AS-TRONOJ*< --f^--' J Digitized by the Internet Arciiive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/elementarypartofOOroutrich THE ELEMENTARY PART OF A TREATISE ON THE DYNAMICS OF A SYSTEM OF EIGID BODIES. BEING PAKT I. OF A TREATISE ON THE WHOLE SUBJECT. THE ELEMENTARY PART OF A TEEATISE ON THE DYNAMICS OF A SYSTEM OF EIGID BODIES. BEING PART I. OF A TREATISE ON THE WHOLE SUBJECT. BY EDWARD JOHN ROUTH, Sc.D., LL.D., F.R.S., &c. HON. FELLOW OF PETERH0U8E, CAMBRIDGE ; FELLOW OF THE SENATE OP THE UNIVERSITY OF LONDON. SEVENTH EDITION, REVISED AND ENLARGED. Hontron : MACMILLAN AND CO., Limited NEW YOEK: THE MACMILLAN COMPANY 1905 \All Rights Reserved.} ^^^^-^^^ ^.l<^ ASTRONOMY LIBRAR); First Edition, 1860. Second Edition, 1868. Third Edition, 1877. Fourth Edition, 1882. Fifth Edition, 1891. Sixth Edition, 1897. Seventh Edition, 1905. ^TT^tJiA^ey^ 19 OS .. tlBRARY PREFACE. rjlHE opportunity of a new edition has enabled the author -*- to make numerous additions to both the volumes of this treatise. To make room for these some less important matter has been omitted. Many of these additions have already appeared in the German translation of this work and this is particularly the case with the additions made to the second volume. In the seven or eight years which have elapsed since the translation was published the progress of the science has not been slow. Much new matter therefore has been introduced into both the volumes and this has been arranged either as new theorems or as examples according to their importance. The dynamical principles of the subject are given in this volume together with the more elementary applications, while the more difficult theories and problems appear in the second. Sometimes one case of a problem supplies an example sufficiently elementary to appear in this volume while the general theory is given in the next. For example, the small oscillations of a vertical top and the motion of a sphere on a rough plane are partly discussed here, but they are more fully treated of in the second volume. In order that the plan of the book may be understood, a short summary of the next volume has been added to the table of contents. Each chapter has been made as far as possible complete in itself This arrangement is convenient for those who are already acquainted with dynamics, as it enables them to direct their attention to those parts in which they may feel most interested. It also enables the student to select his own order of reading. Vlll PREFACE. The student who is just beginning dynamics may not wish to be delayed by a chapter of preliminary analysis before he enters on the real subject of the book. He may therefore begin with D'Alembert's Principle and read only those parts of chapter I. to which reference is made. Others may wish to pass on as soon as possible to the principles of Angular Momentum and Vis Viva. Though a different order may be found advisable for some readers, I have ventured to indicate a list of Articles to which those who are beginning dynamics should first turn their attention. As in the previous editions a chapter has been devoted to the discussion of Motion in Two Dimensions. This course has been adopted because it seemed expedient to separate the difficulties of dynamics from those of solid geometry. Throughout each chapter there will be found numerous ex- amples, many being very easy, while others are intended for the more advanced student. In order to obtain as great a variety of problems as possible, a collection has been added at the end of each chapter, taken from the Examination Papers which have been set in the University and in the Colleges. As these problems have been constructed by many different examiners, it is hoped that this selection will enable the student to acquire facility in solving all kinds of dynamical problems. There are many useful instruments and important experimental researches whose theories require only a knowledge of dynamics and which can be easily understood without any long or intricate description. It will be seen that many of these have been selected as useful examples. Historical sketches have been attempted whenever they could be briefly given. Such notices, if not carried too far, add greatly to the interest of the subject. It is chiefly with the memoirs written since the early part of the last century that we are here concerned, and the number of these is so great that anything more than a slight notice of some of them is impossible. PREFACE. IX A useful theorem is many times discovered and probably each itime with variations. It is thus often difficult to ascertain who is I the real author. It has therefore been found necessary to correct [some of the references given in the former editions and to add references where there were none before. The use of dots and accents for differential coefficients with regard to the time has been continued whenever a short notation was desirable. One objection to this notation is that the mean- ing of the symbol may be greatly changed by a slight error in the number of the dots or accents. As this might increase the difficulties of the subject to a beginner, the use of dots in the earlier chapters has been restricted chiefly to the working of examples, and care has been taken that the results should be clearly stated. EDWARD J. ROUTH. Peterhouse, August 1905. CONTENTS. CHAPTER I. ON MOMENTS OF INERTIA. ABTS. 1- - 2. 3- - 9 10- -11 12- -14 15- -17 18. 19- -32. 33- -39. 40- -44. 45. 46. 48- -51. 52- -55. 56- -59 60- -61. 62- -65. PAGES On finding Moments of Inertia by integration ... 1 Definitions, elementary propositions and reference table . 2 — 8 Method of Differentiation 8 — 9 Theorem of Parallel Axes 9—12 Theorem of the Six Constants of a Body .... 12 — 14 Method of Transformation of Axes 14 — 15 Ellipsoids of Inertia, Invariants, &c 15 — 22 Equimomental Bodies, Triangle, Tetrahedron, &c. See Note, page 423 22—26 Theory of Projections. See Note, pages 423 — 4 . . . 26 — 29 Moments with higher powers. See Note, page 425 . . 29 — 30 Theory of Inversion 30 — 31 Centre of Pressure, &c. 31 — 33 Principal Axes 33 — 35 Foci of Inertia 35—38 Arrangement of Principal Axes ...... 38 — 40 Condition that a Line should be a Principal Axis . . 40 — 41 Locus of Equal Moments, Equimomental Surface, &c. . 41 — 44 CHAPTER II. d'alembert's principle, &c. 66 — 78. D'Alembert's Principle and the Equations of Motion 79 — 82. Independence of Translation and Kotation. 83. General method of using D'Alembert's Principle 84 — 87. Impulsive Forces Examples 45—55 55—58 58—59 59—62 62—63 Xll CONTENTS. CHAPTER III. MOTION ABOUT A FIXED AXIS, ARTS. 88—91. The Fundamental Theorem 92—93. The Pendulum and the Centre of Oscillation 94_96. Change of temperature and of the buoyancy of the air 97. Moments of Inertia found by experiment .... 98—105. Length of the Seconds Pendulum with correction for resistance of the air 106—107. Construction of a Pendulum 108. The Pendulum as a Standard of Length .... 109— ir)&. Oscillation of a watch balance 110—1. J. Pressures on the fixed Axis. Bodies symmetrical and not symmetrical. Impulses 114. Analysis of results. Examples 115—116. Dynamical and Geometrical Similarity .... 117 — 119. Permanent Axes of Eotation, Initial Axes .... 120. The Centre of Percussion 121—125. The Balhstic Pendulum 126—129. The Anemometer 82— s(; 86—^11 89— ••<) 90-1' .2 92— '.i;i 93—117 97- '.''.I CHAPTER IV. MOTION IN TWO DIMENSIONS. 130 — 133. General methods of forming the Equations of Motion 134. Angular Momentum 135—138. Method of Solution by Differentiation 139—143. Vis Viva, Force Function and Work . 144 — 148. Examples of Solution 149. Characteristics of a Body 150 — 152. Stress at any point of a Eod .... 153—157. Laws of Friction 158 — 160. Discontinuity of Friction, and Indeterminate Motion 161 — 163. A Sphere on an imperfectly rough plane 164. Friction Couples 165 — 166. Friction of a carriage and other examples . 167. Kigidity of cords 168 — 169. Impulsive Forces, General Principles . . . 170 — 175. Examples of sudden changes of motion, reel, sphere column, &c. Work of an impulse. Earthquakes 176—178. Impact of Compound Inelastic Bodies, &c. 179—180. Impact of Smooth Elastic Bodies. See Art. 404 181 — 198. The general problem of the Impact of two Bodies, smooth or rough, elastic or inelastic. The representative point 199—202. Initial Motions. Also Examples .... 203—213. Eelative Motion and Moving Axes .... Examples disc CONTENTS. Xlll CHAPTER V. MOTION IN THREE DIMENSIONS. ARTS. PAGES !U — 228. Translation and Kotation. Base Point, Central Axis . 184 — 190 )-2;38— 239. The Velocity of any Point 195—197 240—247. Composition of Screws, &c 197—204 248 — 259. Moving Axes and Euler's Equations 204 — 210 260. The Centrifugal Forces of a Body 210—212 2(31 — 267. Angular Momentum with Fixed or Moving Axes . . 212 — 218 268—270. Examples of Top and Sphere. See Vol. II. . . . 218—222 271 — 281. Finite Rotations. Theorems of Rodrigues and Sylvester. -^ Screws, &c ^ .;2— 229 CHAPTER YI. ON MOMENTUM. 282—283. Fundamental Theorem 230—232 284 — 286 h. Attracting particles. Lagrange. Laplace. Jacobi . 232 — 237 287. Living Things 237—239 288—298. Sudden fixtures and changes 239—245 299. Gradual changes 245—246 300. Motion of a string, &c. See Vol. II 246—248 301—305. The Invariable Plane 248-254 306 — 314. Impulsive forces in three dimensions 254 — 259 315 — 331. The general problem of the Impact of two Bodies in three dimensions, the bodies being smooth or rough, elastic or inelastic. The representative point .... 259 — 268 Examples 269—271 CHAPTER VII. VIS VIVA. 332—341 342. 343. 344. 345. 346. 347. 348. 349. 350- 363- 865- 367- 371. -362. 366. 370. Force-function and Work . Work done by Gravity and Units of Work Work of an Elastic String . Work of Collecting a Body . Work of a Gaseous Pressure Work of an Impulse . Work of a Membrane . Work of a couple Work of Bending a Rod Principle of Vis Viva, Potential and Kinetic Energy Expressions for Vis Viva of a Body Theorems and Examples on Vis Viva Principle of Similitude. Models Fronde's Theorem 272—276 276—277 277—278 278—281 281—282 282-283 283 283—284 284 284—290 290—292 292—294 294—297 297—298 XIV CONTENTS. AKTS. PAGES 372. Savart's Theorem 298 373. Theory of dimensions 298—299 374— 374 a. Imaginary time 299—300 375—376, Clausius' theory of stationary motion. The Virial . . 300—302 377—381. Carnot's theorems 302—304 382— 386. The equation of Virtual Work applied to Impulses . . 304—306 387—388. Kelvin's theorem. Bertrand's theorem. Examples . . 306 — 309 389. Imperfectly elastic and rough bodies 309—310 390—394. Gauss' principle of Least Constraint 310—313 Examples 313 — 316 CHAPTER VIII. Lagrange's equation. Typical Equation for Finite Foi^ces. See Note, page 429 . 317—322 Indeterminate Multipliers . 322 — 323 Lagrange's equations for Impulsive Forces .... 323 — 327 Example on the equivalent pendulum .... 327 — 329 Euler's equations, &c 329 Vis Viva, Liouville's integrals and elliptic coordinates . 329 — 332 Examples on impulses ....... 332 — 333 The Keciprocal Function 333—336 Hamilton's equations . 336 — 338 Eeciprocal Theorems 338—339 The modified Lagrangian Function. Its use in forming Lagrange's and Hamilton's equations .... 339 — 342 Coordinates which appear only as velocities . . . 342 — 344 Non-conservative Forces 344—346 Systems not holonomous. Indeterminate coefficients. Appell's function S 346—353 431—431 h. Change of the independent variable t to r . . . 353—356 Examples 357 — 358 CHAPTER IX. SMALL OSCILLATIONS. 432 — 438. Oscillations with one degree of freedom .... 359 — 363 439—440. Moments about the Instantaneous Axis. See Art. 448 . 363—364 441 — 444, Oscillations of Cylinders, with the use of the circle of stability 364 — 367 445. Oscillations of a body guided by two curves . . . 367 — 368 446. Oscillation when the path of the Centre of Gravity is known 368 — 369 447. Oscillations deduced from Vis Viva 369 — 370 448. Moments about the Central Axis 370 — 371 449 — 452. Oscillations deduced from the ordinary equations of motion 371 — 374 453 — 462. Lagrange's method . 374 — 387 463 — 466. Initial motions 387 390 467 — 469. The energy test of stability 390—393 470 — 476. The Cavendish Experiment 393 399 Examples 400—401 395- -399. 400. 401- -404. 406. 407. 408. 409- -413. 414- -416 a. 417. 418- -421. 422- -425. 426- -428. 429- -430j. CONTENTS. XV CHAPTER X. ON SOME SPECIAL PROBLEMS. PAGES Oscillations of a rocking body in three dimensions . . 402 Relative indicatrix 402—403 Cylinder of stability and the time of oscillation . . . 403 — 405 Oscillations of rough cones rolling on each other to the first order of small quantities . 405 — 408 Large Tautochronous motions ...... 408 — 411 Effect of a resisting medium ...... 411 — 412 Eough cycloid, resisting medium 412 — 413 Historical summary ........ 413 Motion on any rough curve in a resisting medium k'v^, with any forces . . . . . . . . 413 — 415 Euler's theorem . 415 Time of motion 415 With central force Xr, resistance 2kv, the rough tautochrone IS p = ip. Discussion . 415 — 417 Conditions of stability and times of oscillations of rough cylinders to any order of small quantities . . . 417 — 420 Conditions of stability and times of oscillation of rough cones to any order 420 — 422 NOTES. 39. Moment of inertia of a tetrahedron in space of n dimensions 44. The four equimomental points of a body 45. Moments with higher powers 286. Steady motion of four attracting particles 399. The proof of Lagrange's Equations 410, 462. Historical Notes .... 423 423^424 425—426 426—429 429—430 430 The following subjects will be treated of in the second volume. Theory of moving axes, Clairaut's theorem, motion relative to the earth, and gyroscopes. Theory of small oscillations with several degrees of freedom both about a position of equilibrium and about a state of steady motion. Motion of a body about a fixed point under no forces. Motion of a body under any forces, top, sphere, solid of revolution, any solid. Linear equations, conditions (1) for the absence of powers' of the time, and (2) for stability. Theory of free and forced oscillations. Methods of Isolation and of Multipliers. Applications of the calculus of finite differences, chain and network of particles Applications of the calculus of variations, Hamilton, Jacobi, Lagrange, (fee. Precession and Nutation. Motion of the Moon about its centre. Motion of a string or chain, (1) loose, (2) tight. Impact and Vibrations of elastic rods. Motion of a membrane (1) homogeneous, (2) heterogeneous. Conjugate functions applied to vortex motion. The student, to whom the subject is entirely new, is advised to read first the following articles : Chap. I. 1—25, 33—36, 47—52. Chap. II. 66—87. Chap. III. 88—93, 98—104, 110, 112—118. Chap. IV. 130—164, 168—174, 179—186, 199. Chap. V. 214—245, 248—256, 261—269. Chap. VI. 282—285, 287—295, 299—304, 306—309. Chap. VII. 332—373. Chap. VIII. 395—409. Chap. IX. 432— 4G3, 467—476. Chap. X. 483, 488—499. EKRATUM IN VOL. IL Page 458, line 23. For "To these oscillations we add the complementary function" read "with these oscillations we compare those of the unloaded membrane." CHAPTER I. MOMENTS OF INERTIA. 1. In the subsequent pages of this work it will be found that certain integrals continually recur. It is therefore convenient to collect these into a preliminary chapter for reference. Though their bearing on dynamics may not be obvious beforehand, yet the student may be assured that it is as useful to be able to write down moments of inertia with facility as it is to be able to quote the centres of gravity of the elementary bodies. In addition however to these necessary propositions there are many others which are useful as giving a more complete view of the arrangement of the axes of inertia in a body. These also have been included in this chapter though they are not of the same importance as the former. 2. All the integrals used in dynamics as well as those used in statics and some other branches of mixed mathematics are included in the one form jjjccf'y^zydxdydz, where {a, /3, 7) have particular values. In statics two of these three exponents are usually zero, and the third is either unity or zero, according as we wish to find the numerator or denomi- nator of a coordinate of the centre of gravity. In dynamics of the three exponents one is zero, and the sum of the other two is usually equal to 2. The integral in all its generality has not yet been fully discussed, probably because only certain cases have any real utility. In the case in which the body considered is a homogeneous ellipsoid the value of the general integral has been found in gamma functions by Lejeune Dirichlet in Vol. iv. of Liouvilles Journal. His results were afterwards extended by Liouville in the same volume to the case of a heterogeneous ellipsoid in which the strata of uniform density are similar ellipsoids. In this treatise, it is intended chiefly to restrict ourselves to the consideration of moments and products of inertia, as being the only cases of the integral which are useful in dynamics. 2 MOMENTS OF INERTIA. [CHAP. I. 3. Definitions. If the mass of every particle of a material system is multiplied by the square of its distance from a straight line, the sum of the products so formed is called the moment of inertia of the system about that line. If M be the mass of a system and k be such a quantity that Mk^ is its moment of inertia about a given straight line, then k is called the radius of gyration of the system about that line. The term "moment of inertia" was introduced by Euler, and has now come into general use wherever Rigid Dynamics is studied. It will be convenient for us to use the following additional terms. If the mass of every particle of a material system is multi- plied by the square of its distance from a given plane or from a given point, the sum of the products so formed is called the moment of inertia of the system with reference to that plane or that point. If two straight lines Ox, Oy be taken as axes, and if the mass of every particle of the system be multiplied by its two co- ordinates X, y, the sum of the products so formed is called the product of inertia of the system about those two axes. This might, perhaps more conveniently, be called the product of inertia of the system with reference to the two coordinate planes xz, yz. The term moment of inertia with regard to a plane seems to have been first used by M. Binet in the Journal Poly technique, 1813. 4. Let a body be referred to any rectangular axes Ox, Oy, Oz meeting in a point 0, and let x, y, z be the coordinates of any particle m, then according to these definitions the moments of inertia about the axes of x, y, z respectively will be A = tm{y^ + z% B==^X7n(z^-\-a^X G = tm(x^ + y^). The moments of inertia with regard to the planes yz, zx, xy, respectively, will be A' = Xmx^, B' = l^my", C = l,mz\ The products of inertia with regard to the axes yz, zx, xy^ will D — ^myz, E = "Xmzx, F = Imixy, Lastly, the moment of inertia with regard to the origin will H^^m ix^ + 2/' + z^ = Xmr\ where r is the distance of the particle m from the origin. 5. Elementary Propositions. The following propositions may be established without difficulty, and will serve as illustrationj of the preceding definitions. ART. 6.] BY INTEGRATION. 3 (1) The three moments of inertia J., B, G about three rectangular axes are such that the sum of any two of them is greater than the third. For A + B - C =2'Zmz'^ and is positive. (2) The sum of the moments of inertia about any three rectangular axes meeting at a given point is always the same; and is equal to twice the moment of inertia with respect to that point. For A+B+G = 2^711 {x^ + y'^ + z^) = 1'Lmr^, and is therefore independent of the directions of the axes. (3) The sum of the moments of inertia of a system with reference to any plane through a given point and its normal at that point is constant and equal to the moment of inertia of the system with reference to that point. Take the given point as origin and the plane as the plane of xy, then C" + C = 27?ir2, which is independent of the directions of the axes. Hence we infer that A' = \{B-\-G-A\ B' = i(C + A-B), and C =^i{A+B-C). (4) Any product of inertia as D cannot numerically be so great as ^A. (5) li A, B, F are the moments and product of inertia of a lamina about two rectangular axes in its plane, then AB is greater than F\ If t be any quantity we have At^ + 2Ft + B = 'Lm{yt + x)^=& positive quantity. Hence the roots of the quadratic At^ + 2Ft + B = are imaginary, and therefore AB>F'^. (6) Prove that for any body {A-\-B-G){B + G-A)> 4>E\ {A + B - G) {B + G - A){G + A- B)> SDEF. (7) The moment of inertia of the surface of a sphere of radius a and mass M about any diameter is M^a^. Since every element is equally distant from the centre its moment of inertia about the centre is Ma^. Hence by (2) the result follows. (8) The moment of inertia of the surface of a hemisphere of radius a and mass M about every diameter is M|a^. Xhis follows immediately from (7) by completing the sphere, writing 2M for M and halving the result. 6. It is clear that the process of finding moments and products of inertia is merely that of integration. We may illustrate this by the following example. To find the moment of inertia of a uniform triangular plate about an axis in its plane passing through one angular point Let ABG be the triangle, Ay the axis about which the moment is required. Draw Ax perpendicular to Ay and produce BG to meet Ay in D. The given triangle ABG may be regarded as the 1—2 4 MOMENTS OF INERTIA. [chap. difference of the triangles ABD, AGD. Let us then first find the moment of inertia of ABD. Let PQP'Q' be an elementary area whose sides PQ, P'Q' are parallel to the base AD, and let PQ cut Ax in M. Let yS be the distance of the angular point B from the axis Ay, AM = x and AD = l. Then the elementary area PQP'Q is clearly I^—q — dx, and its moment B — X of inertia about Ay is fil —^ dx . x-, where /m is the mass per unit of area. Hence the moment of inertia of the triangle ABD 1 - ^a^dx = -^Ij{ ij^m Similarly if 7 be the distance of the angular point G from the axis Ay, the moment of inertia of the triangle AGD is yV/^^T- Hence the moment of inertia of the given triangle ABG is j\/jLl(l3^ — rf). Now ^1/3 and ^^7 are the areas of the triangles ABD, AGD. Hence if M be the mass of the triangle ABG, the moment of inertia of the triangle about the axis Ay is iif(;6^ + y87 + y). Ex. If each element of the mass of the triangle be multiplied by the wth power of its distance from the straight line through the angle A, then it may be proved in the same way that the sum of the products is 2M ^ +1 _ -^n+l (w + l)(n + 2) /3-7 7. When the body is a lamina the 7noment of inertia about an aoois perpendicular to its plane is equal to the sum of the momenta of inertia about any two rectangular axes in its plane drawn from the point where the former axis meets the plane. For let the axis of z be taken normal to the plane, then, if A, B, G are the moments of inertia about the axes, we have, A = lmy% B = 'Lmx\ G = l^m (x^ + y'), and therefore G = A -\- B. We may apply this theorem to the case of the triangle. Letj ^\ y be the distances of the points B, G from the axis Ax. Thet the moment of inertia of the triangle about a normal to the plane of the triangle through the point A is iM(fi' + ^7 + y + ^'2 ^ ^Y + ^2), ART. 8.] BY INTEGRATION. 5 Ex. Prove that the moment of inertia of the perimeter of a circle of radius a and mass WI about any diameter is ^Ma?. Since every element is equally distant from the axis of the circle, the moment of inertia about that axis is G = Ma^. Since A=B, the result follows at once. 8. Reference Table. The following moments of inertia occur so frequently that they have been collected together for reference. The reader is advised to commit to men5ory the follow- ing table : The moment of inertia of (1) A rectangle whose sides are 2a and 26 about an axis through its centre in its plane per-] __ a^ pendicular to the side 2a J ~ ™^^^ 3" » about an axis through its centre perpendicu-] _ a^ ■{•¥ lar to its plane J ~ ^^^^ ~s~ ' (2) An ellipse semi-axes a and h about the major axis o. = mass2- , about the minor axis b = mass — , about an axis perpendicular to its plane) _ a^-\-¥ through the centre J ~ ^^^^ ~T~ ' In the particular case of a circle of radius a, the moment of inertia about a diameter = mass -r-, and that about a perpen- dicular to its plane through the centre = mass -^ . (3) An ellipsoid semi-axes a, b, c X. ..u • ^' + ^' about the axis a = mass — = — . 5 In the particular case of a sphere of radius a the moment of 2 inertia about a diameter = mass - a^. 5 (4) A right solid whose sides are 2a, 2b, 2c about an axis through its centre perpendicular! _ 6^ + c^ to the plane containing the sides b and c J 3 * These results may be all included in one rule, which the author has long used as an assistance to the memory. Moment of inertia] (sum of squares of perpendicular about an axis I semi-axes) = mass of symmetry ) 3, 4 or 5 The denominator is to be 3, 4 or 5, according as the body is ectangular, elliptical or ellipsoidal. 6 MOMENTS OF INERTIA. [chap. I. Thus, if we require the moment of inertia of a circle of radius a about a diameter, we notice that the perpendicular semi-axis in its plane is the radius a, and that the semi-axis perpendicular to its plane is zero, the moment of inertia required is therefore M — , if M be the, mass. If we require the moment about a per- pendicular to its plane through the centre, we notice that the perpendicular semi-axes are each equal to a and the moment required is therefore M CL' + a 4 =^2 9. As the process for determining these moments of inertia is very nearly the same for all these cases, it will be sufficient to consider only two instances. To determine the moment of inertia of an ellipse about the minor axis. Let the equation of the ellipse be y — -sjd^-x'^. Take any elementary area PQ parallel to the axis of y, then clearly the moment of inertia is [a b fa i/x j x^ydx = 4kfi- I x^sja^-x^dx, Jo ^ J where fi is the mass of a unit of area. To integrate this, put a: = a sin 0, and the integral becomes w rr a^ I cos2 8m2 0d0 = aM ^ — ^ ^0 = ; .". the moment of inertia = u7ra& — = mass — . 4 4 In the same way we may show that the product of inertia of an elliptic quadrant about its axis = mass k— . 2ir To determine the moment of inertia of an ellipsoid about a principal diameter. Let the equation of the ellipsoid , a;2 ^2 ,^2 ^^ ^^ + P + -2 = 1- Take any ele- mentary area PNQ parallel to the plane of yz. Its area is evidently ttPN.QN. Now PN is the value of z when y = 0, and QN the value of y when z = 0, as obtained from the equation of the ellipsoid ; .-. PN=~Jar^\ a ^ QN= - Ja^ - x^ a^ .'. the area of the element : irbc (a^-x^). ART. 9.] BY INTEGRATION. 7 Let /* be the mass of the unit of volume, then the whole moment of inertia = M l_^-2 («'-^') f-^^-d« = /"4^2 /_^(« -^)— ^2-(« -^')^^ 4 , 62+^2 62 + c2 = fi-Trabc — 3 — =ma88 — ^— , do 5 In the same way we may show that the product of inertia of the octant of an ellipsoid about the axes of {x, y) = mass -— . OTT Ex. 1. The moment of inertia of an arc of a circle whose radius is a and which subtends an angle 2a at the centre about an axis (a) through its centre perpendicular to its plane = iTfa^, (6) through its middle point perpendicular to its plane = 21f (1 j a^, (c) about the diameter which bisects the arc = M (1 ^ — j ^ . Ex. 2. The moment of inertia of the part of the area of a parabola cut off by any ordinate at a distance x from the vertex is ^Mx^ about the tangent at the vertex, and ^My^ about the principal diameter, where y is the ordinate corre- sponding to x. Ex. 3. The moment of inertia of the area of the lemniscate r^ = a^ co82d about a line through the origin in its plane and perpendicular to its axis is iW"a2(37r + 8)/48. Ex. 4. A lamina is bounded by four rectangular hyperbolas, two of them have the axes of coordinates for asymptotes, and the other two have the axes for principal diameters. Prove that the sum of the moments of inertia of the lamina about the coordinate axes is ^ (a^ - a'^) (/S^ - jS'^), where a, a' ; j3, j3' are the semi- major axes of the hyperbolas. Take the equations xy = ii, x^-y^ = v, then the two moments of inertia are B = jjx^Jdudv and A=jjy'^Jdudv, where IjJ is the Jacobian of (w, v) with regard to {x, y). This gives at once A-\-B = ^l\dudv, where the limits are clearly u = \a^ to u = ^a"\ v=§^ to v = ^'^. Ex. 5. A lamina is bounded on two sides by two similar ellipses, the ratio of the axes in each being m, and on the other two sides by two similar hyperbolas, the ratio of the axes in each being n. These four curves have their principal diameters along the coordinate axes. Prove that the product of inertia about the coordinate (a2 _ g/2\ lo'Z _ 0'2\ axes is '^ — 7 , L ». > where a, a' ; B, B' are the semi-major axes of the curves. Ex. 6. If da- is an element of the surface of a sphere referred to any rect- angular axes meeting at the centre, prove that j x^"'d .'. moment of inertia of shell = ^Trppq (p^ + q^) a'^da. In the same way the mass of solid ellipsoid = ^irppqa^ ; .*. mass of shell ^A^irppqa^da. Hence the moment of inertia of an indefinitely thin ellipsoidal shell of mass M bounded by similar ellipsoids is ^M {If + c% By reference to Art. 8, it will be seen that this is the same as the moment of inertia of the circumscribing right solid of equal mass. These two bodies therefore have equal moments of inertia about their axes of symmetry at the centre of gravity. ART. 13.] OTHER METHODS. 9 11. The moments of inertia of a heterogeneous body whose boundary is a surface of uniform density may sometimes be found 1)V the method of differentiation. Suppose the moment of inertia of a homogeneous body of density D, bounded by any surface of uniform density, to be known. Let this when expressed in terms of some parameter a be (a) D. Then the moment of inertia of a stratum of density D will be (f> {a) Dda. Replacing D by the variable density p, the moment of inertia required will heJp(j>Xa) da. Ex. 1. Show that the moment of inertia of a heterogeneous ellipsoid about the major axis, the strata of uniform density being similar concentric ellipsoids, and the density along the major axis varying as the distance from the centre, is Ex. 2. The moment of inertia of a heterogeneous ellipse about the minor axis, the strata of uniform density being confocal ellipses and the density along the ,,;,., , ,, , . 3M4a5 + c5-5a3c2 minor axis varying as the distance from the centre, is -^rp- -7^-5 o — it-v 12. Other methods of finding moments of inertia. The moments of inertia given in the table in Art. 8 are only a few of those in continual use. The moments of inertia of an ellipse, for example, about its principal axes are there given, but we shall also frequently want its moments of inertia about other axes. It is of course possible to find these in each separate case by integra- tion. But this is a tedious process, and it may be often avoided by the use of the two following propositions. The moments of inertia of a body about certain axes through its centre of gravity, which we may take as axes of reference, are regarded as given in the table. In order to find the moment of inertia of that body about any other axi» we shall investigate: (1) A method of comparing the required moment of inertia with that about a parallel axis through the centre of gravity. This is the theorem of parallel axes. (2) A method of determining the moment of inertia about this parallel axis in terms of the given moments of inertia about the axes of reference. This is the theorem of the six constants of a body. 13. Theorem of Parallel Axes. Given the moments and products of inertia about all axes through the centre of gravity of a body, to deduce the momeiits and products about all parallel axes. The moment of inertia of a system of bodies about any axis is equal to the moment of inertia about a parallel axis through the centre of gravity plus the moment of inertia of the whole mass collected at the centre of gravity about the original axis. The product of inertia about any two axes is equal to the product of inertia about two parallel axes through the centre of 10 MOMENTS OF INERTIA. [CHAP. I. gravity plus the product of inertia of the whole mass collected at the centre of gravity about the original axes. Firstly, take the axis about which the moment of inertia is required as the axis of z. Let m be the mass of any particle of the body, which generally will be any small element. Let x, y, z be the coordinates of m, x, y, z those of the centre of gravity G of the whole system of bodies, x\ y, z' those of m referred to a system of parallel axes through the centre of gravity. Then since -;^^ — , ^ , -^^^ — are the coordinates of the Zm 2,m 2,711 centre of gravity of the system referred to the centre of gravity as the origin, it follows that ^mx' = 0, ^my = 0, 2m/ = 0. The moment of inertia of the system about the axis of z is . =^m[{x+xj^-{y + y')% = Sm (x'' + 2/') + Sm (^'2 + 1/'2) + 2x . l^mx' + 2y . l^my'. \ Now ^m(x'^ + y^) is the moment of inertia of a mass 2m collected at the centre of gravity, and 1m(x''^-\- y"^) is the moment of inertia of the system about an axis through G, also "^mx' = 0, y,my = ; whence the proposition is proved. J It follows from this theorem, that, of all axes parallel to ai given straight line that one has the least moment of inertia which passes through the centre of gravity. Secondly, take the axes of x, y as the axes about which the product of inertia is requited. The product required is = 2m xy = 2m (x 4- x') {y + y'), = xy . Xm + ^mxy' + x%my' + y%moc , = xyXm + ^mx'y'. Now xy . 