -7^ must be so slight as not to interfere sensibly with 
 >>•" the flexure of the two sheets. Take hold of one 
 corner of one sheet and lift the whole. The two 
 will open out into two sheets of a developable 
 surface, of which the curve, bending into a tor- 
 tuous curve, is the edge of regression. The tan- 
 gent to the curve drawn in one direction from 
 the point of contact, will always lie in one of the 
 sheets, and its continuation on the other side in the 
 other sheet. Of course a double-sheeted developable polyhedron can 
 be constructed by this process, by starting from a polygon instead 
 of a curve. 
 
 134. A flexible but perfectly inextensible surface, altered in form 
 in any way possible for it, must keep any hne traced on it un- 
 changed in length ; and hence any two intersecting lines unchanged 
 in mutual inclination. Hence, also, geodetic lines must remain 
 geodetic lines. 
 
KINEMATICS. 45 
 
 135. We have now to consider the very important kinematical 
 conditions presented by the changes of volume or figure experienced 
 by a soHd or Hquid mass, or by a group of points whose positions 
 with regard to each other are subject to known conditions. 
 
 Any such definite alteration of form or dimensions is called a 
 Straiii. 
 
 Thus a rod which becomes longer or shorter is strained. Water, 
 when compressed, is strained. A stone, beam, or mass of metal, in a 
 building or in a piece of framework, if condensed or dilated in any 
 direction, or bent, twisted, or distorted in any way, is said to ex- 
 perience a strain. A ship is said to 'strain' if, in launching, or 
 when working in a heavy sea, the different parts of it experience 
 relative motions. 
 
 136. If, when the matter occupying any space is strained in any 
 way, all pairs of points of its substance which are initially at equal 
 distances from one another in parallel lines remain equidistant, it 
 may be at an altered distance ; and in parallel lines, altered, it may 
 be, from their initial direction ; the strain is said to be homogeneous. 
 
 137. Hence if any straight line be drawn through the body in its 
 initial state, the portion of the body cut by it will continue to be a 
 straight line when the body is homogeneously strained. For, if 
 ABC be any such line, AB and BC, being parallel to one line in the 
 initial, remain parallel to one line in the altered state; and therefore 
 remain in the same straight Hne with one another. Thus it follows 
 that a plane remains a plane, a parallelogram a parallelogram, and a 
 parallelepiped a parallelepiped. 
 
 138. Hence, also, similar figures, whether constituted by actual 
 portions of the substance, or mere geometrical surfaces, or straight or 
 curved lines passing through or joining certain portions or points of 
 the substance, similarly situated (i. e. having corresponding parameters 
 parallel) when altered according to the altered condition of the body, 
 remain similar and similarly situated among one another. 
 
 139. The lengths of parallel lines of the body remain in the same 
 proportion to one another, and hence all are altered in the same pro- 
 portion. Hence, and from § 137, we infer that any plane figure 
 becomes altered to another plane figure which is a diminished or 
 magnified orthographic projection of the first on some plane. 
 
 The elongation of the body along any line is the proportion which 
 the addition to the distance between any two points in that line bears 
 to their primitive distance. 
 
 140. Every orthogonal projection of an ellipse is an ellipse (the 
 case of a circle being included). Hence, and from § 139, we see 
 that an ellipse remains an ellipse; and an ellipsoid remains a sur- 
 face of which every plane section is an ellipse ; that is, remains an 
 ellipsoid. 
 
46 PRELIMINAR Y. 
 
 141. The ellipsoid which any surface of the body initially spheri- 
 cal becomes in the altered condition, may, to avoid circumlocutions, 
 be called the Strain Ellipsoid. 
 
 142. In any absolutely unrestricted homogeneous strain there are 
 three directions (the three principal axes of the strain ellipsoid), at 
 right angles to one another, which remain at right angles to one 
 another in the altered condition of the body. Along one of these 
 the elongation is greater, and along another less, than along any 
 other direction in the body. Along the remaining one the elongation 
 is less than in any other line in the plane of itself and the first men- 
 tioned, and greater than along any other line in the plane of itself 
 and the second. 
 
 N'ote. — Contraction is to be reckoned as a negative elongation: the 
 maximum elongation of the preceding enunciation may be a mini- 
 mum contraction: the minimum elongation may be a maximum 
 contraction. 
 
 143. The ellipsoid Into which a sphere becomes altered may be 
 an ellipsoid of revolution, or, as it is called, a spheroid, prolate, or 
 oblate. There is thus a maximum or minimum elongation along 
 the axis, and equal minimum or maximum elongation along all lines 
 perpendicular to the axis. 
 
 Or it may be a sphere ; in which case the elongations are equal in 
 all directions. The effect is, in this case, merely an alteration of 
 dimensions without change of figure of any part. 
 
 144. The principal axes of a strain are the principal axes of the 
 ellipsoid into which it converts a sphere. The principal elongations 
 of a strain are the elongations in the direction of its principal axes. 
 
 145. When the positions of the principal axes, and the magnitudes 
 of the principal elongations of a strain are given, the elongation of 
 any line of the body, and the alteration of angle between any two 
 lines, may be obviously determined by a simple geometrical construc- 
 tion. 
 
 146. With the same data the alteration of angle between any tw^o 
 planes of the body may also be easily determined, geometrically. 
 
 147. Let the ellipse of the annexed diagram represent the section 
 of the strain ellipsoid through the greatest and least principal axes. 
 
 Let S'OS, TO The the two diameters of 
 
 this ellipse, which are equal to the mean 
 
 principal axis of the ellipsoid. Every 
 
 plane through O, perpendicular to the 
 
 plane of the diagram, cuts the ellipsoid 
 
 in an ellipse of which one principal axis 
 
 ^' is the diameter in which it cuts the ellipse 
 
 of the diagram, and the other, the mean principal diameter of the 
 
 ellipsoid. Hence a plane through either SS' or TT', perpendicular 
 
KINEMATICS, 
 
 47 
 
 to the plane of the diagram, cuts the ellipsoid in an ellipse of which 
 the two principal axes are equal, that is to say, in a circle. Hence 
 the elongations along all lines in either of these planes are equal to 
 the elongation along the mean principal axis of the strain ellipsoid. 
 
 148. The consideration of the circular sections of the strain ellip- 
 soid is highly instructive, and leads to important views with reference 
 to the analysis of the most general character of a strain. First let us 
 suppose there to be no alteration of volume on the whole, and neither 
 elongation nor contraction along the mean principal axis. 
 
 Let OX and OZ be the directions of maximum elongation and 
 maximum contraction respectively. 
 Let A be any point of the body 
 in its primitive condition, and A^ the 
 same point of the altered body, so 
 that OA=a.OA, 
 
 Now, if we take 0C= OA^^ and 
 if C^ be the position of that point 
 of the body which was in the 
 position C initially, we shall have 
 
 OC,^-OC, and therefore 0C = 
 
 'a ' 
 
 OA. Hence the two triangles COA 
 and CpA^ are equal and similar. 
 
 Hence CA experiences no alteration of length, but takes the altered 
 position C^A^ in the altered position of the body. Similarly, if we 
 measure on XO produced, OA' and OA' equal respectively to OA 
 and OA^^ we find that the line CA' experiences no alteration in length, 
 but takes the altered position C^ A'^. 
 
 Consider now a plane of the body initially through CA perpen- 
 dicular to the plane of the diagram, which will be altered into a plane 
 through Ci^i, also perpendicular to the plane of the diagram. All 
 lines initially perpendicular to the plane of the diagram remain so, 
 and remain unaltered in length. ^C has just been proved to remain 
 unaltered in length. Hence (§ 139) all lines in the plane we have 
 just drawn remain unaltered in length and in mutual inclination. 
 Similarly we see that all lines in a plane through CA'^ perpendicular 
 to the plane of the diagram, altering to a plane through C-^A\^ per- 
 pendicular to the plane of the diagram, remain unaltered in length and 
 in mutual inclination. 
 
 149. The precise character of the strain we have now under con- 
 sideration will be elucidated by the following : — Produce CO, and take 
 OC and 0C\ respectively equal to OC and OC^. Join CA, C'A', 
 C\A^, and C\A\, by plain and dotted lines as in the diagram. 
 Then we see that the rhombus CA CA' (plain lines) of the body in 
 its initial state becomes the rhombus C\ ^1 C^ A'-^ (dotted) in the 
 altered condition. Now imagine the body thus strained to be 
 moved as a rigid body (i. e. with its state of strain kept unchanged) 
 
48 
 
 PRELIMINARY. 
 
 J.;--... 
 
 till A^ coincides with A, and C\ with C\ keeping all the lines of 
 ^ the diagram still in the same plane. A\Cx will 
 
 take a position in CA' produced, as shown in 
 the new diagram, and the original and the altered 
 parallelogram will be on the same base A C\ and 
 between the same parallels AC and CA\, and 
 their other sides will be equally inclined on the 
 two sides of a perpendicular to them. Hence, 
 irrespectively of any rotation, or other absolute 
 motion of the body not involving change of form 
 or dimensions, the strain under consideration 
 may be produced by holding fast and unaltered the plane of the 
 body through A C\ perpendicular to the plane of the diagram, and 
 making every plane parallel to it sHde, keeping the same distance, 
 through a space proportional to this distance (i.e. different planes 
 parallel to the fixed one slide through spaces proportional to their 
 distances). 
 
 150. This kind of strain is called a siinple shear. The plane of 
 a shear is a plane perpendicular to the undistorted planes, and 
 parallel to the lines of the relative motion. It has (i) the property 
 that one set of parallel planes remain each unaltered in itself ; (2) 
 
 that another set of parallel planes remain 
 each unaltered in itself. This other set is 
 got when the first set and the degree or 
 amount of shear are given, thus : — Let 
 CC^ be the motion of one point of one 
 plane, relative to a plane KL held fixed — 
 the diagram being in a plane of the shear. 
 Bisect CCx in N. Draw NA perpendicular 
 to it. A plane perpendicular to the plane 
 of the diagram, initially through AC^ and 
 finally through AC-^^ remains unaltered in 
 its dimensions. 
 
 151. One set of parallel undistorted planes and the amount of 
 their relative parallel shifting having been given, we have just seen 
 how to find the other set. The shear may be otherwise viewed, and 
 considered as a shifting of this second set of parallel planes, relative 
 to any one of them. The amount of this relative shifting is of course 
 equal to that of the first set, relatively to one of them. 
 
 152. The principal axes of a shear are the lines of maximum 
 elongation and of maximum contraction respectively. They may 
 be found from the preceding construction (§ 150), thus: — In the 
 plane of the shear bisect the obtuse and acute angles between the 
 planes destined not to become deformed. The former bisecting line 
 is the principal axis of elongation, and the latter is the principal 
 axis of contraction, in their initial positions. The former angle 
 (obtuse) becomes equal to the latter, its supplement (acute), in the 
 
KINEMATICS. 49 
 
 altered condition of the body, and the ^-'-— ~-«^ 
 
 lines bisecting the altered angles are the /""^ ^^^\ 
 principal axes of the strain in the altered Dj^^^ 4^. --^B 
 
 Otherwise, taking a plane of shear for y ^^^---^^'^^j^^^^^ \| 
 
 the plane of the diagram, let AB be a Jj ^ iT — ^~~j:j 
 
 line in which it is cut by one of either -" 
 
 set of parallel planes of no distortion. On any portion AB of this 
 as diameter, describe a semicircle. Through C, its middle point, 
 draw, by the preceding construction, CD the initial, and C£ the 
 final, position of an unstretched line. Join DA, DB, EA, EB. 
 DA, DB are the initial, and EA, EB the final, positions of the 
 principal axes, 
 
 153. The ratio of a shear is the ratio of elongation and contrac- 
 tion of its principal axes. Thus if one principal axis is elongated 
 in the ratio i : a, and the other therefore (§ 148) contracted in the 
 ratio a : I, a is called the ratio of the shear. It will be convenient 
 generally to reckon this as the ratio of elongation ; that is to say, 
 to make its numerical measure greater than unity. 
 
 In the diagram of § 152, the ratio of DB to EB, or of EA to DA, 
 is the ratio of the shear. 
 
 154. The amount of a shear is the amount of relative motion per 
 unit distance between planes of no distortion. 
 
 It is easily proved that this is equal to the excess of the ratio of 
 the shear above its reciprocal. 
 
 155. The planes of no distortion in a simple shear are clearly the 
 circular sections of the strain ellipsoid. In the ellipsoid of this 
 case, be it remembered, the mean axis remains unaltered, and is a 
 mean proportional between the greatest and the least axis. 
 
 156. If we now suppose all lines perpendicular to the plane of the 
 shear to be elongated or contracted in any proportion, without 
 altering lengths or angles in the plane of the shear, and if, lastly, 
 we suppose every line in the body to be elongated or contracted in 
 some other fixed ratio, we have clearly (§ 142) the most general 
 possible kind of strain. 
 
 157. Hence any strain whatever may be viewed as compounded 
 of a uniform dilatation in all directions, superimposed on a simple 
 elongation in the direction of one principal axis superimposed on a 
 simple shear in the plane of the two other principal axes, 
 
 158. It is clear that these three elementary component strains may 
 be applied in any other order as well as that stated. Thus, if the 
 simple elongation is made first, the body thus altered must get just 
 the same shear in planes perpendicular to the line of elongation 
 as the originally unaltered body gets when the order first stated is 
 followed. Or the dilatation may be first, then the elongation, and 
 finally the shear, and so on. 
 
 T. 4 
 
so PRELIMINARY, 
 
 159. When the axes of the ellipsoid are lines of the body whose 
 direction does not change, the strain is said to be piire^ or unaccom- 
 panied by rotation. The strains we have already considered were 
 pure strains accompanied by rotations. 
 
 160. If a body experience a succession of strains, each unaccom- 
 panied by rotation, its resulting condition will generally be producible 
 by a strain and a rotation. From this follows the remarkable corol- 
 lary that three pure strains produced one after another, in any piece 
 of matter, each without rotation, may be so adjusted as to leave the 
 body unstrained, but rotated through some angle about some axis. 
 We shall have, later, most important and interesting applications to 
 fluid motion, which will be proved to be instantaneously, or differ- 
 entially, irrotational ; but which may result in leaving a whole fluid 
 mass merely turned round from its primitive position, as if it had 
 been a rigid body. [The following elementary geometrical in- 
 vestigation, though not bringing out a thoroughly comprehensive 
 view of the subject, affords a rigorous demonstration of the pro- 
 position, by proving it for a particular case. 
 
 Let us consider, as above (§ 150), a simple shearing motion. A 
 point O being held fixed, suppose the matter of the body in a plane, 
 cutting that of the diagram perpendicularly in CZ>, to move in this 
 plane from right to left parallel to CD ; and in other planes parallel 
 to it let there be motions proportional to their distances from O. 
 Consider first a shear from /* to P^\ then from P^ on to P^ ; and 
 let O be taken in a line through Pj, perpendicular to CD. During 
 
 the shear from P to /\ 
 D a point Q moves of 
 course to Q^ through a 
 distance QQi = PPx- 
 Choose Q midway be- 
 tween P and /\, so that 
 P\Q.^ QiP=\P\P- Now, as we have seen above (§ 152), the 
 line of the body, which is the principal axis of contraction in the 
 shear from ^ to Q^^ is OA^ bisecting the angle QOE at the be- 
 ginning, and OA^, bisecting QyOE at the end, of the whole 
 motion considered. The angle between these two lines is half 
 the angle Q^OQ,', that is to say, is equal to P^OQ. Hence, 
 if the plane CD is rotated through an angle equal to PiOQ, in 
 the plane of the diagram, in the same way as the hands of a watch, 
 during the shear from Q to Q^^ or, which is the same thing, the 
 shear from Pto Pi, this shear will be effected without final rotation 
 of its principal axes. (Imagine the diagram turned round till OA^ 
 lies along OA. The actual and the newly imagined position of CD 
 will show how this plane of the body has moved during such non- 
 rotational shear.) 
 
 Now, let the second step, P^ to P^, be made so as to complete 
 the whole shear, P to P2, which we have proposed to consider. 
 Such second partial shear may be made by the common shearing 
 
kinematics: -^^ 
 
 process parallel to the new position (imagined in the preceding 
 parenthesis) of CD^ and to make it also non-rotational, as its 
 -predecessor has been made, we must turn further round, in the 
 same direction, through an angle equal to QiOF^. Thus in these 
 two steps, each made non-rotational, we have turned the plane CD 
 round through an angle equal to QiOQ. But now, we have a whole 
 shear PF.2 ; and to make this as one non-rotational shear, we must 
 turn CD through an angle I^^ OF only, which is less than QiOQ by 
 the excess of P^OQ above QOP. Hence the resultant of the two 
 shears, PP^^ -^1^2' ^^^^"^ separately deprived of rotation, is a single 
 shear PP^, and a rotation of its principal axes, in the direction of 
 the hands of a watch, through an angle equal to QOP^ — POQ. 
 
 161. Make the two partial shears each non-rotationally. Return 
 from their resultant in a single non-rotational shear: we conclude with 
 the body unstrained, but turned through the angle QOP^-POQj in 
 the same direction as the hands of a watch.] 
 
 162. As there can be neither annihilation nor generation o^ matter 
 in any natural motion or action, the whole quantity of a fluid within 
 any space at any time must be equal to the quantity originally in 
 that space, increased by the whole quantity that has entered it, and 
 diminished by the whole quantity that has left it. This idea, when 
 expressed in a perfectly comprehensive manner for every portion of 
 a fluid in motion, constitutes what is commonly called the ' equation 
 of continuity.^ 
 
 163. Two ways of proceeding to express this idea present 
 themselves, each aff"ording instructive views regarding the properties 
 of fluids. In one we consider a definite portion of the fluid ; follow 
 it in its motions; and declare that the average density of the substance 
 varies inversely as its volume. We thus obtain the equation ot con- 
 tinuity in an integral form. 
 
 The form under which the equation of continuity is most commonly 
 given, or the differential equation of continuity, as we may call it, ex- 
 presses that the rate of diminution of the density bears to the density, 
 at any instant, the same ratio as the rate of increase of the volume of 
 an infinitely small portion bears to the volume of this portion at the 
 same instant. 
 
 164. To find the differential equation of continuity, imagine a 
 space fixed in the interior of a fluid, and consider the fluid 
 which flows into this space, and the fluid which flows out of it, 
 across different parts of its bounding surface, in any time. If the 
 fluid is of the same density and incompressible, the whole quantity of 
 •matter in the space in question must remain constant at all times, and 
 therefore the quantity flowing in must be equal to the quantity flowing 
 out in any time. If, on the contrary, during any period of motion, 
 more fluid enters than leaves the fixed space, there will be condensa- 
 tion of matter in that space ; or if more fluid leaves than enters, there 
 will be dilatation. The rate of augmentation of the average density 
 
$2 PRELIMINARY, 
 
 of the fluid, per unit of time, in the fixed space in question, bears to 
 the actual density, at any instant, the same ratio that the rate of 
 acquisition of matter into that space bears to the whole matter in that 
 space. 
 
 165. Several references have been made in preceding sections to 
 the number of independent variables in a displacement, or to the 
 degrees of freedom or constraint under which the displacement takes 
 place. It may be well, therefore, to take a general (but cursory) view 
 of this part of the subject itself. 
 
 166. A free point has //^r^^ degrees of freedom, inasmuch as the 
 most general displacement which it can take is resolvable into three, 
 parallel respectively to any three directions, and independent of each 
 other. It is generally convenient to choose these three directions of 
 resolution at right angles to one another. 
 
 If the point be constrained to remain always on a given surface, 
 one degree of constraint is introduced, or there are left but two 
 degrees of freedom. For we may take the normal to the surface 
 as one of three rectangular directions of resolution. No displacement 
 can be effected parallel to it : and the other two displacements, at 
 right angles to each other, in the tangent plane to the surface, are 
 independent. 
 
 If the point be constrained to remain on each of two surfaces, it 
 loses two degrees of freedom, and there is left but one. In fact, 
 it is constrained to remain on the curve which is common to both 
 surfaces, and along a curve there is at each point but one direction 
 of displacement. 
 
 167. Taking next the case of a free rigid system, we have evidently 
 six degrees of freedom to consider — three independent displacements 
 or translations in rectangular directions as a point has, and three 
 independent rotations about three mutually rectangular axes. 
 
 If it have one point fixed, the system loses three degrees of free- 
 dom ; in fact, it has now only the rotations above mentioned. 
 
 This fixed point may be, and in general is, a point of a continuous 
 surface of the body in contact with a continuous fixed surface. These 
 surfaces may be supposed * perfectly rough,' so that sliding may be 
 impossible. 
 
 If a second point be fixed, the body loses two more degrees of 
 freedom, and keeps only one freedom to rotate about the line joining 
 the two fixed points. 
 
 If a third point, not in a line with the other two, be fixed, the body 
 is fixed. 
 
 168. If one point of the rigid system is forced to remain on a 
 smooth surface, one degree of freedom is lost ; there remain yfz^^, two 
 displacements in the tangent plane to the surface, and three rotations. 
 As an additional degree of freedom is lost by each successive limita- 
 tion of a point in the body to a smooth surface, six such conditions 
 completely determine the position of the body. Thus if six points 
 
KINEMATICS, 53 
 
 properly chosen on the barrel and stock of a rifle be made to rest on 
 six convex portions of the surface of a fixed rigid body, the rifle may 
 be replaced any number of times in precisely the same position, for 
 the purpose of testing its accuracy. 
 
 A fixed V under the barrel near the muzzle, and another under 
 the swell of the stock close in front of the trigger-guard, give four 
 of the contacts, bearing the weight of the rifle. A fifth (the one 
 to be broken by the recoil) is supplied by a nearly vertical fixed 
 plane close behind the second V, to be touched by the trigger-guard, 
 the rifle being pressed forward in its V's as far as this obstruction 
 allows it to go. This contact may be dispensed with and nothing 
 sensible of accuracy lost, by having a mark on the second V, and a 
 corresponding mark on barrel or stock, and sliding the barrel back- 
 wards or forwards in the V's till the two marks are, as nearly as 
 can be judged by eye, in the same plane perpendicular to the barrel's 
 axis. The sixth contact may be dispensed with by adjusting two 
 marks on the heel and toe of the butt to be as nearly as need be 
 in one vertical plane judged by aid of a plummet. This method 
 requires less of costly apparatus, and is no doubt more accurate and 
 trustworthy, and more quickly and easily executed, than the ordi- 
 nary method of clamping the rifle in a massive metal cradle set on a 
 heavy mechanical slide. 
 
 A geometrical clamp is a means of applying and maintaining 
 six mutual pressures between two bodies touching one another at 
 six points. 
 
 A 'geometrical slide* is any arrangement to apply five degrees 
 of constraint, and leave one degree of freedom, to the relative motion 
 of two rigid bodies by keeping them pressed together at just five 
 points of their surfaces. 
 
 Ex. I. The transit instrument would be an instance if one end 
 of one pivot, made slightly convex, were pressed against a fixed 
 vertical end-plate, by a spring pushing at the other end of the axis. 
 The other four guiding points are the points, or small areas, of con- 
 tact of the pivots on the Y*s. 
 
 Ex. 2. Let two rounded ends of legs of a three-legged stool 
 rest in a straight, smooth, V-shaped canal, and the third on a smooth 
 horizontal plane. Gravity maintains positive determinate pressures 
 on the five bearing points; and there is a determinate distribution 
 and amount of friction to be overcome, to produce the rectilineal 
 translational motion thus accurately provided for. 
 
 Ex. 3. Let only one of the feet rest in a V canal, and let 
 another rest in a trihedral hollow in line with the canal, the third 
 still resting on a horizontal plane. There are thus six bearing points, 
 one on the horizontal plane, two on the sides of the canal, and three 
 on the sides of the trihedral hollow : and the stool is fixed in a 
 determinate position as long as all these six contacts are unbroken. 
 Substitute for gravity a spring, or a screw and nut (of not infinitely 
 rigid material), binding the stool to the rigid body to which these 
 six planes belong. Thus we have a 'geometrical clamp,' which 
 
54, PRELIMINARY. 
 
 clamps two bodies together with perfect firmness in a perfectly 
 definite position, without the aid of friction (except in the screw, 
 if a screw is used) ; and in various practical appUcations gives 
 very readily and conveniently a more securely firm connexion by 
 one screw slightly pressed, than a clamp such as those commonly 
 made hitherto by mechanicians can give with three strong screws 
 forced to the utmost. 
 
 Do away with the canal and let two feet (instead of only one) rest 
 on the plane, the other still resting in the conical hollow. The 
 number of contacts is thus reduced to five (three in the hollow 
 and two on the plane), and instead of a ' clamp ' we have again 
 a slide. This form of sHde, — a three-legged stool with two feet 
 resting on a plane and one in a hollow, — will be found very useful in 
 a large variety of applications, in which motion about an axis is de- 
 sired when a material axis is not conveniently attainable. Its first 
 application was to the 'azimuth mirror,' an instrument placed on 
 the glass cover of a mariner's compass and used for taking azimuths 
 of sun or stars to correct the compass, or of landmarks or other 
 terrestrial objects to find the ship's position. It has also been applied 
 to the ' Deflector,' an adjustible magnet laid on the glass of the 
 compass bowl and used, according to a principle first we believe 
 given by Sir Edward Sabine, to discover the 'semicircular' error 
 produced by the ship's iron. The movement may be made very 
 frictionless when the plane is horizontal, by weighting the move- 
 able body so that its centre of gravity is very nearly over the 
 foot that rests in the hollow. One or two guard feet, not to touch 
 the plane except in case of accident, ought to be added to give 
 a broad enough base for safety. 
 
 The geometrical slide and the geometrical clamp have both been 
 found very useful in electrometers, in the * siphon recorder,' and in 
 an instrument recently brought into use for automatic signalling 
 through submarine cables. An infinite variety of forms may be 
 given to the geometrical slide to suit varieties of application of the 
 general principle on which its definition is founded. 
 
 An old form of the geometrical clamp, with the six pressures pro- 
 duced by gravity, is the three V grooves on a stone slab bearing the 
 three legs of an astronomical or magnetic instrument. It is not 
 generally however so 'well-conditioned' as the trihedral hole, the 
 V groove, and the horizontal plane contact, described above. 
 
 There is much room for improvement by the introduction of 
 geometrical slides and geometrical clamps, in the mechanism of 
 mathematical, optical, geodetic, and astronomical instruments :. 
 which as made at present are remarkable for disregard of geome- 
 trical and dynamical principles in their slides, micrometer screws, 
 and clamps. Good workmanship cannot compensate for bad design, 
 whether in the safety-valve of an ironclad, or the movements and 
 adjustments of a theodolite. 
 
 169. If one point be constrained to remain in a curve, there remain 
 four degrees of freedom. 
 
KINEMATICS, 
 
 55 
 
 If two points be constrained to remain in given curves, there are 
 four degrees of constraint, and we have left two degrees of freedom. 
 One of these may be regarded as being a simple rotation about the 
 line joining the constrained parts, a motion which, it is clear, the 
 body is free to receive. It may be shown that the other possible 
 motion is of the most general character for one degree of freedom ; 
 that is to say, translation and rotation in any fixed proportions, as of 
 the nut of a screw. 
 
 If one line of a rigid system be constrained to remain parallel to 
 itself, as for instance, if the body be a three-legged stool standing on 
 a perfectly smooth board fixed to a common window, shding in its 
 frame with perfect freedom, there remain three displacements and one 
 rotation. 
 
 But we need not farther pursue this subject, as the number of 
 combinations that might be considered is almost endless ; and those 
 already given suffice to show how simple is the determination of the 
 degrees of freedom or constraint in any case that may present itself. 
 
 170. One degree of constraint of the most general character, is not 
 producible by constraining one point of the body to a curve surface ; 
 but it consists in stopping one line of the body from longitudinal 
 motion, except accompanied by rotation round this line, in fixed 
 proportion to the longitudinal motion. Every other motion being 
 left unimpeded, there remains free rotation about any axis perpen- 
 dicular to that line (two degrees of freedom) ; and translation in any 
 direction perpendicular to the same line (two degrees of freedom). 
 These last four, with the one degree of freedom to screw, con- 
 stitute the five degrees of freedom, which, with one degree of con- 
 straint, make up the six elements. This condition is realized in the 
 following mechanical arrangement, which seems the simplest that 
 can be imagined for the purpose : — 
 
 Let a screw be cut on one shaft of a Hooke's joint, and let the 
 other shaft be joined to a fixed shaft by a second Hooke's joint. 
 A nut turning on that screw-shaft has the most general kind of 
 motion admitted when there is one degree of constraint. Or it is 
 subjected to just one degree of constraint of the most general cha- 
 racter. It has five degrees of freedom ; for it may move, ist, by 
 screwing on its shaft, the two Hooke's joints being at rest; 2nd, 
 it may rotate about either axis of the first Hooke's joint, or any axis 
 in their plane (two more degrees of freedom : being freedom to rotate 
 about two axes through one point) ; 3rd, it may, by the two Hooke's 
 joints, each bending, have translation without rotation in any direction 
 perpendicular to the link or shaft between the two Hooke's joints 
 (two more degrees of freedom). But it cannot have a motion of 
 translation parallel to the line of the link without a definite propor- 
 tion of rotation round this line ; nor can it have rotation round this 
 line without a definite proportion of translation parallel to it. 
 
CHAPTER II. 
 
 DYNAMICAL LAWS AND PRINCIPLES. 
 
 ; 171. In the preceding chapter we considered as a subject of pure 
 geometry the motion of points, lines, surfaces, and volumes, whether 
 taking place with or without change of dimensions and form ; and the 
 results we there arrived at are of course altogether independent of the 
 idea of matter^ and of \k\Q forces which matter exerts. We have here- 
 tofore assumed the existence merely of motion, distortion, etc.; we 
 now come to the consideration, not of how we might consider such 
 motion, etc., to be produced, but of the actual causes which in the 
 material world do produce them. The axioms of the present chapter 
 must therefore be considered to be due to actual experience, in the 
 shape either of observation or experiment. How such experience is 
 tp be conducted will form the subject of a subsequent chapter. 
 
 172. We cannot do better, at all events in commencing, than follow 
 Newton somewhat closely. Indeed the introduction to the Principia 
 contains in a most lucid form the general foundations of dynamics. 
 The Definitiofies and Axiomata, sive Leges Motus, there laid down, 
 require only a few amplifications and additional illustrations, suggested 
 by subsequent developments, to suit them to the present state of 
 science, and to make a much better introduction to dynamics than 
 we find in even some of the best modern treatises. 
 
 173. We cannot, of course, give a definition of Matter which will 
 satisfy the metaphysician ; but the naturalist may be content to know 
 matter as that which can he perceived by the senses ^ or as that which can 
 be acted upon by, or can exert, force. The latter, and indeed the 
 former also, of these definitions involves the idea of Force, which, in 
 point of fact, is a direct object of sense ; probably of all our senses, 
 and certainly of the * muscular sense.' To our chapter on Properties 
 of Matter we must refer for further discussion of the question, What 
 is matter! 
 
 174. The Quantity of Matter in a body, or, as we now call it, the 
 Mass of a body, is proportional, according to Newton, to the Volume 
 and the Density conjointly. In reality, the definition gives us the 
 meaning of density rather than of mass ; for it shows us that if twice 
 the original quantity of matter, air for example, be forced into a vessel 
 
DYNAMICAL LA WS AND PRINCIPLES, 57 
 
 of given capacity, the density will be doubled, and so on. But it also 
 shows us that, of matter of uniform density, the mass or quantity is 
 proportional to the volume or space it occupies. 
 
 Let M be the mass, p the density, and V the volume, of a homo- 
 geneous body. Then 
 
 M=Fp; 
 
 if we so take our units that unit of mass is that of unit volume of a 
 body of unit density. 
 
 If the density be not uniform, the equation 
 
 M= Vfj 
 
 gives the Average (§ 26) density; or, as it is usually called, the Mean 
 density, of the body. 
 
 It is worthy of particular notice that, in this definition, Newton 
 says, if there be anything which freely pervades the interstices of all 
 bodies, this is not taken account of in estimating their Mass or 
 Density. 
 
 175. Newton further states, that a practical measure of the mass 
 of a body is its Weight. His experiments on pendulums, by which he 
 establishes this most important remark, will be described later, in our 
 chapter on Properties of Matter. 
 
 As will be presently explained, the unit mass most convenient for 
 British measurements is an imperial pound of matter. 
 
 176. The Qiiantity of Motion, or the Momentum^ of a rigid body 
 moving without rotation is proportional to its mass and velocity con- 
 jointly. The whole motion is the sum of the motions of its several 
 parts. Thus a doubled mass, or a doubled velocity, would correspond 
 to a double quantity of motion ; and so on. 
 
 Hence, if we take as unit of momentum the momentum of a unit 
 of matter moving with unit velocity, the momentum of a mass M 
 moving with velocity v is Mv. 
 
 Yll, Change of Quantity of Motion, or Change of Momentum, is 
 proportional to the mass moving and the change of its velocity 
 conjointly. 
 
 Change of velocity is to be understood in the general sense of § 31. 
 Thus, in the figure of that section, if a velocity represented by OA be 
 changed to another represented by OC, the change of velocity is 
 represented in magnitude and direction by A C 
 
 178. Pate of Change of Momentum, or Acceleration of Momentum, is 
 
 proportional to the mass moving and the acceleration of its velocity 
 
 conjointly. Thus (§ 44) the rate of change of momentum of a 
 
 felhng body is constant, and in the vertical direction. Again (§ 36) 
 
 the rate of change of momentum of a mass M, describing a circle of 
 
 MV^ 
 fadius R, with uniform velocity V, is — 5— , and is directed to the 
 
 centre of the circle ; that is to say, it depends upon a change of di- 
 rection, not a change of speed, of the motion. .... 
 
SS^ PRELIMINARY. 
 
 179. The Vis Viva, or Kinetic Energy, of a moving body is pro- 
 portional to the mass and the square of the velocity, conjointly. If 
 we adopt the same units of mass and velocity as before, there is 
 particular advantage in defining kinetic energy as halftht product of 
 the mass and the square of its velocity. 
 
 180. Rate of Change of Kinetic Energy (when defined as above) is 
 the product of the velocity into the component of acceleration of 
 momentum in the direction of motion. 
 
 Suppose the velocity of a mass M to be changed from v to v^ in 
 any time t j the rate at which the kinetic energy has changed is 
 
 -.\M{vf-'i^) = -M{:v-i),\{v^^v), 
 
 "^ow - M{v^-v) is the rate of change of momentum in the direc- 
 tion of motion, and J (v^ + v) is equal to v, if t be infinitely small. 
 Hence the above statement. It is often convenient to use Newton's 
 Fluxional notation for the rate of change of any quantity per unit of 
 
 time. In this notation (§ 28) v stands for - {v,~v) ; so that the rate 
 of change of^Mv^, the kinetic energy, is Mv . v. (See also §§229, 241.) 
 
 181. It is lo be observed that, in what precedes, with the exception of 
 the definition of density, we have taken no account of the dimensions 
 of the moving body. This is of no consequence so long as it does 
 not rotate, and so long as its parts preserve the same relative positions 
 amongst one another. In this case we may suppose the whole of the 
 matter in it to be condensed in one point or particle. We thus speak 
 of a material particle, as distinguished from 2^ geometrical poiftt. If the 
 body rotate, or if its parts change their relative positions, then we 
 cannot choose any one point by whose motions alone we may de- 
 termine those of the other points. In such cases the momentum and 
 change of momentum of the whole body in any direction are, the 
 sums of the momenta, and of the changes of momentum, of its parts, 
 in these directions ; while the kinetic energy of the whole, being non- 
 directional, is simply the sum of the kinetic energies of the several 
 parts or particles. 
 
 182. Matter has an innate power of resisting external influences, 
 so that every body, so far as it can, remains at rest, or moves uni- 
 formly in a straight Hne. 
 
 This, the Inertia of matter, is proportional to the quantity of matter 
 in the body. And it follows that some cause is requisite to disturb a 
 body's uniformity of motion, or to change its direction from the 
 natural rectilinear path. 
 
 183. Eorce is any cause which tends to alter a body's natural state 
 of rest, or of uniform motion in a straight line. 
 
 Force is wholly expended in the Action it produces ; and the body, 
 after the force ceases to act, retains by its inertia the direction of 
 
DYNAMICAL LA WS AND PRINCIPLES. 59 
 
 motion and the velocity which were given to it. Force may be of 
 divers kinds, as pressure, or gravity, or friction, or any of the attractive 
 or repulsive actions of electricity, magnetism, etc. 
 
 184. The three elements specifying a force, or the three elements 
 which must be known, before a clear notion of the force under con- 
 sideration can be formed, are, its place of application, its direction, 
 and its magnitude. 
 
 ^ {a) The place of application of a force. The first case to be con- 
 sidered is that in which the place of application is a point. It has 
 been shown already in what sense the term ' point ' is to be taken, 
 and, therefore, in what way a force may be imagined as acting at a 
 point. In reality, however, the place of application of a force is 
 always either a surface or a space of three dimensions occupied by 
 matter. The point of the finest needle, or the edge of the sharpest 
 knife, is still a surface, and acts as such on the bodies to which it 
 may be applied. Even the most rigid substances, when brought 
 together, do not touch at a point merely, but mould each other so 
 as to produce a surface of application. On the other hand, gravity 
 is a force of which the place of application is the whole matter of the 
 body whose weight is considered ; and the smallest particle of matter 
 that has weight occupies some finite portion of space. Thus it is to 
 be remarked, that there are two kinds of force, distinguishable by 
 their place of application — force whose place of application is a 
 surface, and force whose place of application is a solid. When a 
 heavy body rests on the ground, or on a table, force of the second 
 character, acting downwards, is balanced by force of the first character 
 acting upwards. 
 
 {b) The second element in the specification of a force is its 
 direction. The direction of a force is the line in which it acts. 
 If the place of application of a force be regarded as a point, a 
 line through that point, in the direction in which the force tends to 
 move the body, is the direction of the force. In the case of a force 
 distributed over a surface, it is frequently possible and convenient 
 to assume a single point and a single line, such that a certain force 
 acting at that point in that Une would produce the same effect as is 
 really produced. 
 
 {c) The third element in the specification of a force is its magnitude. 
 This involves a consideration of the method followed in dynamics for 
 measuring forces. Before measuring anything it is necessary to have 
 a unit of measurement, or a standard to which to refer, and a prin- 
 ciple of numerical specification, or a mode of referring to the standard. 
 These will be supplied presently. See also § 224, below. 
 
 185. The Measure of a Force is the quantity of motion which it 
 produces in unit of time. 
 
 The reader, who has been accustomed to speak of a force of so 
 many pounds, or so many tons, may be reasonably startled when he 
 finds that Newton gives no countenance to such expressions. The 
 method is not correct unless it be specified at what part of the earth's 
 
66^ PRELIMINARY, 
 
 surface the pound, or other definite quantity of matter named, is to 
 be weighed ; for the weight of a given quantity of matter differs in 
 different latitudes. 
 
 It is often, however, convenient to use instead of the absolute 
 unit (§ i88), the gravitation unit — which is simply the weight of unit 
 mass. It must, of course, be specified in what latitude the observation 
 is made. Thus, let W be the mass of a body in pounds; g the 
 velocity it would acquire in falling for a second under the influence 
 of its weight, or the earth's attraction diminished by centrifugal 
 force ; and P its weight measured in kinetic or absolute units. We 
 have p- j^g^ 
 
 The force of gravity on the body, in gravitation units, is W. 
 
 186. According to the system commonly followed in mathe- 
 matical treatises on dynamics till fourteen years ago, when a small 
 instalment of the first edition of the present work was issued for 
 the use of our students, the unit of mass was g times the mass of 
 the standard or unit weight. This definition, giving a varying and a 
 very unnatural unit of mass, was exceedingly inconvenient. By taking. 
 the gravity of a constant mass for the unit of force it makes the unit 
 of force greater in high than in low latitudes. In reality, standards 
 of weight are masses^ not forces. They are employed primarily in 
 commerce for the purpose of measuring out a definite quantity of 
 matter; not an amount of matter which shall be attracted by the 
 earth with a given force. 
 
 A merchant, with a balance and a set of standard weights, would 
 give his customers the same quantity of the same kind of matter 
 however the earth's attraction might vary, depending as he does upon 
 weights for his measurement ; another, using a spring-balance, would 
 defraud his customers in high latitudes, and himself in low, if his 
 instrument (which depends on constant forces and not on the gravity 
 of constant masses) were correctly adjusted in London. 
 
 It is a secondary application of our standards of weight to employ 
 them for the measurement oi forces, such as steam pressures, mus- 
 cular power, etc. In all cases where great accuracy is required, 
 the results obtained by such a method have to be reduced to 
 what they would have been if the measurements of force had been 
 made by means of a perfect spring-balance, graduated so as to 
 indicate the forces of gravity on the standard weights in some con- 
 ventional locality. 
 
 It is therefore very much simpler and better to take the imperial 
 pound, or other national or international standard weight, as, for 
 instance, the gramme (see the chapter on Measures and Instru- 
 ments), as the unit of mass, and to derive from it, according to 
 Newton's definition above, the unit of force. This is the method 
 which Gauss has adopted in his great improvement dl the system of 
 measurement of forces. ^x 
 
 187. The formula, deduced by Clairault from observation, and a 
 certain theory regarding the figure and density of the earth, may be 
 
DYNAMICAL LAWS AND PRINCIPLES. €i 
 
 'employed to calculate the most probable value of the apparent force 
 of gravity, being the resultant of true gravitation and centrifugal force, 
 in any locality where no pendulum observation of sufficient accuracy 
 has been made. This formula, with the two coefficients which it 
 involves, corrected according to modern pendulum observations, is 
 as follows : — 
 
 Let G be the apparent force of gravity on a unit mass at the 
 equator, and g that in any latitude X ; then 
 
 ^=6^(i + -oo5i3sin'X). 
 
 The value of G^ in terms of the absolute unit, to be explained 
 immediately, is 
 
 32*088. 
 
 According to this formula, therefore, polar gravity will be 
 g= 32-088 X 1-00513 = 32-252. 
 
 188. As gravity does not furnish a definite standard, independent 
 of locality, recourse must be had to something else. The principle 
 of measurement indicated as above by Newton, but first introduced 
 practically by Gauss in connexion with national standard masses, 
 furnishes us with what we want. According to this principle, the 
 standard or unit force is that force which^ acti?ig oft a natioftal standard 
 unit of matter during the unit of time, generates the unit of velocity. 
 
 This is known as Gauss' absolute unit ; absolute, because it fur- 
 nishes a standard force independent of the differing amounts of 
 gravity at different localities. It is however terrestrial and incon- 
 stant if the unit of time depends on the earth's rotation, as it does 
 in our present system of chronometry. The period of vibration of 
 a piece of quartz crystal of specified shape and size and at a stated 
 temperature (a tuning-fork, or bar, as one of the bars of glass used 
 in the ' musical glasses ') gives us a unit of time which is constant 
 through all space and all time, and independent of the earth. A 
 unit of force founded on such a unit of time would be better entitled 
 to the designation absolute than is the * absolute unit ' now generally 
 adopted, which is founded on the mean solar second. But this de- 
 pends essentially on one particular piece of matter, and is therefore 
 liable to all the accidents, etc. which affect so-called National 
 Standards however carefully they may be preserved, as well as to 
 the almost insuperable practical difficulties which are experienced 
 when we attempt to make exact copies of them. Still, in the present 
 state of science, we are really confined to such approximations. The 
 recent discoveries due to the Kinetic theory of gases and to Spectrum 
 analysis (especially when it is applied to the light of the heavenly 
 bodies) indicate to us natural standard pieces of matter such as 
 atoms of hydrogen, or sodium, ready made in infinite numbers, all 
 absolutely alike in every physical property. The time of vibration 
 of a sodium particle corresponding to any one of its modes of vibra- 
 tion, is known to be absolutely independent of its position in the 
 universe, and it will probably remain the same so long as the particle 
 
52 PRELIMINARY, 
 
 itself exists. The wave-length for that particular ray, i.e. the space 
 through which light is propagated in vacuo during the time of one 
 complete vibration of this period, gives a perfectly invariable unit of 
 length ; and it is possible that at some not very distant day the mass 
 of such a sodium particle may be employed as a natural standard for 
 the remaining fundamental unit. This, the latest improvement made 
 upon our original suggestion of a Feretmial Springy is due to Clerk 
 Maxwell. 
 
 189. The absolute unit depends on the unit of matter, the unit of 
 time, and the unit of velocity; and as the unit of velocity depends on 
 the unit of space and the unit of time, there is, in the definition, a 
 single reference to mass and space, but a double reference to time ; 
 and this is a point that must be particularly attended to. 
 
 190. The unit of mass may be the British imperial pound, or, 
 better, the gramme : the unit of space the British standard foot, or, 
 better, the centimetre ; and the unit of time the mean solar second. 
 
 We accordingly define the British absolute unit force as ' the force 
 which, acting on one pound of matter for one second, generates a 
 velocity of one foot per second.' 
 
 191. To render this standard intelligible, all that has to be done is 
 to find how many absolute units will produce, in any particular locality, 
 the same effect as the force of gravity on a given mass. The way to 
 do this is to measure the effect of gravity in producing acceleration 
 on a body unresisted in any way. The most accurate method is 
 indirect, by means of the pendulum. The result of pendulum ex- 
 periments made at Leith Fort, by Captain Kater, is, that the velocity 
 acquired by a body falling unresisted for one second is at that place 
 32*207 feet per second. The preceding formula gives exactly 32*2, 
 for the latitude 55° 35', which is approximately that of Edinburgh. 
 The variation in the force of gravity for one degree of difference of 
 latitude about the latitude of Edinburgh is only '0000832 of its own 
 amount. It is nearly the same, though somewhat more, for every 
 degree of latitude southwards, as far as the southern limits of the 
 British Isles. On the other hand, the variation per degree would be 
 sensibly less, as far north as the Orkney and Shetland Isles. Hence 
 the augmentation of gravity per degree from south to north through- 
 out the British Isles is at most about -^^kw^ ^^ its whole amount in 
 any locality. The average for the whole of Great Britain and Ireland 
 differs certainly but little from 32-2. Our present application is, that 
 the force of gravity at Edinburgh is 32*2 times the force which, acting 
 on a pound for a second, would generate a velocity of one foot per 
 second; in other words, 32*2 is the number of absolute units which 
 measures the weight of a pound in this latitude. Thus, speaking 
 very roughly, the British absolute unit of force is equal to the weight 
 of about half an ounce. 
 
 192. Forces (since they involve only direction and magnitude) may 
 be represented, as velocities are, by straight lines in their directions, 
 and of lengths proportional to their magnitudes, respectively. 
 
DYNAMICAL LAWS AND PRINCIPLES. (i^ 
 
 Also the laws of composition and resolution of any number of 
 forces acting at the same point, are, as we shall shov/ later (§ 221), 
 the same as those which we have already proved to hold for velo- 
 cities; so that with the substitution of force for velocity, §§ 30, 31 
 are still true. 
 
 193. The Component of a force in any direction, sometimes 
 called the Effective Component in that direction, is therefore found 
 by multiplying the magnitude of the force by the cosine of the 
 angle between the directions of the force and the component. The 
 remaining component in this case is perpendicular to the other. 
 
 It is very generally convenient to resolve forces into components 
 parallel to three lines at right angles to each other; each such reso- 
 lution being effected by multiplying by the cosine of the angle 
 concerned. 
 
 194. [If any number of points be placed in any positions in space, 
 another can be found, such that its distance from any plane what- 
 ever is the mean of their distances from that plane ; and if one or 
 more of the given points be in motion, the velocity of the mean 
 point perpendicular to the plane is the mean of the velocities of 
 the others in the same direction. 
 
 If we take two points A^, A2, the middle point, P^, of the line 
 joining them is obviously distant from any plane whatever by a 
 quantity equal to the mean (in this case the half sum or difference 
 as they are on the same or on opposite sides) of their distances 
 from that plane. Hence tzaice the distance of P2 from any plane 
 is equal to the (algebraic) sum of the distances of A-^, A from it. 
 Introducing a third point A^, if we join A^P^ and divide it in P^ 
 so that A-iP^- 2P^P2, three times the distance of P^ from any plane 
 is equal to the sum of the distance of A^ and twice that of P^ from 
 the same plane: i. e. to the sum of the distances of A^, A. 2, and A^ 
 from it ; or its distance is the mean of theirs. And so on for any 
 number of points. The proof is exceedingly simple. Thus suppose 
 Pn to be the mean of the first ;/ points A^, A^^.-.A^^; and A^^^ any 
 other point. Divide A^^^P^ in P^^^ so that A^^^P^^^ = nP^ 
 
 Then from P^, -^„+i> ^„+i> draw perpen- 
 diculars to any plane, meeting it in 6*, 7] V. 
 Draw P^QR parallel to STV, Then 
 QK.. : RA^, :: PP ,, : PA^,, :: 1 : n + 1, 
 Hence n + iQP^+^ = RA^^^. Add to these 
 71+ I ^7" and its equal nP^S+RV, and we get 
 
 '^^^{Q^n..^QT)-nP^S + RV+RA^,,, 
 i.e. n+iP^,,T=nP,S + A„^^V, 
 
 In words, n+ 1 times the distance of P^^^ from any plane is equal 
 to that of ^^+1 with n times that of P^, i. e. equal to the sum of the 
 
64 PRELIMINARY. 
 
 distances of A^^ ^^j-'-^n+i ^^^m the plane. Thus if the proposition 
 be true for any number of points, it is true for one more — and so on 
 - — but it is obviously true for two, hence for three, and therefore 
 generally. And it is obvious that the order in which the points are 
 taken is immaterial. 
 
 As the distance of this point from any plane is the mean of the 
 distances of the given ones, the rate of increase of that distance, 
 i. e. the velocity perpendicular to the plane, must be the mean of the 
 rates of increase of their distances — i. e. the mean of their velocities 
 perpendicular to the plane.] 
 
 195. The Centre of Inertia or Mass of a system of equal material 
 points (whether connected with one another or not) is the point 
 whose distance is equal to their average distance from any plane 
 whatever (§ 194). 
 
 A group of material points of unequal masses may always be 
 imagined as composed of a greater number of equal material points, 
 because we may imagine the given material points divided into dif- 
 ferent numbers of very small parts. In any case in which the magni- 
 tudes of the given masses are incommensurable, we may approach as 
 near as we please to a rigorous fulfilment of the preceding statement, 
 by making the parts into which we divide them sufficiently small. 
 
 On this understanding the preceding definition may be applied 
 to define the centre of inertia of a system of material points, whether 
 given equal or not. The result is equivalent to this : — 
 
 The centre of inertia of any system of material points whatever 
 (whether rigidly connected with one another, or connected in any 
 way, or quite detached), is a point whose distance from any plane 
 is equal to the sum of the products of each mass into its distance 
 from the same plane divided by the sum of the masses. 
 
 We also see, from the proposition stated above, that a point whose 
 distance from three rectangular planes fulfils this condition, must 
 fulfil this condition also for every other plane. 
 
 The co-ordinates of the centre of inertia, of masses Wj, Wg, etc., 
 at points {x^y jFi, ^1), (^2) y^^ ^i)i ^tc, are given by the following 
 formulae : — 
 
 _ _ w-^x^ + Te/jy^a + Gtc. _ '^wx _ _ Srqy - _ ^7vz 
 ~ W1 + W2 + etc. ~^w ' ^w * ^w ' 
 
 These formulae are perfectly general, and can easily be put into 
 the particular shape required for any given case. 
 
 The Centre of Inertia or Mass is thus a perfectly definite point in 
 every body, or group of bodies. The term Centre of Gravity is often 
 very inconveniently used for it. The theory of the resultant action of 
 gravity, which will be given under Abstract Dynamics, shows that, 
 except in a definite class of distributions of matter, there is no fixed 
 point which can properly be called the Centre of Gravity of a rigid 
 body. In ordinary cases of terrestrial gravitation, however, an ap- 
 proximate solution is available, according to which, in common par- 
 lance, the term Centre of Gravity may be used as equivalent to 
 
DYNAMICAL LAWS AND PRINCIPLES. 65 
 
 Centre of Inertia ; but it must be carefully remembered that the fun- 
 damental ideas involved in the two definitions are essentially different. 
 The second proposition in § 194 may now evidently be stated 
 thus : — The sum of the momenta of the parts of the system in any 
 direction is equal to the momentum in the same direction of a mass 
 equal to the sum of the masses moving with a velocity equal to the 
 velocity of the centre of inertia. 
 
 196. The mean of the squares of the distances of the centre of 
 
 p inertia, /, from each of the points of a system 
 
 ^^ is less than the mean of the squares of the dis- 
 
 ^yy i tance of any other point, O, from them by the 
 
 ^^^ / \ square of 01. Hence the centre of inertia is 
 
 ^^ y j^ the point the sum of the squares of whose 
 
 ^ i U distances from any given points is a minimum. 
 
 For OP" = or + IP' + 2OIIQ, P being any one of the points 
 and PQ perpendicular to 01. But IQ is the distance of P from 
 a plane through / perpendicular to OQ. Hence the mean of all 
 distances, /ft is zero. Hence 
 
 (mean of IP') = (mean of OP') - 0I\ which is the proposition. 
 
 197. Again, the mean of the squares of the distances of the points 
 of the system from any line, exceeds the corresponding quantity for 
 a parallel line through the centre of inertia, by the square of the 
 distance between these lines. 
 
 For in the above figure, let the plane of the paper represent a 
 plane through / perpendicular to these lines, O the point in which 
 the first line meets it, P the point in which it is met by a parallel 
 line through any one of the points of the system. Draw, as before, 
 PQ perpendicular to 01. Then PI is the perpendicular distance, 
 from the axis through /, of the point of the system considered, PO 
 is its distance from the first axis, 01 the distance between the two 
 axes. 
 
 Then, as before, 
 
 (mean of OP') = OP + (mean of IP')\ 
 since the mean of IQ is still zero, IQ being the distance of a 
 point of the system from the plane through / perpendicular to 01. 
 
 198. If the masses of the points be unequal, it is easy to see (as 
 in § 195) that the first of these theorems becomes — 
 
 The sum of the squares of the distances of the parts of a system 
 from any point, each multiplied by the mass of that part, exceeds the 
 corresponding quantity for the centre of inertia by the product of 
 the square of the distance of the point from the centre of inertia, by 
 the whole mass of the system. 
 
 Also, . the sum of the products of the mass of each part of 
 a system by the square of its distance from any axis is called the 
 Moment of Inertia of the system about this axis ; and the second 
 proposition above is equivalent to — 
 
66 PRELIMINARY, 
 
 The moment of inertia of a system about any axis is equal to the 
 moment of inertia about a parallel axis through the centre of inertia, 
 /, together with the moment of inertia, about the first axis, of the 
 whole mass supposed condensed at /. 
 
 199. The Moment of any physical agency is the numerical mea- 
 sure of its importance. Thus, the moment of inertia of a body 
 round an axis (§ 198) means the importance of its inertia relatively 
 to rotation round that axis. Again, the moment of a force round a 
 point or round a line (§ 46), signifies the measure of its importance as 
 regards producing or balancing rotation round that point or round 
 that line. 
 
 It is often convenient to represent the moment of a force by a line 
 numerically equal to it, drawn through the vertex of the triangle 
 representing its magnitude, perpendicular to its plane, through the 
 front of a watch held in the plane with its centre at the point, and 
 facing so that the force tends to turn round this point in a direction 
 opposite to the hands. The moment of a force round any axis is the 
 moment of its component in any plane perpendicular to the axis, 
 round the point in which the plane is cut by the axis. Here we 
 imagine the force resolved into two components, one parallel to the 
 axis, which is ineffective so far as rotation round the axis is con- 
 cerned; the other perpendicular to the axis (that is to say, having its 
 line in any plane perpendicular to the axis). This latter component 
 may be called the effective component of the force, with reference 
 to rotation round the axis. And its moment round the axis may be 
 defined as its moment round the nearest point of the axis, which is 
 equivalent to the preceding definition. It is clear that the moment 
 of a force round any axis, is equal to the area of the projection on 
 any plane perpendicular to the axis, of the figure rejoresenting its 
 moment round any point of the axis. 
 
 200. [The projection of an area, plane or curved, on any plane, 
 is the area included in the projection of its bounding line. 
 
 If we imagine an area divided into any number of parts, the pro- 
 jections of these parts on any plane make up the projection of the 
 whole. But in this statement it must be understood that the areas 
 of partial projections are to be reckoned as positive if particular 
 sides, which, for brevity, we may call the outside of the projected 
 area and the front of the plane of projection, face the same way, 
 and negative if they face oppositely. 
 
 Of course if the projected surface, or any part of it, be a plane area 
 at right angles to the plane of projection, the projection vanishes. 
 The projections of any two shells having a common edge, on any 
 plane, are equal. The projection of a closed surface (or a shell with 
 evanescent edge), on any plane, is nothing. 
 
 Equal areas in one plane, or in parallel planes, have equal projec- 
 tions on any plane, whatever may be their figures. 
 
 Hence the projection of any plane figure, or of any shell edged 
 by a plane figure, on another plane, is equal to its area, multiplied 
 
DYNAMICAL LAWS AND PRINCIPLES. 67 
 
 by the cosine of the angle at which its plane is inclined to the plane 
 of projection. This angle is acute or obtuse, according as the out- 
 side of the projected area, and the front of the plane of projection, 
 face on the whole towards the same parts, or oppositely. Hence 
 lines representing, as above described, moments about a point in 
 different planes, are to be compounded as forces are. See an 
 analogous theorem in § 107.] 
 
 201. A Couple is a pair of equal forces acting in dissimilar direc- 
 tions in parallel lines. The Motnmt of a couple is the sum of the 
 moments of its forces about any point in their plane, and is therefore 
 equal to the product of either force into the shortest distance between 
 their directions. This distance is called the Ann of the couple. 
 
 The Axis of a Couple is a line drawn from any chosen point of 
 reference perpendicular to the plane of the couple, of such magnitude 
 and in such direction as to represent the magnitude of the moment, 
 and to indicate the direction in which the couple tends to turn. The 
 most convenient rule for fulfilling the latter condition is this: — Hold 
 a watch with its centre at the point of reference, and with its plane 
 parallel to the plane of the couple. Then, according as the motion 
 of the hands is contrary to, or along with the direction in which the 
 couple tends to turn, draw the axis of the couple through the face or 
 through the back of the watch. It will be found that a couple is 
 completely represented by its axis, and that couples are to be resolved 
 and compounded by the same geometrical constructions performed 
 with reference to their axes as forces or velocities, with reference to 
 the lines directly representing them. 
 
 202. By introducing in the definition of moment of velocity (§ 46) 
 the mass of the moving body as .a factor, we have an important 
 element of dynamical science, the Moment of Mometiium. The 
 laws of composition and resolution are the same as those already 
 explained. 
 
 203. [If the point of application of a force be displaced through 
 a small space, the resolved part of the displacement in the direction 
 of the force has been called its Virtual Velocity. This is positive or 
 negative according as the virtual velocity is in the same, or in the 
 opposite, direction to that of the force. 
 
 The product of the force, into the virtual velocity of its point of 
 application, has been called the Virtual Moment of the force. These 
 terms we have introduced since they stand in the history and develop- 
 ments of the science ; but, as we shall show further on, they are 
 inferior substitutes for a far more useful set of ideas clearly laid down 
 by Newton.] 
 
 204. A force is said to do ivork if its place of application has a 
 positive component motion in its direction ; and the work done by it 
 is measured by the product of its amount into this component motion. 
 
 Generally, unit of work is done by unit force acting through unit 
 space. In lifting coals from a pit, the amount of work done is 
 
 5—2 
 
68 PRELIMINARY. 
 
 proportional to the weight of the coals lifted; that is, to the force 
 overcome in raising them ; and also to the height through which they 
 are raised. The unit for the measurement of work adopted in practice 
 by British engineers, is that required to overcome a force equal to the 
 weight of a pound through the space of a foot ; and is called a Foot- 
 Powid. (See § 185.) 
 
 In purely scientific measurements, the unit of work is not the foot- 
 pound, but the kinetic unit force (§ 190) acting through unit of space. 
 Thus, for example, as we shall show further on, this unit is adopted 
 in measuring the work done by an electric current, the units for 
 electric and magnetic measurements being founded upon the kinetic 
 unit force. 
 
 If the weight be raised obliquely, as, for instance, along a smooth 
 inclined plane, the space through which the force has to be overcome 
 is increased in the ratio of the length to the height of the plane ; but 
 the force to be overcome is not the whole weight, but only the resolved 
 part of the weight parallel to the plane; and this is less than the 
 weight in the ratio of the height of the plane to its length. By multi- 
 plying these two expressions together, we find, as we might expect, 
 that the amount of work required is unchanged by the substitution of 
 the oblique for the vertical path. 
 
 205. Generally, for any force, the work done during an indefinitely 
 small displacement of the point of application is the virtual moment 
 of the force (§ 203), or is the product of the resolved part of the force 
 in the direction of the displacement into the displacement. 
 
 From this it appears, that if the motion of the point of application 
 be always perpendicular to the direction in which a force acts, such a 
 force does no work. Thus the mutual normal pressure between a 
 fixed and moving body, the tension of the cord to which a pendulum 
 bob is attached, or the attraction of the sun on a planet if the planet 
 describe a circle with the sun in the centre, are all instances in which 
 no work is done by the force. 
 
 206. The work done by a force, or by a couple, upon a body 
 turning about an axis, is the product of the moment of either into the 
 angle (in circular measure) through which the body acted on turns, if 
 the moment remains the same in all positions of the body. If the 
 moment be variable, the above assertion is only true for indefinitely 
 small displacements, but maybe made accurate by employing the proper 
 average moment of the force or of the couple. The proof is obvious. 
 
 207. Work done on a body by a force is always shown by a cor- 
 responding increase of vis viva, or kinetic energy, if no other forces 
 act on the body which can do work or have work done against them. 
 If work be done against any forces, the increase of kinetic energy is 
 less than in the former case by the amount of work so done. In 
 virtue of this, however, the body possesses an equivalent in the form 
 of Fofe?itial Efiergy (§ 239), if its physical conditions are such that 
 these forces will act equally, and in the same directions, if the motion 
 of the system is reversed. Thus there may be no change of kinetic 
 
DYNAMICAL LAWS AND PRINCIPLES. 69 
 
 energy produced, and the work done may be wholly stored up as 
 potential energy. 
 
 Thus a weight requires work to raise it to a height, a spring requires 
 work to bend it, air requires work to compress it, etc. ; but a raised 
 weight, a bent spring, compressed air, etc., are stores of energy which 
 can be made use of at pleasure. 
 
 208. In what precedes we have given some of Newton's Definitiones 
 nearly in his own words ; others have been enunciated in a form more 
 suitable to modern methods ; and some terms have been introduced 
 which were invented subsequent to the publication of the Principia. 
 But the Axiomata^ sive Leges Mofih, to which we now proceed, are 
 given in Newton's own words. The two centuries which have nearly 
 elapsed since he first gave them have not shown a necessity for any 
 addition or modification. The first two, indeed, were discovered by 
 Galileo : and the third, in some of its many forms, was known to 
 Hooke, Huyghens, Wallis, Wren, and others, before the publication 
 of the Principia. Of late there has been a tendency to divide the 
 second law into two, called respectively the second and third, and to 
 ignore the third entirely, though using it directly in every dynamical 
 problem ; but all who have done so have been forced indirectly to 
 acknowledge the incompleteness of their substitute for Newton's system, 
 by introducing as an axiom what is called D'Alembert's principle, which 
 is really a deduction from Newton's rejected third law. Newton's own 
 interpretation of his third law directly points out not only D'Alembert's 
 principle, but also the modern principles of Work and Energy. 
 
 209. An Axiom is a proposition, the truth of which must be ad- 
 mitted as soon as the terms in which it is expressed are clearly 
 understood. And, as we shall show in our chapter on ' Experience,' 
 physical axioms are axiomatic to those who have sufficient knowledge 
 of physical phenomena to enable them to understand perfectly what 
 is asserted by them. Without further remark we shall give Newton^s 
 Three Laws ; it being remembered that, as the properties of matter 
 might have been such as to render a totally different set of laws 
 axiomatic, these laws must be considered as resting on convictions 
 drawn from observation and experiment, not on intuitive perception. 
 
 210. Lex I. Corpus omne perseuerare in statu suo qtdescendi vel 
 movendi uniformiter i?t directimi^ nisi qtiatenus illud d viribus impressis 
 cogitur statum suum niutare. 
 
 Every body cojitinues in its state of rest or of uniform motion in a 
 straight line, except in so far as it may be compelled by impressed forces 
 to change that state. 
 
 211. The meaning of the term Rest, in physical science, cannot be 
 absolutely defined, inasmuch as absolute rest nowhere exists in nature. 
 If the universe of matter were finite, its centre of inertia might fairly 
 be considered as absolutely at rest ; or it might be imagined to be 
 moving with any uniform velocity in any direction whatever through 
 infinite space. But it is remarkable that the first law of motion 
 
70 PRELIMINARY. 
 
 enables us (§215, below) to explain what may be called directional 
 rest. Also, as will be seen farther on, a perfectly smooth spherical 
 body, made up of concentric shells, each of uniform material and 
 density throughout, if made to revolve about an axis, will, /// spite of 
 impressed forces, revolve with uniform angular velocity, and will main- 
 tain its axis of revolution in an absolutely fixed direction. Or, as will 
 soon be shown (§ 233), the plane in which the moment of momentum 
 of the universe (if finite) round its centre of inertia is the greatest, 
 which is clearly determinable from the actual motions at any instant, 
 is fixed in direction in space. 
 
 212. We may logically convert the assertion of the first law of 
 motion as to velocity into the following statements : — 
 
 The times during which any particular body, not compelled by 
 force to alter the speed of its motion, passes through equal spaces, 
 are equal. And, again — Every other body in the universe, not com- 
 pelled by force to alter the speed of its motion, moves over equal 
 spaces in successive intervals, during which the particular chosen body 
 moves over equal spaces. 
 
 213. The first part merely expresses the convention universally 
 adopted for the measurement of Time. The earth in its rotation 
 about its axis, presents us with a case of motion in which the con- 
 dition of not being compelled by force to alter its speed, is more 
 nearly fulfilled than in any other which we can easily or accurately 
 observe. And the numerical measurement of time practically rests 
 on defining equal intervals of time, as times duri?ig which the earth turns 
 through equal angles. This is, of course, a mere convention, and 
 not a law of nature; and, as we now see it, is a part of Newton's 
 first law. 
 
 214. The remainder of the law is not a convention, but a great 
 truth of nature, which we may illustrate by referring to small and 
 trivial cases as well as to the grandest phenomena we can conceive. 
 
 A curling-stone, projected along a horizontal surface of ice, travels 
 equal distances, except in so far as it is retarded by friction and by 
 the resistance of the air, in successive intervals of time during which 
 the earth turns through equal angles. The sun moves through equal 
 portions of interstellar space in times during which the earth turns 
 through equal angles, except in so far as the resistance of interstellar 
 matter, and the attraction of other bodies in the universe, alter his 
 speed and that of the earth's rotation. 
 
 215. If two material points be projected from one position. A, at 
 the same instant with any velocities in any directions, and each left to 
 move uninfluenced by force, the line joining them will be always 
 parallel to a fixed direction. For the law asserts, as we have seen, 
 that AP : AP' :: AQ : AQ\ i( P, Q, and again P\ Q', are simulta- 
 neous positions ; and therefore P(2 is parallel to P'Q'. Hence if four 
 material points O, P, Q, R are all projected at one instant from one 
 position, OP, OQ, OR are fixed directions of reference ever after. 
 
D YNAMICAL LA WS AND PRINCIPLES. 7 1 
 
 But, practically, the determination of fixed directions in space 
 (§ 233) is made to depend upon the rotation of groups of particles 
 exerting forces on each other, and thus involves the Third Law of 
 Motion. 
 
 216. The whole law is singularly at variance with the tenets of the 
 ancient philosophers, who maintained that circular motion is perfect. 
 
 The last clause, '■nisi quate7ius^ etc., admirably prepares for the 
 introduction of the second law, by conveying the idea that it is force 
 alo7ie ivhich can produce a change of motion. How, we naturally in- 
 quire, does the change of motion produced depend on the magnitude 
 and direction of the force which produces it? The answer is — 
 
 217. Lex II. Mutatio7iem mot us proportion akm esse vi 7notrici i7n' 
 pressae, et fieri secundum Ii7iea77i recta77i qua vis ilia imprimitur. 
 
 Cha7ige of 77iotio7i is proportiojialto the i77ipressed force^ and takes place 
 in the di7'ection of the straight line in which thefo7'ce acts. 
 
 218. If any force generates motion, a double force will generate 
 double motion, and so on, whether simultaneously or successively, 
 instantaneously or gradually, applied. And this motion, if the body 
 was moving beforehand, is either added to the previous motion if 
 directly conspiring with it; or is subtracted if directly opposed ; or 
 is geometrically compounded with it, according to the kinematical 
 principles already explained, if the line of previous motion and the 
 direction of the force are inclined to each other at any angle. (This 
 is a paraphrase of Newton's own comments on the second law.) 
 
 219. In Chapter I. we have considered change of velocity, or 
 acceleration, as a purely geometrical element, and have seen how it 
 may be at once inferred from the given initial and final velocities of a 
 body. By the definition of a quantity of motion (§ 211), we see that, 
 if we multiply the change of velocity, thus geometrically determined, 
 by the mass of the body, we have the change of motion referred to in 
 Newton's law as the measure of the force which produces it. 
 
 It is to be particularly noticed, that in this statement there is nothing 
 said about the actual motion of the body before it was acted on by the 
 force : it is only the cha7ige of motion that concerns us. Thus the 
 same force will produce precisely the same change of motion in a 
 body, whether the body be at rest, or in motion with any velocity 
 whatever. 
 
 220. Again, it is to be noticed that nothing is said as to the body 
 being under the action of one force only ; so that we may logically 
 put a part of the second law in the following (apparently) amplified 
 form : — 
 
 Wheji any forces whatever act 07i a body, the7t, whether the body be 
 07'igi7ially at rest or moving with a7iy velocity a7id i7i a7iy direction, each 
 force produces in the body the exact change of 7notio7i which itivouldhave 
 produced if it had acted singly on the body originally at rest. 
 
 221. A remarkable consequence follows immediately from this view 
 of the second law. Since forces are measured by the changes of 
 
72 . PRELIMINARY. 
 
 motion they produce, and their directions assigned by the directions 
 in which these changes are produced; and since the changes of 
 motion of one and the same body are in the directions of, and pro- 
 portional to, the changes of velocity — a single force, measured by the 
 resultant change of velocity, and in its direction, will be the equivalent 
 of any number of simultaneously acting forces. Hence 
 
 The resultajit of any number of forces {applied at one point) is to he 
 found by the same geometrical process as the resultant of any jtumber of 
 simultaneous velocities, 
 
 222. From this follows at once (§ 31) the construction of the 
 Parallelogram of Forces for finding the resultant of two forces, and 
 the Polygon of Forces for the resultant of any number of forces, in 
 lines all through one point. 
 
 The case of the equilibrium of a number of forces acting at one 
 point, is evidently deducible at once from this ; for if we introduce 
 one other force equal and opposite to their resultant, this will produce 
 a change of motion equal and opposite to the resultant change of 
 motion produced by the given forces ; that is to say, will produce a 
 condition in which the point experiences no change of motion, which, 
 as we have already seen, is the only kind of rest of which we can ever 
 be conscious. 
 
 223. Though Newton perceived that the Parallelogram of Forces, 
 or the fundamental principle of Statics, is essentially involved in the 
 second law of motion, and gave a proof which is virtually the same as 
 the preceding, subsequent writers on Statics (especially in this country) 
 have very generally ignored the fact ; and the consequence has been 
 the introduction of various unnecessary Dynamical Axioms, more or 
 less obvious, but in reality included in or dependent upon Newton's 
 laws of motion. We have retained Newton's method, not only on 
 account of its admirable simplicity, but because we believe it contains 
 the most philosophical foundation for the static as well as for the 
 kinetic branch of the dynamic science. , 
 
 224. But the second law gives us the means of measuring force, 
 and also of measuring the mass of a body. 
 
 For, if we consider the actions of various forces upon the same 
 body for equal times, we evidently have changes of velocity produced 
 which 2XQ proportional to the forces. The changes of velocity, then, 
 give us in this case the means of comparing the magnitudes of different 
 forces. Thus the velocities acquired in one second by the same mass 
 (falling freely) at different parts of the earth's surface, give us the 
 relative amounts of the earth's attraction at these places. 
 
 Again, if equal forces be exerted on different bodies, the changes 
 of velocity produced in equal times must be inversely as the masses 
 of the various bodies. This is approximately the case, for instance, 
 with trains of various lengths started by the same locomotive : it is 
 exactly realized in such cases as the action of an electrified body on 
 a number of solid or hollow spheres of the same external diameter, 
 and of different metals. 
 
DYNAMICAL LAWS AND PRINCIPLES. 73 
 
 Again, if we find a case in which different bodies, each acted on 
 by a force, acquire in the same time the same changes of velocity, 
 the forces must be proportional to the masses of the bodies. This, 
 when the resistance of the air is removed, is the case of falling bodies; 
 and from it we conclude that the weight of a body in any given 
 locality, or the force with which the earth attracts it, is proportional 
 to its mass ; a most important physical truth, which will be treated 
 of more carefully in the chapter devoted to Properties of Matter. 
 
 225. It appears, lastly, from this law, that every theorem of Kine- 
 matics connected with acceleration has its counterpart in Kinetics. 
 Thus, for instance (§ 38), we see that the force under which a par- 
 ticle describes any curve, may be resolved into two components, one 
 in the tangent to the curve, the other towards the centre of curvature ; 
 their magnitudes being the acceleration of momentum, and the pro- 
 duct of the momentum and the angular velocity about the centre of 
 curvature, respectively. In the case of uniform motion, the first of 
 these vanishes, or the whole force is perpendicular to the direction 
 of motion. When there is no force perpendicular to the direction 
 of motion, there is no curvature, or the path is a straight line. 
 
 226. We have, by means of the first two laws, arrived at a defijiitiott 
 and a measure of force ; and have also found how to compound, and 
 therefore also how to resolve, forces : and also how to investigate 
 the motion of a single particle subjected to given forces. But more 
 is required before we can completely understand the more complex 
 cases of motion, especially those in which we have mutual actions 
 between or amongst two or more bodies; such as, for instance, 
 attractions, or pressures, or transferrence of energy in any form. 
 This is perfectly supplied by 
 
 227. Lex III. Actioni contrariam semper et aeqiialem esse reactio- 
 nem: sive corporiim duorum actiones in se mutuo semper esse aequales 
 et in partes contrarias dirigi. 
 
 To every action there is always an equal and contrary reaction: or, the 
 mutual actio7is of any two bodies are always equal a?td oppositely directed. 
 
 228. If one body presses or draws another, it is pressed or 
 drawn by this other with an equal force in the opposite direction. 
 If any one presses a stone with his finger, his finger is pressed with 
 the same force in the opposite direction by the stone. A horse 
 towing a boat on a canal is dragged backwards by a force equal to 
 that which he impresses on the towing-rope forwards. By whatever 
 amount, and in whatever direction, one body has its motion changed 
 by impact upon another, this other body has its motion changed by 
 the same amount in the opposite direction; for at each instant during 
 the impact the force between them was equal and opposite on the 
 two. When neither of the two bodies has any rotation, whether 
 before or after impact, the changes of velocity which they experience 
 are inversely as their masses. 
 
 When one body attracts another from a distance, this other attracts 
 it with an equal and opposite force. This law holds not only for 
 
74 PRELIMINARY. 
 
 the attraction of gravitation, but also, as Newton himself remarked 
 and verified by experiment, for magnetic attractions : also for electric 
 forces, as tested by Otto-Guericke. 
 
 229. What precedes is founded upon Newton's own comments 
 on the third law, and the actions and reactions contemplated are 
 simple forces. In the scholium appended, he makes the following 
 remarkable statement, introducing another specification of actions 
 and reactions subject to his third law, the full meaning of which 
 seems to have escaped the notice of commentators : — 
 
 Si aestitnetur agentis actio ex ejus vi et velocitate conjundim; et 
 similiter resistentis reactio aestimetur conjunctim ex ejus partium singu- 
 lariim velocitatibus et viribiis resistendi ab earwn attritione, cohaesioiie^ 
 pondere^ et acceleratioiie oriiindis ; erunt actio et reactio^ in omni instru- 
 mentorimi usn, sibi invicem semper aeqiiales. 
 
 In a previous discussion Newton has shown what is to be under- 
 stood by the velocity of a force or resistance ; i. e. that it is the 
 velocity of the point of application of the force resolved in the direction 
 of the force, in fact proportional to the virtual velocity. Bearing this 
 in mind, we may read the above statement as follows : — 
 
 If the action of an agent be measured by the product of its force into 
 its velocity; and if similarly, the reaction of the resistaiice be 7neasured 
 by the velocities of its several parts into their several forces, whether 
 these arise from friction, cohesion, weight, or acceleration; — action and 
 reaction, in all combinations of machines, will be equal and opposite. 
 
 To avoid confusion it is perhaps better to use the word Activity as 
 the equivalent of Actio in this second specification. 
 
 Farther on we shall give a full development of the consequences 
 of this most important remark. 
 
 230. Newton, in the passage just quoted, points out that forces 
 of resistance against acceleration are to be reckoned as reactions 
 equal and opposite to the actions by which the acceleration is pro- 
 duced. Thus, if we consider any one material point of a system, 
 its reaction against acceleration must be equal and opposite to the 
 resultant of the forces which that point experiences, whether by the 
 actions of other parts of the system upon it, or by the influence of 
 matter not belonging to the system. In other words, it must be in 
 equilibrium with these forces. Hence Newton's view amounts to this, 
 that all the forces of the system, with the reactions against accelera- 
 tion of the material points composing it, form groups of equilibrating 
 systems for these points considered individually. Hence, by the 
 principle of superposition of forces in equilibrium, all the forces 
 acting on points of the system form, with the reactions against acce- 
 leration, an equilibrating set of forces on the whole system. This 
 is the celebrated principle first explicitly stated, and very usefully 
 applied, by D'Alembert in 1742, and still known by his name. We 
 have seen, however, that it is very distinctly implied in Newton's 
 own interpretation of his third law of motion. As it is usual to inves- 
 
DYNAMICAL LAWS AND PRINCIPLES. 75 
 
 tigate the general equations or conditions of equilibrium, in treatises 
 on Analytical Dynamics, before entering in detail on the kinetic 
 branch of the subject, this principle is found practically most useful 
 in showing how we may write down at once the equations of motion 
 for any system for which the equations of equilibrium have been 
 investigated. 
 