2m is the product of inertia of a mass 2m collected at G and ^mx'y' is the product of the whole system about axes through G ; whence the proposition is proved. Let there be two parallel axes A and B at distances a and h from the centre of gravity of the body. Then, if M be the mass of the material system. moment of inertia] ^ _ jnioment of inertia ^, ^ about A ) \ about B Hence when the moment of inertia of a body about one ax is known, that about any other parallel axis may be found. It is' obvious that a similar proposition holds with regard to the pro- ducts of inertia. ART. 14.] THEOREM OF PARALLEL AXES. 11 14. The preceding proposition may he generalized as follows. Let any system be in motion, and let oo, y, z, be the coordinates at the time t of any particle of mass m. Let also x, y, z\ x, y, z be the resolved velocities and accelerations of the same particle, where the dots represent as usual differentiations with regard to the time. Suppose V = tm(j> {x, X, X, y, y, y, z, i, z) to be a given function depending on the structure and motion of the system, the summation extending throughout the system. Also let (j) be an algebraic function of the first or second order. Thus s. Select some convenient set of axes which we may call x, y, z having the same origin such that the six constants of the body, " P 7^ viz. Zmx^, 'Lmy'^, ^mz'^, "Lmxy, "Lmyz, Smzx, are all known or can **// // "^ ff ^^ easily found. Let the direction-cosines of these axes be given °- P y by the diagram in the margin. We then have ^ = ax + a'y + a"z, 'q = ^x + ^'y + p"z , ^=yx + y'y + y"z. Substitut- ing these values and expanding we obtain an expression for Sm0 (^, rj, f) in terms of the six known constants of the body. ART. 19.] TRANSFORMATION OF AXES. 15 The result may appear at first sight to be rather compHcated, but if the new axes be properly chosen it reduces in most cases to a few terms. Thus if the axes of {x, y, z) are principal axes all the terms 'Zmxy, Hmyz, liVizx are zero. Supposing this choice to be made, the formula reduces to the convenient form 2m0(^, 77, r) = 0(a. ^, 7)2ma;2 + 0(a', ^', y')'Lmy' + [a" , /3", y")'Z.mz^...{l). In using this formula, the coefficient of 'Zmx^ is obtained by substituting for (^» '?» r) ill (f> '?> f) the direction-cosines of the new axis of x, i.e. the cosines in the row of the diagram marked x. The coefficient of Zmy" may be obtained by substituting the direction-cosines of the new axis of y, i.e. the cosines in the row marked y, and so on. If it be required to change the origin of coordinates also, this may be done by an application of the theorem in Art. 14. If the body is a triangular area or a tetrahedral volume, the value of the integral Sm0 may be written down at sight when the coordinates of the corners of the body are given. We have merely to replace the body by any convenient system of equi- momental points, see Art. 36. Ex. 1. The coordinates of the centre of an elliptic area are (/, g, h) and the direction-cosines of its axes are (a, /3, 7) (a', /3', 7'), prove that Zwf^ = M {h^ + ia^y^ + ib^y'^). Ex. 2. Let Ox, Oy, Oz be the principal axes at the origin, prove that the product of inertia F' = 'Em^rj about two rectangular axes 0^, Orj whose directions are (a, a, a") (/3, j3', j3") is given by either of the formulae S w^t; = a^'Zmx^ + a^' Xmy^ + o!'^' ^mz^ = - a/34 - a'p'B - a"/3"C. The first result is seen at once to be true by substituting the values of I, 7/ given above ; and the second result follows immediately from the first since a^+a'^ + a"^"=0. These are vei-y simple formulae to find products of inertia. Ex. 3. Let {7, 7', y") be the direction-cosines of a fixed axis Of. Then as Of, Orj turn round Of, prove that both I)''^ + E'^ and A'B' -F''^ are constant where A', B', C, D', E', F' are the moments and products of inertia of the body referred I to these moving axes. For by Ex.2, -D'= A^y -f B^'y' + C^"y", -E' = Aay + Ba'y' + Cal'y" ; .-. Z>'2 + E'-' = A^y^ (a2 + /32) + 2AByy' {aa' + ^^') + &G. ; since a? + ^^=l-y'^ = y"^ + y"'^ and aa' + ^^'=-yy' we have D'2 ^E"' = {A~BY {yy'f + {B- Cf {y'y"f + {G-Af (7"7)2. Similarly A'B' -F'-^ = BGy' + CAy'^ + ABy"^. 19. The Ellipsoids of Inertia. The expression which has been found in Art. 15 for the moment of inertia / about a straight line whose direction-cosines are (a, y8, 7). I = Aa' + Bff' + Gy' - 2Dl3y - 2Eya - 2Fal3, admits of a very useful geometrical interpretation. Let a radius vector OQ move in any manner about the given point 0, and be of such length that the moment of inertia about OQ may be proportional to the inverse square of the length. Then if R represent the length of the radius vector whose direction- 16 MOMENTS OF INERTIA. [CHAP I. cosines are (a, ^, 7), we have / = Me^/R^, where e is some constant introduced to keep the dimensions correct, and M is the mass. We shall sometimes abbreviate Me* into the single symbol K. ^ Hence the polar equation of the locus of Q is ^ = Aci' + B/3' + Cy' - 2D^y - 2Eya - 2Fa0. Transforming to Cartesian coordinates, we have K=AX'' + BY'+CZ'-2DYZ-2EZX-2FXY, which is the equation of a quadric. Thus to every point of a material body there is a corresponding quadric which possesses the property that the moment of inertia about any radius vector is represented by the inverse square of that radius vector. The convenience of this construction is, that the relations which exist between the moments of inertia about straight lines meeting at any given point may be discovered by help of the known properties of a quadric. Since a moment of inertia is essentially positive, being by definition the sum of a number of squares, it is clear that every radius vector R must be real. Hence the quadric is always an ellipsoid. It is called the momental ellipsoid, and was first used by Cauchy, Exercices de Math. Vol. 1 1. So much has been written on the ellipsoids of inertia that it is difficult to deter- mine what is really due to each of the various authors. The reader will find much information on these points in Prof. Cayley's report to the British Association on the Special Problems of Dynamics, 1862. 20. The Invariants. The momental ellipsoid is defined by a geometrical property, viz. that any radius vector is equal to some constant divided by the square root of the moment of inertia about that radius vector. Hence whatever coordinate axes are taken, we must always arrive at the same ellipsoid. If therefore the momental ellipsoid be referred to any set of rectangular axes, the coefficients of X^ Y\ Z^, -2YZ, - 2ZX, -2Z7 in its equa- tion will still represent the moments and products of inertia about these axes. Since the discriminating cubic determines the lengths of the axes of the ellipsoid, it follows that its coefficients are unaltered by a transformation of axes. But these coefficients are A-\-B-]-G, AB-\-BG+GA-D'-E'-F\ ABC - 2DEF - AD'' - BE' - GF\ Hence for all rectangular axes having the same origin, these are invariable and all are greater than zero. Airr. 23.] ellipsoids of inertia. 17 ^2 y^ /I 1 21. It should be noticed that the constant e is arbitrary, though when once chosen it cannot be altered. Thus we have a series of similar and similarly situated ellipsoids, any one of which may be used as a momental ellipsoid. When the body is a plane lamina, a section of the ellipsoid corresponding to any point in the lamina by the plane of the lamina, is called a momental ellipse at that point. If principal axes at any point of a body be taken as axes of coordinates, the equation of the momental ellipsoid takes the simple form AX"" + BV + GZ^^ = Me^ where M is the mass and e* any constant. Let us now apply this to some simple cases. Ex. 1. To Jind the momental ellipsoid at the centre of a material elliptic disc. Taking the same notation as before, we have A = ^Ml)^, B = ^Ma'^, C = lM{a^-{-h'^). Hence the ellipsoid is 1 3Ib'^X^ + 1 Ma^ Y-^ + ^M (a^ + h'^) Z'^ = 3h\ Since e is any constant, this may be written „ + tt + ( -< a^ b^ '\a' When Z = 0, this becomes an ellipse similar to the boundary of given disc. Hence we infer that the momental ellipse at the centre of an elliptic area is any similar and similarly situated ellipse. This also follows from Art. 17, Ex. 1. Ex. 2. To find the momental ellipsoid at any point O of a material straight rod AB of mass M and length 2a. Let the straight line OAB be the axis of x, O the origin, G the middle point of AB, OG = c. If the material line can be regarded as indefinitely thin, ^=0, B = M {^a^ + c^) = G, hence the momental ellipsoid is Y^ + Z' = €'^, where e' is any constant. The momental ellipsoid is therefore an longated spheroid, which becomes a right cylinder having the straight line for ,xis, when the rod becomes indefinitely thin. Ex. 3. The momental ellipsoid at the centre of a material ellipsoid is (Z>2 + c2) Z2 + {c2 + a-) T' + {a" + h") Z' = e^ where e is any constant. It should be noticed that the longest and shortest axes of he momental ellipsoid coincide in direction with the longest and shortest axes •espectively of the material ellipsoid. 22. Conversely, we may show that any ellipsoid being given, a real material body can be found of which it is the momental ellipsoid provided the sum of the Jtquares of the reciprocals of any two of its axes is greater than the square of 'he reciprocal of the third. For let the moments of inertia about the principal diameters be A=Kla^, B = Kjb^, C = Klc^, then by Art. 5 it is necessary that the sum of any two of the ihree A, B, C should be greater than the third. Again, this condition is sufficient, 'or if we place two particles on each principal diameter, at such distances from the )rigin, ±jp, ±g, ±?-, and of such masses, m, m\ m", that 4.mp'^ = B + G-A, 4mq^-=G + A-B, 4mr- = A + B-C, hese six particles will have the principal diameters for principal axes, and the ven quantities. A, B, G for their principal moments of inertia. 23. Elementary Properties of Principal Axes. By a sonsideration of some simple properties of ellipsoids, the following ropositions are evident : R. D. 2 18 MOMENTS OF INERTIA. [CHAP. I. I. Of the moments of inertia of a body about axes meeting at a given point, the moment of inertia about one of the principal axes is greatest and about another least For, in the momental ellipsoid, the moment of inertia about a radius vector from the centre is least when that radius vector is greatest and vice versa. And it is evident that the greatest and least radii vectores are two of the principal diameters. It follows by Art. 5 that of the moments of inertia with regard to all planes passing through a given point, that with regard to one principal plane is greatest and with regard to another is least. II. If the three principal moments at any point are equal to each other, the ellipsoid becomes a sphere. Every diameter is then a principal diameter, and the radii vectores are all equal. Hence every straight line through is a principal axis at 0, and the moments of inertia about them a,re all equal. For example, the perpendiculars from the centre of gravity of a cube on the three faces are principal axes ; for, the body being referred to them as axes, we clearly have ^mxy = 0, ^myz = 0, ^mzx = 0. Also the three moments of inertia about them are by symmetry equal. Hence every axis through the centre of gravity of a cube is a principal axis, and the moments of inertia about them are all equal. Next suppose the body to be a regular solid. Consider two planes drawn through the centre of gravity each parallel to a face of the solid. The relations of these two planes to the solid are in all respects the same. Hence also the momental ellipsoid at the centre of gravity must be similarly situated with regard to each of these planes, and the same is true for planes parallel to all the faces. Hence the ellipsoid must be a sphere and the moment of inertia will be the same about every axis. Ex. 1. Three equal particles A, B, C are placed at the corners of an equilateral triangle ; prove that the momental ellipse at their centre of gravity G is a circle. By symmetry the diameters GA, GB, GG of the momental ellipse at G must be equal. The ellipse is therefore a circle. Ex. 2. Four equal particles are placed at the corners of a tetrahedron. If the momental ellipsoid at their centre of gravity is a sphere, prove that the tetrahedra is regular. Ex. 3. Any point in a body being given and any plane drawn through i prove that two straight lines at right angles can be drawn in this plane through such that the product of inertia about them is zero. These are the axes of the section of the momental ellipsoid at the point formed by the given plane. 24. At every point of a material system there are always thri principal axes at right angles to each other. Construct the momental ellipsoid at the given point. Then has been shown that the products of inertia about the axes art i ART. 26.] ELLIPSOIDS OF INERTIA. 19 half the coefficients of - XF, - YZ, — ZX in the equation of the momental ellipsoid referred to these straight lines as axes of coordinates. Now if an ellipsoid be referred to its principal diameters as axes, these coefficients vanish. Hence the principal diameters of the ellipsoid are the principal axes of the system. But every ellipsoid has at least three principal diameters, hence every material system has at least three principal axes. 25. Ex. 1. The principal axes at the centre of gravity being the axes of reference, prove that the momental ellipsoid at the point [p, q, r) is (4+5' + »'') -^+ (§ +»"'+^') Y' + (jj+P^ + Q'') ^' - 2?'-Fif - 2rpZX-2pqXY=e\ when referred to its centre as origin. Ex. 2. Show that the cubic equation to find the three principal moments of inertia at any point {p, q, r) may be written in the form of a determinant I-A ^ q--r^ pq rp PQ j^[ -r--p- qr -r--p- I-C = 0. m=0,\ •n=oi n = 0.) If (I, m, n) be proportional to the direction-cosines of the axes corresponding to any one of the values of 7, theii' values may be found from the equations {I-{A+3Iq^ + Mr^) } I + Mpqm + Mipn = 0, Mpql +\I-{B + Mr^ + Mp^) } m + Mqm ■- Mrpl + Mqrm + { I - ( C + iHp^ + Mq'^) ) Thus {I, m, n) are proportional to the minors of the constituents of any row of the determinant. Ex. 3. If S^ = be the equation to the momental ellipsoid at the centre of gravity referred to any rectangular axes written in the form given in Art. 19, hen the momental ellipsoid at the point P whose coordinates are {p, q, r) is S + M{p'^ + q^ + r^){X-^+Y-^ + Z^)-M{pX + qY-rfZf~ = 0. lence show (1) that the conjugate planes of the straight line OP in the momental llipsoids at O and P are parallel and (2) that the sections perpendicular to OP lave their axes parallel. 26. Ellipsoid of Gyration. The reciprocal surface of the nomental ellipsoid is another ellipsoid, which has also been em- >loyed to represent, geometrically, the positions of the principal ixes and the moment of inertia about any line. We shall require the following elementary proposition. The reciprocal surface >f the ellipsoid -^ +^ + -^ = 1 is the ellipsoid a^x^ + bhj- + c^z- = €*. Let ON be the perpendicular from the origin O on the tangent plane at any int P of the first ellipsoid, and let I, m, n be the direction-cosines of ON, then }N^ = a^l^+b-^m^+c^n'^. Produce ON to Q so that OQ = e^jON, then Q is a point n the reciprocal surface. Let OQ = R., :. €^ = {arP-+b'^m'^ + c-n^)R^. Changing his to rectangular coordinates, we get e'^=d^x'^ +}}^y^ + c^z^. '1 9 20 MOMENTS OF INERTIA. [CHAP. I. To each point of a material body there corresponds a series of similar momental ellipsoids. If we reciprocate these we get another series of similar ellipsoids coaxial with the first, and such that the moments of inertia of the body about the perpendiculars on the tangent planes to any one ellipsoid are proportional to the squares of those perpendiculars. It is, however, convenient to call that particular ellipsoid the ellipsoid of gyration which makes the moment of inertia about a perpendicular on a tangent plane equal to the product of the mass into the squai'e of that perpendicular. If M be the mass of the body and A, B, G the principal moments, the equation of the ellipsoid of gyration is ~A'^ B'^ G ~ M' It is clear that the constant on the right-hand side must be IjM, for when Y and Z are put equal to zero, MX"^ must by definition be A. 27. Conversely, the series of momental ellipsoids at any point of a body may be regarded as the reciprocals, with different con- stants, of the ellipsoid of gyration at that point. They are all of an opposite shape to the ellipsoid of gyration, having their longest axes in the direction of the shortest axis and their shortest axes in the direction of the longest axis of the ellipsoid of gyration. The momental ellipsoids however resemble the general shape of the body more nearly than the ellipsoid of gyration. They are protuberant where the body is protuberant and compressed where the body is compressed. The exact reverse of this is the case in the ellipsoid of gyration. See Art. 22, Ex. 3. 28. Ex, 1. To find the ellipsoid of gyration at the centre of a material elliptic disc. Taking the values of A, B, C given in Art. 22, Ex. 1, we see that the ellipsoid of gyration is -j^ + — . + -^ ,0 = 7. Ex. 2. The ellipsoid of gyration at any point of a material rod AB is -r- + T-T, — -o + ,- > 0=1, taking the notation of Art. 21, Ex. 2. It is thus a very ^a^ + c^ ^a^-\-c^ flat spheroid which, when the rod is indefinitely thin, becomes a circular area, whose centre is at 0, whose radius is J^ a^ + c^ and whose plane is perpendicular to the rod. Ex. 3. It may be shown that the general equation of the ellipsoid of gyration referred to any set of rectangular axes meeting at the given point of the body is A -F -E MX 1=0, -F B -D 3IY -E -D C MZ MX MY MZ M or, when expanded, {BG - D2) A'2+ (C^ - £2) Y^ + {AB - F'^) Z-^ + 2{AD + EF) YZ + 2 (BE + FD) ZX + 2{CF + DE) XY=~{ABC -AD^- BE'- - GF^-2DEF). The right-hand side, when multiplied by 31, is the discriminant obtained by leaving out the last row and the last column, and the coefficients of A'^, Y^, Z'-, 2ZX, 2XY, 2YZ are the minors of this discriminant. _J ART. 32.] ELLIPSOIDS OF INERTIA. 21 29. The use of the ellipsoid whose equation referred to the principal axes at the centre of gravity is z^ J[!_ _Z!_^1 ^mx^ ^my"- Sm^'-* if' has been suggested by Legendre in his Fonctions ElUptiques. This ellipsoid is to be regarded as a homogeneous solid of such density that its mass is equal to that of the body. By Art. 8, Ex. 3, it possesses the property that its moments of inertia with regard to its principal axes, and therefore by Art. 15 its moments of inertia with regard to all planes and axes, are the same as those of the body. We may call this ellipsoid the equimomental ellipsoid or Legendre s ellipsoid. Ex. If a plane move so that the moment of inertia with regard to it is always proportional to the square of the perpendicular from the centre of gravity on the plane, then this plane envelopes an ellipsoid similar to Legendre's ellipsoid. 30. There is another ellipsoid which is sometimes used. By Art. 15 the moment of inertia with reference to a plane whose direction-cosines are (a, /3, 7) is r = Sma;2 . a^ + Swi?/^ . ^2 ^ 2»i22 ^ ^2 ^ 21^myz . ,87 + 11.mzx . 7a + 2'2vixy . a^. Hence, as in Art. 19, we may construct the ellipsoid 2ma;2 . Z^ + Sm?/2 . 72 + Sm^^ . ^2 + 2Smt/2 . YZ + 2^mzx . ZX+22mxy . XY= K. Then the moment of .inertia with regard to any plane through the centre is repre- sented by the inverse square of the radius vector perpendicular to that plane. If we compare the equation of the momental ellipsoid with that of this ellipsoid, we see that one may be obtained from the other by subtracting the same quantity from each of the coefficients of Z^, y^, Z^. Hence the two ellipsoids have their circular sections coincident in direction. This ellipsoid may also be used to find the moments of inertia about any straight line through the origin. For we may deduce from Art. 15 that the moment of inertia about any radius vector is represented by the difference between the inverse square of that radius vector and the sum of the inverse squares of the semi-axes. This ellipsoid is a reciprocal of Legendre's ellipsoid. All these ellipsoids have their principal diameters coincident in direction, and any one of them may be used to determine the directions of the principal axes at any point. 31. When the body considered is a lamina, the section of the ellipsoid of gyration at any point of the lamina by the plane of the lamina is called the ellipse of gyration. If the plane of the lamina be the plane of xy, we have 'Zmz^ = 0. The section of the fourth ellipsoid is then clearly the same as an ellipse of gyration at the point. If any momental ellipse be turned round its centre through a right angle it evidently becomes similar and similarly situated to the ellipse of gyration. Thus, in the case of a lamina, any one of these ellipses may be easily changed into the others. 32. Equimomental Cone. A straight line passes through a fixed point and moves about it in such a nuinner that the moment of inertia about the line is alioays the same and equal to a given quantity I. To find the equation of the cone generated by the straight line. Let the principal axes at O be taken as the axes of coordinates, and let (a, j3, 7) be the direction-cosines of the straight line in any position. Then by Art. 16 we have Aa^-irB^-+Cy- = I. Hence the equation of the locus is 22 MOMENTS OF INERTIA. [CHAP. I. or, transforming to Cartesian coordinates, {A - I)x^+{B - I)y^+ {C - 1) z^ = 0. It appears from this equation that the principal diameters of the cone are the principal axes of the body at the given point. The given quantity I must be less that the greatest and greater than the least of the moments A, B, C. Let A, B, C he arranged in descending order of magni- tude ; then if I be less than B, the cone has its concavity turned towards the axis C, if I be greater than B the concavity is turned towards the axis ^, if I=B the cone becomes two planes which are coincident with the central circular sections of the momental ellipsoid at the point 0. The geometrical peculiarity of this cone is that its circular sections in all cases are coincident in direction with the circular sections of the momental ellipsoid at the vertex. This cone is called an equimoviental cone at the point at which its vertex is situated. 33. On Equimomental Bodies. Two bodies or systems of bodies are said to be equimomental when their moments of inertia about all straight lines are equal each to each. 34. If two systems have the same centre of gravity, the same mass, the same principal axes and principal moments at the centre of gravity, it follows from the two fundamental propositions of Arts. 13 and 15 that their moments of inertia about all straight lines are equal, each to each. The converse theorem is also true. If the two bodies have equal moments of inertia about every straight line, it is evident that the axes of maxima and minima moments are the same in the two bodies. Of all straight lines having a given direction that one has the least moment of inertia for either body which passes through the centre of gravity of that body (Art. 13). Consider any direction perpendicular to the straight line joining the two centres of gravity G, G'. The minimum for one body passes through G and for the other through G'. They cannot be the same unless G, G' coincide. Next consider all the directions which pass through the common centre of gravity. The axes of greatest and least moments of inertia for each body are two of the principal axes of that body (Art. 23). These must therefore coincide in the two bodies. The third axis in each body is perpendicular to these two, and they also must coincide. Lastly, consider two parallel axes at a distance p apart, one passing through the common centre of gravity. By the theorem of parallel axes, the difference of the moments of inertia about these for either body is Mpr, where M is the mass of that body. But both the moments of inertia and the distance p are the same for each body. Hence the masses are also equal. It is easy to see that two equimomental systems must have ART. 36.] EQUIMOMENTAL BODIES. 23 the same momental ellipsoid, and therefore the same principal axes at every point. 35. Case of a Triangle. To find the moments and products of inertia of a triangle about any axes whatever. If y8 and 7 be the distances of the angular points B, G, of a triangle ABC from any straight line AX drawn through the angle A^ in the plane of the triangle, it is known that the moment of inertia of the triangle about AX is ^M{^'^-\- ^^ + 'y-\ where M is the mass of the triangle. Let three equal particles, the mass of each being \M, be placed at the middle points of the three sides. Then it is easily seen, that the moment of inertia of the three particles about AX is ftm -©■-©■' which is the same as that of the triangle. The three particles, treated as one system, and the triangle have the same centre of gravity. Let this point be called 0. Draw any straight line OX' through the common centre of gravity parallel to AX, then it is evident that the moments of inertia of the two systems about OX' are also equal. Since this equality exists for all straight lines through in the plane of the triangle, it will be true for two straight lines OX', OV at right angles, and therefore also for a straight line OZ' perpendicular to the plane of the triangle. One of the principal axes at of the triangle, and of the systems of three particles, is normal to the plane, and therefore the same for the two systems. The principal axes at in the plane, are those two straight lines about which the moments of inertia are greatest and least, and therefore by what precedes these axes are the same for the two systems. If at any point two systems have the same principal axes and principal moments, they have also the same moments of inertia about all axes through that point, and the same products of inertia about any two straight lines meeting in that point. And if this point be the centre of gravity of both systems, the same thing will also be true for any other point. If then a particle whose mass is one-third that of the triangle he placed at the middle point of each side, the moment of inertia of the triangle about any straight line, is the same as that of the system of particles, and the product of inertia about any two straight lines meeting one another, is the same as that of the system of particles. 36. The existence of equimomental points is of the greatest utility in finding the moments and products of inertia of a body about any axes. They may also be used for more general integra- , I 24 MOMENTS OF INERTIA. [CHAP, tioti^. Thus suppose any given body to be equimomental to three particles whose coordinates are (a?,, y^, z^), (a?2, 3/3, 2^2), (x^, y^, 2-3). Since the masses placed at these points may not in all cases be equal, let these masses be respectively M^, M.., M^, where of course the sum is equal to the mass of the body. Let (^(a*, y, z) be any function of 00, y, z which does not contain any power higher than the second. Let it be required to find the value of the integral or sum ^m<^ {x, y, z) taken throughout the body, where m is an element of the mass. The required integral is evidently equal to M,(f> {a;,, y,, Zj) + Mo (x^, y.,, -s-o) + J\l,(j) {x.„ y,, z^). By properly choosing the equivalent points we may use a similar rule in which <^ is any cubic or qiiartic function of x, y, z, but as these cases are not wanted in rigid dynamics we shall merely state a few results a little farther on. The same body may be equimomental to several systems of points, and some of these sets may be more convenient than the others. In order that a set of equimomental points may be useful it is necessary (1) that the points should be so conveniently placed in the body that their coordinates can be easily found with regard to any given axes, (2) that the number of points employed in the set should be as small as possible. Of these two requisites the first is by far the more important. Equimomental points have another use besides that of shorten- ing integrations which may otherwise be troublesome. It will be presently seen that they have a dynamical importance. 37. A motut'iital ellipsoid at the centre of gravity of any triangle may he found as follows. Let an ellipse be inscribed in the triangle touching two of the sides AB, BG in their middle points F, D. Then, by Carnot's theorem, it touches the third side CA in its middle point E. Since DF is parallel to CA the tangent at E, the straight line joining E to the middle point N of DF passes through the centre, and therefore the centre of the conic is at the centre of gravity of the triangle. This conic may be shown to be a momental ellipse of the triangle at 0. To prove this, let us find the moment of inertia of the triangle about OE. Let OE — r, and let /' be the semi-conjugate diameter, and w the angle between r and r'. Now ON=^r, and hence from the equation of the ellipse i«W2 = |r'=^, therefore moment of ) .,,, ,« . o M A'^ inertia about 0£ \ = n^ .^r^sm^u,, =^.:^,'. where A' is the area of the ellipse, so that the moments of inertia of the system about OE, OF, OD are proportional inversely to OE'^, OF^, 0D'\ If we take a momental ellipse of the right dimensions, it will cut the inscribed conic in E, F, and D, and therefore also at the opposite ends of the diameters through these points. But two conies cannot cut each other in six points unless they are iden- tical. Hence this conic is a momental ellipse at O of the triangle. A normal at O to the plane of the triangle is a principal axis of the triangle (Art. 16). Hence a momental ellipsoid of the triangle has the inscribed conic for one principal section. If 2a and 26 be the lengths of the axes of this conic, 2c that of the axis of the ellipsoid which is perpendicular to the plane of the lamina, we have, by Arts. 7 and 19, l/c2=l/a2+l/62. ART. 38.] EQUIMOMENTAL BODIES. 25 If the triangle be an equilateral triangle, the momental ellipsoid becomes a spheroid, and every axis through the centre of gravity in the plane of the triangle is a principal axis. Since any similar and similarly situated ellipse is also a momental ellipse, we may take the ellipse circumscribing the triangle, and having its centre at the centre of gravity, as the momental ellipse of the triangle. 88. Ex. 1. A momental ellipse at an angular point of a triangular area touches tlie opposite side at its middle point and bisects the adjacent sides. Ex. 2. A momental ellipse at the middle point F of the side AB of a triangular lamina ABC circumscribes the triangle and has FG, FB for conjugate diameters. Prove also that another momental ellipse at the same point F touches the sides ACy BC at their middle points. Ex. 3. The principal radii of gyration at the centre of gravity of a triangle d^ + b^ + c'^ A^ are the roots of the equation x* o6' ^^ + T7:o = ^> where A is the area of the triangle. Ex. 4. The direction of the principal axes at the centre of gravity O of a triangle may be constructed thus. Draw at the middle point D of any side BG lengths DH = — , BH' — ~~- along the perpendicular, where p is the perpendicular from A on BG and /o, k' are the principal radii of gyration found by the last example. Then OH, OH' are the directions of the principal axes at O, whose moments of inertia are respectively Mk^ and Mk'^. Ex. 5. The directions of the principal axes and the principal moments at the centre of gravity may also be determined thus. Draw at the middle point D of any side BG a perpendicular DK=BGI2iJS. Describe a circle on OK as diameter and join D to the middle point of OK by a line cutting the circle in R and S, then OR, OS are the directions of the principal axes, and the moments of inertia about them are respectively p/. DS'^ and ^M.DR'K Ex. 6. Let four particles each one-sixth of the mass of the area of a parallelo- gram be placed at the middle points of the sides and a fifth particle one-third of the same mass at the centre of gravity, then these five particles and the area of the parallelogram are equirnomental systems. Ex. 7. Let particles each equal to one-twelfth of the mass of a quadrilateral area be placed at each corner and let a fifth particle of negative mass but also one- twelfth be placed at the intersection of the diagonals. Then the centre of gravity of the quadrilateral area is the centre of gravity of these five particles. Let a sixth particle equal to three-quarters of the mass of the quadrilateral be placed at the centre of gravity thus found. Prove that these six particles are equirnomental to the quadrilateral area. Ex. 8. Let particles each equal to one-quarter of the mass of an elliptic area be placed at the middle points of the chords joining the extremities of any pair of con- jugate diameters. Prove that these four particles are equirnomental to the elliptic area. Ex. 9. Let a tenth of the mass of a solid homogeneous ellipsoid be placed at each of the six extremities of a set of conjugate diameters and two-fifths of the mass at the centre, prove that this system of particles is equimomental to the ellipsoid. Ex. 10. Any sphere of radius a and mass M is equimomental to a system of \ four particles each of mass ^ ( } placed so that their distances from the centre ! make equal angles with each other and are each equal to r, and a fifth particle equal i to the remainder of the mass of the sphere placed at the centre. 