 231. Every rigid body may be imagined to be divided into inde- 
 finitely small parts. Now, in whatever form we may eventually 
 find a physical explanation of the origin of the forces which act 
 between these parts, it is certain that each such small part may be 
 considered to be held in its position relatively to the others by mutual 
 forces in lines joining them. 
 
 232. From this we have, as immediate consequences of the second 
 and third laws, and of the preceding theorems relating to centre of 
 inertia and moment of momentum, a number of important propo- 
 sitions such as the following : — 
 
 ia) The centre of inertia of a rigid body moving in any manner, 
 but free from external forces, moves uniformly in a straight line. 
 
 {I)) When any forces whatever act on the body, the motion of the 
 centre of inertia is the same as it would have been had these forces 
 been applied with their proper magnitudes and directions at that 
 point itself. 
 
 {c) Since the moment of a force acting on a particle is the same 
 as the moment of momentum it produces in unit of time, the changes 
 of moment of momentum in any two parts of a rigid body due to 
 their mutual action are equal and opposite. Hence the moment of 
 momentum of a rigid body, about any axis which is fixed in direction, 
 and passes through a point which is either fixed in space or moves 
 uniformly in a straight line, is unaltered by the mutual actions of the 
 parts of the body. 
 
 id) The rate of increase of moment of momentum, when the body 
 is acted on by external forces, is the sum of the moments of these 
 forces about the axis. 
 
 233. We shall for the present take for granted, that the mutual 
 action between two rigid bodies may in every case be imagined as 
 composed of pairs of equal and opposite forces in straight lines. 
 From this it follows that the sum of the quantities of motion, parallel 
 to any fixed direction, of two rigid bodies influencing one another 
 in any possible way, remains unchanged by their mutual action; 
 also that the sum of the moments of momentum of all the particles 
 of the two bodies, round any line in a fixed direction in space, and 
 passing through any point moving uniformly in a straight line in any 
 direction, remains constant. From the first of these propositions we 
 infer that the centre of inertia of any number of mutually influencing, 
 bodies, if in motion, continues moving uniformly in a straight line,, 
 unless in so far as the direction or velocity of its motion is changed 
 by forces acting mutually between them and some other matter not 
 belonging to them ; also that the centre of inertia of any body or 
 
76 PRELIMINARY. 
 
 system of bodies moves just as all their matter, if concentrated in 
 a point, would move under the influence of forces equal and parallel 
 to the forces really acting on its different parts. From the second 
 we infer that the axis of resultant rotation through the centre of 
 inertia of any system of bodies, or through any point either at rest 
 or moving uniformly in a straight line, remains unchanged in direc- 
 tion, and the sum of moments of momenta round it remains constant 
 if the system experiences no force from without. This principle 
 is sometimes called Cofiservation of Areas, a not very convenient 
 designation. From this principle it follows that if by internal action 
 such as geological upheavals or subsidences, or pressure of the winds 
 on the water, or by evaporation and rain- or snow-fall, or by any in- 
 fluence not depending on the attraction of 'Sun or moon (even though 
 dependent on solar heat), the disposition of land and water becomes 
 altered, the component round any fixed axis of the moment of mo- 
 mentum of the earth's rotation remains constant. 
 
 234. The kinetic energy of any system is equal to the sum of the 
 kinetic energies of a mass equal to the sum of the masses of the 
 system, moving with a velocity equal to that of its centre of inertia, 
 and of the motions of the separate parts relatively to the centre of 
 inertia. 
 
 Let 6>/ represent the velocity of the centre of inertia, IP that of 
 
 p any point of the system relative to O. Then 
 
 ^J^ the actual velocity of that point is OP, and the 
 
 ^^^^/ \ proof of § 196 applies at once — it being re- 
 
 ^-^ / \ membered that the mean of 7(2, i. e. the mean 
 
 -^i— r^ i of the velocities relative to th'e centre of inertia 
 
 ^ i « g^^^ parallel to 01, is zero by § 65. 
 
 235. The kinetic energy of rotation of a rigid system about any 
 axis is (§§ 55, 179) expressed by \%mi^<i?, where ;// is the mass of 
 any part, r its distance from the axis, and w the angular velocity of 
 rotation. It may evidently be written in the form \i^^%mt^. The 
 factor ^niT^ is of course (§ 198) the Moment of Inertia of the system 
 about the axis in question. 
 
 It is worth while to notice that the moment of momentum of any 
 rigid system about an axis, being '^mvr=(a%7nr^, is the product of 
 the angular velocity into the moment of inertia; while, as above, the 
 half product of the moment of inertia by the square of the angular 
 velocity is the kinetic energy. 
 
 If we take a quantity k, such that 
 
 k'^m^-^mr^ 
 
 k is called the Radius of Gyration about the axis from which r is 
 measured. The radius of gyration about any axis is therefore the 
 distance from that axis at which, if the whole mass were placed, it 
 would have the same moment of inertia as before. In a fly-wheel, 
 where it is desirable to have as great a moment of inertia with as 
 small a mass as possible, within certain limits of dimensions, the 
 
D YNAMICAL LA WS AND PRINCIPLES. 7 7 
 
 greater part of the mass is formed into a ring of the largest admis- 
 sible diameter, and the radius of this ring is then approximately the 
 radius of gyration of the whole. 
 
 236. The rate of increase of moment of momentum is thus, in New- 
 ton's notation (§ 28), wi%jnr^; and, in the case of a body free to rotate 
 about a fixed axis, is equal to the moment of the couple about that 
 axis. Hence a constant couple gives uniform acceleration of angular 
 
 velocity; or <b= . By § 178 we see that the corresponding 
 
 Force 
 formula for linear acceleration is s = v = — — ^ . 
 
 M 
 
 237. For every rigid body there may be described about any point 
 as centre, an ellipsoid (called Poinsofs Momenta! Ellipsoid) which is 
 such that the length of any radius-vector is inversely proportional to 
 the radius of gyration of the body about that radius- vector as axis. 
 
 The axes of the ellipsoid are the Principal Axes of inertia of the 
 body at the point in question. 
 
 When the moments of inertia about two of these are equal, the 
 ellipsoid becomes a spheroid, and the radius of gyration is the same 
 for every axis in the plane of its equator. 
 
 When all three principal moments are equal, the ellipsoid becomes 
 a sphere, and every axis has the same radius of gyration. 
 
 238. The principal axes at any point of a rigid body are normals 
 to the three surfaces of the second order which pass through that 
 point, and are confocal with an ellipsoid, having its centre at the 
 centre of inertia, and its three principal diameters coincident with the 
 three principal axes through these points, and equal respectively to 
 the doubles of the radii of gyration round them. This ellipsoid is 
 called the Cetitral Ellipsoid. 
 
 239. A rigid body is said to be kinetically symmetrical about its 
 centre of inertia when its moments of inertia about three principal 
 axes through that point are equal ; and therefore necessarily the 
 moments of inertia about all axes through that point equal (§ 237), 
 and all these axes principal axes. About it uniform spheres, cubes, 
 and in general any complete crystalline solid of the first system (see 
 chapter on Properties of Matter) are kinetically symmetrical. 
 
 A rigid body is kinetically symmetrical about an axis when this 
 axis is one of the principal axes through the centre of inertia, and 
 the moments of inertia about the other two, and therefore about any 
 line in their plane, are equal. A spheroid, a square or equilateral 
 triangular prism or plate, a circular ring, disc, or cylinder, or any 
 complete crystal of the second or fourth system, is kinetically sym- 
 metrical about its axis. 
 
 240. The foundation of the abstract theory of energy is laid by 
 Newton in an admirably distinct and compact manner in the sentence 
 of his scholium already quoted (§ 229), in which he points out its 
 
78 PRELIMINARY. 
 
 application to mechanics^. The actio agentis, as he defines it, which 
 is evidently equivalent to the product of the effective component 'of 
 the force, into the velocity of the point on which it acts, is simply, in 
 modern English phraseology, the rate at which the agent works. The 
 subject for measurement here is precisely the same as that for which 
 Watt, a hundred years later, introduced the practical unit of a ^Horse- 
 p07ver^^ or the rate at which an agent works when overcoming 33,000 
 times the weight of a pound through the space of a foot in a minute ; 
 that is, producing 550 foot-pounds of work per second. The unit, 
 however, which is most generally convenient is that which Newton's 
 definition implies, namely, the rate of doing work in which the unit 
 of energy is produced in the unit of time. 
 
 241. Looking at Newton's words (§ 229) in this light, we see that 
 they may be logically converted into the following form : — 
 
 Work done o?i any system of bodies (in Newton's statement, the parts 
 of any machine) has its equivalent in work done against friction^ 
 molecular forces, or gravity, if there be no accele?'ation ; but if there 
 be acceleration, part of the work is expended in overcoming the resistance 
 to acceleration, and the additional kinetic energy developed is equivalent 
 to the work so spent. This is evident from § 180. 
 
 When part of the work is done against molecular forces, as in 
 bending a spring; or against gravity, as in raising a weight; the 
 recoil of the spring, and the fall of the weight, are capable at any 
 future time, of reproducing the work originally expended (§ 207). 
 But in Newton's day, and long afterwards, it was supposed that work 
 was absolutely lost by friction; and, indeed, this statement is still to 
 be found even in recent authoritative treatises. But we must defer 
 the examination of this point till we consider in its modern form the 
 principle of Co?iservation of Energy. 
 
 242. If a system of bodies, given either at rest or in motion, be 
 influenced by no forces from without, the sum of the kinetic energies 
 of all its parts is augmented in any time by an amount equal to the 
 whole work done in that time by the mutual forces, which we may 
 imagine as acting between its points. When the lines in which these 
 forces act remain all unchanged in length, the forces do no work, and 
 the sum of the kinetic energies of the whole system remains constant. 
 If, on the other hand, one of these lines varies in length during the 
 motion, the mutual forces in it will do work, or will consume work, 
 according as the distance varies with or against them. 
 
 243. A limited system of bodies is said to be dynamically con- 
 servative (or simply cotiservative, when force is understood to be the 
 subject), if the mutual forces between its parts always perform, or 
 always consume, the same amount of work during any motion 
 
 1 The reader will remember that we use the word 'mechanics' in its true classical 
 sense, the science of machines, the sense in which Newton himself used it, when he 
 dismissed the further consideration of it by saying (in the scholium referred to), 
 Caeferum viecJianicani h-nctare non rst Jinjiis iusfifuti. 
 
DYNAMICAL LA WS AND PRINCIPLES. 79 
 
 whatever, by which it can pass from one particular configuration 
 to another. 
 
 244. The whole theory of energy in physical science is founded 
 on the following proposition : — 
 
 If the mutual forces between the parts of a material system are 
 independent of their velocities, whether relative to one another, or 
 relative to any external matter, the system must be dynamically 
 conservative. 
 
 For if more work is done by the mutual forces on the different 
 parts of the system in passing from one particular configuration to 
 another, by one set of paths than by another set of paths, let the 
 system be directed, by frictionless constraint, to pass from the first 
 configuration to the second by one set of paths and return by the 
 other, over and over again for ever. It will be a continual source of 
 energy without any consumption of materials, which is impossible. 
 
 245. The potential energy of a conservative system, in the confi- 
 guration which it has at any instant, is the amount of work that its 
 mutual forces perform during the passage of the system from any 
 one chosen configuration to the configuration at the time referred to. 
 It is generally, but not always, convenient to fix the particular con- 
 figuration chosen for the zero of reckoning of potential energy, so 
 that the potential energy, in every other configuration practically 
 considered, shall be positive. 
 
 246. The potential energy of a conservative system, at any instant, 
 depends solely on its configuration at that instant, being, according to 
 definition, the same at all times when the system is brought again 
 and again to the same configuration. It is therefore, in mathematical 
 language, said to be a function of the co-ordinates by which the 
 positions of the different parts of the system are specified. If, for 
 example, we have a conservative system consisting of two material 
 points; or two rigid bodies, acting upon one another with force 
 dependent only on the relative position of a point belonging to one 
 of them, and a point belonging to the other; the potential energy 
 of the system depends upon the co-ordinates of one of these points 
 relatively to lines of reference in fixed directions through the other. 
 It will therefore, in general, depend on three independent co-ordi- 
 nates, which we may conveniently take as the distance between the 
 two points, and two angles specifying the absolute direction of the 
 line joining them. Thus, for example, let the bodies be two uniform 
 metal globes, electrified with any given quantities of electricity, and 
 placed in an insulating medium such as air, in a region of space 
 under the influence of a vast distant electrified body. The mutual 
 action between these two spheres will depend solely on the relative 
 position of their centres. It will consist partly of gravitation, de- 
 pending solely on the distance between their centres, and of electric 
 force, which will depend on the distance between them, but also, in 
 virtue of the inductive action of the distant body, will depend on the 
 absolute direction of the line joining their centres. Or again, if the 
 
8o PRELIMINARY. 
 
 system consist of two balls of soft iron, in any locality of the earth's 
 surface, their mutual action will be partly gravitation, and partly 
 due to the magnetism induced in them by terrestrial magnetic force. 
 The portion of the potential energy depending on the latter cause, 
 will be a function of the distance between their centres and the in- 
 clination of this line to the direction of the terrestrial magnetic force, 
 
 247. In nature the hypothetical condition of § 243 is apparmtly 
 violated in all circumstances of motion. A material system can never 
 be brought through any returning cycle of motion without spending 
 more work against the mutual forces of its parts than is gained from 
 these forces, because no relative motion can take place without 
 meeting with frictional or other forms of resistance; among which 
 are included (i) mutual friction between solids sliding upon one 
 another; (2) resistances due to the viscosity of fluids, or imperfect 
 elasticity of solids; (3) resistances due to the induction of electric 
 currents; (4) resistances due to varying magnetization under the 
 influence of imperfect magnetic retentiveness. No motion in nature 
 can take place without meeting resistance due to some, if not to all, 
 of these influences. It is matter of everyday experience that friction 
 and imperfect elasticity of solids impede the action of all artificial 
 mechanisms; and that even when bodies are detached, and left to 
 move freely in the air, as falling bodies, or as projectiles, they expe- 
 rience resistance owing to the viscosity of the air. 
 
 The greater masses, planets and comets, moving in a less resisting 
 medium, show less indications of resistance ^ Indeed it cannot be said 
 that observation upon any one of these bodies, with the possible excep- 
 tion of Encke's comet, has demonstrated resistance. But the analogies 
 of nature, and the ascertained facts of physical science, forbid us to 
 doubt that every one of them, every star, and every body of any kind 
 moving in any part of space, has its relative motion impeded by the 
 air, gas, vapour, medium, or whatever we choose to call the substance 
 occupying the space immediately round it; just as the motion of a 
 rifle-bullet is impeded by the resistance of the air. 
 
 248. There are also indirect resistances, owing to friction impeding 
 the tidal motions, on all bodies which, like the earth, have portions 
 of their free surfaces covered by liquid, which, as long as these bodies 
 move relatively to neighbouring bodies, must keep drawing ofl" energy 
 from their relative motions. Thus, if we consider, in the first place, 
 the action of the moon alone, on the earth with its oceans, lakes, and 
 rivers, we perceive that it must tend to equalize the periods of the 
 earth's rotation about its axis, and of the revolution of the two bodies 
 about their centre of inertia ; because as long as these periods differ, 
 the tidal action of the earth's surface must keep subtracting energy 
 from their modons. To view the subject more in detail, and, at the 
 same time, to avoid unnecessary complications, let us suppose the 
 
 1 Newton, Principia. (Remarks on the first law of motion.) 'Majora auteni 
 Planetarum et Cometarum corpora motus suos et progressivos et circulares, in 
 spatiis minus resistentibus factos, conservant diutius.' 
 
DYNAMICAL LA WS AND PRINCIPLES. 8i 
 
 moon to be a uniform spherical body. The mutual action and 
 reaction of gravitation between her mass and the earth's, will be 
 equivalent to a single force in some line through her centre • and 
 must be such as to impede the earth's rotation as long as this is 
 performed in a shorter period than the moon's motion round the 
 earth. It must therefore lie in some such direction as the line MQ 
 in the diagram, which represents, necessarily 
 with enormous exaggeration, its deviation, 
 OQ, from the earth's centre. Now the actual 
 force on the moon in the line MQ^ may be 
 regarded as consisting of a force in the line 
 MO towards the earth's centre, sensibly 
 equal in amount to the whole force, and a 
 comparatively very small force in the line 
 MT perpendicular to MO. This latter is 
 very nearly tangential to the moon's path, 
 and is in the direction with her motion. 
 Such a force, if suddenly commencing to act, would, in the first place, 
 increase the moon's velocity; but after a certain time she would have 
 moved so much farther from the earth, in virtue of this acceleration, 
 as to have lost, by moving against the earth's attraction, as much 
 velocity as she had gained by the tangential accelerating force. The 
 integral effect on the moon's motion, of the particular disturbing 
 cause now under consideration, is most easily found by using the prin- 
 ciple of moments of momenta (§ 233). Thus we see that as much 
 moment of momentum is gained in any time by the motions of the 
 centres of inertia of the moon and earth relatively to their common 
 centre of inertia, as is lost by the earth's rotation about its axis. It 
 is found that the distance would be increased to about 347,100 miles, 
 and the period lengthened to 48-36 days. Were there no other body 
 in the universe but the earth and the moon, these two bodies might 
 go on moving thus for ever, in circular orbits round their common 
 centre of inertia, and the earth rotating about its axis in the same 
 period, so as always to turn the same face to the moon, and therefore 
 to have all the liquids at its surface at rest relatively to the solid. But 
 the existence of the sun would prevent any such state of things from 
 being permanent. There would be solar tides — twice high water and 
 twice low water — in the period of the earth's revolution relatively to 
 the sun (that is to say, twice in the solar day, or, which would be the 
 same thing, the month). This could not go on without loss of energy 
 by fluid friction. It is not easy to trace the whole course of the 
 disturbance in the earth's and moon's motions which this cause 
 would produce, but its ultimate effect must be to bring the earth, 
 moon, and sun to rotate round their common centre of inertia, like 
 parts of one rigid body. It is probable that the moon, in ancient 
 times liquid or viscous in its outer layer if not throughout, was thus 
 brought to turn always the same face to the earth. 
 
 249. We have no data in the present state of science for estimating 
 the relative importance of tidal friction, and of the resistance of the 
 T. 6 
 
82 PRELIMINARY. 
 
 resisting medium through which the earth and moon move; but what- 
 ever it may be, there can be but one ultimate result for such a system 
 as that of the sun and planets, if continuing long enough under ex- 
 isting laws, and not disturbed by meeting with other moving masses 
 in space. That result is the falling together of all into one mass, 
 which, although rotating for a time, must in the end come to rest 
 relatively to the surrounding medium. 
 
 250. The theory of energy cannot be completed until we are able 
 to examine the physical influences which accompany loss of energy 
 in each of the classes of resistance mentioned above (§ 247). We 
 shall then see that in every case in which energy is lost by resistance, 
 heat is generated; and we shall learn from Joule's investigations that 
 the quantity of heat so generated is a perfectly definite equivalent for 
 the energy lost. Also that in no natural action is there ever a develop- 
 ment of energy which cannot be accounted for by the disappearance 
 of an equal amount elsewhere by means of some known physical 
 agency. Thus we shall conclude, that if any Hmited portion of the 
 material universe could be perfectly isolated, so as to be prevented 
 from either giving energy to, or taking energy from, matter external 
 to it, the sum of its potential and kinetic energies would be the same 
 at all times: in other words, that every material system subject to no 
 other forces than actions and reactions between its parts, is a dyna- 
 mically conservative system, as defined above (§ 243). But it is only 
 when the inscrutably minute motions among small parts, possibly the 
 ultimate molecules of matter, which constitute light, heat, and mag- 
 netism; and the intermolecular forces of chemical affinity; are taken 
 into account, along with the palpable motions and measurable forces 
 of which we become cognizant by direct observation, that we can 
 recognize the universally conservative character of all natural dynamic 
 action, and perceive the bearing of the principle of reversibility on the 
 whole class of natural actions involving resistance, which seem to 
 violate it. In the meantime, in our studies of abstract dynamics, it 
 will be sufficient to introduce a special reckoning for energy lost in 
 working against, or gained from work done by, forces not belonging 
 
 "palpably to the conservative class. 
 
 251. The only actions and reactions between the parts of a system, 
 not belonging palpably to the conservative class, which we shall con- 
 sider in abstract dynamics, are those of friction between soHds sliding 
 on solids, except in a few instances in which we shall consider the 
 general character and ultimate results of effects produced by viscosity 
 of fluids, imperfect elasticity of solids, imperfect electric conduction, 
 or imperfect magnetic retentiveness. We shall also, in abstract dyna- 
 mics, consider forces as applied to parts of a limited system arbitrarily 
 from without. These we shall call, for brevity, the applied forces. 
 
 252. The law of energy may then, in abstract dynamics, be ex- 
 pressed as follows : — 
 
 The whole work done in any time, on any limited material system, 
 by applied forces, is equal to the whole effect in the forms of potential 
 
DYNAMICAL LAWS AND PRINCIPLES. 83 
 
 and kinetic energy produced in the system, together with the work lost 
 in friction. 
 
 253. This principle may be regarded as comprehending the whole 
 of abstract dynamics, because, as we now proceed to show, the con- 
 ditions of equihbrium and of motion, in every possible case, may be 
 derived from it. 
 
 254. A material system, whose relative motions are unresisted by 
 friction, is in equilibrium in any particular configuration if, and is not 
 in equilibrium unless, the rate at which the applied forces perform 
 work at the instant of passing through it is equal to that at which 
 potential energy is gained, in every possible motion through that 
 configuration. This is the celebrated principle of virtual velocities 
 which Lagrange made the basis of his Mecaniqiie Analytiqiie. 
 
 255. To prove it, we have first to remark that the system cannot 
 possibly move away from any particular configuration except by work 
 being done upon it by the forces to which it is subject: it is therefore 
 in equilibrium if the stated condition is fulfilled. To ascertain that 
 nothing less than this condition can secure the equilibrium, let us 
 first consider a system having only one degree of freedom to move. 
 Whatever forces act on the whole system, we may always hold it in 
 equilibrium by a single force applied to any one point of the system 
 in its line of motion, opposite to the direction in which it tends to 
 move, and of such magnitude that, in any infinitely small motion in 
 either direction, it shall resist, or shall do, as much work as the other 
 forces, whether applied or internal, altogether do or resist. Now, by 
 the principle of superposition of forces in equilibrium, we might, 
 without altering their effect, apply to any one point of the system such 
 a force as we have just seen would hold the system in equilibrium, and 
 another force equal and opposite to it. All the other forces being 
 balanced by one of these two, they and it might again, by the principle 
 of superposition of forces in equilibrium, be removed; and therefore 
 the whole set of given forces would produce the same effect, whether 
 for equilibrium or for motion, as the single force which is left acting 
 alone. This single force, since it is in a line in which the point of its 
 application is free to move, must move the system. Hence the given 
 forces, to which the single force has been proved equivalent, cannot 
 possibly be in equilibrium unless their whole work for an infinitely 
 small motion is nothing, in which case the single equivalent force is 
 reduced to nothing. But whatever amount of freedom to move the 
 whole system may have, we may always, by the application of fric- 
 tionless constraint, limit it to one degree of freedom only; — and this 
 may be freedom to execute any particular motion whatever, possible 
 under the given conditions of the system. If, therefore, in any such 
 infinitely small motion, there is variation of potential energy uncom- 
 pensated by work of the applied forces, constraint limiting the freedom 
 of the system to only this motion will bring us to the case in which we 
 have just demonstrated there cannot be equilibrium. But the applica- 
 
 6—2 
 
84 PRELIMINARY. 
 
 tion of constraints limiting motion cannot possibly disturb equilibrium, 
 and therefore the given system under the actual conditions cannot be 
 in equilibrium in any particular configuration if the rate of doing work 
 is greater than that at which potential energy is stored up in any pos- 
 sible motion through that configuration. 
 
 256. If a material system, under the influence of internal and 
 applied forces, varying according to some definite law, is balanced 
 by them in any position in which it may be placed, its equilibrium is 
 said to be neutral. This is the case with any spherical body of 
 uniform material resting on a horizontal plane. A right cylinder or 
 cone, bounded by plane ends perpendicular to the axis, is also in 
 neutral equilibrium on a horizontal plane. Practically, any mass of 
 moderate dimensions is in neutral equilibrium when its centre of 
 inertia only is fixed, since, when its longest dimension is small in 
 comparison with the earth's radius, gravity is, as we shall see, ap- 
 proximately equivalent to a single force through this point. 
 
 But if, when displaced infinitely little in any direction from a par- 
 ticular position of equilibrium, and left to itself, it commences and 
 continues vibrating, without ever experiencing more than infinitely 
 small deviation in any of its parts, from the position of equilibrium, 
 the equilibrium in this position is said to be stable. A weight sus- 
 pended by a string, a uniform sphere in a hollow bowl, a loaded sphere 
 resting on a horizontal plane with the loaded side lowest, an oblate 
 body resting with one end of its shortest diameter on a horizontal 
 plane, a plank, whose thickness is small compared with its length and 
 breadth, floating on water, are all cases of stable equilibrium; if we 
 neglect the motions of rotation about a vertical axis in the second, 
 third, and fourth cases, and horizontal motion in general, in the fifth, 
 for all of which the equilibrium is neutral. 
 
 If, on the other hand, the system can be displaced in any way from 
 a position of equilibrium, so that when left to itself it will not vibrate 
 within infinitely small limits about the position of equilibrium, but will 
 move farther and farther away from it, the equilibrium in this position 
 is said to be unstable. Thus a loaded sphere resting on a horizontal 
 plane with its load as high as possible, an egg-shaped body standing 
 on one end, a board floating edgewise in water, would present, if they 
 could be realized in practice, cases of unstable equilibrium. 
 
 When, as in many cases, the nature of the equilibrium varies with 
 the direction of displacement, if unstable for any possible displace- 
 ment it is practically unstable on the whole. Thus a circular disc 
 standing on its edge, though in neutral equiHbrium for displacements 
 in its plane, yet being in unstable equilibrium for those perpendicular 
 to its plane, is practically unstable. A sphere resting in equilibrium on 
 a saddle presents a case in which there is stable, neutral, or unstable 
 equilibrium, according to the direction in which it may be displaced 
 by rolling; but practically it is unstable. 
 
 257. The theory of energy shows a very clear and simple test for 
 discriminating these characters, or determining whether the equilibrium 
 
DYNAMICAL LAWS AND PRINCIPLES. 85 
 
 is neutral, stable, or unstable, in any case. If there is just as much 
 potential energy stored up as there is work performed by the applied and 
 internal forces in any possible displacement, the equilibrium is neutral, 
 but not unless. If in every possible infinitely small displacement 
 from a position of equihbrium there is more potential energy stored 
 up than work done, the equilibrium is thoroughly stable, and not 
 unless. If in any or in every infinitely small displacement from a 
 position of equilibrium there is more work done than energy stored 
 up, the equilibrium is unstable. It follows that if the system is in- 
 fluenced only by internal forces, or if the applied forces follow the 
 law of doing always the same amount of work upon the system pass- 
 ing from one configuration to another by all possible paths, the whole 
 potential energy must be constant, in all positions, for neutral equili- 
 brium; must be a minimum for positions of thoroughly stable equili- 
 brium; must be either a maximum for all displacements, or a maximum 
 for some displacements and a minimum for others, when there is 
 unstable equilibrium. 
 
 258. We have seen that, according to D'Alembert's principle, as 
 explained above (§ 230), forces acting on the different points of a 
 material system, and their reactions against the accelerations which 
 they actually experience in any case of motion, are in equiUbrium 
 with one another. Hence in any actual case of motion, not only is 
 the actual work done by the forces equal to the kinetic energy pro- 
 duced in any infinitely small time, in virtue of the actual accelerations; 
 but so also is the work which would be done by the forces, in any 
 infinitely small time, if the velocities of the points constituting the 
 system were at any instant changed to any possible infinitely small 
 velocities, and the accelerations unchanged. This statement, when 
 put into the concise language of mathematical analysis, constitutes 
 Lagrange's application of the * principle of virtual velocities' to ex- 
 press the conditions of D'Alembert's equilibrium between the forces 
 acting, and the resistances of the masses to acceleration. It com- 
 prehends, as we have seen, every possible condition of every case of 
 motion. The 'equations of motion' in any partix:ular case are, as 
 Lagrange has shown, deduced from it with great ease. 
 
 259. When two bodies, in relative motion, come into contact, 
 pressure begins to act between them to prevent any parts of them 
 from jointly occupying the same space. This force commences from 
 nothing at the first point of collision, and gradually increases per unit 
 of area on a gradually increasing surface of contact. If, as is always 
 the case in nature, each body possesses some degree of elasticity, and 
 if they are not kept together after the impact by cohesion, or by some 
 artificial appliance, the mutual pressure between them will reach a 
 maximum, will begin to diminish, and in the end will come to nothing, 
 by gradually diminishing in amount per unit of area on a gradually 
 diminishing surface of contact. The whole process would occupy 
 not greatly more or less than an hour if the bodies were of such 
 dimensions as the earth, and such degrees of rigidity as copper, steel, 
 
86 PRELIMINARY. 
 
 or glass. It is finished, probably, within a thousandth of a second, 
 if they are globes of any of these substances not exceeding a yard 
 in diameter. 
 
 260. The whole amount, and the direction, of the * Impact' Qx^t- 
 rienced by either body in any such case, are reckoned according to 
 the * change of momentum' which it experiences. The amount of 
 the impact is measured by the amount, and its direction by the 
 direction of the change of momentum, which is produced. The 
 component of an impact in a direction parallel to any fixed line is 
 similarly reckoned according to the component change of momentum 
 in that direction. 
 
 261. If we imagine the whole time of an impact divided into 
 a very great number of equal intervals, each so short that the force 
 does not vary sensibly during it, the component change of momentum 
 in any direction during any one of these intervals will (§ 185) be 
 equal to the force multiplied by the measure of the interval. Hence 
 the component of the impact is equal to the sum of the forces in all 
 the intervals, multiplied by the length of each interval. 
 
 262. Any force in a constant direction acting in any circumstances, 
 for any time great or small, may be reckoned on the same principle ; 
 so that what we may call its whole amount during any time, or its 
 
 * time-integral^^ will measure, or be measured by, the whole momentum 
 which it generates in the time in question. But this reckoning is not 
 often convenient or useful except when the whole operation con- 
 sidered is over before the position of the body, or configuration of 
 the system of bodies, involved, has altered to such a degree as to 
 bring any other forces into play, or alter forces previously acting, 
 to such an extent as to produce any sensible effect on the momentum 
 measured. Thus if a person presses gently with his hand, during 
 a few seconds, upon a mass suspended by a cord or chain, he pro- 
 duces an effect which, if we know the degree of the force at each 
 instant, may be thoroughly calculated on elementary principles. No 
 approximation to a full determination of the motion, or to answering 
 such a partial question as ' how great will be the whole deflection 
 produced?' can be founded on a knowledge of the ^time-integral* 
 alone. If, for instance, the force be at first very great and gradually 
 diminish, the effect will be very different from what it would be if the 
 force were to increase very gradually and to cease suddenly, even 
 although the time-integral were the same in the two cases. But if 
 the same body is ' struck a blow,' in a horizontal direction, either by 
 the hand, or by a mallet or other somewhat hard mass, the action 
 of the force is finished before the suspending cord has experienced 
 any sensible deflection from the vertical. Neither gravity nor any 
 other force sensibly alters the effect of the blow. And therefore the 
 whole momentum at the end of the blow is sensibly equal to the 
 
 * amount of the impact,' which is, in this case, simply the time- 
 integral. 
 
DYNAMICAL LAWS AND PRINCIPLES. 87 
 
 263. Such Is the case of Robins' Ballistic Pendulum^ a massive 
 block of wood movable about a horizontal axis at a considerable 
 distance above it — employed to measure the velocity of a cannon or 
 musket-shot. The shot is fired into the block in a horizontal direc- 
 tion perpendicular to the axis. The impulsive penetration is so 
 nearly instantaneous, and the inertia of the block so large compared 
 with the momentum of the shot, that the ball and pendulum are 
 moving on as one mass before the pendulum has been sensibly deflected 
 from the position of equilibrium. This is the essential peculiarity of the 
 ballistic method ; which is used also extensively in electro-magnetic 
 researches and in practical electric testing, when the integral quantity 
 of the electricity which has passed in a current of short duration is to 
 be measured. The ballistic formula (§ 272) is appHcable, with the 
 proper change of notation, to all such cases. 
 
 264. Other illustrations of the cases in which the time-integral 
 gives us the complete solution of the problem may be given without 
 limit. They include all cases in which the direction of the force is 
 always coincident with the direction of motion of the moving body, 
 and those special cases in which the time of action of the force is so 
 short that the body's motion does not, during its lapse, sensibly alter 
 its relation to the direction of the force, or the action of any other 
 forces to which it may be subject. Thus, in the vertical fall of a 
 body, the time-integral gives us at once the change of momentum ; 
 and the same rule applies in most cases of forces of brief duration, 
 as in a * drive ' in cricket or golf. 
 
 265. The simplest case which we can consider, and the one usually 
 treated as an introduction to the subject, is that of the collision of 
 two smooth spherical bodies whose centres before colHsion were 
 moving in the same straight line. The force between them at each 
 instant must be in this line, because of the symmetry of circumstances 
 round it ; and by the third law it must be equal in amount on the 
 two bodies. Hence (Lex II.) they must experience changes of 
 motion at equal rates in contrary directions ; and at any instant of 
 the impact the integral amounts of these changes of motion must be 
 equal. Let us suppose, to fix the ideas, the two bodies to be moving 
 both before and after impact in the same direction in one line : one 
 of them gaining on the other before impact, and either following it 
 at a less speed, or moving along with it, as the case may be, after 
 the impact is completed. Cases in which the former is driven back- 
 wards by the force of the collision, or in which the two moving in 
 opposite directions meet in collision, are easily reduced to dependence 
 on the same formula by the ordinary algebraic convention with regard 
 to positive and negative signs. 
 
 In the standard case, then, the quantity of motion lost, up to any 
 instant of the impact, by one of the bodies, is equal to that gained 
 by the other. Hence at the instant when their velocities are equalized 
 they move as one mass with a momentum equal to the sum of the 
 
88 PRELIMINARY. 
 
 momenta of the two before Impact. That is to say, if v denote the 
 common velocity at this instant, we have 
 
 {M-\-M')v=:MV-^M'V\ 
 
 MV+M'V 
 
 or 
 
 M+M' 
 
 if M, M' denote the masses of the two bodies, and K, V their 
 velocities before impact. 
 
 During this first period of the impact the bodies have been, on 
 the whole, coming into closer contact with one another, through a 
 compression or deformation experienced by each, and resulting, as 
 remarked above, in a fitting together of the two surfaces over a 
 finite area. No body in nature is perfectly inelastic ; and hence, 
 at the instant of closest approximation, the mutual force called 
 into action between the two bodies continues, and tends to separate 
 them. Unless prevented by natural surface cohesion or welding (such 
 as is always found, as we shall see later in our chapter on Properties 
 of Matter, however hard and well polished the surfaces may be), or 
 by artificial appliances (such as a coating of wax, applied in one of 
 the common illustrative experiments; or the coupling applied between 
 two railway-carriages when run together so as to push in the springs, 
 according to the usual practice at railway-stations), the two bodies are 
 actually separated by this force, and move away from one another. 
 Newton found t\i2it, provided the impact is not so violent as to make any 
 sensible permanent indentation in either body, the relative velocity of 
 separation after the impact bears a proportion to their previous 
 relative velocity of approach, which is constant for the same two 
 bodies. This proportion, always less than unity, approaches more 
 and more nearly to it the harder the bodies are. Thus with balls of 
 compressed wool he found it f, iron nearly the same, glass if. The 
 results of more recent experiments on the same subject have con- 
 firmed Newton's law. These will be described later. In any case 
 of the collision of two balls, let e denote this proportion, to which we 
 give the name Coefficient of Restitution^ ; and, with previous nota- 
 tion, let in addition U, U' denote the velocities of the two bodies 
 after the conclusion of the impact ; in the standard case each being 
 positive, but U' > U. Then we have 
 
 U'- u=e{y- V'\ 
 and, as before, since one has lost as much momentum as the other 
 has gained, mU^ M' U' = MV^ M' V, 
 
 From these equations we find 
 
 (J/-f M')U= MF+ M'r-eM'{V- V), 
 with a similar expression for U'. 
 
 1 In most modern treatises this is called a 'coefficient of elasticity;' a 
 misnomer, suggested, it may be, by Newton's words, but utterly at variance with 
 modern language and modern knowledge regarding elasticity. 
 
DYNAMICAL LAWS AND PRINCIPLES. 89 
 
 Also we have, as above, 
 
 Hence, by subtraction, 
 
 (J/+J/')(z/- U)=eM'{y- V')=e{M'V-{M^M')v + MV\, 
 and therefore ^ _ ^= e ( V- v). 
 
 Of course we have also U' -v = e{v- V). 
 
 These results may be put in words thus : — The relative velocity of 
 either of the bodies with regard to the centre of inertia of the two 
 is, after the completion of the impact, reversed in direction, and 
 diminished in the ratio e -. \. 
 
 266. Hence the loss of kinetic energy, being, according to §§ 233, 
 234, due only to change of kinetic energy relative to the centre of 
 inertia, is to this part of the whole as 1 - e^ \ i. 
 
 Thus by § 234, 
 Initial kinetic energy = | {M -^ M')v' + 1 J/ ( F- v)' + ^M' (v - F')\ 
 Final „ „ =i{M+M')v^ + ^M{v-U'y + ^M'(C/'-vy. 
 Loss = ^ (i - ^) { Jf ( F-vf + JkI{v- Vy}. 
 
 267. When two elastic bodies, the two balls supposed above for 
 instance, impinge, some portion of their previous kinetic energy will 
 always remain in them as vibrations. A portion of the loss of energy 
 (miscalled the effect of imperfect elasticity alone) is necessarily due 
 to this cause in every real case. 
 
 Later, in our chapter on the Properties of Matter, it will be shown 
 as a result of experiment, that forces of elasticity are, to a very close 
 degree of accuracy, simply proportional to the strains (§ 135), within 
 the limits of elasticity, in elastic solids which, like metals, glass, etc., 
 bear but small deformations without permanent change. Hence when 
 two such bodies come into collision, sometimes with greater and 
 sometimes with less mutual velocity, but with all other circumstances 
 similar, the velocities of all particles of either body, at corresponding 
 times of the impacts, will be always in the same proportion. Hence 
 the velocity of separation of the centres of inertia after impact will 
 bear a constant proportion to the previous velocity of approach ; 
 which agrees with the Newtonian law. It is therefore probable that 
 a very sensible portion, if not the whole, of the loss of energy in the 
 visible motions of two elastic bodies, after impact, experimented on 
 by Newton, may have been due to vibrations ; but unless some other 
 cause also was largely operative, it is difficult to see how the loss was 
 so much greater with iron balls than with glass. 
 
 268. In certain definite extreme cases, imaginable although not 
 realizable, no energy will be spent in vibrations, and the two bodies 
 will separate, each moving simply as a rigid body, and having in this 
 simple motion the whole energy of work done on it by elastic force 
 during the collision. For instance, let the two bodies be cylinders, 
 or prismatic bars with flat ends, of the same kind -of substance, and of 
 
96 . PRELIMINARY, 
 
 equal and similar transverse sections ; and let this substance have the 
 property of compressibility with perfect elasticity, in the direction of 
 the length of the bar, and of absolute resistance to change in every 
 transverse dimension. Before impact, let the two bodies be placed 
 with their lengths in one line, and their transverse sections (if not 
 circular) similarly situated, and let one or both be set in motion in 
 this line. Then, if the lengths of the two be equal, they will separate 
 after impact with the same relative velocity as that with which they 
 approached, and neither will retain any vibratory motion after the 
 end of the collision. The result, as regards the motions of the two 
 bodies after the collision, will be sensibly the same if they are of any 
 real ordinary elastic solid material, provided the greatest transverse 
 diameter of each is very small in comparison of its length. 
 
 269. If the two bars are of an unequal length, the shorter will, after 
 
 the impact, be in exactly the same state as if it had struck another 
 
 of its own length, and it therefore will move as a rigid body after the 
 
 collision. But the other will, along with a motion of its centre of 
 
 gravity, calculable from the principle that its whole momentum must 
 
 (§ '^Z'i) be changed by an amount equal exactly to the momentum 
 
 gained or lost by the first, have also a vibratory motion, of which the 
 
 whole kinetic and potential energy will make up the deficiency of 
 
 energy which we shall presently calculate in the motions of the centres 
 
 of inertia. For simplicity, let the longer body be supposed to be at 
 
 rest before the collision. Then the shorter on striking it will be left 
 
 at rest ; this being clearly the result in the case of the ^ = i in the 
 
 preceding formulae (§ 265) applied to the impact of one body striking 
 
 another of equal mass previously at rest. The longer bar will move 
 
 away with the same momentum, and therefore with less velocity of its 
 
 centre of inertia, and less kinetic energy of this motion, than the other 
 
 body had before impact, in the ratio of the smaller to the greater 
 
 mass. It will also have a very remarkable vibratory motion, which, 
 
 when its length is more than double of that of the other, will consist 
 
 of a wave running backwards and forwards through its length, and 
 
 causing the motion of its ends, and, in fact, of every particle of it, to 
 
 take place by ' fits and starts,' not continuously. The full analysis of 
 
 these circumstances, though very simple, must be reserved until we 
 
 are especially occupied with waves, and the kinetics of elastic solids. 
 
 It is sufficient at present to remark, that the motions of the centres of 
 
 inertia of the two bodies after impact, whatever they may have been 
 
 previously, are given by the preceding formulae with for e the value 
 
 M' 
 
 -^ , where M and M' are the smaller and larger mass respectively. 
 
 270. The mathematical theory of the vibrations of solid elastic 
 spheres has not yet been worked out ; and its application to the case 
 of the vibrations produced by impact presents considerable difficulty. 
 Experiment, however, renders it certain, that but a small part of the 
 whole kinetic energy of the previous motions can remain in the form 
 of vibrations after the impact of two equal spheres of glass or of 
 
DYNAMICAL LAWS AND PRINCIPLES, 91 
 
 ivory. This is proved, for instance, by the common observation, that 
 one of them remains nearly motionless after striking the other pre- 
 viously at rest; since, the velocity of the common centre of inertia of 
 the two being necessarily unchanged by the impact, we infer that the 
 second ball acquires a velocity nearly equal to that which the first had 
 before striking it. But it is to be expected that unequal balls of the 
 same substance coming into collision will, by impact, convert a very 
 sensible proportion of the kinetic energy of their previous motions 
 into energy of vibrations ; and generally, that the same will be the 
 case when equal or unequal masses of different substances come into 
 collision ; although for one particular proportion of their diameters, 
 depending on their densities and elastic qualities, this effect will be 
 a minimum, and possibly not much more sensible than it is when the 
 substances are the same and the diameters equal. 
 
 271. It need scarcely be said that in such cases of impact as that 
 of the tongue of a bell, or of a clock-hammer striking its bell (or 
 spiral spring as in the American clocks), or of pianoforte-hammers 
 striking the strings, or of the drum struck with the proper implement, 
 a large part of the kinetic energy of the blow is spent in generating 
 vibrations. 
 
 272. The Moment of aft Impact about any axis is derived from the 
 line and amount of the impact in the same way as the moment of 
 a velocity or force is determined from the line and amount of the 
 velocity or force, § 46. If a body is struck, the change of its 
 moment of momentum about any axis is equal to the moment of the 
 impact round that axis. But, without considering the measure of the 
 impact, we see (§ 233) that the moment of momentum round any axis, 
 lost by one body in striking another, is, as in every case of mutual 
 action, equal to that gained by the other. 
 
 Thus, to recur to the ballistic pendulum — the line of motion of the 
 bullet at impact may be in any direction whatever, but the only part 
 which is effective is the component in a plane perpendicular to the 
 axis. We may therefore, for simplicity, consider the motion to be in 
 a line perpendicular to the axis, though not necessarily horizontal. 
 Let m be the mass of the bullet, v its velocity, and / the distance of 
 its line of motion from the axis. Let M be the mass of the pendulum 
 with the bullet lodged in it, and k its radius of gyration. Then if <o 
 be the angular velocity of the pendulum when the impact is complete, 
 
 mvp = Mk^dij 
 from which the solution of the question is easily determined. 
 
 For the kinetic energy after impact is changed (§ 207) into its 
 equivalent in potential energy when the pendulum reaches its position 
 of greatest deflection. Let this be given by the angle 6 : then the 
 height to which the centre of inertia is raised is/i(i - cos 6) if A be its 
 distance from the axis. Thus 
 
 MgA (t- cos 6) ^-iMiW^i"^^, 
 
^""Jw 
 
 9 2 PRELIMINAR Y, 
 
 . 6 m p V m p TTV .^ -, 
 or 2Sin-= 7>.Y.— 7== -r>.^. -^, IX 7^ 
 
 2 M k Jg/i M h gT' 
 
 an expression for the chord of the angle of deflection. In practice 
 the chord of the angle 6 is measured by means of a light tape or 
 cord attached to a point of the pendulum, and slipping with small 
 friction through a clip fixed close to the position occupied by that 
 point when the pendulum hangs at rest. 
 
 273. Work do7ie by an impact is, in general, the product of the 
 impact into half the sum of the initial and final velocities of the point 
 at which it is applied, resolved in the direction of the impact. In the 
 case of direct impact, such as that treated in § 265, the initial kinetic 
 energy of the body is ^MF^ the final ^MZ/^, and therefore the gain 
 by the impact is 
 
 or, which is the same. 
 
 But M{U- V) is (§ 260) equal to the amount of the impact. Hence 
 the proposition : the extension of which to the most general cir- 
 cumstances is not difficult, but requires somewhat higher analysis 
 than can be admitted here. 
 
 274. It is worthy of remark, that if any number of impacts be 
 applied to a body, their whole effect will be the same whether they 
 be applied together or successively (provided that the whole time 
 occupied by them be infinitely short), although the work done by 
 each particular impact is, in general, different according to the order 
 in which the several impacts are applied. The whole amount of 
 work is the sum of the products obtained by multiplying each impact 
 by half the sum of the components of the initial and final velocities 
 of the point to which it is applied. 
 
 275. The effect of any stated impulses, applied to a rigid body, 
 or to a system of material points or rigid bodies connected in any 
 way, is to be found most readily by the aid of D'Alembert's principle; 
 according to which the given impulses, and the impulsive reaction 
 against the generation of motion, measured in amount by the 
 momenta generated, are in equilibrium; and are, therefore, to be 
 dealt with mathematically by applying to them the equations of 
 equilibrium of the system. 
 
 276. [A material system of any kind, given at rest, and subjected 
 to an impulse in any specified direction, and of any given magnitude, 
 moves off so as to take the greatest amount of kinetic energy which 
 the specified impulse can give it. 
 
 277. If the system is guided to take, under the action of a given 
 impulse, any motion different from the natural motion, it will have 
 less kinetic energy than that of the natural motion, by a difference 
 equal to the kinetic energy of the motion represented by the resultant 
 (§ 67) of those two motions, one of them reversed. 
 
DYNAMICAL LA WS AND PRINCIPLES. 93 
 
 Cor. If a set of material points are struck independently by 
 impulses each given in amount, more kinetic energy is generated if 
 the points are perfectly free to move each independently of all the 
 others, than if they are connected in any way. And the deficiency 
 of energy in the latter case is equal to the amount of the kinetic 
 energy of the motion which geometrically compounded with the 
 motion of either case would give that of the other. 
 
 278. Given any material system at rest. Let any parts of it be 
 set in motion suddenly with any specified velocities, possible accord- 
 ing to the conditions of the system; and let its other parts be 
 influenced only by its connexions with those. It is required to 
 find the motion. The solution of the problem is — The motion 
 actually taken by the system is that which has less kinetic energy than 
 any other motion fulfilling the prescribed velocity conditions. And 
 the excess of the energy of any other such motion, above that of the 
 actual motion, is equal to the energy of the motion that would be 
 generated by the action alone of the impulse which, if compounded 
 with the impulse producing the actual motion, would produce this 
 other supposed motion.] 
 
 279. Maupertuis' celebrated principle of Least Action has been, 
 even up to the present time, regarded rather as a curious and some- 
 what perplexing property of motion, than as a useful guide in kinetic 
 investigations. We are strongly impressed with the conviction that 
 a much more profound importance will be attached to it, not only 
 in abstract dynamics, but in the theory of the several branches of 
 physical science now beginning to receive dynamic explanations. 
 As an extension of it. Sir W. R. Hamilton' has evolved his method 
 of Varying Action^ which undoubtedly must become a most valuable 
 aid in future generalizations. 
 
 What is meant by ' Action' in these expressions is, unfortunately, 
 something very different from the Actio Agent is defined by Newton, 
 and, it must be admitted, is a much less judiciously chosen word. 
 Taking it, however, as we find it now universally used by writers on 
 dynamics, we define the Action of a Moving System as proportional 
 to the average kinetic energy, which it has possessed during the time 
 from any convenient epoch of reckoning, multiplied by the time. 
 According to the unit generally adopted, the action of a system 
 which has not varied in its kinetic energy, is twice the amount of the 
 energy multiplied by the time from the epoch. Or if the energy has 
 been sometimes greater and sometimes less, the action at time / 
 is the double of what we may call the time-integral of the energy; 
 that is to say, the action of a system is equal to the sum of the 
 average momenta for the spaces described by the particles from any 
 era each multiplied by the length of its path. 
 
 280. The principle of Least Action is this : — Of all the different 
 sets of paths along which a conservative system may be guided to 
 move from one configuration to another, with the sum of its potential 
 
 1 Phil. Trans., 1834— 1835. 
 
94 PRELIMINARY. 
 
 and kinetic energies equal to a given constant, that one for which the 
 action is the least is such that the system will require only to be 
 started with the proper velocities, to move along it unguided. 
 
 281. [In any unguided motion whatever, of a conservative system, 
 the Action from any one stated position to any other, though not 
 necessarily a minimum, fulfils the stationary condition, that is to say, 
 the condition that the variation vanishes, which secures either a 
 minimum or maximum, or maximum-minimum.] 
 
 282. From this principle of stationary action, founded, as we have 
 seen, on a comparison between a natural motion, and any other 
 motion, arbitrarily guided and subject only to the law of energy, the 
 initial and final configurations of the system being the same in each 
 case; Hamilton passes to the consideration of the variation of the 
 action in a natural or unguided motion of the system produced by 
 varying the initial and final configurations, and the sum of the 
 potential and kinetic energies. The result is, that — 
 
 283. The rate of decrease of the action per unit of increase of 
 any one of the free (generalized) co-ordinates specifying the 
 initial configuration, is equal to the corresponding (generalized) com- 
 ponent momentum of the actual motion from that configuration : 
 the rate of increase of the action per unit increase of any one 
 of the free co-ordinates specifying the final configuration, is equal 
 to the corresponding component momentum of the actual motion 
 towards this second configuration : and the rate of increase of the 
 action per unit increase of the constant sum of the potential and 
 kinetic energies, is equal to the time occupied by the motion of 
 which the action is reckoned. 
 
 284. The determination of the motion of any conservative system 
 from one to another of any two configurations, when the sum of its 
 potential and kinetic energies is given, depends on the determination 
 of a single function of the co-ordinates specifying those configura- 
 tions by means of two quadratic, partial differential equations of the 
 first order, with reference to those two sets of co-ordinates respec- 
 tively, with the condition that the corresponding terms of the two 
 differential equations become separately equal when the values of 
 the two sets of co-ordinates agree. The function thus determined 
 and employed to express the solution of the kinetic problem was 
 called the Characteristic Function, by Sir W. R. Hamilton, to whom 
 the method is due. It is, as we have seen, the 'action' from one 
 of the configurations to the other; but its peculiarity in Hamilton's 
 system is, that it is to be expressed as a function of the co-ordinates 
 and a constant, the whole energy, as explained above. It is evidently 
 symmetrical witli respect to the two configurations, changing only in 
 sign if their co-ordinates are interchanged. 
 
 ■ 285. The most general possible solution of the quadratic, partial 
 differential equation of the first order, satisfied by Hamilton's Cha- 
 
DYNAMICAL LA WS AND PRINCIPLES, 95 
 
 racteristic Function (either terminal configuration alone varying), 
 when interpreted for the case of a single free particle, expresses the 
 action up to any point from some point of a certain arbitrarily 
 given surface, from which the particle has been projected, in the 
 direction of the normal, and with the proper velocity to make the 
 sum of the potential and actual energies have a given value. In other 
 words, the physical problem solved by the most general solution of 
 that partial differential equation, for a single free particle, is this : — 
 
 Let free particles, not mutually influencing one another, be pro- 
 jected normally from all points of a certain arbitrarily given surface, 
 each with the proper velocity to make the sum of its potential and 
 kinetic energies have a given value. To find, for that one of the 
 particles which passes through a given point, the 'action' in its course 
 from the surface of projection to this point. The Hamiltonian 
 principles stated above, show that the surfaces of equal action cut 
 the paths of the particles at right angles; and give also the 
 following remarkable properties of the motion : — 
 
 If, from all points of an arbitrary surface, particles not mutually 
 influencing one another be projected with the proper velocities in 
 the directions of the normals; points which they reach with equal 
 actions lie on a surface cutting the paths at right angles. The 
 infinitely small thickness of the space between any two such surfaces 
 corresponding to amounts of action differing by any infinitely small 
 quantity, is inversely proportional to the velocity of the particle 
 traversing it ; being equal to the infinitely small diflerence of action 
 divided by the whole momentum of the particle. 
 
 286. Irrespectively of methods for finding the 'characteristic 
 function' in kinetic problems, the fact that any case of motion what- 
 ever can be represented by means of a single function in the manner 
 explained in § 284, is most remarkable, and, when geometrically 
 interpreted, leads to highly important and interesting properties of 
 motion, which have valuable applications in various branches of 
 Natural Philosophy ; one of which, explained below, led Hamilton * 
 to a general theory of optical instruments, comprehending the whole 
 in one expression. Some of the most direct applications of the 
 general principle to the motions of planets, comets, etc., considered 
 as free points, and to the celebrated problem of perturbations, known 
 as the Problem of Three Bodies, are worked out in considerable detail 
 by Hamilton {Phil. Trans. ^ 1834-5), and in various memoirs by 
 Jacobi, Liouville, Bour, Donkin, Cayley, Boole, etc. 
 
 The now abandoned, but still interesting, corpuscular theory of 
 light furnishes the most convenient language for expressing the 
 optical application. In this theory light is supposed to consist of 
 material particles not mutually influencing one another ; but subject 
 to molecular forces from the particles of bodies, not sensible at 
 sensible distances, and therefore not causing any deviation from 
 uniform rectilinear motion in a homogeneous medium, except within 
 
 1 On the Theory of Systems of Rays. Trans. R. I. A., 1824, 1830, 1832. 
 
96 PRELIMINARY. 
 
 an indefinitely small distance from its boundary. The laws of reflec- 
 tion and of single refraction follow correctly from this hypothesis, 
 which therefore suffices for what is called geometrical optics. 
 
 We hope to return to this subject, with sufficient detail, in treating 
 of Optics. At present we limit ourselves to state a theorem com- 
 prehending the known rule for measuring the magnifying power of 
 a telescope or microscope (by comparing the diameter of the object- 
 glass with the diameter of pencil of parallel rays emerging from the 
 eye-piece, when a point of light is placed at a great distance in front 
 of the object-glass), as a particular case. 
 
 287. Let any number of attracting or repelling masses, or perfectly 
 smooth elastic objects, be fixed in space. Let two stations, O and O', 
 be chosen. Let a shot be fired with a stated velocity, V, from (7, 
 in such a direction as to pass through O'. There may clearly be 
 more than one natural path by which this maybe done; but, generally 
 speaking, when one such path is chosen, no other, not sensibly di- 
 verging from it, can be found; and any infinitely small deviation in 
 the line of fire from (9, will cause the bullet to pass infinitely near to, 
 but not through O'. Now let a circle, with infinitely small radius r, 
 be described round O as centre, in a plane perpendicular to the line 
 of fire from this point, and let — all with infinitely nearly the same 
 velocity, but fulfilling the condition that the sum of the potential and 
 kinetic energies is the same as that of the shot from O — bullets be 
 fired from all points of this circle, all directed infinitely nearly parallel 
 to the line of fire from O, but each precisely so as to pass through O'. 
 Let a target be held at an infinitely small distance, «', beyond Cf, 
 in a plane perpendicular to the line of the shot reaching it from O. 
 The bullets fired from the circumference of the circle round (9, will, 
 after passing through 0\ strike this target in the circumference of an 
 exceedingly small ellipse, each with a velocity (corresponding of 
 course to its position, under the law of energy) difiering infinitely 
 little from V\ the common velocity with which they pass through O'. 
 Let now a circle, equal to the former, be described round 0\ in the 
 plane perpendicular to the central path through 0\ and let bullets be 
 fired from points in its circumference, each with the proper velocity, 
 and in such a direction infinitely nearly parallel to the central path 
 as to make it pass through O. These bullets, if a target is held to 
 
 y 
 receive them perpendicularly at a distance a = a' -^,, beyond O, will 
 
 strike it along the circumference of an ellipse equal to the former 
 and placed in a corresponding position; and the points struck by the 
 individual bullets will correspond in the manner explained below. 
 Let F and F' be points of the first and second circles, and Q and 
 Q the points on the first and second targets which bullets from them 
 strike ; then if F' be in a plane containing the central path through 
 (y, and the position which Q would take if its ellipse were made 
 circular by a pure strain (§ 159) ; Q and Q are similarly situated on 
 the two ellipses. 
 
DYNAMICAL LAWS AND PRINCIPLES. 97 
 
 288. The most obvious optical application of this remarkable 
 result is, that in the use of any optical apparatus whatever, if the eye 
 and the object be interchanged without altering the position of the 
 instrument, the magnifying power is unaltered. This is easily under- 
 stood when, as in an ordinary telescope, microscope, or opera-glass 
 (Galilean telescope), the instrument is symmetrical about an axis, and 
 is curiously contradictory of the common idea that a telescope 'dimi- 
 nishes' when looked through the wrong way, which no doubt is true 
 if the telescope is simply reversed about the middle of its length, eye 
 and object remaining fixed. But if the telescope be removed from the 
 eye till its eye-piece is close to the object, the part of the object seen 
 will be seen enlarged to the same extent as when viewed with the 
 telescope held in the usual manner. This is easily verified by looking 
 from a distance of a few yards, in through the object-glass of an 
 opera-glass, at the eye of another person holding it to his eye in 
 the usual way. 
 
 The more general application may be illustrated thus: — Let the 
 points, (9, O' (the centres of the two circles described in the preceding 
 enunciation), be the optic centres of the eyes of two persons looking 
 at one another through any set of lenses, prisms, or transparent 
 media arranged in any way between them. If their pupils are of 
 equal sizes in reality, they will be seen as similar ellipses of equal 
 apparent dimensions by the two observers. Here the imagined 
 particles of light, projected from the circumference of the pupil of 
 either eye, are substituted for the projectiles from the circumference 
 of either circle, and the retina of the other eye takes the place of the 
 target receiving them, in the general kinetic statement. 
 
 289. If instead of one free particle we have a conservative system 
 of any number of mutually influencing free particles, the same state- 
 ment may be applied with reference to the initial position of one of 
 the particles and the final position of another, or with reference to the 
 initial positions, or to the final positions, of two of the particles. It 
 thus serves to show how the influence of an infinitely small change in 
 one of those positions, on the direction of the other particle passing 
 through the other position, is related to the influence on the direction 
 of the former particle passing through the former position produced 
 by an infinitely small change in the latter position, and is of immense 
 use in physical astronomy. A corresponding statement, in terms of 
 generalized co-ordinates, may of course be adapted to a system of 
 rigid bodies or particles connected in any way. All such statements 
 are included in the following very general proposition: — 
 
 The rate of increase of the component momentum relative to 
 any one of the co-ordinates, per unit of increase of any other co- 
 ordinate, is equal to the rate of increase of the component momentum 
 relative to the latter per unit increase or diminution of the former 
 co-ordinate, according as the two co-ordinates chosen belong to one 
 configuration of the system, or one of them belongs to the initial 
 configuration and the other to the final. 
 
 r. 7 
 
98 PRELIMINARY. 
 
 290. If a conservative system is infinitely little displaced from a 
 configuration of stable equilibrium, it will ever after vibrate about this 
 configuration, remaining infinitely near it; each particle of the system 
 performing a motion which is composed of simple harmonic vibra- 
 tions. If there are i degrees of freedom to move, and we consider 
 any system of generalized co-ordinates specifying its position at 
 any time, the deviation of any one of these co-ordinates from its 
 value for the configuration of equilibrium will vary according to a 
 complex harmonic function (§ 88), composed in general of i simple 
 harmonics of incommensurable periods, and therefore (§ 85) the whole 
 motion of the system will not recur periodically through the same 
 series of configurations. There are in general, however, i distinct 
 determinate displacements, which we shall call the normal displace- 
 ments, fulfilling the condition, that if any one of them be produced 
 alone, and the system then left to itself for an instant at rest, this 
 displacement will diminish and increase periodically according to 
 a simple harmonic function of the time, and consequently every 
 particle of the system will execute a simple harmonic movement in 
 the same period. This result, we shall see later, includes cases in 
 which there are an infinite number of degrees of freedom; as, for 
 instance, a stretched cord; a mass of air in a closed vessel; waves 
 in water, or oscillations of water in a vessel of limited extent, or an 
 elastic solid; and in these applications it gives the theory of the 
 so-called 'fundamental vibration,' and successive 'harmonics' of the 
 cord, and of all the different possible simple modes of vibration in 
 the other cases. In all these cases it is convenient to give the name 
 * fundamental mode' to any one of the possible simple harmonic 
 vibrations, and not to restrict it to the gravest simple harmonic mode, 
 as has been hitherto usual in respect to vibrating cords and organ- 
 pipes. 
 
 The whole kinetic energy of any complex motion of the system is 
 equal to the sum of the kinetic energies of the fundamental constitu- 
 ents; and the potential energy of any displacements is equal to the sum 
 of the potential energies of its normal components. Corresponding 
 theorems of normal constituents and fundamental modes of motion, 
 and the summation of their kinetic and potential energies in complex 
 motions and displacements, hold for motion in the neighbourhood of 
 a configuration of U7istable equilibrium. In this case, some or all of 
 the constituent motions are fallings away from the position of equi- 
 librium (according as the potential energies of the constituent normal 
 vibrations are negative). 
 
 291. If, as may be in particular cases, the periods of the vibrations 
 for two or more of the normal displacements are equal, any displace- 
 ment compounded of them will also fulfil the condition of a normal 
 displacement. And if the system be displaced according to any one 
 such normal displacement, and projected with velocity corresponding 
 to another, it will execute a movement, the resultant of two simple 
 harmonic movements in equal periods. The graphic representation 
 
DYNAMICAL LA WS AND PRINCIPLES. 99 
 
 of the variation of the corresponding co-ordinates of the system, laid 
 down as two rectangular co-ordinates in a plane diagram, will con- 
 sequently (§ 82) be a circle or an ellipse; which will therefore, of 
 course, be the form of the orbit of any particle of the system which 
 has a distinct direction of motion, for two of the displacements in 
 question. But it must be remembered that some of the principal 
 parts may have only one degree of freedom ; or even that each part 
 of the system may have only one degree of freedom (as, for instance, 
 if the system is composed of a set of particles each constrained to 
 remain on a given line, or of rigid bodies on fixed axes, mutually 
 influencing one another by elastic cords or otherwise). In such a 
 case as the last, no particle of the system can move otherwise than 
 in one line; and the ellipse, circle, or other graphical representation 
 of the composition of the harmonic motions of the system, is merely 
 an aid to comprehension, and not a representation of any motion 
 actually taking place in any part of the system. 
 
 292. In nature, as has been said above (§ 250), every system 
 uninfluenced by matter external to it is conservative, when the 
 ultimate molecular motions constituting heat, light, and magnetism, 
 and the potential energy of chemical affinities, are taken into account 
 along with the palpable motions and measurable forces. But (§ 247) 
 practically we are obliged to admit forces of friction, and resistances 
 of the other classes there enumerated, as causing losses of energy to 
 be reckoned, in abstract dynamics, without regard to the equivalents 
 of heat or other molecular actions which they generate. Hence when 
 such resistances are to be taken into account, forces opposed to the 
 motions of various parts of a system must be introduced into the 
 equations. According to the approximate knowledge which we have 
 from experiment, these forces are independent of the velocities when 
 due to the friction of sofids; and are simply proportional to the 
 velocities when due to fluid viscosity directly, or to electric or magnetic 
 influences, with corrections depending on varying temperature, and 
 on the varying configuration of the system. In consequence of the 
 last-mentioned cause, the resistance of a real liquid (which is always 
 more or less viscous) against a body moving very rapidly through it, 
 and leaving a great deal of irregular motion, such as 'eddies,' in its 
 wake, seems to be nearly in proportion to the square of the velocity; 
 although, as Stokes has shown, at the lowest speeds the resistance 
 is probably in simple proportion to the velocity, and for all speeds 
 may, it is probable, be approximately expressed as the sum of two 
 terms, one simply as the velocity, and the other as the square of the 
 velocity. If a solid is started from rest in an incompressible fluid, 
 the initial law of resistance is no doubt simple proportionality to the 
 velocity, (however great, if suddenly enough given;) until by the 
 gradual growth of eddies the resistance is increased gradually till it 
 comes to fulfil Stokes's law. 
 
 293. The effect of friction of solids rubbing one against another 
 is simply to render impossible the infinitely small vibrations with which 
 
 7—2 
 
loo PRELIMINARY, 
 
 we are now particularly concerned ; and to allow any system in which 
 it is present, to rest balanced when displaced within certain finite 
 limits, from a configuration of frictionless equilibrium. In mechanics 
 it is easy to estimate its effects with sufficient accuracy when any 
 practical case of finite oscillations is in question. But the other 
 classes of dissipative agencies give rise to resistances simply as the 
 velocities, without the corrections referred to, when the motions are 
 infinitely small, and can never balance the system in a configuration 
 deviating to any extent, however small, from a configuration of 
 equilibrium without friction. In the theory of infinitely small vibra- 
 tions, they are to be taken into account by adding to the expressions 
 for the generalized components of force, terms consisting of the 
 generalized velocities each multiplied by a constant, which gives us 
 equations still remarkably amenable to rigorous mathematical treat- 
 ment. The result of the integration for the case of a single degree 
 of freedom is very simple; and it is of extreme importance, both for 
 the explanation of many natural phenomena, and for use in a large 
 variety of experimental investigations in Natural Philosophy. Partial 
 conclusions from it, in the first place, stated in general terms, are 
 as follows: — 
 
 294. If the resistance per unit velocity is less than a certain limit, 
 in any particular case, the motion is a- simple harmonic oscillation, 
 with amplitude decreasing by equal proportions in equal successive 
 intervals of time. But if the resistance exceeds this limit, the system, 
 when displaced from its position of equilibrium and left to itself, 
 returns gradually towards its position of equilibrium, never oscillating 
 through it to the other side, and only reaching it after an infinite 
 time. 
 
 In the unresisted motion, let «' be the rate of acceleration, when 
 
 the displacement is unity; so that (§ 74) we have T=^ — ; and let the 
 
 rate of retardation due to the resistance corresponding to unit velocity 
 
 be k. Then the motion is of the oscillatory or non-oscillatory class 
 
 according as k^ <{2nf or k'^-> {2tif. In the first case, the period of 
 
 n 
 the oscillation is increased, by the resistance, from 7" to Tj~^ — TT^Yi* 
 
 and the rate at which the Napierian logarithm of the amplitude 
 diminishes per unit of time is \k. 
 
 295. An indirect but very simple proof of this important propo- 
 sition may be obtained by means of elementary mathematics as 
 follows : — A point describes a logarithmic spiral with uniform angular 
 velocity about the pole — find the acceleration. 
 
 Since the angular velocity of SP and the inclination of this line 
 to the tangent are each constant, the linear velocity of P is as SP. 
 Take a length PT^ equal to n SP, to represent it. Then the 
 hodograph, the locus of /, where »S/ is parallel and equal to PT, is 
 evidently another logarithmic spiral similar to the former, and de- 
 scribed with the same uniform angular velocity. Hence (§§ 35, 49) 
 
DYNAMICAL LA WS AND PRINCIPLES. loi 
 
 pt, the acceleration required, is equal to n Sp, and makes with Sp an 
 angle Spt equal to SPT. Hence, if Pu be drawn parallel and equal 
 
 to //, and uv parallel to PT, the whole acceleration // or Pu may be 
 resolved into Pv and vu. Now Pvu is an isosceles triangle, whose 
 base angles {v, u) are each equal to the constant angle of the spiral. 
 Hence Pv and vu bear constant ratios to Puj and therefore to SP 
 and /'T' respectively. 
 
 The acceleration, therefore, is composed of a central attractive 
 part proportional to the distance, and a tangential retarding part 
 proportional to the velocity. 
 
 And, if the resolved part of /*'s motion parallel to any line in the 
 plane of the spiral be considered, it is obvious that in it also the 
 acceleration will consist of two parts — one directed towards a point 
 in the line (the projection of the pole of the spiral), and proportional 
 to the distance from it ; the other proportional to the velocity, but 
 retarding the motion. 
 
 Hence a particle which, unresisted, would have a simple harmonic 
 motion, has, when subject to resistance proportional to its velocity, 
 a motion represented by the resolved part of the spiral motion just 
 described. 
 
 296. If a be the constant angle of the spiral, w the angular velocity 
 of SP, we have evidently 
 
 PT, sin a = SP.o), But PT= nSP, so that n = -A- . 
 
 sma 
 
 Hence Pv = Pu =// = nSp = nPT= n' . SP 
 
 and vu = 2Pv . cos a = 2« cos aPT= k . PT (suppose). 
 
 2 
 
 Thus the central force at unit distance is n^= . ^ , and the co- 
 
 sm a 
 
 cc. • I. r ' ^ • r 2(0 cos a 
 
 emcient of resistance \s k= 211 cos a = — -. . 
 
 sma 
 
 27r 
 The time of oscillation in the resolved motion is evidently -- ; but, 
 
 (U 
 
 if there had been no resistance, the properties of simple harmonic 
 
I03 PRELIMINARY. 
 
 motion show that it would have been — : so that it is increased by 
 
 I ^' 
 
 the resistance in the ratio cosec a to i , or f^ to k n^ — . 
 
 V 4 
 
 The rate of diminution of SP is evidently 
 
 PT, C0Sa = ;2 cos a SP^-SP', 
 
 2 
 
 that is, SP diminishes in geometrical progression as time increases, 
 
 k 
 the rate being - per unit of time per unit of length. By an ordinary 
 
 result of arithmetic (compound interest payable every instant) the 
 
 k 
 diminution of log . SP in unit of time is - . 
 ° 2 
 
 This process of solution is only applicable to resisted harmonic 
 
 vibrations when n is greater than - . When ;/ is not greater than - 
 
 the auxiliary curve can no longer be a logarithmic spiral, for the 
 moving particle never describes more than a finite angle about the 
 pole. A curve, derived from an equilateral hyperbola, by a process 
 somewhat resembling that by which the logarithmic spiral is deduced 
 from a circle, may be introduced ; but then the geometrical method 
 ceases to be simpler than the analytical one, so that it is useless to 
 pursue the investigation farther, at least from this point of view. 
 
 297. The general solution of the problem, to find the motion of 
 a system having any number, /, of degrees of freedom, when infinitely 
 little disturbed from a position of equilibrium, and left to move subject 
 to resistances proportional to velocities, shows that the whole motion 
 may be resolved, in general determinately, into 2/ different motions 
 each either simple harmonic with amplitude diminishing according to 
 the law stated above (§ 294), or non-oscillatory, and consisting of 
 equi-proportionate diminutions of the components of displacement 
 in equal successive intervals of time. 
 
 298. When the forces of a system depending on configuration, 
 and not on motion, or, as we may call them for brevity, the forces 
 of position, violate the law of conservatism, we have seen (§ 244) 
 that energy without limit may be drawn from it by guiding it per- 
 petually through a returning cycle of configurations, and we have 
 inferred that in every real system, not supplied with energy from 
 without, the forces of position fulfil the conservative law. But it is 
 easy to arrange a system artificially, in connexion with a source of 
 energy, so that its forces of position shall be non-conservative ; and 
 the consideration of the kinetic effects of such an arrangement, 
 especially of its oscillations about or motions round a configuration 
 of equilibrium, is most instructive, by the contrasts which it presents 
 to the phenomena of a natural system. 
 
 299. But although, when the equilibrium is stable, no possible 
 
DYNAMICAL LA WS AND PRINCIPLES. 103 
 
 infinitely small displacement and velocity given to the system can 
 cause it, when left to itself, to go on moving either farther and farther 
 away till a finite displacement is reached, or till a finite velocity is 
 acquired; it is very remarkable that stability should be possible, 
 considering that even in the case of stability an endless increase of 
 velocity may, as is easily seen from § 244, be obtained merely by 
 constraining the system to a particular closed course, or circuit of 
 configurations, nowhere deviating by more than an infinitely small 
 amount from the configuration of equilibrium, and leaving it at 
 rest anywhere in a certain part of this circuit. This result, and 
 the distinct peculiarities of the cases of stability and instability, are 
 sufticiently illustrated by the simplest possible example, — that of a 
 material particle moving in a plane. 
 
 300. There is scarcely any question in dynamics more important 
 for Natural Philosophy than the stability or instability of motion. We 
 therefore, before concluding this chapter, propose to give some 
 general explanations and leading principles regarding it. 
 
 A 'conservative disturbance of motion' is a disturbance in the 
 motion or configuration of a conservative system, not altering the 
 sum of the potential and kinetic energies. A conservative disturb- 
 ance of the motion through any particular configuration is a change 
 in velocities, or component velocities, not altering the whole kinetic 
 energy. Thus, for example, a conservative disturbance of the motion 
 of a particle through any point, is a change in the direction of its 
 motion, unaccompanied by change of speed. 
 
 301. The actual motion of a system, from any particular con- 
 figuration, is said to be stable if every possible infinitely small con- 
 servative disturbance of its motion through that configuration may 
 be compounded of conservative disturbances, any one of which would 
 give rise to an alteration of motion which would bring the system 
 again to some configuration belonging to the undisturbed path, in 
 a finite time, and without more than an infinitely small digression. 
 If this condition is not fulfilled, the motion is said to be unstable. 
 
 302. For example, if a body. A, be supported on a fixed vertical 
 axis ; if a second, B^ be supported on a parallel axis belonging to 
 the first; a third, C, similarly supported on B^ and so on; and if 
 B, C, etc., be so placed as to have each its centre of inertia as far as 
 possible from the fixed axis, and the whole set in motion with 
 a common angular velocity about this axis, the motion will be 
 thoroughly stable. If, for instance, each of the bodies is a flat 
 rectangular board hinged on one edge, it is obvious that the whole 
 system will be kept stable by centrifugal force, when all are in one 
 plane and as far out from the axis as possible. But if A consists 
 partly of a shaft and crank, as a common spinning-wheel, or the fly- 
 wheel and crank of a steam-engine, and if B be supported on the 
 crank-pin as axis, and turned inwards (towards the fixed axis, or 
 across the fixed axis), then, even although the centres of inertia of C, 
 
I04 PRELIMINARY. 
 
 Z>, etc., are placed as far from the fixed axis as possible, consistent 
 with this position of B^ the motion of the system will be unstable. 
 
 303. The rectilinear motion of an elongated body lengthwise, or 
 of a flat disc edgewise, through a fluid is unstable. But the motion of 
 either body, with its length or its broadside perpendicular to the 
 direction of motion, is stable. Observation proves the assertion we 
 have just made, for real fluids, air and water, and for a great variety 
 of circumstances affecting the motion; and we shall return to the 
 subject later, as being not only of great practical importance, but 
 profoundly interesting, and by no means difficult in theory. 
 
 304. The motion of a single particle affords simpler and not less 
 instructive illustrations of stability and instability. Thus if a weight, 
 hung from a fixed point by a light inextensible cord, be set in motion 
 so as to describe a circle about a veitical line through its position of 
 equilibrium, its motion is stable. For, as we shall see later, if dis- 
 turbed infinitely little in direction without gain or loss of energy, it 
 will describe a sinuous path, cutting the undisturbed circle at points 
 successively distant from one another by definite fractions of the 
 circumference, depending upon the angle of inclination of the string 
 to the vertical. When this angle is very small, the motion is sensibly 
 the same as that of a particle confined to one plane moving under 
 the influence of an attractive force towards a fixed point, simply pro- 
 portional to the distance ; and the disturbed path cuts the undisturbed 
 circle four times in a revolution. Or if a particle confined to one 
 plane, move under the influence of a centre in this plane, attracting 
 with a force inversely as the square of the distance, a path infinitely 
 little disturbed from a circle will cut the circle twice in a revolution. 
 Or if the law of central force be the nxkv power of the distance, and if 
 fz + 3 be positive, the disturbed path will cut the undisturbed circular 
 
 TT 
 
 orbit at successive angular intervals, each equal to . . But the 
 
 V// + 3 
 motion will be unstable if n be negative, and - « > 3. 
 