26 MOMENTS OF INERTIA. [CHAP. I. 39. Case of a Tetrahedron. To find the moments and pro- ducts of inertia of a tetrahedron about any axes whatever, i.e. to find a system of equimomental particles. Let A BCD be the tetrahedron. Through one angular point D draw any plane and let it be taken as the plane of xy. Let D be the area of the base ABC, a, y8, 7 the distances of its angular points from the plane of xy, and p the length of the perpendicular from D on the base ABC. Let PQR be any section parallel to the base ABC and of thickness du, where u is the perpendicular from D on PQE. The moment of inertia of the triangle PQR with respect to the plane of xy is the same as that of three equal particles, each one- third its mass, placed at the middle points of its sides. The volume of the element PQR = -^ J^du. The ordinates of the middle points of the sides AB, BC, GA are respectively ^ (a + 0), i (^ + 7)^ i (7 + «)• Hence, by similar triangles, the ordinates of the middle points of PQ, QR, RP are i (« + ^) Wp. i (^ + 7) ^IP> i (7 + «) ^/P- The moment of inertia of the triangle PQR with regard to the plane xy is therefore Integrating from u = to u=p, we have the moment of inertia of the tetrahedron with regard to the plane xy = TV^{«'+/52 + 7'+/37 + 7a + a^}, where V is the volume. If particles each one-twentieth of the mass of the tetrahedron were placed at each of the angular points and the rest of the mass, viz. four- fifths, were collected at the centre of gravity, the moment of inertia of these five particles with regard to the plane of .2/ would be = f|(«+|±-^J + J«^+ £^^ + £-yS which is the same as that of the tetrahedron. The centre of gravity of these five particles is the centre of gravity of the tetrahedron, and together they make up the mass of the tetrahedron. Hence, by Art. 13, the moments of inertia of the two systems with regard to any plane through the centre of gravity are the same, and by the same article this equality will exist for all planes whatever. It follows, by Art. 5, that the moments of inertia about any straight line are also equal. The two systems are therefore equimomental. 40. Theory of Projections. If the distance of every point in a given figure in space from some fixed plane be increased in a ART. 42.] EQUIMOMENTAL BODIES. 27 fixed ratio, the figure thus altered is called the projection of the given figure. By projecting a figure from three planes at right angles as base planes in succession, the figure may be often much implified. Thus an ellipsoid can always be projected into a sphere, and any tetrahedron into a regular tetrahedron. It is clear that if the base plane from which the figure is projected be moved parallel to itself into a position distant D from its former position, no change of form is produced in the projected figure. If n be the fixed ratio of projection the pro- ected figure has merely been moved through a space nD perpen- dicular to the base plane. We may therefore suppose the base lane to pass through any given point which may be convenient. 41. If two bodies are equimomental, their projections are also Bquimomental. Let the origin be the common centre of gravity, then the two bodies are such that 1m = Sm' ; 2m^ = 0, 2mV = 0, &c., Sm^- = 2mV'^, Imyz = 1my'z, &;c., unaccented letters referring to one body and accented letters to the other. Let both the bodies be projected from the plane of xy in the fixed ratio 1 : n. Then any point whose coordinates are (x, y, z) is transferred to X, y, nz) and {x\ y\ z') to {x\ y\ nz). Also the elements of mass Im, ni' become m/tand nm. It is evident that the above equalities are not affected by these changes, and that therefore the projected bodies are equimomental. The projection of a mo mental ellipse of a plane area is a niomental ellipse of the projection. Let the figure be projected from the axis of x as base line, so that any point (x, y) is transferred to {x, y) where y' — ny, and any element of area m becomes m where m! = nm. Then Imx^ = - %m'x^, '%mxy = -^ ^m'xy', ^my'^ = — Im'y"^. The momental ellipses of the primitive and the projection are l^my^X^ - 21mxyX Y + Imx' Y^ = Me', Im'y'^X'^ - 2Xm'xy'X'Y' + l^m'x'Y' = M'e\ To project the former we put X'=X, Y' = nY. Its equation becomes identical with the latter by virtue of the above equalities when we put e'^ = €'n\ 42. Ex. 1. A momental ellipse of the area of a square at its centre of gravity is easily seen to be the inscribed circle. By projecting this figure first with one side base line, and secondly with a diagonal as base, the square becomes successively a rectangle and a parallelogram. Hence one momental ellipse at the centre of gravity of a parallelogram is the inscribed conic touching the sides at their middle points. Ex. 2. By projecting an equilateral triangle into any triangle, we may infer the results of some of the previous articles, but the method will be best explained by its application to a tetrahedron. 28 MOMENTS OF INERTIA. [CHAP. I. Ex. 3. Since any ellipsoid may be obtained by projecting a sphere, we infer by Art. 38, Ex. 10, that any solid ellipsoid of mass M is equimomental to a system of four particles each of mass ^ -^ placed on a similar ellipsoid whose linear dimen- sions are n times as great as those of the material ellipsoid, so that the eccentric lines of the particles make equal angles with each other, and a fifth particle equal to the remainder of the mass of the ellipsoid placed at the centre of gravity. If this material ellipsoid be the Legendre's ellipsoid of any given body, we see that any body whatever is equimomental to a system of five particles placed as above described on an ellipsoid similar to the Legendre's ellipsoid of the body. Ex. 4. Show that a solid oblique cone on an elliptic base is equimomental to a system of three particles each one-tenth of the mass of the cone placed on the cir- cumference of the base so that the differences of their eccentric angles are equal, a fourth particle equal to three-tenths of the cone placed at the middle point of the straight line joining the vertex to the centre of gravity of the base, and a fifth particle to make up the mass of the cone placed at the centre of gravity of the volume. 43. To find an ellipsoid equimomental to any tetrahedron. The moments of inertia of a regular tetrahedron with regard to all planes through the centre of gravity are equal by Art. 23. If r be the radius of the inscribed sphere, the moment with regard to a plane parallel to one face is easily seen by Art. 39 to be M _ . If then we describe a sphere of radius p= JSr, with its centre at the centre 5 of gravity, and its mass equal to that of the tetrahedron, this sphere and the tetra- hedron will be equimomental. Since the centre of gravity of any face projects into the centre of gravity of the projected face, we infer that the ellipsoid to which any tetrahedron is equimomental is similar and similarly situated to that inscribed in the tetrahedron and touching each face in its centre of gravity, but has its linear dimensions greater in the ratio 1 : ^J'd. It may also be easily seen that the sphere whose radius is p= iJSr, touches each edge of the regular tetrahedron at its middle point. Hence we infer that the ellipsoid equimomental to any tetrahedron touches each edge at its middle point and has its centre at the centre of gravity of the volume. Ex. 1. If £2 be the sum of the squares of the edges of a tetrahedron, F- the sum of the squares of the areas of the faces and V the volume, show that the semi- axes of the ellipsoid inscribed in the .tetrahedron, touching each face in the centre of gravity and having its centre at the centre of gravity of the tetrahedron, are the roots of 6__^ 4 Z! 2_Z!_-n ^ 2^. 3 '^ "^24.32^ 26. 3~' and that, if the roots be ^Pi, ±p.2' ^P3» *^6 moments of inertia with regard to the principal planes of the tetrahedron are M -~ , M -^ , M ~~ . o o o Ex. 2. If a perpendicular EP be drawn at the centre of gravity E of any! face = 4p2/2), where p is the perpendicular from the opposite corner of the tetrahedron " on that face, then P is a point on the principal plane corresponding to the root p of the cubic. 44. Four particles of equal mass can always be found lohich are equimomental to any yiven solid body. Let O be the centre of gravity of the body, Ox, Oy, Oz, the principal axes at 0. Let the moments of inertia with regard to the coordinate planes be Ma^, 31^^, and My"^. By Art. 34, the mass of each particle must be ^M. Let (x-^yiZ^) &c. {.x^y^z^) ART. 45.] EQUIMOMENTAL BODIES. 29 bo the required coordinates of these four points. Then these twelve coordinates must satisfy the nine equations Now if we write .T-^^ = a^-^, .x-2 = a^2 &c. yi^^Vi^ Vi^^V^ *c. ^i==7^i &c. we have nine equations to find the twelve coordinates (Ii7;i^i) &c. (^47/4^4) which differ from those just written down only in having a"^, ^^, y^ each replaced by unity. These modified equations express that the momental ellipsoid at O of the four particles must be a sphere. The equations are therefore satisfied if the four points, whose coordinates are represented by the Greek letters, are the corners of a regular tetra- hedron. (See also Art. 23, Ex. 2.) This tetrahedron may be regarded as inscribed in a sphere whose radius is ,^3. If we project this sphere into an ellipsoid whose semi-axes are a, ]8, 7 the regular tetrahedron will be deformed into an oblique tetra- hedron. The corners of this oblique tetrahedron are the required equimomental points. In the same way we may prove that three particles of equal mass can always be found which are equimomental to any plane area. If 3Ia'^, M^^, and zero are the moments of inertia of the area about the principal planes at the centre of gravity, the result is that these particles must lie on the ellipse ^-x^' + a^y^ = 2a^^". It also follows that, if one of these points, as I>, be taken anywhere on this ellipse, the other two points, E and F, are at the opposite extremities of that chord which is bisected in some point N by the produced radius DO so that ON =1^01). 45. Moments with Higher Powers. These moments are not often wanted in dynamics though they are useful in other subjects. It will therefore be sufficient to state here some general results and to sketch the proofs in a note at the end of this volume. Some generalisations will also be added. Let da and dv be any elementary area and volume as the case may be. Let z be its ordinate referred to any plane of xy. Our object is to find the integral jz'^do- or jzMv for a triangle, quadri- lateral, tetrahedron, &c. Let the coordinates of the corners of the body be {x-^^yiZ^), (x^y^z^, &c. Let SniziZ^, &c.) represent the sum of the different homogeneous products of n dimensions of as many of the ^'s as are included in the bracket. Then for a triangle of area A, For a quadrilateral of area A 1 2 A where z^ is the ordinate of the intersection of the diagonals. For a tetrahedron of volume V ■^^"* = or + !)(,!+% fa +3) ^" ("■"^»"'>- .1 30 MOMENTS OF INERTIA. [CHAP For two tetrahedra joined together, whose united volume is V where / is the ordinate of the point of intersection of the common base with the straight line joining the two vertices. We notice that, except for the factor A or F representing the area or volume, these four expressions are functions of the ordinates only of the corners and are not functions of the differences of the abscissae. When the value of jz^'da- is known that of n\xz'^-^ddv for a tetrahedron can be represented by eight equivalent points. We collect nine- fortieths of the volume at the centre of gravity of each face and one-fortieth at each corner. Other examples may he found in No. 83 Quarterly Journal of Mathematics, 1886. 46. Theory of Inversion. To explain how the theory of invey^sion can he applied to find moments of inertia. Let a radius vector drawn from some fixed origin O to any point P of a figure be produced to P', where the rectangle OP .OP' = k^, k being some given quantity. Then as P travels all over the given figure, P' traces out another which is called the inverse of the given figure. Let {x, y, z) be the coordinates of P, {x', y\ z') those of P'; r, r' the radii vectores, dv, dv' corresponding polar elements of volume; p, p', dm, dm' their respective densities and masses. Let dw be the solid angle subtended at by either dv or dv'. Then dv' = r'^d(j)dr' =i-\ r^du}dr=l~\ dv, x' X //c\^^ and since — = - we have x''^dv'=[ - \ x^dv. Now dm = pdv, dm' = p'dv'. If then p' /r\^*' we take ?- = ( J we have l,x''^dm'=z'SiX^dm, with similar equalities in the case of all th^ other moments and products of inertia. When the body is an area or an arc the ratio of dv' to dv is different. We have dv' ^k\* /k\^ in these cases respectively — = ( - j or ( - J . Similar results however follow which may be all summed up in the following theorem. ART. 47.] EQUIMOMENTAL BODIES. 31 Theor. I. Let any body be changed into another by inversion loith regard to \any point 0. If the densities at corresponding points be denoted by p, p' and their \ distances from by r, r', let p' = p{-A . Then these two bodies have the same moments of inertia with regard to all straight lines through 0. Here n = 10, 8 or 6 I according as the body is a volume, an area or an arc. It also follows that the two bodies have the same principal axes at the point O, and the same ellipsoids of gyration. We may also obtain the following theorem by the use of Kelvin's method of finding the potentials of attracting bodies by Inversion. Theob. II. Let any body be changed into another body by inversion with regard to any point 0. If the densities at corresponding points P, P' be denoted by p, p', and their distances from by r, r', let p' = p(—,\ . Then the moment of inertia of the second body with regard to any point C is equal to that of the first body with -regard to the corresponding point G multiplied by either of the equal quantities or I ^' 7Tn ' -^^^^ w = 8, 6 or 4 according as the body is a volume, area, or arc. To prove this, consider the case in which the body is a volume. By similar triangles CP . r' = C'P' . OG. We then find pdv {GPf (-^j =p'dv' {G'P'f, by pro- ceeding as before. This being true for every element the theorem follows at once. Ex. The density of a solid sphere varies inversely as the tenth power of the distance from an external point 0. Prove that its moment of inertia about any straight line through is the same as if the sphere were homogeneous and its density equal to that of the heterogeneous sphere at a point where the tangent from O meets the sphere. Prove that if the density had varied inversely as the sixth power of the distance from O, the masses of the two spheres would have been equal. What is the condition that they should have a common centre of gravity ? [Math. Tripos. 47. Centre of Pressure. If a plane lamina is immersed in a homogeneous fluid it is proved in treatises on hydrostatics that the pressures on the elements of area act normally to the plane and are proportional to the product of the area of the element by the depth below a fixed horizontal plane often called "the effective surface." It easily follows from statical principles that the centre of these parallel forces lies in the plane of the lamina and is the same however the forces are turned round their points of application provided they remain parallel. This point is called in hydrostatics the centre of pressure. Let the intersection of the lamina with the effective surface be taken as the axis of x and let the axis of y be in the plane of the lamina, the axes being rectangular. Then by the common formulae for the centre of parallel forces ^ _ Product of inertia about Osc, Oy moment of the area about Ox ' ^ _ Moment of inertia about Qx ~~ moment of the area about Ox ' 32 MOMENTS OF INERTIA. [CHAP. 1. Let the given area be equimomental to particles whose masses are 7?ii, m^ &c. and let (^i, 3/1), {x.2, y^, &c. be the coordinates of these particles. Then X = =i , F = v.; . ^ Zmy zmy But these are the formulae to find the centre of gravity of particles whose masses are proportional to m^y^, in^y2 &c. having the same coordinates as before. Hence this rule, If any area he equimomental to a series of particles, the centime of pressure of the area is the centre of gravity of the same particles with their masses increased in the ratio of their depths. For example, the centre of pressure of a triangle wholly im- mersed is the centre of gravity of three weights placed at the middle points of the sides and each proportional to the depth of the point at which it is placed. In this article we confine our attention to the hydrostatical properties of the point, but we may notice that the coordinates X and Y are so useful that in dynamics also names have been given to them. It follows from the formulae (5) of the next article that X is the abscissa of the principal point of the axis of x, so that the projection of the centre of pressure of any area on its intersection with the effective surface is the principal point of that intersection. It will also be shown in Chap. ill. that the ordinate Y is equal to the distance of the centime of oscillation from the axis of suspension. In this way we can translate our hydrostatical results into dynamics, and conversely. Since the coordinates X, Y depend only on the ratio of the moments and products of inertia to the mass and on the position of the centre of gravity, it is clear that two equimomental areas have the same centre of pressure. Ex. 1. If p, q, r be the depths of the corners of a triangular area wholly immersed in a fluid, prove that the areal coordinates of its centre of pressure referred to the sides of the triangle itself are ill+i?/*), i(l + ql^), i {! + ?'/«), where s =p ■rq + r. This may be proved by replacing the triangle by three weights situated at the middle points of the sides proportional to their depths, and taking moments about the sides in succession to find their centre of gravity. Ex, 2. Let any vertical area be referred to Cartesian rectangular axes Ox, Oy, with the origin at the centre of gravity. Let the depth of the centre of gravity be h, and let the intersection of the area with the surface of the fluid make an angle with the axis of x, and let this intersection in the standard case cut the positive side of the axis of y. Let A, B and F be the moments and product of inertia of the area about the axes. Then by taking moments about Ox, Oy we see that the coordinates of the centre of pressure are B Bin 6 - F Qo^ e „ Fsmd-AaoQd "" where a is the area. ART. 48.] CENTRE OF PRESSURE. 33 Ex. 3. If the area turn round its centre of gravity in its own plane the locus of its centre of pressure in the area is an ellipse and in space is a circle. The ellipse has its principal diameters coincident in direction with the principal axes of the area at the centre of gravity. The circle has its centre in the vertical through the centre of gravity. Ex. 4. In a heterogeneous fluid the pressure at any point P referred to a unit of area is given by p=:d + hz^ where z is the depth of P. Prove that the depth of tlie centre of pressure of any triangular area wholly immersed at any inclination to tlie horizon is — ~\ , ,""^^ , where H„ is the arithmetic mean of the different aH^ + hH^ iiomogeneous products of n dimensions of the depths z-^, z.^, z^ of the three corners of the triangle. Ex. 5. In rotating fluids the pressure at any point P is given by p = a + bz + cr^ where r is the distance of P from the axis of z which is vertical. Show that the pressure on any part of the area of the containing vessel is given by (1) whole pressure = j{a + bz + cr'^) da = {a + bz) o- + ccrk^ ^/'here a is the area of the part pressed, z the depth of its centre of gravity, and ak^ ;he moment of inertia about the axis of z. (2) Vertical ^ressuYe = jj{a + bz + cr^)dxdy = aP+bV+cPk"^ vhere P is the projection of . z =z—h J [ence '%mx'z = cos Q^mxz + sin 6^myz\ _ ^ -h{cose^mx + ^\iieXmy)]~^ ^^^' ^my'z' = — sin 6%mxz + cos 62myz[ _ „ . -h{ — sm 62mx + cos 6Smy) J ^ ^' ^mx'y' = 2m (y^ — x^) — h l^mxy cos 2^ = (3). The last equation shows that tan2^ = ^^f^^=/^ (4). ^m{x^-y^) B-A ^ ^ )rding to the previous notation. R. D. 3 34 MOMENTS OF INERTIA. [CHAP. I. The equations (1) and (2) must be satisfied by the same value of h. Eliminating h we get ^mxz^my ^Xmyz^mx as the con- dition that the axis of z should be a principal axis at some point in its length. Substituting in (1) we have , Xmyz 1,mxz ,^. h= ^-^ =-^^ (o). z^my zmx The equation (5) expresses the condition that the axis of z should he a principal axis at some point in its length; and the value of h gives the position of this point. If ^mxz = and ^myz = 0, the equations (1) and (2) are both satisfied by h — 0. These are therefore the sufficient and necessary conditions that the axis of z shoidd he a principal axis at the origin. If the system be a plane lamina and the axis of ^ be a normal to the plane at any point, we have z = 0. Hence the conditions ^mxz = and ^myz = are satisfied. Therefore one of the principal axes at any point of a plane lamina is a normal to the plane at that point. In the case of a surface of revolution bounded by planes perpendicular to the axis, the axis is a principal axis at any point of its length. Again, equation (4) enables us, when one principal axis is given, to find the other two. If 6 = a be the first value of 0, all the others are included in 6 = a + in7r; hence all these values give only the same axes over again. 49. Since (4) does not contain h, it appears that if the axis of ^ be a principal axis at more than one point, the principal axes at those points are parallel. Again, in that case (5) must be satisfied by more than one value of h. But, since h enters only in the first power, this cannot be unless ^mx — 0, l.my = 0, ^mxz = 0, Xmyz — ; so that the axis must pass through the centre of gravity and be a principal axis at the origin, and therefore (since the origin is arbitrary) a principal axis at every point in its length. If the principal axes at the centre of gravity be taken as the axes of X, y, z, (1) and (2) are satisfied for all values of h. Hence, if a straight line be a principal axis at the centre of gravity, it is a principal axis at every point in its length. If the given straight line is parallel to a principal axis at the centre of gravity G, it is easy to see that the given line is a principal axis at the projection of G on itself. For let the origin be taken at the projection, and let dff, Gri, G^ be a parallel system of axes, then since S^iff, ^my^ and z are zero, it follows from Art. 13 that Xmxz and Xmyz are also zero. 50. Let the system be projected on a plane perpendicular to the given straight line, so that the ratios of the elements of mass ART. 52.] POSITIONS OF THE PRINCIPAL AXES OF A SYSTEM. 35 to each other are unaltered. The given straight line, which has been taken as the axis of z, cuts this plane in 0, and will be a principal axis of the projection at 0, because, the projected system being a plane lamina, the conditions ^mxz = 0, %myz = are both satis tied. Since z does not appear in equation (4), it follows that, if the given straight line be a principal axis at some point G in its length, the other two principal axes at G will be parallel to the prhicipal axes of the projected system at 0. These last may often be conveniently found by the next proposition. 51. Ex. 1. The principal axes of a right-angled triangle at the right angle are, one perpendicular to the plane and two others inclined to its sides at the angles - tan~i -^r— , „ , where a and h are the sides of the triangle adjacent to the 2 a- - 6^ [right angle. We have tan 26 = ^^' . , Art. 48, and by Art. 35, ^ = ill ^, B = J\A, F=M^. Ex. 2. The principal axes of a quadrant of an ellipse at the centre are, one fperpendicular to the plane and two others inclined to the principal diameters at the ►angles - tan~i - —^ — — , , where a and h are the semi-axes of the ellipse. Ex. 3. The principal axes of a cube at any point P are, the straight line ^joining P to the centre of gravity of the cube, and any two straight lines at P fperpendicular to PO, and perpendicular to each other. ; Ex. 4. Prove that the locus of a point P at which one of the principal axes is I parallel to a given straight line is a rectangular hyperbola in the plane of which the ' centre of gravity of the body lies, and one of whose asymptotes is parallel to the given straight line. But if the given straight line be parallel to one of the principal axes at the centre of gravity, the locus of P is that principal axis or the perpen- :ilicular principal plane. Take the origin at the centre of gravity, and one axis of coordinates parallel to the given straight line. Ex. 5. The principal point of any side AB of a triangular area ABC bisects ithe distance between the middle point of that side and the foot of the perpendicular tfrom the opposite corner on the side. Ex. 6. An edge of a tetrahedron will be a principal axis at some point in its length only when it is perpendicular to the opposite edge. [Jullien. Conversely, if this condition be satisfied, the edge will be a principal axis at ii point C, such that OC = ^ON, where N is the middle point of the edge and is he foot of the perpendicular distance between it and the opposite edge. Ex. 7. The axes Ox, Oij are so placed that the product of inertia F or Sm-rr/ s zero. If A and B are the moments of inertia about these axes, prove that the iOroduct of inertia about two perpendicular axes Ox', Oy' in the plane xy is F' = ^{A-B)Qin2d where 6 is the angle xOx' measured in the positive direction from Ox, 52. Foci of Inertia. Given^the positions of the principal axes Ox, Oy, Oz at the centre of gravity 0, and the moments of Inertia about them, to find the positions of the principal axes at any ooint P in the plane of xy, and the moments of inertia about them. 3—2 36 MOMENTS OF INERTIA. [chap. I. Let the mass of the body be M, and let A, B be the moments of inertia about the axes Ox, Oy, of which we shall suppose A the greater. Let S, H be two points in the axis of greate^mo- / A -^ Ti ment, one on each side of the origin so that 0S= OH = ^ "^^• These may he called the foci of inertia for that principal plane. Because these points are in one of the principal axes at the centre of gravity, the principal axes at 8 and H are parallel to the axes of coordinates, and the moments of inertia about those in the plane of xy are respectively J. and B + M.OS''=-A. These being equal, any straight line through S or H in the plane of xy is a principal axis at that point, and the moment of inertia about it is equal to A. See Arts. 16 and 23. If P be any point in the plane of xy, then one of the principal axes at P will be perpendicular to the plane xy. For, if p, q be the coordinates of P, the conditions that this line should be a principal axis are Sm (x-p)z = 0, 2m (y-q)2 = 0, which are obviously satisfied, because the centre of gravity is the origin, and the principal axes the axes, of coordinates. The other two principal axes may be found thus. If two straight lines meeting at a point P be such that the moments of inertia about them are equal, then, provided they are in a principal plane, the principal axes at P bi- sect the angles between these two straight lines. For, if with centre P we describe the mo- mental ellipse, the axes of this ellipse bisect the angles between any two equal radii vectores. Join SP and HP ; the moments of inertia about SP, HP are each equal to A. Hence, if PG and PT are the internal and external bisectors of the angle SPH, PG, PT are the principal axes at P. If therefore with S and H as fOci we describe any ellipse or hyperbola, the tangent and normal at any point are the principal axes at that point. 53. Take any straight line MN ^through the origin, making an angle 6 with the axis of x. Draw SM, HN perpendiculars on MN. The moment of inertia about 3IN is =Acos'^e + Bsin^ d = A - {A -B)sm^ d = A-M.{OS sin ef = A-M. S]\P. I ART. 55.] POSITIONS OF THE PRINCIPAL AXES OF A SYSTEM. 37 Through P draw FT parallel to MN, and let SY and HZ be the perpendiculars from S and H on it. The moment of inertia about PT is then ^ moment about MN + M . MY'^ I =A + M{MY- SM) {MY +SM) I =A + M.SY.HZ. In the same way it may be proved that the moment of inertia about a line PG passing between H and S is less than A by the mass into the product of the perpen- diculars from S and H on PG. I If therefore with S and H as foci we describe any ellipse or hyperbola, the moment j of inertia about any tangent to either of these curves is constant. It follows from this that the moments of inertia about the principal axes at P are equal to B + ^M {SP a.HP)^ For if a and b be the axes of the ellipse we have a^-b'^ = 0S^= {A-B)IM, and hence A + M . SY . HZ = A + Mb'^ = B + Ma^ = B + IM {SP + HP)^, and the hyperbola may be treated in a similar manner. 54. This reasoning may be extended to points lying in any given plane passing through the centre of gravity of the body. Let Ox, Oy be the axes in the given plane such that the product of inertia about them is zero (Art. 23). Construct the points S and H as before, so that OS'^ and OH'^ are each equal to the difference of the moments of inertia about Ox and Oy divided by the mass. Draw Sy' a parallel through S to the axis of y, the product of inertia about Sx, Sy' is equal to that about Ox, Oy together with the product of inertia of the whole mass collected at O. Both these are zero, hence the section of the momental ellipsoid at S is a circle, and the moment of inertia about every straight line through >S^ in the plane xOy is the same and equal to that about Ox. We can then show that the moments of inertia about PH and PS are equal ; so that PG, PT, the internal and external bisectors of the angle SPH, are the principal diameters of the section of the mo- mental ellipsoid at P by the given plane. And it also follows that the moments of inertia about the tangents to a conic whose foci are S and H are the same. 55. Ex. 1. To find the foci of inertia of an elliptic area. The moments of inertia about the major and minor axes are ^Mb^ and |Ma-. Hence the minor axis is the axis of greatest moment. The foci of inertia therefore lie in the minor axis at a distance from the centre = ^ fja- - f^, i.e. half the distance of the geometrical foci from the centre. Ex. 2. Two particles each of mass m are placed at the extremities of the minor . axis of an elliptic area of mass M. Prove that the principal axes at any point of I the circumference of the ellipse will be the tangent and normal to the ellipse, pro- m 5 e^ vided that M 8 1 , Ex. 3. At the points which have been called foci of inertia tico of the principal [ moments are equal. Show that it is not in general true that a point exists such that the moments of inertia about all axes through it are the same, and find the conditions that there may be such a point. Such points when they exist in a solid \ body may be called the spherical points of inertia of that solid. [ Refer the body to the principal axes at the centre of gravity. Let P be the point i required, {x, y, z) its coordinates. Since the momental ellipsoid at P is to be a * sphere, the products of inertia about all rectangular axes meeting at P are zero, i Hence, by Art. 13, xy = 0, yz — 0^ zx — 0. It follows that two of the three x, y, z 38 MOMENTS OF INERTIA. [CHAP. I. must be zero, so that the point must be on one of the principal axes at the centre of gravity. Let this be called the axis of z. Since the moments of inertia about three axes at P parallel to the coordinate axes are A+3Iz^, B + Mz"^ and C, we see that these cannot be equal unless A=B and each is less than C. There are then two points on the axis of unequal moment which are equimomental for all axes. [Poisson and Binet. Ex. 4. The spherical points of a hemispherical surface are the centre and a point on the surface. Find also the spherical points of a solid hemisphere. By Art. 5, Ex. 8, the moments of inertia about every axis througVi the centre are the same. Hence the centre is one spherical point. Since the centre of gravity bisects the distance between the points the position of the other follows at once. 56. Arrangement of Principal Axes. Given the positions of the principal axes at the centre of gravity and the moments of inertia about them, to find the positions of the principal axes* and the principal moments at any other point P. Let the body be referred to its principal axes at the centre of gravity 0, let A, B, C be its principal moments, the mass of the body being taken as unity. Construct a quadric confocal with the ellipsoid of gyration, and let the squares of its semi-axes be a^ = A -j-\, ¥ = B + \ c^ = G -\-X. Let us find the moment of inertia with regard to any tangent plane. Let (a, /S, 7) be the direction angles of the perpendicular to any tangent plane. The moment of inertia, with regard to a parallel plane through 0, is i(A+B-\- G)-{A cos^ a-\-B cos^ /3 + cos" 7). The moment of inertia, with regard to the tangent plane, is found by adding the square of the perpendicular distance between the planes, viz. (A + X) cos^ a + (i^ + X) cos- /3 -{-(G + X) cos^ 7. We get moment of inertia with) ti r'\ \ regard to a tangent plane] - 2 {^ + B -{- C) + \ = ^(B-\-G-A)-\-a\ Thus the moments of inei^tia with regard to all tangent planes to any one quadric confocal with the ellipsoid of gyration are the same. These planes are all principal planes at the point of contact. For draw any plane through the point of contact P, then in the case in which the confocal is an ellipsoid, the tangent plane parallel to this plane is more remote from the origin than this plane. Therefore, the moment of inertia with regard to any plane through P is less than the moment of inertia with regard to a tangent plane to the confocal ellipsoid through P. That is, the tangent plane to the ellipsoid is the principal plane of greatest moment. In the same way the tangent plane to the confocal * Some of the following theorems were given by Lord Kelvin and Mr Townsend, in two articles which appeared at the same time in the Mathematical Journal, 1846. Their demonstrations are different from those given in this treatise. The theorem that the principal axes at P are normals to the three confocals is now ascribed in Thomson and Tait's Treatise on Natural Philosophy to Binet, Journal de VEcole Polytechnique, 1811. AUT. 59.] POSITIONS OF THE PRINCIPAL AXES OF A SYSTEM. 39 hyperboloid of two sheets through P is the principal plane of least moment. It follows that the tangent plane to the confocal hyperboloid of one sheet is the principal plane of mean moment. Through a given point P, three confocals can he drawn, and the normals to these confocals are the principal axes at P. By Art. 5, Ex. 3, the pi'incipal axis of least moment is normal to the confocal ellipsoid and that of greatest moment normal to the confocal hyper- boloid of two sheets. 