 305. The case of a particle moving on a smooth fixed surface 
 under the influence of no other force than that of the constraint, and 
 therefore always moving along a geodetic line of the surface, affords 
 extremely simple illustrations of stability and instability. For instance, 
 a particle placed on the inner circle of the surface of an anchor-ring, 
 and projected in the plane of the ring, would move perpetually in that 
 circle, but unstably, as the smallest disturbance would clearly send it 
 away from this path, never to return until after a digression round the 
 outer edge. (We suppose of course that the particle is held to the 
 surface, as if it were placed in the infinitely narrow space between a 
 solid ring and a hollow one enclosing it.) But if a particle is placed 
 on the outermost, or greatest, circle of the ring, and projected in its 
 plane, an infinitely small disturbance will cause it to describe a sinuous 
 path cutting the circle at points round it successively distant by angles 
 
DYNAMICAL LAWS AND PRINCIPLES. 105 
 
 each equal to tt / - , and therefore at intervals of time, each equal to 
 
 - / - , where a denotes the radius of that circle, <o the angular velocity 
 m^ a 
 
 in it, and b the radius of the circular cross section of the ring. This 
 is proved by remarking that an infinitely narrow band from the outer- 
 most part of the ring has, at each point, a and b from its principal 
 radii of curvature, and therefore (§ 134) has for its geodetic lines the 
 great circles of a sphere of radius ^ab, upon which it may be bent. 
 
 306. In all these cases the undisturbed motion has been circular 
 or rectilineal, and, when the motion has been stable, the effect of a 
 disturbance has been pef-iodic, or recurring with the same phases in 
 equal successive intervals of time. An illustration of thoroughly stable 
 motion in which the effect of a disturbance is not * periodic,' is pre- 
 sented by a particle sliding down an inclined groove under the action 
 of gravity. To take the simplest case, we may consider a particle 
 sliding down along the lowest straight line of an inclined hollow 
 cylinder. If slightly disturbed from this straight line, it will oscillate 
 on each side of it perpetually in its descent, but not with a uniform 
 periodic motion, though the durations of its excursions to each side of 
 the straight line are all equal. 
 
 307. A very curious case of stable motion is presented by a particle 
 constrained to remain on the surface of an anchor-ring fixed in a 
 vertical plane, and projected along the great circle from any point of 
 it, with any velocity. An infinitely small disturbance will give rise to 
 a disturbed motion of which the path will cut the vertical circle over 
 and over again for ever, at unequal intervals of time, and unequal 
 angles of the circle ; and obviously not recurring periodically in any 
 cycle, except with definite particular values for the whole energy, some 
 of which are less and an infinite number are greater than that which 
 just suffices to bring the particle to the highest point of the ring. The 
 full mathematical investigation of these circumstances would afibrd an 
 excellent exercise in the theory of differential equations, but it is not 
 necessary for our present illustrations. 
 
 308. In this case, as in all of stable motion with only two degrees 
 of freedom, which we have just considered, there has been stability 
 throughout the motion ; and an infinitely small disturbance from any 
 point of the motion has given a disturbed path which intersects the 
 undisturbed path over and over again at finite intervals of time. 
 But, for the sake of simpHcity, at present confining our attention to 
 two degrees of freedom, we have a Ihnited stability in the motion of an 
 unresisted projectile, which satisfies the criterion of stability only at 
 points of its upward, not of its downward, path. Thus if MOPQ be 
 the path of a projectile, and if at O it be disturbed by an infinitely 
 small force either way perpendicular to its instantaneous direction of 
 motion, the disturbed path will cut the undisturbed infinitely near 
 the point /'where the direction of motion is perpendicular to that at 0\ 
 
ic6 PRELIMINARY. 
 
 as we easily see by considering that the line joining two particles pro- 
 jected from one point at the same instant with equal velocities in the 
 
 directions of any two lines, will always remain perpendicular to the 
 line bisecting the angle between these two. 
 
 309. The principle of varying action gives a mathematical criterion 
 for stability or instability in every case of motion. Thus in the first 
 place it is obvious {§§ 308, 311), that if the action is a true minimum 
 in the motion of a system from any one configuration to the con- 
 figuration reached at any other time, however much later, the motion 
 is thoroughly unstable. For instance, in the motion of a particle con- 
 strained to remain on a smooth fixed surface, and uninfluenced by 
 gravity, the action is simply the length of the path, multiplied by the 
 constant velocity. Hence in the particular case of a particle unin- 
 fluenced by gravity, moving round the inner circle in the plane of an 
 anchor-ring considered above, the action, or length of path, is clearly 
 a minimum for any one point to the point reached at any subsequent 
 time. (The action is not merely a minimum, but is the least possible, 
 from any point of the circular path to any other, through less than half 
 a circumference of the circle.) On the other hand, although the path 
 from any point in the greatest circle of the ring to any other at a dis- 
 tance from it along the circle, less than tt ^ab^ is clearly least possible 
 if along the circumference ; the path of absolutely least length is not 
 along the circumference between two points at a greater circular 
 distance than irjab from one another, nor is the path along the 
 circumference between them a minimum at all in this latter case. On 
 any surface whatever which is everywhere anticlastic, or along a geo- 
 detic of any surface which passes altogether through an anticlastic 
 region, the motion is thoroughly unstable. For if it were stable from 
 any point O, we should have the given undisturbed path, and the 
 disturbed path from O cutting it at some point Q — two difterent 
 
DYNAMICAL LAWS AND PRINCIPLES. 107 
 
 geodetic lines joining two points ; which is impossible on any 
 anticlastic surface, inasmuch as the sum of the exterior angles of 
 any closed figure of geodetic lines exceeds four right angles when 
 the integral curvature of the enclosed area is negative, which 
 is the case for every portion of surface thoroughly anticlastic. 
 But, on the other hand, it is easily proved that if we have an endless 
 rigid band of curved surface everywhere synclastic, with a geodetic 
 line running through its middle, the motion of a particle projected 
 along this line will be stable throughout, and an infinitely slight disturb- 
 ance will give a disturbed path cutting the given undisturbed path again 
 and again for ever at successive distances differing according to the 
 different specific curvatures of the intermediate portions of the surface. 
 
 310. If, from any one configuration, two courses differing infinitely 
 little from one another, have again a configuration in common, this 
 second configuration will be called a kinetic focus relatively to the 
 first: or (because of the reversibility of the motion) these two con- 
 figurations will be called conjugate kinetic foci. Optic foci, if for 
 a moment we adopt the corpuscular theory of light, are included aa 
 a particular case of kinetic foci in general. But it is not difficult 
 to prove that there must be finite intervals of space and time be- 
 tween two conjugate foci in every motion of every kind of system, 
 only provided the kinetic energy does not vanish. 
 
 311. Now it is obvious that, provided only a sufficiently short 
 course is considered, the action, in any natural motion of a system, 
 is less than for any other course between its terminal configurations. 
 It will be proved presently (§ 314) that the first configuration up to 
 which the action, reckoned from a given initial configuration, ceases 
 to be a minimum, is the first kinetic focus; and conversely, that when 
 the first kinetic focus is passed, the action, reckoned from the initial 
 configuration, ceases to be a minimum ; and therefore of course can 
 never again be a minimum, because a course of shorter action, 
 deviating infinitely little from it, can be found for a part, without 
 altering the remainder of the whole, natural course. 
 
 312. In such statements as this it will frequently be convenient 
 to indicate particular configurations of the system by single letters, 
 as (9, P, Q, P ; and any particular course, in which it moves through 
 configurations thus indicated, will be called the course O...P...Q...P. 
 The action in any natural course will be denoted simply by the 
 terminal letters, taken in the order of the motion. Thus OP will 
 denote the action from O to P ; and therefore OP = -PO. When 
 there are more real natural courses from O to P than one, the 
 analytical expression for OP will have more than one real value; 
 and it may be necessary to specify for which of these courses the 
 action is reckoned. Thus we may have 
 
 OP for O...P...P, 
 OP for O...P\..P, 
 OP for O...P"...P, 
 three different values of one algebraic irrational expression. 
 
io8 
 
 PRELIMINARY. 
 
 313. In terms of this notation the preceding statement (§311) 
 may be expressed thus : — If, for a conservative system, moving on 
 a certain course 0...F...0' ...P\ the first kinetic focus conjugate 
 to (9 be 0\ the action OP, in this course, will be less than the action 
 along any other course deviating infinitely little from it: but, on the 
 other hand, OP' is greater than the actions in some courses from 
 O to P' deviating infinitely little from the specified natural course 
 O...P...O'...P'. 
 
 314. It must not be supposed that the action along OP is neces- 
 sarily t/je least possible from O to P. There are, in fact, cases in 
 which the action ceases to be least of all possible, before a kinetic 
 focus is reached. Thus if OEAPO'E'A' be a sinuous geodetic line 
 cutting the outer circle of an anchor-ring, or the equator of an oblate 
 spheroid, in successive points O, A, A' ,\\. is easily seen that 0\ the 
 first kinetic focus conjugate to O, must lie somewhat beyond A. 
 But the length OEAP, although a minimum (a stable position for 
 a stretched string), is not the shortest distance on the surface from 
 O to jP, as this must obviously be a line lying entirely on one side 
 
 of the great circle. From O to any point, Q, short of A^ the distance 
 along the geodetic OEQA is clearly the least possible : but if Q be 
 near enough to A (that is to say, between A and the point in which 
 the envelop of the geodetics drawn from O, cuts OEA), there will 
 also be two other geodetics from O to Q. The length of one of 
 these will be a minimum, and that of the other not a minimum. 
 If Q is moved forward to A, the former becomes OE^A, equal and 
 similar to OEA, but on the other side of the great circle : and the 
 latter becomes the great circle from O to A. If now Q be moved 
 on to P, beyond A, the minimum geodetic OEAP ceases to be the 
 less of the two minima, and the geodetic OEPly'mg altogether on the 
 other side of the great circle becomes the least possible line from 
 O to P. But until Pis advanced beyond the point O', in which it 
 is cut by another geodetic from O lying infinitely nearly along it, 
 the length OEAP remains a minimum according to the general 
 proposition of § 311. 
 
DYNAMICAL LAWS AND PRINCIPLES. 109 
 
 315. As it has been proved that the action from any configuration 
 ceases to be a minimum at the first conjugate kinetic focus, we see 
 immediately that if O' be the first kinetic focus conjugate to (9, reached 
 after passing O, no two configurations on this course from O to O' 
 can be kinetic foci to one another. For, the action from O just 
 ceasing to be a minimum when O' is reached, the action between any 
 two intermediate configurations of the same course is necessarily a 
 minimum. 
 
 316. When there are i degrees of freedom to move there are in 
 general, on any natural course from any particular configuration, (7, 
 at least /- i kinetic foci conjugate to O. Thus, for example, on the 
 course of a ray of light emanating from a luminous point (9, and pass- 
 ing through the centre of a convex lens held obliquely to its path, 
 there are two kinetic foci conjugate to (7, as defined above, being the 
 points in which the line of the central ray is cut by the so-called 
 *focal lines'^ of a pencil of rays diverging from O and made con- 
 vergent after passing through the lens. But some or all of these 
 kinetic foci may be on the course previous to O ', as, for instance, in 
 the case of a common projectile when its course passes obliquely 
 downwards through O. Or some or all may be lost, as when, 
 in the optical illustration just referred to, the lens is only strong 
 enough to produce convergence in one of the principal planes, or 
 too weak to produce convergence in either. Thus also in the 
 case of the undisturbed rectilineal motion of a point, or in the 
 motion of a point uninfluenced by force, on an anticlastic surface 
 (§ 309)? there are no real kinetic foci. In the motion of a pro- 
 jectile (not confined to one vertical plane) there can be only one 
 kinetic focus on each path, conjugate to one given point ; though 
 there are three degrees of freedom. Again, there may be any number 
 more than i—\ of foci in one course, all conjugate to one con- 
 figuration, as for instance on the course of a particle, uninfluenced by 
 force, moving round the surface of an anchor-ring, along either the 
 outer great circle, or along a sinuous geodetic such as we have con- 
 sidered in § 311, in which clearly there are an infinite number of foci 
 each conjugate to any one point of the path, at equal successive dis- 
 tances from one another. 
 
 317. If/- I distinct^ courses from a configuration (9, each difl'ering 
 
 infinitely little from a certain natural course O ..E . . (9, .. O ^ 
 
 <^i-i- • <2j cut it in configurations Oj, O^, O^, . . . 0^_^, and if, besides 
 these, there are not on it any other kinetic foci conjugate to (9, 
 between O and ft and no focus at all, conjugate to E^ between 
 E and Q, the action in this natural course from O to Q \^ the 
 maximum for all courses 0.,.P^, P^...Q; P^ being a configura- 
 
 * In our second volume we hope to give all necessary elementary explanations on 
 this subject. 
 
 2 Two courses are here called not distinct if they differ from one another only in 
 the absolute magnitude, not in the proportions, of the components of the deviations 
 by which they differ from the standard course. 
 
no PRELIMINARY. 
 
 tion infinitely nearly agreeing with some configuration between E 
 
 and O^ of the standard course O . . E .. (9, . . O^ O^^^ . . Q, 
 
 and O . .. R^, P,-" Q denoting the natural courses between O and R^, 
 and R^ and Q, which deviate infinitely little from this standard course. 
 
 318. Considering now, for simplicity, only cases in which there 
 are but two degrees (§ 165) of freedom to move, we see that after 
 any infinitely small conservative disturbance of a system in passing 
 through a certain configuration, the system will first again pass 
 through a configuration of the undisturbed course, at the first con- 
 figuration of the latter at which the action in the undisturbed motion 
 ceases to be a minimum. For instance, in the case of a particle, 
 confined to a surface, and subject to any conservative system of force, 
 an infinitely small conservative disturbance of its motion through any 
 point, O, produces a disturbed path, which cuts the undisturbed path 
 at the first point, 0\ at which the action in the undisturbed path from 
 O ceases to be a minimum. Or, if projectiles, under the influence of 
 gravity alone, be thrown from one point, O, in all directions with 
 equal velocities, in one vertical plane, their paths, as is easily proved, 
 intersect one another consecutively in a parabola, of which the focus 
 is (9, and the vertex the point reached by the particle projected 
 directly upwards. The actual course of each particle from O is the 
 course of least possible action to any point, R, reached before the 
 enveloping parabola, but is not a course of minimum action to any 
 point, Q, in its path after the envelop is passed. 
 
 319. Or again, if a particle slides round along the greatest circle of 
 the smooth inner surface of a hollow anchor-ring, the 'action,' or 
 simply the length of path, from point to point, will be least possible 
 for lengths (§ 305) less than tt Jab. Thus if a string be tied round 
 outside on the greatest circle of a perfectly smooth anchor-ring, it will 
 slip off unless held in position by staples, or checks of some kind, at 
 distances of not less than this amount, Trjab, from one another in 
 succession round the circle. With reference to this example, see also 
 § 314, above. 
 
 Or, if a particle slides down an inclined hollow cylinder, the 
 action from any point will be the least possible along the straight path 
 to any other point reached in a time less than that of the vibration 
 one way of a simple pendulum of length equal to the radius of the 
 cylinder, and influenced by a force equal to g cos /, instead of g the 
 whole force of gravity. But the action will not be a minimum from 
 any point, along the straight path, to any other point reached in a 
 longer time than this. The case in which the groove is horizontal 
 (i = o) and the particle is projected along it, is particularly simple and 
 instructive, and may be worked out in detail with great ease, without 
 assuming any of the general theorems regarding action. 
 
CHAPTER III. 
 
 EXPERIENCE. 
 
 320. By the term Experience, in physical science, we designate, 
 according to a suggestion of Herschel's, our means of becoming 
 acquainted with the material universe and the laws which regulate it. 
 In general the actions which we see ever taking place around us are 
 complex, or due to the simultaneous action of many causes. When, 
 as in astronomy, we endeavour to ascertain these causes by simply 
 watching their effects, we observe; when, as in our laboratories, we 
 interfere arbitrarily with the causes or circumstances of a pheno- 
 menon, we are said to experiment. 
 
 321. For instance, supposing that we are possessed of instru- 
 mental means of measuring time and angles, we may trace out by 
 successive observations the relative position of the sun and earth at 
 different instants; and (the method is not susceptible of any accuracy, 
 but is alluded to here only for the sake of illustration) from the 
 variations in the apparent diameter of the former we may calculate 
 the ratios of our distances from it at those instants. We have thus a 
 set of observations involving time, angular position with reference to 
 the sun, and ratios of distances from it ; sufficient (if numerous 
 enough) to enable us to discover the laws which connect the varia- 
 tions of these co-ordinates. 
 
 Similar methods may be imagined as applicable to the motion of 
 any planet about the sun, of a satellite about its primary, or of one 
 star about another in a binary group. 
 
 322. In general all the data of Astronomy are determined in this 
 way, and the same may be said of such subjects as Tides and Meteor- 
 ology. Isothermal Lines, Lines of Equal Dip or Intensity, Lines of 
 No Declination, the Connexion of Solar Spots with Terrestrial Mag- 
 netism, and a host of other data and phenomena, to be explained 
 under the proper heads in the course of the work, are thus deducible 
 from Observation merely. In these cases the apparatus for the gigantic 
 experiments is found ready arranged in Nature, and all that the 
 philosopher has to do is to watch and measure their progress to its 
 last details. 
 
1 1 2 PRELIMINAR V. 
 
 323. Even in the instance we have chosen above, that of the 
 planetary motions, the observed eifects are complex ; because, unless 
 possibly in the case of a double star, we have no instance of the 
 widistiirbed action of one heavenly body on another; but to a first 
 approximation the motion of a planet about the sun is found to be 
 the same as if no other bodies than these two existed; and the 
 approximation is sufficient to indicate the probable law of mutual 
 action, whose full confirmation is obtained when, its truth being 
 assumed, the disturbing effects thus calculated are allowed for, and 
 found to account completely for the observed deviations from the 
 consequences of the first supposition. This may serve to give an 
 idea of the mode of obtaining the laws of phenomena, which can 
 only be observed in a complex form; and the method can always be 
 directly applied when one cause is known to be pre-eminent. 
 
 324. Let us take a case of the other kind — that in which the effects 
 are so complex that we cannot deduce the causes from the observation 
 of combinations arranged in Nature, but must endeavour to form for 
 ourselves other combinations which may enable us to study the effects 
 of each cause separately, or at least with only slight modification from 
 the interference of other causes. 
 
 A stone, when dropped, falls to the ground; a brick and a boulder, 
 if dropped from the top of a cliff at the same moment, fall side by 
 side, and reach the ground together. But a brick and a slate do not; 
 and while the former falls in a nearly vertical direction, the latter 
 describes a most complex path. A sheet of paper or a fragment of 
 gold-leaf presents even greater irregularities than the slate. But by 
 a slight modification of the circumstances, we gam a considerable 
 insight into the nature of the question. The paper and gold-leaf, if 
 rolled into balls, fall nearly in a vertical line. Here, then, there are 
 evidently at least two causes at work, one which tends to make all 
 bodies fall, and that vertically; and another which depends on the 
 form and substance of the body, and tends to retard its fall and alter 
 its vertical direction. How can we study the effects of the former on 
 all bodies without sensible complication from the latter? The effects 
 of Wind, etc., at once point out what the latter cause is, the air (whose 
 existence we may indeed suppose to have been discovered by such 
 effects); and to study the nature of the action of the former it is 
 necessary to get rid of the complications arising from the presence 
 of air. Hence the necessity for Experhtient. By means of an appa- 
 ratus to be afterwards described, we remove the greater part of the 
 air from the interior of a vessel, and in that we try again our expe- 
 riments on the fall of bodies; and now a general law, simple in the 
 extreme, though most important in its consequences, is at once appa- 
 rent — viz. that all bodies, of whatever size, shape, or material, if 
 dropped side by side at the same instant, fall side by side in a space 
 void of air. Before experiment had thus separated the phenomena, 
 hasty philosophers had rushed to the conclusion that some bodies 
 possess the quality of heaviness^ others that of lightness^ etc Had 
 
EXPERIENCE. 113 
 
 this state of things remained, the law of gravitation, vigorous though 
 its action be throughout the universe, could never have been recog- 
 nized as a general principle by the human mind. 
 
 Mere observation of lightning and its effects could neVer have led 
 to the discovery of their relation to the phenomena presented by 
 rubbed amber. A modification of the course of Nature, such as the 
 bringing down of atmospheric electricity into our laboratories, was 
 necessary. Without experiment we could never even have learned 
 the existence of terrestrial magnetism. 
 
 325. When a particular agent or cause is to be studied, experi- 
 ments should be arranged in such a way as to lead if possible 
 to results depending on it alone; or, if this cannot be done, they 
 should be arranged so as to show differences produced by vary- 
 ing it. 
 
 326. Thus to determine the resistance of a wire against the 
 conduction of electricity through it, we may measure the whole 
 strength of current produced in it by electromotive force between 
 its ends when the amount of this electromotive force is give7t, 
 or can be ascertained. But when the wire is that of a submarine 
 telegraph cable there is always an tmknown and ever varying 
 electromotive force between its ends, due to the earth (produc- 
 ing what is commonly called the "earth-current"), and to deter- 
 mine its resistance the difference in the strength of the current 
 produced by suddenly adding to or subtracting from the terres- 
 trial electromotive force, the electromotive force of a given 
 voltaic battery is to be very quickly measured; and this is to be 
 done over and over again, to eliminate the effect of variation of 
 the earth current during the i^^ seconds of time which must 
 elapse before the electro- static induction permits the current due to 
 the battery to reach nearly enough its full strength to practically 
 annul error on this score. 
 
 327. Endless patience and perseverance in designing and trying 
 different methods for investigation are necessary for the advancement 
 of science : and indeed, in discovery, he is the most likely to succeed 
 who, not allowing himself to be disheartened by the non-success of 
 one form of experiment, judiciously varies his methods, and thus 
 interrogates in every conceivably useful manner the subject of his 
 investigations. 
 
 328. A most important remark, due to Herschel, regards what are 
 called r^^V///^/ phenomena. When, in an experiment, all known causes 
 being allowed for, there remain certain unexplained effects (exces- 
 sively slight it may be), these must be carefully investigated, and every 
 conceivable variation of arrangement of apparatus, etc., tried; until, if 
 possible, we manage so to exaggerate the residual phenomenon as to 
 be able to detect its cause. It is here, perhaps, that in the present 
 state of science we may most reasonably look for extensions of our 
 knowledge ; at all events we are warranted by the recent history of 
 Natural Philosophy in so doing. Thus, to take only a very few 
 
 T. 8 
 
1 14 PRELIMINARY. 
 
 instances, and to say nothing of the discovery of electricity and mag- 
 netism by the ancients, the pecuUar smell observed in a room in 
 which an electrical machine is kept in action, was long ago observed, 
 but called the 'smell of electricity,' and thus left unexplained. The 
 sagacity of Schonbein led to the discovery that this is due to the 
 formation of Ozone, a most extraordinary body, of enormous chem- 
 ical energies ; whose nature is still uncertain, though the attention of 
 chemists has for years been directed to it. 
 
 329. Slight anomalies in the motion of Uranus led Adams and 
 Le Verrier to the discovery of a new planet; and the fact that a 
 magnetized needle comes to rest sooner when vibrating above a 
 copper plate than when the latter is removed, led Arago to what 
 was once called magnetism of rotation, but has since been explained, 
 immensely extended, and applied to most important purposes. In 
 fact, this accidental remark about the oscillation of a needle led 
 to facts from which, in Faraday's hands, was evolved the grand 
 discovery of the Induction of Electrical Currents by magnets or 
 by other currents. We need not enlarge upon this point, as in 
 the following pages the proofs of the truth and usefulness of the 
 principle will continually recur. Our object has been not so much 
 to give applications as methods, and to show, if possible, how to 
 attack a new combination, with the view of separadng and studying 
 in detail the various causes which generally conspire to produce 
 observed phenomena, even those which are apparently the simplest. 
 
 330. If, on repetidon several times, an experiment continually gives 
 different results, it must either have been very carelessly performed, 
 or there must be some disturbing cause not taken account of. And, 
 on the other hand, in cases where no very great coincidence is 
 likely on repeated trials, an unexpected degree of agreement between 
 the results of various trials should be regarded with the utmost 
 suspicion, as probably due to some unnoticed peculiarity of the 
 apparatus employed. In either of these cases, however, careful 
 observation cannot fail to detect the cause of the discrepancies or 
 of the unexpected agreement, and may possibly lead to discoveries 
 in a totally unthought-of quarter. Instances of this kind may be 
 given without limit ; one or two must suffice. 
 
 331. Thus, with a very good achromatic telescope a star appears 
 to have a sensible disc. But, as it is observed that the discs of 
 all stars appear to be of equal angular diameter, we of course suspect 
 some comrnon error. Limiting the aperture of the object-glass 
 increases the appearance in question, which, on full investigation, 
 is found to have nothing to do with discs at all. It is, in fact, a dif- 
 fraction phenomenon, and will be explained in our chapters on Light. 
 
 Again, in measuring the velocity of Sound by experiments con- 
 ducted at night with cannon, the results at one station were never 
 found to agree exactly with those at the other ; sometimes, indeed, 
 the differences were very considerable. But a little consideration 
 
EXPERIENCE. 1 1 5 
 
 led to the remark, that on those nights in which the discordance 
 was greatest a strong wind was blowing nearly from one station 
 to the other. Allowing for the obvious effect of this, or rather 
 eliminating it altogether, the mean velocities on different evenings 
 were found to agree very closely. 
 
 332. It may perhaps be advisable to say a few words here about 
 the use of hypotheses, and especially those of very different gradations 
 of value which are promulgated in the form of Mathematical Theories 
 of different branches of Natural Philosophy. 
 
 333. Where, as in the case of the planetary motions and disturb- 
 ances, the forces concerned are thoroughly known, the mathematical 
 theory is absolutely true, and requires only analysis to work out its 
 remotest details. It is thus, in general, far ahead of observation, and 
 is competent to predict effects not yet even observed — as, for instance, 
 Lunar Inequalities due to the action of Venus upon the Earth, etc. etc., 
 to which no amount of observation, unaided by theory, would ever 
 have enabled us to assign the true cause. It may also, in such 
 subjects as Geometrical Optics, be carried to developments far beyond 
 the reach of experiment ; but in this science the assumed bases of the 
 theory are only approximate, and it fails to explain in all their peculi- 
 arities even such comparatively simple phenomena as Halos and 
 Rainbows ; though it is perfectly successful for the practical purposes 
 of the maker of microscopes and telescopes, and has, in these cases, 
 carried the construction of instruments to a degree of perfection 
 which merely tentative processes never could have reached. 
 
 334. Another class of mathematical theories, based to a certain 
 extent on experiment, is at present useful, and has even in certain 
 cases pointed to new and important results, which experiment has 
 subsequently verified. Such are the Dynamical Theory of Heat, the 
 Undulatory Theory of Light, etc. etc. In the former, which is based 
 upon the experimental fact that heat is motion^ many formulae are 
 at present obscure and uninterpretable, because we do not know 
 what is moving or how it moves. Results of the theory in which 
 these are not involved, are of course experimentally verified. The 
 same difficuldes exist in the Theory of Light. But before this 
 obscurity can be perfectly cleared up, we must know something 
 of the ultimate, or molecular, constitution of the bodies, or groups 
 of molecules, at present known to us only in the aggregate. 
 
 335. A third class is well represented by the Mathematical Theories 
 of Heat (Conduction), Electricity (Statical), and Magnetism (Perma- 
 nent). Although we do not know how Heat is propagated in bodies, 
 nor what Statical Electricity or Permanent Magnetism are, the laws 
 of their forces are as certainly known as that of Gravitation, and 
 can therefore like it be developed to their consequences, by the 
 application of Mathematical Analysis. The works of Fourier', 
 
 ^ Theorie Ajtalytique de la Chaleur. Paris, 1822. 
 
 8—2 
 
1 1 6 PRELIMINAR V. 
 
 Green \ and Poisson'', are remarkable instances of such develop- 
 ment. Another good example is Ampere's Theory of Electro- 
 dynamics. 
 
 336. Mathematical theories of physical forces are, in general, of 
 one of two species. First, those in which the fundamental assump- 
 tion is far more general than is necessary. Thus the celebrated 
 equation of Laplace's Functions contains the mathematical foundation 
 of the theories of Gravitation, Statical Electricity, Permanent Mag- 
 netism, Permanent Flux of Heat, Motion of Incompressible Fluids, 
 etc. etc., and has therefore to be accompanied by limiting consider- 
 ations when applied to any one of these subjects. 
 
 337. Again, there are those which are built upon a few experiments, 
 or simple but inexact hypotheses, only; and which require to be 
 modified in the way of extension rather than limitation. As a notable 
 example, we may refer to the whole subject of Abstract Dynamics, 
 which requires extensive modifications (explained in Division III.) 
 before it can, in general, be applied to practical purposes. 
 
 338. When the most probable result is required from a number of 
 observations of the same quantity which do not exactly agree, we 
 must appeal to the mathematical theory of probabilities to guide us 
 to a method of combining the results of experience, so as to eliminate 
 from them, as far as possible, the inaccuracies of observation. But 
 it must be explained that we do not at present class as inaccuracies 
 of ohsei'vation any errors which may affect alike every one of a series 
 of observations, such as the inexact determination of a zero-point or 
 of the essential units of time and space, the personal equation of the 
 observer, etc. The process, whatever it may be, which is to be 
 employed in the elimination of errors, is applicable even to these, but 
 only when several distinct series of observations have been made, with 
 a change of instrument, or of observer, or of both. 
 
 339. We understand as inaccuracies of observation the whole class 
 of errors which are as likely to lie in one direction as another in suc- 
 cessive trials, and which we may fairly presume would, on the average 
 of an infinite number of repetitions, exactly balance each other in 
 excess and defect. Moreover, we consider only errors of such a 
 kind that their probability is the less the greater they are ; so that 
 such errors as an accidental reading of a wrong number of whole 
 degrees on a divided circle (which, by the way, can in general be 
 probably corrected by comparison with other observations) are not to 
 be included. 
 
 340. Mathematically considered, the subject is by no means an 
 easy one, and many high authorities have asserted that the reasoning 
 employed by Laplace, Gauss, and others, is not well founded ; although 
 the results of their analysis have been generally accepted. As an 
 excellent treatise on the subject has recently been published by Airy, 
 
 1 Essay on the Application of Mathematical Analysis to the Theories oj 
 Electricity and Magnetism. Nottingham, 1828. KidY^vmi^A in Crelle's Journal . 
 
 2 M6 moires snr le Magneiisme. Mem. de VAcad, des Sciences, 181 r. 
 
EXPERIENCE. 117 
 
 it is not necessary for us to do more than sketch in the most cursory 
 manner what is called the Method of Least Squares. 
 
 341. Supposing the zero-point and the graduation of an instrument 
 (micrometer, mural circle, thermometer, electrometer, galvanometer, 
 etc.) to be absolutely accurate, successive readings of the value of a 
 quantity (linear distance, altitude of a star, temperature, potential, 
 strength of an electric current, etc.) may, and in general do, con 
 tinually differ. What is most probably the true value of the observed 
 quantity ? 
 
 The most probable value, in all such cases, if the observations are all 
 equally reliable, will evidently be the simple mean ; or if they are not 
 equally reliable, the mean found by attributing weights to the several 
 observations in proportion to their presumed exactness. But if several 
 such means have been taken, or several single observations, and if 
 these several means or observations have been differently qualified 
 for the determination of the sought quantity (some of them being 
 likely to give a more exact value than others), we must assign theoret- 
 ically the best method of combining them in practice. 
 
 342. Inaccuracies of observation are, in general, as likely to be in 
 excess as in defect. They are also (as before observed) more likely 
 to be small than great ; and (practically) large errors are not to be 
 expected at all, as such would come under the class of avoidable mis- 
 takes. It follows that in any one of a series of observations of the 
 same quantity the probability of an error of magnitude x, must depend 
 upon x^, and must be expressed by some function whose value 
 diminishes very rapidly as x increases. The probability that the 
 error lies between x and x-v'^x^ where Ix is very small, must also be 
 proportional to Sx The law of error thus found is 
 
 I -^nx 
 Jtt h 
 where ^ is a constant, indicating the degree of coarseness or delicacy 
 of the system of measurement employed. The co-efficient —j~ secures 
 
 that the sum of the probabilities of all possible errors shall be unity, 
 as it ought to be. 
 
 343. The Probable Error of an observation is a numerical quantity 
 such that the error of the observation is as Ukely to exceed as to fall 
 short of it in magnitude. 
 
 If we assume the law of error just found, and call P the probable 
 error in one trial, we have the approximate result 
 
 p= 0-477/2. 
 
 344. The probable error of any given multiple of the value of an 
 observed quantity is evidently the same multiple of the probable error 
 of the quantity itself 
 
 The probable error of the sum or difference of two quantities, 
 affected by independent errors, is the square root of the sum of the 
 squares of their separate probable errors. 
 
1 1 8 PRELIMINAR V, 
 
 345. As above remarked, the principal use of this theory is in the 
 deduction, from a large series of observations, of the values of the 
 quantities sought in such a form as to be liable to the smallest pro- 
 bable error. As an instance — by the principles of physical astronomy, 
 the place of a planet is calculated from assumed values of the elements 
 of its orbit, and tabulated in the Nautical Almanac. The observed 
 places do not exactly agree with the predicted places, for two reasons 
 
 ■ — first, the data for calculation are not exact (and in fact the main 
 object of the observation is to correct their assumed values); second, 
 the observation is in error to some unknown amount. Now the 
 difference between the observed, and the calculated, places depends 
 on the errors of assumed elements and of observation. Our methods 
 are applied to eliminate as far as possible the second of these, and the 
 resulting equations give the required corrections of the elements. 
 
 Thus if B be the calculated R. A. of a planet : ha, Be, 8^, etc., the 
 corrections required for the assumed elements : the true R.A. is 
 
 6 + ASa + £Se + nSra- + etc., 
 
 where A, E, 11, etc., are approximately known. Suppose the observed 
 R.A. to be ©, then 
 
 e + ABa + EBe + UBzn +... = ©, 
 or ABa + EBe + HSra- + . . . = © - ^, 
 
 a known quantity, subject to error of observation. Every observation 
 made gives us an equation of the same form as this, and in general 
 the number of observations greatly exceeds that of the quantities Ba, 
 Be, Bw, etc., to be found. 
 
 346. The theorems of § 344 lead to the following rule for com- 
 bining any number of such equations which contain a smaller number 
 of unknown quantities : — 
 
 Make the probable error of the second member the same in each equa- 
 tion, by the employ me^tt of a proper factor : midtiply each equation by the 
 co-efficie?it of x in it and add all, for one of the final equations ; and so, 
 with reference to y, z, etc., for the others. The probable errors of the 
 values of x, y, etc., found from these final equations will be less than 
 those of the values derived from any other linear method of com- 
 bining the equations. 
 
 This process has been called the method of Least Squares, because 
 the values of the unknown quantities found by it are such as to render 
 the sum of the squares of the errors of the original equations a 
 minimum. 
 
 347. When a series of observations of the same quantity has been 
 made at different times, or under different circumstances, the law 
 connecting the value of the quantity with the time, or some other 
 variable, may be derived from the results in several ways — all more 
 or less approximate. Two of these methods, however, are so much 
 more extensively used than the others, that we shall devote a page or 
 
EXPERIENCE. 
 
 119 
 
 two here to a preliminary notice of them, leaving detailed instances 
 of their application till we come to Heat, Electricity, etc. They 
 consist in (i) a Curve, giving a graphic representation of the relation 
 between the ordinate and abscissa, and (2) an Empirical Formula 
 connecting the variables. 
 
 348. Thus if the abscissae represent intervals of time, and the 
 ordinates the corresponding height of the barometer, we may con- 
 struct curves which show at a glance the dependence of barometric 
 pressure upon the time of day; and so on. Such curves may be 
 accurately drawn by photographic processes on a sheet of sensitive 
 paper placed behind the mercurial column, and made to move past 
 it with a uniform horizontal velocity by clockwork. A similar pro- 
 cess is applied to the Temperature and Electricity of the atmosphere, 
 and to the components of terrestrial magnetism. 
 
 349. When the observations are not, as in the last section, con- 
 tinuous, they give us only a series of points in the curve, from which, 
 however, we may in general approximate very closely to the result 
 of continuous observation by drawing, libera manii, a curve passing 
 through these points. This process, however, must be employed 
 with great caution ; because, unless the observations are sufficiently 
 close to each other, most important fluctuations in the curve may 
 escape notice. It is applicable, with abundant accuracy, to all cases 
 where the quantity observed changes very slowly. Thus, for instance, 
 weekly observations of the temperature at depths of from 6 to 24 feet 
 underground were found by Forbes sufficient for a very accurate 
 approximation to the law of the phenomenon. 
 
 350. As an instance of the processes employed for obtaining an 
 empirical formula, we may mention methods of Interpolation, to which 
 the problem can always be reduced. Thus from sextant observations, 
 
 . at known intervals, of the altitude of the sun, it is a common problem 
 of Astronomy to determine at what instant the altitude is greatest, 
 and what is that greatest altitude. The first enables us to find the 
 true solar time at the place, and the second, by the help of the 
 Nautical Almanac, gives the latitude. The calculus of finite differ- 
 ences gives us formulae proper for various data ; and Lagrange has 
 shown how to obtain a very useful one by elementary algebra. 
 In finite differences we have 
 
 f{x + h) =f{x) + hXf(x) + ^-^~> AYix) + . . . 
 
 This is useful, inasmuch as the successive differences, ^/(x\ 
 Ay(^), etc., are easily calculated from the tabulated results of obser- 
 vation, provided these have been taken for equal successive in- 
 crements of ^. 
 
 If for values x^, x^,...Xn, a function takes the values y^, y^, y^,... 
 y„, Lagrange gives for it the obvious expression 
 
1 20 PRELIMINAR V. 
 
 Here is assumed that the function required is a rational and 
 integral one in x of the n-i^^ degree; and, in general, a similar 
 limitation is in practice applied to the other formula above ; for in 
 order to find the complete expression for /{x), it is necessary to 
 determine the values of A/(x), ^y{x), .... If ;2 of the co-efficients be 
 required, so as to give the n chief terms of the general value of/{x)f 
 we must have n observed simultaneous values of x and/(^), and the 
 expression becomes determinate and of the n—i^^ degree in ^. 
 
 In practice it is usually sufficient to employ at most three terms 
 of the first series. Thus to express the length / of a rod of metal as 
 depending on its temperature f, we may assume 
 
 l^ being the measured length at any temperature t^, A and B are to 
 be found by the method of least squares from values of / observed for 
 different given values of t. 
 
 351. These formulae are practically useful for calculating the 
 probable values of any observed element, for values of the in- 
 dependent variable lying within the range for which observation has 
 given values of the element. But except for values of the inde- 
 pendent variable either actually within this range, or not far beyond 
 it in either direction, these formulae express functions which, in 
 general, will differ more and more widely from the truth the further 
 their application is pushed beyond the range of observation. 
 
 In a large class of investigations the observed element is in its 
 nature a periodic function of the independent variable. The har- 
 monic analysis (§ 88) is suitable for all such. When the values of the 
 independent variable for which the element has been observed are 
 not equidiflferent the co-efficients, determined according to the method 
 of least squares, are found by a process which is necessarily very 
 laborious ; but when they are equidifferent, and especially when the 
 difference is a submultiple of the period, the equation derived from 
 the method of least squares becomes greatly simplified. Thus, if Q 
 denote an angle increasing in proportion to /, the time, through four 
 right angles in the period, T^ of the phenomenon; so that 
 
 let /(^) = ^^ + ^,cos^-i-^2Cos 2^+ ... 
 + B^ sin ^ -I- ^2 sin 2^ + . . . 
 where A^^ A^, Ag,...B^, B^,... are unknown co-efficients, to be 
 determined so that /(O) may express the most probable value of the 
 element, not merely at times between observations, but through all 
 time as long as the phenomenon is strictly periodic. By taking as 
 many of these coefficients as there are of distinct data by observation, 
 the formula is made to agree precisely with these data. But in most 
 applications of the method, the periodically recurring part of the phe- 
 nomenon is expressible by. a small number of terms of the harmonic 
 series, and the higher terms, calculated from a great number of data, 
 
EXPERIENCE. 121 
 
 express either irregularities of the phenomenon not likely to recur, 
 or errors of observation. Thus a comparatively small number of 
 terms may give values of the element even for the very times of ob- 
 servation, more probable than the values actually recorded as having 
 been observed, if the observations are numerous but not minutely 
 accurate. 
 
 The student may exercise himself in writing out the equations to 
 determine five, or seven, or more of the coefficients according to the 
 method of least squares ; and reducing them by proper formulae of 
 analytical trigonometry to their simplest and most easily calculated 
 forms where the values of Q for which f{Q) is given are equidifferent. 
 
 He will thus see that when the difference is -v-, i being any integer, 
 
 and when the number of the data is / or any multiple of it, the equa- 
 tions contain each of them only one of the unknown quantities : so 
 that the method of least squares affords the most probable values of 
 the co-efficients, by the easiest and most direct elimination. 
 
 OF THE 
 
 VNIVERSltY 
 
CHAPTER IV. 
 MEASURES AND INSTRUMENTS. 
 
 352. Having seen in the preceding chapter that for the investiga- 
 tion of the laws of nature we must carefully watch experiments, either 
 those gigantic ones which the universe furnishes, or others devised 
 and executed by man for special objects — and having seen that in 
 all such observations accurate measurements of Time, Space, Force, 
 etc., are absolutely necessary, we may now appropriately describe a 
 few of the more useful of the instruments employed for these pur- 
 poses, and the various standards or units which are employed in 
 them. 
 
 353. Before going into detail we may give a rapid resiwie of the 
 principal Standards and Instruments to be described in this chapter. 
 As most, if not all, of them depend on physical principles to be 
 detailed in the course of this work, we shall assume in anticipation 
 the establishment of such principles, giving references to the future 
 division or chapter in which the experimental demonstrations are 
 more particularly explained. This course will entail a slight, but 
 unavoidable, confusion — slight, because Clocks, Balances, Screws, 
 etc., are familiar even to those who know nothing of Natural Phi- 
 losophy ; unavoidable, because it is in the very nature of our subject 
 that no one part can grow alone, each requiring for its full develop- 
 ment the utmost resources of all the others. But if one of our 
 departments thus borrows from others, it is satisfactory to find that 
 it more than repays by the power which its improvement affords 
 them. 
 
 354. We may divide our more important and fundamental instru- 
 ments into four classes — 
 
 Those for measuring Time ; 
 
 „ „ Space, linear or angular ; 
 
 Force; 
 „ „ Mass. 
 
 Other instruments, adapted for special purposes such as the 
 measurement of Temperature, Light, Electric Currents, etc., will 
 come more naturally under the head of the particular physical 
 energies to whose measurement they are applicable. Descriptions of 
 
MEASURES AND INSTRUMENTS. 123 
 
 self-recording instruments such as tide-gauges, and barometers, ther- 
 mometers, electrometers, recording photographically or otherwise the 
 continuously varying pressure, temperature, moisture, electric poten- 
 tial of the atmosphere, and magnetometers recording photographi- 
 cally the continuously varying direction and magnitude of the terres- 
 trial magnetic force, must likewise be kept for their proper places in 
 our work. 
 
 Calculating Machines have also important uses in assisting 
 physical research in 'a great variety of ways. They belong to two 
 classes : — 
 
 I. Purely Arithmetical, dealing with integral numbers of units. 
 All of this class are evolved from the primitive use of the calculuses 
 or little stones for counters (from which are derived the very names 
 calculation and " The Calculus "), through such mechanism as that 
 of the Chinese Abacus, still serving its original purpose well in 
 infant schools, up to the Arithmometer of Thomas of Colmar and the 
 grand but partially realized conceptions of calculating machines by 
 Babbage. 
 
 II. Continuous Calculating Machines. These are not only useful 
 as auxiliaries for physical research but also involve important dy- 
 namical and kinematical principles belonging properly to our subject. 
 
 355. We shall now consider in order the more prominent instru- 
 ments of each of these four classes, and some of their most important 
 applications : — 
 
 Clock, Chronometer, Chronoscope, Applications to Observation 
 
 and to self-registering Instruments. 
 Vernier and Screw-Micrometer, Cathetometer, Spherometer, 
 
 Dividing Engine, Theodolite, Sextant or Circle. 
 Common Balance, Bifilar Balance, Torsion Balance, Pendulum, 
 
 Dynamometer. 
 
 Among Standards we may mention — 
 
 1. Time. — Day, Hour, Minute, Second, sidereal and solar. 
 
 2. Space. — Yard and Metre: Radian, Degree, Minute, Second. 
 
 3. Force. — Weight of a Pound or Kilogramme, etc., in any par- 
 
 ticular locality (gravitation unit); poundal or dyne. Kinetic 
 Unit. 
 
 4. Mass. — Pound, Kilogramme, etc. 
 
 356. Although without instruments it is impossible to procure or 
 apply any standard, yet, as without the standards no instrument could 
 give us absolute measure, we may consider the standards first -^ 
 referring to the instruments as if we already knew their principles 
 and applications. 
 
 357. First we may notice the standards or units of angular 
 measure : 
 
1 24 PRELIMINAR V. 
 
 Radian^ or angle whose arc is equal to radius ; 
 
 Degree, or ninetieth part of a right angle, and its successive 
 subdivisions into sixtieths called Minutes^ Seconds, Thirds, etc. The 
 division of the right angle into 90 degrees is convenient because it 
 makes the half-angle of an equilateral triangle (sin"' |) an integral 
 number (30) of degrees. It has long been universally adopted by all 
 Europe. The decimal division of the right angle, decreed by the 
 French Republic when it successfully introduced other more sweeping 
 changes, utterly and deservedly failed. 
 
 The division of the degree into 60 minutes and of the minute into 
 60 seconds is not convenient; and tables of the circular functions for 
 degrees and hundredths of the degree are much to be desired. 
 Meantime, when reckoning to tenths of a degree suffices for the 
 accuracy desired, in any case the ordinary tables suffice, as 6' is y^- of 
 a degree. 
 
 The decimal system is exclusively followed in reckoning by radians. 
 The value of two right angles in this reckoning is 3-14159... , or tt. 
 Thus IT radians is equal to 180". Hence i8o°-=-7ris 57°-29578 ... , 
 or 57" 17' 44'^*8 is equal to one radian. In mathematical analysis, 
 angles are uniformly reckoned in terms of the radian. 
 
 358. The practical standard of time is the Siderial Day, being the 
 period, nearly constant', of the earth's rotation about its axis (§ 237). 
 From it is easily derived the Mean Solar Day, or the mean interval 
 which elapses between successive passages of the sun across the 
 meridian of any place. This is not so nearly as the former, an abso- 
 lute or invariable unit; secular changes in the period of the earth's 
 
 ^ In our first edition of our larger treatise it was stated that Laplace had calculated 
 
 from ancient observations of eclipses that the period of the earth's rotation about 
 
 its axis had not altered by TTn^iJTrTT-irir of itself since 720 B.C. In § 830 it was 
 
 pointed out that this conclusion is overthrown by farther information from 
 
 Physical Astronomy acquired in the interval between the printing of the two 
 
 sections, in virtue of a correction which Adams had made as early as 1863 upon 
 
 Laplace's dynamical investigation of an acceleration of the moon's mean motion, 
 
 produced by the Sun's attraction, showing that only about half of the observed 
 
 acceleration of the moon's mean motion relatively to the angular velocity of the 
 
 earth's rotation was accounted for by this cause. [Quoting from the first edition, 
 
 § 830.] "In 1859 Adams communicated to Delaunay his final result: — that at 
 
 'the end of a century the moon is 5" 7 before the position she would have, 
 
 'relatively to a meridian of the earth, according to the angular velocities of the 
 
 'two motions, at the beginning of the century, and the acceleration of the 
 
 'moon's motion truly calculated from the various disturbing causes then recog- 
 
 'nized. Delaunay soon after verified this result : and about the beginning of 
 
 '1866 suggested that the true explanation may be a retardation of the earth's 
 
 'rotation by tidal friction. Using this hypothesis, and allowing for the conse- 
 
 *quent retardation of the moon's mean motion by tidal reaction (§ 276), Adams, 
 
 'in an estimate which he has communicated to us, founded on the rough as- 
 
 * sumption that the parts of the earth's retardation due to solar and lunar tides 
 
 'are as the squares of the respective tide-generating forces, finds 22* as the 
 
 'error by which the earth would in a century get behind a perfect clock rated 
 
 ' at the beginning of the century. If the retardation of rate giving this integral 
 
 'effect were uniform {§ 32), the earth, as a timekeeper, would be going slower 
 
 'by *22 of a second per year in the middle, or "44 of a second per year at the 
 
 'end, than at the beginning of a century." 
 
MEASURES AND INSTRUMENTS, 125 
 
 revolution round the sun affect it, though very shghtly. It is divided 
 into 24 hours, and the hour, hke the degree, is subdivided into 
 successive sixtieths, called minutes and seconds. The usual sub- 
 division of seconds is decimal. 
 
 It is well to observe that seconds and minutes of time are distin- 
 guished from those of angular measure by notation. Thus we have 
 for time 13'' 43™ 27'-58, but for angular measure 13" 43' 2"]"-^^. 
 
 When long periods of time are to be measured, the mean solar 
 year, consisting of 366-242203 siderial days, or 365-242242 mean 
 solar days, or the century consisting of 100 such years, may be con- 
 veniently employed as the unit. 
 
 359. The ultimate standard of accurate chronometry must (if the 
 human race live on the earth for a few million years) be founded on 
 the physical properties of some body of more constant character 
 than the earth : for instance, a carefully-arranged metallic spring, 
 hermetically sealed in an exhausted glass vessel. The time of vibra- 
 tion of such a spring would be necessarily more constant from day to 
 day than that of the balance-spring of the best possible chronometer, 
 disturbed as this is by the train of mechanism with which it is con- 
 nected: and it would certainly be more constant from age to age 
 than the time of rotation of the earth, retarded as it now is by tidal 
 resistance to an extent that becomes very sensible in 2000 years; 
 and cooling and shrinking to an extent that must produce a very 
 considerable effect on its time-keeping in fifty million years. 
 
 360. The British standard of length is the Imperial Yard, defined 
 as the distance between two marks on a certain metallic bar, pre- 
 served in the Tower of London, when the whole has a temperature of 
 60° Fahrenheit. It was not directly derived from any fixed quantity 
 in nature, although some important relations wdth natural elements have 
 been measured with great accuracy. It has been carefully compared 
 with the length of a second's pendulum vibrating at a certain station in 
 the neighbourhood of London, so that should it again be destroyed, 
 as it was at the burning of the Houses of Parliament in 1834, and 
 should all exact copies of it, of which several are preserved in various 
 places, be also lost, it can be restored by pendulum observations. A 
 less accurate, but still (unless in the event of earthquake disturbance) 
 a very good, means of reproducing it exists in the measured base-lines 
 of the Ordnance Survey, and the thence calculated distances between 
 definite stations in the British Islands, which have been ascertained 
 in terms of it with a degree of accuracy sometimes within an inch 
 per mile, that is to say, within about g^o'^oTT* 
 
 361. In scientific investigations, we endeavour as much as possible 
 to keep to one unit at a time, and the foot, which is defined to be 
 one-third part of the yard, is, for British measurement, generally 
 adopted. Unfortunately the inch, or one-twelfth of a foot, must 
 sometimes be used, but it is subdivided decimally. The statute mile, 
 or 1760 yards, is unfortunately often used when great lengths on land 
 
126 PRELIMINARY. 
 
 are considered; but the sea-mile, or average minute of latitude, is 
 much to be preferred. Thus it appears that the British measurement 
 of length is more inconvenient in its several denominations than the 
 European measurement of time, or angles. 
 
 362. In the French metrical system the decimal division is exclu- 
 sively employed. The standard, (unhappily) called the Metre, was 
 defined originally as the ten-millionth part of the length of the 
 quadrant of the earth's meridian from the pole to the equator ; but it 
 is now defined practically by the accurate standard metres laid up in 
 various national repositories in Europe. It is somewhat longer than 
 the yard, as the following Table shows : 
 
 Centimetre = '3937043 inch. 
 
 Metre = 3*280869 feet. 
 Kilometre = '6213767 British 
 Statute mile. 
 
 Inch =25'39977 millimetres. 
 Foot= 3*047972 decimetres. 
 British Statute mile 
 
 = 1609*329 metres. 
 
 363. The unit of superficial measure is in Britain the square yard, 
 in France the metre carre. Of course we may use square inches, 
 feet, or miles, as also square millimetres, kilometres, etc., or the 
 Hectare = 1 0,000 square mbtres. 
 
 Square inch= 6*451483 square centimetres. 
 
 „ foot= 9*290135 „ decimetres. 
 
 „ yard= 83*61121 „ decimetres. 
 Acre = '4046792 of a hectare. 
 
 Square British Statute mile = 258*9946 hectare. 
 
 Hectare = 2-471093 acres. 
 
 364. Similar remarks apply to the cubic measure in the two 
 countries, and we have the following Table : — 
 
 Cubic inch= 16*38661 cubic centimetres. 
 
 „ foot= 28*31606 „ decimetres or Z//r^j. 
 Gallon = 4-543808 litres. 
 
 „ =277-274 cubic inches, by Act of Parliament, 
 
 now repealed. 
 Litre = '0353 15 cubic feet. 
 
 365. The British unit of mass is the Pound (defined by standards 
 only) ; the French is the Kilogramme, defined originally as a litre of 
 water at its temperature of maximum density; but now practically 
 defined by existing standards. 
 
 Gramme = 15 '43 23 5 grains. 
 Kilogram. = 2*20362125 lbs. 
 
 Grain =64*79896 milligrammes. 
 Pound = 453*5927 grammes. 
 
 Professor W. H. Miller finds {Phil. Trans., 1857) that the 'kilo- 
 gramme des Archives^ is equal in mass to 15432*349 grains: and 
 the * kilogramme type laiton^ deposited in the Ministere de ITnterieure 
 in Paris, as standard for French commerce, is 15432*344 grains. 
 
 366. The measurement of force, whether in terms of the weight 
 of a stated mass in a stated locality, or in terms of the absolute or 
 
MEASURES AND INSTRUMENTS. 127 
 
 kinetic unit, has been explained in Chapter II. (See §§221 — 227.) 
 From the measures of force and length we derive at once the measure 
 of work or mechanical effect. That practically employed by engi- 
 neers is founded on the gravitation measure of force. Neglecting the 
 difference of gravity at London and Paris, we see from the above 
 Tables that the following relations exist between the London and the 
 Parisian reckoning of work : — 
 
 Foot-pound =0-13825 kilogramme-metre. 
 
 Kilogramme-metre =7*2331 foot-pounds. 
 
 367. A Clock is primarily an instrument which, by means of a 
 train of wheels, records the number of vibrations executed by a 
 pendulum ; a Chronometer or Watch performs the same duty for the 
 oscillations of a flat spiral spring — ^just as the train of wheel-work in 
 a gas-meter counts the number of revolutions of the main shaft 
 caused by the passage of the gas through the machine. As, how- 
 ever, it is impossible to avoid friction, resistance of air, etc., a pendu- 
 lum or spring, left to itself, would not long continue its oscillations, 
 and, while its motion continued, would perform each oscillation in 
 less and less time as the arc of vibration diminished : a continuous 
 supply of energy is furnished by the descent of a weight, or the 
 uncoiling of a powerful spring. This is so applied, through the 
 train of wheels, to the pendulum or balance-wheel by means of a 
 mechanical contrivance called an Escapei?ie?it, that the oscillations are 
 maintained of nearly uniform extent, and therefore of nearly uniform 
 duration. The construction of escapements, as well as of trains of 
 clock-wheels, is a matter of Mechanics, with the details of which we 
 are not concerned, although it may easily be made the subject of 
 mathematical investigation. The means of avoiding errors intro- 
 duced by changes of temperature, which have been carried out in 
 Compensation pendulums and balances, will be more properly described 
 in our chapters on Heat. It is to be observed that there is little 
 inconvenience if a clock lose or gain regularly; that can be easily 
 and accurately allowed for : irregular rate is fatal. 
 
 368. By means of a recent application of electricity, to be after- 
 wards described, one good clock, carefully regulated from time to 
 time to agree with astronomical observations, may be made (without 
 injury to its own performance) to control any number of other less- 
 perfectly constructed clocks, so as to compel their pendulums to 
 vibrate, beat for beat, with its own. 
 
 369. In astronomical observations, time is estimated to tenths of 
 a second by a practised observer, who, while watching the phe- 
 nomena, counts the beats of the clock. But for the very accurate 
 measurement of short intervals, many instruments have been devised. 
 Thus if a small orifice be opened in a large and deep vessel full of 
 mercury, and if we know by trial the weight of metal that escapes 
 say in five minutes, a simple proportion gives the interval which 
 elapses during the escape of any given weight. It is easy to con- 
 
1 28 PRELIMINAR V. 
 
 trive an adjustment by which a vessel may be placed under, and 
 withdrawn from, the issuing stream at the time of occurrence of any 
 two successive phenomena. 
 
 370. Other contrivances are sometimes employed, called Stop- 
 watches, Chronoscopes, etc., which can be read off at rest, started 
 on the occurrence of any phenomenon, and stopped at the oc- 
 currence of a second, then again read off; or which allow of the 
 making (by pressing a stud) a slight ink-mark, on a dial revolving 
 at a given rate, at the instant of the occurrence of each phe- 
 nomenon to be noted. But, of late, these have almost entirely given 
 place to the Electric Chronoscope, an instrument which will be fully 
 described later, when we shall have occasion to refer to experiments 
 in which it has been usefully employed. 
 
 371. We now come to the measurement of space, and of angles, 
 and for these purposes the most important instruments are the Vernier 
 and the Screw. 
 
 372. Elementary geometry, indeed, gives us the means of dividing 
 any straight line into any assignable number of equal parts; but in 
 
 practice this is by no means an accurate 
 or reliable method. It was formerly used 
 in the so-called Diagonal Scale, of which 
 the construction is evident from the dia- 
 gram. The reading is effected by a 
 sliding piece whose edge is perpendicular 
 to the length of the scale. Suppose that 
 it is PQ. whose position on the scale is 
 required. This can evidently cut only one 
 of the transverse lines. Its number gives 
 the number of tenths of an inch (4 in the 
 figure), and the horizontal line next above 
 the point of intersection gives evidently 
 the number of hundredths (in the present case 4). Hence the 
 reading is 7*44. As an idea of the comparative uselessness of 
 this method, we may mention that a quadrant of 3 feet radius, 
 which belonged to Napier of Merchiston, and is divided on the 
 limb by this method, reads to minutes of a degree; no higher 
 accuracy than is now attainable by the pocket sextants made by 
 Troughton and Simms, the radius of whose arc is virtually little 
 more than an inch. The latter instrument is read by the help 
 of a Vernier. 
 
 373. The Vernier is commonly employed for such instruments as 
 the Barometer, Sextant, and Cathetometer, while the Screw is applied 
 to the more delicate instruments, such as Astronomical Circles, 
 Micrometers, and the Spherometer. 
 
 374. The vernier consists of a slip of metal which slides 
 along a divided scale, the edges of the two being coincident. 
 Hence, when it is appUed to a divided circle, its edge is circular, 
 
MEASURES AND INSTRUMENTS. 
 
 129 
 
 and it moves about an axis passing through the centre of the 
 divided Hmb. 
 
 In the sketch let 0, 1, 2, ... 10 denote the divisions on the vernier, 
 o, r, 2, etc., any set of consecutive divisions on the limb or scale 
 
 If, when and o com- 
 
 Wv^AA, 
 
 ^AA/^A/1 
 
 30 
 
 29- 
 
 ■vutv^V^ 
 
 ^rO^ 
 
 along whose edge it slides. 
 
 cide, 10 and n coincide also, then 10 divisions of 
 the vernier are equal in length to 11 on the limb; 
 and therefore each division of the vernier is yjths, 
 or ly^Q- of a division on the limb. If, then, the ver- 
 nier be moved till 1 coincides with i, will be y^yth 
 of a division of the limb beyond o; if 2 coincide 
 with 2, will be xV^''^ beyond o; and so on. Hence 
 to read the vernier in any position, note first the 
 division next to o, and behind it on the limb. This 
 is the integral number of divisions to be read. For 
 the fractional part, see which division of the vernier 
 is in a line with one on the limb; if it be the 4th 
 (as in the figure), that indicates an addition to the 
 reading of y^^ths of a division of the limb; and so on. 
 Thus, if the figure represent a barometer scale divided 
 
 in. 
 
 into inches and tenths, the reading is 30-34, the zero 
 
 line of the vernier being adjusted to the level of the mercury. 
 
 375. If the limb of a sextant be divided, as it usually is, to third- 
 parts of a degree, and the vernier be formed by dividing twenty-one 
 of these into twenty equal parts, the instrument can be read to 
 twentieths of divisions on the limb, that is, to minutes of arc. 
 
 If no Hne on the vernier coincide with one on the limb, then since 
 the divisions of the former are the longer there will be one of the 
 latter included between the two lines of the vernier, and it is usual 
 in practice to take the mean of the readings which would be given 
 by a coincidence of either pair of bounding lines. 
 
 376. In the above sketch and description, the numbers on the 
 scale and vernier have been supposed to run opposite ways. This 
 is generally the case with British instruments. In some foreign ones 
 the divisions run in the same direction on vernier and limb, and in 
 that case it is easy to see that to read to tenths of a scale division we 
 must have ten divisions of the vernier equal to 7iiiie of the scale. 
 
 In general to read to the ;/th part of a scale division, n divisions of 
 the vernier must equal n + \ or n- i divisions on the limb, according 
 as these run in opposite or similar directions. 
 
 377. The principle of the Scre7ef has been already noticed (§ 1 14). 
 It may be used in either of two ways, i.e. the nut may be fixed, 
 and the screw advance through it, or the screw may be prevented 
 from moving longitudinally by a fixed collar, in which case the nut, 
 if prevented by fixed guides from rotating, will move in the direction 
 of the common axis. The advance in either case is evidently pro- 
 portional to the angle through which the screw has turned about its . 
 
 T. 
 
I30 PRELIMINARY. 
 
 axis, and this may be measured by means of a divided head fixed 
 perpendicularly to the screw at one end, the divisions being read oif 
 by a pointer or vernier attached to the frame of the instrument. The 
 nut carries with it either a tracing-point (as in the dividing engine) or 
 a wire, thread, or half the object-glass of a telescope (as in micro- 
 meters), the thread or wire, or the play of the tracing-point, being 
 at right angles to the axis of the screw. 
 
 378. Suppose it be required to divide a line into any number 
 of equal parts. The line is placed parallel to the axis of the screw 
 with one end exactly under the tracing-point, or under the fixed wire 
 of a microscope carried by the nut, and the screw-head is read off. By 
 turning the head, the tracing-point or microscope wire is brought to the 
 other extremity of the line; and the number of turns and fractions of 
 a turn required for the whole line is thus ascertained. Dividing this 
 by the number of equal parts required, we find at once the mnnber 
 of turns and fractional parts corresponding to one of the required 
 divisions, and by giving that amount of rotation to the screw over and 
 over again, drawing a Hne after each rotation, the required division is 
 effected. 
 
 379. In the Micrometer^ the movable wire carried by the nut 
 is parallel to a fixed wire. By bringing them into optical contact 
 the zero reading of the head is known; hence when another reading 
 has been obtained, we have by subtraction the number of turns 
 corresponding to the length of the object to be measured. The 
 absolute value of a turn of the screw is determined by calculation 
 from the number of threads in an inch, or by actually applying the 
 micrometer to an object of known dimensions. 
 
 380. For the measurement of the thickness of a plate, or the cur- 
 vature of a lens, the Spheroineter is used. It consists of a screw nut 
 rigidly fixed in the middle of a very rigid three-legged table, with its 
 axis perpendicular to the plane of the three feet (or finely rounded 
 ends of the legs,) and an accurately cut screw working in this nut. 
 The lower extremity of the screw is also finely rounded. The number 
 of turns, whole or fractional, of the screw, is read off by a divided 
 head and a pointer fixed to the stem. Suppose it be required to 
 measure the thickness of a plate of glass. The three feet of the 
 instrument are placed upon a nearly enough flat surface of a hard 
 body, and the screw is gradually turned until its point touches and 
 presses the surface. The muscular sense of touch perceives resistance to 
 the turning of the screw when, after touching the hard body, it presses 
 on it with a force somewhat exceeding the weight of the screw. The 
 first effect of the contact is a diminution of resistance to the turning, 
 due to the weight of the screw coming to be borne on its fine pointed 
 end instead of on the thread of the nut. The sudderi increase of 
 resistance at the instant when the screw commences to bear part of 
 the weight of the nut finds the sense prepared to perceive it with re- 
 markable delicacy on account of its contrast with the immediately 
 preceding diminution of resistance. The screw-head is now read off, 
 
MEASURES AND INSTRUMENTS. 131 
 
 and the screw turned backwards until room is left for the insertion, 
 beneath its point, of the plate whose thickness is to be measured. 
 The screw is again turned until increase of resistance is again per- 
 ceived; and the screw-head is again read off. The difference of the 
 readings of the head is equal to the thickness of the plate, reckoned 
 in the proper unit of the screw and the division of its head. 
 
 381. If the curvature of a lens is to be measured, the instrument 
 is first placed, as before, on a plane surface, and the reading for the 
 contact is taken. The same operation is repeated on the spherical 
 surface. The difference of the screw readings is evidently the 
 greatest thickness of the glass which would be cut off by a plane 
 passing through the three feet. This is sufficient, with the distance 
 between each pair of feet, to enable us to calculate the radius of the 
 spherical surface. 
 
 In fact if a be the distance between each pair of feet, / the length 
 of screw corresponding to the difference of the two readings, R the 
 
 radius of the spherical surface; we have at once 2i? = — .+ /, or, as / 
 
 is generally very small compared with a^ the diameter is, very ap- 
 
 proximately, — ,. 
 3^ 
 
 382. The Cathetometer is used for the accurate determination of 
 differences of level — for instance, in measuring the height to which a 
 fluid rises in a capillary tube above the exterior free surface. It 
 consists of a long divided metallic stem, turning round an axis as 
 nearly as may be parallel to its length, on a fixed tripod stand : and, 
 attached to the stem, a spirit-level. Upon the stem slides a metallic 
 piece bearing a telescope of which the length is approximately enough 
 perpendicular to the axis. The telescope tube is as nearly as may be 
 perpendicular to the length of the stem. By levelling screws in two 
 feet of the tripod the bubble of the spirit-level is brought to one 
 position of its glass when the stem is turned all round its axis. This 
 secures that the axis is vertical. In using the instrument the 
 telescope is directed in succession to the two objects whose difference 
 of level is to be found, and in each case moved (generally by a 
 delicate screw) up or down the stem, until a horizontal wire in the 
 focus of its eye-piece coincides with the image of the object. The 
 difference of readings on the vertical stem (each taken generally by 
 aid of a vernier sliding piece) corresponding to the two positions of 
 the telescope gives the required difference of level. 
 
 383. The principle of the Balance is generally known. We may 
 note here a few of the precautions adopted in the best balances to 
 guard against the various defects to which the instrument is liable ; 
 and the chief points to be attended to in its construction to secure 
 delicacy, and rapidity of weighing. 
 
 The balance-beam should be very stiff, and as light as possible 
 consistently with the requisite stiffness. For this purpose it is 
 
 9—2 
 
132 PRELIMINARY. 
 
 generally formed either of tubes, or of a sort of lattice-fram6work. 
 To avoid friction, the axle consists of a knife-edge, as it is called ; 
 that is, a wedge of hard steel, which, when the balance is in use, 
 rests on horizontal plates of polished agate. A similar contrivance 
 is appHed in very delicate balances at the points of the beam from 
 which the scale-pans are suspended. When not in use, and just 
 before use, the beam with its knife-edge is lifted by a lever arrange- 
 ment from the agate plates. While thus secured it is loaded with 
 weights as nearly as possible equal (this can be attained by previous 
 trial with a coarser instrument), and the accurate determination is 
 then readily effected. The last fraction of the required weight is 
 determined by a rider, a very small weight, generally formed of wire, 
 which can be worked (by a lever) from the outside of the glass case 
 in which the balance is enclosed, and which may be placed in 
 different positions upon one arm of the beam. This arm is gra- 
 duated to tenths, etc., and thus shows at once the value of the rider in 
 any case as depending on its moment or leverage, § 233. 
 
 384. Qualities of a balance : 
 
 1. Stability. — For stability of the beam alone without pans and 
 weights, its centre of gravity must be below its bearing knife-edge. 
 For stability with the heaviest weights the line joining the points at 
 the ends of the beam from which the pans are hung must be below 
 the knife-edge bearing the whole. 
 
 2. Sensibility. — The beam should be sensibly deflected from a 
 horizontal position by the smallest difference between the weights in 
 the scale-pans. The definite measure of the sensibility is the angle 
 through which the beam is deflected by a stated difference between 
 the loads in the pans. 
 
 3. Quickness. — This means rapidity of oscillation, and consequently 
 speed in the performance of a weighing. It depends mainly upon the 
 depth of the centre of gravity of the whole below the knife-edge and 
 the length of the beam. 
 
 In our Chapter on Statics we shall give the investigation. The 
 sensibiHty and quickness are calculated for any given form and 
 dimensions of the instrument, in § 572. 
 
 A fine balance should turn with about a 500,000th of the greatest 
 load which can safely be placed in either pan. 
 
 The process of Double Weighings which consists in counterpoising a 
 mass by shot, or sand, or pieces of fine wire, and then substituting 
 weights for it in the same pan till equilibrium is attained, is more 
 laborious, but more accurate, than single weighing ; as it eliminates 
 all errors arising from unequal length of the arms, etc. 
 
 Correction is required for the weights of air displaced by the two 
 bodies weighed against one another when their difference is too large 
 to be negligable. 
 
 385. In the Torsion-balance invented, and used with great effect, 
 by Coulomb, a force is measured by the torsion of a fibre of silk, a 
 glass thread, or a metallic wire. The fibre- or wire is fixed at its 
 
MEASURES AND INSTR UMENTS. 1 3 3 
 
 upper end, or at both ends, according to circumstances. In general 
 it carries a very light horizontal rod or needle, to the extremities of 
 which are attached the body on which is exerted the force to be 
 measured, and a counterpoise. The upper extremity of the torsion 
 fibre is fixed to an index passing through the centre of a divided 
 disc, so that the angle through which that extremity moves is directly 
 measured. If, at the same time, the angle through which the needle 
 has turned be measured, or, more simply, if the index be always 
 turned till the needle assumes a different position determined by 
 marks or sights attached to the case of the instrument — we have the 
 amount of torsion of the fibre, and it becomes a simple statical pro- 
 blem to determine from the latter the force to be measured ; its direc- 
 tion, and point of application, and the dimensions of the apparatus, 
 being known. The force of torsion as depending on the angle of 
 torsion was found by Coulomb to follow the law of simple proportion 
 up to the limits of perfect elasticity — as might have been expected 
 from Hooke's Law (see Properties of Matter), and it only remains 
 that we determine the amount for a particular angle in absolute 
 measure. This determination is, in general, simple enough in theory; 
 but in practice requires considerable care and nicety. The torsion- 
 balance, however, being chiefly used for comparative, not absolute, 
 measure, this determination is often unnecessary. More will be said 
 about it when we come to its application. 
 
 386. The ordinary spiral spring-balances used for roughly com- 
 paring either small or large weights or forces, are, properly speaking, 
 only a modified form of torsion-balance S as they act almost entirely 
 by the torsion of the wire, and not by longitudinal extension or by 
 flexure. Spring-balances we believe to be capable, if carefully con- 
 structed, of rivalling the ordinary balance in accuracy, while, for some 
 applications, they far surpass it in sensibility and convenience. They 
 measure directly force, not mass; and therefore if used for deter- 
 mining masses in diff"erent parts of the earth, a correction must be 
 applied for the varying force of gravity. The correction for tem- 
 perature must not be overlooked. These corrections may be avoided 
 by the method of double weighing. 
 
 387. Perhaps the most delicate of all instruments for the measure- 
 ment of force is the Pendulum. It is proved in Kinetics (see Div. II.) 
 that for any pendulum, whether oscillating about a mean vertical 
 position under the action of gravity, or in a horizontal plane, under 
 the action of magnetic force, or force of torsion, the square of the 
 number of small oscillations in a given time is proportional to the 
 magnitude of the force under which these oscillations take place. 
 
 For the estimation of the relative amounts of gravity at different 
 places, this is by far the most perfect instrument. The method of 
 coincidences by which this process has been rendered so excessively 
 delicate will be described later. 
 
 ^ Binet. See also J. Thomson. Cambridge and Dublin Math. Journal, 1848. 
 
1 34 PRELIMINAR Y. 
 
 In fact, the kinetic measure of force, as it is the first and 
 
 most truly elementary, is also far the most easy as well as perfect 
 
 method in many practical cases. It admits of an easy reduction to 
 gravitation measure. 
 
 388. Weber and Gauss, in constructing apparatus for observations 
 of terrestrial magnetism, endeavoured so to modify them as to admit 
 of their being read from some distance. For this purpose each bar, 
 made at that time too ponderous, carried a plain mirror. By means 
 of a scale, seen after reflection in the mirror and carefully read with 
 a telescope, it was of course easy to compute the deviations which the 
 mirror had experienced. But, for many reasons, it was deemed neces- 
 sary that the deflections, even under considerable force, should be 
 very small. With this view the Bifilar suspension was introduced. 
 The bar-magnet is suspended horizontally by two vertical wires or 
 fibres of equal length so adjusted as to share its weight equally 
 between them. When the bar turns, the suspension-fibres become 
 inclined to the vertical, and therefore the bar must rise. Hence, if 
 we neglect the torsion of the fibres, the bifilar actually measures a 
 force by comparing it with the weight of the suspended magnet. 
 
 Let a be the half length of the bar between the points of attach- 
 ment of the wires, ^ the angle through which the bar has been turned 
 (in a horizontal plane) from its position of equilibrium, / the length 
 of one of the wires. 
 
 Then if Q be the couple tending to turn the bar, and W its weight, 
 
 , ^ Wa^ sin^ 
 we have O = — — . , 
 
 2 
 
 which gives the couple in terms of the deflection 0. 
 
 If the torsion of the fibres be taken into account, it will be 
 sensibly equal to Q (since the greatest inclination to the vertical 
 is small), and therefore the couple resulting from it will be EO, 
 where E is some constant. This must be added to the value of Q, 
 just found in order to get the whole deflecting couple. 
 
 389. Ergometers are instruments for measuring energy. White's 
 friction brake measures the amount of work actually performed in 
 
 any time by an engine or other ' prime mover,' by allowing it during 
 the time of trial to waste all its work on friction. Morin!s ergometer 
 measures work without wasting any of it, in the course of its trans- 
 mission from the prime mover to machines in which it is usefully 
 employed. It consists of a simple arrangement of springs, measur- 
 ing at every instant the couple with which the prime mover turns the 
 shaft that transmits its work, and an integrating machine from which 
 the work done by this couple during any time can be read off. 
 
 390. White's friction brake consists of a lever clamped to the 
 shaft, but not allowed to turn with it. The moment of the force 
 required td prevent the lever from going round with the shaft, 
 
 sj ^--j. sin- 
 
MEASURES AND INSTRUMENTS. isS 
 
 multiplied by the whole angle through which the shaft turns, measures 
 the whole work done against the friction of the clamp. The same 
 result is much more easily obtained by wrapping a rope or chain 
 several times round the shaft, or round a cylinder or drum carried 
 round by the shaft, and applying measured forces to its two ends 
 in proper directions to keep it nearly steady while the shaft turns 
 round without it. The difference of the moments of these two forces 
 round the axis, multiplied by the angle through which the shaft turns, 
 measures the whole work spent on friction against the rope. If we 
 remove all other resistance to the shaft, and apply the proper amount 
 of force at each end of the rope or chain (which is very easily done 
 in practice), the prime mover is kept running at the proper speed 
 for the test, and having its whole work thus wasted for the time and 
 measured. 
 
DIVISION II. 
 
 ABSTRACT DYNAMICS. 
 
 CHAPTER v.— INTRODUCTORY. 
 
 391. Until we know thoroughly the nature of matter and the 
 forces which produce its motions, it will be utterly impossible to 
 submit to mathematical reasoning the exact conditions of any phy- 
 sical question. It has been long understood, however, that an ap- 
 proximate solution of almost any problem in the ordinary branches 
 of Natural Philosophy may be easily obtained by a species of ab- 
 straction^ or rather limitation of the data, such as enables us easily 
 to solve the modified form of the question, while we are well assured 
 that the circumstances (so modified) affect the result only in a super- 
 ficial manner. 
 
 392. Take, for instance, the very simple case of a crowbar em- 
 ployed to move a heavy mass. The accurate mathematical investi- 
 gation of the action would involve the simultaneous treatment of the 
 motions of every part of bar, fulcrum, and mass raised; and from our 
 almost complete ignorance of the nature of matter and molecular 
 forces, it is clear that such a treatment of the problem is impossible. 
 
 It is a result of observation that the particles of the bar, fulcrum, 
 and mass, separately, retain throughout the process nearly the same 
 relative positions. Hence the idea of solving, instead of the above 
 impossible question, another, in reality quite different, but, while 
 infinitely simpler, obviously leading to 7iearly the same results as the 
 former. 
 
 393. The new form is given at once by the experimental result 
 of the trial. Imagine the masses involved to be perfectly rigid (i.e. 
 incapable of changing their forms or dimensions), and the infinite 
 multiplicity of the forces, really acting, may be left out of consi- 
 deration ; so that the mathematical investigation deals with a finite 
 (and generally small) number of forces instead of a practically infinite 
 number. Our warrant for such a substitution is established thus. 
 
ABSTRACT DYNAMICS. 137 
 
 394. The only effects of the intermolecular forces would be ex- 
 hibited in molecular alterations of the form or volume of the masses 
 involved. But as these (practically) remain almost unchanged, the 
 forces which produce, or tend to produce, changes in them may be 
 left out of consideration. Thus we are enabled to investigate the 
 action of machinery by supposing it to consist of separate portions 
 whose forms and dimensions are unalterable. 
 
 395. If we go a little farther into the question, we find that the 
 lever bends ^ some parts of it are extended and others compressed. 
 This would lead us into a very serious and difficult inquiry if we had 
 to take account of the whole circumstances. But (by experience) we 
 find that a sufficiently accurate solution of this more formidable case 
 of the problem may be obtained by supposing (what can fiever be 
 realized in practice) the mass to be homogeneous, and the forces 
 consequent on a dilatation, compression, or distortion, to be propor- 
 tional in magnitude, and opposed in direction, to these deformations 
 respectively. By this farther assumption, close approximations may 
 be made to the vibrations of rods, plates, etc., as well as to the statical 
 effects of springs, etc. 
 