57. The moment of inertia with regard to the point P is, by Art. 14, i (J. + 5 + 0) + 0P\ Hence, by Art. 5, Ex. 3, the moments of inertia about the normals to the three confocals through P whose parameters are Xi, X.2» ^3 are respectively 0P^-\^, OP'--X,, OP2-X3. 58. If we describe any other confocal and draw a tangent cone to it whose vertex is P, the axes of this cone are known to be the normals to three confocals through P. This gives another con- struction for the principal axes at P. If the confocal diminish without limit, until it becomes a focal conic, we see that the principal axes of the system at P are the principal diameters of a cone whose vertex is P and base a focal conic of the ellipsoid of gyration at the centre of gravity. Ex. Prove that the moment of inertia about any generator of the cone, vertex P, reciprocal to the tangent cone drawn from P to the ellipsoid of gyration is the same. [Math. Tripos, 1895. 59. If we wish to use only one quadric, w^e may consider the confocal ellipsoid through P. We know* that the normals to the other two confocals are tangents to the lines of curvature on the ellipsoid, and are also parallel to the principal diameters of the diametral section made by a plane parallel to the tangent plane at P. And if A 5 A be these principal semi-diameters, we know that A2 = Xi — JJi , X3 = Xi — -L/2". Hence, if through any point P we describe the quadric x^ y^ z^ A +X 5+X C+X the axes of coordinates being the principal axes at the centre of gravity, then the principal axes at P are the normal to this quadric, and parallels to the axes of the diametral section made by a plane parallel to the tangent plane at P. And if these axes are 2i)i and 21)3, the principal moments at P are 0P'^-\, OP^-\^-D,\ OP'-\ + D,\ * A geometrical proof of the propositions required for this article was given in the former editions, but these results are now too well known to render this necessary. 40 MOMENTS OF INERTIA. [CHAP. I. Ex. If two bodies have the same centre of gravity, the same principal axes at the centre of gravity and the differences of their j^rincipal moments equal, each to each, then these bodies have the same principal axes at all points. 60. Condition that a line should be a principal axis. The axes of coordinates being the principal axes at the centre of gravity it is required to express the condition that any given straight line may be a principal axis at some point in its length and to find that point. Let the equations of the given straight line be ^-f ^y-9 ^^-^' (1)^ I m n then it must be a normal to some quadric 0^ y'^ z^ at the point at which the straight line is a principal axis. Hence comparing the equation of the normal to (2) with (1), we ^^''^ Z^ = ^^' 5TX='^"'' CT"X = '''' ^^^- These six equations must be satisfied by the same values of x, y, z, \ and fjL. Substituting for x, y, z from (3) in (1), we get '^ I m n Equating the values oi fi given by these equations we have f_9_ 1_^ ^_/ I m _7n V _n I ... A^rB-T^'TTTA ^ ^' This clearly amounts to only one equation, and is the required condition that the straight line should be a principal axis at some point in its length. Substituting for x, y, z from (8) in (2), we have \ (l- + m- + n-) = -^-{Al^-\- Bm' + Cn-), which gives one value only to X. The values of X and fi having been found, equations (3) will determine x, y, z the coordinates of the point at which the straight line is a principal axis. The geometrical meaning of this condition may be found by the following considerations, which were given by Townsend in the Mathematical Jownal. The normal and tangent plane at every point of a quadric will meet any principal plane in a point and a straight line, which are pole and polar with regard to the focal conic in that plane. Hence, to find whether any assumed straight line is a principal axis or not, draw any plane perpendicular to the straight line and produce both the straight line and the plane to meet any principal plane at the centre of gravity. If the line of intersection of the plane be parallel to the polar ART. 62.] POSITIONS OF THE PRINCIPAL AXES OF A SYSTEM. 41 line of the point of intersection of the straight line with respect to the focal conic, the straight line will be a principal axis, if otherwise it will not be so. And the point at which it is a principal axis may be found by drawing a plane through the polar line perpendicular to the straight line. The point of intersection is the required point. The analytical condition (4) exactly expresses the fact that the polar line is jparallel to the intersection of the plane. 61. Ex. 1. Show that the straight line a{x-a) = b{y -b)=c [z -c) is at some point in its length a principal axis of an ellipsoid whose semi-axes are a, b, c. Ex. 2. Show that any straight line drawn on a lamina is a principal axis of that lamina at some point. Where is this point if the straight line pass through the centre of gravity? Ex. 3. Given a iplane fx + gy + hz -1 = 0, there is always some point in it at iwhich it is a principal plane. Also this point is its intersection with the straight iline xlf-A=ylg-B = zlh-G. Ex. 4. Let two points P, Q be so situated that a principal axis at P intersects a principal axis at Q. Then if two planes be drawn at P and Q perpendicular to these principal axes, their intersection will be a principal axis at the point where it is cut by the plane containing the principal axes at P and Q. [Townsend. For let the principal axes at P, Q meet any principal plane at the centre of (gravity in _p, q, and let the perpendicular planes cut the same principal plane in LN, MN. Also let the perpendicular planes intersect each other in RN. Then RN is perpendicular to the plane containing the points P, Q, p, q. Also since the polars of ^ and q are LN, MN, it follows that pq is the polar of the point N, Hence the straight line RN satisfies the criterion of the last Article. Ex. 5. If P be any point in a principal plane at the centre of gravity, then every axis which passes through P, and is a principal axis at some point, lies in one of two perpendicular planes. One of these planes is the principal plane at the centre of gravity, and the other is a plane perpendicular to the polar line of P with regard to the focal conic. Also the locus of all the points Q at which QP is a prin- cipal axis is a circle passing through P and having its centre in the principal plane. [Townsend. Ex. 6. The edge of regression of the developable surface which is the envelope of the normal planes of any line of curvature drawn on a confocal quadric is a curve such that all its tangents are principal axes at some point in each. 62. Locus of equal Moments. To find the locus of the points at which two principal moments of inertia are equal to each other. The principal moments at any point P are If we equate I^ and I2 we have A = 0, and the point P must lie on the elliptic focal conic of the ellipsoid of gyration. If we equate /g and Is we have i)i = A, so that P is an um- bilicus of any ellipsoid confocal with the ellipsoid of gyration. The locus of these umbilici is the hyperbolic focal conic. In the first of these cases we have X = — C, and Do is the semi- 42 MOMENTS OF INERTIA. [CHAP. I. diameter of the focal conic conjugate to OP. Hence D^ -{■ 0F^ = sum of squares of semi-axes = A — G-\-B—G. The three principal moments are therefore I^ = 1^= OP'^ -{- C, 1^= A + B - C, and the axis of unequal moment is a tangent to the focal conic. The second case may be treated in the same way by using a confocal hyperboloid, we therefore have Jg = /g = OP- + i?, I^ = A -\- C— B, and the axis of unequal moment is a tangent to the focal conic. These results follow also by combining Arts. 57 and 58. The cone which envelopes the ellipsoid of gyration and has its vertex at P must by these articles be a right cone if two principal moments at P are equal. But we know from solid geometry that this only happens when the vertex lies on a focal conic, and the un- equal axis is then a tangent to that conic. 63. To find the curves on any confocal quadric at which a principal moment of inertia is equal to a given quantity I. Firstly. The moment of inertia about a normal to a confocal quadric is OP'^-X. If this be constant, we have OP constant, and therefore the required curve is the intersection of that quadric with a concentric sphere. Such a curve is a sphero-conic. Secondly. Let us consider those points at which the moment of inertia about a tangent is constant. Construct any two confocals whose semi-major axes are a and a'. Draw any two tangent planes to these which cut each other at right angles. The moment of inertia about their intersection is the sum of the moments of inertia with regard to the two planes, and is therefore B -\- C - A + a'^ + a'^. Thus the moments of inertia about the intersections of perpendicular tangent planes to the same confocals are equal to each other. Let a, a', a" be the semi-major axes of the three confocals which meet at any point P, then since confocals cut at right angles the moment of inertia about a tangent to the intersection of the confocals a', a" is I-^=:B + C - A-\-a''^-\-a"'^. The intersection of these two confocals is a line of curvature on either. Henc the moments of inertia about the tangents to any line of curvature are equal to o another; and these tangents are principal axes at the point of contact. On the quadric a draw a tangent PT making angles and ^tt - (f> with the tangents to the lines of curvature at the point of contact P. If I^, Ig be the moments about the tangents to these lines of curvature, the moment of inertia about the tangent Pr=l2cos2 0-f l3sin-0 =^B + C -A + (a"2 -f a^) cos^ + {a- + a'-) sin^ 0. But, along a geodesic on the quadric a, a'^ sin^ + a"^ cos'-^ is constant. Hence the moments of inertia about the tangents to any geodesic on tht quadric are equal to each other. 64. Ex. 1. If a straight line touch any two confocals whose semi-major axes are a, a', the moment of inertia about it is 7i + C - ^ + a- + a"^. Ex. 2. When a body is referred to its principal axes at the centre of gravity, show how to find the coordinates of the point P at which the three principal moments are equal to the three given quantities 1-^,1^,1.^. [Jullien's Problem. a VllT. 65.] POSITIONS OF THE PRINCIPAL AXES OF A SYSTEM. 43 The elliptic coordinates of P are evidently a'^ = \{I^ + I^-I^-B-C + A), &c. ; II id the coordinates [x^ y, z) may then be found by Salmon's formulae, Ex. 3. Let two planes at right angles touch two confocals whose semi-major ixes are a, a' ; and let a, a' be the values of a, a' for confocals touching the inter- section of the planes; then a^ + a'^ = ar + a.''-, and the product of inertia with regard ;o the two planes is (a-a'^-a^a'^)^. 65. Equixnomental Surface. The locus of all those points at which one of ;he principal moments of inertia of the body is equal to a given quantity is called m equimomental surface. To find the equation to such a surface we have only to put I^ constant, this ,'ives \ = r'^-I. Substituting in the equation of the confocal quadric, the equation )f the surface becomes r2 ?;2 + r + x^ + y'^ + z- + A-I x^ + y- + z^ + B-I x^ + y^ + z'^+C - 1 Through any point P on an equimomental surface describe a confocal quadric ;^ + B^ = A+B-\-G + ^\ therefore D^^>B + C + 1\, which is >^ + 2\. Hence D<^> the greatest radius vector of the ellipsoid, which is impossible. Ex. 2. Find the locus of all those points in a body at which (1) the sum of the principal moments is equal to a given quantity I, (2) the sum of the products of the principal moments taken two and two together is equal to I^, (3) the product of the principal moments is equal to I^. The results are (1) by Art. 13, a sphere whose radius is {{I- A -B- C)IM}^, (2) by Art. 65, the surface {x' + y^ + z')'^ + {A+B + C){x' + y^ + z^) + Ax'^ + By^+Cz^ + AB + BC + CA = r\ (3) the surface A'B'C - A'lfz'^ - B'z-x'^ - C'x'^y^ - 2xhfz^ = r'^, where A' = A + rj^ + z'^, with similar expressions for B', C, CHAPTEH II. D'ALEMBERT'S PRINCIPLE, ETC. 66. The principles, by which the motion of a single particle under the action of given forces can be determined, will be found discussed in any treatise on dynamics of a particle. These prin- ciples are called the three laws of motion. It is shown that if {a:, y, z) be the coordinates of the particle at any time t referred to three rectangular axes fixed in space, m its mass, X, F, Z the forces resolved parallel to the axes, the motion may be found by ,'ing the simultaneous equations, SOIV ™di^=^' ™^=^' '"d«i=^- If we regard a rigid body as a collection of material particles connected by invariable relations, we may write down the equa- tions of the several particles in accordance with the principles just stated. The forces on each particle are however no longer known, some of them being due to the mutual actions of the particles. We assume (1) that the action between two particles is along the line which joins them, (2) that the action and reaction between any two are equal and opposite. Suppose there are n particles, then there will be 'in equations, and, as shown in any treatise on statics, Zn — 6 unknown reactions. To find the motion it will be necessary to eliminate these unknown quantities. We shall thus obtain six resulting equations, and these will be shown, a little further on, to be sufficient to determine the motion of the body. When there are several rigid bodies which mutually act and react on each other the problem becomes still more complicated. But it is unnecessary for us to consider in detail either this or the preceding case, for D'Alembert has proposed a method by which : all the necessary equations may be obtained without writing down tthe equations of motion of the several particles, and without making any assumption as to the nature of the mutual actions i except the following, which may be regarded as a natural conse- fquence of the laws of motion : The internal actions and reactions of any system of ligid bodies in motion are in equilibrium amongst themselves. 46 D'ALEMBERT'S principle. [chap. II. 67. To explain D'Alembert's principle. In the application of this principle it will be convenient to use the term effective force, which may be defined as follows. When a particle is moving as part of a rigid body, it is acted on by the external impressed forces and also by the molecular reactions of the other particles. If we consider this particle to be separated from the rest of the body, and all these forces re- moved, there is some one force which, under the same initial conditions, would make it move in the same way as before. This force is called the effective force on the particle. It is evidently the resultant of the impressed and molecular forces on the particle. Let m be the mass of the particle, (x, y, z) its coordinates referred to any fixed rectangular axes at the time t. The accele- rations of the particle are -^-^ , -~ and -^ . Let / be the resultant of these, then, as explained in dynamics of a particle, the effective force is measured by mf. Let F be the resultant of the impressed forces, R the resultant of the molecular forces on the particle. Then 7w/ is the resultant of F and R. Hence if mf be reversed, the three F, R and mf are in equilibrium. We may apply the same reasoning to every particle of each body of the system. We thus have a group of forces similar to R, a group similar to F, and a group similar to mf the three groups forming a system of forces in equilibrium. Now by D'Alembert's principle the group R will itself form a system of forces in equili- brium. Whence it follows that the group F will be in equilibrium with the group mf Hence If forces equal to the effective forces hut acting in exactly opposite directions were applied at each point of the system these would he in equilibrium with the impressed forces. By this principle the solution of a dynamical problem is reduced to that of a problem in statics. The process is as follows. We first choose some quantities by means of which the position of the system in space may be determined. We then express the effective forces on each element in terms of these quan- tities. These, when reversed, will be in equilibrium with the given impressed forces. Lastly, the equations of motion for each body may be formed, as is usually done in statics, by resolving in three directions and taking moments about three straight lines. 68. Before the publication of D'Alembert's principle a vast number of dynamical problems had been solved. These may be found scattered through the early volumes of the Memoirs of St Petersburg, Berlin and Paris, in the works of John Bernoulli and the Opuscula of Euler. They require for the most part the determi- AiiT. 69.] d'alembert's principle. 47 iKition of the motions of several bodies with or without weight which push or pull each other by means of threads or levers to which they are fastened or along which they can glide, and which having a certain impulse given them at first are then left to themselves or are compelled to move in given lines or surfaces. The postulate of Huyghens, " that if any weights are put in motion by the force of gravity they cannot move so that the centre of gravity of them all shall rise higher than the place from which it descended," was generally one of the principles of the solution : but other principles were always needed in addition to this, and it required the exercise of ingenuity and skill to detect the most suitable in each case. Such problems were for some time a sort of trial of strength among mathe- maticians. The Traite de dynamique published by D'Alembert in 1743 put an end to this kind of challenge by supplying a direct and general method of resolving, or at least throwing into equations, any imaginable problem. The mechanical diffi- culties were in this way reduced to difficulties of pure mathematics. See Montucla, Vol. III. page 615, or Whewell's version in his History of the Inductive Sciences. D'Alembert uses the following words : — " Soient A, B, C, &c. les corps qui com- posent le systeme, et supposons qu'on leur ait imprim^ les mouvemens, a, b, c, &c. qu'ils soient forces, a cause de leur action mutuelle, de changer dans les mouvemens a, b, c, &c. II est clair qu'on pent regarder le mouvement a imprime au corps A comme compose du mouvement a, qu'il a pris, et d'un autre mouvement a ; qu'on pent de meme regarder les mouvemens b, c, &c. comme composes des mouvemens b, /3; c, 7 ; &c., d'ou il s'ensuit que le mouvement des corps A, B, C, &g. entr'eux auroit ete le meme, si au lieu de leur donner les impulsions a, b, c, on leur eut donne a-la-fois les doubles impulsions a, a ; b, ^ ; &c. Or par la supposition les corps A, B, C, &c. ont pris d'eux-memes les mouvemens a, b, c, &c. done les mouve- mens o, j8, 7, &c. doivent §tre tels qu'ils ne derangent rien dans les mouvemens a, b, c, &c. c'est-a-dire que si les corps n'avoient re(?u que les mouvemens a, /3, y, &c. ces mouvemens auroient du se detruire mutuellement, et le systeme demeurer en repos. De la resulte le principe suivant pour trouver le mouvement de plusieurs corps qui agissent les uns sur les autres. Decomposez les mouvemens a, b, c, &c. imprimes a chaque corps, chacun en deux autres a, a ; b, /3 ; c, 7 ; &c. qui soient tels que si Ton n'eut imprime aux corps que les mouvemens a, b, c, &c. ils eussent pu conserver les mouvemens sans se nuire reciproquement ; et que si on ne leur eut imprime que les mouvemens a, ^, 7, &c. le systeme fut demeure en repos ; il est clair que a, b, c, &c. seront les mouvemens que ces corps prendront en vertu de leur action. Ce qu'il falloit trouver." 69. The following remarks on D'Alembert's principle have been supplied by Sir G. Airy: I have seen some statements of or remarks on this principle which appear to me to be erroneous. The principle itself is not a new physical principle, nor any addition to existing physical principles ; but is a convenient principle of combination of mechanical considerations, which results in a comprehensive process of great elegance. The tacit idea, which dominates through the investigation, is this : — That every mass of matter in any complex mechanical combination may be conceived as containing in itself two distinct properties :— one that of connexion in itself, of susceptibility to pressure-force, and of connexion with other such masses, but not of inertia nor of impressions of momentum : — the other that of discrete molecules of matter, held in their places by the connexion-frame, susceptible to externally impressed momentum, and possessing inertia. The unioii produces an imponderable skeleton, carrying ponderable particles of matter. I 48 d'alembert's principle. [chap. II. Now the action of external momentum-forces on any one particle tends to produce a certain momentum-acceleration in that particle, which (generally) is not allowed to produce its full effect. And what prevents it from producing its full effect? It is the pressure of the skeleton-frame, which pressure will be measured by the difference between the impressed momentum-acceleration and the actual momentum-acceleration for the same. Thus every part of the skeleton sustains a pressure-force depending on that difference of momenta. And the whole mechanical system, however complicated, may now be conceived as a system of skeletons, each sustaining pressure-forces, and (by virtue of their combination) each impressing forces on the others. And what will be the laws of movement resulting from this connexion ? The forces are pressure-forces, acting on imponderable skeletons, and they must balance according to the laws of statical equihbrium. For if they did not, there would be instantaneous change from the understood motion, which change would be accompanied with instantaneous change of momentum-acceleration of the mole- cules, that would produce different pressures corresponding to equilibrium. (It is to be remarked that momentum cannot be changed instantaneously, but momentum-acceleration can be changed instantaneously.) We come thus to the conclusion that, taking for every rnolecule the dif- ference between the impressed momentum-acceleration and the actual momentum- acceleration, those differences through the entire machine will statically balance. And — combining in one group all the impressed momentum-accelerations, and in another group all the actual momentum-accelerations — it is the same as saying that the impressed momentum-accelerations through the entire machine will balance the actual momentum-accelerations through the entire machine. This is the usual expression of D'Alembert's principle. 70. The ordinar}^ notation for the successive differential co- efficients of a function is very convenient when we are not always using the same independent variable. In a treatise on dynamics the time is usually the independent variable, and it is unnecessary to be continually calling attention to that fact. For this reason it is usual to represent the successive differential coefficients with regard to the time by accents or dots or some other marks placed over the dependent variable. It will be convenient to restrict the dot notation to represent differentiations with regard to the time dx dj~ 00 solely, thus x and x will be simply abbreviations for -^ and -5— . Dots will never be used to represent differentiations with regard to any quantity other than the time. When any other abbreviations are used for differential coefficients they will be preceded by an explanation. This abbreviated notation is very convenient in working examples or w^henever mistakes cannot be produced by an occasional error in the dots. But in stating results to which reference has afterwards to be made, or in which it is important that there should be no misconception as to the meaning, it will be found better to use the more extended notation. 71. Example of D^Alembert^s principle. A light rod ART. 71.] d'alembert's principle. 4>9 OAB can turn freely in a vertical plane about a smooth fixed hinge at 0. Two heavy particles whose masses are m and m! are attached to the rod at A and B and oscillate with it. It is required to find the motion. The oscillatory motion of a single particle is usually discussed in treatises on elementary dynamics. It is proved that the time 3f a small oscillation is proportional to the square root of the radius of the circle described. In our problem we have two particles describing circular arcs of different radii in the same 3ime. Each particle must therefore modify the motion of the 3ther. The particle with the shorter radius hastens the motion 3f the other and is itself retarded by the slower motion of that )ther. Our object is to find the resulting motion. By using D'Alembert's principle we are able to change this iynamical problem into an ordinary statical question, which when solved by the rules of statics gives the differential equations of 3he motion. Let OA = a, OB = b, and let the angle the rod OAB makes vith the vertical O2 be 6. The particle A describes a circular arc, lence its effective forces are known by elementary dynamics to )e mad and ma6'\ the former being directed along a tangent to the 'jrcular arc in the direction in which 6 increases and the latter blong the radius AO inwards. Similarly the effective forces of he particle B are m^bd and mfbv'^ along its tangent and radius especti vely. The directions of these effective forces are represented n figure 1 by the double-headed arrows, while the single-headed Fig. (1). Fig. (2) Kg. (3) rrows indicate the directions of the weights mg and m'g of the •articles. By D'Alembert's principle the four effective forces when versed are in equilibrium with the weights of the particles. o avoid introducing the unknown reaction at and those j^etween the particles and the rod, let us take moments for the Vhole system about 0. The forces maQ"^ and m'bO'^ being directed 'long BAO have no moments. The moments of the other two re ma? 6 and m'b'^d. Reversing these and adding the moments of he weights we have {ma^ + mb")d -\- {ma -{■ m'b)gsm6 = (1). K. D. 4 50 d'aLEMBERT'S principle. [chap. II. This is the dififerential equation of motion. When it has been solved and the two arbitrary constants determined by the initial conditions we shall have 6 expressed as a function of the time. But without entering here into the analytical solution we may shortly obtain the result. We notice that if we put m' = and write I for a, the equation (1) must give the motion of a single particle oscillating in a circle of radius l. This motion is therefore given by W+gsme = (2). This is of the same form as the equation (1). Hence the rod OAB oscillates as if the two particles were joined together into a single particle and placed at a distance I = t^ from the ° ^ ^ ma -^ mb hinge 0. As a variation on this problem, let us find the motion when the rod OAB moves round the vertical as a conical pendidum with uniform angular velocity, the angle 6 which OAB makes with the vertical being constant. In this problem also the particles describe circles, but their planes are horizontal and their centres are at E and F as repre- sented in fig. 2. The motion round the vertical being uniform, the effective force of A resolved along the tangent to its path is zero, while the effective force along its radius AE inwards is m.a sin 64>^, being the angle made by the plane zOA with any fixed plane passing through Oz. Similarly the whole effective force on B is directed along its radius BF and is equal to m'6sin d^. The directions of these effective forces are represented by the double-headed arrows in fig. 2. Reversing these and taking moments as before about 0, we have — {ma^ -f- m¥) sin 6 cos 6^'^ -H {ma + m'b)g sin ^ = 0. Hence the angular velocity cj) of the plane zOA round the vertical is given by . ^ (ma-\-m'b)g ^ (7na' + m'b^)cose ^ ^' except when the rod is vertical. In this case again the result shows that the motion of the rod OAB round the vertical is the same as if the particles were collected into a single particle and placed at the same distance from as in the first problem. In these problems we have followed the rule given in Art. 67. We first express the effective forces by using the results given in treatises on dynamics of a particle. We reverse these effective forces and express by equations the conditions of equilibrium. These equations are the equations of motion. ART. 72.] GENERAL EQUATIONS. 51 Ex. 1. If three particles are attached to the rod at different distances from O, find the motion, (1) when the system oscillates in a vertical plane, and (2) when it revolves uniformly round the vertical. Ex. 2. If the two particles are attached to O by two strings OA, AB as shown in fig. 3, and the system revolves round the vertical with a uniform angular velocity ^, show that (m . AE . OE + m' .BF. OF) ip^=(rn.AE + m'. BF) g. 72. General Equations of Motion. To apply D'Alemhert's principle to obtain the equations of motion of a system of rigid bodies. Let {x, y, z) be the coordinates of the particle m at the time t referred to any set of rectangular axes fixed in space. Then -T-1 > -rr , and -7— will be the accelerations of the particle. Let dt^ dt^ dt^ ^ X, Y, Z be the impressed accelerating forces on the same particle resolved parallel to the axes. By D'Alembert's principle the forces m(X-5), "^(F-g), ™(^-g). together with similar forces on every particle, will be in equi- librium. Hence by the principles of statics we have the equation and two similar equations for y and z ; these are obtained by resolving parallel to the axes. Also we have iand two similar equations for zx and xy ; these are obtained by Staking moments about the axes. These equations may be written in the more convenient forms d ^ dx ^ ^\ -TTZm^7 = ZmX \ dt dt (A), dt dt d ^ dz ^ dt dt In a precisely similar manner, by taking the expressions for ihe accelerations in polar coordinates, we should have obtained mother but equivalent set of equations of motion. 4—2 ■(B). 52 D'ALEMBERT'S principle. [chap. II. 73. Coordinates of a body. The equations of motion of Art. 72 are the general equations of motion of any dynamical system. They are, however, extremely inconvenient in their present form. When the system considered is a rigid body and not merely a finite number of separate particles, the S's are all definite inte- grals. There are also an infinite number of ^'s, y's and z's all connected together by an infinite number of geometrical equations. It will be necessary, as suggested in Art. 67, to find some finite number of quantities which determine the position of the body in space and to express the effective forces in terms of these quantities. These are called the coordinates of the body *. It is most important in theoretical dynamics to choose the coordinates properly. They should be (I) such that a knowledge of them in terms of the time determines the motion of the body in a convenient manner, and (2) such that the dynamical equations when expressed in terms of them may be as little complicated as possible. Let us first enquire how many coordinates are necessary to fix the position of a body. The position of a body in space is given when we know the coordinates of some point in it and the angles w^hich two straight lines fixed in the body make with the axes of coordinates. There are three geometrical relations existing between these six angles, so that the position of a body may be made to depend on six independent variables, viz. three coordinates and three angles. These migh. be taken as the coordinates of the body. It is evident that we may express the coordinates {cc, y, z) of any particle m of a body in terms of the coordinates of that body and quantities which are known and remain constant during the motion. First let us suppose the system to consist only of a single body, then if we substitute these expressions for x, y, z in the equations (A) and (B) of Art. 72, we shall have six equations to determine the six coordinates of the body in terms of the time. Thus the motion will be found. If the system consist of several bodies, we shall, by considering each separately, have six equations for each body. If there be any unknown reactions between the bodies, these will be included in X, F, Z. For each reaction there will be a corresponding geometrical relation con- necting the motion of the bodies. Thus on the whole we shall have sufficient equations to determine the motion of the system. When the motion is in two dimensions these six coordinates are reduced to three. These are the two coordinates of the point fixed in the body, and the angle some straight line fixed in the body makes with a straight line fixed in space. * Sir W. Hamilton uses the phrase " marks of position," but subsequent writers have adopted the term coordinates. See Cayley's Report to the Brit. Assoc, 1857. ART. 75.] GENERAL PRINCIPLES. 53 74. Let us next consider how the equations of motion (A) formed by resolution can be simplified by a proper choice of coordinates. We must find the resolved part of the momentum and the re- solved part of the effective forces of a system in any direction. Let the given direction be taken as the axis of x. Let {x, y, z) be the coordinates of any particle vi^hose mass is m. The re- (j X solved part of its momentum in the given direction is 'W^ -,- . Hence the resolved part of the momentum of the whole system is (1 X Sm -J- . Let {x, y, z) be the coordinates of the centre of gravity of the system and M the whole mass. Then Mx = %mx ; Hence the resolved part of the momentum of a system in any direction is equal to the whole mass midtiplied into the resolved part of the velocity of the centre of gravity. That is, the linear momentum of a system is the same as if the whole mass were collected into its centre of gravity. In the same way, the resolved part of the effective forces of a system in any direction is equal to the whole mass multiplied into the resolved part of the acceleration of the centre of gravity. It appears from this proposition that it will be <;onvenient to take the coordinates of the centre of gravity of ea^ rigid body in the system as three of the coordinates of that body. We can then express in a simple form the resolved part of the efifective forces in any direction. 