 396. We may pursue the process farther. Compression, in general, 
 develops heat, and extension, cold. These alter sensibly the elas- 
 ticity of a body. By introducing such considerations, we reach, 
 without great difficulty, what may be called a third approximation 
 to the solution of the physical problem considered. 
 
 397. We might next introduce the conduction of the heat, so 
 produced, from point to point of the solid, with its accompanying 
 modifications of elasticity, and so on ; and we might then consider 
 the production of thermo-electric currents, which (as we shall see) 
 are always developed by unequal heating in a mass if it be not per- 
 fectly homogeneous. Enough, however, has been said to show,yf/'j/, 
 our utter ignorance as to the true and complete solution of any 
 physical question by the only perfect method, that of the consideration 
 of the circumstances which affect the motion of every portion, sepa- 
 rately, of each body concerned ; and, second, the practically sufficient 
 manner in which practical questions may be attacked by limiting their 
 generality, the li?7titations introduced being themselves deduced from ex- 
 perience, and being therefore Nature's own solution (to a less or 
 greater degree of accuracy) of the infinite additional number of 
 equations by which we should otherwise have been encumbered. 
 
 398. To take another case : in the consideration of the propa- 
 gation of waves on the surface of a fluid, it is impossible, not only 
 on account of mathematical difficulties, but on account of our igno- 
 rance of what matter is, and what forces its particles exert on each 
 other, to form the equations which would give us the separate motion 
 of each. Our first approximation to a solution, and one sufficient 
 for most practical purposes, is derived from the consideration of the 
 
138 INTRODUCTORY, 
 
 motion of a homogeneous, incompressible, and perfectly plastic mass; 
 a hypothetical substance which, of course, nowhere exists in nature. 
 
 399. Looking a little more closely, we find that the actual motion 
 differs considerably from that given by the analytical solution of the 
 restricted problem, and we introduce farther considerations, such as 
 the comprssibility of fluids, their internal friction, the heat generated 
 by the latter, and its effects in dilating the mass, etc. etc. By such 
 successive corrections we attain, at length, to a mathematical result 
 which (at all events in the present state of experimental science) 
 agrees, within the limits of experimental error, with observation. 
 
 400. It would be easy to give many more instances substantiating 
 what has just been advanced, but it seems scarcely necessary to do 
 so. We may therefore at once say that there is no question in 
 physical science which can be completely and accurately investigated 
 by mathematical reasoning (in which, be it carefully remembered, 
 it is not necessary that symbols should be introduced), but that there 
 are different degrees of approximation, involving assumptions more 
 and more nearly coincident with observation, which may be arrived 
 at in the solution of any particular question. 
 
 401. The object of the present division of this work is to deal with the 
 first and secojid of these approxi^nations. In it we shall suppose all 
 
 solids either rigid, i.e. unchangeable in form and volume, or elastic; 
 but in the latter case, we shall assume the law, connecting a com- 
 pression or a distortion with the force which causes it, to have a 
 particular form deduced from experiment. And we shall also leave 
 out of consideration the thermal or electric effects which compression 
 or distortion generally produce. We shall also suppose fluids, whether 
 liquids or gases, to be either incompressible or compressible ac- 
 cording to certain known laws; and we shall omit considerations 
 of fluid friction, although we admit the consideration of friction 
 between solids. Fluids will therefore be supposed perfect^ i.e. such 
 that any particle may be moved amongst the others by the slightest 
 force. 
 
 402. When we come to Properties of Matter and the Physical 
 Forces, we shall give in detail, as far as they are yet known, the 
 modifications which farther approximations have introduced into the 
 previous results. 
 
 403. The laws of friction between solids were very ably investi- 
 gated by Coulomb; and, as we shall require them in the succeeding 
 chapters, we give a brief summary of them here ; reserving the more 
 careful scrutiny of experimental results to our chapter on Properties 
 of Matter. 
 
 404. To produce sliding of one solid body on another, the sur- 
 faces in contact being plane, requires a tangential force which 
 depends, — (i) upon the nature of the bodies; (2) upon their polish, 
 or the species and quantity of lubricant which may have been applied; 
 
ABSTRACT DYNAMICS. 139 
 
 (3) upon the normal pressure between them, to which it is in general 
 directly proportional; (4) upon the length of time during which they 
 have been suffered to remain in contact. 
 
 It does not (except in extreme cases where scratching or abrasion 
 takes place) depend sensibly upon the area of the surfaces in contact. 
 This, which is called Statical Friction, is thus capable of opposing a 
 tangential resistance to motion which may be of any requisite amount 
 up to ^R\ where R is the whole normal pressure between the bodies; 
 and /x (which depends mainly upon the nature of the surfaces in 
 contact) is the co-efficient of Statical Friction. This co-efficient varies 
 greatly with the circumstances, being in some cases as low as ©'03, in 
 others as high as o'8o. Later we shall give a table of its values. 
 Where the applied forces are insufficient to produce motion, the 
 whole amount of statical friction is not called into play; its amount 
 then just reaches what is sufficient to equiUbrate the other forces, and 
 its direction is the opposite of that in which their resultant tends to 
 produce motion. When the statical friction has been overcome, and 
 sliding is produced, experiment shows that a force of friction con- 
 tinues to act, opposing the motion, sensibly proportional to the 
 normal pressure, and independent of the velocity. But for the same 
 two bodies the co-efficient of Kinetic Friction is less than that of Sta- 
 tical Friction, and is approximately the same whatever be the rate of 
 motion. 
 
 405. When among the forces acting in any case of equilibrium, 
 there are frictions of solids on solids, the circumstances would not 
 be altered by doing away with all friction, and replacing its forces by 
 forces of mutual action supposed to remain unchanged by any in- 
 finitely small relative motions of the parts between which they act. 
 By this artifice all such cases may be brought under the general 
 principle of Lagrange (§ 254). 
 
 406. In the following chapters on Abstract Dynamics we will 
 confine ourselves chiefly to such portions of this extensive subject 
 as are Hkely to be useful to us in the rest of the work. 
 
CHAPTER VI. 
 
 STATICS OF A PARTICLE.— ATTRACTION. 
 
 407. We naturally divide Statics into two parts — the equilibrium 
 of a Particle^ and that of a rigid or elastic Body or System of Fariicles 
 whether solid or fluid. The second law of motion suffices for one 
 part — for the other, the third, and its consequences pointed out by 
 Newton, are necessary. In the succeeding sections we shall dispose 
 of the first of these parts, and the rest of this chapter will be devoted 
 to a digression on the important subject of Attraction. 
 
 408. By § 2 21, forces acting at the same point, or on the same 
 material particle, are to be compounded by the same laws as velo- 
 cities. Therefore the sum of their resolved parts in any direction 
 must vanish if there is equilibrium j whence the necessary and sufii- 
 cient conditions. 
 
 They follow also directly from Newton's statement with regard to 
 work, if we suppose the particle to have any velocity, constant in 
 direction and magnitude (and § 211, this is the most general sup- 
 position we can make, since absolute rest has for us no meaning). 
 For the work done in any time is the product of the displacement 
 during that time into the algebraic sum of the effective components 
 of the applied forces, and there is no change of kinetic energy. 
 Hence this sum must vanish for every direction. Practically, as any 
 displacement may be resolved into three, in any three directions not 
 coplanar, the vanishing of the work for any one such set of three 
 suffices for the criterion. But, in general, it is convenient to assume 
 them in directions at right angles to each other. 
 
 Hence, for the equilibrium of a material particle, it is necessary^ and 
 sufficient^ that the (algebraic) sums of the applied forces, resolved in 
 any one set of three rectangular directions, should vanish. 
 
 409. We proceed to give a detailed exposition of the results 
 which follow from the first clause of § 408. For three forces only we 
 have the following statement. 
 
 The resultant of two forces, acting on a material point, is repre- 
 
STATICS OF A PARTICLE. -^ATTRACT! ON, 141 
 
 sented in direction and magnitude by the diagonal, through that 
 point, of the parallelogram described upon lines representing the 
 forces. 
 
 410. Parallelogram of forces stated symmetrically as to the three 
 forces concerned^ usually called the Triangle of Forces. If the lines 
 
 representing three forces acting on a material point be equal and 
 parallel to the sides of a triangle, and in directions similar to those 
 of the three sides when taken in order round the triangle, the three 
 forces are in equilibrium. 
 
 Let GEF be a triangle, and 
 let MA, MB, MC, be respectively 
 equal and parallel to the three 
 sides EF, FG, GE of this trian- 
 gle, and in directions similar to 
 the consecutive directions of 
 these sides in order. The point 
 Mis in equilibrium. 
 
 411. [True Triangle of Forces. Let three 
 tive directions round a triangle, DEE, and be 
 represented respectively by its sides : they are 
 not in equilibrium, but are equivalent to a 
 couple. To prove this, through D draw DH, 
 equal and parallel to EF, and in it introduce 
 a pair of balancing forces, each equal to EF. 
 Of the five forces, three, DE, DH and ED, 
 are in equilibrium, and may be removed ; 
 and there are then left two forces, EF and HD, equal, parallel, and 
 in dissimilar directions, which constitute a couple.] 
 
 412. To find the resultant of any number of forces in lines through 
 one point, not necessarily in one plane — 
 
 forces act in consecu- 
 
 Let MA^, MA^,MA. 
 
 MA. 
 
 ^ repre- 
 sent four forces acting on M, in one 
 plane; required their resultant. 
 
 Find by the parallelogram of forces, 
 the resultant of two of the forces, MA^ 
 and MA^. It will be represented by 
 MD'. Then similarly, find MD", the 
 resultant of MD' (the first subsidiary 
 resultant), and MA^, the third force. 
 Lastly, find MD'", the resultant of 
 MD" and MA^, MD" represents the 
 resultant of the given forces. 
 
 Thus, by successive applications of the fundamental proposition, 
 the resultant of any number of forces in lines through one point can 
 be found. 
 
 413. In executing this construction, it is not necessary to describe 
 
142 
 
 ABSTRACT DYNAMICS. 
 
 the successive parallelograms, or even to draw their diagonals. It is 
 
 enough to draw through the given point 
 a line equal and parallel to the repre- 
 sentative of any one of the forces ; 
 through the point thus arrived at, to 
 draw a line equal and parallel to the 
 representative of another of the forces, 
 and so on till all the forces have been 
 taken into account. In this way we get 
 such a diagram as the annexed. 
 
 The several given forces may be taken 
 in any order, in the construction just 
 described. The resultant arrived at is 
 necessarily the same, whatever be the order in which we choose to 
 take them, as we may easily verify by elementary geometry. 
 
 In the fig. the order is MA^, ^^5> 
 MA^, MA^, MA^. 
 
 414. If, by drawing lines equal and 
 parallel to the representatives of the forces, 
 a closed figure is got, that is, if the line 
 last drawn leads us back to the point 
 from which we started, the forces are in 
 equilibrium. If, on the other hand, the 
 figure is not closed (§ 413), the resultant 
 is obtained by drawing a line from the 
 starting-point to the point finally reached; 
 (from M to T>) : and a force represented by DAf will equilibrate the 
 system. 
 
 415. Hence, in general, a set of forces represented by lines equal 
 and parallel to the sides of a complete polygon, are in equilibrium, 
 provided they act in lines through one point, in directions similar to 
 the directions followed in going round the polygon in one way. 
 
 416. Polygon of Forces. The construction we have just con- 
 sidered, is sometimes called the polygon of forces; but the true 
 polygon of forces, as we shall call it, is something quite different. 
 In it the forces are actually along the sides of a polygon, and repre- 
 sented by them in magnitude. Such a system must clearly have a 
 turning tendency, and it may be demonstrated to be reducible to one 
 couple. 
 
 417. In the preceding sections we have explained the principle 
 involved in finding the resultant of any number of forces. We have 
 now to exhibit a method, more easy than the parallelogram of forces 
 affords, for working it out in actual cases, and especially for obtaining 
 a convenient specification of the resultant. The instrument employed 
 for this purpose is Trigonometry. 
 
 418. A distinction may first be pointed out between two classes 
 of problems, direct and inverse. Direct problems are those in which 
 the resultant of forces is to be found ; inverse, those in which com- 
 
STATICS OF A FARIYCLE.-^ATTR ACTION. 143 
 
 ponents of a force are to be found. The former class is fixed and 
 determinate ; the latter is quite indefinite, without limitations to be 
 stated for each problem. A system of forces can produce only one 
 effect; but an infinite number of systems can be obtained, which 
 shall produce the same effect as one force. The problem, therefore, 
 of finding components must be, in some way or other, limited, 'lliis 
 may be done by giving the lines along which the components are to 
 act. To find the components of a given force, in any three given 
 directions, is, in general, as we shall see, a perfectly determinate 
 problem. 
 
 Finding resultants is called Composition of Forces. 
 
 Finding components is called Resolution of Forces. 
 
 419. Co7?tpositio7i of Forces. 
 
 Required in position and magnitude the resultant of two given 
 forces acting in giving lines on a material point. 
 
 Let MA, MB represent two forces, 
 F and Q, acting on a material point M. 
 Let the angle BMA be denoted by i. 
 Required the magnitude of the resultant, 
 and its inclination to the line of either 
 force. M P 
 
 Let F denote the magnitude of the resultant; let a denote the 
 angle FMA, at which its line MD is inclined to MA, the line of the 
 first force F; and let ^ denote the angle DMB, at which it is inclined 
 to MB, the direction of the force (9. 
 
 Given F, Q, and t : required F, and a or p. 
 We have 
 
 MB' = MA"" + MB"" - 2MA.MB X cos MAD. 
 
 Hence, according to our present notation, 
 
 R'=.F' + Q'- 2FQ cos (180"- 0, 
 or F^ = F^+Q^ + 2FQco?>i. 
 
 Hence F ={F''+ Q' ^- 2FQC0?, i)K (i) 
 
 To determine a and ^ after the resultant has been found ; we have 
 
 sin DMA = -rp^ sin MAD, 
 MD ' 
 
 or sma = ^smi, (2) 
 
 and similarly, 
 
 p 
 sin/3=-^sint. (s) 
 
 420. These formulae are useful for many applications ; but they 
 have the inconvenience that there may be ambiguity as to the angle, 
 whether it is to be acute or obtuse, which is to be taken when either 
 sin a or sin /8 has been calculated. If t is acute, both a and /? are 
 acute, and there is no ambiguity. If i is obtuse, one of the two 
 

 2MD . MA 
 
 or 
 
 ^^^'^- 2RF^ 
 
 and similarly, 
 
 
 144 ABSTRACT DYNAMICS. 
 
 angles, a, yS, might be either acute or obtuse ; but as they cannot be 
 both obtuse, the smaller of the two must, necessarily, be acute. If, 
 therefore, we take the formula for sin a, or for sin )8, according as the 
 force P, or the force Q, is the greater, we do away with all ambiguity, 
 and have merely to take the value of the angle shown in the table of 
 sines. And by subtracting the value thus found, from the given 
 value of t, we find the value, whether acute or obtuse, of the other of 
 the two angles, a, ^. 
 
 421. To determine a and /8 otherwise. After the magnitude of the 
 resultant has been found, we know the three sides, ]\IA^ AD^ MD^ 
 of the triangle DMA, then we have 
 
 ^,,, MD' + MA'-AD' 
 
 (4) 
 
 (5) 
 
 by successive applications of the elementary trigonometrical formula 
 used above for finding MD. Again, using this last-mentioned for- 
 mula for AID' or R^ in the numerators of (4) and (5), and reducing, 
 
 we have cosa = ^ -, (o) 
 
 r, + RcosL . . 
 
 co5/3 = ^^-^ ; (7) 
 
 formulae which are convenient in many cases. There is no am- 
 biguity in the determination of either a or ;8 by any of the four 
 equations (4), (5), (6), (7). 
 
 Remark. — Either sign (4- or -) might be given to the radical 
 in (i), and the true line of action and the direction of the force in it 
 would be determined without ambiguity by substituting in (2) and 
 (3) the value of R with either sign prefixed. Since, however, there 
 can be no doubt as to the direction of the force indicated, it will be 
 generally convenient to give the positive sign to the value of R. But 
 in special cases, the negative sign, which with the proper interpre- 
 tation of the formulae will lead to the same result as the positive, 
 will be employed. 
 
 422. Another method of treating the general problem, which is 
 useful in many cases, is this : 
 Let 
 
 which implies that 
 
 P=F^G, 
 Q^F^G. 
 
STATICS OF A PARTICLE. -^ATTRACTION. 145 
 
 ^and G will be both positive if F> Q. Hence, instead of the two 
 given forces, F and Q. we may suppose that we have on the point M 
 four forces; — two, each equal to F, acting in 
 the same directions, MK, ML, as the given 
 forces, and two others, each equal to G, of 
 which one acts in the same direction, MK, as 
 
 F, and the other in ML\ the direction opposite 
 to Q. Now the resultant of the two equal 
 forces, F, bisects the angle between them, 
 KML) and by the investigation of § 423 
 below, its magnitude is found to be 2FC0S J t. 
 Again, the resultant of the two equal forces, 
 
 G, is similarly seen to bisect the angle, KMF , 
 between the line of the given force, F, and 
 the continuation through M of the line of the given force, Q; and to 
 be equal to 2 6^ sin \ i, since the angle KLM' is the supplement of i. 
 Thus, instead of the two given forces in lines inclined to one another 
 at the angle t, which may be either an acute, an obtuse, or a right 
 angle, we have two forces, 2^cos \ i and 2 6^ sin J t, acting in lines, 
 MS, MT, which bisect the angles LMK and KML!, and therefore 
 are at right angles to one another. Now, according to § 429 below, 
 we find the resultant ^ of these two forces by means of the following 
 formulae : — 
 
 tan SMD = —= ^ , 
 
 2I1 cos \ I 
 
 and R = 2jFcos \ i sec SMD, 
 
 F— 
 
 or tan(it-a)=-^-j-->tanJt, <8) 
 
 and R = {F^ Q)co?>\i sec (J t - a) 
 
 = (/'+0cosi(a + ^)cosi(a-^). (9) 
 
 These formulae might have been derived from the standard formulae 
 for the solution of a plane triangle when two sides {P and 0, and 
 the contained angle (tt - i) are given. 
 
 423. We shall now investigate some cases of the general formulae. 
 Case I. Let the forces be equal, that is, let Q. = F in the preceding 
 formulae. 
 
 Then, by (i), R^ ^ 2F' + 2F' cos i=2F'{i + cos i) 
 
 = 4/''cos'Jt. 
 
 Hence i?=2/'cosJt, 
 
 an important result which might, of course, have been obtained 
 directly from the proper geometrical construction in this case. Also 
 
 by (2), 
 
 1 In the diagram the direction of the balancing force is shown by the arrow- 
 head in the line DM. 
 
 T. 10 
 
146 ABSTRACT DYNAMICS. 
 
 ^ sin I C^ sin t . , 
 
 sin a = — -— = -^ — = sin \ t, 
 
 which agrees with what we see intuitively, that a = /5 = J i. 
 
 424. Case II. Let F= Q ; and let t = 120^ Then 
 
 cos J t = cos 60° = I, and (§ 423) i? = P. 
 The resultant, therefore, of two equal forces inclined at an angle 
 of 120" is equal to each of them. This result is interesting, because 
 it can be obtained very simply, and quite independently of this 
 investigation. A consideration of the symmetry of the circumstances 
 will show that if three equal forces in one plane be applied to a 
 material point in lines dividing the space around it into three equal 
 angles, they must be in equilibrium; which is perfectly equivalent to 
 the preceding conclusion. 
 
 425. Case III. Let t = 0% cos t = i ; 
 then R = {I^+Q' + 2FQf, 
 
 R = F-^Q. 
 
 426. Case IV. Let t = 180"; cos t = - i ; 
 then R = {F' + Q'-2FQ)K 
 
 F = F-Q. 
 This is also one of the cases in which it is convenient to give some- 
 times the negative sign, sometimes the positive to the expression for 
 the resultant force : for if Q be greater than F, the preceding expression 
 will be negative, and the interpretation will be found by considering 
 that the force which vanishes when F= Q, is in the direction of F 
 when F is the greater, and in the contrary direction, or in that of Qy 
 when F is the less of the two forces. 
 
 427. Case F. Forces nearly conspiring. Let the angle t be very 
 small, then sintwt;^ cosi«i. 
 
 The general expressions {§419) therefore become, 
 R^F+Q, 
 Qi 
 
 sm^«^p^^. 
 
 a«:- 
 
 To the same degree of approximation 
 
 p^ 
 
 Hence „ + ^«^. «i^^ =.. 
 
 * The sign «s is used to denote approximate equality. 
 
 (.0) 
 
STATICS OF A PARTICLE.— ATTRACTION. 147 
 
 This shows that the errors in the values of a and P obtained ap- 
 proximately by this method compensate; one being as much above, 
 as the other is below, the true value. 
 
 We therefore conclude that the resultant of two forces very nearly 
 conspiring is approximately equal to their sum, and approximately 
 divides the angle between them into parts inversely as the forces. 
 
 When the angle between the forces is infinitely small, they may 
 either conspire in acting on one point in one line ; or they may act 
 on different points in parallel Hues. In either case the resultant is 
 precisely equal to their sum. Actually conspiring forces we have 
 already considered ; parallel forces we shall consider more particularly 
 when we treat of the equilibrium of a rigid body. We may briefly 
 examine the case here however. Suppose the actual points of appli- 
 cation of the forces to be ^ . . 
 and B, but let their lines ^^__^_,-^ 
 
 meet in a point M; join ^ " ^ 
 
 AB, and let MAB be an M ■''^^s^i:!!^:::^^^^^-^^ 
 
 isosceles triangle. Let this 
 point M be removed grad- 
 ually to an infinite distance in the direction of a perpendicular, OMy 
 bisecting the line AB. The resultant will still divide the angle in- 
 versely as the forces : and as the circular measure of the angle is any 
 arc described from M as centre divided by the radius, every such arc 
 will be divided in the same proportion. Now, if M be infinitely 
 distant, that is if the lines of the forces be parallel, the arc will become 
 a straight line, and will be divided into parts inversely as the forces. 
 
 In actual cases of forces acting on a point, and very nearly con- 
 spiring, the following approximate equations show how nearly the 
 resultant approaches the sum of the forces : — 
 
 smO^O] cos6'«i-i^'. 
 
 BT 
 
 
 Ji^(P^Q)-l^Qc\ (12) 
 
 that is, the resultant of two forces very nearly conspiring falls short of 
 their sum by the square of the angle between them multiplied into a 
 quarter of their harmonic mean^ 
 
 428. Case VI. Forces nearly opposed. 
 
 1*'. Let the angle t be very obtuse, and the two forces exactly equal. 
 
 1 The Harmonic Mean of two numbers is the reciprocal of the mean of their 
 
 2PQ 
 reciprocals. Thus the harmonic mean of P and Q is „ „ . 
 
 10 — 2 
 
148 ABSTRACT DYNAMICS. 
 
 Let t = TT - ^, where 6 is very small, 
 
 then J ' = i 7^ ~ i ^> 
 
 cos J t = sin I ^, 
 
 7?=2/'sin|^, 
 
 and since the sine of a very small angle is equal to the angle, in 
 circular measure R-^PB. 
 
 Hence the resultant of two equal very nearly opposed forces is 
 proportional to the defalcation from direct opposition: being ap- 
 proximately equal to either of the forces multiplied into the supple- 
 ment of the angle between them. 
 
 2°. If the forces are neither equal nor nearly equal, the resultant 
 will be approximately equal to their difference. 
 
 We have as before, 
 
 cos t«- I, 
 
 R^^F'+Q'-2FQ. 
 
 Therefore R^P-Q, 
 
 The ambiguity as to whether the acute angle, shown in the table, 
 or its supplement, is to be chosen in either case, may be removed by 
 considering which of the two forces is the greater. 
 
 Thus, as we suppose P to be greater than Q^ a is acute, and there- 
 
 fore sma«aw-!^:^ — -— ^ 
 
 and p is obtuse. 
 
 Therefore ^8 « tt - ^j^'J^ 
 
 r. Pl-Qit 
 P^-PZ^' 
 
 We find, by addition, 
 
 a + /3 = -^--^t = i, 
 
 and conclude, as in the former case, that the errors in the approximate 
 values of a and p compensate, one being as much above, as the other 
 is below, the true value. 
 
 It is only when R is comparable in magnitude with P and Q, that 
 the foregoing solution is applicable. 
 
 But if P exceeds Q, or if Q exceeds P, by any difference which is 
 considerable in comparison with either, the formulae hold. 
 
 Let us suppose now that, while P remains of any constant mag- 
 nitude, Q is made to increase from nothing, gradually, until it becomes 
 
STATICS OF A PARTICLE.— ATTRACTION. 149 
 
 first equal to, and then greater than P, the angle t remaining constant. 
 The angle a will increase very slowly, according to the approximate 
 formula (10), until Q becomes nearly equal to P. Then as the value 
 of Q is increased until it becomes greater than P, the value of a will 
 •increase very rapidly through nearly two right angles, until it falls 
 but little short of t, when its supplement will be approximately ex- 
 pressed by the formula (10). 
 
 In this transition, from Q<P to Q>P, the direction and magni- 
 tude of the resultant are most conveniently found by means of (§ 422) 
 the last of the three general methods given above for determining the 
 resultant of two forces. 
 
 Thus, instead of the two given forces we may substitute two 
 forces in lines bisecting respectively the obtuse angle LMK, or t, 
 and the acute angle KML! and of magnitudes which approximate 
 to \(^P-\-Q) (tt-i), and P-Q, respectively, when t is nearly two 
 right angles. 
 
 We infer, finally, that, however nearly P and Q are equal to one 
 another, the approximate formulae of § 428, 2° hold, provided only 
 \ {P+ Q){tt- i) is a small fraction oi P~ Q. 
 
 429. Case VII. Let t = 90"; cos 1 = 0, sin t = 1 ; 
 
 then R = (P'+Q')K (13) 
 
 and sin a = ^ 
 
 J I. 04) 
 
 sm/3 = -^ 
 
 In this case, p being the complement of a, sin p = cos a. 
 Hence cos « = "d • 
 
 _ , . sin a 
 
 Lastly, since tan a = , 
 
 •' cosa 
 
 we deduce tan a = -p, (15) 
 
 and R = Pseca. (16) 
 
 Remark. — These formulae have thus been derived from the general 
 expression (§ 419); but they can also be very readily got from a 
 special geometrical construction, corresponding to the case in which 
 the lines of the forces are at right angles to one another, the prin- 
 ciples to be used being (i) the parallelogram of forces; (2) Euchd L, 
 XLVIL; and (3) the trigonometrical definitions of sine, cosine, and 
 tangent. 
 
 430. This case is of importance, for It affords us the formulae for 
 rectangular resolution ; by the aid of which we shall, a little later, 
 proceed to calculate the resultant of any number of forces in one 
 plane. We might calculate the resultant by applying the elementary 
 
15° ABSTRACT DYNAMICS. 
 
 formulae (§§ 419, 420, 421) to repetitions of the parallelogram of forces. 
 But this process would be very complicated and tedious, if the forces 
 were numerous, and their magnitudes and angles given in numbers; 
 and we shall see that it may be avoided by resolving all the forces 
 along two lines at right angles to one another, and thus obtaining 
 as equivalent to them, two forces along these lines. 
 
 We shall first consider the general inverse problem (§ 418), or the 
 resolution of forces. 
 
 431. If a force acting on a material point, and two lines in one 
 plane with the line of that force, be given, it is possible to find deter- 
 minately two forces along those lines, of which the given force is the 
 resultant. 
 
 The two forces thus determined are called the components of the 
 given force along the given lines, and if we substitute these two forces 
 for the given force, we are said to resolve the given force into two 
 forces along the given lines; or, to resolve the force along the given 
 lines. 
 
 Geometrical Solution. Let M be the given point ; 7?, the 
 given force acting on it in the line, MK', 
 and J/T^and MG the given fines. 
 
 It is required to find the components 
 along J/T^and MG of R in MK. 
 
 Take any convenient length MD to 
 represent the magnitude of the given 
 force, R. Through D draw DA parallel 
 to GM, and let it cut MF in A ; and 
 also through D draw DB parallel to FM^ and let it cut MG in B ; 
 MA and MB represent the required magnitudes of the components. 
 
 433. Trigonometrical Solution. If the angle KMF be given = a, 
 and KMG = /?, and if the required component of the given force R 
 along MF be denoted by P^ and the component along MG by Q, we 
 deduce from equations (2) and (3) (§ 420), the following : — 
 
 ^~sin(a + ^)' ^'^^ 
 
 ^-sin(a + )8)- ('^> 
 
 434. When the given lines of resolution are at right angles to one 
 another, these expressions are modified in the manner shown above 
 
 (§ 429, Case VII), or we may find them 
 at once from the geometrical construc- 
 tion proper for the case, thus : — 
 
 Let MX, MY be the given lines; 
 XMY = 9o^ and MD = R. Also, as be- 
 fore, DMA = a, and DMB = jS. Draw 
 DA parallel to YM, or perpendicular to 
 MX, and make MB = AD. Then in the 
 
STA TICS OF A PAR TICLE.-^A TTRA CTION. 1 5 1 
 
 right-angled triangle MAD, MA =M£> co^ DMA, and AD^MD 
 sin DMA. 
 
 Hence, since MA represents the component along MX, and MB 
 the component along MY, 
 
 F=Rcosa, (19) 
 
 (2 = i? sin a, or (2 = i? cos /?. (20) 
 
 Hence, in rectangular resolution, the component, along any line, 
 of a given force, is equal to the product of the number expressing the 
 given force, into the cosine of the angle at which its direction is in- 
 clined to that line. 
 
 435. Application of the Resolution of Forces. 
 number of forces jP^, P^, P^,F^, P^, 
 acting respectively in lines ML^^ 
 ML,^, ML^, ML^, ML^, on a ma- 
 terial point M\ required their re- 
 sultant. 
 
 Through M, draw at right angles 
 to each other, and in the same plane 
 as the given forces, two lines, XX' 
 and YY', which may be called lines 
 or axes of resolution. Let the angle 
 which the resultant forms with the 
 line of resolution MX, be denoted 
 by 0, and let the angles, which the 
 lines of the forces make respectively 
 with the lines of resolution, be denoted by a^, ^^) a^, (3^; a^, p^; &c.; 
 that is, L^MX=a^, L^MY=^p^, and so on. 
 
 The angles /3j p„, &c., are merely the complements of a^ a^, &c., 
 and, except for the"" sake of symmetry, they need not have been intro- 
 duced into our notation. 
 
 Resolve (§ 434) the first force P^, into two components, one along 
 MX, and one along MY. These are 
 
 P^ cos ttj along MX, which force may be denoted by X^, 
 and P^ sin a^ along MY, which force may be denoted by Y^. 
 
 Treat all the other forces in like manner, thus reducing them to 
 components along MX and MY] and add together the components 
 along each of the lines of resolution. Then if X denote the sum of 
 the components along MX, and y the sum of the components along 
 MY, we have 
 
 X=P^ cos a, + P^ cos a^ + P^ cos a^ + P^ COS a^ + P^ COS a^, 
 F= P^ sin ttj + P"^ sin a^ + P^ sin a^ + P^ sin a^+ P^ sin a^. 
 
 Lastly, to find the resultant of X and K 
 
 (§429). i?=v'(jr'+n 
 
 and cos ^--bj 
 
 (2,) 
 (22) 
 
152 ABSTRACT DYNAMICS. 
 
 or, as is in general better for calculation, 
 
 Y 
 tan^ = -, (23) 
 
 whence we derive the magnitude of the resultant, 
 
 Ji = XsQce. (24) 
 
 The calculation will in general be facilitated by the use of log-' 
 arithms; for which purpose equations (25) and (24) are to be modified 
 in the following manner : — 
 
 tab. log. tan 6 = log. Y- log. X+ 10, (25) 
 
 log. R = log. X+ tab. log. sec. ^ - 10. (26) 
 
 Remark i. — It is to be observed that the sums X of the different 
 components X^., X^^ &c., and Fof Y^, Y^, &c., are got by an algebraic 
 addition, whatever may be the algebraic signs of the several terms. 
 If the given forces act all round the point M, it will happen in the 
 resolution that the different components do not all act in the same 
 directions along XX' and YY\ It will be necessary, therefore, to 
 fix upon one direction as positive. Thus, if MX and MY be posi- 
 tive directions, MX\ MY' will be negative; and absolute values of 
 the components, which act from M to X', and from M to F', 
 must be subtracted from, instead of added to, those along MX 
 2.ndMY. 
 
 Remark 2. — In choosing the axes of resolution, it simplifies the 
 problem to fix on one of the lines which represent the forces, as one 
 of the axes, and a line perpendicular to it, as the other. 
 
 Let J/Zj, the line of the first force P^, be the axis JOT, and MY, 
 a line perpendicular to it, the other, 
 
 a^ in this case is nothing; and the angle F^ MP^ = a^. 
 Hence, if a, = o, the resolution of the first force is 
 'X^=P^COSa^=Pj.f 
 P^ sin ttj = o, 
 
 that is, P^ requires no resolution. 
 
 If two of the forces happen to be at right angles, it will be con- 
 venient to choose the axes along them, and then neither requires 
 resolution. 
 
 Actual cases may often be simplified by observing if any two of the 
 forces are opposite, in which case, one force, equal to the excess of 
 the greater above the less, and acting in the direction of the greater, 
 may be taken instead of them. 
 
 Remark 3. — When the direction of the resultant is known, and its 
 magnitude is required, it is most convenient to make it one of the 
 axes of resolution. 
 
 •'{% 
 
STATICS OF A PARTICLE.— ATTRACTION. 153 
 
 Let MK be the direction of the 
 resultant of F^, F^, F^, F^, the dif- 
 ferent forces. Resolve each force 
 into two, one along MK, and one 
 in a hne perpendicular to it. Add 
 the components along MK. The 
 sum must be the magnitude of the 
 resultant; and the components along 
 the other line must balance one an- 
 other. Hence, 
 
 X=^R = F^ cos A, MKh F^ cos A^MK-v &c., 
 and Y= F^ sin A^ MK+ F^ sin A^ MK+ &c. - o. 
 
 Remark 4. — Equations (23) and (24) may be employed with ad- 
 vantage in all cases where the numbers of significant figures in the 
 values to be used for X and Fare large. 
 
 By equations (23) and (24) the direction of the resultant is first 
 determined, and then its magnitude, not as in equations (21) and (22), 
 the magnitude first, and then the direction. 
 
 436. For the better understanding of what follows a slight digres- 
 sion (§§ 437, 464) upon projections and geometrical co-ordinates is 
 now inserted. 
 
 437. The projection of a point on a straight line, is the point in 
 which the latter is cut by a perpendicular to it from the former. 
 
 438. Any line, joining two points, is called an arc. It is not 
 necessary to confine this expression to its most usual signification of 
 a continuous curve line. It may be appfied to a straight line joining 
 two points, as an extreme case; or it may be applied to a zigzag or 
 angular path from one point to the other; or to a self-cutting path, 
 whether curved or polygonal; in short, to any track whatever, from 
 one point to the other. 
 
 ♦ 
 
 439. The projection of an arc on a straight line, is the portion of 
 the latter intercepted between the projections of the extremities of the 
 former. 
 
 440. If we imagine an arc divided into any number of parts, the 
 projections of these parts, taken consecutively on any straight line, 
 make up consecutively the projection of the whole. Hence, the sum 
 of the projections of the parts is equal to the projection of the whole. 
 But in this statement, it must be understood that, of such partial 
 projections laid down in order, those which are drawn in one di- 
 rection, or forwards, being reckoned as positive, those which are 
 drawn in the other direction, or backwards, must be reckoned as 
 negative. 
 
 441. The projection of an arc on any straight line, is equal to the 
 length of the straight line joining the extremities of the former, mul- 
 
154* ABSTRACT DYNAMICS. 
 
 tiplied by the cosine of the angle* at which it is inclined to the latter. 
 This angle, if not a right angle, will be acute or obtuse, according to 
 the convention which is understood as to the direction reckoned 
 positive in the line of projection ; and the extremity of the arc which 
 is Xd^tn first in drawing 2, positive line from one extremity of it to the 
 other. 
 
 442. The orthogonal projection of a line, straight or curved, 
 closed or not closed, on a plane, is the locus of the points in which 
 the latter is cut by perpendiculars to it from all points of the former. 
 
 Other kinds of projections are also used in geometry; but when no 
 other designation is applied or understood, the simple ttrm projectio?i 
 will always mean orthogonal projection. 
 
 443. A circuit is a line returning into itself, or a line without ends 
 in a finite space. It is (if a continuous curve) often called a closed 
 curve; or if made up altogether of rectilinear parts, a closed polygoii. 
 A circuit in one plane may be either simple or self-cutting. The latter 
 variety has been called by De Morgan, autotomic. But whether simple 
 or autotomic, there is just one definite course to go round a circuit; 
 and at double or multiple points, this course must be distinctly 
 indicated^ (arrow-heads being generally used for the purpose on 
 a diagram, like the finger-posts where two or more roads cross). A 
 circuit not confined to one plane need never be considered to be 
 autotomic, unless as an extreme case. Thus, if we take any thread or 
 wire, however fine, and bend it into any curve or broken line, or tie it 
 into the most complicated knot or succession of knots, but attach its 
 ends together; any geometrical line drawn altogether within it, from 
 any one point of it, round through its length back to the same point, 
 constitutes essentially a simple or not self-cutting circuit. 
 
 444. 'The area enclosed by,' or *the area of a simple plane 
 circuit, is an expression which requires no explanation. But, as has 
 been shown by De Morgan^, a peculiar rule of interpretation is 
 necessary to apply the same expression to an autotomic plane circuit, 
 and it has no apphcation, hitherto defined, to a circuit not confined to 
 one plane. 
 
 445. The area of an autotomic plane circuit, is the sum of the 
 areas of all its parts each multiplied by zero with unity as many times 
 added as the circuit is crossed'* from right to left, and unity as many 
 
 1 The angle at which one line is inclined to another, is the angle between two 
 lines drawn parallel to them from any point, in directions similar to the directions 
 in the given lines which are reckoned positive. 
 
 " 'A curve which has double or multiple points, may be in many different 
 ways a circuit, or mode of proceeding from one point to the same again. Thus the 
 figure of 8 may be traced as a self -cutting circuit, in the way in which it is natural 
 if the curve be a continuous lemtiiscate, or it may be traced as a circuit presenting 
 two coincident salient points. A determinate area requires a determinate mode of 
 making the circuit.' De Morgan, Cambridge and Dublin Mathematical yournal^ 
 May, 1850. 
 
 ^ * Extension of the word area,' Cambridge and Dublin Mathematical yournal^ 
 May, 1850. 
 
 ^ A moving point is said to cross a plane circuit from right to left, if it crosses 
 
STATICS OF A PARTICLE.— ATTRACTION. 155 
 
 times subtracted as the circuit is crossed from left to right, when a 
 point is carried in the plane from the outside to any position within 
 the enclosed area in question. The diagram, which is that given by De 
 Morgan, will show more clearly what is meant by this use of the word 
 area. The reader, with this as a model, may exercise himself by 
 drawing autotomic circuits and numbering the different portions of 
 the enclosed area according to the rule, which he will then find no 
 difficulty in understanding. 
 
 446. Any portion of surface, edged or bounded by a circuit, is 
 called a skeiL 
 
 A plane area may be regarded as an extreme case, but generally 
 the surface of a shell will be supposed to be curved. 
 
 A simple shell is a shell of which the surface is single throughout. 
 One side of the shell must always be distinguished from the other, 
 whatever may be the convolutions of its surface. Thus we shall have 
 a marked and unmarked side, or an outside and an inside, to dis- 
 tinguish from one another. 
 
 447. The projection of a shell on any plane, is the area included 
 in the projection of its bounding line. 
 
 448. If we imagine a shell divided into any number of parts, the 
 projections of these parts on any plane make up the projection of the 
 whole. But in this statement it must be understood that the areas of 
 partial projections are to be reckoned as positive only if the marked 
 side, or, as we shall call it, the outside, of the projected area, and a 
 marked side, which we shall call the front, of the plane of projection, 
 face the same way. 
 
 If the outside of any portion of the projected area faces on the 
 whole backwards, relatively to the front of the plane of projection, the 
 projection of this portion is to be reckoned as negative in the sum. 
 
 from the right side to the left side as regarded by a person looking from any point 
 of the circuit in the direction reckoned positive. 
 
1 56 ABSTRA CT D YNAMICS. 
 
 Of course if the projected surface, or any part of it, be a plane area 
 at right angles to the plane of projection, the projection vanishes. 
 
 Cor. The projections of any two shells having a common edge, on 
 any plane, are equal. The projection of a closed surface (or a shell 
 with evanescent edge), on any plane, is nothing. 
 
 449. Equal areas in one plane or in parallel planes, have equal 
 projections on any plane, whatever may be their figures. [The proof 
 is easily found.] 
 
 Hence the projection of any plane figure, or of any shell, edged by 
 a plane figure, on another plane, is equal to its area, multiplied by the 
 cosine of the angle at which its plane is inclined to the plane of pro- 
 jection. This angle is acute or obtuse, according as the marked sides 
 of the projected area, and of the plane of projection face, on the whole, 
 towards the same parts, or on the whole oppositely. 
 
 450. Two rectangles, with a common edge, but not in one plane, 
 have their projection on any other plane, equal to that of one rect- 
 angle, having their two remote sides for one pair of its opposite sides. 
 For, the sides of this last-mentioned rectangle constitute the edge of 
 a shelly which we may make by applying two equal and parallel 
 triangular areas to the sides of the given rectangles; and the sum of 
 the projections of these two triangles on any plane, according to the 
 rule of § 448, is nothing. 
 
 Hence (as is shown by a very simple geometrical proof, which is 
 left as an exercise to the student), we have the following construction 
 to find a single plane area whose projection on any plane is equal to 
 the sum of the projections of any two given plane areas. 
 
 From any convenient point of reference draw straight lines per- 
 pendicular to the two given plane zx^z.^ forward^ relatively to their 
 marked sides considered as fronts. Make these lines numerically 
 equal to the two areas respectively. On these describe a parallel- 
 ogram, and draw the diagonal of this parallelogram through the point 
 of reference. Place an area with one side marked as front, in any 
 position perpendicular to this diagonal, facing forwards, and relatively 
 to the direction in which it is drawn from the point of reference. 
 Make this area equal numerically to the diagonal. Its projection on 
 any plane will be equal to the sum of the projections of the two given 
 areas, on the same plane. 
 
 The same construction maybe continued; just as, in § 413, the 
 geometrical construction to find the resultant of any number of 
 forces; and thus we find a single plane area whose projection on any 
 plane is equal to the sum of the projections on the same plane of any 
 given plane areas. And as any shell may (if it be not composed of a 
 finite) be regarded as composed of an infinite number of plane areas, 
 the same construction is applicable to a shell. Hence the projection 
 of a shell on any plane is equal to the projection on the same plane, 
 of a certain plane area, determined by the preceding construction. 
 
 From this it appears that the projection of a shell is nothing on 
 
STATICS OF A PARTICLE.—ATTRACTION. 157 
 
 any plane perpendicular to the one plane on which its projection 
 is greater than on any other; and that the projection on any inter- 
 mediate plane is equal to the greatest projection multiplied by the 
 cosine of the inclination of the plane of the supposed projection to 
 the plane of greatest projection. 
 
 451. To specify a point is to state precisely its position. As we 
 have no conception of position, except in so far as it is relative, the 
 specification of a point requires definite objects of reference, that is, 
 objects to which it may be referred. The means employed for this 
 purpose are certain elements called co-ordinates, from the system of 
 specification which Descartes first introduced into mathematics. This 
 system seems to have originated in the following method, for de- 
 scribing a curve by a table of numbers, or by an equation. 
 
 452. Given a plane curve, a fixed line in its plane, and a fixed 
 point in this line, choose as many points in the curve as are required 
 to indicate sufficiently its form: draw perpendiculars from them to 
 the fixed line, and measure the distances along it, cut oft' by these 
 lines, reckoning from the fixed point. In this way any number of 
 points in the curve were specified. The parts thus cut off along the 
 fixed line, were termed li7ieae abscissae^ and the perpendiculars, lineae 
 ordinaiim applicatae. The system was afterwards improved by draw- 
 ing through the point of reference a line at right angles to the first, 
 and measuring off along it the ot'dmafes of the curve. The two lines 
 at right angles to one another are called the axes of reference, or the 
 lines of reference. The ordinate and abscissa of any point are termed 
 its co-ordinates; and an equation between them, by which either may 
 be calculated when the other is given, expresses the curve in a per- 
 fectly full and precise manner. 
 
 453. It is not necessary that the lines of reference be chosen at 
 right angles to each other. But when they are chosen, inclined at 
 any other angle than a right angle, the co-ordinates of the point 
 specified are not its perpendicular distances from them, but its 
 distances from either, measured parallel to the other. Such oblique 
 co-ordinates are sometimes convenient, but rectangular co-ordinates 
 are, in general, the most useful; these we shall now consider. 
 
 454. If the points to be specified are all in one plane, the objects 
 of reference are two lines at right angles to 
 
 one another in that plane. Thus, let P 
 be a point in a plane XOY; and let OX, 
 OY,he two lines in the plane, cutting each 
 other at right angles in the point O. Then 
 will the position of the point F be known, 
 if the perpendicular distance of the point 
 F from the line OX, namely, the length 
 of the line FA, and the perpendicular dis- 
 tance from OY, namely, the length of the 
 Hne FB, be known. 
 
158 
 
 ABSTRACT DYNAMICS. 
 
 455. Again, let points, not all in one 
 plane, but in any positions through space 
 be considered. To specify each point now, 
 three co-ordinates are required, and the 
 objects of reference chosen may be three 
 planes at right angles to one another; thus, 
 the point P is specified by the lines FK^ 
 FH, FI, drawn perpendicular to the planes 
 '^ YZ, ZX, XY, respectively. 
 
 In our standard diagrams the positive 
 directions OX, O Y, OZ, are so taken that 
 if a watch is held in the plane XO Y, with its face towards OZ, an 
 angular motion against the hands would carry a line from OX to O K, 
 through the right angle XO Y. 
 
 456. When the objects to be specified are Hnes all passing through 
 one point, the specifying elements employed, are angles standing in 
 definite relation to them, and to the objects of reference. There are 
 two chief modes in which this kind of specification is carried out : 
 the polar and the symmetrical. 
 
 457. Fo/ar Method. 1°. Lines all in one plane. In this case the 
 object of reference is any fixed line through their common point of 
 intersection, and lying in their plane. 
 
 Let O be the common point of intersection, 
 ^^^__^^^ OX the fixed line, and OF the line to be 
 
 ^^""^^ X specified. Then the position of OF will be 
 
 known, if the angle XOF^ which the line OF 
 makes with OX, be known. 
 
 2°. Lines in space, all passing through one point, may be specified 
 by reference to a plane and a line in it, both passing through their 
 common point of intersection. 
 
 Let OF be one of a number of lines, all 
 passing through O, to be specified with refer- 
 ence to the plane XO K, and the line OX in it. 
 Through OF let a plane be drawn, cutting the 
 plane XO Y at right angles in OE. Then the 
 line OF will be specified, if the angles XOE, 
 EOF are given. 
 
 Corollary. Similarly, if the line OF be the 
 locus of a series of points, any one of these 
 points will be specified, if its distance from O 
 and the two angles specifying the line OFj are 
 known. 
 
 458. Symmetrical Method. In this method the objects of reference 
 are three lines at right angles to each other, through the common 
 point of intersection of the lines to be specified, and the specifying 
 elements are the three angles which each line makes with these three 
 lines of reference. 
 
 Thus, if O be the common point of intersection, OK one of the 
 
STATICS OF A PARTICLE.— ATTRACTION. 159 
 
 lines to be specified, and OX, OV, OZ, the lines of reference; then 
 the angles XOK, YOK, ZOK, are the specifying elements. 
 
 459. From what has now been said, it will be seen that the pro- 
 jections of a given line on other three at right angles to each other 
 are immediately expressible, if its direction is specified by either of 
 the two methods. 
 
 1°. Fo/ar Method. Let OK be the given line, 
 and OX, OV, OZ, the lines along which it is to be 
 projected. Through OZ and OX let a plane pass, 
 cutting the plane XOV in OF. Through X draw 
 another plane, KEA, cutting OX perpendicularly in 
 A and KEB cutting O Y perpendicularly in B. 
 Then KE, being the intersection of two planes 
 each perpendicular to XOY, is perpendicular to 
 every line in this plane. Hence, OEK is a right 
 angle. 
 
 Hence, 
 
 OE^OKq.q^KOE. 
 
 Again, since the plsine XAE was drawn perpendicular to OX, 
 OAE is a right angle. 
 
 Hence, OA = OF cos FOX= OX cos KOF cos EOX, 
 
 and similarly, OB = OF cos FO Y= OX cos XOF cos EO Y, 
 or if we put 
 
 OX=r, XOF = <i>, FOK=i, KOZ=Q = \Tr-i, 
 and let the required lines be denoted by x, y, z, then 
 ^ = r sin ^ cos <^,) 
 
 y = rsm 6s\n ^,> (i) 
 
 z = r cos 6. ) 
 
 2°. Synwtetrical Method. Let the line be referred to rectangular 
 axes by the three angles, a = ^(9X, 13 = XOY, y = XOZ. 
 Then the required projections are 
 
 ^ = ^cosa, y = r cos P, z = r cosy. 
 
 460. Referring again to the diagram, we have 
 
 OF'=OA'+OB', 
 and OX' = 'OF'+OC', 
 
 therefore, OX' = OA' + OB' + 0C\ 
 
 or r' = x' -^y' + z'. (2) 
 
 Substituting here for x, y, z, their values, in terms of r, a, p, y, 
 found above, and dividing both members of the resulting equation by 
 f-^, we have 
 
 I = cos^ a ■\- cos^ P + cos^ y. (3) 
 
 461. In the symmetrical method, three angles are used; but, as we 
 have seen, only two are necessary to fix the position of the line. We 
 
i6o ABSTRACT DYNAMICS, 
 
 now see that, if two of the three angles, a, ft y, are given, the third 
 can be found. Suppose a and ^ given, then by § 460, 
 cos^ y = I - cos* a - cos^ ft 
 
 For logarithmic calculation, the following modification of the pre- 
 ceding formula is useful, 
 
 cos^ y = sin^ a - cos^ i^ = - cos (a + ft x cos (a - ft, 
 whence cos y = ^{- cos (a + ft x cos (« - ft} 
 = \/{cos (tt - a - ft X cos (a - /8)}, 
 Tab. Log. cos y = 1 {T. L. cos (tt - a - ^) + T. L. cos (a - ft}. (4) 
 
 462. The following comparison will show in what way the two 
 systems are related, and how it is possible to derive the specifying 
 elements of either from those of the other. In the polar method, the 
 
 ■ fixed line in the equatorial plane, corresponds with one of the three 
 lines of reference in the symmetrical. A line in the equatorial plane, 
 drawn at right angles to the fixed line of the polar system, constitutes 
 a second line of reference in the symmetrical system. The third line 
 in the symmetrical system, is the axis of the polar system, from 
 which the polar distance \B) is measured. A comparison of pre- 
 ceding formulae shows that . 
 
 cos a = sin ^ cos ^,) 
 
 cos y8 = sin ^ sin <^, > . (5) 
 
 cos y = cos B, ) 
 
 463. The cosines of the three angles, a, ft y, of the symmetrical 
 system, are commonly called the direction cosines of the line specified. 
 If we denote them by /, w, n^ we have as above, 
 
 P^■m^ + n^=\. (6) 
 
 A line thus specified is for brevity called the line (/, ;;?, n). 
 
 \il^ 7n, ?t, are the direction cosines of a certain line; it is clear that 
 - /, - m, — ;/, are the direction cosines of the line in the opposite 
 direction from O. Thus it appears that the direction cosines of the 
 line, specify not only the straight line in which it lies, but the direction 
 in it which is reckoned as positive. 
 
 464. We conclude this digression with some applications of the 
 principles explained in it, which are useful in many dynamical in- 
 vestigations. 
 
 {a) To find the mutual inclination, 6, of two lines, (/, ;«, «), 
 (/, 7n', n'). Measure off any length 0K= r, along the first line (see 
 fig. of § 459). We have, as above, 
 
 OA=lr, AE = mr, EK=nr. 
 
 Now (§ 441), the projection of OK on the second line, is equal to 
 the sum of the projections of OA^ AE, EK, on the same. But the 
 cosines of the angles at which these several lengths are inclined to the 
 line of projection, are respectively cos ^, /, m', n'. Hence 
 
 OK cos 6 = OAf + AEm' + EKn'. 
 
STATICS OF A PARTICLE.— ATTRACTION. i6i 
 
 If we substitute in this, for OK, OA, AE, EK, their values shown 
 above, and divide both members by r, it becomes 
 
 cos 9 = 11' + mm' + ;?;/, (7) 
 
 a most important and useful formula. 
 
 Sometimes it is useful to have the sine instead of the cosine of 0. 
 To find it we have of course, 
 
 sin^ ^ = I - (// + mm! + ni^Y. 
 
 This expression may be modified thus: — instead of i, take what is 
 equal to it, {P + m'^ + //'') (/' + m'-\- tf), 
 
 and the second member of the preceding becomes 
 
 (P + m' + n') {P + 7n'' + n"') - (// + mm' + nn'Y 
 
 = {mji'Y + {nm'Y - 2mm' nn' + &c. = {mn' - nm'Y + &c. 
 
 Hence, sin = {{mn' - nm'Y + (nr - /n'Y + {/m' - mfY}^, (8) 
 
 (d) To find the direction cosines. A, fj., v, of the common perpen- 
 dicular to two lines, (/, pi, n), (/, m', n'). 
 
 The cosine of the inclination of (X, /j., v) to (/, m, n) is, according 
 to (7) above, /X + mfx + nv, and therefore 
 
 /X + mix + nv = o,) 
 similarly A + m'fi + nv =o,y (9) 
 
 also (§463) X' + tJi' + v' = i.) 
 
 These three equations suffice to determine the three unknown 
 quantities. A, fx, v. Thus, from the first two of them, we have 
 
 X _ /A _ V , > 
 
 mn' — nm' nl' — In' Im' - I'm ' ^ 
 
 From these and the third of (9), we conclude 
 
 ^ mn' - nm' „ 
 
 {{mn'-nm'Y^ Inl'-ln'Y ■¥ {Im' -m^Y)^ 
 
 or if we denote, as above, by 6, the mutual inclination of (/, m, n) 
 
 . {mn' — nm') {nV — hi) {Int — ml') . . 
 
 A= -. J , IK — : -^ , V= ^ J. . (11) 
 
 sm 6> ^ sm ^ ' sm ^ ^ 
 
 The sign of each of these three expressions may be changed, in as 
 much as either sign may be given to the numerical value found for 
 sin by (8). But as they stand, if sin is taken positive, they express 
 the direction cosines of the perpendicular drawn from O through the 
 face of a watch, held in the plane (/, m, n), {/', m', n'), and so facing 
 that angular motion, against or with the hands, would carry a line 
 from the direction, (/, m, 71), through an angle less than 180° to the 
 direction, (/', m', n'), according as angular motion, through a right 
 angle from OX to (9 F is against or ivith the ha?ids of a watch, held in 
 
 T. II 
 
1 62 ABSTRACT DYNAMICS. 
 
 the plane XOY, and facing towards OZ. This rule is proved by 
 supposing, as a particular case, the lines (/, ;;z, n), (/', m\ n'), to 
 coincide with OX and (9 F respectively; and then supposing them 
 altered in their mutual inclination to any other angle between o and 
 TT, and their plane turned to any position whatever. 
 
 If we measure off any lengths, OK-^r, and OK' = r'^ along the 
 two lines, (/, m^ n) and (/', 7n', n'\ and describe a parallelogram 
 upon them, its area is equal to r/ sin 6^ since r' sin 6 is the length of 
 the perpendicular from K' to OK. Hence, using the preceding 
 expression (8) for sin 6, and taking 
 
 Ir = X, mr = j, nr = 0, 
 // = x\ m'r' = y , « V = z\ 
 
 we conclude the following propositions. 
 
 if) The area of a parallelogram described upon lines from the 
 origin of co-ordinates to points (^, 7, 0), (x', y^ z') is equal to 
 
 {{y^ -y'^y + {^^' - ^'^y + {^y - ^'yy^- (^ 2) 
 
 And, as X, ja, v, are the cosines of the angles at which the plane of 
 this area is inclined to the planes of YZj ZX, XY, respectively, its 
 projections on these planes are 
 
 yz' -y'z, zx' -z'x, xy'-xy. (13) 
 
 The figures of these projections are parallelograms in the three 
 planes of reference; that in the plane YZ, for instance, being de- 
 scribed on lines drawn from the origin to the points {y, z) and {y', z). 
 It is easy to prove this (and, of course, the corresponding expressions 
 for the two other planes of reference,) by elementary geometry. 
 Thus, it is easy to obtain a simple geometrical demonstration of the 
 equations (8) and (11). It is sufficient here to suggest this investi- 
 gation as an exercise to the student. It essentially and obviously 
 includes the rule of signs] stated above (§ 464 {a)). 
 
 {d) The volume of a parallelepiped described on OK^ OK, 0K\ 
 three lines drawn from O to three points 
 
 {x, y, z), {x\ y\ z\ (x'\ y'\ z'% is equal to 
 x" {yz' -y'z) +y" {zx' - z'x) + z" (x/ - x'y), (14) 
 
 an expression which is essentially positive, if OK, OK, OK", are 
 arranged in order similarly to OX, OY, OZ {see § 455 above). The 
 proof is left as an exercise for the student. 
 
 In modern algebra, this expression is called a determinant, and is 
 written thus : — 
 
 X, y, z, 
 
 oo\ y, z', (15) 
 
 465. To find the resultant of three forces acting on a material 
 point in lines at right angles to one another. 
 
^^^ 
 
 X- 
 
 STATICS OF A PARTICLE.— ATTRACTION. 163 
 
 1° To find the magnitude of the resultant. F j^ 
 
 Let the forces be given numerically, X, Y, Z, 
 and let them be represented respectively by 
 the lines MA, MB, MC at right angles to 
 one another. 
 
 First determine the resultant of X and Y 
 in magnitude. If we denote it by R\ we 
 have (§ 429) 
 
 R'=j{X' + Y'). (i) ^r A 
 
 This resultant, represented by ME, lies in the plane BMA\ 
 and since the Hues of the forces X and Y are perpendicular to the 
 Hne MC, the Hne ME must also be perpendicular to it; for, if a line 
 be perpetidicular to two other lines, it is perpendicular to every other line 
 in their plane; hence R' acts perpendicularly to Z. 
 
 Next, find the resultant of R' and Z, the third force. If we denote 
 it by R, we have 
 
 R=J{R''+Z'), 
 
 and substituting for R'^ its value, we have 
 
 R=J{X' + Y' + Z'). (2) 
 
 2°. To find the direction of the resultant. Determine first the 
 inclination of the subsidiary resultant R' to MA or MB. Let the 
 angle EMA be denoted by <^ ; then we have 
 
 Next, let 7 denote the angle at which the line MZ> is inclined to 
 MC; that is, the angle CMD\ we have 
 
 cos7 = -^. (3) 
 
 Thus, by means of the two angles y and <^, the position of the line 
 MD, and, consequently, that of the resultant is found. 
 
 466. In the numerical solution of actual cases, it will generally be 
 found most convenient to calculate the three elements in the following 
 order: 1°, </>, 2°, y, 3°, R. 
 
 1°. To calculate ^, the formula already given, may be taken 
 
 Y 
 Un<j> = ^. (4) 
 
 (5) 
 
 (6) 
 
 (7) 
 II — 2 
 
 2°. 
 
 To calculate y. 
 
 We have 
 tany = -^. 
 
 But 
 
 
 R' = Xsec<f>. 
 
 Hence 
 
 
 X sec <j> 
 
 3^^. 
 
 To calculate R. 
 
 R = Zsecy. 
 
i64 ABSTRACT DYNAMICS. 
 
 467. The angles determined by these equations specify the line of 
 the resultant, by what was called in previous sections (^ 457, 459) 
 the Polar Method. 
 
 The symmetrical specification of the resultant is to be found thus : 
 Let (in fig. of § 465) the angles at which the line of the resultant, 
 MD^ is incUned to those of the forces be respectively denoted by a, ^, 
 and y. Then, as above (equation (3)), 
 
 Z 
 
 cosy=-^. . (8) 
 
 By the same method we shall find 
 
 cos a = - , (9) 
 
 Y 
 
 and cos/8^^. (10) 
 
 If, therefore, there are three forces at right angles to one another, 
 the cosine of the inclination of their resultant to any one of them is 
 equal to- this force divided by the resultant. 
 
 This method requires that the magnitude of the resultant be known 
 before its position is determined. For the latter purpose, any two of 
 the angles, as was shown in Chapter V, are sufficient. 
 
 468. We shall now consider the resolution of forces along three 
 specified lines. The most important case of all is that in which the 
 lines are at right angles to one another. 
 
 Let the force R^ given to be resolved, be represented by MD^ 
 and let the angles which it forms with the lines of resolution be 
 given, either a, ^, y, or y, <^. Required the components X^ K, Z. 
 
 1°. Suppose a, ^, y are given, then we deduce 
 from equation (9) X-Rq,o^ a; 
 
 from equation (10) Y=R cos p ; 
 
 and from equation (8) Z=R cos y. 
 
 2°. Suppose the data are R, y, <^, that is, the magnitude of the 
 resultant, its inclination to one of the axes of resolution, and the 
 inclination of the plane of the resultant and that axis to either of the 
 other axes. 
 
 To find the components X and V : resolve the force R in the 
 vertical plane CMED into two rectangular components along MC 
 and ME. Let the angle CMD be denoted by y. Then we have 
 for the component along MC^ 
 
 Z=R cosy^ (11) 
 
 and for the component along ME, 
 
 ME = R sin y. 
 
 Next, resolve the component along ME in the horizontal plane 
 BMAEy into two, one along MAy and the other along MB. Let 
 
STATICS OF A PARTICLE.— ATTRACTION, 165 
 
 Hie angle EMA be denoted by <^. Then we have for the com- 
 ponent along MA, 
 
 X =- ME cos cl> = R sin y cos 4>, (12) 
 
 and for the component along MB, 
 
 V= ME sin <\i = R sm y sin <^. (13) 
 
 469. We are now prepared to solve the general problem : — Given, 
 any number of forces acting on one point, in lines which lie in 
 different planes, required their resultant in position and magni- 
 tude. 
 
 Through the point acted on, draw three lines or axes of resolution 
 at right angles to one another. Resolve each force, by § 468, 1°, or 
 by § 468, 2°, into three components, acting respectively along the 
 three lines. When all the forces have been thus treated, add severally 
 the sets of components : by this means, all the forces are reduced to 
 three at right angles to one another. Find, by equation (2), their 
 resultant: the single force thus obtained is the resultant of the given 
 forces, which was to be found. 
 
 Remark. — All the remarks made with reference to the resolution 
 and composition of forces along two axes (§ 435) apply, with the ne- 
 cessary extension, to that of forces along three. 
 
 470. We are now prepared to answer the question which forms 
 the first general head of Statics ; What are the conditions of Equi- 
 librium of a material point 1 The answer may be put in one or other 
 of two forms. 
 
 1°. If a set of forces acting on a material point be in equilibrium, 
 any one of them must be equal and opposite to the resultant of the 
 others: or, 
 
 2°. If a set of forces acting on a material point be in equilibrium, 
 the resultant of the whole set must be equal to nothing. 
 
 471. Let us consider the first of these statements. 
 
 Given, a set of forces, F^, F^, F^, &c., in equilibrium : the force 
 /*,, for example, is equal and opposite to the resultant of F^, F.^, &c. ; 
 or, the resultant of F^, F^, &c., is —F^. Omitting F^, find the 
 resultant of the remaining forces by the general method ; the com- 
 ponents of this resultant will be 
 
 /*3 cos a^ + F3 cos a, + &c. along MX. 
 F^ cos p^ + F^ cos /?3 -t- &c. along MY. 
 F^ cos y^ + F^ cos yg + &c. along MZ. 
 
 Now, if — F^ be the resultant, the components of - F^ will be 
 equivalent respectively to the components of this resultant, there- 
 fore 
 
 - Pj cos a.^ = F^ cos a^ -I- F^ cos ttg + &c. 
 
 - /*j cos j8j = F^ cos ^2 + F.^ cos /?3 + &c. 
 
 - F^ cos y, -^- F^^ cos y, 4- P,^ COS y., + &c. 
 
i66 ABSTRACT DYNAMICS. 
 
 Which equations, in the following more general form, express the 
 required conditions: 
 
 F^ cos a^ + P^ cos a^ + P.^ COS ttg + &c. = o. 
 
 P^ cos ^j + P^ cos /Sg + P^ cos ^3 + &c. = o. 
 P^ cos yj + /'g cos y^ 4- 7^3 cos y3 + &c. = o. 
 
 472. The second form of the answer may be illustrated either a^ 
 dynamically, or b^ algebraically. 
 
 {a) Suppose all the forces reduced to three, X, F, Z, acting at right 
 angles to each other. Under what circumstances will three forces 
 give a vanishing resultant .? Substitute for X and Y their resultant 
 R\ and consider R' and Z at right angles to one another. If they 
 give a vanishing resultant, that is, if Z and R! balance, they must 
 either be equal and directly opposed, or else they must each be equal 
 to nothing. But they are not directly opposed, therefore each is 
 equal to nothing. Now, since R' = o, X and Y, v/hich are equi- 
 valent to R\ must also each be equal to nothing: in order, therefore, 
 that the resultant of forces acting along three lines at right angles to 
 one another may vanish, we have 
 
 P^ cos a^ + P^ COS a^ + &C. = o. 
 Pi COS ySi + P^ COS 1^1 + &C. = O. 
 
 P^ COS yi 4- P^ COS y^ + &c. = o. 
 
 {b) The general expression for the resultant is 
 
 R' = X' + Y' + Z\ 
 
 Now, for equilibrium, R = o, 
 
 and therefore, X' + Y' + Z' = o. 
 
 But the sum of three positive quantities can be equal to nothing, only 
 when each of them is nothing : hence 
 
 X=o, 
 Y=o, 
 Z=o. 
 
 473. We may take one or two particular cases as examples of the 
 general results above. Thus, 
 
 1. If the particle rest on a smooth curve, the resolved force along 
 the curve must vanish. 
 
 2. If the curve be rough, the resultant force along it must be 
 balanced by the friction. 
 
 3. If the particle rest on a smooth surface, the resultant of the 
 applied forces must evidently be perpendicular to the surface. 
 
 4. If it rest on a rough surface, friction will be called into play, 
 resisting motion along the surface; and there will be equilibrium at 
 any point within a certain boundary, determined by the condition 
 that at z/ the friction is fx times the normal pressure on the surface, 
 while within it the friction bears a less ratio to the normal pressure. 
 When the only applied force is gravity, we have a very simple result, 
 which is often practically useful. Let 6 be the angle between the 
 
STATICS OF A PARTICLE.— ATTRACTION. 167 
 
 normal to the surface and the vertical at any point; the normal 
 pressure on the surface is evidently W co^ 6^ where Wh the weight 
 of the particle ; and the resolved part of the weight parallel to the 
 surface, which must of course be balanced by the friction, is W sin 0. 
 In the limiting position, when sliding is just about to commence, the 
 greatest possible amount of statical friction is called into play, and 
 we have J^ sin 6 = ijlW cos $, 
 
 or tan 6 = 11. 
 
 The value of 6 thus found is called the Angle of Repose, and may 
 be seen in nature in the case of sand-heaps, and slopes formed by 
 debris from a disintegrating cliff (especially of a flat or laminated 
 character), on which the lines of greatest slope are inclined to the 
 horizon at an angle determined by this consideration. 
 
 474. A most important case of the composition of forces acting 
 at one point is furnished by the consideration of the attraction of a 
 body of any form upon a material particle anywhere situated. Experi- 
 ment has shown that the attraction exerted by any portion of matter 
 upon another is not modified by the neighbourhood, or even by the 
 interposition, of other matter; and thus the attraction of a body on a 
 particle is the resultant of the several attractions exerted by its parts. 
 To treatises on applied mathematics we must refer for the examina- 
 tion of the consequences, often very curious, of various laws of 
 attraction; but, dealing with Natural Philosophy, we confine our- 
 selves to the law of gravitation, which, indeed, furnishes us with an 
 ample supply of most interesting as well as useful results. 
 
 475. This law, which (as a property of matter) will be carefully 
 considered in the next Division of this Treatise, may be thus 
 enunciated. 
 
 Every particle of matter in the universe attracts every other particle 
 with a force, whose direction is that of the line joijiing the two, and 
 whose magnitude is directly as the product of their masses, and inversely 
 as the square of their distance from each other. 
 
 Experiment shows (as will be seen further on) that the same law 
 holds for electric and magnetic attractions ; and it is probable that it 
 is the fundamental law of all natural action, at least when the acting 
 bodies are not in actual contact. 
 
 476. For the special applications of Statical principles to which 
 we proceed, it will be convenient to use a special unit of mass, or 
 quantity of matter, and corresponding units for the measurement of 
 electricity and magnetism. 
 
 Thus if, in accordance with the physical law enunciated in § 475, 
 we take as the expression for the forces exerted on each other by 
 masses M and m^ at distance D, the quantity 
 
 Mm ^ 
 
 it is obvious that our unit force is the mutual attraction of two units 
 of mass placed at unit of distance from each other. 
 
1 6^ ABSTRA CT D YNAMICS. 
 
 477. It is convenient for many applications to speak of the density 
 of a distribution of matter, electricity, etc., along a line, over a sur- 
 face, or through a volume. 
 
 Here density of line is the quantity of matter per unit of length. 
 „ „ surface „ „ „ „ area. 
 
 „ ,, volume „ - „ „ „ volume. 
 
 478. In applying the succeeding investigations to electricity or 
 
 magnetism, it is only necessary to premise that M and m stand for 
 
 quantities of free electricity or magnetism, whatever these may be, and 
 
 that here the idea of mass as depending on inertia is not necessarily 
 
 Mm 
 involved. The formula -jr^ will still represent the mutual action, if 
 
 we take as unit of imaginary electric or magnetic matter, such a quan- 
 tity as exerts unit force on an equal quantity at unit distance. Here, 
 however, one or both of M, m may be negative ; and, as in these 
 applications like kinds repel each other, the mutual action will be 
 attraction or repulsion, according as its sign is negative or positive. 
 With these provisos, the following theory is applicable to any of the 
 above-mentioned classes of forces. We commence with a few simple 
 cases which can be completely treated by means of elementary geo- 
 metry. 
 
 479. If the different points of a spherical surface attract equally 
 with forces varying inversely as the squares of the distances, a particle 
 placed witJwi the surface is not attracted in any direction. 
 
 Let HIKL be the spherical surface, and P the particle within it. 
 Let two hues HK^ IL, intercepting very small arcs HI, KL, be 
 drawn through F; then, on account of the 
 similar triangles IIFI, KFL, those arcs will 
 be proportional to the distances HF, IF; 
 and any small elements of the spherical sur- 
 face atZT/and KI, each bounded all round 
 by straight lines passing through F [and very 
 nearly coinciding with IIK\ will be in the 
 duplicate ratio of those lines. Hence the 
 forces exercised by the matter of these ele- 
 ments on the particle F are equal ; for they 
 are as the quantities of matter directly, and the squares of the 
 distances, inversely; and these two ratios compounded give that of 
 equality. The attractions therefore, being equal and opposite, de- 
 stroy one another: and a similar proof shows that all the attractions 
 due to the whole spherical surface are destroyed by contrary attrac- 
 tions. Hence the particle Z' is not urged in any direction by these 
 attractions. 
 
 480. The division of a spherical surface into infinitely small ele- 
 ments, will frequently occur in the investigations which follow: and 
 Newton's method, described in the preceding demonstration, in which 
 the division is effected in such a manner that all the parts may be 
 taken together m pairs of opposite eletnents with reference to an internal 
 
STATICS OF A PARTICLE.— ATTRACTION. 169 
 
 point ; besides other methods deduced from it, suitable to the special 
 problems to be examined ; will be repeatedly employed. The follow- 
 ing digression (§§ 481, 486), in which some definitions and elemen- 
 tary geometrical propositions regarding this subject are laid down, 
 will simplify the subsequent demonstrations, both by enabling us, 
 through the use of convenient terms, to avoid circumlocution, and 
 by affording us convenient means of reference for elementary prin- 
 ciples, regarding which repeated explanations might otherwise be 
 necessary. 
 
 481. If a straight line which constantly passes through a fixed 
 point be moved in any manner, it is said to describe, or generate, 
 a conical surface of which the fixed point is the vertex. 
 
 If the generating line be carried from a given position continuously 
 through any series of positions, no two of which coincide, till it is 
 brought back to the first, the entire line on the two sides of the fixed 
 point will generate a complete conical surface, consisting of two 
 sheets, which are called vertical or opposite cones. Thus the elements 
 ZT/and KL, described in Newton's demonstration given above, may 
 be considered as being cut from the spherical surface by two opposite 
 cones having P for their common vertex. 
 
 482. If any number of spheres be described from the vertex of a 
 cone as centre, the segments cut from the concentric spherical sur- 
 faces will be similar, and their areas will be as the squares of the 
 radii. The quotient obtained by dividing the area of one of these 
 segments by the square of the radius of the spherical surface from 
 which it is cut, is taken as the measure of the solid attgle of the cone. 
 The segments of the same spherical surfaces made by the opposite 
 cone, are respectively equal and similar to the former. Hence the 
 solid angles of two vertical or opposite cones are equal : either may 
 be taken as the solid angle of the complete conical surface, of which 
 the opposite cones are the two sheets. 
 
 483. Since the area of a spherical surface is equal to the square of 
 its radius multiplied by 477, it follows that the sum of the solid angles 
 of all the distinct cones which can be described with a given point as 
 vertex, is equal to 477. 
 
 484. The solid angles of vertical or opposite cones being equal, 
 we may infer from what precedes that the sum of the solid angles 
 of all the complete conical surfaces which can be described with- 
 out mutual intersection, with a given point as vertex, is equal 
 
 to 27r. 
 
 485. The solid angle subtended at a point by a superficial area of 
 any kind, is the solid angle of the cone generated by a straight line 
 passing through the point, and carried entirely round the boundary 
 of the area. 
 
 486. A very small cone, that is, a cone such that any two posi- 
 tions of the generating line contain but a very small angle, is said to 
 be cut at right angles, or orthogonally, by a spherical surface de- 
 
lyo ABSTRACT DYNAMICS. 
 
 scribed from its vertex as centre, or by any surface, whether plane or 
 curved, which touches the spherical surface at the part where the cone 
 is cut by it. 
 
 A very small cone is said to be cut obliquely, when the section is 
 inclined at any finite angle to an orthogonal section ; and this angle 
 of inclination is called the obliquity of the section. 
 
 The area of an orthogonal section of a very small cone is equal to 
 the area of an oblique section in the same position, multiplied by the 
 cosine of the obliquity. 
 
 Hence the area of an oblique section of a small cone is equal to 
 the quotient obtained by dividing the product of the square of its 
 distance from the vertex, into the solid angle, by the cosine of the 
 obliquity. 
 
 487. Let E denote the area of a very small element of a spherical 
 surface at the point E (that is to say, an element every part of which 
 is very near the point E), let w denote the solid angle subtended by E 
 at any point P, and let PE^ produced if necessary, meet the surface 
 again in E' \ then a denoting the radius of the spherical surface, we 
 have 
 
 ^_ 2a.i^.PE^ 
 ^ " EE' ' 
 
 For, the obliquity of the element E, considered as a section of the 
 cone of which P is the vertex and the element 
 E a section (being the angle between the 
 given spherical surface and another described 
 from P as centre, with PE as radius), is 
 equal to the angle between the radii EP and 
 EC^ of the two spheres. Hence, by con- 
 sidering the isosceles triangle ECEf, we find 
 that the cosine of the obliquity is equal to 
 
 hEE' EE' 
 
 j-,^ or to , 
 
 EC 2a 
 
 and we arrive at the preceding expression for E. 
 
 488. The attradioji of a uniform spherical surface on an external 
 point is the same as if the whole ?nass were collected at the centre^. 
 
 Let P be the external point, C the centre of the sphere, and CAP 
 a straight line cutting the spherical surface in A. Take / in CP, 
 so that CPf CA, CI may be continual proportionals, and let the 
 
 1 This theorem, which is more comprehensive than that of Newton in his first 
 proposition regarding attraction on an external point (Prop. LXXI.), is fully es- 
 tablished as a corollary to a subsequent proposition (LXXIII. cor. 2). If we had 
 considered the proportion of the forces exerted upon two external points at 
 different distances, instead of, as in the text, investigating the absolute force on 
 one point, and if besides we had taken together all the pairs of elements which 
 would constitute two narrow annular portions of the surface, in planes perpen- 
 dicular to PC, the theorem and its demonstration would have coincided precisely 
 with Prop. LXXI. of the Pnncipia. 
 
STATICS OF A PARTICLE,— ATTRACTION. 171 
 
 whole spherical surface be di- 
 vided into pairs of opposite ele- 
 ments with reference to the poijit 
 L 
 
 Let H and H' denote the 
 magnitudes of a pair of such 
 elements, situated respectively 
 at the extremities of a chord 
 HH' \ and let 00 denote the 
 magnitude of the solid angle 
 subtended by either of these elements at the point /. 
 