75. Lastly, let us consider how the equations of motion (B) formed by taking moments can be simplified by a proper choice of the three remaining coordinates. We must find the moment of the momentum and the moment of the effective forces about any straight line. Let the given straight line be taken as the axis of x, then just as in statics yZ — zY is the moment of a force about the axis of x, so, replacing Y and Z by ij and i, the moment of the momentum about the axis of x is Sm [y t: ~ ^ ~ii] - Now this is an expression of the second degree. If, then, we substitute y = y-\-y\ z — z ^ z\ we get as in Art. 14 dz' , dy'\ -.^ / dz _ dy"" ^'^^yi[t-''ii)+^[ydt-'dt -^ dz' _^ dy' . dz ^ dy ^ 54 D'ALEMBERT'S principle. [chap. II. Exactly as in Art. 14, all the terms in the second line are zero because Sm^ = 0, Sm/ = 0, where M is the mass of the system or body under consideration. The second term is the moment about the axis of x of the momentum of a mass M moving with the centre of gravity. The first term is the moment about a straight line parallel to the axis of x, not of the actual momenta of all the several particles but of their momenta relatively to the centre of gravity. In the case of any particular body it therefore depends only on the motion of the body relatively to its centre of gravity. In finding its value we may suppose the centre of gravity reduced to rest by applying to every particle of the system a velocity equal and opposite to that of the centre of gravity. Hence we infer that The moment of the momentum of a system about any straight line is equal to the moment of the momentum of the whole mass supposed collected at its centre of gravity and moving with it, together with the moment of the momentum of the system relative to its centre of gravity about a straight line drawn parallel to the given straight line through the centre of gravity. In the same way, this proposition will be also true if for the " momentum " of the system we substitute its " effective force." By taking the axis Ox through the centre of gravity, we see that the moment of the relative momenta about any straight line through the centre of gravity is equal to that of the actual momenta. It appears from this proposition that it will be convenient to refer the angular motion of a body to a system of coordinate axes meeting at the centre of gravity. A general expression for the moment of the effective forces about any straight line through the centre of gravity cannot be conveniently investigated at this stage. Different expressions will be found advantageous under different circumstances. There are three cases to which attention should be particularly directed : (1) that of a body turning about an axis fixed in the body and fixed in space ; (2) that of motion in two dimensions, and (3) Euler's expression when the body is turning about a fixed point. These will be found at the beginnings of the third and fourth chapters and in the fifth chapter respectively. 76. Let a rigid body be turning about any point fixed in the body, such as the centre of gravity. Let 0|, Orj, 0^ be a new set of rectangular axes fixed in the body. Then the ordinary formulae for transformation of axes give y = l^ + mr} + n^, z = \^ + firi + v^ where the direction-cosines {Imn) (kjuv) are functions of the time. We see therefore that the angular momentum ART. 79.] INDEPENDENCE OF TRANSLATION AND ROTATION. 55 where A = l\- X/, and B, C &c. are similar functions of the direction-cosines. Now S/»^^ SwT/^, (fee, and also the coefficients A, B, &c. would be the same for any system of particles equimomental to the given body. We therefore infer that the moment of the effective forces of a rigid body about any straight line is the same as that for any equimomental system which moves with the body. In the same way we may show that the resolved parts of the effective forces are the same. Hence in calailatirig tJie effective forces of a rigid body ice may replace it by any convenient equimomental system ichich is rigidly connected ivith it. 77. The quantity ^m{xy — yx) expresses the moment of the momentum about the axis of z. It is called the angular momentum of the system about the axis of z. There is another interpretation which can be given to it. If we transform to polar coordinates, we have xij — yx — r^Q. Now \T^dQ is the elementary area described round the origin in the time dt by the projection of the particle on the plane of xy. If twice this polar area be mtdtiplied by the mass of the particle, it is called the area conserved by the particle in the time dt round the axis of z. Hence ^ ( dy dx is called the area conserved by the system in a unit of time, or more simply the area conserved. 78. Three Important Propositions. Summing up the results of the articles from 72 onwards, we see that we have established three important propositions. Since any straight line fixed in space may be taken as an axis of coordinates, the three equations (A) of Art. 72 may be written in the typical form d /Linear Momentum in any\ _ /Resolved impressed\ dt \ fixed direction / V force / ' For the same reason, the three equations (B) of the same article may be written in the typical form d /Angular Momentum about \ _ /Moment of im- dt\ a fixed straight line I ~\ pressed forces Thirdly, we see by Art. 74, that the typical expression for the linear momentum may be written /Linear Momentum in\ _ /Mass x resolved velocityX \ any fixed direction / V of centre of gravity / " The corresponding typical expression for the angular momentum is deferred for the present. 79. Independence of Translation and Rotation. We may now enunciate two important propositions, which follow at b 56 d'alembert's principle. [chap. II. once from the preceding results. It will, however, be more useful to deduce them from first principles. (1) The motion of the centre of gravity of a system acted on hy any forces is the same as if all the mass were collected at the centre of gravity and all the forces were applied at that point parallel to their former directions. (2) The motion of a body, acted on hy any forces, about its centre of gravity is the same as if the centre of gravity were fixed arid the same forces acted on the body. Taking any one of the equations (A) we have 27?i -^ = 2mX. If X, y, z be the coordinates of the centre of gravity, then 'xZm = SmA' ; .*. -^ Sm = 2mX, and the other equations may be treated in a similar manner. Since these are the equations which give the motion of a mass 2m acted on by forces SmX, &c., the first priociple is proved. Taking any one of the equations (B) we have Let x — x-\-x', y = y -\-y', z = z -\- z', then proceeding as in Art. 14 or Art. 75 this equation becomes ^ / , dh/ , d^x\ /_ d^y _ d'^x\ ^ ^^ / t- v. S™ [x^-y ^-j + (^0. ^ -2/ ^ j 2m = 2m (^I - yX). Now the axes of coordinates are quite arbitrary, let them be so chosen that the centre of gravity is passing through the origin at the moment under consideration. Then i^ = 0, ^ = 0, but dxjdt, dyjdt are not necessarily zero. The equation then becomes This equation does not contain the coordinates of the centre of gravity and holds at every separate instant of the motion and therefore is always true. But this and the two similar equations obtained from the other two equations of (B) are exactly the equations of moments we should have had if we had regarded the centre of gravity as a fixed point and taken it as the origin of moments. 80. These two important propositions are called respectively the principles of the conservation of the motions of translation and rotation. The first was given by Newton in the fourth corollary to the third law of motion, and was afterwards generalized by mi \x "■ ART. 82.] METHOD OF USE. 57 D'Alembert and Montucla. The second is more recent and seems to have been discovered about the same time by Euler, Bernoulli and the Chevalier d'Arcy. Another name has also been given to these results. Together they constitute the principle of the independence of the motions of translation and rotation. The motion of the centre of gravity is the same as if the whole mass were collected at that point, and is therefore quite independent of the rotation. The motion round the centre of gravity is the same as if that point were fixed, and is therefore independent of the motion of that point. 81. By the first principle the problem of finding the motion of the centre of gravity of a system, however complex the system may be, is reduced to the problem of finding the motion of a single particle. By the second the problem of finding the angular motion of a free body in space is reduced to that of determining the motion of that body about a fixed point. Example of the first principle. In using the first principle it should be noticed that the impressed forces are to be applied at the centre of gravity parallel to their former directions. Thus, if a rigid body be moving under the influence of a central force, the motion of the centre of gravity is not generally the same as if the whole mass were collected at the centre of gravity and it were then acted on by the same central force. What the principle asserts is, that, if the attraction of the central force on each element of the body be found, the motion of the centre of gravity is the same as if these forces were applied at the centre of gravity parallel to their original directions. If the impressed forces act always parallel to a fixed straight line, or if they tend to fixed centres and vary as the distance from those centres, the magnitude and direction of their resultant are jhe same whether we suppose the body collected into its centre of gravity or not. But in most cases care must be taken to find the .^esultant of the impressed forces as they really act on the body before it has been collected into its centre of gravity. 82. Example of the second principle. Let us next con- sider an example of the second principle. Suppose the earth to 3e in rotation about some axis through its centre of gravity and to be acted on by the attractions of the sun and moon. Then we learn, from the second principle, that if the resultant attraction of these bodies pass through the centre of gravity of the earth, the rotation about the axis will not be in any way affected. In what- 3ver way the centre of gravity of the earth may move in space, the axis of rotation will have its direction fixed in space and the angular velocity will be constant. Two important consequences folIoAv immediately from this result. The centre of gravity of the 58 D'ALEMBERT'S principle. [chap. II. earth is known to describe an orbit round the sun, which is very nearly in one plane, and the changes of the seasons chiefly depend on the inclination of the earth's axis to the plane of motion of the centre of the earth. The permanence of the seasons is therefore established. Secondly, since the angular velocity is constant, it follows that the length of the sidereal day is invariable. Strictly speaking the resultant attraction due to any particle of the sun and moon does not pass through the centre of gravity of the earth. The reason is that the earth is not a perfect sphere whose strata of equal density are concentric spheres. But since the ellipticities of these strata are all small the motion of rotation of the earth will be but slightly affected. Nevertheless the sun (for instance) will act with unequal forces on those parts of the earth's equator which are nearer to it and on those more remote. Thus the sun's attraction will tend to turn the earth about an axis lying in the plane of the equator and which is perpen- dicular to the radius vector of the sun. The general effect of this couple on the rotation of the earth is very remarkable. It will be proved in a later chapter (1) that the period of rotation of the earth is unaltered, (2) that though the direction of the earth's axis is no longer fixed in space, yet the axis still preserves, on the whole, the same inclination to the plane of the earth's motion round the sun. Thus the per- manence of the seasons, as far as these causes are concerned, remains unaffected. 83. General Method of using D'Alembert^s principle. The general problem in dynamics to be solved may be stated thus. Any number of rigid bodies press both against each other and against fixed points, curves, or surfaces and are acted on by given forces ; find their motion. The mode of using D'Alembert's principle for the solution may be stated thus. Let X, y, z be the coordinates of the centre of gravity of any one of these bodies referred to three rectangular axes fixed in space. Let three other coordinates of this body be chosen so that the three moments of the momentum of the body about three rectangular axes fixed in direction and meeting at the centre of gravity may be found conveniently in terms of them. Let /ii, h^,h^ be these three moments of the momentum, and let M be the mass. Then the effective forces of the body are equivalent d^x d?v (Pz to the three effective forces M-j-, -^77^' ^'^/2 ^^^ ^^® three effective couples -^ , -j^ , -j^ . The three effective forces act ^ at at at at the centre of gravity parallel to the axes of x, y, z respectively, and the three couples act round the three axes about which the moments of the momentum were taken. The effective forces of all the other bodies of the system may be expressed in a similar manner. Then all these effective forces and couples being reversed will ART. 84] IMPULSIVE FORCES. 59 36 in equilibrium with the impressed forces. The equations of 3(]uilibrium may be found by resolving in such directions and making moments about such straight lines as may be convenient. Instead of reversing the effective forces it is usually found more convenient to write the impressed and effective forces on opposite •iides of the equations. Taking the bodies separately we may thus obtain, by three esolutions and three moments, six equations of motion for each oody. If two rigid bodies press against each other or against a fixed obstacle there may be one or more unknown reactions. But there will also be in general as many equations to express the conditions Df contact. The mode of writing down these conditions of contact will be explained in the chapters which follow. Thus we shall have as many equations as there are coordinates md reactions. But sometimes by a judicious choice of the direc- tions in which we resolve, or of the straight lines about which we 3ake moments, we may (exactly as in statics) avoid introducing 5ome of these reactions into the equations. This will reduce the aumber of equations which have to be formed. We may also sometimes avoid these reactions by resolving or taking moments br two of the bodies as if they formed for an instant one single Dody. These differential equations will then have to be solved. The iifferent methods of proceeding will be explained further on. 3renerally we can find one integral by a method called the Drinciple of Vis Viva. A rule will be given to write down this ntegral without previously forming the equations of motion. We have here limited ourselves to the method of forming the equation by resolving and taking moments. But we may proceed )therwise. Thus Lagrange has given a method of writing down ;he equations of motion by which, amongst other advantages, the abour of eliminating the reactions is avoided. Application of UAlemberfs Principle to impulsive forces. 84. If a force F act on a particle of mass m always in the ^ame direction, the equation of motion is dv ^ Arhere v is the velocity of the particle at the time t. Let T be the nterval during which the force acts, and let v, v' be the velocities it the beginning and end of that interval. Then rT {v'-v)=\ 1 Jo 60 d'alembert's principle. [chap. II. Now suppose the force F to increase without limit while the interval T decreases without limit. Then the integral may have a finite limit. Let this limit be P. The equation then becomes m {v —v) = P. The velocity in the interval T has increased or decreased from V to v'. Supposing the velocity to have remained finite, let V be its greatest value during this interval. Then the space described is less than VT. Since this vanishes in the limit the particle has not moved during the action of the force F. It has not had time to move, but its velocity has been changed from v to v' . We may consider that a proper measure has been found for a force when from that measure we can deduce all the effects of the force. In the case of finite forces we have to determine both the change of place and the change in the velocity of the particle. It is therefore necessary to divide the whole time of action into elementary times and determine the effect of the force during each of these. But in the case of infinite forces which act for an indefinitely short time, the change of place is zero, and the change of velocity is the only element to be determined. It is therefore more convenient to collect the whole force expended into one measure. A force so measured is called an impulse. It may be defined as the limit of a force which is infinitely great, but acts only during an infinitely short time. There are of course no such forces in nature, but there are forces which are very great, and act only during a very short time. The blow of a hammer is a force of this kind. They may be treated as if they were im- pulses, and the results will be more or less correct according to the magnitude of the force and the shortness of the time of action. They may also be treated as if they were finite forces, and the small displacement of the body during the short time of action of the force may be found. The quantity P may be taken as the measure of the force. An impulsive force is measured by the whole momentum generated by the impulse. 85. In determining the effect of an impulse on a body, the effect of all finite forces which act on the body at the same time may be omitted. For let a finite force / act on a body at the same time as an impulsive force F. Then as before we have V — v = f Jo Fdt r/dt m m mm But in the limit fT vanishes. Similarly the force / may be omitted in the equation of moments. JIT. 87.] IMPULSIVE FORCES. 61 86. To obtain the general equations of motion of a system acted '11 hy any number of impulses at once. Let u, V, w, u, V, w be the velocities of a particle of mass m );uallel to the axes just before and just after the action of the mpulses. Let X' , Y\ Z' be the resolved parts of the impulse on n parallel to the axes. Taking the same notation as before, we lave the equation Smir = "SmX, or integrating tm{n:-u) = Xm rXdt = %X' (1). Jo Similarly we have the equations 2m {v' -v) = SF ...(2), tm (w' -w) = IZ' ...(3). Again the equation Sm (xy — yx) = 2m {xY — yX) becomes on ntegration 2m {xy — yx) = 2m {xjYdt — yfXdt). In this integration x, y are regarded as constants, because the Uiration T of the impulse is so short that the body has not time ,0 move (Art. 84), i.e. the changes of x, y during this interval may )e neglected. Taking the equation between limits, %m{x(v'-v) -y(u'-u)] =t(xY'-yX') (4). T'he other two equations become l.m{y(w'-w)-2(v' -v)\=X(yZ-zY') (5), tm{z(u' - u) - X (w' - w)} = ^ {zX'- xZ' ) (6). In the following investigations it will be found convenient to ise accented letters to denote the states of motion after impact kvhich correspond to those denoted by the same letters unaccented Defore the action of the impulse. Since the changes in direction ind magnitude of the velocities of the several particles of the bodies are the only objects of investigation, it will be found convenient to express the equations of motion in terms of these velocities. 87. In applying D'Alembert's Principle to impulsive forces the only change which must be made is in the mode of measuring the effective forces. If (u, v, w), (u, v\ w') be the resolved part of the velocity of any particle just before and just after the impulse, and if m be its mass, the effective forces will be measured by m (u — u\ m {v' — v), and m {w' — w). The quantity mf in Art. 67 is to be regarded as the measure of the impulsive force which, if the particle were separated from the rest of the body, would produce these changes of momentum. In this case, if we follow the notation of Arts. 74 and 75, the resolved part of the effective force in the direction of the axis of z is the difference of the values of Xmdz/dt just before and jusfc after this action of the impulses, and this is the same as the 62 d'alembert's principle. [chap. II. difference of the values of Mdzjdt at the same instants. In the same way the moment of the effective forces about the axis of z will be the difference of the values of vdt y dt) just before and just after the action of the impulses. We may therefore extend the general proposition of Art. 83 to impulsive forces in the following manner. Let {u, V, w), {u, v', w') be the velocities of the centre of gravity of any rigid body of mass TIf just before and just after the action of the impulses resolved parallel to any three fixed rectangular axes. Let (Ai, hr^, h.^), {hi, h^, 4/) be the moments of momentum relative to the centre of gravity about three rectangular axes fixed in direction and meeting at the centre of gravity, the moments being taken respectively just before and just after the impulses. Then the effective forces of the body are equivalent to the three effective forces M {u — u), M {v' — v), M{w' — w), acting at the centre of gravity parallel to the rectangular axes, together with the three effective couples (/?/ — AJ, {h^ — h^), Qi^ — K) about those axes. These effective forces and couples being reversed will be in equilibrium with the impressed forces. The equations of equili- brium may then be formed according to the rules of statics. Examples. Ex. 1. Two particles moving in the same plane are projected in parallel but opposite directions with velocities inversely proportional to their masses. Find the motion of their centre of gravity. Ex. 2. A person is placed on a perfectly smooth table, show how he may get off. Ex. 3. Explain how a person sitting on a chair is able to move the chair across the room by a series of jerks, without touching the ground with his feet. Ex. 4. A person is placed at one end of a perfectly rough board which rests on a smooth table. Supposing he walks to the other end of the board, determine how far the board has moved. If he steps off the board, show how to determine its subsequent motion. Ex. 5. The motion of the centre of gravity of a shell shot from a gun in vacuo is a parabola, and its motion is unaffected by the bursting of the shell. Ex. 6. A rod revolving uniformly in a horizontal plane round a pivot at its extremity suddenly snaps in two : determine the motion of each part. Ex. 7. A cube slides down a perfectly smooth inclined plane with four of its edges horizontal. The middle point of the lowest edge comes in contact with a small fixed obstacle and is reduced to rest. Determine whether the cube is also reduced to rest, and show that the resultant impulsive action along the edge will not act along the inclined plane. Ex. 8. Two persons A and B are situated on a perfectly smooth horizontal plane at a distance a from each other. A throws a ball to B which reaches B after EXAMPLES. 63 a time t. Show that A will begin to slide along the plane with a velocity malMt, where M is his own mass and 7/i that of the ball. If the plane had been perfectly rough, explain in general terms the nature of the pressures between ^'s feet and the plane which would have prevented him from sliding. Would these pressures have had a single resultant? Ex. 9. A cannon rests on an imperfectly rough horizontal plane and is fired with such a charge that the relative velocity of the ball and cannon at the moment when the ball leaves the cannon is F. If M be the mass of the cannon, m that of the ball, and //, the coefficient of friction, show that the cannon will recoil a distance - — on the plane. Bl+mJ 2/j,g Ex. 10. A spherical cavity of radius a is cut out of a cubical mass so that the centre of gravity of the remaining mass is in the vertical through the centre of the cavity. The cubical mass rests on a perfectly smooth horizontal plane, but the interior of the cavity is perfectly rough. A sphere of mass m, and radius b, rolls down the side of the cavity starting from rest with its centre on a level with the centre of the cavity. Show that when the sphere next comes to rest, the cubical mass will have moved, through a space ,: , where 31 is the mass of the M + m remaining portion of the cube. Would the result be the same if the cavity were smooth or imperfectly rough? Ex. 11. Two railway engines drawing the same train are connected by a loose chain and come several times in succession into collision with each other; the leading engine being a little top-heavy and the buffers of both rather low. The fore-wheels of the first engine are observed to jump up and down. What dynamical explanation can be given of this rocking motion ? At what level should the buffers be placed that it may not occur? Camb. Trans. Vol. vii. 1841. Ex. 12. Sir C. Lyell in his account of the earthquake in Calabria in 1783, mentions two obelisks each of which was constructed of three great stones laid one on the top of the other. After the earthquake, the pedestal of each obelisk was found to be in its original place, but the separate stones above were turned partially round and removed several inches from their position without falling. The shock which agitated the building was therefore described as having been horizontal and vorticose. Show that such a displacement would be produced by a simple rectilinear shock, if the resultant blow on each stone did not pass through its centre of gravity. See Mallet's Dynamics of Earthquakes. Milne in his Earthquakes, 1886, page 196, discusses the latter explanation and refers to some similar cases which occurred in the earthquake at Yokohama in 1880. 1. CHAPTER III. MOTION ABOUT A FIXED AXIS. 88. The Fundamental Theorem. A rigid body can turn freely about an axis fixed in the body and in space, to find the moment of the effective forces about the axis of rotation. Let any plane passing through the axis and fixed in space be taken as a plane of reference, and let be the angle which any- other plane through the axis and fixed in the body makes with the first plane. Let m be the mass of any element of the body, r its distance from the axis, and let cj) be the angle made by a plane through the axis and the element m with the plane of reference. The velocity of the particle m is r(j) in a direction perpen- dicular to the plane containing the axis and the particle. The moment of the momentum of this particle about the axis is clearly mr^cj). Hence the moment of the momenta of all the particles is X {mr^^). Since the particles of the body are rigidly connected with each other, it is^ obvious that <^ is the same for every particle, and equal to 6. Hence the moment of the momenta of all the particles of the body about the axis is Xtnr'^d, i.e. the moment of inertia of the body about the axis multiplied into the angular velocity. The accelerations of the particle m are r^ and - r^ perpen- dicular to, and along the direction in which r is measured, the moment of the effective forces on m about the axis is mr^if), hence the moment of the effective forces on all the particles of the body about the axis is 2 (mr^if)). By the same reasoning as before this is equal to %mr~Q, i.e. the moment of inertia of the body about the axis into the angular acceleration. 89. To determine the motion of a body about a fixed axis under the actioii of any forces. By D'Alembert's principle the effective forces when reversed will be in equilibrium with the impressed forces. To avoid intro- ducing the unknown reactions at the axis, let us take moments about the axis. ART. 91.] THE FUNDAMENTAL THEOREM. 65 Firstly, let the forces he impulsive. Let w, w be the angular velocities of the body just before and just after the action of the forces. Then, following the notation of the last article, w . Sm7'^ — ft) . Xmr^ = L, where L is the moment of the impressed forces about the axis ; , moment of forces about axis .'. CD — (O = —. ; -, 7-. moment oi inertia about axis This equation will determine the change in the angular velocity produced by the action of the forces. Secondly, let the forces he finite. Then taking moments about the axis, we have -j— . Xmr^ = L ; d?6 _ moment of forces about axis df^ moment of inertia about axis ' This equation when integrated will give the values of 6 and ddjdt at any time. Two undetermined constants will make their appearance in the course of the solution. These are to be deter- mined from the given initial values of S and ddjdt. Thus the whole motion can be found. 90. It appears from this proposition that the motion of a rigid body about a fixed axis depends on (1) the moment of the forces about that axis and (2) the moment of inertia of the body about the axis. Let Mk^ be this moment of inertia, so that k is the radius of gyration of the body. Then if the whole mass of the body were collected into a particle and attached to the fixed axis by a rod without inertia, whose length is the radius of gyration k, and if this system be acted on by forces having the same moment as before, and be set in motion with the same initial values of d and cWldt, then the whole subsequent angular or gyratory motion of the rod will be the same as that of the body. We may say briefly, that a body turning about a fixed axis is dynamically given, when ive knoiv its mass and radius of gyration. 91. Ex. 4 perfectly rough circular horizontal board is capable of revolving freely round a vertical axis through its centre. A man whose iveight is equal to that of the board ivalks on and round it at the edge : when he has completed the circuit what will be his position in space ? Let a be the radius of the board, Mk^ its moment of inertia about the vertical axis. Let w be the angular velocity of the board', w' that of the man about the vertical axis at any time. And let F be the action between the feet of the man and the board. The equation of motion of the board is by Art. 89, Mk^C3= - Fa (1). The equation of motion of the man is by Art. 79, 3Iata' = F (2). Eliminating F and integrating, we get k^o} + a^ — (Ij- n{l-ih) + c Let the linear expansion of the substance which forms the rod and jar be denoted by a and that of mercury by /3 for each degree of the thermometer. If the thermometer used be Fahrenheit's, we have a = -0000065668, j3 = -00003336, accord- ing to some experiments of Dulong and Petit. Thus we see that a and ^ are so small that their squares may be neglected. In calculating the height of the mercury it must be remembered that the jar expands laterally, and thus the relative vertical expansion of the mercury is 3/3 - 2a, which we shall represent by y. If then the temperature of every part be increased f^, we have a, I, k, c, all increased in the ratio 1-j-at : 1, while h is increased in the ratio 1 + 7^ : 1. Since L is to be unaltered, we have rIT. rlT. iJT \ /IT. {dL dL^ dL^ dL \ dL dl dk dc J dh hy = Q But L is a homogeneous function of one dimension, hence dL da' dL^ dL """-Ti'-^dk , dL dc dL^ dh^- = L. The condition becomes therefore by substitution = — — - . a- 7 L dh Let A, B he the numerator and denominator of the expression for L given by equation (1). Then taking the logarithmic differential 1 dL _ n{^h-l ) in_n f^h-l 1 L dh ~ A "^ "^ ~ ^ V ir~ "^ 2 nf%h- b\ L Hence the required condition is ^ , " — r= — r- . i -^ tA (2). ^{^-a) h c \ L 2) ^"2 + n This calculation has more theoretical than practical importance, for the nu- merical values of a and /3 depend a good deal on the purity of the metals and on the mode in which they have been worked. The adjustment must therefore be ART. 96.] THE PENDULUM. 7l finally made by experiment. If the rate of the clock is found to be affected by a change of temperature it is usual to alter slightly the quantity of mercury in the jar until by trial the adjustment is found to be satisfactory. In the investigation we have supposed a and /3 to be absolutely constant, but this is only a very near approximation. Thus a change of 80° Fah. would alter /3 by less than a fiftieth of its value. When the adjustment is made the compensation is not strictly correct, for the iron jar and mercury have been supposed to be of uniform temperature. Now the different materials of which the pendulum is composed absorb heat at different rates, and therefore while the temperature is changing there will be some slight , error in the clock. The whole length of a seconds pendulum of this construction is about 44 inches, the expansion and contraction of which is corrected by a column of mercury in the jar about 7 inches long. The radius of the jar is usually about one inch. The weight of the mercury is then about 10 to 12 pounds which, added to that of the jar, frame, and rod, brings the total weight to about 14 pounds. Ex. If, as a first approximation, we regard the mercury as the weight, the jar and the rod being only of sufficient mass to hold up the mercury, and if we also suppose h and a to be so much less than L that we may reject the squares of their ratios to L, prove that the equation (1) gives L = l-^h and that the equation (2) gives h — \L. 95. Buoyancy of Air. Another cause of error in a clock pendulum is the buoyancy of the air. This produces an upioard force acting at the centre of gravity of the volume of the pendulum equal to the weight of the air displaced. A very slight modification of the fundamental investigation in Art. 92 will enable us to take this into account. Let V be the volume of the pendulum, D the density of the air ; h^, h^, the distances of the centres of gravity of the mass and volume respectively from the axis of suspension, Mk'^ the moment of inertia of the mass about the axis of suspension. Let us also suppose the pendulum to be symmetrical about a plane through the axis and either centre of gravity. The equation of motion is then Mk'^ d = - Mg\ sin d + VDgho sin ^ (1). By the same reasoning as before we infer that if I be the length of the equivalent pendulum j = h^-h^ — (2). The density D of the air is continually changing, the changes being indicated by variations in the height of the barometer. Let h be the value of the right-hand side of this equation for any standard density D. Suppose the actual density to be D + 8D and let I + 81 be the corresponding length of the seconds pendulum, then we , , ,.^ ,. ^. k^dl , VdD . ,^ , dl h^ VD dD have by differentiation — — - = he, ~^ , and therefore — = -f -— r -=- . L^ ' M L h M D This formula gives in a convenient form the change in the length of the equi- valent pendulum due to a change in the density of the air. 96. Ex. 1. If the centres of gravity of the mass and volume were very nearly coincident and the weight of the air displaced were ^^^j,- of the weight of the pendulum, show that a rise of one inch in the barometer would cause an error in the rate of going of the seconds pendulum of nearly one-fifth of a second per day. This example will enable us to estimate the general effect of a rise of the barometer on the rate of going of an iron pendulum. Ex. 2. If a barometer were attached to the pendulum show that the rise or fall 72 MOTION ABOUT A FIXED AXIS. [CHAP. III. of the mercury as the density of the air changed could be so arranged as to keep the time of vibration unaltered. This method was suggested first by Dr Eobinson of Armagh in 1831 in the fifth vohime of the memoirs of the Astronomical Society, and afterwards by Mr Denison in the Astronomical Notices for Jan. 1873. In the Armagh Places of Stars published in 1859, Dr Eobinson described the difficulties he found in practice before he was satisfied with the working of the clock. The jar of mercury in Graham's mercurial pendulum might be used as the cistern of the barometer, as Mr Denison remarks. The theory of the construction is that in differentiating equation (2) we are to suppose k'^, &c. variable and I constant. Prof. Eankine read a paper to the British Association in 1853 in which he proposed to use a clock with a centrifugal or revolving pendulum, part of which should consist of a siphon barometer. The rising and falling of the barometer would affect the rate of going of the clock so that the mean height of the mercurial column during any long period would register itself. Ex. 3. If the pendulum be supposed to drag a quantity of air with it which bears a constant ratio to the density D of the surrounding air and adds yD to the moment of inertia of the pendulum without increasing the moving power, show that the change produced in the simple equivalent pendulum by a change of density 8D is given by 5l = y5DI3Ili^ (see Art. 105). 97. Moments of Inertia found by experiment. In many experimental investigations it is necessary to determine the moment of inertia of the body experimented on about some axis. If the body be of regular shape and be so far homogeneous that the errors of this assumption are of the order to be neglected, we can determine the moment of inertia by calculation. But sometimes this cannot be done. If we can make the body oscillate under gravity about any axis parallel to the given axis placed in a horizontal position, we can determine by equation (4) of Art. 92 the radius of gyration about a parallel axis through the centre of gravity. This requires however that the distances of the centre of gravity from the axes should be very accurately found. Sometimes it is more convenient to attach the body to a pendulum of known mass whose radius of gyration about a fixed horizontal axis has been previously found by observing the time of oscillation. Then by a new determination of the time of oscillation, the moment of inertia of the compound body, and therefore that of the given body, may be found, the masses being known. If the body be a lamina, we may thus find the radii of gyration about three axes passing through the centre of gravity. By measuring three lengths along these axes inversely proportional to these radii of gyration, we have three points on a momental ellipse at the centre of gravity. The ellipse may then be con- structed. The directions of its principal diameters are the principal axes, and the reciprocals of their lengths represent on the same scale as before the principal radii of gyration. ART. 98.] LENGTH OF THE SECONDS PENDULUM. 78 If the body be a solid, six observed radii of gyration will de- termine the principal axes and moments at the centre of gravity. But in most cases some of the circumstances of the particular problem under consideration will simplify the process. The following example illustrates the use of the method in determining or eliminating the unknown moments of inertia which occur in some experimental researches. Other examples are given in Arts. 99, 122, &c. Ex. A symmetrical magnet can turn freely about a vertical axis which passes through its middle point, and the effect of the earth's magnetism on it is represented by a couple whose moment is F sin 9, where d is the angle the axis of the magnet makes with the meridian. The extremities of the magnet can be loaded at pleasure with two equal spherical brass weights which rest on the magnet by sharp points so that the weights do not partake of the rotatory motion of the magnet. If I be the moment of inertia of the magnet, fx the mass of either sphere, 2c the distance between their centres, prove that the times of oscillation without and with the spheres are T=27r{I/F}^ r' = 27r{(Z + 2Atc2)/F}*, whence I and F can be found when T and T' have been observed. If the weights were rigidly attached to the magnet, we must increase lixc^ by |iue- where e is the radius (see Art. 148). In this case e must be measured as well as c, but the error due to friction at the point of attachment is avoided. This method of finding the value of F is commonly ascribed to Weber. See Taylor's translations of Scientific Memoirs, and Airy's Magnetinm. 98. On the length of the Seconds Pendulum. The oscillations of a rigid body may be used to determine the numerical value of the accelerating force of gravity. Let r be the half time of a small oscillation of a body made in vacuo about a horizontal axis, h the distance of the centre of gravity from the axis, k the radius of gyration about a parallel axis through the centre of gravity. Then we have by Art. 92, k^ + Jf^ = \hT^ (1), where \= ~, so that X is the length of the simple pendulum whose complete time of oscillation is two seconds. I We might apply this formula to any regular body for which ' k and h could be found by calculation. Experiments have thus been made with a rectangular bar, drawn as a wire and suspended J^-2 ^ J^2 from one end. In this case — j — , which is the length of the simple equivalent pendulum, is easily seen to be two-thirds of the ' length of the rod. The preceding formula then gives \ or g as soon as the ' time of oscillation has been observed. By inverting the rod and taking the mean of the results in the two positions any error arising from want of uniformity in density or figure may be partially obviated. It has, however, been found impracticable to obtain a rod sufficiently uniform to give results in accordance k with each other. 74 MOTION ABOUT A FIXED AXIS. [CHAP. III. 99. If we make a body oscillate in succession about two parallel axes not at the same distance from the centre of gravity, we get two equations similar to (1), viz. ]^^^k2 = XliT\ k'-^h'^-^Xlir'^' (2). Between these two we may now eliminate k^, thus /j2 _ 7/2 ^^A=/,^2_/,V2 (3)^ A, This equation gives \. Since h^ has disappeared, the form and structure of the body is now a matter of no importance. Let a body be constructed with two apertures into which knife edges can be fixed. The apertures may be triangular to prevent slipping. Resting on these knife edges, the body can be made to oscillate through small arcs. The perpendicular distances h, h' of the centre of gravity from the axes must then be measured with great care. The formula will then give X. 100. In Capt. Rater's method (PhiL Trans. 1818) the body has a sliding vveight in the form of a ring which can be moved up and down by means of a screw. The body itself has the form of a bar and the apertures are so placed that the centre of gravity lies between them. The ring weight is then moved until the two times of oscillation are exactly equal. The equation (3) then becomes — - — = t- (4), A, which determines \. The advantage of this construction is that the position of the centre of gravity, which is not found without difficulty by experiment, is not required. All we want is h + h\ the exact distance between the knife edges. The disadvantage is that the ring weight has to be moved until two times of oscillation, each of which it is difficult to observe, are made equal. 101. The equation (3) can be written in the form h+J^_r^ + T^ h + h' J \ - 2 ^^h-h'^"^ ^ '■ \ We now see that, if the body be so constructed that the times of oscillation about the two axes of suspension are very nearly equal, t^ — t'^ will be small, and therefore it will be sufficient in the last term to substitute for h and h' their approximate values. The position of the centre of gravity is of course to be found as accurately as possible, but any small error in its position is of no very great consequence, for such an error is multiplied by the small quantity t'^ — t"^. The advantage of this construction over Rater's is that the ring weight may be dispensed with and yet the only element which must be measured with extreme accuracy is h + h\ the distance between the knife ed^es. ART. 103.] LENGTH OF THE SECONDS PENDULUM. 75 102. In order to measure the distance between the knife edges, Captain Kater first compared the different standards of length then in use, in terms of each of which he expressed the length of his pendulum. Since then a much more complete comparison of these and other standards has been made under the direction of the Committee appointed for that purpose in 1843. Phil. Trans., 1857. Having settled his unit of length, Captain Kater proceeded to measure the distance between the knife edges by means of microscopes. Two different methods were used, which however cannot be described here. As an illustration of the extreme care necessary in these measurements, the following fact may be mentioned. Though the images of the knife edges were always perfectly sharp and well defined, their distance when seen on a black ground was -000572 of an inch less than when seen on a white ground. This difference appeared to be the same, whatever the relative illumination of the object and ground might be, so long as the difference of character was preserved. Three sets of measurements were taken, two at the be- ginning of the experiments, and the third after some time. The object of the last set was to ascertain if the knife edges had suffered from use. The mean results of these three differed by less than a ten-thousandth of an inch from each other, the distance to be measured being 39*44085 inches. 103. The time of a single vibration cannot be observed directly, because this would require the fraction of a second of time as shown by the clock to be estimated either by the eye or ear. The difficulty may be overcome by observing the time, say of a thousand vibrations, and thus the error of the time of a single vibration is divided by a thousand. The labour of so much counting may however be avoided by the use of the method of coincidences. The pendulum is placed in front of a clock pendulum whose time of vibration is slightly different. Certain marks made on the two pendulums are observed by a telescope at the lowest point of their arcs of vibration. The field of view is limited by a diaphragm to a narrow aperture across which the marks are seen to pass. At each succeeding vibration one pendulum follows the other more closely, and at last its mark is completely covered by the other during the passage across the field of view of the telescope. After a few vibrations it appears again preceding the other. In the interval from one disappearance to the next, one pendulum has made, as nearly as possible, one complete oscillation more than the other. We have therefore to count the number of vibrations made by either pendulum in the interval. At the beginning of the counting let one pendulum coincide with the other as nearly as we can judge. Suppose that after n half vibrations of the clock pendulum the next coincidence has not quite arrived, but that after n+ 1 half vibrations the coincidence has passed. If the clock pendulum be the slower of the two, the other must have made ?i -f 2 or n + S half vibrations in the interval. Thus the time of one half vibration of the pendulum lies between the 71 72/ -|- 1 fractions — — - and ~ of the period of the clock vibration. 71+ 2 71 + 3 ^ Taking either of these estimates as the real time of a half 76 MOTION ABOUT A FIXED AXIS. [CHAP. III. vibration of the pendulum the error is less than the fraction 2 ,- ^r^^ -; of the time of a half vibration of the clock {n-\-2)(n-\-S) pendulum. It appears from this that the error varies nearly in- versely as the square of the number of vibrations between two coincidences. In this manner 530 half vibrations of a clock pendulum, each equal to a second, were found to correspond to 532 of Captain Kater's pendulum. The error of this estimate is so small that in twenty-four hours it would accumulate only to about three-fifths of a second. The ratio of the times of vibration of the pendulum and the clock pendulum may thus be calculated with extreme accuracy. The rate of going of the clock must then be found by astronomical means. The reader should notice the resemblance between this process of comparing two clocks with the use of the vernier in comparing lengths. Of course there are differences, because the vernier is applied to space, and we have here to do with time. But the general principle is the same. In some more recent experiments the observation of the coincidences was assisted by the use of a momentary electrical illumination of the slit, Nature 1898, Feb. 10. 104. The Reductions. The time of vibration thus obtained will require several corrections which are called " reductions." For instance, if the oscillation be not so small that we can put sin ^ = ^ in Art. 92, we must make a reduction to infinitely small arcs. The general method of effecting this will be considered in the chapter on Small Oscillations. Another reduction is necessary if we wish to reduce the result to what it would have been at the level of the sea. The attraction of the intervening land may be allowed for by Dr Young's rule (Phil. Trans. 1819). We may thus obtain the force of gravity at the level of the sea, supposing all the land above this level were cut off and the sea constrained to keep its present level. As the level of the sea is altered by the attraction of the land, further corrections are still necessary if we wish to reduce the result to the surface of that spheroid which most nearly represents the earth. See Camb. Phil. Trans. Vol. viii. On the vai^iation of gravity at the surface of the eai'th, by Sir G. Stokes. Mr Baily gives as the length of the pendulum whose half time of vibration is a mean solar second in the open air in the latitude of London 39'133 inches, and as the length of a similar pendulum vibrating sidereal seconds 38*919 inches. 105. Correction for Resistance of the Air. The observations must be made in the air. To correct for this we have to make a reduction to a vacuum. This reduction consists of three parts: (1) The correction for buoyancy, (2) Du Buat's correction for the air dragged along by the pendulum, (3) The resistance of the air. m ART. 105.] LENGTH OF THE SECONDS PENDULUM. 77 The volume V of the pendulum may be found by measuring the dimensions of the body. As the "reduction to a vacuum" is only a correction, any small un- avoidable errors in calculating the dimensions will produce an effect only of the second order on the value of X. Let p be the density of the air when the body is oscillating about one knife edge, p' the density when oscillating about the other. If the observation be made within an hour or two hours, we may put p = p'. The effect of buoyancy is allowed for by supposing a force Vpg to act upwards at the centre of gravity of the volume of the body. If the body be made as nearly as possible symmetrical about the two knife edges this centre of gravity will be half way between the knife edges, see Art. 95. Du Buat discovered by experiment that a pendulum drags ivith it to and fro a certain mass of air ichich increases the inertia of the body toithout adding to the moving force of gravity. This result has been confirmed by Bessel and Stokes. The mass dragged bears to the mass of air displaced by the body a ratio which depends on the external shape of the body. Let us represent it by fxVp. If the body be symmetrical about the knife edges, so that the external shape is the same whichever edge is made the axis of suspension, p, will be the same for each oscilla- tion. We must add to the k^ of equation (1) in Art. 92 and therefore also in Art. 98 the term p^Vph'^jm, where k' is the radius of gyration of the dragged air about either axis of suspension and m is the mass of the pendulum. Taking these two corrections the equation (1) of Art. 98 will now become \ m 2 J Similarly for the oscillation about the other knife edge, ,., ,,, p.Vp'k'-^ ^ ,,/,, Vp' h + h'\ m \ m 2 J We must eliminate />;- as before. If the observations about the two knife edges succeed each other at a short interval we may put p — p, and then Du Buat's correction will disappear. This is of course a very great advantage. We then have h + h' T^ + r'\^h+h' ( Vp\ the last term being very small, because r and t' are nearly equal. The resistance of the air will be some function of the angular velocity ddjdt of the pendulum. Since the angular velocity is very small we may expand this function and take only the first power. Supposing that Maclaurin's theorem does not fail, and that no coe£&cient of a higher power than the first is very great, this gives a resistance proportional to ddjdt. The equation of motion will therefore take the form df^ ^ dt' where 27r/ri is the time of a complete oscillation in a vacuum, and the term on the right-hand side is that due to the resistance of the air. The discussion of this equation will be found in the chapter on Small Oscillations. When the density of the air is increased, the three corrections (buoyancy, the addition to the inertia, and the resistance of the air) combine to increase the time of oscillation of a pendulum and therefore to make a clock go a little slower. The reader may consult, Du Buat, Principes d'hydraulique 1786 ; F. W. Bessel, Royal Academy of Sciences, Berlin 1826, Baily "On the correction of the pendulum," Phil. Trans. 1832, Account of the operations of the great trigonometrical survey in India by Capt. Heaviside 1879, Gen. Walker's Account of recent pendulum opera- tions <&c., Phil. Trans. 1890. 78 MOTION ABOUT A FIXED AXIS. [CHAP. III. 106. Construction of a pendulum. In constructing a reversible pendulum to measure the force of gravity, the following are points of importance. 1. The axes of suspension, or knife edges, must not be at the same distance from the centre of gravity of the mass. They should be parallel to each other. 2. The times of oscillation about the two knife edges should be nearly equal. 3. The external form of the body must be symmetrical, and the same about the two axes of suspension. 4. The pendulum must be of such a regular shape that the dimensions of all the parts can be readily calculated. These conditions are satisfied if the pendulum be of rect- angular shape with two cylinders placed one at each end. The external forms of these cylinders should be equal and similar, but one solid and the other hollow, and such that the distance between the knife edges is to be as nearly as possible equal to the length of the simple equivalent pendulum found by calculation. This is called Repsold's pendulum. 5. The pendulum should be made, as far as possible, of one metal, so that as the temperature changes it may be always similar to itself. In this case since the times of oscillation of similar bodies vary as the square root of their linear dimensions, it is easy to reduce the observed time of oscillation to a standard tem- perature. The knife edges however must be made of some strong substance not likely to be easily injured. 107. Ex. 1. If the knife edges be not perfectly sharp, let r be the difference of their radii of curvature ; show that A very nearly, when the pendulum vibrates in vacuo. It appears that the correction vanishes if the knife edges are equally sharp. By interchanging the knife edges we have the same equation with the sign of r changed. By making a few observa- tions we may thus determine r. A proposition similar to this has been ascribed to Laplace by Dr Young. Let p, p' be the radii of curvature of the knife edges. Then by taking moments about the instantaneous axis we may show (as in Art. 98) that k^ + h' = \{h-\-p)T^. Since p is small we may write this in the form k^ + Ji^ - {k^ + h^) t = \Jit^. The times of vibration r, r' are nearly equal, hence by Art. 92 we have k^ = hh' very nearly. Substituting this value of k in the small terms we get k^ +h^-{h + h') p = Xhr^. There is a similar equation for the pendulum when it vibrates about the other knife edge, which may be obtained from this by interchanging h, h' and r, r'. Eliminating k^ as in Art. 99, and remembering that r=p' - p, we obtain the result to be proved. Ex. 2. A heavy spherical ball is suspended by a very fine wire successively from two points of support A and B, whose vertical distance b has been carefully I ART. 108.] A STANDARD OF LENGTH. 79 measured, thus forming two pendulums. The lowest point of the ball is, on each suspension, made to be as exactly as possible on the same level, which level is approximately at depths a and a' below A and B respectively. If r be the radius of the ball, which is small compared with a or a', and I, V the lengths of the simple I— I' 2 r^ equivalent pendulums, prove that — — = 1 - ^ -. ry-^ very nearly. By count- ing the number of oscillations performed in a given time by each pendulum, show how to find the ratio of I to l'. Thence show how to find g and point out which lengths must be most carefully measured and which need only be approximately found, so as to render this method effective. This method is mentioned in Grant's History of Physical Astronomy, page 155, as having been used by Bessel in 1826. 108. A Standard of Ziengtli. The length of the seconds pendulum has been used as a national standard of length. By an Act of Parliament passed in 1824, it was declared that tlie distance between the centres of two points in the gold studs in the straight brass rod then in the custody of the clerk of the House of Commons, whereon the words and figures " standard yard, 1760" were engraved, should be the original and genuine standard of length called a yard, the brass being at the tem- peiature of 62° Fahr. And as it was expedient that the said standard yard if injured should be restored to the same length by reference to some invariable natural standard, it was enacted that the new standard yard should be of such length that the pendulum vibrating seconds of mean time in the latitude of London in a vacuum at the level of the sea should be 39-1393 inches. On Oct. 16, 1834, occurred the fire at the Houses of Parliament, in which the standards were destroyed. The bar of 1760 was recovered, but one of its gold pins bearing a point was melted out and the bar was otherwise injured. In 1838 a commission was appointed to report to the Government on the course best to be pursued under the peculiar circumstances of the case. In 1841 the com- mission reported that they were of opinion that the definition by which the standard yard is declared to be a certain brass rod was the best which it was possible to adopt. With respect to the provision for restoration they did not recommend a reference to the length of the seconds pendulum. " Since the passing of the act of 1824 it has been ascertained that several elements of reduction of the pendulum experiments therein referred to are doubtful or erroneous : thus it was shown by Dr Young, Phil. Trans. 1819, that the reduction to the level of the sea was doubtful; by Bessel, Astron. Nachr. No. 128, and by Sabine, Phil. Trans. 1829, that the reduction for the weight of air was erroneous ; by Baily, Phil. Trans. 1832, that the specific gravity of the pendulum was erroneously estimated and that the faults of the agate planes introduced some elements of doubt ; by Kater, Phil. Tram. 1830, and by Baily, Astron. Soc. Memoirs, Vol. ix., that very sensible errors were introduced in the operation of comparing the length of the pendulum with Shuckburgh's scale used as a representative of the legal standard. It is evident, therefore, that the course prescribed by the act would not necessarily reproduce the length of the original yard." The commission stated that there were several measures which had been formerly accurately compared with the original standard yard, and that by the use of these the length of the original yard could be determined without sensible error. In 1843 another commission was appointed to compare all the existing measures and to construct from them a new Parliamentary standard. Unexpected difficulties occurred in the course of the comparison, which cannot be described here. A full account of the proceedings of the commission will be found in a paper contributed by Sir G. Airy to the Royal Society in 1857. A standard bar of gun metal was finally produced which was legalised as the standard by the act of 1855. Copies 80 MOTION ABOUT A FIXED AXIS. [CHAP. III. are kept at the Mint, the Eoyal Observatory, the Eoyal Society and many other places. A new Imperial yard has been in course of construction since 1897. The weights and measures act of 1878 regulates the law on this subject. In France the standard of length is the m^tre. This, like our standard yard, was originally defined by reference to a length given in nature. The ten millionth part of the length of a meridian of the earth measured from the pole to the equator was declared to be the legal metre. But when new and more accurate measurements were subsequently made, it became evident that the length of the legal metre could not be altered for each improvement in the measure of the earth. Practically there- fore the definition of the metre is a certain length preserved in Paris. The use of the seconds pendulum as a standard of length assumes that a standard of time has already been obtained. In this case we must have recourse to some natural standard, and the one usually chosen is the time of rotation of the earth on its axis. This is recommended by its simplicity, for the interval between two successive transits of the same star across the meridian is very nearly equal to the time of rotation of the earth. But other natural standards may also be used to check the clock. For an account of the recommendations made in the two reports (1873 and 1874) by the Units Committee of the British Association, see Prof. Everett's treatise on Units and Physical Constants. 109. Oscillation of a Watch Balance. A rod B'OB can turn freely about its centre of gravity which is fixed, and is acted on by a very fine spiral spring GPB. The spring has one end C fixed in position in such a manner that the tangent at C is also fixed, and has the other end B attached to the rod so that the tangent at B makes a constant angle with the rod. The rod being turned through any angle, it is required to find the time of oscillation. This is the construction used in watches, just as the pendulum is used in clocks, to regulate the motion. In many watches the rod is replaced by a wheel whose centre is 0. Let Ox be the position of the rod when in equilibrium, and let 6 be the angle the rod makes with Ox at any time t, Afk^ the moment of inertia of the rod about 0. Let p be the radius of curvature at any point P of the spring, p^ the value of p when in equilibrium. Let {x, y) be the coordinates of P referred to as origin and Ox as axis of x. Let us consider the forces which act on the rod and the portion BP of the spring. The forces on the rod are X, Y the resolved parts of the action at parallel to the axes of co- ordinates, and the reversed effective forces which are equivalent to a couple Mk'd^dldt^ The forces on the spring are, the reversed effective forces which, owing to the fineness of the spring, are so small that they may be neg- lected, and the resultant action across the section of the spring at P. This resultant action is produced by the tensions of the innumerable fibres which make up the. ART. 109.] OSCILLATION OF A WATCH BALANCE. 81 spring, and these are equivalent to a force at P and a couple. When an elastic spring is bent so that its curvature is changed, it is proved both by experiment and theory, that this couple is pro- poitional to the change of curvature at P. We may therefore represent it hy E i J , where E depends only on the material of which the spring is made and on the form of its section. To avoid introducing the unknown force at P, we take moments about P. This gives This equation is true whatever point P may be chosen. Con- sidering the left side constant at any moment and {w, y) variable, this becomes the intrinsic equation to the form of the spring. Let BP = s. Multiplying this equation by ds and integrating along the whole length I of the spiral spriui^, we have Now ds/p is the angle between two consecutive normals, hence Jds/p is the angle between the extreme normals. At C the normal to the spring is fixed throughout the motion, therefore is the angle between the normals at B in the two Mk^%^^-E(--^)-Xy-^ Yx. /{' P Po positions in which 6 = and 6 = 0. But since the normal at B makes a constant angle with the rod, this angle is the angle which the rod makes with its position of equilibrium. Also if X, y be the coordinates of the centre of gravity of the spring at the time t, we have foods — xl, Jyds = yl. Hence the equation of motion becomes Mk'^ ri: — —~r^ -^ Y^ — ^V- dt^ I Let us suppose that in the position of equilibrium there is no pressure on the axis 0, then, if the oscillation is small, X and Y will, throughout the motion, be small quantities of the order 6. Let us also suppose that the fulcrum is placed over the centre of gravity of the spring when at rest. Then if the number of spiral turns cf the spring be numerous and if each turn be nearly circular, the centre of gravity will never deviate far from G. Thus the terms Yx and Xy are each the product of two small quantities, and are therefore at least of the second order. Neglecting these terms we have Mk^ -r- = — ,6. dt^ I 'MkH Hence the time of oscillation is 27rA / - j-j 4- R. D. 82 MOTION ABOUT A FIXED AXIS. [CHAP. III. It appears that to a first approximation the time of oscillation is independent of the form of the spring in equilibrium, and depends only on its length and on the form of its section. This brief discussion of the motion of a watch balance is taken from a memoir presented to the Academy of Sciences. The reader is referred to an article in Liouville's Journal, 1860, for a further investigation of the conditions necessary for isochronism and for a determination of the best forms for the spring. 109 a. If the length I of the spring is too long the time of oscillation is increased and the watch goes too slow. To correct this when necessary the clamp by which the point C is held is attached to a rod Ox which can turn stiffly round 0. The spring being held tight at D, let the rod Ox be moved from D, the spring slides through the clamp C and thus the length of CB, which is the effective length I of the spring, is shortened. Similarly by moving the rod Ox towards D, the effective length of the spring is increased. When the temperature rises, the length I of the spring is increased. For this and other reasons the watch will lose time. The compensation for a change of temperature is now usually effected by altering the moment of inertia of the oscil- lating body. The circumference of the balance wheel instead of being a complete circle consists of two arcs each less than a semi-circumference. An extremity of each is attached to one extremity of the rod BOB' , and each carries a small mass which is attached to it near its free extremity. Each arc is constructed of two thin slips of different metals lying side by side, the outer of which is made of brass and the inner of steel. As the temperature increases the brass slip expands more than the steel slip so that the arcs bend inward. The distances of the small masses from the axis are decreased and the moment of inertia of the whole balance is diminished. The proper positions of the masses on the circular arcs are determined by trial and this is usually a troublesome process. As thus constructed the instrument corrects the error only to a near approxi- mation. The changes in MkH, and in the coefficient of elasticity E, due to changes of temperature, follow somewhat different laws, and cannot be made to neutralize each other throughout the whole of any large range of temperature. What remains is called the "secondary error" and the modes of correcting it are described in treatises on clocks and watches. 109 h. The effect of the pressure and resistance of the air on the balance has not here been allowed for. According to Du Buat's theory (Art. 105) the general effect is to increase the moment of inertia MP of the balance by a small quantity R which is proportional to the density or pressure of the air. The time T of oscillation is therefore increased by ^TRjMk'^. The watch therefore goes a little slower, the change of rate being proportional to the pressure. A short summary of some ex- periments made to test this result is given in the Bulletin de la Societe Astronomique de France, April 1904. When great accuracy is required the chronometer might be enclosed in an air- tight case so that the density of the air inside might be kept constant. 