 We have (§ 486), 
 
 Zr= 7TT7>5 and II'= — 777777- • 
 
 cos CHI cos CHI 
 
 Hence, if p denote the density of the surface, the attractions of the 
 two elements H and H' on F are respectively 
 
 CO IH' io IH' 
 
 P cos CHI ' FH' ' P cos CHI ' FH" ' 
 Now the two triangles FCH, HCI have a common angle at C, and, 
 since FC : CH v. CH \ CI^ the sides about this angle are propor- 
 tional. Hence the triangles are similar; so that the angles CFH 
 and CHI ^XQ equal, and 
 
 IH _CH_ a^ 
 
 'HF~ CF ~CF' 
 In the same way it may be proved, by considering the triangles 
 FCH\ HCI, that the angles CFH and CHIzxq equal, and that 
 
 IH _ CH _ _a 
 
 HF~ CF ~ CF' 
 Hence the expressions for the attractions of the elements ^and H 
 on F become 
 
 0) «^ , o) a* 
 
 ^'^^^CHI' CP' ^ cos CHI' CF' ' 
 which are equal, since the triangle HCH is isosceles; and, for the 
 same reason, the angles CFH, CFH, which have been proved to 
 be respectively equal to the angles CHI, CHI, are equal. We 
 infer that the resultant of the forces due to the two elements is in 
 the direction FC, and is equal to 
 
 a' 
 20). p. -^-3. 
 
 To find the total force on F, we must take the sum of all the 
 forces along /'C due to the pairs of opposite elements; and, since 
 the multiplier of w is the same for each pair, we must add all the 
 values of w, and we therefore obtain (§ 483), for the required re- 
 sultant, 
 
 47rp^^ 
 
172 ABSTRACT DYNAMICS. 
 
 The numerator of this expression (being the product of the density 
 
 into the area of the spherical surface) is equal to the mass of the 
 
 entire charge; and therefore the force on P is the same as if the 
 
 whole mass were collected at C 
 
 Cor. The force on an external point, infinitely near the surface, 
 
 is equal to 47rp, and is in the direction of a normal at the point. 
 
 The force on an internal point, however near the surface, is, by 
 
 a preceding proposition, equal to nothing. 
 
 489. Let o- be the area of an infinitely small element of the surface 
 at any point jP, and at any other point H of 
 the surface let a small element subtending a 
 solid angle co, at jP, be taken. The area of 
 this element will be equal to 
 
 \P cos CHP' 
 
 and therefore the attraction along HP, which 
 it exerts on the element <t at P, will be equal 
 to pi». pa- 0) 2 
 
 cos CNP' ^^ cosCBP^"^' 
 
 Now the total attraction on the element at T'is in the direction CP; 
 the component in this direction of the attraction due to the element 
 H, is 
 
 0) . p V ; 
 and, since all the cones corresponding to the different elements of the 
 spherical surface lie on the same side of the tangent plane at P, we 
 deduce, for the resultant attraction on the element o-, 
 
 27rp^(r. 
 From the corollary to the preceding proposition, it follows that this 
 attraction is half the force which would be exerted on an external 
 point, possessing the same quantity of matter as the element o-, and 
 placed infinitely near the surface. 
 
 490. In some of the most important elementary problems of the 
 theory of electricity, spherical surfaces with densities varying inversely 
 as the cubes of distances from excentric points occur: and it is of 
 fundamental importance to find the attraction of such a shell on an 
 internal or external point. This may be done synthetically as follows ; 
 the investigation being, as we shall see below, virtually the same 
 as that of § 479, or § 488. 
 
 491. Let us first consider the case in which the given point S and 
 the attracted point P are separated by the spherical surface. The 
 two figures represent the varieties of this case in which, the point S 
 being without the sphere, P is within; and, S being within, the 
 attracted point is external. The same demonstration is applicable 
 literally with reference to the two figures; but, for avoiding the con- 
 sideration of negative quantities, some of the expressions may be 
 conveniently modified to suit the second figure. In such instances 
 
STATICS OF A PARTICLE.— ATTRACTION. 173 
 
 the two expressions are given in a double line, the upper being that 
 which is most convenient for the first figure, and the lower for the 
 second. 
 
 Let the radius of the sphere be denoted by a, and let / be the 
 distance of S from C, the centre of the sphere (not represented in 
 the figures). 
 
 Join SF and take T in this line (or its continuation) so that 
 (fig. i) SF.ST=r-a\ 
 (fig. 2) SF.TS=a!'-f\ 
 Through T draw any line cutting the spherical surface at K, K', 
 Join SK^ SK\ and let the fines so drawn cut the spherical surface 
 again in E, E'. 
 
 Let the whole spherical surface be divided into pairs of opposite 
 elements with reference to the point T. Let K and K' be a pair of 
 such elements situated at the extremities of the chord KK\ and 
 subtending the solid angle m at the point T\ and let elements E and 
 E' be taken subtending at 6* the same solid angles respectively as the 
 elements K and K'. By this means we may divide the whole 
 spherical surface into pairs of conjugate elements, E^ E', since it is 
 easily seen that when we have taken every pair of elements, K, K\ 
 
 P 
 
 ^" 
 
 K 
 
 the whole surface will have been exhausted, without repetition, by the 
 deduced elements, E^ E' . Hence the attraction on P will be the 
 final resultant of the attractions of all the pairs of elements, E^ E'. 
 
 Now if p be the surface density at E^ and if i^ denote the attraction 
 of the element E on F^ we have 
 
 E=P^ 
 EF'' 
 
 According to the given law of density we shall have 
 
 X 
 ^~ SE'' 
 where X is a constant. Again, since SEK is equally inclined to the 
 spherical surface at the two points of intersection, we have 
 P^SE^ A--— 2ao>.TK\ 
 SK' SX'' KK' ' 
 
 and hence 
 
 X SE' 2ai^.TK^ 
 ^ SE''SK'' KK' , 2a TK' 
 
 EF* ' KK' ' SE . SK' . EF' 
 
174 ABSTRACT DYNAMICS, 
 
 Now, by considering the great circle in which the sphere is cut by a 
 plane through the line SK, we find that 
 
 (fig. i) SK.SE=f'-a\ 
 
 (fig. 2) KS.SE = a^-f\ 
 and hence SK. SE = ST. ST, firom which we infer that the triangles 
 XST, TSE are similar; so that TK -. SK v. PE -. ST. Hence 
 
 TK^ I 
 
 2 CiD2 > 
 
 SK\PE^ ST 
 
 and the expression for /^becomes 
 
 KK''SE.ST'''^' 
 Modifying this by preceding expressions we have 
 
 Similarly, if E' denote the attraction of -£' on T, we have 
 
 Now in the triangles which have been shown to be similar, the 
 angles TKS, ETS are equal; and the same may be proved of the 
 angles K'ST, TSE'. Hence the two sides SK, SK' of the triangle 
 KSK' are inclined to the third at the same angles as those between 
 the line TS and directions TE, TE' of the two forces on the point 
 T; and the sides SK, SK' are to one another as the forces, E, E', 
 in the directions TE, TE', It follows, by ' the triangle of forces,' 
 that the resultant of F and E' is along TS, and that it bears to the 
 component forces the same ratios as the side KK' of the triangle 
 bears to the other two sides. Hence the resultant force due to the 
 two elements E and E' on the point T, is towards 6", and is equal to 
 
 KK\f~a^).ST'' ' ' {/'--a') ST' ' 
 
 The total resultant force will consequently be towards ^S"; and we 
 find, by summation (§ 466) for its magnitude, 
 
 X . 47ra 
 {r~a')ST'' 
 
 Hence we infer that the resultant force at any point T, separated 
 from S by the spherical surface, is the same as if a quantity of 
 
 matter equal to '^^^^ were concentrated at the point S, 
 
STATICS OF A PARTICLE.— ATTRACTION. 175 
 
 492. To find the attraction when S and F are either both without 
 or both within the spherical surface. 
 
 Take in CS, or in CS produced through S, a point S^, such that 
 
 CS. CS^ = a\ 
 
 Then, by a well-known geometrical theorem, if E be any point on 
 the spherical surface, we have 
 
 SE f 
 
 
 
 S^E a' 
 
 
 
 Hence 
 
 we have 
 
 SE'~/\S,E'' 
 
 
 
 Hence, 
 if 
 
 p being 
 
 the electrical density at E, 
 Xa' 
 
 r K 
 
 ^ S^E' S^E' ' 
 Ai = . 
 
 we 
 
 have 
 
 Hence, by the investigation in the preceding section, the attraction 
 on F is towards Si , and is the same as if a quantity of matter equal 
 
 Xi . A7ra 
 
 to T^i 2 were concentrated at that pomt ; 
 
 Ii ~ ^ 
 
 /i being taken to denote CS^. If for /i and X^ we substitute their 
 values, -p and -^ , we have the modified expression 
 
 A -2. . 47ra 
 
 for the quantity of matter which we must conceive to be collected 
 atSy 
 
 493. If a spherical surface be electrified in such a way that the 
 electrical density varies inversely as the cube of the distance from 
 an internal point S, or from the corresponding external point S-^, it 
 will attract any external point, as if its whole electricity were con- 
 
176 ABSTRACT DYNAMICS. 
 
 centrated at S^ and any internal point, as if a quantity of electricity 
 greater than its own in the ratio of ^ to /were concentrated at S^ 
 
 Let the density at E be denoted, as before, by -^^^ . Then, if we 
 
 consider two opposite elements at £ and E\ which subtend a solid 
 
 angle w at the point S, the areas of these elements being —^ — ^^r, — 
 
 and ' ' , the quantity of electricity which they possess will be 
 
 ££' 
 
 \,2a.tti/ I I \ \.2a.(i> 
 or 
 
 2a. W/ I I \ 
 
 ££^\S£'^S£') ^^ S£,S£ 
 
 Now S£ . SE' is constant (Euc. III. 35) and its value is a^ -/^ 
 Hence, by summation, we find for the total value of electricity on 
 the spherical surface 
 
 Hence, if this be denoted by m^ the expressions in the preceding 
 paragraphs, for the quantities of electricity which we must suppose to 
 be concentrated at the point S or *Sj, according as F is without or 
 within the spherical surface, become respectively 
 
 nif and ^ m. 
 
 494. The direct analytical solution of such problems consists in 
 the expression, by § 408, of the three components of the whole at- 
 traction as the sums of its separate parts due to the several particles 
 of the attracting body; the transformation, by the usual methods, of 
 these sums into definite integrals; and the evaluation of the latter. 
 This is, in general, inferior in elegance and simplicity to the less 
 direct mode of solution depending upon the determination of the 
 potential energy of the attracted particle with reference to the forces 
 exerted upon it by the attracting body, a method which we shall 
 presently develop with peculiar care, as it is of incalculable value in 
 the theories of Electricity and Magnetism as well as in that of 
 Gravitation. But before we proceed to it, we give some instances of 
 the direct method. 
 
 {a) A useful case is that of the attraction of a circular plate of 
 uniform surface density on a point in a line through its centre, and 
 perpendicular to its plane. 
 
 All parallel slices, of equal thickness, of any cone attract equally 
 (both in magnitude and direction) a particle at the vertex. 
 
 For the proposition is true of a cone of infinitely small angle, the 
 masses of the slices being evidently as the squares of their distances 
 from the vertex. If / be the thickness, p the volume density, and w 
 the angle, the attraction is w/p. 
 
 All slices of a cone of infinitely small angle, if of equal thickness 
 
STATICS OF A PARTICLE.— ATTRACTION. 177 
 
 and equally inclined to the axis of the cone, exert equal forces on a 
 particle at the vertex. For the area of any inclined section, whatever 
 be its orientation, is greater than that of the corresponding transverse 
 section in the ratio of unity to the cosine of the angle of inclination. 
 
 Hence if a plane touch a sphere at a 
 point B, and if the plane and sphere have 
 equal surface density at corresponding 
 points P and / in a line drawn through 
 A^ the point diametrically opposite to By 
 corresponding elements at P and / exert 
 equal attraction on a particle at A. 
 
 Thus the attraction on A, of any part 
 of the plane, is the same as that of the 
 corresponding part of the sphere, cut out 
 by a cone of infinitely small angle whose vertex is A. 
 
 Hence if we resolve along the line AB the attraction of pq on A, 
 the component is equal to the attraction along Ap of the -transverse 
 section pr, i.e. /qw, where co is the angle subtended at A by the 
 element pq, and p the surface density. 
 
 Thus any portion whatever of the sphere attracts A along AB 
 with a force proportional to its spherical opening as seen from ^; 
 and the same is, by what was proved above, true of a flat plate. 
 
 Hence as a disc of radius a subtends at a point distant h from it, 
 in the direction of the axis of the disc, a solid angle 
 
 \ J/i' + aV' 
 
 the attraction of such a disc is 
 
 ''"'{'- JW^)' 
 
 which for an infinite disc becomes, whatever the distance h^ 
 
 27: p. 
 
 From the preceding formula many useful results may easily be 
 deduced : thus, 
 
 {b) A uniform cylinder of length /, and diameter a, attracts a point 
 in its axis at a distance x from the nearest end with a force 
 
 27rp {/- J{x + lY + a' + J^^T?]. 
 
 When the cylinder is of infinite length (in one direction) the at- 
 traction is therefore 
 
 27rp [Jx^ +a^ -x); 
 
 and, when the attracted particle is in contact with the centre of the 
 end of the infinite cylinder, this is 
 
 2-7? pa. 
 
 (c) A right cone, of semivertical angle a, and length /, attracts a 
 
 T. 12 
 
178 ABSTRA CT D YNAMICS. 
 
 particle at its vertex. Here we have at once for the attraction, the 
 expression 
 
 2irpl{l -COS a), 
 
 which is simply proportional to the length of the axis. 
 
 It is of course easy, when required, to find the necessarily less 
 simple expression for the attraction on any point of the axis. 
 
 (d) For magnetic and electro-magnetic applications a very useful 
 case is that of two equal uniform discs, each perpendicular to the line 
 joining their centres, on any point in that line — their masses (§ 478) 
 being of opposite sign — that is, one repelling and the other attracting. 
 
 Let a be the radius, p the mass of a superficial unit, of either, c 
 their distance, x the distance of the attracted point from the nearest 
 disc, The whole force is evidently 
 
 {X + ^ X ^ 
 
 J{x + cy + a'~ Jl^^T^j' 
 
 In the particular case when c is diminished without limit, this 
 becomes 
 
 27rp{: 
 
 (^ + «^)l 
 
 495. Let P and P' be two points infinitely near one another on 
 two sides of a surface over which matter is distributed ; and let p be 
 the density of this distribution on the surface in the neighbourhood 
 of these points. Then whatever be the resultant attraction, i?, at /*, 
 due to all the attracting matter, whether lodging on this surface, or 
 elsewhere, the resultant force, R\ on P' is the resultant of a force 
 equal and parallel to R, and a force equal to 4Trp, in the direction 
 from P' perpendicularly towards the surface. For, suppose PP' to 
 be perpendicular to the surface, which will not limit the generality of 
 the proposition, and consider a circular disc, of the surface, having 
 its centre in Pp\ and radius infinitely small in comparison with the 
 radii of curvature of the surface but infinitely great in comparison 
 with PjP\ This disc will [§ 494] attract -P and P with forces, 
 each equal to 27rp and opposite to one another in the line FF'. 
 Whence the proposition. It is one of much importance in the theory 
 of electricity. 
 
 496. It may be shown that at the southern base of a hemispherical 
 hill of radius a and density p, the true latitude (as measured by the 
 aid of the plumb-line, or by reflection of starlight in a trough of 
 mercury) is diminished by the attraction of the mountain by the 
 angle 
 
 G-ipa^ 
 where G is the attraction of the earth, estimated in the same units. 
 
STATICS OF A PARTICLE,— ATTRACTION. 179 
 
 Hence, if R be the radius and o- the mean density of the earth, the 
 angle is 
 
 ^Trpa ^ pa . , 
 
 ^.<rli-%pa '°'i^ approximately. 
 
 Hence the latitudes of stations at the base of the hill, north and 
 
 south of it, diifer by ^ ^2 + — h instead of by ^, as they would 
 
 do if the hill were removed. 
 
 In the same way the latitude of a place at the southern edge of a 
 
 hemispherical caz'ify is increased on account of the cavity by ^ ^-= 
 
 where p is the density of the superficial strata. 
 
 497. As a curious additional example of the class of questions 
 we have just considered, a deep crevasse, extending east and west, 
 increases the latitude of places at its southern edge by (approx- 
 imately) the angle f ^ where is the density of the crust of the 
 
 CTjCV 
 
 earth, and a is the width of the crevasse. Thus the north edge of 
 
 the crevasse will have a lower latitude than the south edge if f - > i, 
 
 which might be the case, as there are rocks of density f x 5*5 or 
 3 '67 times that of water. At a considerable depth in the crevasse, 
 this change of latitudes is nearly doubled^ and then the southern side 
 has the greater latitude if the density of the crust be not less than 
 1*83 times that of water. 
 
 498. It is interesting, and will be useful later, to consider as a 
 particular case, the attraction of a sphere whose mass is composed of 
 concentric layers, each of uniform density. Let <r be, as above, the 
 mean density of the whole globe, and t the density of the upper crust. 
 
 The attraction at a depth ^, small compared with the radius, is 
 
 ^i:<T^{R-h)^G, 
 
 where otj denotes the mean density of the nucleus remaining when a 
 shell of thickness h is removed from the sphere. Also, evidently, 
 
 ^i?<T^ {R - Hf + 47rT (i? - h^h = Itto-J?', 
 
 or G,(R-/iy + 4'^T{R-/iy/i=:GR% 
 
 whence G^ = G(i + -^j-47rTk 
 
 The attraction is therefore unaltered at a depth /i if 
 
 _=*,r(T=27rT, I.e. T = |cr. 
 
 499. Some other simple cases may be added here, as their results 
 will be of use to us subsequently. 
 
 12^2 
 
i8o 
 
 ABSTRACT DYNAMICS. 
 
 (a) The attraction of a circular arc, AB, of uniform density, on a 
 particle at the centre, C, of the circle, lies 
 evidently in the line CD bisecting the arc. 
 Also the resolved part parallel to CD of 
 the attraction of an element at F is 
 
 mass of element at /* „^ ^ 
 ^7^2 cos . z BCD. 
 
 Now suppose the density of the chord AB 
 to be the same as that of the arc. Then 
 for (mass of element at F x cos l BCD) 
 we may put (mass of projection of element 
 on AB at 0; since, if BT be the tan- 
 gent at B, I BTQ = L BCD, 
 
 Hence attraction along CD = 
 
 sum of projected elements 
 CD^ 
 
 pAB 
 CD"' 
 
 if p be the density of the given arc, 
 
 _ 2psin L ACD 
 CD 
 
 It is therefore the same as the attraction of a mass equal to the 
 chord, with the arc's density, concentrated at the point D. 
 
 (p) Again, a limited straight line of uniform density attracts any 
 
 external point in the 
 \C same direction and with 
 the same force as the 
 corresponding arc of a 
 circle of the same den- 
 sity, which has the point 
 for centre, and touches 
 the straight line. 
 
 For if CpB be drawn 
 cutting the circle in p 
 and the line in B; ele- 
 
 CB 
 
 ment at p : element Sit B :: Cp : CB ^^; that is, as Cp' : CB\ 
 
 Hence the attractions of these elements on C are equal and in the 
 same line. Thus the arc ab attracts C as the line AB does ; and, by 
 the last proposition, the attraction oi AB bisects the angle -^C5, and 
 is equal to 
 
 -^sin^z^C^. 
 
STATICS OF A PARTICLE.— ATTRACTION. i8i 
 
 {c) This may be 
 put into other use- 
 ful forms — thus, let 
 CKF bisect the an- 
 gle ACB, and let 
 Aa^ Bb, EF^ be 
 drawn perpendicular 
 to CF from the 
 ends and middle 
 point of AB, We 
 have 
 
 . „_^ KB . -__- AB CD 
 sm L KCB^-^sm l CKD^-j^-^^-^, 
 
 Hence the attraction, which is along CK, is 
 2pAB pAB 
 
 CF.. 
 
 (I). 
 
 (AC+ CB) CK 2 (^C+ CB) (AC-^r CB'-AB') 
 For evidently, 
 
 bK : Ka :: BK : KA :: BC : CA :: bC : (G?, 
 
 i.e., ^/^ is divided, externally in C, and internally in X, in the same 
 ratio. Hence, by geometry, 
 
 XC. CF=aC. Cb = i{A C+ CB' - AB'}, 
 
 which gives the transformation in (i). 
 
 (d) CF is obviously the tangent at C to a hyperbola, passing 
 through that point, and having A and B as foci. Hence, if in 
 any plane through AB any hyperbola be described, with foci A 
 and Bj it will be a line of force as regards the attraction of the 
 line AB ; that is, as will be more fully explained later, a curve which 
 at every point indicates the direction of attraction. 
 
 (e) Similarly, if a prolate spheroid be described with foci A and B, 
 and passing through C, Ci^will evidently be the normal at C; thus 
 the force on a particle at C will be perpendicular to the spheroid ; 
 and the particle would evidently rest in equilibrium on the surface, 
 even if it were smooth. This is an instance of (what we shall pre- 
 sently develop at some length) a surface of equilibrium, a level 
 surface, or an equipotential surface. 
 
 (/) We may further prove, by a simple application of the 
 preceding theorem, that the lines of force due to the attraction 
 of two infinitely long rods in the line AB produced, one of which 
 is attractive and the other repulsive, are the series of ellipses 
 described from the extremities, A and B, as foci, while the surfaces 
 of equilibrium are generated by the revolution of the confocal 
 hyperbolas. 
 
i83 , ABSTRACT DYNAMICS. 
 
 500. As of immense importance, in the theory not only of gra- 
 vitation but of electricity, of magnetism, of fluid motion, of the 
 conduction of heat, etc., we give here an investigation of the most 
 important properties of the Potential. 
 
 501. This function was introduced for gravitation by Laplace, 
 but the name was first given to it by Green, who may almost be 
 said to have created the theory, as we now have it. Green's work 
 was neglected till 1846, and before that time most of its important 
 theorems had been re-discovered by Gauss, Chasles, Sturm, and 
 Thomson. 
 
 In § 245, the potential energy of a conservative system in any con- 
 figuration was defined. When the forces concerned are forces acting, 
 either really or apparently, at a distance, as attraction of gravitation, 
 or attractions or repulsions of electric or magnetic origin, it is in 
 general most convenient to. choose, for the zero configuration, infinite 
 distance between the bodies concerned. We have thus the following 
 definition : — 
 
 502. The mutual potential energy of two bodies in any relative 
 position is the amount of work obtainable from their mutual repulsion, 
 by allowing them to separate to an infinite distance asunder. When 
 the bodies attract mutually, as for instance when no other force than 
 gravitation is operative, their mutual potential energy, according to 
 the convention for zero now adopted, is negative, or their exhaustion 
 of potential energy is positive. 
 
 503. The Potential at any point, due to any attracting or repelling 
 body, or distribution of matter, is the mutual potential energy between 
 it and a unit of matter placed at that point. But in the case of 
 gravitation, to avoid defining the potential as a negative quantity, 
 it is convenient to change the sign. Thus the gravitation potential, 
 at any point, due to any mass, is the quantity of work required to 
 remove a unit of matter from that point to an infinite distance. 
 
 504. ^ Hence, if V be the potential at any point P, and V^ that at 
 a proximate point (2, it evidently follows from the above definition 
 that V- Fj is the work required to remove an independent unit of 
 matter from P\.q Q-, and it is useful to note that this is altogether 
 independent of the form of the path chosen between these two points, 
 as it gives us a preliminary idea of the power we acquire by the 
 introduction of this mode of representation. 
 
 Suppose Q to be so near to P that the attractive forces exerted on 
 unit of matter at these points, and therefore at any point in the line 
 jP^, may be assumed to be equal and parallel. Then if F represent 
 the resolved part of this force along PQ, F. PQ is the work required 
 to transfer unit of matter from P to Q. Hence 
 
 V-V^=F.PQ, 
 ""—PQ^ 
 
STATICS OF A PARTICLE.— ATTRACTION. 183 
 
 that is, the attraction on unit of matter at P in any direction PQ^ 
 is the rate at which the potential at P increases per unit of length 
 oiPQ. 
 
 505. A surface, at every point of which the potential has the same 
 value, and therefore called an Equipotential Surface^ is such that the 
 attraction is everywhere in the direction of its normal. For in no 
 direction along the surface does the potential change in value, and 
 therefore there is no force in any such direction. Hence if the 
 attracted particle be placed on such a surface (supposed smooth and 
 rigid), it will rest in any position, and the surface is therefore some- 
 times called a Surface of Eqicilibrium. We shall see later, that the 
 force on a particle of a liquid at the free surface is always in the 
 direction of the normal, hence the term Level Surface, which is often 
 used for the other terms above. 
 
 506. If a series of equipotential surfaces be constructed for values 
 of the potential increasing by equal small amounts, it is evident from 
 § 504 that the attraction at any point is inversely proportional to 
 the normal distance between two successive surfaces close to that 
 point; since the numerator of the expression for F is, in this case, 
 constant. 
 
 507. A line drawn from any origin, so that at every point of its 
 length its tangent is the direction of the attraction at that point, is 
 called a Line of Force; and it obviously cuts at right angles every 
 equipotential surface which it meets. 
 
 These three last sections are true whatever be the law of attraction ; 
 in the next we are restricted to the law of the inverse square of the 
 distance. 
 
 508. If, through every point of the boundary of an infinitely 
 small portion of an equipotential surface, the corresponding lines of 
 force be drawn, we shall evidently have a tubular surface of infinitely 
 small section. The resultant force, being at every point tangential 
 to the direction of the tube, is inversely as its normal transverse 
 section. 
 
 This is an immediate consequence of a most important theorem, 
 which will be proved later. The surface integral of the attraction 
 exerted by any distribution of matter i?t the direction of the normal at 
 every poi7it of any closed suiface is ^irM; where M is the amount of 
 matter withiii the surface., while the attraction is considered positive or 
 negative according as it is inibards or outwards at any poi?tt of the 
 surface. 
 
 For in the present case the force perpendicular to the tubular 
 part of the surface vanishes, and we need consider the ends only. 
 VVhen none of the attracting mass is within the portion of the tube 
 considered, we have at once 
 
 F^uT - Fvs' = o, 
 i^ being the force at any point of the section whose area is zsr. 
 
 This is equivalent to the celebrated equation of Laplace. 
 
i84 ABSTRACT DYNAMICS. 
 
 When the attracting body is symmetrical about a point, the lines 
 of force are obviously straight lines drawn from this point. Hence 
 the tube is in this case a cone, and, by § 486, ra- is proportional to 
 the square of the distance from the vertex. Hence F is inversely 
 as the square of the distance for points external to the attracting 
 mass. 
 
 When the mass is symmetrically disposed about an axis in in- 
 finitely long cylindrical shells, the lines of force are evidently perpen- 
 dicular to the axis. Hence the tube becomes a wedge, whose section 
 is proportional to the distance from the axis, and the attraction is 
 therefore inversely as the distance from the axis. 
 
 When the mass is arranged in infinite parallel planes, each of 
 uniform density, the lines of force are obviously perpendicular to 
 these planes; the tube becomes a cylinder; and, since its section is 
 constant, the force is the same at all distances. 
 
 If an infinitely small length / of the portion of the tube considered 
 pass through matter of density p, and if w be the area of the section 
 of the tube in this part, we. have 
 
 F'us — F'w = 47r/o)p, 
 
 This is equivalent to Poisson's extension of Laplace's equation. 
 
 509. In estimating work done against a force which varies in- 
 versely as the square of the distance from a fixed point, the mean 
 force is to be reckoned as the geometrical mean between the forces 
 at the beginning and end of the path: and, whatever may be the 
 path followed, the effective space is to be reckoned as the difference 
 of distances from the attracting point. Thus the work done in any 
 course is equal to the product of the difference of distances of the 
 extremities from the attracting point, into the geometrical mean of 
 the forces at these distances; or, if O be the attracting point, and 7/1 
 its force on a unit mass at unit distance, the work done in moving a 
 particle, of unit mass, from any position F to any other position 
 F\ is 
 
 n' mm 
 
 (,OP-OP) J 
 
 or 
 
 Qp2 Qp,2^ -- Qp Qp,' 
 
 To prove this it is only necessary to remark, that for any infinitely 
 small step of the motion, the effective space is clearly the difference 
 of distances from the centre, and the working force may be taken as 
 the force at either end, or of any intermediate value, the geometrical 
 mean for instance: and the preceding expression applied to each 
 infinitely small step shows that the same rule holds for the sum 
 making up the whole work done through any finite range, and by 
 any path. 
 
 Hence, by § 503, it is obvious that the potential at F, of a mass m 
 
 m 
 situated at (9, is -7^] and thus that the potential of any mass at a 
 
 point F is to be found by adding the quotients of every portion of 
 the mass, each divided by its distance from /*. 
 
STATICS OF A PARTICLE.— ATTRACTION. 185 
 
 510. Let ^ be any dosed surface, and let 6^ be a point, either 
 external or internal, where a mass, m, of matter is collected. Let N 
 be the component of the attraction of m in the direction of the 
 normal drawn inwards from any point P, of S. Then, if da denotes 
 an element of 6", and // integration over the whole of it, 
 
 // N d<T = 47rw, or = o, 
 
 according as O is internal or external. 
 
 Case /, O internal. Let OP^PJP^... be a straight line drawn in 
 any direction from O, cutting 6* in P^^ P^, P^, etc., and therefore 
 passing out at P^, in at P^^ out again at P^, in again at P^, and so 
 on. Let a conical surface be described by lines through O, all in- 
 finitely near OP^P„..., and let w be its solid angle (§ 482). The 
 portions of jjNda- corresponding to the elements cut from S by 
 this cone will be clearly each equal in absolute magnitude to lonij 
 but will be alternately positive and negative. Hence as there is an 
 odd number of them, their sum is + irnn. And the sum of these, for 
 all solid angles round O, is (§ 483) equal to ^irm; that is to say, 
 jjNda-^ 4irm. 
 
 Case II, O external. Let OP^P^P^... be aline drawn from O pass- 
 ing across S, inwards at P^, outwards at P„, and so on. Drawing, 
 as before, a conical surface of infinitely small solid angle, w, we have 
 still uitn for the absolute value of each of the portions of jjNdcr 
 corresponding to the elements which it cuts from S', but their signs 
 are alternately negative and positive ; and therefore as their number 
 is even, their sum is zero. Hence 
 
 jjNdiT = 0. 
 
 From these results it follows immediately that if there be any con- 
 tinuous distribution of matter, partly within and partly without a 
 closed surface S, and N and da- be still used with the same signifi- 
 cation, we have 
 
 J5Nd(T = 4TrM 
 
 if M denote the whole amount of matter within S, 
 
 511. From this it follows that the potential cannot have a maxi- 
 mum or minimum value at a point in free space. For if it were so, 
 a closed surface could be described about the point, and indefinitely 
 near it, so that at every point of it the value of the potential would be 
 less than, or greater than, that at the point ; so that N would be 
 negative or positive all over the surface, and therefore jjNda- would 
 be finite, which is impossible, as the surface contains none of the 
 attracting mass. 
 
 512. It is also evident that iVmust have positive values at some 
 parts of this surface, and negative values at others, unless it is zero 
 all over it. Hence in free space the potential, if not constant round 
 any point, increases in some directions from it, and diminishes in 
 
i86 ABSTRACT DYNAMICS. 
 
 others; and therefore a material particle placed at a point of 
 zero force under the action of any attracting bodies, and free 
 from all constraint, is in unstable equilibrium, a result due to 
 Earnshaw\ 
 
 513. If the potential be constant over a closed surface which 
 contains none of the attracting mass, it has the same constant value 
 throughout the interior. For if not, it must have a maximum or 
 minimum value somewhere within, which is impossible. 
 
 514. The mean potential over any spherical surface, due to matter 
 entirely without it, is equal to the potential at its centre ; a theorem 
 apparently first given by Gauss. See also Cambridge Mathematical 
 
 Journal^ Feb. 1845 (vol. iv. p. 225). This proposition is merely an 
 extension, to any masses, of the converse of the following statement, 
 which is easily seen to follow from the results of §§ 479, 488 expressed 
 in potentials instead of forces. The potential of an uniform spherical 
 shell at an external point is the same as if its mass were condensed 
 at the centre. At all internal points it has the same value as at the 
 surface. 
 
 515. If the potential of any masses has a constant value, F, 
 through any finite portion, K, of space, unoccupied by matter, it is 
 equal to F through every part of space which can be reached in any 
 way without passing through any of those masses : a very remarkable 
 proposition, due to Gauss. For, if the potential difier from V in 
 space contiguous to K, it must (§513) be greater in some parts and 
 less in others. 
 
 From any point C within K, as centre, in the neighbourhood of a 
 place where the potential is greater than V, describe a spherical 
 surface not large enough to contain any part of any of the attracting 
 masses, nor to include any of the space external to K except such 
 as has potential greater than V. But this is impossible, since we 
 have just seen (§ 514) that the mean potential over the spherical 
 surface must be V. Hence the supposition that the potential is 
 greater than V in some places and less in others, contiguous to K 
 and not including masses, is false. 
 
 516. Similarly we see that in any case of symmetry round an 
 axis, if the potential is constant through a certain finite distance, 
 however short, along the axis, it is constant throughout the whole 
 space that can be reached from this portion of the axis, without 
 crossing any of the masses. 
 
 517. Let 5 be any finite portion of a surface, or complete closed 
 surface, or infinite surface, and let E be any point on S. (a) It is 
 possible to distribute matter over S so as to produce potential equal 
 to T {£), any arbitrary function of the position of £, over the whole 
 of S. {b) There is only one whole quantity of matter, and one 
 distribution of it, which can satisfy this condition. For the proof of 
 
 ^ Cambridge Phil. Trans., March, 1839. 
 
STATICS OF A PARTICLE.—ATTRACTION. 187 
 
 this and of several succeeding theorems, we refer the reader to our 
 larger work. 
 
 518. It is important to remark that, if S consist, in part, of a 
 closed surface, <2) the determination of U^ the potential at any point, 
 within it will be independent of those portions of 6", if any, which 
 lie without it; and, vice versa, the determination of C/ through external 
 space will be independent of those portions of S, if any, which lie 
 within the part Q. Or if S consist, in part, of a surface Q, extend- 
 ing infinitely in all directions, the determination of C/ through all 
 space on either side of Q, is independent of those portions of S, if 
 any, which lie on the other side. 
 
 519. Another remark of extreme importance is this: — If F(E) 
 be the potential at E of any distribution, M, of matter, and if S be 
 such as to separate perfectly any portion or portions of space, ZT, 
 from all of this matter; that is to say, such that it is impossible to 
 pass into ZTfrom any part of J/ without crossing -5"; then, throughout 
 JI, the value of U will be the potential of M. 
 
 520. Thus, for instance, if .S consist of three detached surfaces, 
 5j, S^, 6*3, as in the diagram, of which S^, S^ are closed, and S^ is 
 an open shell, and if F {E) be 
 the potential due to M, at any 
 point, E, of any of these portions 
 of S', then throughout H^ and 
 
 Z^, the spaces within S^ and witli- \^ f ^^ ^^^^( //< W //, 
 out 6*2, the value of U is simply 
 the potential of M. The value of 
 U through K, the remainder of 
 space, depends, of course, on the 
 character of the composite sur- 
 face S. 
 
 521. From § 518 follows the grand proposition : — // is possible to 
 fitid one, but 710 other thati one, distribution of matter over a surface S 
 
 which shall produce over S, and throughout all space H separated by S 
 from every part of M, the same potential as any given mass M. 
 
 Thus, in the preceding diagram, it is possible to find one, and but 
 one, distribution of matter over S^, S^, S.^ which shall produce over 
 6*3 and through H^ and H^ the same potential as M. 
 
 The statement of this proposition most commonly made is : // is 
 possible to distribute matter over any surface, S, completely enclosing a 
 mass M, so as to produce the same potential as M through all space 
 outside M ; which, though seemingly more limited, is, when inter- 
 preted with proper mathematical comprehensiveness, equivalent to the 
 foregoing. 
 
 522. If S consist of several closed or infinite surfaces, S^, S^, S^, 
 respectively separating certain isolated spaces H^, H^, H^, from H, 
 the remainder of all space, and if F {E) be the potential of masses 
 m^, m^, m^, lying in the spaces H^, H^, H^\ the portions of C/"due to 
 
i88 
 
 ABSTRACT DYNAMICS, 
 
 Sxt -5*2, ^g, respectively will throughout H be equal respectively to 
 the potentials oi m^^ m^, m^, separately. 
 
 For, as we have just seen, it is possible to find one, but only one, 
 distribution of matter over S, which shall produce the potential oi m^^ 
 jj throughout all the space H, H^, 
 
 j^g, etc., and one, but only one, 
 distribution over S^ which shall 
 produce the potential of m^ 
 throughout ZT, H^, H^, etc. ; and 
 so on. But these distributions 
 on ^Sj, vSg, etc., jointly constitute 
 a distribution producing the po- 
 tential F {E) over every part of 
 S^ and therefore the sum of the 
 potentials due to them all, at any point, fulfils the conditions pre- 
 sented for U. This is therefore (§ 5 18) the solution of the problem. 
 
 523. Considering still the case in which F {E) is prescribed to be 
 the potential of a given mass, M : let S be an equipotential surface 
 enclosing M, or a group of isolated surfaces enclosing all the parts of 
 M^ and each equipotential for the whole of M. The potential due to 
 the supposed distribution over S will be the same as that of J/", 
 through all external space, and will be constant (§ 514) through each 
 enclosed portion of space. Its resultant attraction will therefore be 
 the same as that of M on all external points, and zero on all internal 
 points. Hence we see at once that the density of the matter dis- 
 
 tributed over it, to produce F (F), is equal to — where F denotes 
 
 4Tr 
 
 the resultant force of M, at the point F. 
 
 524. When M consists of two portions m^ and pi separated by an 
 equipotential S^, and 6* consists of two portions, S^ and S\ of which 
 the latter separates the former perfectly from m' ; we see, by § 522, 
 that the distribution over S^ produces through all space on the side 
 of it on which S' lies, the same potential, P\, as m^, and the dis- 
 tribution on S' produces through space on the side of it on which S^ 
 lies, the same potential, F', as ?//. But the supposed distribution on 
 the whole of 6" is such as to produce a -constant potential, Ci, over S^, 
 
STATICS OF A PARTICLE,— ATTRACTION. 189 
 
 and consequently the same at every point within S^. Hence the in- 
 ternal potential, due to S^ alone, is C^ - V. 
 
 Thus, passing from potentials to attractions, we see that the re- 
 sultant attraction of 6*1 alone, on all points on one side of it, is the 
 same as that of m^ ; and on the other side is equal and opposite to 
 that of the remainder m' of the whole mass. The most direct and 
 simple complete statement of this result is as follows : — 
 
 If masses m, 7n\ in portions of space, H^ II\ completely separated 
 from one another by one continuous surface S, whether closed or 
 infinite, are known to produce tangential forces equal and in the 
 same direction at each point of S, one and the same distribution 
 of matter over S will produce the force of m throughout II\ and 
 that of m' throughout II. The density of this distribution is equal to 
 
 — , if i? denote the resultant force due to one of the masses, and 
 
 the other with its sign changed. And it is to be remarked that the 
 direction of this resultant force is, at every point, E, of S, perpen- 
 dicular to S, since the potential due to one mass, and the other with 
 its sign changed, is constant over the whole of S. 
 
 525. Green, in first publishing his discovery of the result stated 
 in § 523, remarked that it shows a way to find an infinite variety of 
 closed surfaces for any one of which we can solve the problem of 
 determining the distribution of matter over it which shall produce 
 a given uniform potential at each point of its surface, and con- 
 sequently the same also throughout its interior. Thus, an example 
 which Green himself gives, let J/ be a uniform bar of matter, AA\ 
 The equipotential surfaces round it are, as we have seen above 
 (§ 499 W)j prolate ellipsoids of revolution, each having A and A' for 
 its foci; and the resultant force at Cwas found to be 
 
 ^ CR 
 
 the whole mass of the bar being denoted by m^ its length by 2dJ, and 
 ^'C+^Cby 2/. We conclude that a distribution of matter over 
 the surface of the ellipsoid, having 
 
 I m.CF 
 
 for density at C, produces on all external space the same resultant 
 force as the bar, and zero force or a constant potential through the 
 internal space. This is a particular case of the general result re- 
 garding ellipsoidal shells, proved below, in §§ 536, 537. 
 
 526. As a second example, let J/" consist of two equal particles, 
 at points /, /'. If we take the mass of each as unity, the potential at 
 
 Fisj-p + -jTp'} and therefore 
 
 IP^ FP ^ 
 
190 
 
 ABSTRACT DYNAMICS. 
 
 is the equation of an equipotential surface ; it being understood that 
 negative values of IP and I'P are inadmissible, and that any con- 
 stant value, from 00 to o, may be given to C. The curves in the 
 annexed diagram have t)een drawn, from this equation, for the cases 
 of C equal respectively to 10, 9, 8, 7, 6, 5, 4-5, 4-3, 4-2, 4*1, 4, 3-9, 
 3*8, 37, 3*5, 3, 2-5, 2; the value of //'being unity. 
 
 The corresponding equipotential surfaces are the surfaces traced 
 by these curves, if the whole diagram is made to rotate round II' as 
 
 axis. Thus we see that for any values of C less than 4 the equi- 
 potential surface is one closed surface. Choosing any one of these 
 
 surfaces, let R denote the resultant of forces equal to ^^ and -frSi 
 
 in the lines PI and PI'. Then if matter be distributed over this 
 
 surface, with density at P equal to — , its attraction on any internal 
 
 4t 
 
 point will be zero; and on any external point, will be the same as 
 that of /and/'. 
 
 527. For each value of C greater than 4, the equipotential surface 
 consists of two detached ovals approximating (the last three or four 
 in the diagram, very closely) to spherical surfaces, with centres lying 
 between the points / and /', but approximating more and more 
 closely to these points, for larger and larger values of C. 
 
 Considering one of these ovals alone, one of the series enclosing 
 /', for instance, and distributing matter over it according to the same 
 
 r> 
 
 law of density, — , we have a shell of matter which exerts (§ 525) 
 47r 
 
E 
 
 STATICS OF A PARTICLE.- ATTRACTION. 191 
 
 on external points the same force as /'; and on internal points a 
 force equal and opposite to that of /. 
 
 528. As an example of exceedingly great importance in the theory 
 of electricity, let M consist of a positive mass, m, concentrated 
 at a point /, and a negative 
 
 mass, — ;//, at /' j and let ^S" 
 be a spherical surface cutting 
 //' and //' produced in points 
 A, A^, such that 
 lA :AI'::IA, iTA/.-.m : m'. 
 Then, by a well-known geo- 
 metrical proposition, we shall 
 have m : T£ v. m -. m' \ and 
 therefore 
 
 m _ in 
 
 lE~ TE' 
 
 Hence, by what we have just seen, one and the same distribution of 
 
 matter over »S will produce the same force as m! through all external 
 
 space, and the same as m through all the space within S. And, 
 
 ffi . in' 
 
 finding the resultant of the forces y^ in EI^ and -77^^ in I'E^ pro- 
 
 duced, which, as these forces are inversely as IE to I'E^ is (§222) 
 equal to 
 
 m j.^, m^ir I 
 
 IE\I'E ' °^ 1^ IE'' ' 
 we conclude that the density in the shell at E is 
 
 m^ir I 
 
 47rw' ' lE^ ' 
 
 That the shell thus constituted does attract external points as if its 
 mass were collected at /', and internal points as a certain mass col- 
 lected at /, was proved geometrically in § 491 above. 
 
 529. If the spherical surface is given, and one of the points, /, /', 
 
 CA^ 
 for instance /, the other is found by taking CI' = -?^ ; and for the 
 
 mass to be placed at it we have 
 
 TA CA cr 
 
 '^='^AI = '^-Cl = ''' CA- 
 
 Hence, if we have any number of particles m^^ m^^ etc., at points 
 /j, /g, etc., situated without S, we may find in the same way cor- 
 responding internal points /'„ I\, etc., and masses m\^ m\, etc. ; 
 and, by adding the expressions for the density at E given for each 
 pair by the preceding formula, we get a spherical shell of matter 
 which has the property of acting on all external space with the same 
 force as —in\, -^'25 ^tc, and on all internal points with a force 
 equal and opposite to that of Wj, m^, etc. 
 
192 ABSTRACT DYNAMICS. 
 
 530. An infinite number of such particles may be given, con- 
 stituting a continuous mass M\ when of course the corresponding 
 internal particles will constitute a continuous mass, —M'^ of the 
 opposite kind of matter; and the same conclusion will hold. If S is 
 the surface of a solid or hollow metal ball connected with the earth 
 by a fine wire, and M an external influencing body, the shell of matter 
 we have determined is precisely the distribution of electricity on S 
 called out by the influence of M: and the mass - M\ determined as 
 above, is called the Electric Image of M in the ball, since the electric 
 action through the whole space external to the ball would be 
 unchanged if the ball were removed and — M' properly placed in 
 the space left vacant. We intend to return to this subject under 
 Electricity. 
 
 531. Irrespectively of the special electric application, this method 
 of images gives a remarkable kind of transformation which is often 
 useful. It suggests for mere geometry what has been called the 
 transformation by reciprocal radius-vectors ; that is to say, the sub- 
 stitution for any set of points, or for any diagram of lines or surfaces, 
 another obtained by drawing radii to them from a certain fixed point 
 or origin, and measuring off lengths inversely proportional to these 
 radii along their directions. We see in a moment by elementary 
 geometry that any line thus obtained cuts the radius-vector through 
 any point of it at the same angle and in the same plane as the line 
 from which it is derived. Hence any two lines or surfaces that cut 
 one another give two transformed lines or surfaces cutting at the 
 same angle : and infinitely small lengths, areas, and volumes trans- 
 form into others whose magnitudes are altered respectively in the 
 ratios of the first, second, and third powers of the distances of the 
 latter from the origin, to the same powers of the distances of the 
 former from the same. Hence the lengths, areas, and volumes in 
 the transformed diagram, corresponding to a set of given equal 
 infinitely small lengths, areas, and volumes, however situated, at 
 different distances from the origin, are inversely as the squares, the 
 fourth powers and the sixth powers of these distances. Further, it is 
 easily proved that a straight line and a plane transform into a circle 
 and a spherical surface, each passing through the origin ; and that, 
 generally, circles and spheres transform into circles and spheres. 
 
 532. In the theory of attraction, the transformation of masses, 
 densities, and potentials has also to be considered. Thus, according 
 to the foundation of the method (§ 530), equal masses, of infinitely 
 small dimensions at different distances from the origin, transform into 
 masses inversely as these distances, or directly as the transformed 
 distances : and, therefore, equal densities of lines, of surfaces, and of 
 solids, given at any stated distances from the origin, transform into 
 densities directly as the first, the third, and the fifth powers of those 
 distances ; or inversely as the same powers of the distances, from 
 the origin, of the corresponding points in the transformed system. 
 The usefulness of this transformation in the theory of electricity, 
 
STATICS OF A PARTICLE.— ATTRACTION. 193 
 
 and of attractian in general, depends entirely on the following 
 theorem : — 
 
 Let <f> denote the potential at F due to the given distribution, and <^' 
 the potential at F' due to the transformed distribution : then shall 
 
 <i> = - <^ = - ^. 
 a r 
 
 Let a mass m collected at / be any part of the given distri- 
 bution, and let m at /' be 
 the corresponding part in y^ 
 
 the transformed distribution, -^ 
 
 We have / \ 
 
 a'=Or.OI^OF\OF, / 
 
 and therefore / 
 
 01: OFv. OF' : OF; ^ 
 which shows that the triangles //^(9, /*'/'(? are similar, so that 
 
 IF iFF:: J 01 OF : JOF\Or - OIOF : a\ 
 We have besides 
 
 m : m \ : 01 : a, 
 and therefore 
 
 mm 
 Jp--rF.::a:OP. 
 
 Hence each term of <^ bears to the corresponding term of <^' 
 the same ratio; and therefore the sum, </>, must be to the sum, 
 <^', in that ratio, as was to be proved. 
 
 533. As an example, let the given distribution be confined to a 
 spherical surface, and let O be its centre and a its own radius. The 
 transformed distribution is the same. But the space within it becomes 
 transformed into the space, without it. Hence if ^ be the potential 
 due to any spherical shell at a point P, within it, the potential due 
 
 8 
 
 to the same shell at the point F in OF produced till OF' = -y^ , is 
 
 equal to yrpt ^ (which is an elementary proposition in the spherical 
 
 harmonic treatment of potentials, as we shall see presently). Thus, 
 for instance, let the distribution be uniform. Then, as we know 
 there is no force on an interior point, <^ must be constant; and 
 therefore the potential at F\ any external point, is inversely propor- 
 tional to its distance from the centre. 
 
 Or let the given distribution be a uniform shell, 5, and let O be 
 any eccentric or any external point. The transformed distribution 
 becomes (§§ 531, 532) a spherical shell, 6", with density varying 
 inversely as the cube of the distance from O. If O is within 6", it is 
 also enclosed by .S", and the whole space within S transforms into 
 
 T. 13 
 
194 ABSTRACT DYNAMICS. 
 
 the whole space without S'. Hence (§ 532) the potential of S' at 
 any point without it js inversely as the distance from 0^ and is there- 
 fore that of a certain quantity of matter collected at O. Or if O is 
 external to »S, and consequently also external to S\ the space within 
 S transforms into the space within S' . Hence the potential of S' 
 at any point within it is the same as that of a certain quantity of 
 matter collected at (9, which is now a point external to it. Thus, 
 without taking advantage of the general theorems (§§ 517, 524), we 
 fall back on the same results as we inferred from them in § 528, and 
 as we proved synthetically earlier (§§ 488, 491, 492). It may be 
 remarked that those synthetical demonstrations consist merely of 
 transformations of Newton's demonstration, that attractions balance 
 on a point within a uniform shell. Thus the first of them (§ 488) 
 is the image of Newton's in a concentric spherical surface ; and the 
 second is its image in a spherical surface having its centre external 
 to the shell, or internal but eccentric, according as the first or the 
 second diagram is used. 
 
 534. We shall give just one other application of the theorem 
 of § 532 at present, but much use of it will be made later in the 
 theory of Electricity. 
 
 Let the given distribution of matter be a uniform solid sphere, j5, 
 and let O be external to it. The transformed system will be a solid 
 sphere, B\ with density varying inversely as the fifth power of the 
 distance from (9, a point external to it. The potential of B is the 
 same throughout external space as that due to its mass, w, collected 
 at its centre, C. Hence the potential of B' through space external 
 to it is the same as that of the corresponding quantity of matter 
 collected at C\ the transformed position of C. This quantity is of 
 course equal to the mass of B'. And it is easily proved that C is 
 the position of the image of O in the spherical surface of B\ We 
 conclude that a solid sphere with density varying inversely as the 
 fifth power of the distance from an external point, O, attracts any 
 external point as if its mass were condensed at the image of O in its 
 external surface. It is easy to verify this for points of the axis by 
 direct integration, and thence the general conclusion follows ac- 
 cording to § 508. 
 
 535. The determination of the attraction of an ellipsoid, or of an 
 eUipsoidal shell, is a problem of great interest, and its results will be 
 of great use to us afterwards, especially in Magnetism. We have 
 left it till now, in order that we may be prepared to apply the pro- 
 perties of the potential, as they afford an extremely elegant method 
 of treatment. A few definitions and lemmas are necessary. 
 
 Corresponding points on two confocal ellipsoids are such as coincide 
 when either ellipsoid by a pure strain is deformed so as to coincide 
 with the other. 
 
 And it is easily shown, that if any two points, /*, Q, be assumed on 
 one shell, and their corresponding points, /, ^, on the other, we have 
 Pq-Qp. 
 
STATICS OF A PARTICLE.—ATTRACTION. 195 
 
 The species of shell which it is most convenient to employ in the 
 subdivision of a homogeneous ellipsoid is bounded by similar, simi- 
 larly situated, and concentric ellipsoidal surfaces; and it is evident 
 from the properties of pure strain (§ 141) that such a shell may be 
 produced from a spherical shell of uniform thickness by unijform 
 extensions and compressions in three rectangular directions. Unless 
 the contrary be specified, the word 'shell' in connexion with this 
 subject will always signify an infinitely thin shell of the kind now 
 described. 
 
 536. Since, by § 479, a homogeneous spherical shell exerts no 
 attraction on an internal point, a homogeneous shell (which need 
 not be infinitely thin) bounded by similar, and similarly situated, and 
 concentric ellipsoids, exerts no attraction on an internal point. 
 
 For suppose the spherical shell of § 479, by simple extensions and 
 compressions in three rectangular directions, to be transformed into 
 an ellipsoidal shell. In this distorted form the masses of all parts 
 are reduced or increased in the proportion of the mass of the eUipsoid 
 to that of the sphere. Also the ratio of the lines HP^ PK is un- 
 altered, § 139. Hence the elements IH^ KL still attract /'equally, 
 and the proposition follows as in § 479. 
 
 Hence inside the shell the potential is constant. 
 
 537. Two confocal shells (§ 535) being given, the potential of the 
 first at any point, P^ of the surface of the second, is to that of the 
 second at the corresponding point,/, on the surface of the first, as 
 the mass of the first is to the mass of the second. This beautiful 
 proposition is due to Chasles. 
 
 To any element of the mass of the outer shell at Q corresponds an 
 element of mass of the inner at q, and these bear the same ratio to 
 the whole masses of their respective shells, that the corresponding 
 element of the spherical shell from which either may be derived bears 
 to its whole mass. Whence, since Pq = 0^, the proposition is true 
 for the corresponding elements at Q and ^, and therefore for the 
 entire shells. 
 
 Also, as the potential of a shell on an internal point is constant, 
 and as one of two confocal ellipsoids is wholly within the other : it 
 follows that the external equipotential surfaces for any such shell are 
 confocal ellipsoids, and therefore that the attraction of the shell on an 
 external point is normal to a confocal ellipsoid passing through the 
 point. 
 
 538. Now it has been shown (§ 495) that the attraction of a shell 
 on an external point near its surface exceeds that on an internal 
 point infinitely near it by 47rp where p is the surface-density of the 
 shell at that point. Hence, as (§ 536) there is no attraction on an 
 internal point, the attraction of a shell on a point at its exterior 
 surface is 47rp: or 47rp/ if p be now put for the volume-density, and t 
 for the (infinitely small) thickness of the shell, § 495. From this 
 it is easy to obtain by integration the determination of the whole 
 attraction of a homogeneous ellipsoid on an external particle. 
 
 13 — 2 
 
196 ABSTRACT DYNAMICS. 
 
 539. The following splendid theorem is due to Maclaurin : — 
 The attractions exerted by two homogeneous and confocal ellipsoids 
 
 on the same poifit external to each, or external to one and on the sur- 
 face of the other, are in the same directioft and proportional to their 
 masses. 
 
 540. Ivory's theorem is as follows : — 
 
 Let correspondtJig points F, p, be taken on the surfaces of two homo- 
 geneous co?ifocal ellipsoids, E, e. The x compo7ient of the attraction of E 
 on p, is to that of e on P as the area of the section of E by the plane of 
 yz is to that of the coplanar section of e. 
 
 Poisson showed that this theorem is true for any law of force 
 whatever. This is easily proved by employing in the general ex- 
 pressions for the components of the attraction of any body, after one 
 integration, the properties of corresponding points upon confocal 
 ellipsoids (§ 535). 
 
 541. An ingenious application of Ivory's theorem, by Duhamel, 
 must not be omitted here. Concentric spheres are a particular case 
 of confocal ellipsoids, and therefore the attraction of any sphere on 
 a point on the surface of an internal concentric sphere, is to that of 
 the latter upon a point in the surface of the former as the squares of 
 the radii of the spheres. Now if the law of attraction be such that a 
 homogeneous spherical shell of tmiforjfi thick7iess exerts 7io attraction on 
 a7i inter7ial poi7tt, the action of the larger sphere on the internal point 
 is reducible to that of the smaller. Hence the law is that of the i7i- 
 verse square of the dista7ice, as is easily seen by making the smaller 
 sphere less and less till it becomes a mere particle. This theorem is 
 due originally to Cavendish. 
 
 542. {Def7uti07z.) If the action of terrestrial or other gravity on 
 a rigid body is reducible to a single force in a line passing always 
 through one point fixed relatively to the body, whatever be its position 
 relatively to the earth or other attracting mass, that point is called its 
 centre of gravity, and the body is called a ce7itrobaric body. 
 
 543. One of the most startling result-s of Green's wonderful theory 
 of the potential is its establishment of the existence of centrobaric 
 bodies; and the discovery of their properties is not the least curious 
 and interesting among its very various applications. 
 
 544. If a body {B) is centrobaric relatively to any one attracting 
 mass {A), it is centrobaric relatively to every other: and it attracts 
 all matter external to itself as if its own mass were collected in its 
 centre of gravity.' 
 
 545. Hence §§ 510, 515 show that — 
 
 {a) The ce7itre of gravity of a ce7itrobaric body necessarily lies in its 
 interior; or in other words, ca7t only be reached from external space by 
 a path cutting through some of its 77iass. And 
 
 {b) No centrobaric body can co7isist ofpa7'ts isolated frofn one another, 
 
 ^ Thomson. Proc. R.S.E., Feb. 1864. 
 
STATICS OF A PARTICLE.— ATTRACTION, 197 
 
 each in space externa! to all: in other words, the outer boundary of every 
 centroharic body is a si7igle closed surface. 
 
 Thus we see, by {a)^ that no symmetrical ring, or hollow cylinder 
 with open ends, can have a centre of gravity; for its centre of 
 gravity, if it had one, would be in its axis, and therefore external to 
 its mass. 
 
 546. If any 77tass whatever^ M^ and any single surface, *S, com- 
 pletely enclosing it be given, a distributioJi of any given amount, M\ 
 of matter on this surface may be found which shall make the whole 
 centrobaric with its centre of gravity in any given position (G) within 
 that surface. 
 
 The condition here to be fulfilled is to distribute M' over S, so as 
 by it to produce the potential 
 
 AI^-M' _ 
 EG 
 at any point, E, of S\ V denoting the potential of M at this point. 
 The possibiUty and singleness of the solution of this problem were 
 stated above (§ 517). It is to be remarked, however, that if M' be 
 not given in sufficient amount, an extra quantity must be taken, but 
 neutralized by an equal quantity of negative matter, to constitute the 
 required distribution on S. 
 
 The case in which there is no given body M to begin with is 
 important; and yields the following: — 
 
 547. A given quantity of matter may he distributed in one way, but 
 in only one way, over any given closed surface, so as to constitute a 
 centrobaric body with its centre of gravity at any given point within it. 
 
 Thus we have already seen that the condition is fulfilled by making 
 the density inversely as the distance from the given point, if the 
 surface be spherical. From what was proved in §§ 519, 524 above, 
 it appears also that a centrobaric shell may be made of either half of 
 the lemniscate in the diagram of § 526, or of any of the ovals within 
 it, by distributing matter with density proportional to the resultant 
 force oim 2X I and ;;/ at /' ; and that the one of these points which 
 is within it is its centre of gravity. And generally, by drawing the 
 equipotential surfaces relatively to a mass m collected at a point /, 
 and any other distribution of matter whatever not surrounding this 
 point ; and by taking one of these surfaces which encloses / but 
 no other part of the mass, we learn, by Green's general theorem, 
 and the special proposition of § 524, how to distribute matter 
 over it so as to make it a centrobaric shell with / for centre of 
 gravity. 
 
 548. Under hydrokinetics the same problem will be solved for a 
 cube, or a rectangular parallelepiped in general, in terms of con- 
 verging series; and under electricity (in a subsequent volume) it will 
 be solved in finite algebraic terms for the surface of a lens bounded 
 by two spherical surfaces cutting one another at any sub-multiple of 
 two right angles, and for either part obtained by dividing this surface 
 
198 ABSTRACT DYNAMICS. 
 
 in two by a third spherical surface cutting each of its sides at right 
 angles. 
 
 549. Matter may be distributed in an iftfi?iite number of ways 
 throughout a given closed space, to constitute a centrobaric body with its 
 centre of gravity at any given poi?it within it. 
 
 For by an infinite number of surfaces, each enclosing the given 
 point, the whole space between this point and the given closed surface 
 may be divided into infinitely thin shells; and matter may be dis- 
 tributed on each of these so as to make it centrobaric with its centre 
 of gravity at the given point. Both the forms of these shells and the 
 quantities of matter distributed on them, may be arbitrarily varied in 
 an infinite variety of ways. 
 
 Thus, for example, if the given closed surface be the pointed oval 
 constituted by either half of the lemniscate of the diagram of § 526, 
 and if the given point be the point / within it, a centrobaric solid 
 may be built up of the interior ovals with matter distributed over 
 them to make them centrobaric shells as above (§ 547). From what 
 was proved in § 534, we see that a solid sphere with its density 
 varying inversely as the fifth power of the distance from an external 
 point, is centrobaric, and that its centre of gravity is the image (§ 530) 
 of this point relatively to its surface. 
 
 550. The centre of gravity of a centrobaric body composed of 
 true gravitating matter is its centre of inertia. For a centrobaric 
 body, if attracted only by another infinitely distant body, or by matter 
 so distributed round itself as to produce (§ 517) uniform force in 
 parallel lines throughout the space occupied by it, experiences (§ 544) 
 a resultant force always through its centre of gravity. But in this 
 case this force is the resultant of parallel forces on all the particles of 
 the body, which (see Properties of Matter, below) are rigorously pro- 
 portional to their masses: and it is proved that the resultant of 
 such a system of parallel forces passes through the point defined in 
 § 195, as the centre of inertia. 
 
 551. The moments of inertia of a centrobaric body are equal 
 round all axes through its centre of inertia. In other words (§ 239), 
 all these axes are principal axes, and the body is kinetically sym- 
 metrical round its centre of inertia. 
 
CHAPTER VII. 
 
 STATICS OF SOLIDS AND FLUIDS. 
 
 552. Forces whose lines meet. Let ABC be a rigid body acted 
 on by two forces, F and (2, applied to it 
 at different points, D and E respectively, in 
 lines in the same plane. 
 
 Since the lines are not parallel, they will 
 meet if produced; let them be produced and 
 meet in O. Transmit the forces to act on 
 that point; and the result is that we have 
 simply the case of two forces acting on a 
 material point, which has been already con- 
 sidered. 
 
 553. The preceding solution is applicable to every case of non- 
 parallel forces in a plane, however far removed the point may be in 
 which their Hnes of action meet, and the resultant will of course be 
 found by the parallelogram of forces. The limiting case of parallel 
 forces, or forces whose lines of action, however far produced, do not 
 meet, was considered above, and the position and magnitude of 
 the resultant were investigated. The following is an independent 
 demonstration of the conclusion arrived at. 
 
 554. Parallel forces in a plane. The resultant of two parallel 
 forces is equal to their sum, and is in the parallel line which divides 
 any line drawn across their Hnes of action into parts inversely as their 
 magnitudes. 
 
 1°. Let P and Q be two parallel forces acting on a rigid body in 
 similar directions in lines AB and CD. Draw any line AC across 
 their lines. In it introduce any 
 pair of balancing forces. Sin AG 
 and 5 in CH. These forces will 
 not disturb the equilibrium of 
 the body. Suppose the forces 
 F and S'm. AG, and Q and S in 
 CH^ to act respectively on the 
 points A and C of the rigid body. 
 The forces F and S, in AB and 
 AGy have a single resultant in 
 some line AM, within the angle 
 
200 ABSTRACT DYNAMICS. 
 
 GAB; and Q and S in CD and CH have a resultant in some line 
 CN, within the angle DCff. 
 
 The angles MAC, NCA are together greater than two right . 
 angles, hence the lines MA^ NC will meet if produced. Let them 
 meet in O. Now the two forces P and S may be transferred to 
 parallel lines through O. Similarly the forces g and 5 may be also 
 transferred. Then there are four forces acting on (9, two of which, 
 S in OK and 6* in OL^ are equal and directly opposed. They may, 
 therefore, be removed, and there are left two forces equal to P and 
 (2 in one line on O^ which are equivalent to a single force P-\- Q'ln 
 the same line. 
 
 2°. If, for a moment, we suppose OE to represent the force P, 
 then the force representing S must be equal and parallel to EA^ since 
 the resultant of the two is in the direction OA. That is to say, 
 
 S\P'.\EA\ 0E\ 
 
 and in like manner, by considering the forces S in OL and Q in OE, 
 we find that 
 
 Q'.SwOE'.EC. 
 
 Compounding these analogies, we get at once 
 
 Q: P'.'.EA'.ECy 
 
 that is, the parts into which the line is divided by the resultant are 
 inversely as the forces. 
 
 555. Forces in dissimilar directions. The resultant of two parallel 
 forces in dissimilar directions^, of which one is greater than the other, 
 is found by the following rule : Draw any line across the lines of the 
 forces and produce it across the line of the greater, until the whole 
 line is to the part produced as the greater force is to the less ; a force 
 equal to the excess of the greater force above the less, applied at the 
 extremity of this line in a parallel line and in the direction similar to 
 that of the greater, is the resultant of the system. 
 
 Let /'and Q in KK' and LL\ be the 
 contrary forces. From any point A^ in 
 • the line of P, draw a line AB across the 
 line of Q cutting it in B, and produce the 
 -p line to E, so that AE : BE :: Q \ P. 
 Through E draw a line MM' parallel to 
 K li iW KK' or LL'. 
 
 In MM' introduce a pair of balancing forces each equal to Q- P. 
 Then P in AK' and Q~ P in EM have a resultant equal to their 
 
 1 In future the word 'contrary' will be employed instead of the phrase 
 'parallel and in dissimilar directions' to designate merely directional opposiiiotty 
 while the unqualified word 'opposite' will be understood to signify contrary and 
 in one line. 
 
STATICS OF SOLIDS AND FLUIDS. 201 
 
 sum, or Q. This resultant is in the line LL' j for, from the ana- 
 logy* 
 
 AEvBEv. Q:F, 
 
 we have AE-BE \ BE w Q- P '. F, 
 
 or AB :BEy. Q~F\F. 
 
 Hence F in AK\ Q in BL', and Q-F in EM are in equilibrium 
 and may be removed. There remains only Q-Fm EM\ which is 
 therefore the resultant of the two given forces. This fails when the 
 forces are equal 
 
 556. Any number of parallel forces in a plane. Let F^, F^, F^, 
 etc., be any number of parallel forces 
 
 acting on a rigid body in one plane. j I I I / 
 
 To find their resultant in position and ^ L^ If^'L^ L^ U^ 
 magnitude, draw any line across their r»J f 1 i if 
 
 lines of action, cutting them in points, ^J \pjfa -?[ ^f 
 
 denoted respectively by ^ J, ^2, ^3, etc., 
 
 and in it choose a point of reference O. Let the distances of the 
 lines of the forces from this point be denoted by a^, a^^ ^3, etc.; as 
 OA^ = a^, OA^ = a^, etc. Also let F denote the resultant, and x its 
 distance from O. 
 
 Find the resultant of any two of the forces, as F^ and F^, by 
 § 554. Then if we denote this resultant by F', we have 
 
 F'^F, + F^. 
 Divide A^ A^ in E' into parts inversely as the forces, so that 
 
 F,xA,E' = F^y<E'A^. 
 Hence if we denote OE' by x' we have 
 
 F,x{x'-a,) = F^x{a^-x') 
 or (F, + F^)x'==F,a, + F^a^, 
 
 that is F' x' = F^a^-¥ F^ a^ . 
 
 Similarly we shall find the resultant of R' and F^ to- be 
 F'^R'^F, = F,^F,^F,', 
 and R"x" = R'xf + F^a^^F^a^+F^a^ + F^a^. 
 
 Hence, finally we have 
 
 R^F,^F,^F^^ + /'„ (i), 
 
 and Rx=^F^a^-\-F^a^ + F^a^+ ■.+^„^,. •••(2). 
 
 In this method negative forces or negative values of any of the 
 quantities a^,a^, ..., maybe included, provided the generalized rules of 
 multiplication and addition in algebra are followed. 
 
 557. Any number of parallel forces not in one plane. To find the 
 resultant, let a plane cut the lines of all the forces, and let the points 
 

 
 n 
 
 N, 
 
 K 
 
 J^'JV^ 
 
 202 ABSTRACT DYNAMICS. 
 
 in which they are cut be specified by reference to two rectangular 
 axes in the plane. Let the plane be YOX\ OX^ OV, the axes of 
 reference, O the origin of co-ordinates, and A^, A^, A^, etc., the points 
 in which the plane cuts the lines of the forces, 1*^^, T^, T^, etc. Thus 
 each of these points will be specified by perpendiculars drawn from 
 
 it to the axis. Let the co-ordinates 
 of the point A^ be denoted by x^ yyy 
 of A^, by x^ y^; and so on; that is, 
 OJV, = x„ N, A, =y,; OJV^ = x^, N^ A^ 
 =y^, etc.; let also the final resultant be 
 ^j denoted by R, and its co-ordinates by 
 
 X and y. 
 
 Find the resultant of /\ and P^ by 
 joining Aj^ A„, and dividing the line 
 -jV^-^ inversely as the forces. Suppose E' 
 the point in which this resultant cuts 
 the plane of reference. Then 
 
 T,xA,E' = F,xE^A,. 
 
 To find the co-ordinates, which may be denoted by x'y', of the 
 point E' with reference to OX and O V; draw E'JV' perpendicular 
 to OX and cutting it in JV', and from A^ draw A^ K parallel to OX, 
 or perpendicular to A^N^, and cutting it in ^ and E'N' in M. 
 Then (Euclid vi. 2) 
 
 A,E' :E'A^::A,M:MX. 
 
 Hence T^ x A,M= P^ x MK, 
 
 or P^{x'-x:)--P^Kx^-x'\ 
 
 whence we get {P^ + P^ x' = P^x^ -1- P^x^ ; 
 
 and since P' = P^ + P^^ 
 
 we have P'x' = P^x^ + P^x^ , 
 
 and similarly, P'y' = P^y^ -{- P^y^. 
 
 We may find the resultant of P' and P^ in like manner, and so 
 with all the forces. Hence we have for the final resultant, 
 
 ji==A+p,+j',+ +j>„.. 
 
 Py = P,y, + P,y, + P,j>,+ ... +P,y, 
 
 (3), 
 (4), 
 
 (s). 
 
 These equations may include negative forces, or negative co- 
 ordinates. 
 
 558. Conditions of equilibrium of any number of parallel forces. 
 In order that any given parallel forces may be in equilibrium, it is 
 not sufificient alone, that their algebraic sum be equal to zero. 
 
 For, let P^P^ + P^ + etc. - o. 
 
STATICS OF SOLIDS AND FLUIDS. 203 
 
 From this equation it follows that if the forces be divided into two 
 groups, one consisting of the forces reckoned positive, the other of 
 those reckoned negative, the sum, or resultant (§ 556), of the former 
 is equal to the resultant of the latter; that is, \i ^R and 'F denote the 
 resultants of the positive and negative groups respectively, 
 
 But unless these resultants are directly opposed they do not balance 
 one another; wherefore, if {x^y) and (x'y) be the co-ordinates of 
 ^F and 'F respectively, we must have for equilibrium 
 
 and ,y = 'y', 
 
 whence we get ,F ^x - 'F'x = o 
 
 and ,Fj-'Fy = o. 
 
 But ^F ^x is equal to the sum of those of the terms Fy_x^, F^^, etc., 
 which are positive, and 'F'x is equal to the sum of the others each 
 with its sign changed : and so for ^F^y and 'F'y. Hence the pre- 
 ceding equations are equivalent to 
 
 F^x^-¥ F^^+ +Fjx:^^=^o. 
 
 ■^lJl + -^2j2+ +F^j^=o. 
 
 We conclude that, for equilibrium, it is necessary and sufficient that 
 each of the following three equations be satisfied : — 
 
 F, + F, + F^+ +F„=o (6), 
 
 F^x^ + Fjc^ + F^x.^ + + F^x^ = o (7), 
 
 F,y,+F,y, + F,y,+ +^„y. = o (8). 
 
 559. If equation (6) do not hold, but equations (7) and (8) do, the 
 forces have a single resultant through the origin of co-ordinates. If 
 equation (6) and either of the other two do not hold, there will be a 
 single resultant in a hne through the corresponding axis of reference, 
 the co-ordinates of the other vanishing. If equation (6) and either of 
 the other two do hold, the system is reducible to a single couple in a 
 plane through that hne of reference for which the sum of the products 
 is not equal to nothing. If the plane of reference is perpendicular to 
 the lines of the forces, the moment of this couple is equal to the sum 
 of the products not equal to nothing. 
 
 560. In finding the resultant of two contrary forces in any case in 
 which the forces are unequal — the smaller the difference of magnitude 
 between them, the farther removed is the point of application of the 
 resultant. When the difference is nothing, the point is removed to 
 an infinite distance, and the construction (§ 555) is thus rendered 
 nugatory. The general solution gives in this case F = 0; yet the 
 forces are not in equilibrium, since they are not directly opposed. 
 Hence two equal contrary forces neither balance, nor have a single 
 resultant. It is clear that they have a tendency to turn the body to 
 
204 ABSTRA CT D YNAMICS. 
 
 which they are applied. This system was by Poinsot denominated 
 a couple. 
 
 In actual cases the direction of a couple is generally reckoned 
 positive if the couple tends to turn contrary to the hands of a watch 
 as seen by a person looking at its face, negative when it tends to 
 turn with the hands. Hence the axis, which may be taken to repre- 
 sent a couple, will show, if drawn according to the rule given in § 201, 
 whether the couple is positive or negative, according to the side of its 
 plane from which it is regarded. 
 
 561. Proposition I. Any two couples in the same or in parallel 
 planes are in equilibrium if their moments are equal and they tend to 
 turn in contrary directions. 
 
 1°. Let the forces of the first couple be parallel to those of the 
 second, and let all four forces be in one plane. 
 
 n T A^ n^ -^^^ ^^ forces of the first couple be 
 
 o xr A o ^ .^ ^^ ^^^ ^j^^ ^^^ ^^ ^^^ second 
 
 F' in A'B' and CD'. Draw any line 
 EF' across the lines of the forces, cut- 
 ting them respectively in points F, Fy 
 E' and F' 'y then the moment of the 
 B BR B JD' first couple IS Z'. ^i^ and of the second 
 
 F', E'F' ; and since the moments are equal we have 
 F.EF=F'.E'F'. 
 
 Of the four forces, P m AB and P' in CFf act in similar direc- 
 tions, and F in CD and F' in A'B' also act in similar directions; 
 and their resultants respectively can be determined by the general 
 method (§ 556). The resultant of F in AB, and F' in CD', is thus 
 found to be equal to /*+ F'^ and if HL is the line in which it acts, 
 
 F.EK=F'.KF'. 
 Again, we have F. EF= F' , E'F'. 
 
 Subtract the first member of the latter equation from the first 
 member of the former, and the second member of the latter from the 
 second member of the former : there remains 
 
 F.FK=F'.KE', 
 from which we conclude, that the resultant of F in CD and F' in 
 A'B' is in the line LH. Its magnitude is /*+ F', Thus the given 
 system is reduced to two equal resultants acting in opposite directions 
 in the same straight line. These balance one another, and therefore 
 the given system is in equilibrium. 
 
 Corollary. A couple may be transferred from its own arm to any 
 other arm in the same line, if its moment be not altered. 
 
 562. Proposition I. 2". All four forces in one plane, but those of 
 one couple not parallel to those of the other. 
 
 Produce their lines to meet in four points; and consider the paral- 
 lelogram thus formed. The products of the sides, each into its per- 
 pendicular distance from the side parallel to it, are equal, each product 
 
STATICS OF SOLIDS AND FLUIDS. 205 
 
 being the area of the parallelogram. Hence, since the moments of the 
 two couples are equal, their forces are proportional to the sides of the 
 parallelogram along which they act. And, since the couples tend to 
 turn in opposite directions, the four forces represented by the sides of 
 a parallelogram act in similar directions relatively to the angles, and 
 dissimilar directions in the parallels, and therefore balance one 
 another. 
 
 Corollary. The statical effect of a couple is not altered, if its arm 
 be turned round any point in the plane of the couple. 
 
 563. Proposition I. 3^ The two couples not in the same plane, 
 but the forces equal and parallel. 
 
 Let there be two couples, acting re- •M' pp, 
 
 spectively on arms EF and E'F\ which E^-^rrr^^ — L"*" ^ . . F ' 
 
 are parallel but not in the same plane. .-------r-C.''* *^ |p/ 
 
 Join EF' and E'F. These lines bisect ^r"^ |^p7"'' -^ 
 
 one another in O. \p 
 
 Of the four forces, F on F and F' on 
 E' act in similar directions, and their resultant, equal io F+ F\ may 
 be substituted for them. It acts in a parallel line through O. Simi- 
 larly F on E and F' on F'' have also a resultant equal to F+ F' 
 through O; but these resultants being equal and opposite, balance, 
 and therefore the given system is in equilibrium. 
 
 Remark i. — A corresponding demonstration may be appHed to 
 every case of two couples, the moments of which are equal, though 
 the forces and arms may be unequal. When the forces and arms are 
 unequal, the lines EF'j E'F cut one another in O into parts inversely 
 as the forces. 
 
 Remark 2. — Hence as an extreme case, Proposition I, 1°, may be 
 brought under this head. Let EF be the arm of one couple, EF' 
 of the other, both in one straight line. Join FE' , and divide it 
 inversely as the forces. Then FK : KE' w EF \ E'F' and EF' is 
 divided in the same ratio. 
 
 Corollary. Transposition of couples. Any two couples in the 
 same or in parallel planes, are equivalent, provided their moments 
 are equal, and they tend to turn in similar directions. 
 
 564. Proposition IL Any number of couples in the same or in 
 parallel planes, may be reduced to a single 
 resultant couple, whose moment is equal to the L 
 algebraic sum of their moments, and whose 
 plane is parallel to their planes. 
 
 Reduce all the couples to forces acting on 
 
 one arm AB^ which may be denoted by a. A^ 
 
 Then if /\, F^, P,, etc., be the forces, the mo- yi\ 
 
 ments of the couples will be F^a^ F^a, F^a, ,,^ 
 
 etc. Thus we have /\, F^, F^, etc., in AK, p 
 
 reducible to a single force, their sum, and ^^ 
 similarly, a single force F^ + F^-^ etc., in FL. 
 
2o6 ABSTRACT DYNAMICS. 
 
 These two forces constitute a couple whose moment is (/\ + P. 
 + i^3 + etc.) a. But this product is equal to P^a-\- P^a + P^a + etc., 
 the sum of the moments of the given couples, and therefore any 
 number of couples, etc. If any of the couples act in the direction 
 opposite to that reckoned positive, their moments must be reckoned 
 as negative in the sum. 
 