110. Pressures on the fixed axis. A body moves about a fixed aods under the action of any forces, to find the pressures on the axis. Firstly. Suppose the body and the forces to be symmetrical about the plane through the centre of gravity perpendicular to ART. 110.] PRESSURES ON THE FIXED AXIS. 83 the axis. Then it is evident that the pressures on the axis are reducible to a single force at C the centre of suspension. Let Fy G be the actions of the point of support on the body resolved along and perpendicular to CO, where is the centre of gravity. Let X, F be the sum of the resolved parts of the impressed forces in the same directions, and L their moment round G. Let GG = h and 6 — angle which GG makes with any straight line fixed in space. Taking moments about G, we have gg^ ^ (1) The motion of the centre of gravity is the same as if all the forces acted at that point. Since it describes a circle round G, we have, by taking the tangential and normal resolutions, d^e F+(? ^W = M ...(2), de\^ dt) d^e X-^F ...(3). and then the M dO Equation (1) gives the values of ^ and -r- pressures may be found by equations (2) and (3). If the only force acting on the body be that of gravity, and 6 be measured from the vertical, we have X = Mg cos 6, Y= -Mg sin 0, L = -MghsmO; d^_ gh^ ' ' dt' 'd6\^_ dij ~ sin 6 ,(4). Integrating, w^e have .(5). Itf + h? If the angular velocity of the body be fl when GO is horizontal, we have co = fl when cos^ = 0. We find G=il^. Substituting these values in (2) and (3) we get j^ = n^h -{-gcosd G_ M g sin k' k^ + h^ , where 6 is the angle the perpendicular drawn from the centre of gravity of the body on the axis makes with the vertical measured downwards. It appears from these results that the component of the pressure which is perpendicular to the plane containing the axis and the centre of gravity is independent of the initial conditions. As the 6—2 84 MOTION ABOUT A FIXED AXIS. [CHAP. III. body oscillates this component varies as the distance of the centre of gravity from the vertical plane through the axis. On the other band the component of pressure in the plane containing the axis and the centre of gravity does depend on the initial angular velocity of the body. If the forces are impulsive, the equations (1), (2), (3) are only slightly altered. Let co, w! be the angular velocities of the body just before and just after the action of the impulses. The equa- tions then become "'-" = S7FTI^)' ^("'-<») = ^. = .Y + F, where all the letters have the same meaning as before, except that F, G, X, Y are now impulses instead of finite forces. 111. Ex. 1. A circular area of weight W can turn freely about a horizontal axis perpendicular to its plane which passes through a point C on its circumference. If it start from rest with the diameter through C vertically above C, show that the resultant pressures on the axis when that diameter is horizontal and vertically below C are respectively ^^ITW and ^W. Ex. 2. A thin uniform rod, one end of which is attached to a smooth hinge, is allowed to fall from a horizontal position ; prove that when the horizontal strain is the greatest possible, the vertical strain on the hinge is to the weight of the rod as 11 : 8. [Math. Tripos. Ex. 3. Let a = g . ., — ,~^ , b = g .^ . „ , and let R be the resultant of - i^' - M^"h and G. Construct an ellipse with C for centre and axes equal to 2a and 26 measured along and perpendicular to CO. Let this ellipse be fixed in the body and oscillate with it. Prove that the pressure R varies as the diameter along which it acts. And the direction may be found thus ; let the auxiliary circle cut the vertical through G in V, and let the perpendicular from V on CO cut the ellipse in R. Then CR is the direction of the pressure R, 112. Secondly. Suppose either the body or the forces not to be symmetrical. Let the fixed axis be taken as the axis of z with any origin and plane of xz. These we shall afterwards so choose as to simplify our process as much as possible. Let x, y, z be the coordinates of the centre of gravity at the time t Let co be the angular velocity of the body, / the angular acceleration, so that /= w. Now every element m of the body describes a circle about the axis, hence its accelerations along and perpendicular to the radius vector r from the axis are — wV and fr. Let 6 be the angle which r makes with the plane of ^^ at any time, then from the resolution of forces it is clear that ~x x = — (o^r cos -fr sin d = — (o^w—Jy, and similarly y= — (o'^y ■\-fx. ART. 112.] ' PRESSURES ON THE FIXED AXIS. 85 These equations may also be obtained by differentiating the equations a? = r cos 0, y — r sin 6 twice, remembering that r is constant. Collecting the effective forces of all the elements and com- bining them in Poinsot's manner, we see that they are equivalent to a force acting at the origin and a couple whose six com- ponents are X, = Imx = Sm (- 0)2^ -fy) = - (dHIx -/My (1), Fi = Imy = tm (- co^y -\-fx) = - co^My +fMx .(2), Z,==2mz =0 (3), Lj=^m (yz — zy) = — Xmzy = oyl^myz —f^mxz (4), M^ = 2m (zic — xz) = ^mzx — — co^lmxz —f^myz. . .(5), N^ = ^m(xy-yx) = 2mr^Q) = Mk''f (6). Since z = 0, the right-hand sides of (4) and (5) may obviously be obtained by merely introducing z into the 2 of (2) and (1). Let the body be fixed to the axis at two points distant a, a' from the origin and let the reactions of the points on the body resolved parallel to the coordinate axes be respectively F, G, H] F\ G', H'. Let X, F, Z be the accelerating impressed forces acting on the particle m. Then by D'Alembert's principle, Art. 72, tmX + F+F' = -' N = MW'ai' Here F' , G' are, as before, resolved parts of the pressure at A, and OA = a'. Putting F' = 0, G' — 0, these equations give the couples which must act on the body to produce rotation about Oz. Sub- stituting the values of L, M, N in (1), the equation to the plane of the couple is — ^mxz^ — l^myzr) + i/F^f = (2). Let the momental ellipsoid at the fixed point be constructed and let its equation be ^f + Brj' + C^' -2Drj^~ 2^r| - 2Ff77 = K. The diametral plane of the axis of f is -E^-Dr)-hC^=0 (3). Comparing (2) and (3) we see that the plane of the resultant couple must be the diametral plane of the axis of revolution. If then a body at rest with one point fixed be acted on by any couple it will begin to rotate about the diametral line of the plane of the couple with regard to the momental ellipsoid at the fi^ed point. Thus a body will begin to rotate about a perpendicular to the plane of the couple only when the plane of the couple is parallel to a principal plane of the body at a fixed point. 119. Ex. 1. If a body at rest have one point fixed and be acted on by any couple whose axis is a radius vector OP of the eUipsoid of gyration at 0, the body will begin to turn about a perpendicular from O on the tangent plane at P. Ex. 2. A solid homogeneous ellipsoid is fixed at its centre, and is acted on by a couple in a plane whose direction-cosines referred to the principal diameters are {I, m, n). Prove that the direction-cosines of the initial axis of rotation are pro- portional to -^ — -„, -„- „ *^^ Ex. 3. Any plane section being taken of the momental ellipsoid of a body at a fixed point, the body may be made to rotate uniformly about either of the principal 92 MOTION ABOUT A FIXED AXIS. [CHAP. III. diameters of this section as a fixed axis by the application of a couple of the proper magnitude whose axis is the other principal diameter. For assume the body to be turning uniformly about the axis of z. Then the couples which must act on the body to produce this motion are L = w^ Ztnyz, M = - (J^'Lmxz, N = 0, Art. 112. Then by taking the axis of x such that 'Zmxz = we see that the axis of the couple must be the axis of x and the magnitude of the couple will be L = o}^1,rnyz. Ex. 4. A body having one point O fixed in space is made to rotate uniformly about any proposed straight line by the application of the proper couple. The position of the axis of rotation when the magnitude of the couple is a maximum, has been called an axis of maximum reluctance. Show that there are six axes of maximum reluctance, two in each principal plane, each two bisecting the angles between the principal axes in the plane in which they are. Let the axes of reference be the principal axes of the body at the fixed point, let (Z, VI, n) be the direction-cosines of the axis of rotation, (X, fi, v) those of the axis of the couple G. Then by the last question and the second and third examples of Art. 18, we have ^^ _ ^^^ ^^^^^ = ^^-^--^^ = ^^^^^ , We have then to make G a maximum by variation of {bun) subject to the condition l^ + m'^ + n^=l. The positions of these axes were first investigated by Mr Walton in the Quarterly Journal of Mathematics, 1866, Vol. vii. p. 376. 120. The Centre of Percussion. When the fixed axis is given and the body can be so struck that there is no impulsive pressure on the axis, any point in the line of action of the force is called a centre of percussion. When the line of action of the blow is given, the axis about which the body begins to turn is called the axis of spontaneous rotation. It obviously coincides with the position of the fixed axis in the first case. Let us begin by considering the motion in two dimensions. Imagine a lamina at rest and suspended from a point G with the centre of gravity G vertically under G. Let it be struck by a horizontal blow Y which we may suppose to act in the plane of the lamina at some point A in GG produced. Let GA = a. Let F and G be the impulsive reactions at the fixed point G. Let co' be the angular velocity of the body round (7 just after the blow Y has been given. The equations of motion, exactly as in Art. 110, are therefore „'=^^^^^, ^"'=^' = ^- If the pressure G on the fixed point is zero, we have by eliminating F, ¥ + A- = ah. By Art. 92 this shows that A must be the centre of oscillation of the body. The centre of oscillation is therefore a centre of per- cussion. i ART. 121.] THE CENTRE OF PERCUSSION. 93 Prop. A body is capable of turning freely about a fixed axis. To determine the conditions that there shall be a centre of percussion and to find its position. Take the fixed axis as the axis of z, and let the plane of xz pass through the centre of gravity of the body. Let X, Y, Z be the resolved parts of the impulse, and let ^, 7j, f be the coordinates of any point in its line of action. Let Mk"^ be the moment of inertia of the body about the fixed axis. We have now to find the pressures on the axis, and by equating these to zero we shall discover the conditions for a centre of percussion. The process is virtually the same as that already explained in Art. 113 and again in Art. 117. It seems unnecessary to repeat the steps. Putting y = and omitting the impulsive pressures on the axis because by hypothesis they are to be equated to zero, the six equations of motion of Art. 113 become X=0, Y=AIx(o}' - u,), Z = (1). r}Z - iY= - (w' - w) ^7nxz\ ^X-^Z=-{o}'-u})i:myz I (2). ^Y-rjX= {u}' - w) Mk"^ J From these equations we may deduce the following conditions. I. From (1) we see that X=0, Z = 0, and therefore the force must act perpen- dicular to the plane containing the axis and the centre of gravity. II. Substituting from (1) in the first two equations of (2) we have l/myz = and f = „_ . Since the origin may be taken anywhere in the axis of rotation, let it be so chosen that f=0. Then the axis of z must be a principal axis at the point where a plane passing through the line of action of the blow perpendicular to the axis cuts the axis. Thus there can be no centre of percussion unless the axis be a principal axis at some point in its length. III. Substituting from (1) in the last equation of (2) we have ^ = ^-. By Art. 92 this is the equation to determine the centre of oscillation of the body about the fixed axis treated as an axis of suspension. Hence the perpendicular distance between the line of action of the impulse and the fixed axis must be equal to the distance of the centre of oscillation from the axis. If the fixed axis be parallel to a principal axis at the centre of gravity, the line of action of the blow will pass through the centre of oscillation. Ex. 1. A circular lamina rests on a smooth horizontal table ; how should it be struck that it may begin to turn round a point on its circumference ? The line of action of the blow should divide the perpendicular diameter in the ratio 3 : 1. Ex. 2. A pendulum is constructed of a sphere (radius a, mass M) attached to the end of a thin rod (length b, mass m). Where should it be struck at each oscil- lation that there may be no impulsive pressures to wear out the point of support ? The point is at a distance I from the point of support, where {M {a + b) + ^mb} 1= M {ia^ + {a + bf\ + ^mb^. 121. The Ballistic Pendulum. It is a matter of con- siderable importance in the Theory of Gunnery to determine the velocity of a bullet as it issues from the mouth of a gun. By means of it we obtain a complete test of any theory we have reason to form concerning the motion of the bullet in the gun. We may thus find by experiment the separate effects produced by varying the length of the gun, the charge of powder, or the weight of the ball. By determining the velocity of a bullet at different 94 MOTION ABOUT A FIXED AXIS. [CHAP. III. distances from the gun we may discover the laws which govern the resistance of the air. It was to determine this initial velocity that Robins about 1743 invented the Ballistic Pendulum. Before his time but little progress had been made in the true theory of military projectiles. His New Principles of Gunnery was soon translated into several languages, and Euler added to his translation of it into German an extensive commentary. The work of Euler was again trans- lated into English in 1784. The experiments of Robins were all conducted with musket balls of about an ounce weight, but they were afterwards continued during several years by Dr Hutton, who used cannon balls of from one to nearly three pounds in weight. There are two methods of applying the ballistic pendulum, both of which were used by Robins. In the first method, the gun is attached to a very heavy pendulum ; when the gun is fired the recoil causes the pendulum to turn round its axis and to oscillate through an arc which can be measured. The velocity of the bullet can be deduced from the magnitude of this arc. In the second method, the bullet is fired into a heavy pendulum. The velocity of the bullet is itself too great to be measured directly, but the angular velocity communicated to the pendulum may be made as small as we please by increasing its bulk. The arc of oscillation being measured, the velocity of the bullet can be found by calculation. The initial velocity of a small bullet may also be determined by the use of some rotational apparatus. Two circular discs of paper are attached perpendicularly to the straight line joining their centres, and are made to rotate about this straight line with a great but known angular velocity. Instead of two discs, a cylinder of paper might be used. The bullet being fired through at least two of the moving surfaces, its velocity can be calculated when the situations of the two small holes made by the bullet have been observed. This was originally an Italian invention, but it was much improved and used by Olinthus Gregory in the early part of last century. The electric telegraph is now used to determine the instant at which a bullet passes through any one of a number of screens through which it is made to pass. The bullet severs a fine wire stretched across the screen and thus breaks an electric circuit. This causes a record of the time of transit to be made by an instrument expressly prepared for this purpose. By using several screens the velocities of the same bullet at several points of its course may be found. The ballistic pendulum is thus more of theoretical and historical interest than of practical importance. The two instruments now chiefly used for observations on the ART. 122.] THE BALLISTIC PENDULUM. 95 velocities of bullets are, the chronograph invented by Bashforth and used by the English government, and the chronograph in- vented by Major Le Bouleng^ of the Belgian artillery. 122. A rifle is attached in a horizontal position to a large block of wood which can turn freely about a horizontal axis. The rifle being flred, the recoil causes the pendulum to turn round its axis until brought to rest by the action of gravity. A piece of tape is attached to the pendulum, and is drawn out of a reel during the backward motion of the pendulum, and thus serves to measur^e the amount of the angle of recoil. It is required to find the velocity of the bullet. The initial velocity of the bullet is so much greater than that of the pendulum that we may suppose the ball to have left the rifle before the pendulum has sensibly moved from its initial position. The initial momentum of the bullet may be taken as a measure of the impulse communicated to the pendulum. Let h be the distance of the centre of gravity from the axis of suspension ; / the distance from the axis of the rifle to the axis of suspension ; c the distance from the axis of suspension to the point of attachment of the tape, m the mass of the bullet ; M that of the pendulum and rifle, and n the ratio of ilf to m ; b the chord of the arc of the recoil which is measured by the tape. Let k' be the radius of gyration of the rifle and pendulum about the axis of suspension, v the initial velocity of the bullet. The explosion of the gunpowder generates equal impulsive actions on the bullet and on the rifle. Since the initial velocity of the bullet is v, this action is measured by mv. The initial angular velocity generated in the pendulum by the impulse is by Art. 89 mvf o) = — ^ . The subsequent motion is given (Art. 92) by k'^d^-ghsme; .'. k'^d'' = C + 2gh cos 6 : when ^ = we have 6 = co, and if a is the angle of recoil, when 6 — a, t7 = 0. Hence k'^o)^ = 2gh (1 — cos a). Eliminating &> we have vf=nk'.2sm.\oi\Jgh. But the chord of the arc of the recoil is b = 2c sin ^a. Hence the initial velocity of the bullet is given by mv . cf= Mbk'slgh. The magnitude of k' may be found experimentally by ob- serving the time of a small oscillation of the pendulum and rifle. If r be a half-time we have T=irx/ ~. (Art. 97.) This is the formula given by Poisson in the second volume of his Mecanique. The reader will find in the Philosophical Magazine for June. 1854, an account of some experiments conducted by Dr S. Haughton from which, by the use of this formula, the initial velocities of rifle bullets were calculated. 96 MOTION ABOUT A FIXED AXIS. [CHAP. III. 123. The formula must however be regarded only as a first approximation, for the recoil due to the inflamed powder has been neglected. To make some allowance for this Hutton assumed that the effect of a given charge of powder on the recoil of the gun was the same with as without a ball. Let p be the unknown momen- tum generated by the powder. By trying the experiment, with equal charges of powder, first with and then without a ball, and writing mv +p and p for mv in the two experiments he was able to eliminate p and deduce the value of v. With large charges of powder, the results thus obtained did not agree sufficiently with those obtained by firing the ball into a pendulum (Art. 124), The assumption was therefore not altogether justified by the experiments and further corrections were made. 124. A gun is placed in front of a heavy pendulum, which can turn freely about a horizontal axis. The ball strikes the pendulum horizontally, penetrates into the wood a short distance, and communicates a momentum to the pendulum. The chord of the arc being measured as before by a piece of tape, find the velocity of the bullet. The time, which the bullet takes to penetrate, is so short that we may suppose it completed before the pendulum has sensibly moved from its initial position. Let i be the distance of the ball from the axis of suspension at the moment when the penetration ceases ; let j be the perpendicular distance between the axis and the dnection of motion of the bullet ; let ^ be the angle the length j makes with the length represented by i, so that j = i cos ^. Then if we follow the same notation as before we have at the moment when the impact is concluded mvi cos j8 = {Mk'^ + mi^) u ; also proceeding as before we may prove {Mk'^ + mi^) u}^ = 2Mgh (1 - cos a) + 2mgi {cos /3 - cos (a - /3)}. If the gun be placed as nearly as possible opposite the centre of gravity of the pendulum we have h=j nearly, and if the pendulum be rather long /3 will be very small. Hence, since m is small compared with M, we may as an approximation put i = h and j8 = in the terms which contain m as a factor ; we thus find M + m bh /— v = : \jgi, m cj where l is the distance of the centre of oscillation of the pendulum and ball from the axis of suspension. The inconvenience of this construction as compared with the former is that the balls remain in the pendulum during the time of making one whole set of experi- ments. The weight, and the positions of the centres of gravity and oscillation, will be changed by the addition of each ball which is lodged in the wood. Even then the changes produced in the pendulum itself by each blow are omitted. A great improvement was made by the French in conducting their experiments at Metz in 1839, and at L'Orient in 1842. Instead of a mass of wood, requiring frequent renewals, as in the English pendulum, a permanent recepteur was substi- tuted. This receiver is shaped within as a truncated cone, which is sufficiently long to prevent the shot from passing entirely through the sand with which it is filled. The front is covered with a thin sheet of lead to prevent the sand from being shaken out. This sheet is marked by a horizontal and by a vertical line, the intersection corresponding to the axial line of the cone, so that the actual position of the shot when entering the receiver can be readily determined by these lines. 125. Ex. 1. Show that after each bullet has been fired into a ballistic pen- dulum constructed on the English plan, h must be increased by {j-h)mlM and I by {j - 1) mjM nearly in order to prepare the formula for the next shot. ART. 126.] THE BALLISTIC PENDULUM. 97 Ex. 2. Dr Haughton found that, for rifles fired with a constant charge, the initial velocity of the bullet varies as the square root of the mass of the bullet inversely and as the square root of the length of the gun directly. Show from this that the force developed by the explosion of the powder, diminished by the friction of the barrel, is constant as the ball traverses the rifle. Dr Hutton found that in smooth bores the velocity increases in a ratio some- what less than the square root of the length of the gun, but greater than the cube root of the length. Ex. 3. If the velocity of a bullet issuing from the mouth of a gun 30 inches long be 1000 feet per second, show that the time the bullet takes to traverse the gun is about ^-fy of a second. Ex. 4. It has been found by experiment that, if a bullet be fired into a large fixed block of wood, the depth of penetration of the bullet into the wood varies nearly as the square of the velocity, though as the velocity is very much increased the depth falls short of that given by this rule. Assuming this rule, show that the resistance to penetration is constant and that the time of penetration is the ratio of twice the depth to the initial velocity of the bullet. In an experiment of Dr Button's a ball fired with a velocity of 1500 feet per second was found to penetrate about 14 inches into a block of sound dry elm : show that the time of penetration was ^i^ of a second. 126. The Anemometer. The Anemometer called a "Kobinson" consists of four hemispherical cups attached to four horizontal arms which turn round a vertical axis. The wind blows into the hollows on one side of the axis and against the convex surfaces of the cups on the other. If the anemometer start from rest, it will turn quicker and quicker until the moment of the pressures of the wind balances the moment of the resistances. Let V be the velocity of the wind and v the velocity of the centres of the cups. Let 6 be the angle between the direction of motion of any one cup and that of the wind. Then the velocity of the centre of that cup relatively to the wind will be v\ where v'^ = v'^-2Vvcos0+V^ (1). The determination of the pressure of the wind on the cups is properly a problem in hydrodynamics, but no solution has yet been found. In the meantime we may assume as an approximation the law, suggested by numerous experiments, that the resistance to a body moving in a straight line in a fluid varies as the square of the relative velocity. In any one position of the anemometer the parts of any one cup have different velocities relative to the wind. We shall therefore take as our expression for the moment about the axis of the anemometer of the resultant pressure of the wind some quadratic function of V and v, such as aV^ + 2^Vv + yv^ (2), where a, j3, y depend in some manner as yet unknown on the position of the cups relatively to the wind. Thus a, j8, 7 are functions of 6 and will change as the cups turn round the axis.. What we want however is the average effect on the anemometer. The mean for space is found by multiplying this expression by dd and integrating from ^ = to 27r and finally dividing by 2t. If F be the mean moment about the axis of th& anemometer of the wind pressure, we have F=AV^-2BVv-Cv'^ (3), where A, B, C are constants which depend on the pattern of the anemometer. The signs of these coefficients may be determined by the following reasoning. When the anemometer starts from rest, the initial moment of the wind pressure is R. D. 7 98 MOTION ABOUT A FIXED AXIS. [CHAP, III. regarded as positive. When the cups begin to move, the pressure begins to decrease, so that — must be negative when v is small ; it follows that the sign of the coefficient of Vv in (3) must be negative. Finally, if the wind cease when the cups are in motion so that V= 0, the resistance of the quiescent air must tend to stop the cups. It follows that the coefficient of v^ in (3) must be also negative. 127. When the anemometer has attained its final state of motion, we must have F equal to the mean moment of the friction on the supports. The instru- ment should be so arranged that the friction due to its weight is as small as possible. We may then omit this friction, as our formula is only an approximation. The supports of the anemometer have also to sustain the lateral pressure of the wind. Probably the greater part of the friction thus produced is proportional to the pressure of the wind, and may be included in the formula (3) by an alteration of the constants. As these constants are determined by experiment, we may suppose all forces which are quadratic functions of the velocities to be included in the expression for F. In the Observatory at Greenwich an inverted cup rotating in oil on a fixed conical point is used for the vertical bearing. No further correction is made for friction. This arrangement appears to be very successful, the instrument is very sensitive and exhibits a slow rotation with a very slight movement of the air. When F is equated to zero, we have a quadratic to determine the ratio of V to V. Let m be the positive root thus found. Then the velocity of the centre of any cup being observed, the velocity of the wind is found by simply multiplying this observed quantity by m. We may notice that m is independent of the speed of the wind, and of the size of the machine. It depends however on the pattern of the machine. 128. A variety of experiments have been made to determine the numerical value of m. In some of these the anemometer is attached to the outer edge of a whirling-machine. The axis of the anemometer is thus made to move round with a constant velocity V. If the experiment be made on a calm day, this will represent the effects of a wind of the same velocity on a fixed anemometer. The value of v can be found by counting the number of revolutions of the anemometer in space. In a paper in 1850, published in the Irish Transactions, Dr Eobinson gives m = 3 as the mean value of the ratio as determined by experiments of this kind. This value of m has been generally adopted. Other experiments made in Greenwich Park in 1860 led to the same value of m. These results were considered as confirming in a very high degree the accuracy of this ratio. See the Greenwich Observations for 1862. About 1872 further experi- ments were made with a steam merry-go-round for a whirling machine. These are described by Sir G. Stokes in the Proceedings of the Royal Society for May, 1881. According to some experiments conducted by W. H. Dines in 1889 the value m = B for anemometers of the Kew pattern is too high, and if these results are con- firmed the registered wind velocities are in excess of the truth. See the report of the wind-force committee on the factor of the Kew-pattern Eobinson anemometer, Meteorological Society, Dec. 1889. Another method of conducting the experiments is to have two similar anemo- meters rotating about fixed axes and to apply to one of them a known retarding force of some kind which may diminish its v. Thus we have two different machines moving with different, but known, velocities round their respective axes, from each of which we should deduce the same velocity for the wind. This leads to two equa- tions between which we may eliminate the unknown velocity of the wind. We thus ART. 129.] THE ANEMOMETER. 99 obtaiu an equation connecting the constants A,B, C and the known retarding force. Repeating the experiment, we may obtain a sufficient number of equations to find these constants. The value of m may then be found in the manner explained in Art. 127. The practical difficulty in this method of conducting the experiments is that of finding a known uniform retarding force which may be conveniently applied to the anemometer. The reader may consult a paper by Dr Robinson in the Phil. Trans, for 1880. 129. Ex. 1. Supposing the value of F to be represented by AV^-2BVv, as indicated by some experiments, show that, if an anemometer start from rest, the velocity v of the cups will continually increase and tend to a certain finite limit. Show also that the time, at which the actual velocity of the cups is any given fraction of the limiting velocity, varies as the moment of inertia of the anemometer about its axis, and inversely as the velocity of the wind. Ex. 2. When the anemometer was attached to the outer edge of a merry-go- round, as described above, it was impossible to find a perfectly calm day. If W be the velocity of the wind, which is supposed to be small, then allowance may be made for JV if in the formula F=AV^-2BVv we write V+kW'^IV for V, where k is ^ or f according as the moment of inertia of the anemometer about its axis is very small or very great. The anemometer is supposed to be without friction. This theorem is due to Sir G. Stokes : a demonstration is given in the Proceedings of the Royal Society for May, 1881. Ex. 3. An anemometer without friction is acted on by a gusty wind whose velocity may be represented by the formula V {1 + a sin nt), where a is so small that its square can be neglected. Show that the velocity of any cup will be represented by an expression of the form v {1+ a cos n^ sin n{t- ^)}, so that the anemometer follows all the changes in the force of the wind after an interval ^. Here In AV^-2BVv-Cv^=0, and tan?ii3= ^ .^.-^ — p^^ , where a is the distance of the ' ^ 2a{BV+Cv) centre of a cup from the axis, and I is the moment of inertia of the machine about the axis. The velocities of the currents of air in mines are usually determined by the aid of anemometers of a somewhat different construction. The principle of these is similar to that of Whe.well's anemometer. They are formed of several light vanes placed on a horizontal axis like the sails of a windmill on a small scale but more numerous. The axis is attached to a dial or some other apparatus by which the number of revolutions made by the little windmill can be read off. If V be the velocity of the wind and v the reading of the anemometer it is found by experiment that between certain limits V—av + b, where a and 6 are two constants which depend on the pattern of the anemometer and the friction which the wind has to overcome. The reader may consult a paper by Mr Snell in the Engineer, June 23, 1882. The Annals of the Astronomical Society of Harvard College, Vol. xl. contains an appendix by S. P. Fergusson on anemometer comparisons made in the years 1892 — 94 in Massachusetts. There is also a paper by C. Chree on the theory of the Robinson cup anemometer, Phil. Mag., 1895. 7—2 CHAPTER lY. MOTION IN TWO DIMENSIONS. On the Equations of Motion, 130. The position of a body in space of two dimensions may be determined by the coordinates of its centre of gravity, and the angle some straight line fixed in the body makes with some straight line fixed in space. These three have been called the coordinates of the body, and it is our object to determine them in terms of the time. It will be necessary to express the effective forces of the body in terms of these coordinates. The resolved parts of these effective forces parallel to the axes have been already found in Art. 79, all that is now necessary is to find their moment about the centre of gravity. If {x\ y') be the coordinates of any particle of mass m referred to rectangular axes meeting at the centre of gravity and parallel to the axes fixed in space, this moment has been shown in Art. 76 to be equal to h, where h = ^m {x'y — y'x). Let be the angular coordinate of the body, i.e. the angle some straight line fixed in the body makes with some straight line fixed in space. Let (/, c^') be the polar coordinates of any particle in referred to the centre of gravity of the body as origin. Then r is constant throughout the motion, and ' is the same for every particle of the body and equal to 6. Thus the angular momentum A, exactly as in Art. 88, is h=^tm {x'y' - y'x') = ^m (r''j>') = (Xmr'')4>' = Mm, where Mk"^ is the moment of inertia of the body about its centre of gravity. The angle 6 is the angle some straight line fixed in the body makes with a straight line fixed in space. Whatever straight lines are chosen dOjdt is the same. If this is not obvious, it may be shown thus. Let OA, O'A' be any two straight lines fixed in the body inclined at an angle a to each other. Let OB, O'B' be two straight lines fixed in space inclined at an angle y8 to each ART. 131.] ON THE EQUATIONS OF MOTION. 