 565. Proposition III. Any two couples not in parallel planes 
 may be reduced to a single resultant couple, whose axis is the 
 diagonal through the point of reference of the parallelogram de- 
 scribed upon their axes. 
 
 1°. Let the planes of the two couples cut the plane of the diagram 
 
 perpendicularly in the lines AA' and 
 BB' respectively; let the planes of 
 the couples also cut each other in a 
 line cutting the plane of the diagram 
 in O. Through O, as a point of re- 
 ference, draw OK the axis of the first 
 couple, and OL the axis of the se- 
 cond. On OK and OL construct the 
 ^ ^ parallelogram OKML. Its diagonal 
 
 " OM is the axis of the resultant couple. 
 
 Let the moment of the couple acting in the plane BB\ be denoted 
 by G, and of that in AA\ by H. For the given couples, substitute 
 two others, with arms equal respectively to G and H\, and therefore 
 with forces equal to unity. 
 
 From OB and OA measure off OE = G, and 0F= H, and let these 
 lines be taken as the arms of the two couples respectively. The 
 forces of the couples will thus be perpendicular to the plane of the 
 diagram : those of the first, acting outwards at E^ and inwards at O \ 
 and those of the second, outwards at (7, and inwards at F. Thus, of 
 the four equal forces which we have in all, there are two equal and 
 opposite at (9, which therefore balance one another, and may be 
 removed; and there remain two equal parallel forces, one acting 
 outwards at E, and the other inwards at F^ which constitute a couple 
 on an arm EF. 
 
 This single couple is therefore equivalent to the two given couples. 
 
 2°. It remains to be proved that its axis is OM. Join EF, As, 
 by construction, OL and OK are respectively perpendicular to OA^ 
 and OB, the angle KOL is equal to the angle A OB'. Hence, MLO 
 the supplement of the former is equal to EOF, the supplement of the 
 latter. But OK is equal to OE ; each being equal to the moment of 
 the first of the given couples ; and therefore LM, which is equal to 
 the former, is equal to OE. Similarly OL is equal to OF. Thus there 
 are two triangles, MLO and EOF, with two sides of one respectively 
 equal to two sides of the other, and the contained angles equal : there- 
 fore the remaining sides OM, EF are equal, and the angles LOM\ 
 OFE are equal. But since OL is perpendicular to OF, OM is 
 
STATICS OF SOLIDS AND FLUIDS. 207 
 
 perpendicular to EF. Hence OM is the axis of the resultant 
 couple. 
 
 566. Proposition IV. Any number of couples whatever are either 
 in equilibrium with one another, or may be reduced to a single couple, 
 under precisely the same conditions as those already investigated for 
 forces acting on one point, the axes of the couples being now taken 
 everywhere instead of the lines formerly used to represent the forces. 
 
 1°. Resolve each couple into three components having their axes 
 along three rectangular lines of reference, OX^ O F, OZ. Add all 
 the components corresponding to each of these three lines. Then 
 if the resultant of all the couples whose axes are along the line 
 
 OX^ be denoted by Z, 
 
 OY, „ „ M, 
 
 OZ, „ „ iV^ 
 
 and if G be the resultant of these three, we have 
 
 G^ J{L' + M' + N'): 
 
 and if ^, -q, 0, be the angles which the axis of this couple G, makes 
 with the three axes OX, O V, OZ, respectively, we have 
 
 . Z M.N 
 
 cos 4= 7,; CO^f]=-^\ cos^ = ^. 
 Lr Lr Lr 
 
 567. 2°. Conditions of equilibrium of any number of couples. For 
 equilibrium the resultant couple must be equal to nothing : but as it is 
 compounded of three subsidiary resultant couples in planes at right 
 angles to one another, they also must each be equal to nothing. The 
 remarks already made, and the equations already given in §§ 471, 472, 
 apply with the necessary modification to couples also. Thus, for 
 instance, the equations of equilibrium are 
 
 G^ cos ^j + G^ cos ^3 + G.^ cos t,^ + etc. = o, 
 
 G^ cosr7j + G^ cos 173 + 6^3 cos t]^ + etc. = o, 
 
 C?i cos ^1 + 6^2 cos ^2 + G^ cos ^3 + etc. ^ o. 
 
 568. Before investigating the conditions of equilibrium of any 
 number of forces acting on a rigid body, we shall establish some 
 preliminary propositions. 
 
 1°. A force and a couple in the same or in parallel planes may be 
 reduced to a single force. Let the plane of the couple be the plane 
 of the diagram, and let its moment be 
 denoted by G. Let R, acting in the 
 line OA in the same plane, be the force. 
 Transfer the couple to an arm (which may 
 be denoted by a) through the point O, such 
 that each force shall be equal to R; and let 
 its position be so chosen, that one of the 
 forces shall act in the same straight line with 
 R in OA, but in the opposite direction to it. 
 
 G 
 
208 
 
 ABSTRACT DYNAMICS. 
 
 R and G being known, the length of this arm can be found, for 
 since the moment of the transposed couple is 
 
 Ra^G 
 G 
 
 we have 
 
 a- 
 
 R' 
 
 Through O then, draw a line OCf perpendicular to OA^ making it 
 equal to a. On this arm apply the couple, a force equal to R^ acting 
 on O' in a line perpendicular to 00\ and another in the opposite 
 direction at the other extremity. There are now three forces, two of 
 which, being equal and opposite to one another, in the line AA\ may 
 be removed. One, acting on the point 0\ remains, which is there- 
 fore equivalent to the given system. 
 
 569. 2°. A couple and a force in a given line inclined to its plane 
 may be reduced to a smaller couple in a plane perpendicular to the 
 force, and a force equal and parallel to the given force. 
 
 Let OA be the line of action of the force /?, 
 and let OK be the axis of the couple. Let 
 the moment be denoted by G : and let A OK, 
 the inclination of its axis to the line of the 
 force, be 0. Draw OB perpendicular to OA. 
 By Prop. IV. (§ 566) resolve the couple into 
 two components, one acting round OA as 
 axis, and one round OB. Thus the compo- 
 
 nent round OA will be 
 
 G cos B, 
 
 and the component round OB^ 
 
 G sin Q. 
 Now as G sin B acts in the same plane as the given force -^, this com- 
 ponent together with R may be reduced by § 568 to one force. This 
 force which is equal to R, will act not at O in the line OA^ but in a 
 parallel line through a point O' out of the plane of the diagram. Thus 
 the given system is reduced to a smaller couple G cos ^, and to a 
 force in a line which, by Poinsot, was denominated the central axis of 
 the system. 
 
 670. 3' 
 a couple. 
 
 Let /*, acting 
 
 Any number of forces may be reduced to a force and 
 
 on J/j be one of a number of forces acting in 
 different directions on different points of a rigid 
 body. Choose any point of reference O, for the 
 different forces, and through it draw a line AA' 
 parallel to the line of the first force P^. Through (?, 
 draw 00' perpendicular to A A or the line of the 
 force P^. In the line A A' introduce two equal op- 
 posite forces, each equal to P^. There are now 
 three forces, producing the same effect as the given 
 force, and they may be grouped differently : P^ acting 
 
STATICS OF SOLIDS AND FLUIDS. 209 
 
 in O in the line OA^ and a couple, Py^ acting at O' , and P^ at O in 
 the line 0A\ on an arm 00'. Reduce similarly all. the other forces, 
 each to a force acting on (9, and to a couple. But all the couples 
 thus obtained are equivalent to a single couple, and all the forces are 
 equivalent to one force. Hence, &c. 
 
 571. Reduction of any number of forces to their simplest equi- 
 valent system. 
 
 Suppose any number of forces acting in any directions on different 
 points of a rigid body. Choose three rectangular planes of reference 
 meeting in a point (9, the origin of co-ordinates. In order to effect 
 the reduction it is necessary to bring in all the forces to the point O. 
 This may be done in two different ways — either in two steps, or 
 directly. 
 
 572. 1°. Let the magnitudes of the forces be jPi, P^, &c., and the 
 co-ordinates, with reference to the rectangular planes, of the points at 
 which they act respectively, be (^1, Ji, 2J, {x^., y^, z.^, &c. Let also 
 the direction cosines be (Z^, m^, n^, (4, m,, wj, &c. Resolve each 
 force into three components, parallel to OX^ O V, OZ, respectively. 
 Thus, if (Xj, Vi, Z,), &c., be the components of /\, &c., we shall 
 have 
 
 X, = P,/,; X,= PJ,; &c. (1) 
 
 V,=^P,m,; V,^P„m,; &c. (2) 
 
 Z, = P,n,;Z, = P,n,',&:c. (3) 
 
 To transfer these components to the point O. Let ^i, in AfJ^, be 
 the component, parallel to OX, of the force P^^ acting on the point Af. 
 From M transmit it along its line to a 
 point JV in the plane ZO Y: the co-or- 
 dinates of this point will bejj, z^. From 
 iVdraw a perpendicular NB to OY, 
 and through B draw a line parallel to 
 MK or OX. Introducing in this line 
 a pair of balancing forces each equal 
 to Xj, we have a couple acting on an ^^ q 
 
 
 arm z^ in a plane parallel to XOZ, '^'"^y^ 27 
 and a single force X^ parallel to OX Y'^ 
 in the plane XOY. The moment of 
 this couple is X^z^, and its axis is along 
 OY. Next transfer the force X^ from B to O, by introducing a pair 
 of balancing forces in X' OX, one of which, with the force X^ in the 
 line through B parallel to XX and the direction similar to OX, form 
 a couple acting on an arm y^ . This couple, wKen y^ and X^ are both 
 positive, tends to turn in the plane XOY from O Fto OX. Therefore 
 by the rule, § 201, its axis must be drawn from O in the direction 0Z\ 
 Hence its moment is to be reckoned as- X^y^. Besides this couple 
 there remains a single force equal to X^, in the direction OX, through 
 the point O. Similarly by successive steps transfer the forces Y^, Z^, 
 
 T. 14 
 
2IO ABSTRACT DYNAMICS. 
 
 to the origin of co-ordinates. In this way six couples of transference 
 are got, three tending to turn in one direction round the axes respec- 
 tively, and three in the opposite direction; and three single forces at 
 right angles to one another, acting at the point O. Thus for the force 
 P^, at the point (^i, Ji, z^, we have as equivalent to it at the point 
 O, three forces X^, Y^, Z^, and three couples; 
 
 Zj^ -Y^z^\ moment of the couple round 0X\ (4) 
 
 X^z^-Z^x^\ moment of the couple round OY \ (5) 
 
 Y^x^-X^y^', moment of the couple round OZ. (6) 
 
 All the forces may be brought in to the origin of co-ordinates in a 
 similar way. 
 
 573. 2°. Otherwise: Let P be one of the forces acting in the line 
 
 MT on a point Moi a rigid body. Let 
 O be the origin of co-ordinates; OX, 
 OY, OZ, three rectangular lines of re- 
 ference. Join OMa-nd produce the line 
 to S. From O draw OJV, cutting at 
 right angles in the point JV, the line 
 MT produced through Af. Let OJV 
 be denoted by/, and the angle TMS 
 by K. In a line through O parallel to 
 MT (not shown in diagram) suppose 
 introduced a pair of balancing forces each equal to P. We have 
 thus a single force equal to P acting at O, and a couple, whose 
 moment is Pj>, in the plane ONM. The direction cosines of this 
 plane, or, which is the same thing, the direction cosines of a per- 
 pendicular to it, that is, the axis of the couple are (§ 464), if we denote 
 them by <^, Xj '/'> respectively, 
 
 y z 
 -n--m 
 
 sm K 
 
 z , X 
 -I-- n 
 r r 
 
 r r 
 
 sm K 
 Now in the triangle ONM^ 
 
 ON^ OMsm OMN, 
 that is / = r sin k. 
 
STATICS OF SOLIDS AND FLUIDS. 211 
 
 Hence, if we substitute / for its value in the three preceding equations, 
 the expression for the direction cosines are reduced to 
 
 (7) 
 
 (8) 
 
 p (9) 
 
 To find the component couples round OX, O Y, OZ, multiply these 
 direction cosines respectively by Fj>', whence we get 
 
 Fp .f^=^F{7iy - mz), moment of couple round OX, ( i o) 
 
 Fp .x-F{lz- 7tx), moment of couple round OY, (11) 
 
 Fp.\^ — F {mx - ly)j moment of couple round OZ. (12) 
 
 That this result is the same as that got by the other method will be 
 evident, by considering that (equations i, 2, 3), 
 
 Fl=X- Fm=.Y', Fn = Z. 
 
 574. When by either of the methods all the forces have been re- 
 ferred to 6>, there is obtained a set of couples acting round OX, O Y, 
 OZ; and a set of forces acting along OX, O Y, OZ. Find then the 
 resultant moments of all the couples ; and the sums of all the forces : 
 if L, M, N be the resultant moments round OX, O Y, OZ respectively, 
 we have 
 
 Z = (Z, y, - Y^ z,) + (4^, - F, z^) + &c. (13) 
 
 M= (Xj z^ -Z^ x^ + (Xj z^ -Z2 x^ + &c. (14) 
 
 N= ( y; X, -X,y,) +( y, x, -X,y,) + &c. (15) 
 and if X, Y, Z be the resultant forces, 
 
 X=X,+X, + X^ + 8zc. (i6> 
 
 Y=Y,+ Y,+ Y, + &c. (17). 
 
 Z=Z, + Z^+ Z^ + &LC. (18) 
 
 575. Finally, find the resultant of the three forces by the formulae 
 of Chap. VI, and the resultant of the three couples by Prop. IV 
 (§ 566). Thus, if /, m, n be the direction cosines of the resultant force 
 i?, we have (§§ 463, 467) 
 
 , X Y Z , . 
 
 and if X, fi, v be the direction cosines of the axis of the resultant 
 couple, we have (§ 566) 
 
 ^ L M N . . 
 
 X=g; /x = ^; v=^. (20) 
 
 14—2 
 
2 1 2 ABSTRA CT D YNAMICS. 
 
 676. Conditions of Equilibrium. The conditions of equilibrium 
 of three forces at right angles to one another have been already stated 
 in § 470; and the conditions for three rectangular couples in § 567. 
 
 If a body be acted on by three forces and three couples simul- 
 taneously, all the conditions applicable when they act separately, must 
 also be satisfied when they act conjointly, since a force cannot balance 
 a couple. Six Equations of Equilibrium therefore are necessary and 
 sufficient for a rigid body acted on by any number of forces. These 
 are 
 
 /*! cos a^ + P^ cos a^ + &c. = o, 
 
 P^ cos ft + P^ cos ft + &c. = o, 
 P^ cos y^ + P^ cos 72 + &c. = o, 
 G^ cos 4 + G^ cos ^3 + &c. = o, 
 G^ cos T7i + G^ cos 772 + &c. = o, 
 G^ cos ^1 + G^ cos ^a + &c. = o. 
 
 577. If the line of the resultant found by § 575, is perpendicular 
 to the plane of the couple, that is, if 
 
 X = /, iL-m^ v=n; 
 
 X=Y = z^ (21) 
 
 the system cannot be reduced to another with a force and a smaller 
 couple, and in this case the line found for the resultant force is the 
 central axis of the system. 
 
 578. If, on the other hand, the plane of the couple is parallel to the 
 line of the force, or the axis of the couple perpendicular to the line of 
 the force, that is, if 
 
 /A. + mix + nv = Oj 
 
 or ZX + My+JVZ=o, (22) 
 
 the force and couple may (§ 568) be reduced to one force : and this 
 
 G 
 
 force is parallel to the former, at a distance from it equal to -^ , in 
 
 the plane of it and the couple. Thus, X(7 being the foot of the 
 perpendicular from the origin on the line of action of the resultant 
 force, 0(7 wiW be perpendicular to the Hne of the resultant force, and 
 to the axis of the resultant couple, and therefore its direction cosines 
 
 are (§464, ^); 
 
 mv — /I IX, nX — Ivj //x - m\, (23) 
 
 each of which will be positive when O' lies within the solid angle 
 
 Q 
 
 edged by OX^ OY, OZ. Hence, remembering that 00' = -^^ and 
 
 using the expressions (19) and (20), we find for the co-ordinates of (7 
 
 YN-ZM ZL-XN XM-YL 
 
 7?' ' ^» ' R^ 
 
 (24) 
 
STATICS OF SOLIDS AND FLUIDS. 213 
 
 and we thus complete the specification of the single force to which 
 the system is reduced when (22) holds. 
 
 679. If the line of the force is inclined at any angle to the plane 
 
 of the couple, the resultant system can be further reduced by § 569, 
 
 to a smaller couple and a force in a determinate line, the * central 
 
 axis.' This couple is G cos 6, and according to the notation, may be 
 
 thus expressed by § 464, (7), if we substitute the values given in (19) 
 
 and (20), 
 
 . XL+YM+ZN , . 
 
 Gcose = . (25) 
 
 The other component couple, G sin 9, lies in the same plane as R, 
 and with it may be reduced by §568 to one force, which will be 
 parallel to R, that is, in the direction (/, m, n)^ at a distance from it 
 
 equal to — ^ — . Hence the direction cosines of 00' will be 
 
 mv — n\k n\ — lv l}x. — m\ , r\ 
 
 sin (9 ' sin^ ' sin ^ * ' ^ ' 
 
 Substituting in each of these for /, X, &c., their respective values, 
 
 and multiplying each member by — ^ — , we have for the co-ordinates 
 
 of the point 6>', as in § 578, 
 
 YN-ZM ZL-XN XM-YL . . 
 
 R' ' R' ' R' ' ^^'^ 
 
 A single force, R, through the point thus specified in the direction 
 (/, My n)j with a couple in a plane perpendicular to it, and having . 
 
 XL + YM+ZN 
 R 
 
 for its moment, is consequently the system oi force along central axis 
 and mifitmum couple, to which the given set of forces is determinately 
 reducible by Poinsot's beautiful method. 
 
 580. The position of the central axis may be determined other- 
 wise j thus, instead of in the first place bringing the forces to O, bring 
 them to any point T, of which let (x, y, z) be the co-ordinates. Then 
 instead of Y^z^+Y^z^ + ^c, which we had before (§ 574), we have now 
 
 Y,{z,-z)+Y,{z,-z) + &ic., 
 
 or Y^ z^ + Y^z^ -f &c. - ( y; + Fa -f- &c.) z, 
 
 and so for the others. Then for the moments of the couples of trans- 
 ference we have 
 
 a = Z -{Zy-Yz\ 
 
 0i = M-(Xz-Zx), 
 M = N-(Yx^Xy). 
 
 Now, let T be chosen, if possible, so as to make the resultant 
 
214 ABSTRACT DYNAMICS. 
 
 eouple lie in a plane perpendicular to it. The condition to be ful- 
 filled in this case is 
 
 X Y Z' 
 
 which, when for 3£, &c., we substitute their values, becomes, 
 L-{Zy-Yz) _ M-{Xz -Zx) _ N-(Yx-Xy) 
 X ~ Y ~ Z ' 
 
 which is the equation of the central axis of the system. 
 
 To show that O', the point determined in §§ 578, 579, is in the 
 central axis thus found ; we have, substituting for ^, y, z, the values 
 given in (24), 
 
 Z {ZL - XN) + Y (XM- YL ) 
 
 1 =^- 
 
 Reducing, and remarking that 
 
 LR'-LY'-LZ' = LX\ 
 we find that the first member becomes 
 
 X 
 and is therefore equal to each of the two others. Thus is verified the 
 comparison of the two methods. 
 
 581. In one respect, this reduction of a system of forces to a 
 couple, and a force perpendicular to its plane, is the best and simplest, 
 especially in having the advantage of being determinate, and it gives 
 very clear and useful conceptions regarding the effect of force on 
 a rigid body. The system may, however, be farther reduced to two 
 equal forces acting symmetrically on the rigid body, but whose po- 
 sition is indeterminate. Thus, supposing the central axis of the 
 system has been found, draw a line AA', at right angles through any 
 point C in it, so that CA may be equal to CA\ For R, acting along 
 the central axis, substitute \R at each end of A A'. Thus, choosing 
 this line A A' as the arm of the couple, and calling it a, we have at 
 
 each extremity of it two forces, — perpendicular to the central axis, 
 
 and ^R parallel to the central axis. Compounding these, we get 
 
 two forces, each equal to (\R^-\--y) , through A and A' re- 
 
 spectively, perpendicular to AA\ and equally inclined at the angle 
 
 tan~^ — p on the two sides of the plane through A A' and the central 
 
 axis. 
 
 582. It is obvious, from the formulae of § 195, that if masses pro- 
 portional to the forces be placed at the several points of application 
 of these forces, the centre of inertia of these masses will be the same 
 
STATICS OF SOLIDS AND FLUIDS. 215. 
 
 point in the body as the centre of parallel forces. Hence the re- 
 actions of the different parts of a rigid body against acceleration in 
 parallel lines are rigorously reducible to one force, acting at the centre 
 of inertia. The same is true approximately of the action of gravity 
 on a rigid body of small dimensions relatively to the earth, and hence 
 the centre of inertia is sometimes (§ 195) called the Centre of Gravity. 
 But, except on a centrobaric body (§ 543), gravity is not in general 
 reducible to a single force : and when it is so, this force does not pass 
 through a point fixed relatively to the body in all positions. 
 
 583. The resultant of a system of parallel forces is not a single 
 force when the algebraic sum of the given forces vanishes. In 
 this case the resultant is a couple whose plane is parallel to the 
 common direction of the forces. A good example of this is furnished 
 by a magnetized mass, of steel, of moderate dimensions, subject to the 
 influence of the earth's magnetism only. As will be shown later, the 
 amounts of the so-called north and south magnetisms in each element 
 of the mass are equal, and are therefore subject to equal and opposite 
 forces, all parallel to the line of dip. Thus a compass-needle expe- 
 riences from the earth's magnetism merely a couple or directive action, 
 and is not attracted or repelled as a whole. 
 
 584. If three forces, acting on a rigid body, produce equilibrium, 
 their directions must lie in one plane ; and must all meet in one point, 
 or be parallel. For the proof, we may introduce a consideration 
 which will be very useful to us in investigations connected with the 
 statics of flexible bodies and fluids. 
 
 If ajiy forces^ acting on a solid or fluid body, produce equilibrium, we 
 may suppose any portions of the body to becoffie fixed, or rigid, or rigid 
 aftd fixed, without destroying the equilibrium. 
 
 Applying this principle to the case above, suppose any two points 
 of the body, respectively in the lines of action of two of the forces, to 
 be fixed — the third force must have no moment along the line joining 
 these points; that is, its direction must pass through the line joining 
 them. As any two points in the lines of action may be taken, it 
 follows that the three forces are coplanar. And three forces in one 
 plane cannot equilibrate, unless their- directions are parallel or pass 
 through a point. 
 
 585. It is easy and useful to consider various cases of equilibrium 
 when no forces act on a rigid body but gravity and the pressures, 
 normal or tangential, between it and fixed supports. Thus, if one 
 given point only of the body be fixed, it is evident that the centre of 
 gravity must be in the vertical line through this point — else the weight 
 and the reaction of the support would form an unbalanced couple. 
 Also for stable equilibrium the centre of gravity must be below the 
 point of suspension. Thus a body of any form may be made to 
 stand in stable equilibrium on the point of a needle if we rigidly 
 attach to it such a mass as to cause the joint centre of gravity to be 
 below the point of the needle. 
 
2l6 
 
 ABSTRACT DYNAMICS. 
 
 586. An interesting case of equilibrium is suggested by what are 
 called Rocking Stones, where, whether by natural or by artificial pro- 
 cesses, the lower surface of a loose mass of rock is worn into a convex 
 form which may be approximately spherical, while the bed of rock on 
 which it rests in equilibrium is, whether convex or concave, also ap- 
 proximately spherical, if not plane. A loaded sphere resting on a 
 spherical surface is therefore a type of such cases. 
 
 Let O, O' be the centres of curvature of the fixed and rocking 
 bodies respectively, when in the position of equilibrium. 
 Take any two infinitely small equal arcs FQ, Pp ; and 
 at Q make the angle O'QR equal to FOp. When, by 
 displacement, Q and p become the points in contact, 
 QR will evidently be vertical ; and, if the centre of 
 gravity G, which must be in OPO' when the movable 
 body is in its position of equilibrium, be to the left of 
 QR, the equilibrium will obviously be stable. Hence, 
 if it be below i?, the equilibrium is stable, and not 
 unless. 
 
 Now if p and o- be the radii of curvature OF, O' P 
 
 of the two surfaces, and 6 the angle POp, the angle 
 f\ 
 
 QO'R will be equal to — ; and we have in the triangle 
 
 (?<^'^(§ii9) 
 
 Hence 
 
 i?6>':cr::sin^:sin('^-f-^^ 
 
 : : o- : cr + p (approximately). 
 
 PR = <T- 
 
 cr + p 
 and therefore, for stable equilibrium. 
 
 p + a 
 
 PG< 
 
 pa_ 
 p + 0- 
 
 If the lower surface be plane, p is infinite, and the condition becomes 
 (as in § 256) 
 
 PG<(T. 
 
 If the lower surface be concave, the sign of p must be changed, and 
 the condition becomes 
 
 PG 
 
 pa- 
 
 which cannot be negative, since p must be numerically greater than o- 
 in this case. 
 
 587. If two points be fixed, the only motion of which the system is 
 capable is one of rotation about a fixed axis. The centre of gravity 
 must then be in the vertical plane passing through those points, and 
 below the line adjoining them for stable equilibrium. 
 
 588. If a rigid body rest on a fixed surface, there will in general be 
 only three points of contact, § 380 ; and the body will be in stable 
 
STATICS OF SOLIDS AND FLUIDS 217 
 
 equilibrium if the vertical line drawn from its centre of gravity cuts 
 the plane of these three points within the triangle of which they form 
 the corners. For if one of these supports be removed, the body will 
 obviously tend to fall towards that support. Hence each of the three 
 prevents the body from rotating about the line joining the other two. 
 Thus, for instance, a body stands stably on an inclined plane (if the 
 friction be sufficient to prevent it from sliding down) when the vertical 
 line drawn through its centre of gravity falls within the base, or area 
 bounded by the shortest line which can be drawn round the portion in 
 contact with the plane. Hence a body, which cannot stand on a 
 horizontal plane, may stand on an inclined plane. 
 
 589. A curious theorem, due to Pappus, but commonly attributed 
 to Guldinus, may be mentioned here, as it is employed with advantage 
 in some cases in finding the centre of gravity of a body — though it is 
 really one of the geometrical properties of the Centre of Inertia. It is 
 obvious from § 195. If a plane dosed curve revolve through any a7igle 
 about an axis in its plane ^ the solid content of the surface gefierated is 
 equal to the product of the area of either end into the length of the path 
 described by its centre of gravity ; and the area of the curved surface is 
 equal to the product of the length of the curve into the length of the path 
 described by its centre of gravity. 
 
 590. The general principles upon which forces of constraint and 
 friction are to be treated have been stated above (§§ 258, 405). We 
 add here a few examples, for the sake of illustrating the application 
 of these principles to the equilibrium of a rigid body in some of the 
 more important practical cases of constraint. 
 
 591. The application of statical principles to the Mechanical Powers^ 
 or elementary machines, and to their combinations, however complex, 
 requires merely a statement of their kinematical relations (as in §§ 91, 
 97, 113, &c.) and an immediate translation into Dynamics by New- 
 ton's principle (§241); or by Lagrange's Virtual Velocities (§ 254), 
 with special attention to the introduction of forces of friction, as in 
 § 405. In no case can this process involve further difficulties than 
 are implied in seeking the geometrical circumstances of any infinitely 
 small disturbance, and in the subsequent solution of the equations to 
 which the translation into dynamics leads us. We will not, therefore, 
 stop to discuss any of these questions ; but will take a few examples 
 of no very great difficulty, before for a time quitting this part of the 
 subject. The principles already developed will be of constant use to 
 us in the remainder of the work, which will furnish us with ever- 
 recurring opportunities of exemplifying their use and mode of appli- 
 cation. 
 
 Let us begin with the case of the Balance, of which we promised 
 (§ 384) to give an investigation. 
 
 592. Ex. 1. We will assume the line joining the points of attach- 
 ment of the scale-pans to the arms to be at right angles to the line 
 joining the centre of gravity of the beam with the fulcrum. It is 
 
2i8 ABSTRACT DYNAMICS, 
 
 obvious that the centre of gravity of the beam must not coincide with 
 
 the knife-edge, else the beam would rest indifferently in any position. 
 
 We will suppose, in the first place, that the arms are not of equal length. 
 
 Let O be the fulcrum, G the 
 centre of gravity of the beam, 
 M its mass ; and suppose that 
 with loads P and Q, in the pans 
 the beam rests (as drawn) in a 
 position making an angle B with 
 the horizontal line. 
 
 Taking moments about (?, 
 and, for convenience (see § 185), 
 
 using gravitation measurement of the forces, we have 
 
 Q {AB cos e+OA sin 6) + M. OG sin 6 = F{AC cos 6 - OA sin 0). 
 
 From this we find 
 
 P.AC-Q.AB 
 
 {F+Q)OA + M.OG' 
 If the arms be equal we have 
 
 tang (i--0^^ 
 
 {F+Q) OA + M.OG' 
 
 Hence the Sensibility (§ 384) is greater, (i) as the arms are longer, 
 (2) as the mass of the beam is less, (3) as the fulcrum is nearer to the 
 line joining the points of attachment of the pans, (4) as the fulcrum is 
 nearer to the centre of gravity of the beam. If the fulcrum be in the 
 line joining the points of attachment of the pans, the sensibility is the 
 same for the same difference of loads in the pan. 
 
 To determine the Stability we must investigate the time of oscilla- 
 tion of the balance when slightly disturbed. It will be seen, by refer- 
 ence to a future chapter, that the equation of motion is approximately 
 
 {Mk' + {F+Q) OB'} e + Qg (AB cos 6 + OA sin 0) 
 
 + MgOG sin d - Fg {AC cos $ - OA sin 6) = o, 
 
 ^ being the radius of gyration (§ 235) of the beam. If we suppose 
 the arms and their loads equal, we have for the time of an infinitely 
 small oscillation 
 
 7; 
 
 M/e'+2F.0B' 
 
 {2F.0A+M.0G)g' 
 
 Thus the stability is greater for a given load, (i) the less the length of 
 the beam, (2) the less its mass, (3) the less its radius of gyration, (4) 
 the further the fulcrum from the beam, and from its centre of gravity. 
 With the exception of the second, these adjustments are the very 
 opposite of those required for sensibility. Hence all we can do is to 
 effect a judicious compromise; but the less the mass of the beam, the 
 better will the balance be, in ^of/i respects. 
 
 The general equation, above written, shows that if the length, and 
 the radius of gyration, of one arm be diminished, the corresponding 
 
STATICS OF SOLIDS AND FLUIDS. 
 
 219 
 
 load being increased so as to maintain equilibrium — a form of balance 
 occasionally useful — the sensibility is increased. 
 
 Fx. II. Find the position of equilibrium of a rod AB resting on a 
 smooth horizontal rail D, its lower end pressing against a smooth 
 vertical wall A C parallel to the rail. 
 
 The figure represents a vertical section through the rod, which 
 must evidently be in a plane perpendicular to the wall and rail. 
 
 The only forces acting are three, R the pressure of the wall on the 
 rod, horizontal; S that of the 
 rail on the rod, perpendicular 
 to the rod; W the weight of 
 the rod, acting vertically down- 
 wards at its centre of gravity. 
 If the half-length of the rod be 
 a, and the distance of the rail 
 from the wall b^ these are given 
 — and all that is wanted to fix 
 the position of equilibrium is the angle the rod makes with the wall. 
 
 Call CAB, e. Then we see at once that AD = -r— 7: . 
 
 sm 
 
 Resolving horizontally R- S cos ^ = o, . ( i ) 
 
 vertically W- *Ssin ^ = o. (2) 
 
 Taking moments about A, 
 
 S.AD- JVa sin 0*0, 
 
 or Sb-JVasm'0 = o. (3) 
 
 As there are only three unknown quantities, R, S, and 0, these three 
 equations contain the complete solution of the problem. By (2) 
 and (3) 
 
 sin'^ = -, which gives 0. 
 
 W 
 Hence by (2) S=- — ;-, 
 
 ^ ^ ' sm ^ ' 
 
 and by (i) R = Scos 6 = Wcot 0. 
 
 Fx. III. As an additional example, suppose the wall and rail to be 
 rough, and /x to be the co-efficient of statical friction for both. If the 
 rod be placed in the position of equilibrium just investigated for the 
 case of no friction, none will be called into play, for there will be no 
 tendency to motion to be overcome. If the end A be brought lower 
 and lower, more and more friction will be called into play to over- 
 come the tendency of the rod to fall between the wall and the rail, 
 until we come to a limiting position in which motion is about to 
 commence. At that instant the friction at y^ is /x, times the pressure 
 
ABSTRACT DYNAMICS, 
 
 on the wall, and acts upwards. That at Z> is /* times the pressure 
 on the rod, and acts in the direction DB. CaUing CAD = 0^ in this 
 case, our three equations become 
 
 B^ + IxS^smO^- S^cosO^ =o, (ii) 
 
 IV-fXjR^ — Si sin 0^ - fxS^ cos ^^ = o, (21) 
 
 S.d-Wasm'e^ =0. (3i) 
 
 The directions of both the friction-forces passing through A, neither 
 
 appears in (31). This is why A is preferable to any other point about 
 
 which to take moments. 
 
 By eliminating B^ and S^ from these equations we get 
 
 I - - sm^ 0^ = fJL - sin^ 6^ {2 cosO^-fx sin ^1), (4 ) 
 
 from which 0^ is to be found. Then S^ is known from (31), and B^ 
 from either of the others. 
 
 If the end A be raised above the position of equilibrium without 
 friction, the tendency is for the rod to fall outside the rail ; more and 
 more friction will be called into play, till the position of the rod (^2) 
 is such that the friction reaches its greatest value, /x times the pressure. 
 We may thus find another limiting "position for stability; and between 
 these the rod is in equilibrium in any position. 
 
 It is useful to observe that in this second case the direction of each 
 friction is the opposite to that in the former, and the same equations 
 will serve for both if we adopt the analytical artifice of changing the 
 sign of [J.. Thus for 0^, by (41), 
 
 I -Tsin^^2 = 
 
 fjL T sin^^2 (2 cos O^ + fi sin 6^. 
 
 M 
 
 Ex. IV. A rectangular block lies on a rough horizontal plane, and 
 is acted on by a horizontal force whose line of action is midway be- 
 tween two of the ver- 
 tical sides. Find the 
 magnitude of the force 
 when just sufficient to 
 produce motion, and 
 whether the motion will 
 be of the nature of slid- 
 ing or overturning. 
 
 If the force B tends to 
 overturn the body, it is 
 evident that it will turn about the edge A, and therefore the pressure, B^ 
 of the plane and the friction, 5, act at that edge. Our statical condi- 
 tions are, of course B= JV 
 
 S=B 
 Wb = Ba 
 where b is half the length of the solid, and a the distance of P from 
 
 the plane. From these we have 5= - W. 
 
 a 
 
 c 
 
 \ 
 
 
 D 
 
 
 
 A 
 
 P 
 
 
 
 G 
 
 \ 
 
 
 
 t 
 
 , 
 
 ( 
 
 A 
 
 
 
STATICS OF SOLIDS AND FLUIDS. 
 
 221 
 
 Now ^S* cannot exceed ijlR, whence we must not have - greater 
 
 than /A, if it is to be possible to upset the body by a horizontal force 
 in the line given for F. 
 
 . A simple geometrical construction enables us to solve this and similar 
 problems, and will be seen at once to be merely a graphic representa- 
 tion of the above process. Thus if we produce the directions of 
 the applied force, and of the weight, to meet in ZT, and make at A the 
 angle BAK whose co-tangent is the co-efficient of friction : there will 
 be a tendency to upset, or not, according as ZTis above, or below, AK. 
 
 Ex. V. A mass, such as a gate, is supported by two rings, A and 
 B, which pass loosely round a rough 
 vertical post. In equilibrium, it is ob- 
 vious that at A the part of the ring 
 nearest the mass, and at B the farthest 
 from it, will be in contact with the post. 
 The pressures exerted on the rings, R 
 and 6", will evidently have the directions 
 AC^ CB, indicated in the diagram. If 
 no other force besides gravity act on the 
 mass, the line of action of its weight, IV, 
 must pass through the point C (§ 584). 
 And it is obvious that, however small be 
 the co-efficient of friction, provided there 
 be friction at all, equilibrium is always 
 possible if the distance of the centre of gravity from the post be great 
 enough compared with the distance between the rings. 
 
 When the mass is just about to slide down, the full amount of 
 friction is called into play, and the angles which R and .S make with 
 the horizon are each equal to the angle of repose. If we draw A C, 
 BC according to this condition, then for equilibrium the centre of 
 gravity G must not lie between the post and the vertical line through 
 the point C thus determined. If, as in the figure, G lies in the ver- 
 tical line through C, then a force applied upwards at Q^, or down- 
 wards at Q^, will remove the tendency to fall; but a force applied 
 upwards at Q^, or downwards at Q^, will produce sliding at once. 
 
 A similar investigation is easily applied to the jamming of a sliding 
 piece or drawer, and to the determination of the proper point of appli- 
 cation of a force to move it. This we leave to the student. 
 
 As an illustration of the use of friction, let us consider a cord 
 wound round a rough cylinder, and on the point of sHding. 
 
 Neglecting the weight of the cord, which is small in practice com- 
 pared with the other forces; and con- 
 sidering a small portion AB of the 
 cord, such that the tangents at its 
 extremities include a very small angle 
 6] let T' be the tension at one end, 
 
22 2 ABSTRACT DYNAMICS. 
 
 T' at the other, p the pressure of the rope on the cylinder per unit 
 of length. 
 
 Then/.^^ = 2rsin -= 2"^ approximately. Also \y.p.AB= T - T 
 
 when the rope is just about to slip, i.e. 
 
 fx,Te=r-T, 
 
 or r={i+fie)T. 
 
 Hence, for equal small deflections, 0, of the rope, the tension 
 increases in the geometrical ratio (i + fx6) : i ; and thus by a common 
 theorem (compound interest payable every instant) we have T= £'**7^, 
 if T, T^ be the tensions at the ends of a cord wrapped on a cylinder, 
 when the external angle between the directions of the free [ends is a. 
 [c is the base of Napier's Logarithms.] We thus obtain the singular 
 result, that the dimensions of the cylinder have no influence on the 
 increase of tension by friction, provided the cord is perfectly flexible. 
 
 593. Having thus briefly considered the equilibrium of a rigid 
 body, we propose, before entering upon the subject of deformation 
 of elastic solids, to consider certain intermediate cases, in each of 
 which a particular assumption is made the basis of the investiga- 
 tion — thereby avoiding a very considerable amount of analytical 
 difficulties. 
 
 594. Very excellent examples of this kind are furnished by the 
 statics of a flexible and inextensible cord or chain, fixed at both ends, 
 and subject to the action of any forces. The curve in which the 
 chain hangs in any case may be called a Catenary, although the term 
 is usually restricted to the case of a uniform chain acted on by gravity 
 only. 
 
 595. We may consider separately the conditions of equilibrium of 
 each element; or we may apply the general condition (§ 257) that the 
 whole potential energy is a minimum, in the case of any conservative 
 system of forces; or, especially when gravity is the only external 
 force, we may consider the equilibrium of a 7f;2//<? portion of the chain 
 treated for the time as a rigid body (§ 584). 
 
 596. The first of these methods gives immediately the three follow- 
 ing equations of equilibrium, for the catenary in general : — 
 
 (i) The rate of variation of the tension per unit of length along 
 the cord is equal to the tangential component of the applied force, 
 per unit of length. 
 
 (2) The plane of curvature of the cord contains the normal com- 
 ponent of the applied force, and the centre of curvature is on the 
 opposite side of the arc from that towards which this force acts. 
 
 (3) The amount of the curvature is equal to the normal component 
 of the applied force per unit of length at any point divided by the ten- 
 sion of the cord at the same point. 
 
 The first of these is simply the equation of equilibrium of an 
 infinitely small element of the cord relatively to tangential motion. 
 The second and third express that the component of the resultant 
 
STATICS OF SOLIDS AND FLUIDS. 223 
 
 of the tensions at the two ends of an infinitely small arc, along the 
 normal through its middle point, is directly opposed and is equal to 
 the normal applied force, and is equal to the whole amount of it on 
 the arc. For the plane of the tangent Hnes in which those tensions 
 act is (§ 12) the plane of curvature. And if ^ be the angle between 
 them (or the infinitely small angle by which the angle between their 
 positive directions falls short of tt), and T the arithmetical mean of 
 their magnitudes, the component of their resultant along the line 
 bisecting the angle between their positive directions is 2 Z'sin ^0, rigor- 
 ously: or TO, since 6 is infinitely small. Hence T0=^N6s if hs be 
 the length of the arc, and N^s the whole amount of normal force 
 
 applied to it. But (§ 9) ^ = — if p be the radius of curvature; and 
 
 therefore —= -^. 
 
 P T 
 
 which is the equation stated in words (3) above. 
 
 597. From (1) of § 596, we see that if the appHed forces on any 
 particle of the cord constitute a conservative system, and if any equal 
 infinitely small lengths of the string experience the same force and 
 in the same direction when brought into any one position by motion 
 of the string, the difference of the tensions of the cord at any two 
 points of it when hanging in equilibrium, is equal to the difference 
 of the potential (§ 504) of the forces between the positions occupied 
 by these points. Hence, whatever the position where the potential is 
 reckoned zero, the tension of the string at any point is equal to the 
 potential at the position occupied by it, with a constant added. 
 
 598. From § 596 it follows immediately that if a material particle 
 of unit mass be carried along any catenary with a velocity, s, equal 
 to T, the numerical measure of the tension at any point, the force 
 upon it by which this is done is in the same direction as the resultant 
 of the applied force on the catenary at this point, and is equal to 
 the amount of this force per unit of length, multiplied by T. For 
 denoting by S the tangential, and (as before) by N the normal 
 component of the applied force per unit of length at any point P 
 of the catenary, we have, by § 596 (i), 6" for the rate of variation of 
 s per unit length, and therefore Ss for its variation per unit of time. 
 
 That is to say, 's^Ss^ ST, 
 
 or (§ 225) the tangential component force on the moving particle 
 is equal to ST. Again, by § 596 (3), 
 
 T^ i' 
 NT=^- = -, 
 P P 
 
 or the centrifugal force of the moving particle in the circle of cur- 
 vature of its path, that is to say, the normal component of the 
 force on it, is equal to JVT. And lastly, by (2) this force is in 
 the same direction as JV. We see therefore that the direction of the 
 
224 ABSTRACT DYNAMICS. 
 
 whole force on the moving particle is the same as that of the 
 resultant of 6" and N) and its magnitude is T times the magnitude 
 of this resultant. 
 
 599. Thus we see how, from the more familiar problems of the 
 kinetics of a particle, we may immediately derive curious cases 
 of catenaries. For instance : a particle under the influence of a 
 constant force in parallel lines moves in a parabola with its axis 
 vertical, with velocity at each point equal to that generated by 
 the force acting through a space equal to its distance from the 
 directrix. Hence, if z denote this distance, and / the constant 
 force, T= J 2fz 
 
 in the allied parabolic catenary; and the force on the catenary is 
 parallel to the axis, and is equal in amount per unit of length, to 
 
 J2fz V 2^ 
 
 Hence if the force on the catenary be that of gravity, it must have 
 its axis vertical (its vertex downwards of course for stable equili- 
 brium) and its mass per unit length at any point must be inversely 
 as the square root of the distance of this point above the directrix. 
 From this it follows that the whole weight of any arc of it is 
 proportional to its horizontal projection. 
 
 600. Or, if the question be, to find what force towards a given 
 fixed point, will cause a cord to hang in any given plane curve with 
 this point in its plane; it may be answered immediately from the 
 solution of the coresponding problem in 'central forces.' 
 
 601. When a perfectly flexible string is stretched over a smooth 
 surface, and acted on by no other force throughout its length than 
 the resistance of this surface, it will, when in stable equilibrium, 
 lie along a line of minimum length on the surface, between any 
 two of its points. For (§ 584) its equilibrium can be neither 
 disturbed nor rendered unstable by placing staples over it, through 
 which it is free to slip, at any two points where it rests on the 
 surface: and for the intermediate part the energy criterion of stable 
 equilibrium is that just stated. 
 
 There being no tangential force on the string in this case, and the 
 normal force upon it being along the normal to the surface, its oscu- 
 lating plane (§ 596) must cut the surface everywhere at right angles. 
 These considerations, easily translated into pure geometry, establish 
 the fundamental property of the geodetic lines on any surface. The 
 analytical investigations of the question, when adapted to the case 
 of a chain of not given length, stretched between two given points on 
 a given smooth surface, constitute the direct analytical demonstration 
 of this property. 
 
 In this case it is obvious that the tension of the string is the same 
 at every point, and the pressure of the surface upon it is [§ 596 (3)] 
 at each point proportional to the curvature of the string. 
 
STATICS OF SOLIDS AND FLUIDS. 225 
 
 602. No real surface being perfectly smooth, a cord or chain may 
 rest upon it when stretched over so great a length of a geodetic on a 
 convex rigid body as to be not of minimum length between its 
 extreme points : but practically, as in tying a cord round a ball, 
 for permanent security it is necessary, by staples or otherwise, to 
 constrain it from lateral slipping at successive points near enough 
 to one another to make each free portion a true minimum . on the 
 surface. 
 
 603. A very important practical case is supplied by the con- 
 sideration of a rope wound round a rough cylinder. We may 
 suppose it to lie in a plane perpendicular to the axis, as we thus 
 simplify the question very considerably without sensibly injuring 
 the utility of the solution. To simplify still further, we shall suppose 
 that no forces act on the rope but tensions and the reaction of 
 the cylinder. In practice this is equivalent to the supposition that 
 the tensions and reactions are very large compared with the weight 
 of the rope or chain ; which, however, is inadmissible in some 
 important cases, especially such as occur in the application of the 
 principle to brakes for laying submarine cables, to ergometers, and 
 to windlasses (or capstans with horizontal axes). 
 
 By § 592 we have r- T^cm®, 
 
 showing that, for equal successive amounts of integral curvature 
 (§ 14), the tension of the rope augments in geometrical progression. 
 To give an idea of the magnitudes involved, suppose /u, = -5, ^ = tt, then 
 
 r = r.c -5^ = 4 . 8 1 7; roughly. 
 Hence if the rope be wound three times round the post or cylinder 
 the ratio of the tensions of its ends, when motion is about to com- 
 mence, is 
 
 5" : I or about 15,000 : i. 
 
 Thus we see how, by the aid of friction, one man may easily check 
 the motion of the largest vessel, by the simple expedient of coiling a 
 rope a few times round a post. This application of friction is 
 of great importance in many other applications, especially to ergome- 
 ters (§§ 389, 390). 
 
 604. With the aid of the preceding investigations, the student 
 may easily work out for himself the solution of the general problem 
 of a cord under the action of any forces, and constrained by a 
 rough surface; it is not of sufficient importance or interest to find 
 a place here. 
 
 605. An elongated body of elastic material, which for brevity 
 we shall generally call a wire, bent or twisted to any degree, sub- 
 ject only to the condition that the radius of curvature and the reci- 
 procal of the twist are everywhere very great in comparison with 
 the greatest transverse dimension, presents a case in which, as we 
 
 T. 15 
 
2 26 ABSTRACT DYNAMICS. 
 
 shall see, the solution of the general equations for the equilibrium 
 of an elastic solid is either obtainable in finite terms, or is reducible 
 to comparatively easy questions agreeing in mathematical conditions 
 with some of the most elementary problems of hydrokinetics, elec- 
 tricity, and thermal conduction. And it is only for the determination 
 of certain constants depending on the section of the wire and the 
 elastic quality of its substance, which measure its flexural and 
 torsional rigidity, that the solutions of these problems are required. 
 When the constants of flexure and torsion are known, as we shall 
 now suppose them to be, whether from theoretical calculation or 
 experiment, the investigation of the form and twist of any length 
 of the wire, under the influence of any forces which do not produce 
 a violation of the condition stated above, becomes a subject of 
 mathematical analysis involving only such principles and fornmlae 
 as those that constitute the theory of curvature (§§ 9-15) and twist 
 in geometry or kinematics. 
 
 606. Before entering on the general theory of elastic solids, we 
 shall therefore, according to the plan proposed in § 593, examine 
 the dynamic properties and investigate the conditions of equilibrium 
 of a perfectly elastic wire, without admitting any other condition or 
 limitation of the circumstances than what is stated in § 605, and 
 without assuming any special quality of isotropy, or of crystalline, 
 fibrous or laminated structure in the substance. 
 
 607. Besides showing how the constants of flexural and tor- 
 sional rigidity are to be determined theoretically from the form of 
 the transverse section of the wire, and the proper data as to the 
 elastic qualities of its substance, the complete theory simply in- 
 dicates that, provided the conditional limit of deformation is not 
 exceeded, the following laws will be obeyed by the wire under 
 stress : — 
 
 Let the whole mutual action between the parts of the wire on the 
 two sides of the cross section at any point (being of course the action 
 of the matter infinitely near this plane on one side, upon the matter 
 infinitely near it on the other side), be reduced to a single force 
 through any point of the section and a single couple. Then — 
 
 I. The twist and curvature of the wire in the neighbourhood of 
 this section are independent of the force, and depend solely on the 
 couple. 
 
 II. The curvatures and rates of twist producible by any several 
 couples separately, constitute, if geometrically compounded, the curva- 
 ture and rate of twist which are actually produced by a mutual action 
 equal to the resultant of those couples. 
 
 608. It may be added, although not necessary for our present 
 purpose, that there is one determinate point in the cross section 
 such that if it be chosen as the point to which the forces are trans- 
 ferred, a higher order of approximation is obtained for the fulfilment 
 
STATICS OF SOLID S AND FLUIDS 227 
 
 of these laws than if any other point of the section be taken. That 
 point, which in the case of a wire of substance uniform through its 
 cross section is the centre of inertia of the area of the section, we 
 shall generally call the elastic centre, or the centre of elasticity, of 
 the section. It has also the following important property : — The line 
 of elastic centres, or, as we shall call it, the elastic central line, remains 
 sensibly unchanged in length to whatever stress within our conditional 
 limits (§ 605) the wire be subjected. The elongation or contraction 
 produced by the neglected resultant force, if this is in such a direction 
 as to produce any, will cause the line of rigorously no elongation to 
 deviate only infinitesimally from the elastic central line, in any part 
 of the wire finitely curved. It will, however, clearly cause there to 
 be no line of rigorously unchanged lengthy in any straight part of the 
 wire : but as the whole elongation would be infinitesimal in com- 
 parison with the effective actions with which we are concerned, this 
 case constitutes no exception to the preceding statement. 
 
 609. In the most important practical cases, as we shall see later, 
 those namely in which the substance is either ' isotropic,' which is 
 sensibly the case with common metallic wires, or has an axis of 
 elastic symmetry along the length of the piece, one of the three 
 normal axes of torsion and flexure coincides with the length of the 
 wire, and the two others are perpendicular to it ; the first being an 
 axis of pure torsion, and the two others axes of pure flexure. Thus 
 opposing couples round the axis of the wire twist it simply without 
 bending it ; and opposing couples in either of the two principal planes 
 of flexure, bend it into a circle. 
 
 610. In the more particular case in which two principal rigidities 
 against flexure are equal, every plane through the length of the wire 
 is a principal plane of flexure, and the rigidity against flexure is equal 
 in all. This is clearly the case with a common round wire, or rod, or 
 with one of square section. It can be shown to be the case for a 
 rod of isotropic material and of any form of normal section which 
 is ' kinetically symmetrical' (§ 239) round all axes in its plane through 
 its centre of inertia. 
 
 611. In this case, if one end of the rod or wire be held fixed, and 
 a couple be applied in any plane to the other end, a uniform spiral 
 form will be produced round an axis perpendicular to the plane 
 of the couple. The lines of the substance parallel to the axis of 
 the spiral are not, however, parallel to their original positions: and 
 lines traced along the surface of the wire parallel to its length when 
 straight, become as it were secondary spirals, circling round the 
 main spiral formed by the central line of the deformed wire. Lastly, 
 in the present case, if we suppose the normal section of the wire 
 to be circular, and trace uniform spirals along its surface when 
 deformed in the manner supposed (two of which, for instance, are 
 the lines along which it is touched by the inscribed and the circum- 
 
 15—2 
 
2 28 ABSTRACT DYNAMICS, 
 
 scribed cylinder), these lines do not become straight, but become 
 spirals laid on as it were round the wire, when it is allowed to take 
 its natural straight and untwisted condition. 
 
 612. A wire of equal flexibility in all directions may clearly be held 
 in any specified spiral form, and twisted to any stated degree, by a 
 determinate force and couple applied at one end, the other end being 
 held fixed. The direction of the force must be parallel to the axis 
 of the spiral, and, with the couple, must constitute a system of which 
 this line is (§ 579) the central axis : since otherwise there could not 
 be the same system of balancing forces in every normal section of 
 the spiral. All this may be seen clearly by supposing the wire to 
 be first brought by any means to the specified condition of strain ; 
 then to have rigid planes rigidly attached to its two ends perpendicular 
 to its axis, and these planes to be rigidly connected by a bar lying 
 in this line. The spiral wire now left to itself cannot but be in 
 equilibrium : although if it be too long (according to its form and 
 degree of twist) the equilibrium may be unstable. The force along 
 the central axis, and the couple, are to be determined by the condition 
 that, when the force is transferred after Poinsot's manner to the elastic 
 centre of any normal section, they give two couples together equiva- 
 lent to the elastic couples of flexure and torsion. 
 
 613. A wire of equal flexibility in all directions may be held in 
 any stated spiral form by a simple force along its axis between rigid 
 pieces rigidly attached to its two ends, provided that, along with its 
 spiral form a certain degree of twist be given to it. The force is 
 determined by the condition that its moment round the perpendicular 
 through any point of the spiral to its osculating plane at that point, 
 must be equal and opposite to the elastic unbending couple. The 
 degree of twist is that due (by the simple equation of torsion) to the 
 moment of the- force thus determined, round the tangent at any point 
 of the spiral. The direction of the force being, according to the 
 preceding condition, such as to press together the ends of the spiral, 
 the direction of the twist in the wire is opposite to that of the tortuosity 
 (§ 13) of its central curve. 
 
 614. The principles with which we have just been occupied are 
 immediately applicable to the theory of spiral springs; and we 
 shall therefore make a short digression on this curious and im- 
 portant practical subject before completing our investigation of elastic 
 curves. 
 
 A common spiral spring consists of a uniform wire shaped per- 
 manently to have, when unstrained, the form of a regular helix, with 
 the principal axes of flexure and torsion everywhere similarly situated 
 relatively to the curve. When used in the proper manner, it is 
 acted on, through arms or plates rigidly attached to its ends, by forces 
 such that its form as altered by them is still a regular helix. This 
 condition is obviously fulfilled if (one terminal being held fixed) an 
 
STATICS OF SOLIDS AND FLUIDS. 229 
 
 infinitely small force and infinitely small eouple be applied to the 
 other terminal along the axis, and in a plane perpendicular to it, and 
 if the force and couple be increased to any degree, and always kept 
 along and in the plane perpendicular to the axis of the altered spiral. 
 It would, however, introduce useless complication to work out the 
 details of the problem except for the case (§ 609) in which one of 
 the principal axes coincides with the tangent to the central line, and 
 is therefore an axis of pure torsion, as spiral springs in practice 
 always belong to this case. On the other hand, a very interesting 
 complication occurs if we suppose (what is easily realized in practice, 
 though to be avoided if merely a good spring is desired) the normal 
 section of the wire to be of such a figure, and so situated relatively 
 to the spiral, that the planes of greatest and least flexural rigidity 
 are obHque to the tangent plane of the cylinder. Such a spring when 
 acted on in the regular manner at its ends must experience a certain 
 degree of turning through its whole length round its elastic central 
 curve in order that the flexural couple developed may be, as we 
 shall immediately see it must be, precisely in the osculating plane of 
 the altered spiral. All that is interesting in this very curious effect is 
 illustrated later in full detail (§ 624 of our larger work) in the case of an 
 open circular arc altered by a couple in its own plane, into a circular 
 arc of greater or less radius ; and for brevity and simplicity we shall 
 confine the detailed investigation of spiral springs on which we now 
 enter, to the cases in which either the wire is of equal flexural rigidity 
 in all directions, or the two principal planes of (greatest and least or 
 least and greatest) flexural rigidity coincide respectively with the 
 tangent plane to the cylinder, and the normal plane touching the 
 central curve of the wire, at any point. 
 
 615. The axial force, on the movable terminal of the spring, trans- 
 ferred according to Poinsot to any point in the elastic central 
 curve, gives a couple in the plane through that point and the axis 
 of the spiral. The resultant of this and the couple which we suppose 
 applied to the terminal in the plane perpendicular to the axis of the 
 spiral is the effective bending and twisting couple : and as it is in a 
 plane perpendicular to the tangent plane to the cylinder, the com- 
 ponent of it to which bending is due must be also perpendicular to 
 this plane, and therefore is in the osculating plane of the spiral. This 
 component couple therefore simply maintains a curvature different 
 from the natural curvature of the wire, and the other, that is, the 
 couple in the plane normal to the central curve, pure torsion. 
 The equations of equilibrium merely express this in mathematical 
 language. 
 
 616. The potential energy of the strained spring is 
 
 l[^(t^-^/ + ^TT/, 
 
 if A denote the torsional rigidity, B the flexural rigidity in the plane 
 of curvature, w and tzr^ the strained and unstrained curvatures, and t 
 
230 ABSTRACT DYNAMICS. 
 
 the torsion of the wire in the strained condition, the torsion being 
 reckoned as zero in the unstrained condition. The axial force, and 
 the couple, required to hold the spring to any given length reckoned 
 along the axis of the spiral, and to any given angle between planes 
 through its ends and the axes, are of course (§ 244) equal to the rates 
 of variation of the potential energy, per unit of variation of these 
 co-ordinates respectively. It must be carefully remarked, hovv^ever, 
 that, if the terminal rigidly attached to one end of the spring be held 
 fast, so as to fix the tangent at this end, and the motion of the other 
 terminal be so regulated as to keep the figure of the intermediate 
 spring always truly spiral, this motion will be somewhat complicated ; 
 as the radius of the cylinder, the inclination of the axis of the spiral 
 to the fixed direction of the tangent at the fixed end, and the position 
 of the point in the axis in which it is cut by the plane perpendicular 
 to it through the fixed end of the spring, all vary as the spring changes 
 in figure. The effective components of any infinitely small motion of 
 the movable terminal are its component translation along, and rota- 
 tion round, the instantaneous position of the axis of the spiral [two 
 degrees of freedom], along with which it will generally have an 
 infinitely small translation in some direction and rotation round some 
 line, each perpendicular to this axis, and determined from the two 
 degrees of arbitrary motion, by the condition that the curve remains a 
 true spiral. 
 
 617. In the practical use of spiral springs, this condition is not 
 rigorously fulfilled : but, instead, one of two plans is generally fol- 
 lowed : — (i) Force, without any couple, is applied pulling out or 
 pressing together two definite points of the two terminals, each as 
 nearly as may be in the axis of the unstrained spiral; or (2) One 
 terminal being held fixed, the other is allowed to slide, without any 
 turning, in a fixed direction, being as nearly as may be the direction 
 of the axis of the spiral when unstrained. The preceding investiga- 
 tion is applicable to the infinitely small displacement in either case : 
 the couple being put equal to zero for case (i), and the instantaneous 
 rotatory motion round the axis of the spiral equal to zero for 
 case (2). 
 
 618. In a spiral spring of infinitely small inclination to the plane 
 perpendicular to its axis, the displacement produced in the movable 
 terminal by a force applied to it in the axis of the spiral is a simple 
 rectilineal translation in the direction of the axis, and is equal to the 
 length of the circular arc through which an equal force carries one 
 end of a rigid arm or crank equal in length to the radius of the 
 cylinder, attached perpendicularly to one end of the wire of the spring 
 supposed straightened and held with the other end absolutely fixed, 
 and the end which bears the crank, free to turn in a collar. This 
 statement is due to J. Thomson \ who showed that in pulling out 
 a spiral spring of infinitely small incHnation the action exercised and 
 
 1 Camb. ami Dub. Math. Jour. 1848. 
 
STATICS OF SOLIDS AND FLUIDS. 231 
 
 the elastic quality used are the same as in a torsion-balance with the 
 same wire straightened (§ 386). This theory is, as he proved ex- 
 perimentally, sufficiently approximate for most practical applications ; 
 spiral springs, as commonly made and used, being of very small 
 inclination. There is no difficulty in finding the requisite correction, 
 for the actual inclination in any case. The fundamental principle that 
 spiral springs act chiefly by torsion seems to have been first discovered 
 by Binet in i8i4\ 
 
 619. Returning to the case of a uniform wire straight and untwisted 
 (that is, cylindrical or prismatic) when free from stress; let us suppose 
 one end to be held fixed in a given direction, and no other force 
 from without to influence it except that of a rigid frame attached to 
 its other end acted on by a force, i?, in a given line, AB, and a 
 couple, G, in a plane perpendicular to this line. The form and twist 
 it will have when in equilibrium are determined by the condition that 
 the torsion and flexure at any point, F, of its length are those due to 
 the couple G compounded with the couple obtained by bringing R 
 to F. 
 
 620. Kirchhoflf has made a very remarkable comparison between 
 the static problem of bending and twisting a wire, and the kinetic 
 problem of the rotation of a rigid body. We can give here 
 but one instance, the simplest of all — the Elastic Curve of James 
 Bernoulli, and the common pendulum. A uniform straight wire, 
 either equally flexible in all planes through its length, or having its 
 directions of maximum and minimum flexural rigidity in two planes 
 through its whole length, is acted on by a force and couple in one 
 of these planes, applied either directly to one end, or by means of an 
 arm rigidly attached to it, the other end being held fast. The force 
 and couple may, of course (§ 568), be reduced to a single force, the 
 extreme case of a couple being mathematically included as an in- 
 finitely small force at an infinitely great distance. To avoid any 
 restriction of the problem, we must suppose this force applied to an 
 arm rigidly attached to the wire, although in any case in which the 
 line of the force cuts the wire, the force may be applied directly at 
 the point of intersection, without altering the circumstances of the 
 wire between this point and the fixed end. The wire will, in these 
 circumstances, be bent into a curve lying throughout in the plane 
 through its fixed end and the line of the force, and (§ 609) its curva- 
 tures at difl"erent points will, as was first shown by James Bernoulli, 
 be simply as their distances from this line. The curve fulfilling this 
 condition has clearly just two independent parameters, of which one 
 is conveniently regarded as the mean proportional, a, between the 
 radius of curvature at any point and its distance from the line of force, 
 and the other, the maximum distance, b, of the wire from the line of 
 force. By choosing any value for each of these parameters it is easy 
 to trace the corresponding curve with a very high approximation to 
 accuracy, by commencing with a small circular arc touching at one 
 
 ^ St Venant, Coviptes Rcndtts, Sept. 1864. 
 
232 ABSTRACT DYNAMICS. 
 
 extremity a straight line at the given maximum distance from the line 
 of force, and continuing by small circular arcs, with the proper 
 increasing radii, according to the diminishing distances of their middle 
 points from the line of force. The annexed diagrams are, however, 
 not so drawn, but are simply traced from the forms actually assumed 
 by a flat steel spring, of small enough breadth not to be much 
 disturbed by tortuosity in the cases in which different parts of it cross 
 one another. The mode of application of the force is sufficiently 
 explained by the indications in the diagram. 
 
 621. As we choose particularly the common pendulum for the 
 corresponding kinetic problem, the force acting on the rigid body in 
 the comparison must be that of gravity in the vertical through its 
 centre of gravity. It is convenient, accordingly, not to take imity as 
 the velocity for the point of comparison along the bent wire, but the 
 velocity which gravity would generate in a body falling through a height 
 equal to half the constant, a, of § 620: and this constant, a^ will 
 then be the length of the isochronous simple pendulum. Thus if 
 an elastic curve be held with its Hne of force vertical, and if a point, 
 jP, be moved along it with a constant velocity equal to Jga^ {a de- 
 noting the mean proportional between the radius of curvature at any 
 point, and its distance from the line of force,) the tangent at P will 
 keep always parallel to a simple pendulum, of length a^ placed at 
 any instant parallel to it, and projected with the same angular 
 velocity. Diagrams i to 5, correspond to vibrations of the pendu- 
 lum. Diagram 6 corresponds to the case in which the pendulum 
 would just reach its position of unstable equilibrium in an infinite 
 time. Diagram 7 corresponds to cases in which the pendulum flies 
 round continuously in one direction, with periodically increasing and 
 diminishing velocity. The extreme case, of the circular elastic 
 curve, corresponds to an infinitely long pendulum flying round with 
 finite angular velocity, which of course experiences only infinitely 
 small variation in the course of the revolution. A conclusion worthy 
 of remark is, that the rectification of the elastic curve is the same 
 analytical problem as finding the time occupied by a pendulum in 
 describing any given angle. 
 
 622. For the simple and important case of a natually straight 
 wire, acted on by a distribution of force, but not of couple, through 
 its length, the condition fulfilled at a perfectly free end, acted on by 
 neither force nor couple, is that the curvature is zero at the end, and 
 its rate of variation from zero, per unit of length from the end, is, 
 at the end, zero. In other words, the curvatures at points infinitely 
 near the end are as the squares of their distances from the end in 
 general (or, as some higher power of these distances, in singular 
 cases). The same statements hold for the change of curvature pro- 
 duced by the stress, if the unstrained wire is not straight, but the 
 other circumstances the same as those just specified. 
 
 623. As a very simple example of the equilibrium of a wire sub- 
 ject to forces through its length, let us suppose the natural form to 
 
^ or THE 
 UNIVERSITY 
 
 Of . 
 
 STATICS OF SOLIDS AND FLUIDS. 
 
 333 
 
234 ABSTRACT DYNAMICS. 
 
 be straight, and the appHed forces to be in lines, and the couples 
 to have their axes, all perpendicular to its length, and to be not great 
 enough to produce more than an infinitely small deviation from the 
 straight line. Further, in order that these forces and couples may- 
 produce no torsion, let the three flexure-torsion axes be perpendicular 
 to and along the wire. But we shall not hmit the problem further 
 by supposing the section of the wire to be uniform, as we should thus 
 exclude some of the most important practical applications, as to 
 beams of balances, levers in machinery, beams in architecture and 
 engineering. It is more instructive to investigate the equations of 
 equilibrium directly for this case than to deduce them from the 
 equations worked out above for the much more comprehensive 
 general problem. The particular principle for the present case is 
 simply that the rate of variation of the rate of variation, per unit of 
 length along the wire, of the bending couple in any plane through 
 the length, is equal, at any point, to the applied force per unit of 
 length, with the simple rate of variation of the applied couple sub- 
 tracted. This, together with the direct equations (§ 607) between 
 the component bending couples, gives the required equations of 
 equilibrium. 
 
 624. If the directions of maximum and minimum flexural rigidity 
 lie throughout the wire in two planes, the equations of equilibrium 
 become simplified when these planes are chosen as planes of re- 
 ference, XOY^ XOZ. The flexure in either plane then depends 
 simply on the forces in it, and thus the problem divides itself into 
 two quite independent problems of integrating the equations of 
 flexure in the two principal planes, and so finding the projections 
 of the curve on two fixed planes agreeing with their position when 
 the rod is straight. 
 
 625. When a uniform bar, beam, or plank is balanced on a single 
 trestle at its middle, the droop of its ends is only f of the droop 
 which its middle has when the bar is supported on trestles at its 
 ends. From this it follows that the former is f and latter f of the 
 droop or elevation produced by a force equal to half the weight of 
 the bar, applied vertically downwards or upwards to one end of it, 
 if the middle is held fast in a horizontal position. For let us first 
 suppose the whole to rest on a trestle under its middle, and let two 
 trestles be placed under its ends and gradually raised till the pressure 
 is entirely taken oft" from the middle. During this operation the 
 middle remains fixed and horizontal, while a force increasing to half 
 the weight, applied vertically upwards on each end, raises it through 
 a height equal to the sum of the droops in the two cases above 
 referred to. This result is of course proved directly by comparing 
 the absolute values of the droop in those two cases as found above, 
 with the deflection from the tangent at the end of the cord in the 
 elastic curve. Fig. 2, § 623, which is cut by the cord at right angles. 
 It may be stated otherwise thus : the droop of the middle of a 
 uniform beam resting on trestles at its ends is mcreased in the 
 
STATICS OF SOLIDS AND FLUIDS. 235 
 
 ratio of 5 to 13 by laying a mass equal in weight to itself on its 
 middle : and, if the beam is hung by its middle, the droop of the 
 ends is increased in the ratio of 3 to 11 by hanging on each of 
 them a mass equal to half the weight of the beam. 
 
 626. The important practical problem of finding the distribution 
 of the weight of a solid on points supporting it, when more than 
 two of these are in one vertical plane, or when there are more than 
 three altogether, which (§ 588) is indeterminate^ if the soHd is 
 perfectly rigid, may be completely solved for a uniform elastic beam, 
 naturally straight, resting on three or more points in rigorously fixed 
 positions all nearly in one horizontal line, by means of the preceding 
 results. 
 
 If there are i points of support, the /— i parts of the rod between 
 them in order and the two end parts will form / + i curves expressed 
 by distinct algebraic equations, each involving four arbitrary con- 
 stants. For determining these constants we have 4/+ 4 equations in 
 all, expressing the following conditions : — 
 
 I. The ordinates of the inner ends of the projecting parts of the 
 rod, and of the two ends of each intermediate part, are respectively 
 equal to the given ordinates of the corresponding points of support 
 [2/ equations]. 
 
 II. The curves on the two sides of each support have coincident 
 tangents and equal curvatures at the point of transition from one 
 to the other [2/ equations]. 
 
 III. The curvature and its rate of variation per unit of length 
 along the rod, vanish at each end [4 equations]. 
 
 Thus the equation of each part of the curve is completely de- 
 termined: and by means of it we find the shearing force in any 
 normal section. The difference between these in the neighbouring 
 portions of the rod on the two sides of a point of support, is of 
 course equal to the pressure on this point. 
 
 627. The solution for the case of this problem in which two 
 of the points of support are at the ends, and the third midway 
 between them either exactly in the line joining them, or at any 
 given very small distance above or below it, is found at once, without 
 analytical work, from the particular results stated in § 625. Thus 
 if we suppose the beam, after being first supported wholly by trestles 
 at its ends, to be gradually pressed up by a trestle under its 
 middle, it will bear a force simply proportional to the space through 
 which it is raised from the zero point, until all the weight is taken 
 off the ends, and borne by the middle. The whole distance through 
 
 which the middle rises during this process is, as we found, ^ . -^ ; 
 
 and this whole elevation is f of the droop of the middle in the 
 
 1 It need scarcely be remarked that indeterminateness does not exist in nature. 
 How it may occur in the problems of abstract dynamics, and is obviated by taking 
 something more of the properties of matter into account, is instructively illustrated 
 by the circumstances referred to in the text. 
 
236 ABSTRACT DYNAMICS. 
 
 first position. If therefore, for instance, the middle trestle be fixed 
 exactly in the line joining those under the ends, it will bear f of 
 the whole weight, and leave y\ to be borne by each end. And 
 if the middle trestle be lowered from the line joining the end 
 ones by -/^ of the space through which it would have to be lowered 
 to relieve itself of all pressure, it will bear just \ of the whole weight, 
 and leave the other two-thirds to be equally borne by the two ends. 
 
 628. A wire of equal flexibility in all directions, and straight 
 when freed from stress, offers, when bent and twisted in any manner 
 whatever, not the slightest resistance to being turned round its elastic 
 central curve, as its conditions of equilibrium are in no way affected 
 by turning the whole wire thus equally throughout its length. The 
 useful application of this principle, to the maintenance of equal 
 angular motion in two bodies rotating round different axes, is 
 rendered somewhat difficult in practice by the necessity of a perfect 
 attachment and adjustment of each end of the wire, so as to have 
 the tangent to its elastic central curve exactly in line with the 
 axis of rotation. But if this condition is rigorously fulfilled, and 
 the wire is of exactly equal flexibility in every direction, and exactly 
 straight when free from stress, it will give, against any constant 
 resistance, an accurately uniform motion from one to another of 
 two bodies rotating round axes which may be inclined to one 
 another at any angle, and need not be in one plane. If they are 
 in one plane, if there is no resistance to the rotatory motion, and 
 if the action of gravity on the wire is insensible, it will take some 
 of the varieties of form (§ 620) of the plane elastic curve of James 
 Bernoulli. But however much it is altered from this, whether by 
 the axes not being in one plane, or by the torsion accompanying 
 the transmission of a couple from one shaft to the other, and 
 necessarily, when the axes are in one plane, twisting the wire out 
 of it, or by gravity, the elastic central curve will remain at rest, 
 the wire in every normal section rotating round it with uniform 
 angular velocity, equal to that of each of the two bodies which it 
 connects. Under Properties of Matter, we shall see, as indeed 
 may be judged at once from the performances of the vibrating 
 spring of a chronometer for twenty years, that imperfection in the 
 elasticity of a metal wire does not exist to any such degree as to 
 prevent the practical application of this principle, even in mechanism 
 required to be durable. 
 
 It is right to remark, however, that if the rotation be too rapid, the 
 equilibrium of the wire rotating round its unchanged elastic central 
 curve may become unstable, as is immediately discovered by experi- 
 ments (leading to very curious phenomena), when, as is often done in 
 illustrating the kinetics of ordinary rotation, a rigid body is hung by 
 a steel wire, the upper end of which is kept turning rapidly. 
 
 629. The definitions and investigations regarding strain, §§ 135- 
 161, constitute a kinematical introduction to the theory of elastic 
 solids. We must now, in commencing the elementary dynamics 
 
STATICS OF SOLIDS AND FLUIDS 
 
 237 
 
 of the subject, consider the forces called into play through the 
 interior of a solid when brought into a condition of strain. We 
 adopt, from Rankine', the term stress to designate such forces, 
 as distinguished from strain defined (§ 135) to express the merely 
 geometrical idea of a change of volume or figure. 
 
 630. When through any space in a body under the action of force, 
 the mutual force between the portions of matter on the two sides of 
 any plane area is equal and parallel to the mutual force across any 
 equal, similar, and parallel plane area, the stress is said to be homo- 
 geneous through that space. In other words, the stress experienced 
 by the matter is homogeneous through any space if all equal, similar, 
 and similarly turned portions of matter within this space are similarly 
 and equally influenced by force. 
 
 631. To be able to find the distribution of force over the surface 
 of any portion of matter homogeneously stressed, we must know the 
 direction, and the amount per unit area, of the force across a plane 
 area cutting through it in any direction. Now if we know this for 
 any three planes, in three different directions, we can find it for a 
 plane in any direction as we see in a moment by considering what 
 is necessary for the equilibrium of a tetrahedron of the substance. The 
 resultant force on one of its sides must be equal and opposite to the 
 resultant of the forces on the three others, which is known if these sides 
 are parallel to the three planes for each of which the force is given. 
 
 632. Hence the stress, in a body homogeneously stressed, is com- 
 pletely specified when the direction, and the amount per unit area, 
 of the force on each of three distinct planes is given. It is, in the 
 analytical treatment of the subject, generally convenient to take these 
 planes of reference at right angles to one another. But we should 
 immediately fall into error did we not remark that the specification 
 here indicated consists not of nine but in reality only of six, inde- 
 pendent elements. For if the equilibrating forces on the six faces of 
 a cube be each resolved into three components parallel to its three 
 edges, OX, OV, OZ, we have in 
 all 18 forces; of which each pair 
 acting perpendicularly on a pair of 
 
 opposite faces, being equal and ^^ 
 
 directly opposed, balance one an- ^,^^1 - ^k1 — ^-> 
 
 other. The twelve tangential com- 
 ponents that remain constitute three 
 pairs of couples having their axes . , 
 
 in the direction of the three edges, I i ..-Y 
 
 each of which must separately be in T 
 equilibrium. The diagram shows Q 
 
 the pair of equilibrating couples 
 having OV for axis; from the con- 
 sideration of which we infer that the 
 
 1 Canibrixigc and Dublin Mathematical Journal y 1850. 
 
238 ABSTRACT DYNAMICS, 
 
 forces on the faces {zy)^ parallel to OZ, are equal to the forces on the 
 faces {yx)^ parallel to OX. Similarly, we see that the forces on the 
 faces {yx)^ parallel to OY, are equal to those on the faces (jcs), parallel 
 to OZ) and that the forces on {xz), parallel to OX^ are equal to 
 those on {zy), parallel to OY. 
 
 633. Thus, any three rectangular planes of reference being chosen, 
 we may take six elements thus, to specify a stress : T', Q, ^ the 
 normal components of the forces on these planes; and S, T, U 
 the tangential components, respectively perpendicular to OX, of the 
 forces on the two planes meeting in OX, perpendicular to OY, of 
 the forces on the planes meeting in 6^ 1^ and perpendicular to OZ, 
 of the forces on the planes meeting in OZ; each of the six forces 
 being reckoned, per unit of area. A normal component will be 
 reckoned as positive when it is a traction tending to separate 
 the portions of matter on the two sides of its plane. P, Q, R are 
 sometimes called simple longitudinal stresses, and »S, T, U simple 
 shearing stresses. 
 
 From these data, to find in the manner explained in § 631, the 
 force on any plane, specified by /, m, 71, the direction-cosines of 
 its normal ; let such a plane cut OX, OY, OZ in the three points 
 X, Y, Z. Then, if the area XYZ be denoted for a moment by A, 
 the areas YOZ, ZOX, XO Y, being its projections on the three rec- 
 tangular planes, will be respectively equal to Al, Am, An. Hence, 
 for the equilibrium of the tetrahedron of matter bounded by those 
 four triangles, we have, if F, G, H denote the components of the 
 force experienced by the first of them, XYZ, per unit of its area, 
 
 F.A =P.lA-v U. mA + T.nA, 
 and the two symmetrical equations for the components parallel to 
 6>Fand OZ. Hence, dividing by A, we conclude 
 
 F = Fl + Um-hTn] 
 
 G=Ul+Q?n + Sn\. (i) 
 
 Fr= Tl + Sm + Rn\ 
 These expressions stand in the well-known relation to the ellipsoid 
 
 Px" + (2/ + i?^' + 2 {Syz + Tzx + Uxy) = i, (2) 
 
 according to which, if we take 
 
 X = lr, y = mr, z = nr, 
 and if X, /x, v denote the direction-cosines and p the length of the 
 perpendicular from the centre to the tangent plane at {x, y, z) of the 
 ellipsoidal surface, we have 
 
 ^-Jr^ ^-Jr^ ^=Jr' 
 We conclude that 
 
 634. For any fully specified state of stress in a solid, a quadratic 
 surface may always be determined, which shall represent the stress 
 graphically in the following manner : — 
 
 To find the direction and the amount per unit area, of the force 
 
STATICS OF SOLIDS AND FLUIDS. 239 
 
 acting across any plane in the solid, draw a line perpendicular to 
 this plane from the centre of the quadratic to its surface. The 
 required force will be equal to the reciprocal of the product of the 
 length of this line into the perpendicular from the centre to the 
 tangent plane at the point of intersection, and will be perpendicular 
 to the latter plane. 
 
 635. From this it follows that for any stress whatever there are 
 three determinate planes at right angles to one another such that the 
 force acting in the solid across each of them is precisely perpendicular 
 to it. These planes are called the principal or normal planes of the 
 stress; the forces upon them, per unit area, — its principal or normal 
 tractions; and the Hues perpendicular to them, — its principal or 
 normal axes, or simply its axes. The three principal semi-diameters 
 of the quadratic surface are equal to the reciprocals of the square 
 roots of the normal tractions. If, however, in any case each of the 
 three normal tractions is negative, it will be convenient to reckon 
 them rather as positive pressures; the reciprocals of the square roots 
 of which will be the semi-axes of a real stress-ellipsoid representing 
 the distribution of force in the manner explained above, with pressure 
 substituted throughout for traction. 
 
 636. When the three normal tractions are all of one sign, the 
 stress-quadratic is an ellipsoid; the cases of an ellipsoid of revolution 
 and a sphere being included, as those in which two, or all three, are 
 equal. When one of the three is negative and the two others posi- 
 tive, the surface is a hyperboloid of one sheet. When one of the 
 normal tractions is positive and the two others negative, the surface 
 is a hyperboloid of two sheets. 
 
 637. When one of the three principal tractions vanishes, while 
 the other two are finite, the stress-quadratic becomes a cylinder, 
 circular, elliptic, or hyperbolic, according as the other two are equal, 
 unequal of one sign, or of contrary signs. When two of the three 
 vanish, the quadratic becomes two planes; and the stress in this case 
 is (§ ^Z2)) called a simple longitudinal stress. The theory of prin- 
 cipal planes, and normal tractions just stated (§ 635), is then equiva- 
 lent to saying that any stress whatever may be regarded as made up 
 of three simple longitudinal stresses in three rectangular directions. 
 The geometrical interpretations are obvious in all these cases. 
 
 638. The composition of stresses is of course to be effected by 
 adding the component tractions thus: — If (F^, Q^, F^, S^, T^, U^), 
 (F^j (2^, F^, S^, 7;, U^), etc., denote, according to § 633, any given 
 set of stresses acting simultaneously in a substance, their joint effect 
 is the same as that of a simple resultant stress of which the specifica- 
 tion in corresponding terms is (IF, 2(2, '^F, 2^, 27", 26^. 
 
 639. Each of the statements that have now been made (§§ 630, 
 638) regarding stresses, is applicable to i7ifinitely small strains, if for 
 traction perpendicular to any plane, reckoned per unit of its area, 
 we substitute elongation, in the lines of the traction, reckoned per 
 unit of length; and for half the tangential traction parallel to any 
 
240 
 
 ABSTRACT DYNAMICS. 
 
 direction, shear in the same direction, reckoned in the manner ex- 
 plained in § 154. The student will find it a useful exercise to study 
 in detail this transference of each one of those statements, and to 
 justify it by modifying in the proper manner the results of §§ 150, 151, 
 152, 153, i54» 161, to adapt them to infinitely small strains. It 
 must be remarked that the strain-quadratic thus formed according to 
 the rule of § 634, which may have any of the varieties of character 
 mentioned in §§ 6t^(), 637, is not the same as the strain-ellipsoid of 
 § 141, which is always essentially an ellipsoid, and which, for an in- 
 finitely small strain, differs infinitely little from a sphere. 
 
 The comparison of § 151, with the result of § 632 regarding tan- 
 gential tractions is particularly interesting and important. 
 
 640. The following tabular synopsis of the meaning of the 
 elements constituting the corresponding rectangular specifications of 
 a strain and stress explained in preceding sections, will be found 
 convenient : — 
 
 co^r 
 
 Strain. 
 
 )onents 
 the 
 stress. 
 
 € 
 f 
 
 P 
 
 Q 
 
 a 
 
 s 
 
 b 
 
 T 
 
 Planes; of which 
 
 relative motion, or 
 
 across which force 
 
 is reckoned. 
 
 Direction 
 of relative 
 motion or 
 of force. 
 
 yz 
 zx 
 xy 
 
 
 X 
 
 y 
 
 z 
 
 (yx 
 \zx 
 
 
 y 
 
 z 
 
 (zy 
 [xy 
 
 
 z 
 
 X 
 
 (xz 
 
 
 X 
 
 \yz 
 
 
 y 
 
 u 
 
 641. If a unit cube of matter under any stress {P^ Q, P, S, T, U) 
 experience the infinitely small simple longitudinal strain e alone, the 
 work done on it will be Fe; since, of the component forces, F, U, T 
 parallel to OX^ U and T do no work in virtue of this strain. Simi- 
 larly, (2/, Fg are the works done if, the same stress acting, the simple 
 longitudinal strains f ox g are experienced, either alone. Again, if 
 the cube experiences a simple shear, a, whether we regard it (§ 151) 
 as a differential sliding of the planes yx, parallel to y, or of the planes 
 zx, parallel to z, we see that the work done is Sa: and similarly, 
 Tb if the strain is simply a shear b, parallel to OZ, of planes zy, or 
 parallel to OX, of planes xy. and Uc if the strain is a shear c, parallel 
 to OX, of planes xz, or parallel to OY, of planes yz. Hence the 
 whole work done by the stress {F, Q, F, S, T, U') on a, unit cube 
 taking the strain (e, f, g, a, b, c), is 
 
 Fe+ Q/+ Fg+Sa + Tb + Uc. (3) 
 
 It is to be remarked that, inasmuch as the action called a stress is 
 a system of forces which balance one another if the portion of 
 matter experiencing it is rigid, it cannot do any work when the 
 
STATICS OF SOLIDS AND FLUIDS. 241 
 
 matter moves in any way without change of shape : and therefore no 
 amount of translation or rotation of the cube taking place along with 
 the strain can render the amount of work done different from that 
 just found. 
 
 If the side of the cube be of any length/, instead of unity, each 
 force will be /^ times, and each relative displacement/ times, and, 
 therefore, the work done p^ times the respective amounts reckoned 
 above. Hence a body of any shape, and of cubic content C, sub- 
 jected throughout to a uniform stress (P, Q, F, S, T, U) while taking 
 uniformly throughout a strain {e,f,g, a, b, c), experiences an amount 
 of work equal to 
 
 {Fe+ Q/+ Fg-hSa+n+ Uc)C. (4) 
 
 It is to be remarked that this is necessarily equal to the work done 
 on the bounding surface of the body by forces applied to it from 
 without. For the work done on any portion of matter within the 
 body is simply that done on its surface by the matter touching it all 
 round, as no force acts at a distance from without on the interior 
 substance. Hence if we imagine the whole body divided into any 
 number of parts, each of any shape, the sum of the work done on 
 all these parts is, by the disappearance of equal positive and negative 
 terms expressing the portions of the work done on each part by the 
 contiguous parts on all its sides, and spent by these other parts in 
 this action, reduced to the integral amount of work done by force 
 from without applied all round the outer surface. 
 
 642. If, now, we suppose the body to yield to a stress {P, Q, F^ 
 S, T, U), and to oppose this stress only with its innate resistance to 
 change of shape, the differential equation of work done will [by (4) 
 with de, dfy etc., substituted for ^,/, etc.] be 
 
 dw = Fde + Qdf+ Fdg + Sda + Tdb + Udc. (5) 
 
 If w denote the whole amount of work done per unit of volume in 
 any part of the body while the substance in this part experiences a 
 strain (f,/, g, a, b, c) from some initial state regarded as a state of 
 no strain. This equation, as we shall see later, under Properties of 
 Matter, expresses the work done in a natural fluid, by distorting 
 stress (or difference of pressure indifferent directions) working against 
 its innate viscosity; and w is then, according to Joule's discovery, 
 the dynamic value of the heat generated in the process. The equa- 
 tion may also be applied to express the work done in straining an 
 imperfectly elastic sohd, or an elastic solid of which the temperature 
 varies during the process. In all such applications the stress will 
 depend partly on the speed of the straining motion, or on the varying 
 temperature, and not at all, or not solely, on the state of strain at any 
 moment, and the system will not be dynamically conservative. 
 
 643. Definition. — A perfectly elastic body is a body which, when 
 brought to any one state of strain, requires at all times the same 
 stress to hold it in this state; however long it be kept strained, or 
 however rapidly its state be altered from any other strain, or from 
 no strain, to the strain in question. Here, according to our plan 
 
 T. 16 
 
242 ABSTRACT DYNAMICS. 
 
 (§§ 396) 4°^) ^or Abstract Dynamics, we ignore variation of tempera- 
 ture in the body. If, however, we add a condition of absolutely no 
 variation of temperature, or of recurrence to one specified temperature 
 after changes of strain, we have a definition of that property of perfect 
 elasticity towards which highly elastic bodies in nature approximate; 
 and which is rigorously fulfilled by all fluids, and may be so by some 
 real soHds, as homogeneous crystals. But inasmuch as the elastic 
 reaction of every kind of body against strain varies with varying 
 temperature, and (a thermodynamic consequence of this, as we shall 
 see later) any increase or diminution of strain in an elastic body is 
 necessarily* accompanied by a change of temperature; even a per- 
 fectly elastic body could not, in passing through different strains, 
 act as a rigorously conservative system, but on the contrary, must 
 give rise to dissipation of energy in consequence of the conduction 
 or radiation of heat induced by these changes of temperature. 
 
 But by making the changes of strain quickly enough to prevent 
 any sensible equilization of temperature by conduction or radiation 
 (as, for- instance, Stokes has shown, is done in sound of musical 
 notes travelling through air); or by making them slowly enough to 
 allow the temperature to be maintained sensibly constant' by proper 
 appliances; any highly elastic, or perfectly elastic body in nature may 
 be got to act very nearly as a conservative system. 
 
 644. In nature, therefore, the integral amount, w, of work defined 
 as above, is for a perfectly elastic body, independent (§ 246) of the 
 series of configurations, or states of strain, through which it may have 
 been brought from the first to the second of the specified conditions, 
 provided it has not been allowed to change sensibly in temperature 
 during the process. 
 
 When the whole amount of strain is infinitely small, and the stress- 
 components are therefore all altered in the same ratio as the strain- 
 components if these are altered all in any one ratio; w must be a 
 homogeneous quadratic function of the six variables e, f, g, a, b, «f, 
 which, if we denote by {e, e), {/,/).,. (e,/). . . constants depending 
 on the quality of the substance and on the directions chosen for the 
 axes of co-ordinates, we may write as follows : — 
 
 W = U{e, e)^ + {/J)f ■^{g,g) g' + {a,a)a' + (3,b) d'+ (Cc)^ 
 
 + 2 (e,/) e/+ 2 {e,g) eg +2 {e,a)ea+2{e,b)eb+2 (e^ c)ec 
 
 + 2 {f,g)fg^ 2 {f,a)fa + 2 {/,b)/b + 2 (/ c)/c 
 
 + 2(g,a)ga + 2 (g, b)gb + 2 (g, c)gc 
 
 + 2 [a, b)ab + 2 (a^c) ac 
 
 + 2 {b^c)bc\ 
 
 The 21 co-efficients (^, ^), (/,/)... (<^, <r), in this expression con- 
 stitute the 21 'co-efficients of elasticity,' which Green first showed to 
 be proper and essential for a complete theory of the dynamics of an 
 elastic sohd subjected to infinitely small strains. The only condition 
 
 ^ *0n the Thermoelastic and Therm omagnetic Properties of Matter' (W. 
 Thomson). Quarterly Journal of Mathematics, K'^xW i^ll, ^ Ibid, 
 
STATICS OF SOLIDS AND FLUIDS. 243 
 
 that can be theoretically imposed upon these co-efficients is that they 
 must not permit w to become negative for any values, positive or 
 negative, of the strain-components e, /,. . . c. Under Properties of 
 Matter, we shall see that a false theory (Boscovich's), falsely worked 
 out by mathematicians, has led to relations among the co-efficients of 
 elasticity which experiment has proved to be false. 
 
 645. The average stress, due to elasticity of the solid, when 
 strained from its natural condition to that of strain (e, /, g, «, b, c) is 
 (as from the assumed applicability of the principle of superposition 
 we see it must be) just half the stress required to keep it in this 
 state of strain. 
 
 646. A body is called homogeneous when any two equal, similar 
 parts of it, with corresponding lines parallel and turned towards 
 the same parts, are undistinguishable from one another by any 
 difference in quality. The perfect fulfilment of this condition with- 
 out any limit as to the smallness of the parts, though conceivable, 
 is not generally regarded as probable for any of the real solids or 
 fluids known to us, however seemingly homogeneous. It is, we 
 believe, held by all naturalists that there is a molecular structure^ 
 according to which, in compound bodies such as water, ice, rock- 
 crystal, etc., the constituent substances lie side by side, or arranged 
 in groups of finite dimensions, and even in bodies called si77iple 
 (i.e. not known to be chemically resolvable into other substances) 
 there is no ultimate homogeneousness. In other words, the prevail- 
 ing belief is that every kind of matter with which we are acquainted 
 has a more or less coarse-grained texture, whether having visible 
 molecules, as great masses of solid stone or brick-building, or natural 
 granite or sandstone rocks; or, molecules too small to be visible 
 or directly measurable by us (but not infinitely small) ^ in seemingly 
 homogeneous metals, or continuous crystals, or liquids, or gases. 
 We must of course return to this subject under Properties of Matter ; 
 and in the meantime need only say that the definition of homogeneous- 
 ness may be applied practically on a very large scale to masses of 
 building or coarse-grained conglomerate rock, or on a more moderate 
 scale to blocks of common sandstone, or on a very small scale to 
 seemingly homogeneous metals^; or on a scale of extreme, undis- 
 covered fineness, to vitreous bodies, continuous crystals, sohdified 
 gums, as India rubber, gum-arabic, etc., and fluids. 
 
 647. The substance of a homogeneous solid i^ called isotropic 
 when a spherical portion of it, tested by any physical agency, exhibits 
 no difference in quality however it is turned. Or, which amounts 
 to the same, a cubical portion cut from any position in an isotropic 
 body exhibits the same qualities relatively to each pair of parallel 
 faces. Or two equal and similar portions cut from any positions 
 
 * Probably not undiscoverably smz\\, although of dimensions not yet known to us. 
 ' Which, however, we know, as recently proved by Deville and Van Troost, are 
 porous enough at high temperature to allow very free percolation of gases. 
 
 16 — 2 
 
244 ABSTRACT DYNAMICS. 
 
 in the body, not subject to the condition of parallelism (§ 646), 
 are undistinguishable from one another. A substance which is not 
 isotropic, but exhibits differences of quality in different directions, 
 is called aeolotropic. 
 
 648. An individual body, or the substance of a homogeneous 
 solid, may be isotropic in one quality or class of qualities, but 
 aeolotropic in others. 
 
 Thus in abstract dynamics a rigid body, or a group of bodies 
 rigidly connected, contained within and rigidly attached to a rigid 
 spherical surface, is kinetically symmetrical {§ 239) if its centre of 
 inertia is at the centre of the sphere, and if its moments of inertia 
 are equal round all diameters. It is also isotropic relatively to gravi- 
 tation if it is centrobaric (§ 542), so that the centre of figure is not 
 merely a centre of inertia, but a true centre of gravity. Or a trans- 
 parent substance may transmit light at different velocities in different 
 directions through it (that is, be doubly refracting), and yet a cube 
 of it may (and generally does in natural crystals) absorb the same 
 part of a beam of white hght transmitted across it perpendicularly 
 to any of its three pairs of faces. Or (as a crystal which exhibits 
 dichroisin) it may be aeolotropic relatively to the latter, or to either, 
 optic quality, and yet it may conduct heat equally in all directions. 
 
 649. The remarks of § 646 relative to homogeneousness in the 
 aggregate, and the supposed ultimately heterogeneous texture of all 
 substances however seemingly homogeneous, indicate corresponding 
 limitations and non-rigorous practical interpretations of isotropy. 
 
 650. To be elastically isotropic, we see first that a spherical or 
 cubical portion of any solid, if subjected to uniform normal pressure 
 (positive or negative) all round, must, in yielding, experience no 
 deformation : and therefore must be equally compressed (or dilated) 
 in all directions. But, further, a cube cut from any position in it, 
 and acted on by tange7itial or distorting stress (§ 633) in planes 
 parallel to two pairs of its sides, must experience simple deformation, 
 or shear (§ 150), in the same direction, unaccompanied by condensa- 
 tion or dilatation', and the same in amount for all the three ways 
 in which a stress may be thus applied to any one cube, and for 
 different cubes taken from any different positions in the solid. 
 
 651. Hence the elastic quality of a perfectly elastic, homogeneous, 
 isotropic solid is fully defined by two elements; — its resistance to 
 compression, and its resistance to distortion. The amount of uni- 
 form pressure in all directions, per unit area of its surface, required 
 to produce a stated very small compression, measures the first of 
 
 ^ It must be remembered that the changes of figure and volume we are con- 
 cerned with are so small that the principle of superposition is applicable; so that 
 if any distorting stress produced a condensation, an opposite distorting stress 
 would produce a dilatation, which is a violation of the isotropic condition. But it 
 is possible that a distorting stress may produce, in a truly isotropic solid, conden- 
 sation or dilatation in proportion to the square of its value : and it is probable that 
 such effects may be sensible in India x-ubber, or cork, or other bodies susceptible 
 of great deformations or compressions, with persistent elasticity. 
 
STATICS OF SOLIDS AND FLUIDS. 
 
 245 
 
 these, and the amount of the distorting stress required to produce 
 a stated amount of distortion measures the second. The numerical 
 measure of the first is the compressing pressure divided by the 
 diminution of the bulk of a portion of the substance which, when 
 uncompressed, occupies the unit volume. It is sometimes called 
 the elasticity of volume^ or the resistance to compression. Its reciprocal, 
 or the amount of compression on unit of volume divided by the 
 compressing pressure, or, as we may conveniently say, the compression 
 per unit of volume, per unit of compressing pressure, is commonly 
 called the compressibility. The second, or resistance to change of shape, 
 is measured by the tangential stress (reckoned as in § 62,2,) divided by 
 the amount of the distortion or shear (§ 154) which it produces, and 
 is called the rigidity of the substance, or its elasticity of figure. 
 
 652. From § 148 it follows that a strain compounded of a simple 
 extension in one set of parallels, and a simple contraction of equal 
 amount in any other set perpendicular to those, is the same as a 
 simple shear in either of the two sets of planes cutting the two 
 sets of parallels at 45°. And the numerical measure (§ 154) of this 
 shear, or simple distortion, is equal to dojible the amount of the 
 elongation or contraction (each measured, of course, per unit of 
 length). Similarly, we see (§ 639) that a longitudinal traction (or 
 negative pressure) parallel to one line, and an equal longitudinal 
 positive pressure parallel to any line at right angles to it, is equivalent 
 to a distorting stress of tangential tractions (§ 632) parallel to the 
 planes which cut those lines at 45°. And the numerical measure 
 of this distorting stress, being (§ 633) the amount' of the tangential 
 traction in either set of planes, is equal to the amount of the positive 
 or negative normal pressure, 7iot doubled. 
 
 653. Since then any stress whatever may be made up of simple 
 longitudinal stresses, it follows that, to find the relation between any 
 stress and the strain produced by 
 it, we have only to find the strain 
 produced by a single longitudinal 
 stress, which we may do at 
 once thus : — A simple longitudinal 
 stress, jP, is equivalent to a uni- 
 form dilating tension ^F in all 
 directions, compounded with two 
 distorting stresses, each equal to 
 ^F, and having a common axis 
 in the line' of the given longitu- 
 dinal stress, and their other two 
 axes any two lines at right angles 
 to one another and to it. The 
 diagram, drawn in a plane through 
 one of these latter lines, and the 
 former, sufficiently indicates the synthesis; the only forces not shown 
 being those perpendicular to its plane. 
 
246 ABSTRACT DYNAMICS. 
 
 Hence if n denote the rigidity, and k the reststa?ice to dilatation 
 [being the same as the reciprocal of the compressibility (§ 651)], the 
 effect will be an equal dilatation in all directions, amounting, per 
 unit of volume, to 
 
 1 P 
 
 compounded with two equal distortions, each amounting to 
 
 1 P 
 
 and having (§ 650) their axes in the directions just stated as those 
 of the distorting stresses. 
 
 654. The dilatation and two shears thus determined may be 
 conveniently reduced to simple longitudinal strains by still following 
 the indications of § 652, thus: — 
 
 The two shears together constitute an elongation amounting to 
 
 - — in the direction of the given force, jP, and equal contraction 
 
 amounting to 2— in all directions perpendicular to it. And the 
 
 \ p ^ 
 cubic dilatation 2_ implies a lineal dilatation, equal in all directions, 
 
 \ p 
 
 amounting to ~- . On the whole, therefore, we have 
 
 linear elongation = P T — + — j , in the direction of the applied 
 stress, and 
 
 linear contraction =/* (-7 a ) » ^^ ^ directions perpendicular 
 
 to the applied stress. 
 
 (3) 
 
 655. Hence when the ends of a column, bar, or wire, of isotropic 
 material, are acted on by equal and opposite forces, it experiences 
 
 a lateral lineal contraction, equal to ---t r o^ the longitudinal 
 
 dilatation, each reckoned as usual per unit of lineal measure. One 
 specimen of the fallacious mathematics above referred to (§ 644), is 
 a celebrated conclusion of Navier's and Poisson's that this ratio 
 is J, which requires the rigidity to be f of the resistance to com- 
 pression, for all solids : and which was first shown to be false by 
 Stokes' from many obvious observations, proving enormous discre- 
 pancies from it in many well-known bodies, and rendering it most 
 improbable that there is any approach to a constancy of ratio between 
 
 ^ ' On the Friction of Fluids in Motion, and the Equilibrium and Motion of 
 Elastic Solids.' Trans. Camb. Phil. Soc, April 1845. See also Camb. and 
 Dub. Math. Jour., March 1848. 
 
STATICS OF SOLIDS AND FLUIDS 247 
 
 rigidity and resistance to compression in any class of solids. Thus 
 clear elastic jellies, and India rubber, present familiar specimens of 
 isotropic homogeneous solids, which, while differing very much from 
 one another in rigidity ('stiffness'), are probably all of v6ry nearly 
 the same compressibiHty as water. This being -g-^gVcru P^'^ pound 
 per square inch; the resistance to compression, measured by its 
 reciprocal, or, as we may read it, '308000 lbs. per square inch,* 
 is obviously many hundred times the absolute amount of the 
 rigidity of the stiffest of those substances. A column of any of 
 them, therefore, when pressed together or pulled out, within its limits 
 of elasticity, by balancing forces applied to its ends (or an India 
 rubber band when pulled out), experiences no sensible change of 
 volume, though a very sensible change of length. Hence the pro- 
 portionate extension or contraction of any transverse diameter must 
 be sensibly equal to J the longitudinal contraction or extension : and 
 for all ordinary stresses, such substances may be practically regarded 
 as incompressible elastic solids. Stokes gave reasons for believing 
 that metals also have in general greater resistance to compression, in 
 proportion to their rigidities, than according to the fallacious theory, 
 although for them the discrepancy is very much less than for the 
 gelatinous bodies. This probable conclusion was soon experimentally 
 demonstrated by Wertheim, who found the ratio of lateral to longi- 
 tudinal change of lineal dimensions, in columns acted on solely by 
 longitudinal force, to be about ^ for glass or brass; and by Kirchhoff, 
 who, by a very well-devised experimental method, found '387 as the 
 value of that ratio for brass, and '294 for iron. For copper we find 
 that it probably lies between "226 and '441, by recent experiments^ 
 of our own, measuring the torsional and longitudinal rigidities (§§ 609, 
 657) of a copper wire. 
 
 656. All these results indicate rigidity less in proportion to the 
 compressibility than according to Navier's and Poisson's theory. 
 And it has been supposed by many naturalists, who have seen the 
 necessity of abandoning that theory as inapplicable to ordinary solids, 
 that it may be regarded as the proper theory for an ideal perfect solid, 
 and as indicating an amount of rigidity not quite reached in any 
 real substance, but approached to in some of the most rigid of 
 natural solids (as, for instance, iron). But it is scarcely possible 
 to hold a piece of cork in the hand without perceiving the fallacious- 
 ness of this last attempt to maintain a theory which never had any 
 good foundation. By careful measurements on columns of cork 
 of various forms (among them, cylindrical pieces cut in the ordinary 
 way for bottles) before and after compressing them longitudinally in a 
 Brahmah's press, we have found that the change of lateral dimensions 
 is insensible both with small longitudinal contractions and return 
 dilatations, within the limits of elasticity, and with such enormous 
 longitudinal contractions as to ^ or J of the original length. It is 
 thus proved decisively that cork is much more rigid, while metals, 
 
 1 'On the Elasticity and Viscosity of Metals' (W. Thomson), Froc, R.S., May 
 1865. 
 
248 ABSTRACT DYNAMICS. 
 
 glass, and gelatinous bodies are all less rigid, in proportion to 
 resistance to compression, than the supposed 'perfect solid'; and the 
 utter worthlessness of the theory is experimentally demonstrated. 
 
 657. The modulus of elasticity of a bar, wire, fibre, thin filament, 
 band, or cord of any material (of which the substance need not be 
 isotropic, nor even homogeneous within one normal section), as a 
 bar of glass or wood, a metal wire, a natural fibre, an India rubber 
 band, or a common thread, cord, or tape, is a term introduced 
 by Dr. Thomas Young to designate what we also sometimes call its 
 longitudinal rigidity: that is, the quotient obtained by dividing the 
 simple longitudinal force required to produce any infinitesimal 
 elongation or contraction by the amount of this elongation or con- 
 traction reckoned as always per unit of length. 
 
 658. Instead of reckoning the modulus in units of weight, it is 
 sometimes convenient to express it in terms of the weight of the unit 
 length of the rod, wire, or thread. The modulus thus reckoned, 
 or, as it is called by some writers, the length of the modulus, is 
 of course found by dividing the weight-modulus by the weight of 
 the unit length. It is useful in many applications of the theory of 
 elasticity; as, for instance, in this result, which will be proved 
 later: — the velocity of transmission of longitudinal vibrations (as of 
 sound) along a bar or cord, is equal to the velocity acquired by 
 a body in falling from a height equal to half the length of the 
 modulus*. 
 
 659. The specific modulus of elasticity of an isotropic substance^ or, 
 as it is most often called, simply the modulus of elasticity of the sub- 
 sta?ice, is the modulus of elasticity of a bar of it having some definitely 
 specified sectional area. If this be such that the weight of unit 
 length is unity, the modulus of the substance will be the same as the 
 length of the modulus of any bar of it; a system of reckoning which, 
 as we have seen, has some advantages in application. It is, how- 
 ever, more usual to choose a common unit of area as the sectional 
 area of the bar referred to in the definition. There must also be a 
 definite understanding as to the unit in terms of which the force is 
 measured, which may be either the absolute u?iit (§ i88): or the 
 gravitation unit for a specified locality; that is (§ 191), the weight in 
 that locality of the unit of mass. Experimenters hitherto have stated 
 their results in terms of the gravitation unit, each for his own locality; 
 the accuracy hitherto attained being scarcely in any cases sufficient to 
 
 ^ It is to be understood that the vibrations in question are so much spread out 
 through the length of the body, that inertia does not sensibly influence the trans- 
 verse contractions and dilatations which (unless the substance have in this respect 
 the peculiar character presented by cork, § 656) take place along with them. Also, 
 under Thermodynamics, we shall see that changes of temperature produced by the 
 varying strains cause changes of stress which, in ordinary solids, render the velocity 
 of transmission of longitudinal vibrations sensibly greater than that calculated by 
 the rule stated in the text, if we use the static viodtdus as understood from the 
 definition there given; and we shall learn to take into account the thermal effect 
 ])yusing a definite static modulus, ox kinetic modulus, according to the circumstances 
 of any case that may occur. ' 
 
STATICS OF SOLIDS AND FLUIDS, 249 
 
 require corrections for the different forces of gravity in the different 
 places of observation. 
 
 660. The most useful and generally convenient specification of 
 the modulus of elasticity of a substance is in grammes-weight per 
 square centimetre. This has only to be divided by the specific 
 gravity of the substance to give the length of the modulus. British 
 measures, however, being still unhappily sometimes used in practical 
 and even in high scientific statements, we may have occasion to refer 
 to reckonings of the modulus in pounds per square inch or per square 
 foot, or to length of the modulus in feet. 
 
 661. The reckoning most commonly adopted in British treatises 
 on mechanics and practical statements is pounds per square inch. 
 The modulus thus stated must be divided by the weight of 1 2 cubic 
 inches of the soHd, or by the product of its specific gravity into '4337 \ 
 to find the length of the modulus, in feet. 
 
 To reduce from pounds per square inch to grammes per square 
 centimetre, multiply by 70-3 1, or divide by -014223. French engineers 
 generally state their results in kilogrammes per square millimetre, 
 and so bring them to more convenient numbers, being x^j-oVuo- ^^ ^^ 
 inconveniently large numbers expressing moduli in grammes-weight 
 per square centimetre. 
 
 662. The same statements as to units, reducing factors, and nominal 
 designations, are applicable to the resistance to compression of any 
 elastic solid or fluid, and to the rigidity (§ 651) of an isotropic body; 
 or, in general, to any one of the 2 1 co-efficients in the expressions 
 for terms in stresses of strains, or to the reciprocal of any one of 
 the 21 co-efficients in the expressions for strains in terms of stresses, 
 as well as to the modulus defined by Young. 
 
 663. In §§ 652, 653 we examined the effect of a simple longitudinal 
 stress, in producing elongation in its own direction, and contraction 
 
 ^ This decimal being the weight in pounds of 12 cubic inches of water. The 
 one great advantage of the French metrical system is, that the mass of the unit 
 volume (i cubic centimetre) of water at its temperature of maximum density 
 (3° '945 C-) is unity (i gramme) to a sufficient degree of approximation for almost 
 all practical purposes. Thus, according to this system, tlie density of a body and 
 its specific gravity mean one and the same thing ; whereas on the British no-system 
 the density is expressed by a number found by multiplying the specific gravity by 
 one number or another, according to the choice of a cubic inch, cubic foot, cubic 
 yard, or cubic mile that is made for the unit of volume; and the grain, scruple, 
 gunmaker's drachm, apothecary's drachm, ounce Troy, ounce avoirdupois, pound 
 Troy, pound avoirdupois, stone (Imperial, Ayrshire, Lanarkshire, Dumbarton- 
 shire), stone for hay, stone for corn, quarter (of a hundredweight), quarter (of 
 corn), hundredweight, or ton, that is chosen for unit of mass. It is a remarkable 
 phenomenon, belonging rather to moral and social than to physical science, that 
 a people tending naturally to be regulated by common sense should voluntarily 
 condemn themselves, as the British have so long done, to unnecessary hard labour 
 in every action of common business or scientific work related to measurement, from 
 which all the other nations of Europe have emancipated themselves. We have 
 been informed, through the kindness of Professor W. H. Miller, of Cambridge, 
 that he concludes, from a very tmstworthy comparison of standards by Kupflfer, of 
 St. Petersburgh, that the weight of a cubic decimetre of water at temperature of - 
 maximum density is 1000*013 grammes. 
 
250 
 
 ABSTRACT DYNAMICS. 
 
 in lines perpendicular to it. With stresses substituted for strains, and 
 strains for stresses, we may apply the same process to investigate the 
 longitudinal and lateral tractions required to produce a simple longi- 
 tudinal strain (that is, an elongation in one direction, with no change 
 of dimensions perpendicular to it) in a rod or solid of any shape. 
 
 Thus a simple longitudinal strain e is equivalent to a cubic dilata- 
 tion e without change of figure (or linear dilatation \e equal in all 
 directions), and two distortions consisting each of dilatation \e in the 
 given direction, and contraction \e in each of two directions perpen- 
 dicular to it and to one another. To produce the cubic dilatation, <?, 
 alone requires (§ 651) a normal traction ke equal in all directions. 
 And, to produce either of the distortions simply, since the measure 
 (§ 154) of each is \e^ requires a distorting stress equal to « x §<?, which 
 consists of tangential tractions each equal to this amount, positive (or 
 drawing outwards) in the line of the given elongation, and negative (or 
 pressing inwards) in the perpendicular direction. Thus we have in all 
 
 normal traction = (>^ + 1^«) ^, in the direction of the given*] 
 strain, and I 
 
 normal traction = (^-f/2) <f, in every direction perpen-| 
 dicular to the given strain. J 
 
 (4) 
 
 664. If now we suppose any possible infinitely small strain (^,/, g, 
 a, d, c), according to the specification of § 640, to be given to a body, 
 the stress {!*, Qj R, S, T, U) required to maintain it will be expressed 
 by the following formulae, obtained by successive applications of 
 § ddTy (4) to the components e^ f, g separately, and of § 651 to 
 a, b, c— 
 
 S= na, T== nb, U= nc^ 
 
 F=^%e + ^{f+g\ 
 
 Q = ni/+^(g + e), 
 
 R = Ug+^(e+f), 
 
 where 
 
 (5) 
 
 665. Similarly, by § 651 and § 654 (3), we have 
 a=^-S,b=-T,c = - C/,' 
 
 Mg={R-^{F+Q)}, 
 gnk 
 
 where 
 and 
 
 M= 
 
 
 (6) 
 
STATICS OF SOLIDS AND FLUIDS. 251 
 
 as the formulae expressing the strain (^, /, g, a, b, c) in terms of 
 the stress (F, (2, F, S, T, U). They are of course merely the 
 algebraic inversions of (5) ; and they might have been found by 
 solving these for e, / g^ a, b, c, regarded as the unknown quantities. 
 M is here introduced to denote Young's modulus. 
 
 666. To express the equation of energy for an isotropic substance, 
 we may take the general formula, 
 
 'W = l{Fe-¥ Q/+Fg+Sa + Tb + L/c), 
 and eliminate from it F, Q, etc., by (5) of § 664, or, again, e, /, etc., 
 by (6) of § 665, we thus find 
 
 667. The mathematical theory of the equilibrium of an elastic 
 solid presents the following general problems : 
 
 A solid of any given shape, when undisturbed, is acted on in its 
 substance by force distributed through it in any given manner, and dis- 
 placements are arbitrarily produced, or forces arbitrarily applied, over 
 its bounding surface. It is required to find the displacement of every 
 poi?tt of its substance. 
 
 This problem has been thoroughly solved for a shell of homo- 
 geneous isotropic substance bounded by surfaces which, when undis- 
 turbed, are spherical and concentric ; but not hitherto for a body 
 of any other shape. The limitations under which solutions have 
 been obtained for other cases (thin plates and rods), leading, as we 
 have seen, to important practical results, have been stated above 
 (§ 605). To demonstrate the laws (§ 607) which were taken in 
 anticipation will also be one of our applications of the general 
 equations for interior equilibrium of an elastic solid, which we now 
 proceed to investigate. 
 
 668. Any portion in the interior of an elastic solid may be 
 regarded as becoming perfectly rigid (§ 584) without disturbing the 
 equilibrium either of itself or of the matter round it. Hence the traction 
 exerted by the matter all round it, regarded as a distribution of force 
 applied to its surface, must, with the applied forces acting on the sub- 
 stance of the portion considered, fulfil the conditions of equilibrium of 
 forces acting on a rigid body. This statement, applied to an infinitely 
 small rectangular parallelepiped of the body, gives the general differ- 
 ential equations of internal equilibrium of an elastic solid. It is to be 
 remarked that three equations suffice ; the conditions of equilibrium 
 for the couples being secured by the relation estabHshed above (§632) 
 among the six pairs of tangential component tractions on the six 
 faces of the figure. 
 
 669. One of the most beautiful applications of the general equa- 
 tions of internal equilibrium of an elastic solid hitherto made is 
 
252 ABSTRACT DYNAMICS. 
 
 that of M. de St. Venant to 'the torsion of prisms ^' To one 
 end of a long straight prismatic rod, wire, or solid or hollow cylinder 
 of any form, a given couple is applied in a plane perpendicular to 
 the length, while the other end is held fast : it is required to find 
 the degree of twist produced, and the distribution of strain and 
 stress throughout the prism. The conditions to be satisfied here 
 are that the resultant action between the substance on the two sides 
 of any normal section is a couple in the normal plane, equal to the 
 given couple. Our work for solving the problem will be much 
 simplified by first establishing the following preliminary propo- 
 sitions : — 
 
 670. Let a solid (whether aeolotropic or isotropic) be so acted 
 on by force applied from without to its boundary, that throughout its 
 interior there is no normal traction on any plane parallel or per- 
 pendicular to a given plane, XOY^ which implies, of course, that 
 there is no distorting stress with axes in or parallel to this plane, and 
 that the whole stress at any point of the solid is a simple distorting 
 stress of tangential forces in some direction in the plane parallel to 
 XO Vy and in the plane perpendicular to this direction. Then — 
 
 (i) The interior distorting stress must be equal, and similarly 
 directed, in all parts of the solid lying in any line perpendicular 
 to the plane XO Y. 
 
 (2) It being premised that the traction at every point of any 
 surface perpendicular to the plane XOY'is, by hypothesis, a distribu- 
 tion of force in lines perpendicular to this plane ; the integral amount 
 of it on any closed prismatic or cylindrical surface perpendicular to 
 XO Y, and bounded by planes parallel to it, is zero. 
 
 (3) The matter within the prismatic surface and terminal planes of 
 (2) being supposed for a moment (§ 584) to be rigid, the distribution 
 
 of tractions referred to in 
 (2) constitutes a couple 
 whose moment, divided by 
 the distance between those 
 terminal planes, is equal to 
 the resultant force of the 
 tractions on the area of 
 either, and whose plane is 
 parallel to the lines of these 
 resultant forces. In other 
 
 ^ words, the moment of the 
 
 O ~~ X" distribution of forces over 
 
 the prismatic surface referred to in (2) round any line (OY or OX) in 
 the plane XOY, is equal to the sum of the components {T or *S), 
 perpendicular to the same line, of the traction in either of the 
 terminal planes multiplied by the distance between these planes. 
 
 ^ Memoires des Savants ^Irangas, 1855. ' De 1^ Torsion des Prismes, avec des 
 Considerations sur leur Flexion,' etc. 
 
STATICS OF SOLIDS AND FLUIDS 253 
 
 To prove (i) consider for a moment as rigid (§ 584) an infinite- 
 simal prism, AB (of sectional area to), per- ^ 
 
 pendicular to XOY, and having plane ends, f"^ ^-lu) 
 
 A^ B, parallel to it. There being no forces 
 on its sides (or cylindrical boundary) per- 
 pendicular to its length, its equilibrium so far 
 as motion in the direction of any line {OX), 
 perpendicular to its length, requires that the 
 components of the tractions on its ends be 
 equal and in opposite directions. Hence, 
 in the notation § 633, the distorting - stress 
 components, T, must be equal at A and B -, Yu--^- 
 
 and so must the stress components S, for the B 
 
 same reason. 
 
 To prove (2) and (3) we have only to remark that they are 
 required for the equilibrium of the rigid prism referred to 
 in (3). 
 
 671. For a soUd or hollow circular cylinder, the solution of § 669 
 (given first, we believe, by Coulomb) obviously is that each circular 
 normal section remains unchanged in its own dimensions, figure, 
 and internal arrangement (so that every straight line of its particles 
 remains a straight line of unchanged length), but is turned round 
 the axis of the cylinder through such an angle as to give a uniform 
 rafe of twist equal to the applied couple divided by the product 
 of the moment of inertia of the circular area (whether annular or 
 complete to the centre) into the rigidity of the substance. 
 
 672. Similarly, we see that if a cylinder or prism of any shape 
 be compelled to take exactly the state of strain above specified (§671) 
 with the line through the centres of inertia of the normal sections, 
 taken instead of the axis of the cylinder, the mutual action between 
 the parts of it on the two sides of any normal section will be a couple 
 of which the moment will be expressed by the same formula, that is, 
 the product of the rigidity, into the rate of twist, into the moment 
 of inertia of the section round its centre of inertia. 
 
 673. But for any other shape of prism than a solid or symmetrical 
 hollow circular cylinder, the supposed state of strain will require, 
 besides the terminal opposed couples, force parallel to the length 
 of the prism, distributed over the prismatic boundary, in proportion 
 to the distance along the tangent, from each point of the surface, 
 to the point in which this Hne is cut by a perpendicular to it from the 
 centre of inertia of the normal section. To prove this let a normal 
 section of the prism be represented in the annexed diagram (page 254). 
 Let PK representing the shear at any point, P, close to the prismatic 
 boundary, be resolved into PN and FT respectively along the nor- 
 mal and tangent. The whole shear, PK, being equal to rr, its 
 component, /'iV^, is equal to rr sin w or t./^^. The corresponding 
 component of the required stress is nr.PE, and involves (§ 632) 
 equal forces in the plane of the diagram, and in the plane through 
 
254 ABSTRACT DYNAMICS. 
 
 TP perpendicular to it, each amounting to nr.PE per unit of 
 area. 
 
 An application of force 
 equal and opposite to the 
 distribution thus found 
 over the prismatic boun- 
 dary, would of course alone 
 r produce in the prism, other- 
 wise free, a state of strain 
 which, compounded with 
 that supposed above,would 
 give the state of strain ac- 
 tually produced by the sole 
 application of balancing 
 couples to the two ends. 
 The result, it is easily seen (and it will be proved below), consists of 
 an increased twist, together with a warping of naturally plane normal 
 sections, by infinitesimal displacements perpendicular to themselves, 
 into certain surfaces of anticlastic curvature, with equal opposite 
 curvatures in the principal sections (§ 122) through every point. 
 This theory is due to St. Venant, who not only pointed out the falsity 
 of the supposition admitted by several previous writers, that Cou- 
 lomb's law holds for other forms of prism than the solid or hollow 
 circular cylinder, but discovered fully the nature of the requisite 
 correction, reduced the determination of it to a problem of pure 
 mathematics, worked out the solution for a great variety of important 
 and curious cases, compared the results with observation in a manner 
 satisfactory and interesting to the naturalist, and gave conclusions 
 of great value to the practical engineer. 
 
 674. We take advantage of the identity of mathematical conditions 
 in St. Venant's torsion problem, and a hydrokinetic problem first 
 solved a few years earlier by Stokes \ to give the following statement, 
 which will be found very useful in estimating deficiencies in torsional 
 rigidity below the amount calculated from the fallacious extension 
 of Coulomb's law : — 
 
 675. Conceive a liquid of density n completely filling a closed 
 infinitely light prismatic box of the same shape within as the given 
 elastic prism and of length unity, and let a couple be applied to the 
 box in a plane perpendicular to its length. The effective moment 
 of inertia of the liquid^ will be equal to the correction by which the 
 torsional rigidity of the elastic prism calculated by the false extension 
 of Coulomb's law, must be diminished to give the true torsional 
 rigidity. 
 
 Further, the actual shear of the solid, in any infinitely thin plate of 
 
 * 'On some cases of Fluid Motion,' Cambridge Philosophical Transactions , 1843. 
 
 ' That is the moment of inertia of a rigid solid which, as will be proved in 
 Vol. II., may be fixed within the box, if the liquid be removed, to make its 
 motions the same as they are with the liquid in it. 
 
STATICS OF SOLIDS AND FLUIDS. 
 
 255 
 
 it between two normal sections, will at each point be, when reckoned 
 as a differential sliding (§ 151) parallel to their planes, equal to and 
 in the same direction as the velocity of the liquid relatively to the 
 containing box. 
 
 676. St. Venant's treatise abounds in beautiful and instructive 
 graphical illustrations of his results, from which we select the 
 following : — 
 
 (i) Elliptic cylinder. The plain and dotted curvilineal arcs are 
 * contour lines ' {coupes topographiques) of the section as warped by 
 
 torsion ; that is to say, lines in which it is cut by a series of parallel 
 planes, each perpendicular to the axis. These hues are equilateral 
 hyperbolas in this case. The arrows indicate the direction of rotation 
 in the part of the prism above the plane of the diagram. 
 
 (2) Equilateral triangular prism, — The contour lines are shown 
 
 as in case (i); the dotted curves being those where the warped 
 section falls below the plane of the diagram, the direction of rotation 
 
256 
 
 ABSTRACT DYNAMICS. 
 
 of the part of the prism above the plane being indicated by the 
 bent arrow. 
 
 (3) This diagram shows a series of lines given by St. Venant, 
 and more or less resembling squares, their common equation 
 containing only one constant a. It is remarkable that the values 
 
 fl! = o-5 and a = -\{j2-\) give similar but not equal curvi- 
 lineal squares (hollow sides and acute angles), one of them turned 
 through half a right angle relatively to the other. Everything in the 
 diagram outside the larger of these squares is to be cut away as 
 irrelevant to the physical problem; the series of closed curves 
 remaining exhibits figures of prisms, for any one of which the torsion 
 problem is solved algebraically. These figures vary continuously from 
 a circle, inwards to one of the acute-angled squares, and outwards to 
 the other : each, except these extremes, being a continuous closed 
 curve with no angles. The curves for a = 0-4 and a = -0-2 approach 
 remarkably near to the rectilineal squares, partially indicated in the 
 diagram by dotted lines. 
 
 (4) This diagram shows the contour lines, in all respects as in the 
 cases (i) and (2) for the case of a prism having for section the figure 
 indicated. The portions of curve outside the continuous closed curve 
 are merely indications of mathematical extensions irrelevant to the 
 physical problem. 
 
STATICS OF SOLIDS AND FLUIDS. 
 Y 
 
 257 
 
 (5) This shows, as in the other cases, the contour lines for the 
 warped section of a square prism under torsion. 
 
 r 
 
 T. 
 
 17 
 
258 
 
 ABSTRACT DYNAMICS. 
 
 (6), (7), (8). These are shaded drawings, showing the appear- 
 ances presented by elliptic, square, and flat rectangular bars under 
 
 exaggerated torsion, as may be realized with such a substance as 
 India rubber. 
 
 677. Inasmuch as the moment of inertia of a plane area about 
 an axis through its centre of inertia perpendicular to its plane is 
 obviously equal to the sum of its moments of inertia round any two 
 axes through the same point, at right angles to one another in its 
 plane, the fallacious extension of Coulomb's law, referred to in § 673, 
 
 71 
 
 would make the torsional rigidity of a bar of any section equal to -j-l. 
 
 (§ 665) multiplied into the sum of its flexural rigidities (see below, 
 § 679) in any two planes at right angles to one another through 
 its length. The true theory, as we have seen (§ 675), always 
 gives a torsional rigidity less than this. How great the deficiency 
 may be expected to be in cases in which the figure of the section 
 presents projecting angles, or considerable prominences (which may 
 be imagined from the hydrokinetic analogy we have given in § 675), 
 has been pointed out by M. de St. Venant, with the important 
 practical application, that strengthening ribs, or projections (see, for 
 instance, the fourth annexed diagram), such as are introduced in 
 engineering to give stiffness to beams, have the reverse of a good 
 effect when torsional rigidity or strength is an object, although they 
 are truly of great value in increasing the flexural rigidity, and giving 
 
STATICS OF SOLIDS. AND FLUIDS. 
 
 259 
 
 strength to bear ordinary strains, which are always more or less 
 flexural. With remarkable ingenuity and mathematical skill he has 
 drawn beautiful illustrations of this important practical principle from 
 his algebraic and transcendental solutions. Thus for an equilateral 
 
 
 (2) 
 
 (3) 
 
 (4) 
 
 
 (l) 
 
 Square with curved 
 
 Square with acute 
 
 Star with four 
 
 (5) 
 
 Rectilineal 
 
 corners and hollow 
 
 angles and hollow 
 
 rounded points, 
 
 Equilateral 
 
 square. 
 
 sides. 
 
 sides. 
 
 being a curve of 
 the eighth degree. 
 
 triangle. 
 
 •84346. 
 •88326. 
 
 •8186. 
 •8666. 
 
 •7783. 
 •8276. 
 
 •5874. 
 •6745. 
 
 •60000. 
 •72552. 
 
 triangle, and for the rectilineal and three curvilineal squares shown 
 in the annexed diagram, he finds for the torsional rigidities the 
 values stated. The number immediately below the diagram indicates 
 in each case the fraction which the true torsional rigidity is of the 
 old fallacious estimate (§ 673); the latter being the product of the 
 rigidity of the substance into the moment of inertia of the cross 
 section round an axis perpendicular to its plane through its centre 
 of inertia. The second number indicates in each case the fraction 
 which the torsional rigidity is of that of a solid circular cylinder 
 of the same sectional area. 
 
 678. M. de St. Venant also calls attention to a conclusion from 
 his solutions which to many may be startling, that in his simpler cases 
 the places of greatest distortion are those points of the boundary 
 which are nearest to the axis of the twisted prism in each case, and 
 the places of least distortion those farthest from it. Thus in the 
 elliptic cylinder the substance is most strained at the ends of the smaller 
 principal diameter, and least at the ends of the greater. In the 
 equilateral triangular and square prisms there are longitudinal lines of 
 maximum strain through the middles of the sides. In the oblong 
 rectangular prism there are two lines of greater maximum strain 
 through the middles of the broader pair of sides, and two lines of less 
 maximum strain through the middles of the narrow sides. The 
 strain is, as we may judge from (§ 675) the hydrokinetic analogy, 
 excessively small, but not evanescent, in the projecting ribs of a prism 
 of the figure shown in (4) § 677. It is quite evanescent infinitely near 
 the angle, in the triangular and rectangular prisms, and in each other 
 case as (3) of § 677, in which there is a finite angle, whether acute 
 or obtuse, projecting outwards. This reminds us of a general 
 remark we have to make, although consideration of space may 
 
 17- 
 
26o ABSTRACT DYNAMICS. 
 
 oblige us to leave it without formal proof. A solid of any elastic 
 substance, isotropic or aeolotropic, bounded by any surfaces pre- 
 senting projecting edges or angles, or re-entrant angles or edges, 
 however obtuse, cannot experience any finite stress or strain in the 
 neighbourhood oi 2. projecting 2iVi<^Q (trihedral, polyhedral, or conical); 
 in the neighbourhood of an edge, can only experience simple longi- 
 tudinal stress parallel to the neighbouring part of the edge; and 
 generally experiences infinite stress and strain in the neighbourhood 
 of a re-entrant edge or angle; when influenced by any distribution 
 of force, exclusive of surface tractions infinitely near the angles or 
 edges in question. An important application of the last part of 
 this statement is the practical rule, well known in mechanics, that 
 every re-entering edge or angle ought to be rounded to prevent risk 
 of rupture, in solid pieces designed to bear stress. An illustration 
 of these principles is afforded by the complete mathematical solution 
 of the torsion problem for prisms of fan-shaped sections, such as 
 the annexed figures. In the cases corresponding to figures (4), (5), 
 (6) below, the distortion at the centre of the circle vanishes in (4), 
 is finite and determinate in (5), and infinite in (6). 
 
 (i) (2) (3) (4) (5) (6) 
 
 679. Hence in a rod of isotropic substance the principal axes 
 of flexure (§ 609) coincide with the principal axes of inertia of the 
 area of the normal section ; and the corresponding flexural rigidities 
 are the moments of inertia of this area round these axes multi- 
 plied by Young's modulus. Analytical investigation leads to the 
 following results, due to St. Venant. Imagine the whole rod di- 
 vided, parallel to its length, into infinitesimal filaments (prisms when 
 the rod is straight). Each of these contracts or swells laterally with 
 sensibly the same freedom as if it were separated from the rest of 
 the substance, and becomes elongated or shortened in a straight line 
 to the same extent as it is really elongated or shortened in the circular 
 arc which it becomes in the bent rod. The distortion of the cross 
 section by which these changes of lateral dimensions are necessarily 
 accompanied is illustrated in the annexed diagram, in which either the 
 whole normal section of a rectangular beam, or a rectangular area in 
 the normal section of a beam of any figure, is represented in its strained 
 and unstrained figures, with the central point O common to the two. 
 The flexure is in planes perpendicular to YO Y^ , and concave upwards 
 (or towards X); G the centre of curvature, being in the direction 
 indicated, but too far to be included in the diagram. The straiglit 
 
STATICS OF SOLIDS AND FLUIDS. 
 
 261 
 
 sides AC, BD, and all straight lines parallel to them, of the unstrained 
 rectanojular area become concentric arcs of circles concave in the 
 
 16^ 
 
 A 
 
 
 
 F 
 
 
 c 
 
 T — J-— 
 
 
 F' 
 
 3-J T^ 
 
 V 
 
 \\\\i 
 
 
 
 ZAAit 
 
 7 
 
 1- \ 
 
 \ — — V — ^ — 
 
 
 f\ 
 
 
 / T 
 
 ^ \ 
 
 — \ \ \ 
 
 
 U 
 
 jjTn 
 
 / ^ 
 
 
 mtr 
 
 
 
 
 B 
 
 
 E^ 
 
 ~itztii 
 
 D 
 
 t 
 
 y- 
 
 
 B' 
 
 ^^^'d 
 
 H 
 
 opposite direction, their centre of curvature, H, being for rods of 
 gelatinous substance, or of glass or metal, from 2 to 4 times as far 
 from O on one side as G is on the other. Thus the originally plane 
 sides AC, BD of a rectangular bar become anticlastic surfaces, of 
 
 curvatures - and — , in the two principal sections. A flat rectangular, 
 
 9 P 
 
 or a square, rod of India-rubber [for which a- amounts (§ 655) to very 
 nearly ^, and which is susceptible of very great amounts of strain 
 without utter loss of corresponding elastic action], exhibits this 
 phenomenon remarkably well, 
 
 680. The conditional limitation (§605) of the curvature to being very 
 small in comparison with that of a circle of radius equal to the greatest 
 diameter of the normal section (not obviously necessary, and indeed 
 not generally known to be necessary, we believe, when the greatest 
 diameter is perpendicular to the plane of curvature) now receives its 
 full explanation. For unless the hread^/i, AC, of the bar (or diameter 
 perpendicular to the plane of flexure) be very small in comparison 
 with the mean proportional between the radius, OH, and the thick- 
 ness, AB, the distances from OV to the corners A', C" would fall 
 short of the half thickness, OF, and the distances to B', D' would 
 exceed it by diflerences comparable with its own amount. This 
 would give rise to sensibly less and greater shortenings and stretchings 
 
262 ABSTRACT DYNAMICS. 
 
 in the filaments towards the corners, and so vitiate the solution. 
 Unhappily mathematicians have not hitherto succeeded in solving, 
 possibly not even tried to solve, the beautiful problem thus presented 
 by the flexure of a broad very thin band (such as a watch spring) into 
 a circle of radius comparable with a third proportional to its thickness 
 and its breadth. 
 
 681. But, provided the radius of curvature of the flexure is not 
 only a large multiple of the greatest diameter, but also of a third 
 proportional to the diameters in and perpendicular to the plane of 
 flexure; then however great may be the ratio of the greatest diameter 
 to the least, the preceding solution is applicable; and it is remarkable 
 that the necessary distortion of the normal section (illustrated in the 
 diagram of § 679) does not sensibly impede the free lateral con- 
 tractions and expansions in the filaments, even in the case of a broad 
 thin lamina (whether of precisely rectangular section, or of unequal 
 thicknesses in different parts). 
 
 682. In our sections on hydrostatics, the problem of finding the 
 deformation produced in a spheroid of incompressible liquid by a 
 given disturbing force will be solved ; and then we shall consider the 
 application of the preceding methods to an elastic solid sphere in their 
 bearing on the theory of the tides and the rigidity of the earth. This 
 proposed application, however, reminds us of a general remark of 
 great practical importance, with which we shall leave elastic solids for 
 the present. Considering different elastic solids of similar substance 
 and similar shapes, we see that if by forces applied to them in any 
 way they are similarly strained, the surface tractions in or across 
 similarly situated elements of surface, whether of their boundaries 
 or of surfaces imagined as cutting through their substances, must be 
 equal, reckoned as usual per unit of area. Hence; the force across, 
 or in, any such surface, being resolved into components parallel to 
 any directions; the whole amounts of each such component for 
 similar surfaces of the different bodies are in proportion to the squares 
 of their lineal dimensions. Hence, if equilibrated similarly under the 
 action of gravity, or of their kinetic reactions (§ 230) against equal 
 accelerations (§ 32), the greater body would be more strained than the 
 less ; as the amounts of gravity or of kinetic reaction of similar 
 portions of them are as the mbes of their linear dimensions. Defi- 
 nitively, the strains at similarly situated points of the bodies will 
 be in simple proportion to their linear dimensions, and the displace- 
 ments will be as the squares of these lines, provided that there is no 
 strain in any part of any of them too great to allow the principle 
 of superposition to hold with sufficient exactness, and that no part is 
 turned through more than a very small angle relatively to any other 
 part. To illustrate by a single example, let us consider a uniform 
 long, thin, round rod held horizontally by its middle. Let its 
 substance be homogeneous, of density p, and Young's modulus, M'y 
 and let its length, /, be p times its diameter. Then (as the moment 
 of inertia of a circular area of radius r round a diameter is |7rr'*) the 
 
STATICS OF SOLinS AND FLUIDS 263 
 
 flexural rigidity of the rod will (§ 679) be — tt ( — J . This gives us 
 
 for the curvature at the middle of the rod the elongation and con- 
 traction where greatest, that is, at the highest and lowest points of the 
 normal section through the middle point ; and the droop of the ends ; 
 the following expressions 
 
 M ' M' ^^ SM' 
 
 Thus, for a rod whose length is 200 times its diameter, if its substance 
 be iron or steel, for which p= 775, and M= 194 x 10^ grammes per 
 square centimetre, the maximum elongation and contraction (being 
 at the top and bottom of the middle section where it is held) are 
 each equal to '8 x 10"^ x /, and the droop of its ends 2 x io~^x /^ 
 Thus a steel or iron wire, ten centimetres long, and half a millimetre 
 in diameter, held horizontally by its middle, would experience only 
 •000008 of maximum elongation and contraction, and only "002 of 
 a centimetre of droop in its ends : a round steel rod, of half a centi- 
 metre in diameter, and one metre long, would experience '00008 of 
 maximum elongation and contraction, and '2 of a centimetre of 
 droop : a round steel rod, of ten centimetres diameter, and twenty 
 metres long, must be of remarkable temper (see Properties of Matter) 
 to bear being held by the middle without taking a very sensible per- 
 manent set : and it is probable that no temper of steel is high enough 
 in a round shaft forty metres long, if only two decimetres in dia- 
 meter, to allow it to be held by its middle without either bending 
 it to some great angle, and beyond all appearance of elasticity, or 
 breaking it. 
 
 683. In passing from the dynamics of perfectly elastic solids to 
 abstract hydrodynamics, or the dynamics of perfect fluids, it is con- 
 venient and instructive to anticipate slightly some of the views as to 
 intermediate properties observed in real solids and fluids, which, 
 according to the general plan proposed (§ 402) for our work, will be 
 examined with more detail under Properties of Matter. 
 
 By induction from a great variety of observed phenomena, we are 
 compelled to conclude that no change of volume or of shape can be 
 produced in any kind of matter without dissipation of energy (§ 247); 
 so that if in any case .there is a return to the primitive configuration, 
 some amount (however small) of work is always required to com- 
 pensate the energy dissipated away, and restore the body to the same 
 physical and the same palpably kinetic condition as that in which it 
 was given. We have seen (§ 643), by anticipating something of 
 thermodynamic principles, how such dissipation is inevitable, even in 
 dealing with the absolutely perfect elasticity of volume presented by every 
 fluid, and possibly by some solids, as, for instance, homogeneous 
 crystals. But in metals, glass, porcelain, natural stones, wood, India- 
 rubber, homogeneous jelly, silk fibre, ivory, etc., a d\^\\xiQXfrictional 
 
264 ABSTRACT DYNAMICS. 
 
 resistance^ against every change of shape is, as we shall see later, 
 under Properties of Matter^ demonstrated by many experiments, and 
 is found to depend on the speed with which the change of shape is 
 made. A very remarkable and obvious proof of frictional resistance 
 to change of shape in ordinary solids, is afforded by the gradual, 
 more or less rapid, subsidence of vibrations of elastic solids; mar- 
 vellously rapid in India-rubber, and even in homogeneous jelly; less 
 rapid in glass and metal springs, but still demonstrably, much more 
 rapid than can be accounted for by the resistance of the air. This 
 molecular friction in elastic solids may be properly called viscosity of 
 solids, because, as being an internal resistance to change of shape 
 depending on the rapidity of the change, it must be classed with 
 fluid molecular friction, which by general consent is called viscosity of 
 fluids. But, at the same time, we feel bound to remark that the word 
 viscosity, as used hitherto by the best writers, when solids or hetero- 
 geneous semisolid-semifluid masses are referred to, has not been 
 distinctly applied to molecular friction, especially not to the molecular 
 friction of a highly elastic solid within its hmits of high elasticity, but 
 has rather been employed to designate a property of slow, continual 
 yielding through very great, or altogether unlimited, extent of change 
 of shape, under the action of continued stress. It is in this sense 
 that Forbes, for instance, has used the word in stating that 'Viscous 
 Theory of Glacial Motion' which he demonstrated by his grand 
 observations on glaciers. As, however, he, and many other writers 
 after him, have used the words plasticity and plastic, both with refer- 
 ence to homogeneous solids (such as wax or pitch, even though also 
 brittle; soft metals; etc.), and to heterogeneous semisohd-semifluid 
 masses (as mud, moist earth, mortar, glacial ice, etc.), to designate 
 the property ^ common to all those cases, of experiencing, under 
 continued stress either quite continued and unlimited change of shape, 
 or gradually very great change* at a diminishing (asymptotic) rate 
 through infinite time ; and as the use of the term plasticity implies no 
 more than does viscosity, any physical theory or explanation of the 
 property, the word viscosity is without inconvenience left available 
 for the definition we have given of it above. 
 
 684. A perfect fluid, or (as we shall call it) a fluid, is an unrealizable 
 conception, like a rigid, or a smooth, body : it is defined as a body 
 incapable of resisting a change of shape : and therefore incapable of 
 experiencing distorting or tangential stress (§ 640). Hence its pres- 
 sure on any surface, whether of a solid or of a contiguous portion of 
 
 1 See Proceedings of the Royal Society, May 1865, * On the Viscosity and 
 Elasticity of Metals ' (W. Thomson). 
 
 2 Some confusion of ideas might have been avoided on the part of writers who 
 have professedly objected to Forbes' theory while really objecting only (and we 
 believe groundlessly) to his usage of the word viscosity, if they had paused to 
 consider that no one physical explanation can hold for those several cases; and 
 that Forbes' theory is merely the proof by observation that glaciers have the 
 property that mud (heterogeneous), mortar (heterogeneous), pitch (homogeneous)., 
 water (homogeneous), all have of changing shape indefinitely and continuously 
 under the action of continued stress. 
 
STATICS OF SOLIDS AND FLUIDS. 265 
 
 the fluid, is at every point perpendicular to the surface. In equi- 
 librium, all common liquids and gaseous fluids fulfil the definition. 
 But there is finite resistance, of the nature of friction, opposing change 
 of shape at a finite rate; and, therefore, while a fluid is changing 
 shape, it exerts tangential force on every surface other than normal 
 planes of the stress (§ 635) required to keep this change of shape 
 going on. Hence; although the hydrostatical results, to which we 
 immediately proceed, are verified in practice; in treating of hydro- 
 kinetics, in a subsequent chapter, we shall be obliged to introduce the 
 consideration of fluid friction, except in cases where the circumstances 
 are such as to render its effects insensible. 
 
 685. With reference to a fluid the pressure at any point in any 
 direction is an expression used to denote the average pressure per unit 
 of area on a plane surface imagined as containing the point, and 
 perpendicular to the direction in question, when the area of that 
 surface is indefinitely diminished. 
 
 686. At any point in a fluid at rest the pressure is the same in 
 all directions: and, if no external forces act, the pressure is the same 
 at every point. For the proof of these and most of the following 
 propositions, we imagine, according to § 584, a definite portion of 
 the fluid to become solid, without changing its mass, form, or 
 dimensions. 
 
 Suppose the fluid to be contained in a closed vessel, the pressure 
 within depending on the pressure exerted on it by the vessel, and not 
 on any external force such as gravity. 
 
 687. The resultant of the fluid pressures on the elements of any 
 portion of a spherical surface must, like each of its components, pass 
 through the centre of the sphere. Hence, if we suppose (§ 584) a 
 portion of the fluid in the form of a plano-convex lens to be solidified, 
 the resultant pressure on the plane side must pass through the centre 
 of the sphere; and, therefore, being perpendicular to the plane, must 
 pass through the centre of the circular area. From this it is obvious 
 that the pressure is the same at all points of any plane in the fluid. 
 Hence the resultant pressure on any plane surface passes through 
 its centre of inertia. 
 
 Next, imagine a triangular prism of the fluid, with ends perpen- 
 dicular to its faces, to be solidified. The resultant pressures on its 
 ends act in the line joining the centres of inertia of their areas, 
 and are equal since the resultant pressures on the sides are in 
 directions perpendicular to this line. Hence the pressure is the same 
 in all parallel planes. 
 
 But the centres of inertia of the three faces, and the resultant 
 pressures applied there, lie in a triangular section parallel to the ends. 
 The pressures act at the middle points of the sides of this triangle, 
 and perpendicularly to them, so that their directions meet in a 
 point. And, as they are in equilibrium, they must be proportional 
 to the respective sides of the triangle; that is, to the breadths, or 
 areas, of the faces of the prism. Thus the resultant pressures on the 
 
266 ABSTRACT DYNAMICS. 
 
 faces must be proportional to the areas of the faces, and therefore 
 the pressure is equal in any two planes which meet. 
 
 Collecting our results, we see that the pressure is the same at all 
 points, and in all directions, throughout the fluid mass. 
 
 688. Hence if a force be appHed at the centre of inertia of each 
 face of a polyhedron, with magnitude proportional to the area of 
 the face, the polyhedron will be in equiUbrium. For we may suppose 
 the polyhedron to be a solidified portion of the fluid. The resultant 
 pressure on each face will then be proportional to its area, and will 
 act at its centre of inertia; which, in this case, is the Centre of 
 Pressure. 
 
 689. Another proof of the equality of pressure throughout a mass 
 of fluid, uninfluenced by other external force than the pressure of the 
 containing vessel, is easily furnished by the energy criterion of equi- 
 librium, § 254; but, to avoid complication, we will consider the fluid 
 to be incompressible. Suppose a number of pistons fitted into 
 cylinders inserted in the sides of the closed vessel containing the fluid. 
 Then, if A be the area of one of these pistons, / the average pressure 
 on it, X the distance through which it is pressed, in or out; the energy 
 criterion is that no work shall be done on the whole, i.e. that 
 
 as much work being restored by the pistons which are forced out, as 
 is done by those forced in. Also, since the fluid is incompressible, it 
 must have gained as much space by forcing out some of the pistons 
 as it lost by the intrusion of the others. This gives 
 
 A^x^+A^x^ + ...=t{Ax) = o. 
 The last is the only condition to which x^^ x^^, etc., in the first equa* 
 tion, are subject; and therefore the first can only be satisfied if 
 
 A=A=/^a=etc., 
 that is, if the pressure be the same on each piston. Upon this pro- 
 perty depends the action of Bramah's Hydrostatic Press. 
 
 If the fluid be compressible, the work expended in compressing it 
 from volume VXo F-BF, at mean pressure/, isJfSF. 
 
 If in this case we assume the pressure to be the same throughout, 
 we obtain a result consistent with the energy criterion. 
 
 The work done on the fluid is 5 {Apx)^ that is, in consequence of 
 the assumption, p% {Ax). 
 
 But this is equal to /8 F^ 
 
 for, evidently, S {Ax) = BF. 
 
 690. When forces, such as gravity, act from external matter upon 
 the substance of the fluid, either in proportion to the density of its 
 own substance in its different parts, or in proportion to the density 
 of electricity, or of magnetic polarity, or of any other conceivable 
 accidental property of it, the pressure will still be the same in all 
 directions at any one point, but will now vary continuously from 
 point to point. For the preceding demonstration (§ 687) may still 
 
STATICS OF SOLIDS AND FLUIDS 267 
 
 be applied by simply taking the dimensions of the prism small 
 enough; since the pressures are as the squares of its linear dimen- 
 sions, and the effects of the applied forces such as gravity, as the 
 cubes. 
 
 691. When forces act on the whole fluid, surfaces of equal pressure, 
 if they exist, must be at every point perpendicular to the direction of 
 the resultant force. For, any prism of the fluid so situated that the 
 whole pressures on its ends are equal must experience from the 
 applied forces no component in the direction of its length; and, 
 therefore, if the prism be so small that from point to point of it the 
 direction of the resultant of the applied forces does not vary sensibly, 
 this direction must be perpendicular to the length of the prism. 
 From this it follows that whatever be the physical origin, and the law, 
 of the system of forces acting on the fluid, and whether it be con- 
 servative or non-conservative, the fluid cannot be in equilibrium unless 
 the lines of force possess the geometrical property of being at right 
 angles to a series of surfaces. 
 
 692. Again, considering two surfaces of equal pressure infinitely 
 near one another, let the fluid between them be divided into columns 
 of equal transverse section, and having their lengths perpendicular to 
 the surfaces. The difference of pressure on the two ends being the 
 same for each column, the resultant applied forces on the fluid masses 
 composing them must be equal. Comparing this with § 506, we see 
 that if the applied forces constitute a conservative system, the density 
 of matter, or electricity, or whatever property of the substance they 
 depend on, must be equal throughout the layer under consideration. 
 This is the celebrated hydrostatic proposition that in a fluid at rest , 
 surfaces of equal pressure are also surfaces of equal density and of equal 
 potential, 
 
 693. Hence when gravity is the only external force considered, 
 surfaces of equal pressure and equal density are (when of moderate 
 extent) horizontal planes. On this depends the action of levels, 
 siphons, barometers, etc.; also the separation of liquids of different 
 densities (which do not mix or combine chemically) into horizontal 
 strata, etc., etc. The free surface of a liquid is exposed to the pressure 
 of the atmosphere simply; and therefore, when in equilibrium, must 
 be a surface of equal pressure, and consequently level. In extensive 
 sheets of water, such as the American lakes, differences of atmo- 
 spheric pressure, even in moderately calm weather, often produce con- 
 siderable deviations from a truly level surface. 
 
 694. The rate of increase of pressure per unit of length in the 
 direction of the resultant force, is equal to the intensity of the force 
 reckoned per unit of volume of the fluid. Let F be the resultant 
 force per unit of volume in one of the columns of § 692; / and/' 
 the pressures at the ends of the column, / its length, S its section. We 
 have, for the equilibrium of the column, 
 
 {p'-p)S=SlF. 
 Hence the rate of increase of pressure per unit of length is F> 
 
268 ABSTRACT DYNAMICS. 
 
 If the applied forces belong to a conservative system, for which 
 V and V are the values of the potential at the ends of the column, 
 we have (§ 504) 
 
 r-V=-lFp, 
 where p is the density of the fluid. This gives 
 
 p'-t=-p{V-V), 
 or dp= -pdV. 
 
 Hence in the case of gravity as the only impressed force the rate 
 of increase of pressure per unit of depth in the fluid is p, in gravitation 
 measure (usually employed in hydrostatics). In kinetic or absolute 
 measure (§ 189) it is gp. 
 
 If the fluid be a gas, such as air, and be kept at a constant tem- 
 perature, we have p = ^, where c denotes a constant, the reciprocal of 
 H^ the 'height of the homogeneous atmosphere,* defined (§ 695) 
 below. Hence, in a calm atmosphere of uniform temperature we 
 have 
 
 and from this, by integration, 
 
 where p^ is the pressure at any particular level (the sea-level, for 
 instance) where we choose to reckon the potential as zero. 
 
 When the differences of level considered are infinitely small in 
 comparison with the earth's radius, as we may practically regard them, 
 in measuring the heights of mountains, or of a balloon, by the baro- 
 meter, the force of gravity is constant, and therefore differences of 
 potential (force being reckoned in units of weight) are simply equal 
 to diflerences of level. Hence if x denote height of the level of 
 pressure/ above thatof/^, we have, in the preceding formulae, V=x, 
 and therefore 
 
 /=^„€--j that is, 
 
 695. If the air be at a constant temperature, the pressure 
 diminishes in geometrical progression as the height increases in 
 arithmetical progression. This theorem is due to Halley. Without 
 formal mathematics we see the truth of it by remarking that dif- 
 ferences of pressure are (§ 694) equal to differences of level multiplied 
 by the density of the fluid, or by the proper mean density when the 
 density differs sensibly between the two stations. But the density, 
 when the temperature is constant, varies in simple proportion to 
 the pressure, according to Boyle's law. Hence difl"erences of pres- 
 sure between pairs of stations diff"ermg equally in level are pro- 
 portional to the proper mean values of the whole pressure, which is 
 the well-known compound interest law. The rate of diminution of 
 pressure per unit of length upwards in proportion to the whole 
 pressure at any point, is of course equal to the reciprocal of the height 
 above that point that the atmosphere must have, if of constant 
 density, to give that pressure by its weight. The height thus defined 
 is commonly called 'the height of the homogeneous atmosphere,* a 
 
STATICS OF SOLIDS AND FLUIDS 269 
 
 very convenient conventional expression. It is equal to the product 
 of the volume occupied by the unit mass of the gas at any pressure 
 into the value of tliat pressure reckoned per unit of area, in terms of 
 the weight of the unit of mass. If we denote it by H^ the expo- 
 nential expression of the law is 
 
 which agrees with the final formula of § 694. 
 
 The value of H for dry atmospheric air, at the freezing tem- 
 perature, according to Regnault, is, in the latitude of Paris, 799,020 
 centimetres, or 26,215 feet. Being inversely as the force of gravity 
 in different latitudes (§ 187), it is 798 533 centimetres, or 26,199 feet, 
 in the latitude of Edinburgh and Glasgow. 
 
 696. It is both necessary and sufficient for the equilibrium of an 
 incompressible fluid completely filling a rigid closed vessel, and 
 influenced only by a conservative system of forces, that its density be 
 uniform over every equipotential surface, that is to say, every surface 
 cutting the lines of force at right angles. If, however, the boundary, 
 or any part of the boundary, of the fluid mass considered, be not 
 rigid ; whether it be of flexible solid matter (as a membrane, or a thin 
 sheet of elastic solid), or whether it be a mere geometrical boundary, 
 on the other side of which there is another fluid, or 7iothing [a case 
 which, without believing in vacuum as a reality, we may admit in 
 abstract dynamics (§ 391)], a farther condition is necessary to secure 
 that the pressure from without shall fulfil the hydrostatic equation 
 at every point of the boundary. In the case of a bounding membrane, 
 this condition must be fulfilled either through pressure artificially 
 applied from without, or through the interior elastic forces of the 
 matter of the membrane. In the case of another fluid of different 
 density touching it on the other side of the boundary, all round or 
 over some part of it, with no separating membrane, the condition 
 of equilibrium of a heterogeneous fluid is to be fulfilled relatively 
 to the whole fluid mass made up of the two; which shows that at the 
 boundary the pressure must be constant and equal to that of the fluid 
 on the other side. Thus water, oil, mercury, or any other liquid, in 
 an open vessel, with its free surface exposed to the air, requires for 
 equilibrium simply that this surface be level. 
 
 697. Recurring to the consideration of a finite mass of fluid 
 completely filling a rigid closed vessel, we see, from what precedes, 
 that, if homogeneous and incompressible, it cannot be disturbed from 
 equilibrium by any conservative system of forces; but we do not 
 require the analytical investigation to prove this, as we should have 
 'the perpetual motion' if it were denied, which would violate the 
 hypothesis that the system of forces is conservative. On the other 
 hand, a non-conservative system of forces cannot, under any circum- 
 stances, equilibrate a fluid which is either uniform in density through- 
 out, or of homogeneous substance, rendered heterogeneous in density 
 only through difference of pressure. But if the forces, though not 
 
270 ABSTRACT DYNAMICS. 
 
 conservative, be such that through every point of the space occupied 
 by the fluid a surface may be drawn which shall cut at right angles 
 all the lines of force it meets, a heterogeneous fluid will rest in 
 equilibrium under their influence, provided (§ 692) its density, from 
 point to point of every one of these orthogonal surfaces, varies in- 
 versely as the product of the resultant force into the thickness of 
 the infinitely thin layer of space between that surface and another of 
 the orthogonal surfaces infinitely near it on either side. (Compare 
 § 506). 
 
 698. If we imagine all the fluid to become rigid except an infinitely 
 thin closed tubular portion lying in a surface of equal density, and if 
 the fluid in this tubular circuit be moved any length along the tube 
 and left at rest, it will remain in equilibrium in the new position, 
 all positions of it in the tube being indifferent because of its homo- 
 geneousness. Hence the work (positive or negative) done by the 
 force {X, V, Z) on any portion of the fluid in any displacement 
 along the tube is balanced by the work (negative or positive) done on 
 the remainder of the fluid in the tube. Hence a single particle, acted 
 on only by X, K, Z, while moving round the circuit, that is moving 
 along any closed curve on a surface of equal density, has, at the end 
 of one complete circuit, done just as much work against the force in 
 some parts of its course, as the forces have done on it in the re- 
 mainder of the circuit. 
 
 699. The following imaginary example, and its realization in a 
 subsequent section (§ 701), show a curiously interesting practical 
 application of the theory of fluid equilibrium under extraordinary 
 circumstances, generally regarded as a merely abstract analytical 
 tlieory, practically useless and quite unnatural, 'because forces in 
 nature follow the conservative law.' 
 
 700. Let the lines of force be circles, with their centres all in one 
 line, and their planes perpendicular to it. They are cut at right 
 angles by planes through this axis ; and therefore a fluid may be in 
 equilibrium under such a system of forces. The system will not be 
 conservative if the intensity of the force be according to any other law 
 than inverse proportionality to distance from this axial line; and the 
 fluid, to be in equilibrium, must be heterogeneous, and be so dis- 
 tributed as to vary in density from point to point of every plane 
 through the axis, inversely as the product of the force into the 
 distance from the axis. But from one such plane to another it may 
 be either uniform in density, or may vary arbitrarily. To particularize 
 farther, we may suppose the force to be in direct simple proportion 
 to the distance from the axis. Then the fluid will be in equilibrium 
 if its density varies from point to point of every plane through the 
 axis, inversely as the square of that distance. If we still farther 
 particularize by making the force uniform all round each circular line 
 of force, the distribution of force becomes precisely that of the kinetic 
 reactions of the parts of a rigid body against accelerated rotation. 
 The fluid pressure will (§ 691) be equal over each plane through the 
 
STATICS OF SOLIDS AND FLUIDS. 271 
 
 axis. And in one such plane, which we may imagine carried round 
 the axis in the direction of the force, the fluid pressure will increase in 
 simple proportion to the angle at a rate per unit angle (§55) equal to 
 the product of the density at unit distance into the force at unit distance. 
 Hence it must be remarked, that if any closed line (or circuit) can be 
 drawn round the axis, without leaving the fluid, there cannot be 
 equilibrium without a firm partition cutting every such circuit, and 
 maintaining the diflerence of pressures on the two sides of it, corre- 
 sponding to the angle 2ir. Thus, if the 
 axis pass through the fluid in any part, 
 there must be a partition extending from 
 this part of the axis continuously to the 
 outer bounding surface of the fluid. Or if 
 the bounding surface of the whole fluid be 
 annular (like a hollow anchor-ring, or of 
 any irregular shape), in other words, if the 
 fluid fills a tubular circuit; and the axis 
 (A) pass through the aperture of the ring 
 (without passing into the fluid); there must be a firm partition (CD) 
 extending somewhere continuously across the channel, or passage, or 
 tube, to stop the circulation of the fluid round it; otherwise there 
 could not be equilibrium with the supposed forces in action. If we 
 further suppose the density of the fluid to be uniform round each of 
 the circular lines of force in the system we have so far considered (so 
 that the density shall be equal over every circular cylinder having the 
 line of their centres for its axis, and shall vary from one such 
 cylindrical surface to another, inversely as the squares of their radii), 
 we may, without disturbing the equiHbrium, impose any conservative 
 system of force in lines perpendicular to the axis; that is (§ 506), any 
 system of force in this direction, with intensity varying as some 
 function of the distance. If this function be the simple distance, the 
 superimposed system of force agrees precisely with the reactions 
 against curvature, that is to say, the centrifugal forces, of the parts of 
 a rotating rigid body. 
 
 701. Thus we arrive at the remarkable conclusion, that if a rigid 
 closed box be completely filled with incompressible heterogeneous 
 fluid, of density varying inversely as the square of the distance from 
 a certain line, and if the box be movable round this line as a fixed 
 axis, and be urged in any way by forces applied to its outside, the 
 fluid will remain in equilibrium relatively to the box; that is to say, 
 will move round with the box as if the whole were one rigid body, 
 and will come to rest with the box if the box be brought again to 
 rest: provided always the preceding condition as to partitions be 
 fulfilled if the axis pass through the fluid, or be surrounded by 
 continuous lines of fluid. For, in starting from rest, if the fluid 
 moves like a rigid solid, we have reactions against acceleration, 
 tangential to the circles of motion, and equal in amount to wr per 
 unit of mass of the fluid at distance r from the axis, w being the rate 
 
272 ABSTRACT DYNAMICS. 
 
 of acceleration (§ 57) of the angular velocity; and (see Vol. II.) we 
 have, in the direction perpendicular to the axis outwards, reaction 
 against curvature of path, that is to say, 'centrifugal force,' equal to 
 o)V per unit of mass of the fluid. Hence the equilibrium which we 
 have demonstrated in the preceding section, for the fluid supposed 
 at rest, and arbitrarily influenced by two systems of force (the circular 
 non-conservative and the radial conservative system) agreeing in law 
 with these forces of kinetic reaction, proves for us now the D'Alem- 
 bert (§ 230) equilibrium condition for the motion of the whole fluid as 
 of a rigid body experiencing accelerated rotation: that is to say, 
 shows that this kind of motion fulfils for the actual circumstances the 
 laws of motion, and, therefore, that it is the motion actually taken by 
 the fluid. 
 
 702. In § 688 we considered the resultant pressure on a plane 
 surface, when the pressure is uniform. We may now consider 
 (briefly) the resultant pressure on a plane area when the pressure 
 varies from point to point, confining our attention to a case of 
 great importance ;^that in which gravity is the only applied force, 
 and the fluid is a nearly incompressible liquid such as water. In this 
 case the determination of the position of the Centre of Pressure is 
 very simple ; and the whole pressure is the same as if the plane area 
 were turned about its centre of inertia into a horizonal position. 
 
 The pressure at any point at a depth z in the liquid may be ex- 
 pressed by 
 
 where p is the (constant) density of the liquid, and /^ the (atmo- 
 spheric) pressure at the free surface, reckoned in units of weight per 
 unit of area. 
 
 Let the axis of x be taken as the intersection of the plane of the 
 immersed plate with the free surface of the liquid, and that of y 
 perpendicular to it and in the plane of the plate. Let a be the 
 inclination of the plate to the vertical. Let also A be the area of the 
 portion of the plate considered, and x, y, the co-ordinates of its centre 
 of inertia. 
 
 Then the whole pressure is 
 
 jjj^dxdy = // (/„ + py cos a) dxdy 
 
 = Ap^ 4 ^p^ cos a. 
 
 The moment of the pressure about the axis of Jt: is 
 
 jjpydxdy = Ap^y + Ak^p cos a, 
 
 k being the radius of gyration of the plane area about the axis of x. 
 For the moment about y we have 
 
 jjpxdxdy = Ap^x + p cos a jjxydxdy. 
 The first terms of. these three expressions merely give us again the 
 results of § 688; we may therefore omit them. This will be equi- 
 valent to introducing a stratum of additional liquid above the free 
 surface such as to produce an equivalent to the atmospheric pressure. 
 
STATICS OF SOLIDS AND FLUIDS. 273 
 
 If the origin be now shifted to the upper surface of this stratum we 
 have 
 
 Pressure = Apy cos a, 
 Moment about Ox = Ak^p cos a, 
 
 Distance of centre of pressure from axis oi x = —. 
 
 But if /', be the radius of gyration of the plane area about a horizontal 
 axis in its plane, and passing through jts centre of inertia, we have 
 
 Hence the distance, measured parallel to the axis of j, of the centre 
 of pressure from the centre of inertia is 
 
 and, as we might expect, diminishes as the plane area is more and 
 more submerged. If the plane area be turned about the line through 
 its centre of inertia parallel to the axis of x^ this distance varies as 
 the cosine of its inclination to the vertical; supposing, of course, that 
 by the rotation neither more nor less of the plane area is submerged. 
 
 703. A body, wholly 6t partially immersed in any fluid influenced 
 by gravity, loses, through fluid pressure, in apparent weight an amount 
 equal to the weight of the fluid displaced. For if the body were 
 removed, and its place filled with fluid homogeneous with the sur- 
 rounding fluid, there would be equilibrium, even if this fluid be sup- 
 posed to become rigid. And the resultant of the fluid pressure upon 
 it is therefore a single force equal to its weight, and in the vertical 
 line through its centre of gravity. But the fluid pressure on the 
 originally immersed body was the same all over as on the solidified 
 portion of fluid by which for a moment we have imagined it replaced, 
 and therefore must have the same resultant. This proposition is of 
 great use in Hydrometry, the determination of specific gravity, etc., 
 etc. 
 
 704. The following lemma, while in itself interesting, is of great 
 use in enabling us to simplify the succeeding investigations regarding 
 the stability of equilibrium of floating bodies : — 
 
 Let a homogeneous solid, the weight of unit of volume of which 
 we suppose to be unity, be cut by a horizontal plane in XYX'Y. 
 Let O be the centre of inertia, 1 
 
 and let XX\ YY' be the principal 1' 
 
 axes, of this area. 
 
 Let there be a second plane 
 section of the solid, through YY\ 
 inclined to the first at an infinitely X'\ 
 small angle, B. Then (i) the 
 volumes of the two wedges cut 
 from the solid by these sections 
 are equal; (2) their centres of 
 inertia lie in one plane perpen- 
 
 18 
 
2 74 ABSTRACT DYNAMICS. 
 
 dicular X.o YY \ and (3) the moment of the weight of each of these, 
 round YY^ is equal to the moment of inertia about it of the corre- 
 sponding portion of the area multipHed by Q. 
 
 Take OX^ (9 F as axes, and let Q be the angle of the wedge : the 
 thickness of the wedge at any point P, (^, y)^ is Ox, and the volume 
 of a right prismatic portion whose base is the elementary area dxdy 
 at P is Bxdxdy. 
 
 Now let [ ] and ( ) be employed to distinguish integrations extended 
 over the portions of area to the right and left of the axis of j re- 
 spectively, while integrals over the whole area have no such distin- 
 guishing mark. Let v and v' be the volumes of the wedges ; (x, 7), 
 (^', y') the co-ordinates of their centres of inertia. Then 
 
 v-B\ ffxdxdy] 
 -v' = 0{jjxdxdy), 
 whence v-v' -0 jjxdxdy = o since O is the centre of inertia. Hence 
 v = v', which is (i). 
 Again, taking moments about XX' ^ 
 
 vy = [ffxydxdy] , 
 and — vy = 6 (ffxydxdy). 
 
 Hence vy - v'y' = ffxydxdy. 
 
 But for a principal axis '%xydm vanishes. Hence ly — v'y' -o^ 
 whence, since v - v'^ we have 
 
 y-y'-) which proves (2). 
 
 And (3) is merely a statement in words of the obvious equation 
 
 [ffx.xOdxdy] = 6 [ffx^.dxdy]. 
 
 705. If a positive amount of work is required to produce any 
 possible infinitely small displacement of a body from a position of 
 equilibrium, the equilibrium in this position is stable (§ 256). To 
 apply this test to the case of a floating body, we may remark, first, 
 that any possible infinitely small displacement may (§§ 30, 106) be 
 conveniently regarded as compounded of two horizontal displacements 
 in lines at right angles to one another, one vertical displacement, and 
 three rotations round rectangular axes through any chosen point. If 
 one of these axes be vertical, then three of the component displace- 
 ments, viz. the two horizontal displacements and the rotation about 
 the vertical axis, require no work (positive or negative), and therefore, 
 so far as they are concerned, the equilibrium is essentially neutral. 
 But so far as the other three modes of displacement are concerned, 
 the equilibrium may be positively stable, or may be unstable, or may 
 be neutral, according to the fulfilment of conditions which we now 
 proceed to investigate. 
 
 706. If, first, a simple vertical displacement, downwards, let us 
 suppose, be made, the work is done against an increasing resultant 
 of upward fluid pressure, and is of course equal to the mean increase 
 of this force multiplied by the whole space. If this space be denoted 
 by z, the area of the plane of flotation by A, and the weight of unit 
 bulk of the liquid by 7e>, the increased bulk of immersion is clearly Az, 
 
STATICS OF SOLIDS AND FLUIDS. 
 
 275 
 
 and therefore the increase of the resultant of fluid pressure is wAz, 
 and is in a Hne vertically upward through the centre of gravity of A. 
 The mean force against which the work is done is therefore \7vAz, 
 as this is a case in which work is done against a force increasing 
 from zero in simple proportion to the space. Hence the work done 
 is ^wAz^. We see, therefore, that so far as vertical displacements 
 alone are concerned, the equilibrium is necessarily stable, unless the 
 body is wholly immersed, when the area of the plane of flotation 
 vanishes, and the equilibrium is neutral. 
 
 707. The lemma of § 704 suggests that we should take, as the 
 two horizontal axes of rotation, the principal axes of the plane of 
 flotation. Considering then rotation through an infinitely small angle 
 
 6 round one of these, let G and E be the displaced centres of gravity 
 of the solid, and of the portion of its volume which was immersed 
 when it was floating in equilibrium, and G', FJ the positions which 
 they then had; all projected on the plane of the diagram which we 
 suppose to be through / the centre of inertia of the plane of flotation. 
 The resultant action of gravity on the displaced body is W, its weight, 
 acting downwards through G\ and that of the fluid pressure on it is 
 W upwards through E corrected by the amount (upwards) due to the 
 additional immersion of the wedge AIA', and the amount (down- 
 wards) due to the extruded wedge B'IB. Hence the whole action of 
 
 18—2 
 
2 76 ABSTRACT DYNAMICS. 
 
 gravity and fluid pressure on the displaced body is the couple of 
 forces up and down in verticals through G and E^ and the correction 
 due to the wedges. This correction consists of a force vertically 
 upwards through the centre of gravity of A' I A, and downwards 
 through that of BIB'. These forces are equal [§ 704 (i)], and 
 therefore constitute a couple which [704 (2)] has the axis of the 
 displacement for its axis, and which [§ 704 (3)] has its moment equal 
 to Qwk~A if A be the area of the plane of flotation, and k its radius 
 of gyration (§ 235) round the principal axis in question. But since 
 GE^ which was vertical {G'E') in the position of equilibrium, is 
 incHned at the infinitely small angle 6 to the vertical in the displaced 
 body, the couple of forces W in the verticals through G and E has 
 for moment WhS, if h denote GE] and is in a plane perpendicular 
 to the axis, and in the direction tending to increase the displacement, 
 when G is above E. Hence the resultant action of gravity and fluid 
 pressure on the displaced body is a couple whose moment is 
 
 {wAk^ - Wh)e, or w {Ak^ - Vh)e, 
 
 if V be the volume immersed. It follows that when Ak~> Vh the 
 equilibrium is stable, so far as this displacement alone is concerned. 
 
 Also, since the couple worked against in producing the displace- 
 ment increases from zero in simple proportion to the angle of dis- 
 placement, its mean value is half the above; and therefore the whole 
 amount of work done is equal to 
 
 Iw^Ak""- Vh)e\ 
 
 708. If now we consider a displacement compounded of a vertical 
 (downwards) displacement z, and rotations through infinitely small 
 angles 6, 6' round the two horizontal principal axes of the plane of 
 flotation, we see (§§ 706, 70^) that the work required to produce it is 
 equal to 
 
 \7v [Az' + {Ak' - Vh) 0' + {Ak" -^ Vh) $"1 
 
 and we conclude that, for complete stability with reference to all pos- 
 sible displacements of this kind, it is necessary and sufficient that 
 
 ^ Ak' ^ Ak" 
 n < -p.- , and < — ^- . 
 
 709. When the displacement is about any axis through the centre 
 of inertia of the plane of flotation, the resultant of fluid pressures is 
 equal to the weight of the body; but it is only when the axis is a 
 principal axis of the plane of flotation that this resultant is in the 
 plane of displacement. In such a case the point of intersection of 
 the resultant with the Hne originally vertical, and through the centre 
 of gravity of the body, is called the Metacenti'e. And it is obvious, 
 from the above investigation, that for either of these planes of dis- 
 placement the condition of stable equilibrium is that the metacentre 
 shall be above the centre of gravity. 
 
 710. We shall conclude with the consideration of one case of the 
 
STATICS OF SOLIDS AND FLUIDS. 
 
 277 
 
 equilibrium of a revolving mass of fluid subject only to the gravitation 
 of its parts, which admits of a very simple synthetical solution, without 
 any restriction to approximate sphericity; and for which the following 
 remarkable theorem was discovered by Newton and Maclaurin : — 
 
 711. An oblate ellipsoid of revolution, of any given eccentricity, is 
 a figure of equilibrium of a mass of homogeneous incompressible 
 fluid, rotating about an axis with determinate angular velocity, and 
 subject to no forces but those of gravitation among its parts. 
 
 The angular velocity for a given eccentricity is independent of the 
 bulk of the fluid, and proportional to the square root of its density. 
 
 712. The proof of this proposition is easily obtained from the 
 results already deduced with respect to the attraction of an ellipsoid 
 and the properties of the free surface of a fluid. 
 
 We know, §538, that li AFB he. a meridian section of a homo- 
 geneous oblate spheroid, A C the polar axis, CF an equatorial radius, 
 and /'any point on the surface, the attraction of the spheroid may be 
 resolved into two parts; 
 one, Fp, perpendicular to 
 the polar axis, and vary- 
 ing as the ordinate FM; 
 the other, Fs, parallel to 
 the polar axis, and vary- 
 ing as FN. These com- 
 ponents are not equal 
 when MF and FN are 
 equal, else the resultant 
 attraction at all points in 
 the surface would pass 
 through C; whereas we 
 know that it is in some 
 such direction as Ff, cutting the radius BC between B and C, but at 
 a point nearer to C than n the foot of the normal at F. Let then 
 
 Fp=^a.FM, 
 
 and Fs = p.FN, 
 
 where a and ^ are known constants, depending merely on the density 
 (p), and eccentricity (^), of the spheroid. 
 
 Also, we know by geometry that Nn = (t - /) CN. 
 
 Hence; to find the magnitude of a force Fq perpendicular to the 
 axis of the spheroid, which, when compounded with the attraction, 
 will bring the resultant force into the normal Fn : make pr = Fq, and 
 we must have 
 
 PFp 
 
 Fr _ Nn 
 Fs ~ FN 
 
 = (i 
 
 
 O 
 
 aFs 
 
 Hence 
 
 Fr 
 
 e^)^Fp, 
 
2)8 ABSTRACT DYNAMICS. 
 
 Pp-Pq = (^-e)^^Pp, 
 
 or 
 
 ^^ = {i-(i-^)f}^/ 
 
 Now if the spheroid were to rotate with angular velocity <o about AC, 
 the centrifugal force, §§ 39, 42, 225, would be in the direction Pq, 
 and would amount to 
 
 Hence, if we make 
 
 aj' = a-(i-^^)/?, 
 
 the whole force on P, that is, the resultant of the attraction and 
 centrifugal force, will be in the direction of the normal to the sur- 
 face, which is the condition for the free surface of a mass of fluid in 
 equilibrium. 
 
 Now, (§522 of our larger work) 
 
 l^^-e' . _, I- A 
 
 _ /I Ji-e' . _, \ 
 /3 = 47rp\^~, ^^- sm ej, 
 
 XT 2 f(3 - 2^') Ji--e^ . _, I - /) / V 
 
 Hence ui=27rp r'? — — {^- sm V- 3 — ^ V . (i) 
 
 This determines the angular velocity, and proves it to be proportional 
 to Jp. 
 
 713. If, after Laplace, \ve introduce instead of ^ a quantity « 
 defined by the equation 
 
 I 
 
 -2 > 
 
 i-e- = 
 
 I + e' 
 
 \ (') 
 
 or g = — . = tan (sin V), | 
 
 n/i-^ J 
 
 the expression (i) for w^ is much simplified, and 
 
 When e, and therefore also c, is small, this formula is most easily 
 calculated from 
 
 ^^ = A^'-A'' + etc. (4) 
 
 of which the first term is sufficient when we deal with spheroids so 
 little oblate as the earth. 
 
 The following table has been calculated by means of these simpli- 
 fied formulae. The last figure in each of the four last columns is 
 given to the nearest unit. The two last columns will be explained a 
 few sections later : — 
 
STATICS OF SOLIDS AND FLUIDS. 
 
 279 
 
 e. i 
 
 I 
 
 1 
 
 — when = 3-68x10"'^. 
 
 (i+e--^)^*^'. 
 
 j 
 
 1 
 
 e 
 
 2'irp' 
 
 w '^ ^ 
 
 lirp 
 
 O'l 
 
 9'95o 
 
 0-0027 
 
 79,966 
 
 0*0027 
 
 •2 1 
 
 4-899 1 
 
 •OTO7 
 
 39»397 
 
 •Olio 
 
 •3 
 
 3-180 
 
 •0243 
 
 26,495 
 
 •0258 
 
 •4 
 
 2-291 
 
 •0436 
 
 19,780 
 
 •0490 
 
 •5 
 
 1-732 
 
 •0690 
 
 15.730 
 
 •0836 
 
 •6 
 
 i'333 
 
 •1007 
 
 13,022 
 
 •1356 
 
 •7 
 
 1-020 
 
 •1387 
 
 11,096 
 
 •2172 
 
 •8 
 
 0-750 
 
 •1816 
 
 9,697 
 
 •3588 
 
 •9 
 
 •4843 
 
 •2203 . 
 
 8,804 
 
 •6665 
 
 •91 
 
 •4556 
 
 •2225 
 
 8,759 
 
 •7198 
 
 •92 
 
 •4260 
 
 •2241 
 
 8,729 
 
 •7813 
 
 •93 
 
 •3952 
 
 •2247 
 
 8,718 
 
 •8533 
 
 •94 
 
 •3629 
 
 •2239 
 
 8,732 
 
 •9393 
 
 •95 
 
 •3287 
 
 •2213 
 
 8,783 
 
 I -045 
 
 1 -96 
 
 •2917 
 
 •2160 
 
 8,891 
 
 1-179 
 
 •97 
 
 •2506 
 
 •2063 
 
 9,098 
 
 i'359 
 
 •98 
 
 •2030 
 
 •1890 
 
 9,504 
 
 1-627 
 
 •99 
 
 •1425 
 
 '1551 
 
 10,490 
 
 2-113 
 
 i-oo 
 
 0-0000 
 
 O'OOOO 
 
 00 
 
 00 
 
 From this we see that the value of — • increases gradually from 
 
 27rp 
 
 zero to a maximum as the eccentricity e rises from zero to about 0-93, 
 and then (more quickly) falls to zero as the eccentricity rises from 
 o*93 to unity. The values of the other quantities corresponding to 
 this maximum are given in the table. 
 
 714. If the angular velocity exceed the value calculated from 
 
 ^— = 0*2247, 
 27rp 
 
 (5) 
 
 when for p is substituted the density of the liquid, equilibrium Is im- 
 possible in the form of an ellipsoid of revolution. If the angular 
 velocity fall short of this limit there are always two ellipsoids of 
 revolution which satisfy the conditions of equilibrium. In one of 
 these the eccentricity is greater than 0*93, in the other less. 
 
 715. It may be useful, for special apphcations, to indicate briefly 
 how p is measured in these formulae. In the definitions of §§ 476, 
 477, on which the attraction formulae are based, unit mass is defined 
 as exerting unit force on unit mass at unit distance; and unit volume- 
 density is that of a body which has unit mass in unit volume. Hence, 
 with the foot as our linear unit, we have for the earth's attraction on 
 a particle of unit mass at its surface 
 
 ^^<jR^ 
 
 R' 
 
 rR 
 
 32 
 
28o ABSTRA CT D YNAMICS. 
 
 where R Is the radius of the earth (supposed spherical) in feet, and 
 o- its mean density, expressed in terms of the unit just defined. 
 Taking 20,900,000 feet as the value of ^, we have 
 
 c = o'oooooo368 = 3*68 x lo"'^. (6) 
 
 As the mean density of the earth is somewhere about 5*5 times that 
 of water, the density of water in terms of our present unit is 
 
 ^ 10- = 67x10- 
 5-5 ' 
 
 716. The fourth column of the table above gives the time of rota- 
 tion in seconds, corresponding to each value of the eccentricity, p 
 being assumed equal to the mean density of the earth. For a mass 
 of water these numbers must be multiplied by sJs'S'} ^s the time of 
 rotation to give the same figure is inversely as the square root of the 
 density. 
 
 For a homogeneous liquid niass, of the earth's mean density, 
 rotating in 23^^ 46"^ 4^ we find ^=0*093, which corresponds to an 
 ellipticity of about ^io- 
 
 717. An interesting form of this problem, also discussed by Laplace, 
 is that in which the moment of momentum and the mass of the fluid 
 are given, not the angular velocity; and it is required to find what is 
 the eccentricity of the corresponding ellipsoid of revolution, the result 
 proving that there can be but one. 
 
 It is evident that a mass of any ordinary liquid (not 2l perfect fluids 
 § 684), if left to itself in any state of motion, must preserve unchanged 
 its moment of momentum, § 202. But the viscosity, or internal 
 friction, § 684, will, if the mass remain continuous, ultimately destroy 
 all relative motion among its parts ; so that it will ultimately rotate as 
 a rigid solid. If the final form be an ellipsoid of revolution, we can 
 easily show that there is a single definite value of its eccentricity. 
 But, as it has not yet been discovered whether there is any other 
 form consistent ^\\ki stable equilibrium, we do not know that the mass 
 will necessarily assume the form of this particular ellipsoid. Nor in 
 fact do we know whether even the ellipsoid of rotation may not become 
 an unstable form if the moment of momentum exceed some limit de- 
 pending on the mass of the fluid. We shall return to this subject in 
 Vol. II., as it afl'ords an excellent example of that difficult and delicate 
 question Kinetic Stability^ § 300. 
 
 If we call a the equatorial semi-axis of the ellipsoid, e its eccen- 
 tricity, and to its angular velocity of rotation, the giz>€n quantities are 
 the mass M-^-rrpa^ J 1 —e^, 
 
 and the moment of momentum 
 
 A = -^^Trpoia' J I - e\ 
 
 I'hese equations, along with (3), determine the three quantities, a, r, 
 and (o. 
 
 Eliminating a between the two just written, and expressing e as 
 before in terms of €, we have 
 
STATICS OF SOLIDS AND FLUIDS. 281 
 
 This gives 
 
 O)^ k 
 
 ^""P (l + 6^)^ 
 
 where y^ is a gk'en multiple of ph Substituting in 771 (2) we have 
 
 ^ = (1+ ^) ^^ (^^-^tan-U - pj 
 
 Now the last column of the table in § 713 shows that the value of this 
 function of € (which vanishes with e) continually increases with e, and 
 becomes infinite when c is infinite. Hence there is always one, and 
 only one, value of c, and therefore of e, which satisfies the conditions 
 of the problem. 
 
 718. All the above results might without much difficulty have been 
 obtained analytically, by the discussion of the equations; but we have 
 preferred, for once, to show by an actual case that numerical calcula- 
 tion may sometimes be of very great use. 
 
 719. No one seems yet to have attempted to solve the general 
 problem of finding all the forms of equilibrium which a mass of 
 homogeneous incompressible fluid rotating with uniform angular 
 velocity may assume. Unless the velocity be so small that the figure 
 differs but little from a sphere, the problem presents difficulties of an 
 exceedingly formidable nature. It is therefore of some importance 
 to know that we can by a synthetical process show that another form, 
 besides that of the elHpsoid of revolution, may be compatible with 
 equilibrii^m; vi?;. an ellipsoid with three unequal ajces, of which the 
 least is the axis of rotation. This curious theorem was discovered by 
 Jacobi in 1834, and seems, simple as it is, to have been enunciated 
 by him as a challenge to the French mathematicians \ For the proot 
 we must refer to our larger work. 
 
 ^ See a Paper by Liouville, Journal de VAcqle Poly technique^ cahier xxiii., foot- 
 note to p. 290. 
 
APPENDIX. 
 
 KINETICS. 
 
 {a) In the case of the Simple Pendidum, a heavy particle is sus- 
 pended from a point by a hght inextensible string. If we suppose it 
 to be drawn aside from the vertical position of equilibrium and 
 allowed to fall, it will oscillate in one plane about its lowest position. 
 When the string has an inclination Q to the vertical, the weight mg of 
 the particle may be resolved into mg cos B which is balanced by the 
 tension of the string, and mg sin 6 in the direction of the tangent to 
 the path. If / be the length of the string, the distance (along the arc) 
 from the position of equilibrium is 10. 
 
 Now if the angle of oscillation be small (not above 3° or 4° say), the 
 sine and the angle are nearly equal to each other. Hence the acce- 
 leration of the motion (which is rigorously g sin 0) may be written gB. 
 Hence we have a case of motion in which the acceleration is propor- 
 tional to the distance from a point in the path, that is, by § 74, Simple 
 Harmonic Motion, The square of the angular velocity in the cor- 
 
 responding circular motion is -^. — , = - , and the period of the 
 
 displacement / 
 
 harmonic motion is therefore 27r* /-. In the case of the pendulum, 
 
 the time of an oscillation from side to side of the vertical is usually 
 
 taken — and is therefore ^a / - • 
 
 ((^) Thus the times of vibration of different pendulums are as the 
 square roots of their lengths, for any arcs of vibration, provided only 
 these be small. 
 
 Also the times of vibration of the same pendulum at different 
 places are inversely as the square roots of the apparent force of gravity 
 on a unit mass at these places. 
 
 (c) It was found experimentally by Newton that pendulums of the 
 same length vibrate in equal times at the same place whatever be the 
 material of which their bobs are formed. This would evidently not 
 be the case unless the weight were in every case proportional to the 
 amount of matter in the bob. 
 
APPENDIX. 283 
 
 {i{) If the simple pendulum be slightly disturbed in any way from 
 its position of equilibrium, it will in general describe very nearly an 
 eUipse about its lowest position as centre. This is easily seen from 
 §82. 
 
 ii) If the arc of vibration be considerable, the motion will not be 
 simple harmonic, and the time of vibration will be greater than that 
 above stated; since the acceleration being as the sine of the dis- 
 placement, is in less and less ratio to the displacement as the latter is 
 greater. 
 
 In this case, the motion for any disturbance is, for one revolution, 
 approximately elliptic as before; but the ellipse slowly turns round 
 the vertical, in the direction in which the bob moves. 
 
 (/) The bob may, however, be so projected as to revolve uniformly 
 in a horizontal circle, in which case the apparatus is called a Conical 
 Penduhun. Here we have /sin B for the radius of the circle, and the 
 •force in the direction of the radius is T^sin ^, where J'is the tension of 
 the string. T'cos B balances mg — and thus the force in the radius of 
 the circle is w^tan 0. The square of the angular velocity in the circle 
 
 is therefore t-^^-tt, and the time of revolution 27r . / : or 
 
 /cos 6 V g ' 
 
 where /i is the height of the point of suspension above the 
 
 
 plane of the circle. Thus all conical pendulums with the same height 
 revolve in the same time. 
 
 (g) A rigid mass oscillating about a horizontal axis, under the 
 action of gravity, constitutes what is called a Compound Pmdulwn. 
 
 When in the course of its motion the body is inclined at any angle 
 Q to the position in which it hangs, when in equilibrium, it experiences 
 from gravity, and the resistance of the supports of its axis, a couple, 
 which is easily seen to be equal to 
 
 gWh sin e, 
 where Wis the mass and h the distance of its centre of gravity fronl 
 the axis. This couple produces (§§ 232, 235) acceleration of angular 
 velocity, calculated by dividing the moment of the couple by the 
 moment of inertia of the body. Hence, if / denote the moment of 
 inertia about the supporting axis, the angular acceleration is equal to 
 
 gWsvcv 6 
 I ' 
 Its motion is, therefore, identical (§ {a)) with that of the simple pen- 
 dulum of length equal to 77^ . 
 
 If a rigid body be supported about an axis, which either passes 
 very nearly through the centre of gravity, or is at a very great dis- 
 tance from this point, the length of the equivalent simple pendulum 
 will be very great : and it is clear that some particular distance for 
 the point of support from the centre of gravity will render the length 
 
284 APPENDIX, 
 
 of the corresponding simple pendulum, and, therefore, the time of 
 vibration, least possible. 
 
 To investigate these circumstances for all axes parallel to a given 
 line, through the centre of gravity, let k be the radius of gyration 
 round this line, we have (§ 198), 
 
 and, therefore, if / be the length of the isochronous simple pendulum, 
 k^ 4- k' _ {h-kY + 2hk _ ^^^ (/i-ky 
 ~ h ~ h h ' 
 
 The second term of the last of these forms vanishes when h = k, and 
 is positive for all other values of h. The smallest value of / is, 
 therefore, 2k, and this, the shortest length of the isochronous simple 
 pendulum, is realized when the axis of support is at the distance k 
 from the centre of inertia. 
 
 To find at what distance //, from the centre of inertia the axis must 
 be fixed to produce a pendulum isochronous with the simple pen- 
 dulum, of given length /, we have the quadratic equation 
 
 For the solution to be possible we have seen that / must be greater 
 than, or at least equal to, 2k. If /= 2k, the roots of this equation are 
 equal, k being their common value. For any value of / greater than 
 2k, the equation has two real roots whose sum is equal to /, and pro- 
 duct equal to k^ : hence, for any distance from the centre of inertia 
 less than k, another distance greater than k, which is a third propor- 
 tional to it and k, gives the same time of vibration ; and the length of 
 the simple pendulum corresponding to either case, is equal to the sum 
 of the distances of the two axes from the centre of inertia. This sum 
 is equal to the distance between them if the two axes are in one 
 plane, through the; centre of inertia, and on opposite sides of this 
 point; and, therefore, for axes thus placed, g,nd not equidistant from 
 the centre of inertia, if the times of oscillation of the body when 
 successively supported upon them are found to be equal, it may be 
 inferred that the distance between them is equal to the length of 
 the isochronous simple pendulum. As a simple pendulum exists only 
 in theory, this proposition was taken advantage of by Kater for the 
 practical determination of the force of gravity at any station. 
 
 {h) A uniformly heavy and perfectly flexible cord, placed in the in- 
 terior of a smooth tube in the form of any plane curve, and subject to 
 no external forces, 7vill exert no pressure on the tube if it have every- 
 where the same tension, and move with a certain definiie velocity. 
 
 For, as in § 592, the statical pressure due to the curvature of the 
 
 Q 
 
 rope per unit of length is J'- (where <j is the length of the arc AB 
 
 in that figure) directed inwards to the centre of curvature. Now, the 
 element a-, whose mass is nitr, is moving in a curve whose curvature is 
 
 -with velocity v (suppose). The requisite force is - =mv'0\ 
 
APPENDIX. 285 
 
 Q 
 
 and for unit of length mt^-. Hence if T= mv^ the theorem is true. 
 
 If we suppose a portion of the tube to be straight, and the whole to 
 be moving with velocity v parallel to this line, and against the motion 
 of the cord, we shall have the straight part of the cord reduced to 
 rest, and an undulation, of any^ but unvarying^ form and dimensions, 
 
 .It 
 
 running along it with the linear velocity . / — . 
 
 Suppose the cord stretched by an appended mass of W'^pounds, and 
 suppose its length / feet and its own mass w pounds. Then T=^ IVg, 
 hn = Wy and the velocity of the undulation is 
 
 / 
 
 IV/e- 
 
 — ^ feet per second. 
 
 (J) When a?i incompressible liquid escapes from an orifice^ the velocity 
 is the same as ivould be acquired by falling from the free surface to the 
 level of the orifice. 
 
 For, as we may neglect (provided the vessel is large compared with 
 the orifice) the kinetic energy of the bulk of the liquid; the kinetic 
 energy of the escaping liquid is due to the loss of potential energy 
 of the whole by the depression of the free surface. Thus the pro- 
 position at once. 
 
 {k) The small oscillations of a liquid in a U tube follow the 
 harmonic law. 
 
 The tube being of uniform section S^ a depression of level, x, 
 from the mean, on one side, leads to a rise, x^ on the other; and if 
 the whole column of fluid be of length 2a^ we have the mass 2aSp 
 disturbed through a space x^ and acted on by a force 2Sxgp tending to 
 
 bring it back. The time of oscillation is therefore (§ {a)) 27r \l - 
 and is the same for all liquids whatever be their densities. 
 
INDEX. 
 
 Aberration gives hodograph of Earth's 
 
 orbit 53 
 Abscissae 452 
 Absolute acceleration 64 
 
 — motion 63 
 
 — unit of force, Gauss's, 188; British 190 
 Acceleration, definition 34; uniform 32; 
 
 variable 33; average 33; angular 57; 
 composition and resolution 34, 37 
 
 — directed to a fixed centre 45 
 
 — in a fixed direction 44 
 
 — in logarithmic spiral with uniform 
 angular velocity about the pole 295 
 
 — in Simple Harmonic Motion 74 
 
 — in straight line, uniform 43 
 
 — in uniform circular motion 36, 39, 42 
 
 — of momentum 178 
 
 Accurate measurements, necessity for 
 
 352 
 Action, Least 279 
 
 — Maximum 317 
 
 — Minimum 3 1 1 
 
 — Stationary 281 
 
 — Varying 282 
 Aeolotropic substance, an 647 
 Alteration of latitude by hemispherical 
 
 hill, or cavity 496; by a crevasse 497 
 Ampere's Theory of Electrodynamics 
 
 336 
 Amplitude of S. H. M. 71 
 Angle between two lines, definition of 
 
 441 note 
 Angle of repose 473 
 Angle, solid 482 ; round a .point 483 ; 
 
 subtended at a point 485 
 Angular acceleration 57 
 Angular measure, standard of 357 
 Angular velocity 54; unit of 55; com- 
 position of 107, 108 
 Anticlastic surface 120 
 Approximate treatment of physical 
 
 questions 391 
 Arc, definition of 438; projection of 
 
 an 439 
 Area of an autotomic plane circuit 445 
 Argument of S. H. M. 71 
 Atmosphere Homogeneous 695 ; sec 
 
 Homogeneous 
 
 Attraction not modified by interposition 
 of other matter 474 
 
 — is normal to equipotential surfaces 
 506 
 
 — integral of normal, over a closed 
 surface 510 
 
 — direct analytical calculation of 494 
 
 — law of, when a uniform spherical 
 shell exerts no attraction on an in- 
 ternal point 541 
 
 ■ — law of gravitation 475 
 
 — of gravitating, electric, or magnetic 
 masses 478 
 
 — variation of, in crossing an attracting 
 surface 495 
 
 — of a circular arc for a particle at its 
 centre 499 
 
 — of a right cone for a particle at its 
 vertex 494 {c) 
 
 — of a cylinder on a particle in its axis 
 
 494 {^) 
 
 ■ — of a cylindrical distrihttion of matter 
 508 
 
 • — of a uniform circttlar disc on a par- 
 ticle in its axis 494 {a) 
 
 — of an infinite disc ^g^ 
 
 — of t7vo equal uniform discs, one posi- 
 tive, other negative 494 {d) 
 
 — of an Ellipsoid 535, 537 ; of homo- 
 geneous ellipsoid 538 ; Maclaurin's 
 Theorem 539; Ivory's Theorem 540; 
 Duhamel's application of Ivory's The- 
 orem 54 1 
 
 — of an ellipsoidal shell 535 ; on an in- 
 ternal particle 536 
 
 — of a uniform limited straight line on 
 an external particle 499 {h) 
 
 — of a viouniain on a plumb line 
 496- (a) 
 
 — at the top and the bottom of a pit 
 496 {b) 
 
 — oi infinite parallel planes 508 
 
 — of a sphere composed of concentric 
 shells of uniform density 498 
 
 — of a uniform sphere on an external 
 particle infinitely near its surface 488 
 cor. 
 
 — of a uniform sphere 534, 541 
 
INDEX. 
 
 387 
 
 Attraction of an uninsulated sphere 
 under the influence of an electrified 
 particle 493 
 
 — of a uniform spherical shell on an 
 internal point 479 ; converse proposi- 
 tion 541 
 
 — of a uniform spherical shell on an 
 external point 488 
 
 — oi a spherical surface whose density 
 varies as D~^ from exccTitric points 
 490 et seq.; excentric point inside at- 
 tracted point outside, and vice versa 
 491 ; excentric and attracted point 
 both within or both without 492 
 
 Autotomic circuit 443 
 Average curvature 14 
 Average stress 645 
 Average velocity 26 
 Axiom, physical 209 ; regarding the 
 equilibrium of a non-rigid body 
 
 584 
 Axis of a couple 201 
 Axis, central 579 
 
 Balance, Coulomb's Torsion 385 
 
 — requisites for a good 383 
 
 • — sensibility, stability and constancy 
 of a 384 
 
 — statical principles of 592 
 Balance, spring 386 
 Ballistic pendulum 263, 272 
 Bending of a supported beam or uniform 
 
 bar 625; supported at ends or middle 
 
 625; at ends and middle 627 
 Bending, effect of, on cross section of 
 
 body 679 
 Bifilar suspension 388 
 Body, motion of a rigid 106 
 Body, a perfectly rigid, defined 393, 
 
 401 
 Bramah Press, hydrostatic principle of 
 
 689 
 British system of units of mass 661 note 
 British absolute unit of force 1 90 
 
 Cardioid 105 
 
 Catenary 594 ; a parabola 599 ; kinetic 
 question relative to 598 ; inverse pro- 
 blem 600 
 
 Cathetometer 382 
 
 Central axis 579 
 
 Central ellipsoid 238 
 
 Centre of gravity, and centre of inertia 
 ^95* 542, 582; centrobaric bodies 
 542 ; if it exist is centre of inertia 550 ; 
 position of in stable equilibrium 585, 
 in rocking stones 586; of a body in 
 equilibrium about an axis 587, on a 
 fixed surface 588; Pappus' theorem 
 concerning 589 
 
 Centre of pressure 688, 702 
 
 Centre of mass or inertia 195, 582 ; 
 motion of centre of inertia of a rigid 
 body 232, 550; moments of inertia 
 of centrobaric body round axes 
 through centre of inertia 551 
 
 Centrobaric body 542, proved possible 
 by Green 543, properties of 545 ; 
 centrobaric shell 547 ; centrobaric 
 solid 549 ; moments of inertia of a 
 centrobaric body round axes through 
 centre of inertia 551 
 
 Change of velocity 177, of momen- 
 tum 177 
 
 Characteristic function, Hamilton's 283 
 
 Chasles on confocal ellipsoids 537 
 
 Chronometer 367 
 
 Chronoscope 369 
 
 Circuit, linear 443 ; autotomic 443 
 
 Circular measure, unit of 357 
 
 Clairault's formula for the amount of 
 gravity at a place 187 
 
 Clocks 367 
 
 Closed curve 443 
 
 — polygon 443 
 
 Closed surface, ^7V(/(r, over a 510 
 
 Coarsegrainedness 646 
 
 Coefficient of elasticity 265 note, 644 
 
 Coefficient of restitution 265 ; of glass, 
 iron, wool 265 
 
 Comet, hodograph of orbit of 49 
 
 Component velocity 29 ; acceleration 
 37; of a force, effective 193 
 
 Composition of Velocities 31 ; Accelera- 
 tions 34 ; Simple Harmonic Motions in 
 same direction 75, in different direc- 
 tions 80; Angular velocities 107, 
 about axes meeting in a point 108 ; 
 Rotations 107, successive finite rota- 
 tions 109; Forces 221, of two acting 
 on a point 419, 422, special cases of 
 423 et seq.; nearly conspiring 427, 
 nearly opposed 428, at right angles 
 429, of any set of forces acting on a 
 rigid body 570 ; Couples in same plane 
 or in parallel planes 561, 562, 563, any 
 number, 564 ; not in parallel planes 
 565, any number of 566, and a force 
 568 
 
 Compound pendulum, Appendix g. 
 
 Compressibility 65 1 
 
 Conditions of equilibrium of a particle 
 408 ; a material point 470 ; of parallel 
 forces 558 ; of floating bodies 702—9 ; 
 of any number of couples 567 et seq. 
 
 Cone, orthogonal and oblique section of 
 very small 486 ; solid angle of 482; 
 area of segment cut from spherical 
 surface by a small cone 487 
 
 Cones opposite or vertical 481 
 
288 
 
 INDEX, 
 
 Confocal ellipsoids, correspondingpoints 
 on 535; Chasles' proposition 537 
 
 Conical pendulum, Appendix/" 
 
 Conical surface 480 
 
 Conservation of energy 250 
 
 Conservative system 243 
 
 Constancy of a balance 384 
 
 Constraint of a point 165, of a body 
 167 ; one degree of constraint of the 
 most general character 170 
 
 Contrary forces 555 note 
 
 Continuity, equation of 16-2 
 
 Conversion of units : — pounds per sq. 
 inch to grammes per sq. centimetre 
 661 ; other units 362—366 
 
 Co-ordinates 452; propositions in co- 
 ordinate , geometry 459 
 
 Cord round cylinder 592, 603 
 
 Corresponding points in confocal ellip- 
 soids 535 
 
 Cosines, sum of the squares of the direc- 
 tion, of a line, equal to unity 460 
 
 Couple 201, axis of 201, moment of 
 201, direction of 560 
 
 — composition of in same or parallel 
 planes 561; any number 564; any 
 number not in parallel planes 566 ; 
 conditions of equilibrium of 567; 
 and a force, composition of 568 et seq. 
 
 Curvature of a plane curve 9 ; integral 
 14; average 14; of a surface 120; of 
 oblique sections, Meunier's Theorem, 
 121; principal, Euler's Theorem 122 
 
 Curvature of a lens, how to measure 381 
 
 Curve, plane 11; tortuous 13; of double 
 curvature i r ; continuous 35 ; closed 
 
 443 
 
 Curves use of, in representing experi- 
 mental results 347 
 
 Cycloid 6()^ 103; properties of 104; 
 prolate 103; curtate 103 
 
 D'Alembert's Principle 230 
 
 Day, Sidereal and Mean Solar 357 
 
 Degrees of freedom and constraint 165, 
 of a point 165, of a body 167; one 
 degree of freedom of most general 
 character 170 
 
 Density 1 74 ; linear, surface, volume, 
 477 ; mean density of the e&rth ex- 
 pressed in attraction units 715 
 
 Developable surface 125; practical con- 
 struction of a, from its edge 133 
 
 Diagonal scale 372 
 
 Direction of motion 8 
 
 Direction of rotation, positive 455 
 
 Direction cosine 463 ; sum of squares 
 of, equal to unity 4*60 ; of the common 
 perpendicular to two lines 464 
 
 Displacement of a plane figure in its 
 
 plane 91, examples 96; of a rigid 
 solid 100 
 
 Dissipation of Energy, instances 247, 
 292, 683 
 
 Dissipative systems 292 
 
 Distortion, places of maximum, in a 
 cylinder 678 
 
 Distribution of the weight of a solid on 
 points supporting it 626 
 
 Double-v^'eighing 384 
 
 Duhamel's application of Ivory's theo- 
 rem 541 
 
 Dynamics i 
 
 Dynamometer 389, Morin's 389 
 
 Edge of regression 132 
 Elastic body, a perfectly 643 
 
 — centre of a section of a wire 608; 
 line of elastic centres, 608; rotation 
 of a wire about 628 
 
 — curve transmitting force and couple 
 619, properties of 620; Kirchoffs ki- 
 netic comparison, common pendulum 
 and elastic curve 620 
 
 — solid equilibrium of 667 
 
 — wire or fibre 605 
 
 Elasticity, co-efficient of 265 note', of 
 volume 65 1 ; of figure ^i, i 
 
 Electric images 528; definition 530; 
 transformation by reciprocal radius 
 vectors 531; electric image of a 
 straight line, an angle, a circle, a 
 sphere, a plane 531 ; application to the 
 potential 532; of any distribution of 
 attracting matter on a spherical shell 
 533; uniform shell eccentrically re- 
 flected 533 ; uniform solid sphere 
 eccentricrtliy reflected 534 
 
 Elemeiits of a force 184 
 
 Ellipse, how to draw an 19 
 
 Ellipsoid, central 237 
 
 Ellipsoid, attraction of a, 535 ; corre- 
 sponding points on two 535 ; Ellips- 
 oidal shell defined 535 ; attractibri of 
 homogeneous ellipsoidal shell on in- 
 ternal point 536; Potential constant 
 inside 536; Chasles' Proposition con- 
 cerning 537 ; equipotential surfaces of 
 ^ 537; Maclaurin's Theorem 539; 
 Ivory's 540; comparison of the po- 
 tentials of two 537 
 
 Ellipsoid, Strain 141; principal axes 
 of 142 
 
 Empirical formulae, use of 350 
 
 Energy, kinetic 179; kinetic energy of 
 a system 234; energy in abstract 
 dynamics 241, 251; foundation of 
 the theory of energy 244; potential 
 energy of a conservative system 245 ; 
 conservation of E. 250; inevitable loss 
 
INDEX. 
 
 289 
 
 of energy of visible motion 247 ; po- 
 tential energy of a perfectly elastic 
 body strained 644 ; energy of a 
 strained isotropic substance 6^6 
 
 Epicycloid, integral curvature of 14, 
 motion in 105 
 
 Epoch in simple harmonic motion 71 
 
 Equation of continuity 162 ; integral and 
 differential 163 
 
 Equations of motion of any system 258 
 
 Equilibrium of z. particle, conditions of 
 408, 470, on smooth and rough curves 
 and surfaces 473; conditions of equi- 
 librium offerees acting at a point 470 ; 
 conditions of equilibrium oi three forces 
 acting at a point 584; graphic test 
 o{ forces in equilibrium 414; condi- 
 tions for stable equilibrium of a body 
 585, rocking stones 586, body move- 
 able about an axis 587, body on a 
 fixed surface 588; neutral, stable, and 
 unstable equilibrium, tested by the 
 principle of virtual velocities 256, 
 energy criterion of 257; conditions 
 of equilibrium of parallel forces 558; 
 conditions of equilibrium of forces 
 acting on a rigid body 576; equi- 
 librium of a non-rigid body not af- 
 fected by additional fixtures 584, of 
 a flexible and inextensible cord 594 ; 
 position of equilibrium of a flexible 
 string on a smooth surface 601, rough 
 surfaces 602 ; equilibrium of elastic 
 solid 667, of incompressible fluid com- 
 pletely filling rigid vessel 696, under 
 any system of forces 697 ; equilibrium 
 of a floating body 704 et seq., of a 
 revolving mass of fluid 710 
 
 Equipotential surfaces, examples of 499, 
 505, 526, of ellipsoidal shell 537 
 
 Equivalent of pounds per square inch 
 in grammes per square centimetre 
 661 ; other units 362 — 366. 
 
 Experience 320 
 
 Experiment and observation 324; rules 
 for the conduct of experiment 325 ; use 
 of empirical formulae in exhibiting 
 results of experiment 347 
 
 Euler's theorem on curvature 122, on 
 Impact 276 
 
 E volute 20, 22 
 
 Flexible and inextensible line, Kine- 
 matics of a t6; flexible and inexten- 
 sible surface, flexure of 125, general 
 property of 134; flexible string on 
 smooth surface, position of equili- 
 brium of 601, on rough surface 602 
 
 Flexure of flexible and inextensible sur- 
 face 125, of a wire 605 ; laws of flexure 
 
 and torsion 607 ; axes of pure flexure 
 609 ; case in which the elastic central 
 line is a normal axis of torsion 609 ; 
 where equal flexibility in all directions 
 610; wire strained to any given spiral 
 and twist 612; spiral spring 014; 
 principal axes of 679 ; distortion of the 
 cross section of a bent rod 679 
 
 Floating bodies, stable equilibrium of, 
 lemma 704 ; stability of 705 et seq. ; 
 see Fluid 
 
 Fluid, properties of perfect 401, 684; 
 fluid pressure 685, equal in all di- 
 rections 686, proved by energy cri- 
 terion 689 ; fluid pressure as depending 
 on external forces 690; surfaces of 
 equal pressure are perpendicular to 
 lines of force 691, are surfaces of 
 equal density and equal potential 692 ; 
 rate of increase of pressure 694, in 
 a calm atmosphere of uniform tem- 
 perature 695 (free surface in open 
 vessel is level 696) ; resultant pres- 
 sure on a plane area 702 ; moment of 
 pressure 702 ; loss of apparent weight 
 by immersion 703 ; conditions of equi- 
 librium of a fluid completely filling a 
 closed vessel 696, under non-con- 
 servative system of forces 697, im- 
 aginary example 699, actual case 
 701 ; equilibrium of a floating body, 
 lemma 704, stability 705, work done 
 in a displacement 705, metacentre, 
 condition of its existence 709; oblate 
 spheroid is a figure of equilibrium of 
 a rotating incompressible homoge- 
 neous fluid mass 711; relation be- 
 tween angular velocity of rotation 
 and density with given ellipticity 712 ; 
 table of eccentricities and correspond- 
 ing angular velocities and moments 
 of momentum for a liquid of the 
 earth's mean density 717; equilibrium 
 of rotating ellipsoid of three unequal 
 axes 719 
 
 Fluxion 28 
 
 Forbes's use of Viscous in connection 
 with glacier motion 683 
 
 Force, moment of 46, about a point 
 199, source of the idea of 173, de- 
 fined 183, specification of a 184, 
 measure of a 185, measurement of 
 224, by pendulum 387 ; fo7-ce of grav- 
 ity, Clairault's formula for 187, in 
 absolute units 187, average, in Britain 
 191; unit of force, gravitation 185, 
 absolute 188; British absolute imit 
 191; attraction unit of force 476; 
 representation of forces by lines 192 ; 
 component of force 193 ; composition 
 
 19 
 
290 
 
 INDEX. 
 
 of forces ii\, 418, parallelogram and 
 polygon of -2 22, true polygon of 416, 
 triangle of 410; forces through one 
 point, resultant of 412 ; forces in equi- 
 librium, graphic test of 414; resultant 
 of three forces acting at a point 465 ; 
 resolution of force along three speci- 
 fied lines 468 ; resultant of any number 
 of forces acting at a point 469 ; con- 
 ditions of equilibrium of forces acting 
 at a point 470; resultant of forces 
 whose lines meet 552 ; two parallel 
 forces in a plane 554, in dissimilar 
 directions 555, of any number 556, 
 not in one plane 557, conditions of 
 equilibrium of 558 et seq. ; forces and 
 a couple 568 ; forces may be reduced 
 to one force and one couple 570 ; re- 
 duction of forces to simplest system 
 571; parallel forces whose algebraic 
 sum is zero exert a directive action 
 only 583; conditions of equilibrium 
 of three, acting on a rigid body 584 ; 
 conservative system of 243 
 
 Force in terms of the potential 504 ; at 
 any point, due to attraction of a 
 spherical distribution of matter, a 
 cylindrical distribution, or a distri- 
 bution in infinite planes 508, where 
 it varies as Z>~^ 509 
 
 Force of gravity, Clairault's formula for 
 187, in absolute units 187, average 
 value in Britain r9i, in Edinburgh 
 191, law of 475 
 
 Force, line of, definition 507, instance 
 of 499 ; variation of intensity along 
 a 508 
 
 Force, tube of 508 
 
 Form of equilibrium of a rotating mass 
 of fluid 711 — 719 
 
 Formulae, use of empirical, in exhibit- 
 ing results of experiment 347 
 
 Foucault's pendulum 87 
 
 Fourier's theorem 88 
 
 Freedom of a point, degrees of 165; of 
 a rigid system 167 
 
 Friction brake, White's 390 
 
 Friction, laws of statical 403, kinetic 
 404 ; effect of tidal friction 248 ; fric- 
 tion of liquids varies as the velocity 
 292 ; friction of solids 293 ; of a cord 
 round a cylinder 592, 603 
 
 Gauss's absolute unit of force i88j 
 
 theorem relating to potential 5r5 
 Geodetic line 124, properties of 601 
 Glacier Motion, Forbes's Viscous The- 
 ory of, meaning of Viscous in 683 
 Gravitation, law of 475 ; potential 50:5 
 Gravity, force of, Clairault's formu'la 
 
 for 187, at Edinburgh, in Britain 
 191, in absolute units 187, work 
 done against 509 
 
 Gravity, centre of, and centre of inertia 
 I95> 582 ; centre of gravity 542 ; pro- 
 perties of a body possessing a centre of 
 gravity 544 ; centre of gravity where 
 it exists coincides with centre of inertia 
 550; position of centre of gravity in 
 a body for stable equilibrium 585, in 
 rocking stones 586, in a body with 
 one point fixed: with two points 
 fixed 587, on a surface 588; Pappus' 
 theorem concerning, sometimes called 
 Guldinus' theorem 589 
 
 Green 501; problem in potential 517; 
 the general problem of electric in- 
 fluence possible and determinate 521 
 
 Gyration, radius of 235 
 
 Gyroscopes, motion of 116 . 
 
 Hamilton's Characteristic Function 283 
 
 Harmonic motion 69 ; simple harmonic 
 motion 70, amplitude, argument, 
 epoch, period, phase, 71, instances of 
 72; velocity in simple harmonic mo- 
 tion 73, cuceleration in 74 ; composi- 
 tion of two simple harmonic motions 
 in one line 75, examples 77 ; graphical 
 representation of simple harmonic 
 motion in one line 79; cotnposition 
 of simple harmonic motion in different 
 directions 80; of different kinds in 
 different directions 84; in two rect- 
 angular directions 85 
 
 Harton coal mine experiment 498 
 
 Hodograph 49, of a planet or comet 
 49, of a projectile 50, of motion in 
 a conic section 51, of path where 
 acceleration is directed to a fixed 
 point and varies as D'"^ 6r 
 
 Homogeneous atmosphere defined 695 ; 
 height of 694, at Paris, at Edinburgh 
 695. 
 
 Homogeneous body 646 
 
 Homogeneous strain 135; see Strain 
 
 Horsepower 240 
 
 Hydrodynamics 683 ; see Fluid 
 
 Hydrostatics 685 ; see Fluid 
 
 Hyperbola, how to draw a 19 
 
 Hypocycloid and hypotrochoid 105 
 
 Hypothesis, use of 332 
 
 Image, electric 528; see Electric im- 
 ages 
 
 Impact 259, duration of 259 ; time in- 
 tegral 262 ; ballistic pendulum 263 ; 
 direct impact of spheres, Newton's 
 experiments on 265, loss of kinetic 
 energy in 266, due to 267, case with 
 
INDEX, 
 
 291 
 
 no loss of kinetic energy 268 ; mo- 
 ment of an impact 272; work done 
 by 273; Euler's theorem 276 
 
 Impressed force 183; see Force 
 
 Inclination of two given lines in terms 
 of their direction cosines 464 
 
 Inertia 182 
 
 Inertia, centre of 195; see Centre of 
 Mass 
 
 Inertia, moment of 198, 235 — 239, of 
 a centrobaric body 551 
 
 Inextensible line 16, surface 125, gen- 
 eral property of 134 
 
 Instability of motion 300; instances 
 302, 303, 304 
 
 Interpolation in physical experiments 
 
 350 
 
 Involute 20 
 
 Isotropic substance 647 
 
 Isotropy, conditions fulfilled in elastic 
 650, in one quality and aeolotropy 
 in others 648 
 
 Ivory's theorem on homogeneous con- 
 focal ellipsoids 540 
 
 Kepler's first law a consequence of ac- 
 celeration directed to a fixed point 45 
 
 Kilogramme 365 
 
 Kinematics 4, of a point 7, of an in- 
 extensible and flexible line 16, of a 
 plane figure 91, flexible and inexten- 
 sible surface 127 
 
 Kinetic energy 179, rate of change of 
 180; gain in kinetic energy equiva- 
 lent to work done 207 ; kinetic energy 
 of a system 234; loss of kinetic en- 
 ergy in direct impact 266 
 
 Kinetic foci 310 — 319, number of, in 
 any case 316 
 
 Kinetic friction 404 
 
 Kinetics 2, 3, 4 
 
 Kinetic stability 300; kinetic stability 
 or instability discriminated 301 ; cases 
 of kinetic stability 302, 303, 304; 
 kinetic stability in a circular orbit 
 304; oscillatory kinetic stability 308; 
 general criterion of kinetic stability 
 309 ; motion on anticlastic surfaces is 
 unstable, synclastic stable 309 
 
 Kinetic symmetry 239 
 
 Kirchoff's kinetic comparison between 
 twisting a wire and the motion of a 
 pendulum 620 
 
 Latitude altered by attraction of a 
 mountain, or hemispherical hill, or 
 cavity 496, by a crevasse 497 
 Laws of energy, dynamical, 252 
 Laws of friction 403 ; see Friction 
 Laws of motion, history of 208 j first 
 
 law 210, second 217, third 227, 
 
 Scholium 229, 241 
 Least action 2 79 
 Least squares, method of 340 
 Lemniscate, integral curvature of 14 
 Lengthening of a spiral spring due to 
 
 torsion 618 
 Level surface 505 
 
 Limitation of dynamical problems 391 
 Line density 477 
 Line, expression for a, in co-ordinates 
 
 .459 
 
 Line of elastic centres remains un- 
 changed in length 608 ; see Elastic 
 
 Line of force def. 507, instances of 499, 
 variation of intensity along a 508 
 
 Line, orthogonal projection of a 442 
 
 Liquid, effective moment of inertia of 
 675, note 
 
 Locus of centre of curvature 22 
 
 Longitudinal vibrations, velocity of 
 transmission of 658 
 
 Longitudinal rigidity 657 
 
 Loss of weight of body immersed in 
 fluid 703; see Fhdd 
 
 Lunar tides 77 
 
 Machines, science of i 
 
 Maclaurin's theorem on homogeneous 
 confocal ellipsoids 539 
 
 Mass 174; measurement of 175, 224, 
 unit of 190, 365, 476, 715, British 
 unit of 190; mass v. weight 17 = , 
 186 
 
 — centre of 196; see Centre of Mass 
 
 Matter 173 
 
 Maximum action 317 
 
 Mean angular velocity 58 
 
 Mean density 174; of Earth, Sche- 
 hallien experiment 496, Harton coal 
 mine experiment 498 
 
 Mean solar day 357 
 
 Measure of time 358, 371, of length 
 360, of surface 363, of volume 364, 
 of mass 365, of force 366, of work 
 366, of angles 357, of pressure 661 
 
 Measurement of force 185, 224, of 
 masses 224 
 
 Mechanical powers 591; balance 592 
 
 Mechanics i 
 
 Mechanism 4 
 
 Metacentre 709 ; conditions for its ex- 
 istence ; see Fluid 
 
 Method of least squares 340 
 
 Method of representing experimental 
 results 347 
 
 Metre 362 
 
 Meunier's theorem on curvature 121 
 
 Micrometer 379 
 
 Minimum action 311; two or more 
 
292 
 
 INDEX. 
 
 courses of minimum action possible 
 
 Modulus of elasticity, Young's 657 ; 
 weight and length of modulus 658; 
 specific modulus of isotropic body 
 
 Moment about a pomt, of a velocity or 
 a force 46, representation of 199, of 
 a couple 201, of an impact 272, of 
 pressure 702 
 
 Moment of inertia 198, 235, of a cen- 
 trobaric body 551 
 
 Moment of momentum 202, of a rigid 
 body 232 
 
 Momentum 176, change of 177, accele- 
 ration of 178 
 
 Motion of a material particle 7; rela- 
 tive motion ()i\ simple harmonic mo- 
 tion 69; of troops on suspension 
 bridge 78 ; of point of vibrating string 
 79 ; of a plane figure in its own plane 
 19 ; of a rigid body about a fixed point 
 106 ; general motion of a rigid body 
 112; of a screw in its nut 113; 
 quantity of motion 176; Newton's 
 laws of motion 208, see Laws ; re- 
 sistances to motion 247 ; motion in 
 a resisting medium 292, in a logarith- 
 mic spiral 295 ; of a system slightly 
 disturbed from a position of equilib- 
 rium 290 
 
 Neap tides 77 
 
 Neutral equilibrium 256; of floating 
 
 bodies 705 
 Newton's laws of motion 208, seeZazfj; 
 
 experiments on impact 265 
 Non-conservative system 298 
 Normal 22 
 Normal attraction over a closed surface, 
 
 integral of 510 
 
 Oblique coordinates 453 
 Observation and experiment 320 
 Opposite cones 481 
 Opposite forces 5^5, note 
 Ordinates 452 
 Orthogonal projection 442 
 Oscillation in U tube, Appendix k 
 
 Parallel forces in a plane, resultant of 
 two 554, in dissimilar directions 555, 
 of any number 556, not in one plane 
 557, equilibrium of 558 
 
 Parallelogram of velocities 31, of forces 
 219 
 
 Particle material v. geometrical point 
 7, 181 
 
 Pendulum, Robins' ballistic 263, 272 ; 
 pendulum as a measurer of force 387 ; 
 
 simple pendulum Appendix (a) ; com- 
 pound pendulum Appendix \g) 
 
 Perfect fluid 401, 684 
 
 Perfect solid, ideal 656 
 
 Perfectly elastic body 643; potential 
 energy of perfectly elastic body held 
 strained 644 
 
 Period of simple harmonic motion 7 1 
 
 Periodic disturbance 306 
 
 Periodic function, Fourier's theprem 
 regarding 88 
 
 Perpetual motion the, is impossible 244 
 
 Phase of simple harmonic motion 71 
 
 Physical axiom 209; concerning equi- 
 librium 584 
 
 Plane, osculating 1 2 ; motion of plane 
 figure in its own plane 91 
 
 Planet, path of 45 ; hodograph of 49, 51, 
 61 
 
 Plasticity 683 
 
 Polar coordinates 457, 459 
 
 Polygon plane ii, gauche 11, closed 
 443, of velocities 31, of forces 219 
 
 Potential 500; the mutual potential 
 energy of two bodies 502, at a point 
 503, force in terms of potential 504 ; 
 equipotential surface 505 ; potential 
 due to an attracting particle 509, to 
 any mass 509, potential cannot have 
 a maximum or minimum value at a 
 point in free space 511, cases of this 
 515, 516; has same value throughout 
 the interior as at the surface of a closed 
 space 513; mean value of potential 
 throughout a sphere equal to the value 
 at centre 514; Gauss's Theorem 515; 
 Green's problem 517; potential due 
 to a uniform spherical shell 514) 533 ; 
 how to distribute matter so as to get 
 a given potential 517 — 521 ; potential 
 due to uniform sphere 534; due to 
 ellipsoidal shell 536 
 
 Potential energy due to work done 207, 
 of a conservative system 245 ; the 
 mutual potential energy of two bodies 
 502, of elastic solid held strained 644 
 
 Precession 117 
 
 Precessional rotation 116 
 
 Pressure, centre of 688, 702 ; pressure 
 at a point in a fluid same in every 
 direction 685, 687, 689; surfaces of 
 equal pressure are level surfaces 691 j 
 whole pressure 709 
 
 Principal axes of a strain 144; st&Sirain 
 
 Principal axes of inertia 237 
 
 Probable error 343 
 
 Probable result from a number of obser- 
 vations, deduction of the 338; method 
 of least squares 340 ; practical appli- 
 cation 345 
 
INDEX. 
 
 293 
 
 Projectile, path of 44; hodograph 50 
 
 Projection of areas 200 ; of a point on a 
 straight line 437; orthogonal projec- 
 tion 442, of a shell 447, of any two 
 shells, of a closed surface 448, of 
 equal areas in parallel planes 449, of 
 a plane figure 449 
 
 Pulley, kinematics of 18 
 
 Pure strain 159; see Strain 
 
 Radius of curvature 9 
 
 Radius of g>Tation 235 
 
 Regression, edge of 132 
 
 Relative motion 63 et seq. ; acceleration 
 of 64 
 
 Repose, angle of 404, 473 
 
 Residual phenomena 328 
 
 Resistance to motion 247, 250; varying 
 as the velocity in fluids 293; to 
 change of shape, frictional 683 
 
 Resisting medium 247 
 
 Resolution of velocity 30, of forces 431, 
 geometrical solution 432, trigonomet- 
 rical solution 433, in directions at 
 right angles 434 ; application to find 
 the resultant of a number of forces 
 acting on a point 435 ; resolution of 
 forces alng three specified lines 468 
 
 Rest 211 
 
 Restitution, co-efficient of 265 
 
 Resultant velocity 31 ; resultant of forces 
 on a point 412, 419; three forces act- 
 ing on a point 465, any number 470 
 
 Revolving mass of fluid, equilibrium of 
 710; see Fluid 
 
 Rigid body, displacement of 90, motion 
 of 106, general motion of 112, rigid 
 body defined 393, 401 
 
 Rigidity 651; longitudinal 65 7: rigidity 
 and resistance to compression 655; 
 rigidity as depending on form 677 
 
 Rocking stones 586 
 
 Rolling of bodies 109; of curve upon 
 curve 100 
 
 Rolling motion 118, 119 
 
 Rope round cylinder 592, 603 
 
 Rotation, positive direction of 455 
 
 Rotations about parallel axes, compo- 
 sition of 98 ; composition of rotation 
 and translation in one plane 99 ; ro- 
 tations of a rigid body, composition of 
 106; successive finite rotations 109 
 
 Rotation of a wire round its elastic 
 central line 628 ; see Elastic 
 
 Schehallien experiment 496 
 Scholium to law ill 229, 241 
 Screw, motion of a, in its nut 113, 
 
 337 
 Sea mile 361 
 
 i4» 
 
 Section of a small cone, oblique 486 
 Sensibility and stability of a balance 384, 
 
 592 
 
 Shape, change of, involves dissipation 
 of energy 683, 247 
 
 Shear, simple 150, axes of a 152, 
 measure of a 153, combined with a 
 simple elongation and expansion 156 
 
 Shell def 446 
 
 Siderial day 358 
 
 Simple linear circuit 443 
 
 Simple harmonic motion 70, in me- 
 chanism 72, composition of, in one 
 line 75, examples 77, composition of, 
 in different directions 80, of diflerent 
 kinds in different directions 84, in two 
 rectangular directions 85 ; see Har^ 
 monic 
 
 Simple pendulum, Appendix {a) 
 
 Simple shear 150 ; axes of a shear 152 ; 
 ratio of a shear 153, amount of a 
 154; planes of no distortion in a 155 
 
 Solar system, ultimate tendency of the 
 249 
 
 Solar tides 77 
 
 Sohd angle 482; round a point 483; 
 subtended at a point 485 
 
 Solid, elastic 643, 651; potential energy 
 of elastic solid held strained 644 ; fun- 
 damental problems of the mathemati- 
 cal theory of the equilibrium of an 
 elastic solid 667; equations of internal 
 equilibrium of 668; St Venant's ap- 
 plication to torsion problems 669 ; 
 small bodies stronger than large ones 
 in proportion to their weight 682 ; 
 imperfectness of elasticity in solids 
 683 
 
 Space described under uniform accele- 
 ration in direction of motion 43 
 
 Space, British unit of 190 
 
 Specific modulus of elasticity 689 
 
 Sphere, attraction of, composed of con- 
 centric shells of uniform density 498 ; 
 attraction of uniform sphere and po- 
 tential due to 634 ; see Attraction and 
 Potential 
 
 Spherical shell, uniform, attraction on 
 internal point 479, external point 
 
 488, on an element of the surface 
 
 489, potential due to 533; see At- 
 traction and Potential 
 
 Spherometer 380 
 
 Spinning motion 118 
 
 Spiral, motion in logarithmic 295, 296 
 
 Spiral springs 386, as measurers of force 
 386,614, kinetic energy of 616, length- 
 ening of, due to torsion 6i8 
 
 Spring balance; see Spiral springs 
 
 Spring tides 77 
 
294 
 
 INDEX. 
 
 Stable equilibrium 256, 257; see Centre 
 of Gravity and Floating Bodies 
 
 Stability of motion 300 
 
 Static friction 404 
 
 Statical problems, examples of 591; 
 balance 592; rod with frictionless 
 constraint 592 ; rod constrained by 
 rough surfaces 592 ; block on rough 
 plane 592 ; mass supported by rings 
 round rough post 592; cord wound 
 round cylinder 592 
 
 Statics 2, 3, of a particle 408, of a 
 rigid body, 552 
 
 Stationary action 281 
 
 Straight beam infinitely little bent 
 623 
 
 Strain 135 ; homogeneous strain 136 ; 
 properties of homogeneous strain 137 ; 
 strain ellipsoid 141 ; axes of strain 
 ellipsoid 144 ; elongation and change 
 of direction of any line of a body in 
 condition of strain 145 ; distortion 
 in parallel planes without change of 
 volume 148 ; simple shear 150 ; axes 
 of a shear 152, ratio of a 153, amount 
 of a 154, planes of no distortion in 
 a 155, is a simple elongation and ex- 
 pansion combined with a shear 156; 
 analysis of strain 157 ; pure strain 
 159; composition of pure 160 
 
 Stress 629, homogeneous 630, specifi- 
 cation of a 632, components of a 
 633, simple longitudinal and shearing 
 stress 633 ; stress quadratic 634 ; 
 normal planes and axes of a stress 
 quadratic 635 ; varieties of stress 
 quadratic 636 ; laws of strain and 
 stress compared 639; rectangular ele- 
 ments of strain and stress 640 ; work 
 done by a strain 641 ; a physical ap- 
 plication 642 ; stress produced by a 
 single longitudinal stress 653 ; ratio 
 of lateral contraction to longitudinal 
 extension different for different sub- 
 stances 655 ; stress required to pro- 
 duce a simple longitudinal strain 663 ; 
 stress components in terms of strain 
 for isotropic body 664 ; strain compo- 
 nents in terms of stress 665 ; funda- 
 mental problems in mathematical 
 theory of equilibrium of elastic solid 
 667 ; equation of energy of isotropic 
 body 666 ; equations of internal 
 equilibrium 668 ; comparative strain 
 of similar bodies as depending on 
 dimensions 682 
 
 St Venant on torsion of prisms 669 ; 
 see Torsion 
 
 Surface density 477 
 
 Surface of equilibrium ^05 ; relative in- 
 
 tensity of force at different points of 
 
 a 506 
 Surfaces of equal pressure in a fluid at 
 
 rest are also surfaces of equal potential 
 
 and equal density 692 
 Symmetry, kinetic 239 
 Symmetrical co-ordinates 458, 459 
 Synclastic surface 120 
 System, conservative 243 
 — non-conservative 298 
 
 Tidal friction 247, effect of Tides in 
 lengthening the period of the Earth's 
 rotation about her axis 248, 249 
 
 Tides 77 
 
 Time, unit of 190, measurement of 213, 
 
 .358, 371 
 
 Time integral 262 
 
 Time of rotation of the earth round its 
 axis increased by friction 248 
 
 Time of oscillation of fluid in a U tube 
 Appendix k', of a simple pendulum 
 Appendix b, c, d, e, compound pen- 
 dulum Appendix g; wave running 
 along a stretched cord Appendix h 
 
 Tops, motion of spinning 118 
 
 Torsion, laws of 607 
 
 Torsion balance 383 
 
 Torsion of a wire 605 ; laws of 607 
 
 Torsion of prisms, St Venant on 669, 
 lemma 670; torsion of circular cy- 
 linder 671 ; prism of any shape 672, 
 623 ; hydrokinetic analogue 675 ; con- 
 tour lines for normal sections of 
 prisms &c. under torsion ; elliptic 
 cylinder, equilateral triangular prism, 
 curvilinear square prisms, square 
 prisms: bars elliptic, square, flat, 
 rectangular 676 ; relation of tor- 
 sional rigidity to flexural rigidity 
 677; ratio of torsional rigidity to 
 those of circular rods of same mo- 
 ment of inertia, or of sariie quantity 
 of material 677; places of greatest 
 distortion in twisted prisms 678 
 
 Tortuosity 11 
 
 Tortuous curve I r, 13 
 
 Transformation electrical, by reciprocal 
 radius vectors 531 
 
 Transmission of force through elastic 
 solid 629 ; transmission of homo- 
 geneous stress 630 ; force trans- 
 mitted across any surface in elastic 
 solid 631 
 
 Triangle of forces 410, equivalent to a 
 couple 411 
 
 Triangle of velocities 3 1 
 
 Trochoid 103 
 
 Tubes of force 508 
 
 Turning, positive direction of 455 
 
INDEX. 
 
 295 
 
 Uniform acceleration 36, 43; space 
 
 described 43 
 Uniform circular motion, acceleration 
 
 in 37; composition of two 86 
 Uniform velocity 23 
 Unit angle n ; of angular velocity 55 ; 
 
 — of angular measure 357 
 
 — of cubic measure 364 
 
 — of force 188, 366, 476 
 
 — length 360 — 362 
 
 — mass, space, time 190, 565 
 
 — work (s9ientific) 204, gravitation 204 
 
 — surface 363 
 
 Units, tables for conversion of 362 — 366, 
 
 66r 
 Unstable equilibrium 256, 257; see 
 
 Centre of Gravity and Fluid 
 
 Varying action 2 79 ; optical illustration 
 286 ; a criterion for kinetic stability 
 
 309 
 
 Velocities, parallelogram, triangle, poly- 
 gon of 31 ; examples of velocities 41 
 
 Velocity 23. uniform 23, variable 26, 
 component 29, resolution of 29, 
 resultant 31, moment of 46, angular 
 54, relative d^, change of 177, 
 virtual 203, 254 
 
 Velocity of a planet at any point of its 
 orbit 48 ; in simple harmonic motion 
 
 Velocity of escape of fluid from an ori- 
 fice Appendix g 
 
 — of longitudinal vibrations along a 
 rod 658 
 
 — of wave along stretched cord Ap- 
 pendix h 
 
 Venant (St) on torsion 669; see Tor- 
 sion 
 
 Vernier 373 
 
 \'ertical cones 481 
 
 Vibrations produced by impact 220, 
 269; in a resisting medium 293; 
 along stretched cord Appendix h, 
 velocity of transmission of, through a 
 rod 658 
 
 Virtual velocity 203, 254, moment of 
 .203 
 
 Viscosity of solids 683 ; of fluids 683 
 
 Vis viva 179 
 
 Volume, change of involves dissipation 
 of energy 683 
 
 Volume, density 477, 715 
 
 \'olume, elasticity of 651 
 
 Weber's electrical theory 336 
 
 Weight V. mass 175 ; a measure of mass 
 175, 186 
 
 White's friction brake 390 
 
 Whole pressure on a submerged surface 
 702 
 
 Wire, flexure of a 622 ; see Flexure 
 
 Work 204, unit of 204, against force 
 varying inversely as square of distance 
 509, independent of path pursued 
 under conservative system of force 
 509, done in straining a perfectly 
 elastic body 644 ; transformations of 
 work 207 
 
 Yard 360 
 
 Young's modulus 657 
 
 CAMBEIOGK: FRINTEU by C. J. clay, M.A., AT THE UNIVERSITY PRF.SS. 
 
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