101 Other. Let AOB = e, A'D'E^O', then e'-\-^ = e + a. Since a and /3 are independent of the time, 6 = 6'. By this proposition we learn that the angular velocities of a body in two dimensions are the same about all points. 131. The general method of proceeding will be as follows. Let (x, y) be the coordinates of the centre of gravity of any body of the system referred to rectangular axes fixed in space, M the mass of the body. Then the effective forces of the body dj^x 6}y are together equivalent to two forces measured by M-^-, M -j—^ acting at the centre of gravity and parallel to the axes of co- d'^0 ordinates, together with a couple measured by Mk"^ -j— tending to turn the body about its centre of gravity in the direction in which 6 is measured. By D'Alembert's principle the effective forces of all the bodies, if reversed, will be in equilibrium with the impressed forces. The dynamical equations may then be formed according to the ordinary rules of statics. See Art. 83. Suppose we wish to resolve the forces parallel to the axes of X and y and to take moments about the centre of gravity. Let the impressed forces acting on the body, together with the re- actions due to the other bodies if any, be equivalent to the forces X and Y acting at the centre of gravity and a couple L. The equations of motion of that body are evidently It is found useful in statics to be able to resolve in other directions besides the axes and to be able to take moments about any point we please. In this way we often greatly shorten and simplify the solution. Thus if we wish to avoid the introduction into our equations of some unknown reaction we take moments about the point of application or use the principle of virtual velocities. So in dynamics we are at liberty to resolve our forces and take moments at pleasure. For example, if we take moments about a point G whose coordinates are (f, rj) we have an equation of the form where L' is the moment about C of the impressed forces. In this equation (f, tj) may be the coordinates of any point whatever, whether fixed or moving. In resolving our forces we may replace the Cartesian ex- pressions by the polar forms ^ ] j^ - ^ f;;^] [ ^^^ ^~~iTt\ dt) 102 MOTION IN TWO DIMENSIONS. [CHAP. IV. for the resolved parts parallel and perpendicular to the radius vector. If V be the velocity of the centre of gravity, p the radius of curvature of its path, we may sometimes also use with advantage the forms M -r: and M- for the resolved parts of the effective forces along the tangent and radius of curvature of the path of the centre of gravity. As a guide to a proper choice of the directions in which to resolve the forces or of the points about which we should take moments we may mention two important cases. 132. First, we should search if there be any direction fixed in space in which the resolved part of the impressed forces vanishes. By resolving in this direction we get an equation which can be immediately integrated. Suppose the axis of x to be taken in this direction ; let M, M\ &c. be the masses of the several bodies, w, x', &c. the abscissae of their centres of gravity, then by Art. 78 or 131, we have M^^+M'^ + ... =0, djc dor which by integration gives M-j~ + M' -j- + = (7, where C is some constant to be found from the initial conditions. This equation may be again integrated if necessary. This result might have been derived from the general principle of the conservation of the motion of translation of the centre of gravity laid down in Art. 79. For, since there is no impressed force parallel to the axis of x, the velocity of the centre of gravity of the whole system resolved in that direction is constant. 133. Next, we should search if there be any point fixed in space about luhich the moment of the impressed forces vanishes. By taking moments about that point we again have an equation which admits of immediate integration. Suppose the point to be taken as origin, and the letters to have their usual meaning, then by the first article of this chapter we have V (i^/ d'y d''x\ j.^j,d'd] 0, the S referring to summation for all the bodies of the system. Integrating we have M"(4-»i)+»'f}-<'. 4 where G is some constant to. be determined by the initial con- ditions of the question. This equation expresses the fact that if the impressed forces ART. 134.] ON THE EQUATIONS OF MOTION. 103 have no moment about any fixed point, the angular momentum about that point is constant throughout the motion. This result follows at once from the reasoning in Art. 78. 134. Angular Momentum. As we shall have so frequently to use the equation formed by taking moments, it is important to consider other forms into which it may be put. Let the point about which we are to take moments be fixed in space, so that it may be chosen as the origin of coordinates. Then the moment of the effective forces on the body M is where ^ and ^ are the coordinates of the centre of gravity. The attention of the reader is directed to the meaning of the several parts of this expression. We see that, as explained in Art. 78, the moment of the effective forces is the differential coefficient of the moment of the momentum about the same point. The moment of the momentum by Art. 75 is the same as the moment about the centre of gravity together with the moment of the whole mass collected at the centre of gravity, and moving with the velocity of the centre of gravity. The moment round the centre of gravity is by the first article either of Chap. ill. or Chap. iv. equal to Mk^ -J- a,nd the moment of the collected mass is M (x~- — y-^]. dt \ dt ^ dtj Hence in space of two dimensions we have for any body of mass M angular momentum round] _ nj f dy dx\ ^, ^ ^^ the origin J ~ '^ T ^ ~ ^ d~t) ^'^^ Tf If we prefer to use polar coordinates, we can put this into another form. Let (r, 0) be the polar coordinates of the centre of gravity, then angular momentum round) t.^ .d4> n/n^dO the origin J dt dt If V be the velocity of the centre of gravity, and p the per- pendicular from the origin on the tangent to the direction of its motion, the moment of momentum of the mass collected at the centre of gravity is Mvp, so that we have again angular momentum round) ,, n/n^dd the origin J ^ dt It is clear from Art. 75 that this is the instantaneous angular momentum of the body about the origin whether it is fixed or moving, though in the latter case its differential coefficient with regard to t is not the moment of the effective forces. Since the instantaneous centre of rotation may be regarded as 104 MOTION IN TWO DIMENSIONS. [CHAP. IV. a fixed point, when we have to deal only with the coordinates and with their first differential coefficients with regard to the time, we have ansfular momentum round the) ,, , « T.\dd instantaneous centre j at If Mk'^ be the moment of inertia about the instantaneous do centre, this last moment may be written Mk'^-j-. In taking moments about any point, whether it be the centre of gravity or not, it should be noticed that the Mk'^ in all these formula? is the moment of inertia with regard to the centre of gravity, and not with regard to the point about which we are taking moments. It is only when we are taking moments about the instantaneous centre or about a fixed point that we can use the moment of inertia about that point instead of the moment of inertia about the centre of gravity, and in these cases our expression for the angular momentum includes the angular mo- mentum of the mass collected at the centre of gravity. 135. General Mode of Solution. Suppose we form the equations of motion of each body by resolving parallel to the axes of coordinates and by taking moments about the centre of gravity. We shall get three equations for each body of the form Mx = i^ cos (/> + E cos -l/r + ...\ My = F sin , and whose moment about the centre of gravity is Fp, and R is any one of the reactions. These we shall call the dynamical equations of the body. Besides these there will be certain geometrical equations expressing the connections of the system. As every such forced connection is accompanied by a reaction, and every reaction by some forced connection, the number of geometrical equations will be the same as the number of unknown reactions in the system. Having obtained the proper number of equations of motion we proceed to their solution. Two general methods have been proposed. First Method of Solution. Differentiate the geometrical equa- tions twice with respect to t, and substitute for x, y, 6 from the dynamical equations. We shall then have a sufficient number of equations to determine the reactions. This method will be of great advantage whenever the geometrical equations are of the form Ax + By + Cd = D (2), ART. l;37.] ON THE EQUATIONS OF MOTION. 105 A, B, C, D being constants. Suppose also that the dynamical equations are such that when written in the form (1) they contain only the reactions and constants on the right-hand side without any X, y, or 6. Then, when we substitute in the equation Ax + By + Ce = 0, obtained by differentiating (2), we have an equation containing only the reactions and constants. This being true for all the geon\etrical relations, it is evident that all the reactions will be constant throughout the motion and their values may be found. Hence, when these values are substituted in the dynamical equa- tions (1), their right-hand members will all be constants and the values of x, y, and 6 may be found by an easy integration. If however the geometrical equations are not of the form (2), this method of solution will usually fail. Thus suppose a geo- metrical equation to take the form x' + f = c\ containing squares instead of first powers, then its second differ- ential equation will be XX + yy + x^ + y'^ = ] and, though we can substitute for x, y, we cannot in general eliminate the terms x^ and if. 136. The reactions in a dynamical problem are in many cases produced by the pressures of some smooth fixed obstacles which are touched by the moving bodies. Such obstacles can only push, and therefore if the equations show that such a reaction changes sign at any instant, it is clear that the body will leave the obstacle at that instant. This will occasionally introduce discon- tinuity into our equations. At first the system moves under certain constraints, and our equations are found on that suppo- sition. At some instant to be determined by the vanishing of a reaction one of the bodies leaves its constraints, and the equations of motion have to be changed by the omission of that reaction. Similar remarks apply if the reaction be produced by the pressure of one body against another. It is important to notice that when this first method of solu- tion applies, the reactions are constant throughout the motion, so that the above- discontinuity can never occur. In this case, then, if one body he in contact with another, they will either separate at the beginning of their motion or will always continue in contact. Such reactions are also independent of the initial conditions, and are the same as if the system were placed in any position at rest 137. Suppose that in a dynamical system we have two bodies which press on each other with a reaction R; let us consider how we are to form the corresponding geometrical equation. 106 MOTION IN TWO DIMENSIONS. [CHAP. IV. We have clearly to express the fact that the velocities of the points of contact of the two bodies resolved along the direc- tion of R are equal. The following proposition will he often useful. Let a body be turning about a point Q with an angular velocity 6 = (t> in a direction opposite to the hands of a watch, and let G be moving in the direc- tion GA with a velocity V. It is required to find the velocity of any point P re- solved in any direction PQ making an angle <^ with GA. In the time dt the whole body, and therefore also the point P, is moved through a space Vdt parallel to GA, and during the same time P is moved perpendicular to GP through a space w . GP . dt. Resolving parallel to PQ, the whole displacement of P = ( Fcos initial vis viva. In order that the centre of gravity should reach this altitude it is necessary that the vis viva of the body should vanish, i.e. both the velocity of translation of the centre of gravity and the angular velocity of the body must simultaneously vanish. This cannot in general occur if the body jump off the surface, for the angular velocity and the horizontal velocity of the centre of gravity will not usually both vanish at the moment of the jump^ and both will remain constant, as explained above, during the parabolic motion. After the subsequent impact a new motion may be supposed to begin with a diminished vis viva and therefore a diminished superior limit to the altitude of the centre of gravity. 143. Sometimes there is only one way in which the system can move. In such a case all we have to find is the velocity of the motion. The geometry of the system will determine the x, y, 6 of each body in terms of some one quantity which we may call . The vis viva of the body M, as given by Art. 139, will now take the form ART. 144.] ON THE EQUATIONS OF MOTION. 113 where P is a known function of the coordinates of M. The equation of vis viva will therefore take the form and thus d(l)/dt can be found for any given position of the system. It follows that, if there is only one way in which the system can move, that motion will he determined by the equation of vis viva. But, if there be more than one possible motion, we must find another integral of the equations of the second order. What should be done will depend on the special case under considera- tion. The discovery of the proper treatment of the equations is often a matter of great difficulty. The difficulty will be increased if, in forming the operations, care has not been taken to give them the simplest possible forms. 144. Examples of these Principles. The following ex- amples have been constructed to illustrate the methods of applying the above principles to the solution of dynamical problems. In some cases more solutions than one have been given, to enable the reader to compare different methods. The mode of forming each equation has been «ninutely explained. Running remarks have been made which it is hoped will clear up those difficulties which generally trouble a beginner. The attention of the student is therefore particularly directed to the different principles used in the follow- ing solutions. A homogeneous sphere rolls directly down a perfectly rough inclined plane under the action of gravity. It is required to find the motion. Let a be the inclination of the plane to the horizon, a the radius of the sphere, wi/r its moment of inertia about a horizontal diameter. Let be that point of the inclined plane which was initially touched by the sphere, and N the point of contact at the time t. Then it is obviously convenient to choose for origin, and ON for axis of X. The forces which act on the sphere are, first, the reaction jR perpendicularly to ON, secondly, the friction F acting at N along NO and N^ y vig acting vertically at G the centre. The effective ^^^^ ^ '^/o forces are nix, my acting at G parallel to the axes of a; /^ / x "r and y, and a couple mlc^d tending to turn the sphere round G in the direction NA. Here 6 is the angle y ^/^~~--/N which any straight line fixed in the body makes with a straight line fixed in space. We shall take the fixed straight line in the body to be the radius GA, and the fixed straight line in space the normal to the inclined — plane. Then d is the angle turned through by the sphere. Eesolving along and perpendicular to the inclined plane we have mx = mg sin a - F (1), my= -mgcosa + R (2). Taking moments about N to avoid the reactions, we have max + mk^d = mgaama (3). R. D. 8 114 MOTION IN TWO DIMENSIONS. [CHAP. IV. Since there are two unknown reactions F and E, we shall require two geome- trical relations. Because there is no slipping at N we have x = ae (4). Also, because there is no jumping, y = a (5). Both these equations are of the form required in the first method. Differ- entiating (4) we get x = ad. Joining this to (3) we have x= ., -.^^g sin a (6). 2 5 Since the sphere is homogeneous, k^ = ~a^, and we have if^-^'sina. o 7 If the sphere had been sliding down a smooth plane, the equation of motion would have been a; =^ sin a, so that two-sevenths of gravity is used in turning the sphere, and Jive-sevenths in urging the sphere dowmcards. 1 5 Supposing the sphere to start from rest we have clearly x = -. ^gsin a . t'-, and 2 7 the whole motion is determined. In the above solutions only a f^w of the equations of motion have been used, and if the motion only had been required it would have been unnecessary to write down any equations except (3) and (4). If the reactions also are required; we must use the remaining equations. From (1), (2) and (5) we have 2 F=-rngsina, R = mg cos a. It is usual to delay the substitution of the value of k^ in the equations until the end of the investigation, for this value is often very complicated. But there is another advantage. It serves as a verification of the signs in our original equations, for if equation (6) had been we should have expected some error to exist in the solution. It seems clear that the acceleration could not be made infinite by any alteration of the internal structure of the sphere. Ex. If the plane were imperfectly rough with a coefficient of friction fi less than f tan a, show that the angular velocity of the sphere after a time t from rest would be -^ — t. 2 a 145. A homogeneous sphere rolls down another perfectly rough fixed sphere. Find the motion. Let a and 6 be the radii of the moving and fixed spheres, respectively, C and the two centres. Let OB be the vertical radius of the fixed sphere, and = z BOC. Let F and R be the friction and the normal reaction at N. Then, resolving tangentially and normally to the path of C, we have m {a + 6)

-F. (1), m (a + b) -R (2) . Let A be that point of the moving sphere which originally coincided with B. Then if 6 be the angle which any fixed line, as CA, in the body makes with any fixed line in space, as the vertical, we have by taking moments about C mk^e = Fa (3). It should be observed that we cannot take 6 as the angle ACO because, though CA is fixed in the body, CO is not fixed in space. The geometrical equation is clearly a(d-, w' be the angular velocities of AB, BC at any instant t. The angular momentum of BC about ^ is • m{xi/-yx + k^ia'), Art. 134. The angular momentum of AB is by the same article in{k^ + a^)u}, since AB is turning about A as a fixed point. The initial values of these are respectively m [Sa^ + k^) O, and vi{k^ + a^)Q, since w, w' and 6 are each initially equal to 12 and r is initially equal to the perpendicular from A on the oppo- site side of the equilateral triangle formed by the system. Hence m {k^ + a^) io + m {xy -yx + k^u') = m{2k^ + 4a^)Q (1). We may obtain another equation by the use of the principle of vis viva. The vis viva of the rod BC is m {x^ + y^ + k^w'^), Art. 139. The vis viva of AB is by the same article m{k^ + a^) u'^ since it is turning round -4 as a fixed point. The initial values of these are respectively m {Sa^ + k^) Q"^ and 7ii(k^ + a^)Q^. If T be the tension of the string, p its length at time t, the force function of the tension is p ( - T) dp. According to the rule given in statics to calculate virtual moments, /: the minus sign is given to the tension because it acts so as to diminish p ; and the limits are 2a to p because the string has stretched from its initial length 2a to p. Ey Hooke'slaw T=E ^ - so that, by integration, the force function = -E — • 2a ' './-—- . 4^ Since the reaction at A does not appear. Art. 141, the equation of vis viva is (p-2a)2 m (A;2 + a2) oj^ + m {x'^ + y^+k^io"'} = m {2k'' + 4:a^) n'^-E 2a (2). There are only two possible independent motions of the rods. We can turn AB about A and BC about B, all motions, not compounded of these, being incon- sistent with the geometrical conditions of the question. Two dynamical equations are sufficient to determine these, and we have just obtained two. All the other equations which may be wanted must be derived from geometrical considerations. Let i/', \f/' be the inclinations of the rods AB, BC to the axis of x and let = ^'- i/'. We have x = 2a cos \f/ + acos ^\ ?/ = 2a sin i/' + a sin ;/^', i = - 2a sin \}/(i}-a sin ^'w', y = 2a cos \p(i) + a cos xp'u'. The equations of angular momentum and vis viva then become m (fe2 + 5a2 + 2a2 cos 0)w + m(/c2 + a2 4- 2a2 cos 0)aj' = m(2i'2 + 4a^)« (3), m(/c2 + 5a2)w-'-fm(/c2 + a2)a;'2-f4wa2«a;'cos0 = m(2A;2-f4a2)fi2-E'-^^^ ...(4). These equations determine w, w' in terms of the subsidiary angle 0. It is required to find the greatest length of the elastic string during the motion. At the moment when p is a maximum p = and the whole system is therefore ART. 147.] ON THE EQUATIONS OF MOTION. 119 moving as if it were a rigid body. We therefore have for a single moment w = a;'. The equations (3) and (4) become, when we have substituted for k^ its value |a"-^, (lO + 6cos0)a; = 7i2, (lO + 6cos0) a>'-^=:7fi'^- ^^.j ()t)-2a)2. Eliminating w and remembering that /) = 4acos^0, we have I . E (3/)2 + 16a2) (p - 2a) = 28mn^a^ {p + 2a). This cubic has one positive root greater than 2a. It is also required to find the motion at the instant when the rods are at right angles. At this moment (p = h'rr, and hence (3) and (4) become 3E ma ^^ ' From these equations we easily find w and a>'. It is clear that the values of a>, w' are not real unless 7fi"^ > 10 (^^2 - 1)2 E\ma. Another solution. We may often save ourselves the trouble of some elimination if we form the equations derived from the principles of angular momentum and vis viva in a slightly different manner. The rod BG is turning round B with an angular velocity w', while at the same time B is moving perpendicularly to AB with a velocity 2aw. The velocity of E is therefore the resultant of aw' perpendicular to BC and 2aw perpendicular to AB, both velocities, of course, being applied to the point E. When we wish our results to be expressed in terms of w, ta' we may use these velocities to express the motion of E instead of the coordinates [x, y). Thus in applying the principle of angular momentum, we have to take the moment of the velocity of E about A. Since the velocity 2a w is perpendicular to AB, the length of the perpendicular from A on its direction is ^JB together with the projection of BE on AB, which is 2a + a cos 0. Since the velocity aw' is perpen- dicular to BE, the length of the perpendicular from A on its line of action is BE together with the projection of AB on BE, which is a + 2a cos ). The principle of angular momentum for the two rods gives therefore m (F + 5a2 + 2a2 cos 0) w + m ( k^ + a^ + 2a^ cos ) w' = m {2k'^ + ia^) ft. The right-hand side of this equation, being the initial value of the angular momen- tum, is derived from the left-hand side by putting cos0=: -^ and w = w' = ft. In applying the principle of vis viva, we require the velocity of E. Eegarding it as the resultant of 2aw and aw' we see that, if v be its value, v^={2ab})" + (aw')2 + 2 . 2aw . aw' cos ')2 = 2(l + f cosa) \ W -t 0} J 148. The hob of a heavy pendulum contains a spherical cavity which is filled with water. It is required to determine the motion. Let be the point of suspension, G the centre of gravity of the solid part of the pendulum, MK^ its moment of inertia about 0, and let OG = h. Let G be the centre of the sphere of water, a its radius and OC = c. Let m be the mass of the water. If we suppose the water to be a perfect fluid, the action between it and the case must, by the definition of a fluid, be normal to the spherical boundary. There will therefore be no force tending to turn the fluid round its centre of gravity. As the pendulum oscillates to and fro the centre of the sphere will partake of its motion, but there will be no rotation of the water. The effective forces of the water are by Art. 131 equivalent to the effective force of the whole mass collected at its centre of gravity together with a couple mk-u, where w is the angular velocity of the water, and mk^ its moment of inertia about a diameter. But u has just been proved zero, hence this couple may be omitted. It follows that in all problems of this kind where the body does not turn, or turns with uniform angular velocity, we may collect the body into a single particle placed at its centre of gravity. The pendulum and the collected fluid now form a rigid body turning about a fixed axis, hence if 6 be the angle made by CO a fixed line in the body with the vertical, the equation of motion by Art. 89 is {MIO + mc^) e + {Mh + mc) g sin ^ = 0, where, in finding the moment of gravity, 0, G and C have been supposed to lie in a straight line. The length U of the simple equivalent pendulum is, by Art. 92, 3IK-' + vic^ L': Mh + mc ART. 149.] CHARACTERISTICS OF A BODY. 121 Let mk^ be the moment of inertia of the sphere of water about a diameter. Then, if the water were to become solid and to be rigidly connected with the case, the length L of the simple equivalent pendulum would be, by similar reasoning, MK^ + mjc^ + k^) ~ Mh + mc It appears that L'- . Now if the section of the rod be very small p/a will be large. It appears therefore that the couple, when it exists, will generally have much more effect in breaking the rod than the force. This couple is therefore often taken to measure the whole effect of the forces to break the rod. The tendency of the forces to break a rod OA at any point P is measured numerically by the moment about P of all the forces which act on either of the segments OP, PA of the rod. The resolved part of the force R perpendicular to the rod is called the shear. This is equal to all the forces which act on either of the segments OP, PA, resolved perpendicular to the rod. If the rod be in motion the same reasoning will, by D'Alem- bert's principle, be applicable ; provided that we include the re- versed effective forces among the forces which act on the rod. In most cases the rod will be so little bent that in finding the moment of the impressed forces we may neglect the effects of curvature. If the section of the rod be not very small, this measure of the " tendency to break " becomes inapplicable. It then becomes necessary to consider both the force and the couple. The case does not come within the limits of the present treatise, and the reader is referred to works on elastic solids. In the case of a string the couple vanishes and the force acts 124 MOTION IN TWO DIMENSIONS. [CHAP. IV. along a tangent to the string. The stress at any point is therefore simply measured by the tension. 151. Ex. 1. A rod OA, of length 2a, and inass m, lohich can turn freely about one extremity 0, falls in a vertical plane under the action of gravity. Find the tendency to break at any point P. Let du be any element of the rod distant ?t from P and on the same side of P as the end A of the rod, and let OP = x. Let d be the angle the rod makes with the vertical at the time t. The effective forces on du are du, ,d^d ^ du , JddY respectively perpendicular to and along the rod. The impressed force is m ^ g acting vertically downwards. Let L be the stress-couple at P measured clockwise when acting on PA. By D'Alembert's principle, the moment of the effective forces on PA about P is equal to the moment of gravity plus that of the couple L. Hence - du , ^ d^d I- du • a , T the limits being from m = to u — la-x. This equation may also be obtained by equating - L to the moment of gravity plus that of the reversed effective forces on PA. Also, taking moments about 0, the equation of motion is XT 1 £ J 7-^.7 sin d ,^ .„ Hence we easily find L = „ ,^ x (2a - xf. To find where the rod, supposed equally strong throughout, is most likely to break, we must make L a maximum. This gives dLldx — Q and therefore 3a; = 2a. The point required is at a distance from the fixed end equal to one-third of the length of the rod. Its position is independent of the initial conditions. To find the shear at P we resolve perpendicularly to the rod. We have I- du , , d^6 r du . . -. ■^''^2a^^ + ")d^=-^"^2a^'^^^-^' where Y is the shear and the limits are the same as before. This gives ^ 7ng sin ^ 16a2 (2«-a^)(2a-3a:), which vanishes when the tendency to break is a maximum, and is a maximum at a distance from the fixed end equal to two-thirds of the length of the rod. To find the tension at P we resolve along the rod. If the tension X when acting on PA be measured in the direction OA, we have r du , . fdd\^ . du ^ -^ If the rod start from rest at an inclination a to the vertical, we find, by integrating the equation of motion, ( ;.- ) = ^"- (cos a - cos d). Hence X=^,^{1a-x) { -4a cos ^-1-3 (cos a-cos ^)(2a-fic)}. From these equations we may deduce the following results. (1) The magnitudes of the stress couple and the shear are independent of the initial conditions. ART. 152.] ON THE STRESS AT ANY POINT OF A ROD. 125 (2) The magnitude of either the couple or the shear at any given point of the rod varies as the sine of the inclination of the rod to the vertical. (3) The ratio of the magnitudes of the stress couples at any two given points of the rod is always the same, and the same proposition is also true of the shears. (4) The tension depends on the initial conditions, and, unless the rod starts from rest in the horizontal position, the ratio of the tensions at any two given points varies with the position of the rod. When a tall chimney has to be taken down, it is usual to remove some bricks at the base on one side, replacing each by a wooden prop. When these have been set on fire the chimney, being unsupported on one side, falls like the rod OA and usually breaks at some point of its length. If the chimney were equally strong throughout its length the point of fracture should be one-third up. In an instan- taneous photograph seen by the author this was nearly true. Ex. 2. AV is a fixed smooth vertical rod, AC a rod freely jointed to ^F at the fixed point A, BC a rod freely jointed to ^C and arranged so that B can slide on AV, and a string is attached to the joint C, carrying a mass M. The system rotates with a uniform angular velocity w about AV. Obtain equations to find the inclinations {9, ) of AC and the string to the vertical, and show that the bending moment at a point P on ^C at a distance x from A is ^^^^^^l^sin d {3^-0,2 (x + a) cos d], where a is the length, m the mass of each of the rods AC and BC. [Coll. Exam., 1904. The system is turning uniformly round A V, hence the only effective force on an element dm of mass of the rod AC distant u from ^F is u^udm. All these parallel forces are evidently equivalent to a resultant ^mw^a sin ^ tending from AV and acting at a point distant 2a/3 from A. See also Arts. 47, 114. Let X, Y be the horizontal and vertical components of the reaction at A, let x = AP. By taking moments about P for the portion AP of the rod AC we find (if A is above B) L= - mur^ sin d cos d^^ + mg sin ^ ;. — Xx cos d+Yx 8\nd = 0. 6a 2a This is also numerically equal to the moment of the effective forces on PC together with that of the reactions at the end C of the rod taken with proper signs. But both these moments vanish when P is at C, hence L =0 when x = a. Putting a; = a, we have two equations which immediately lead to the given value of L. The result is not independent of M, for this mass enters into the equation implicitly through d which has not been determined above. 152. Ex. 1. A rigid hoop completely cracked at one point rolU on a perfectly rough horizontal plane and is acted on by no forces but gravity. Prove that the lorench couple at the point of the hoop most remote from the crack ivill be a maximum whenever, the crack being lower than the centre^ the inclination of the diameter through the crack to the horizon is tan''^ 2/7r. [Math. Tripos, 1864. Let w be the angular velocity of the hoop, a its radius. The velocity of any point P of the hoop is the resultant of a velocity aw parallel to the horizontal plane and an equal velocity au along a tangent to the hoop. The first is constant in direction and magnitude and therefore gives nothing to the acceleration of P. The latter is constant in magnitude but variable in direction and gives aw^ as the acceleration, which is directed along a radius of the hoop. Let A be the cracked point, B the other end of the diameter, C the centre, 6 the inclination of ACB to 126 MOTION IN TWO DIMENSIONS. [CHAP. IV. tbe horizon. Let PP' be any element on the upper half of the chcle, BCP = (f> Then the wrench couple, or tendency to break, at B is proportional to / [- aur^a sin (f> + g {a cos d - a cos {(p + 6)]] ad Sg. [Coll. Exam. Let B, C be corners of the square ; A, D the ends of the rod ; 3Ig the weight of the square. First, place the rod ABCD in a horizontal position, the stress couple is greatest at the middle point and is equal to ^Mga, the weight of the portion OA being collected at its centre of gravity B. This is therefore the breaking stress. Next, place the rod in a vertical position, the moment of its own weight about any " point P in the rod being zero, the stress couple L at P is equal to the moment of some of the reactions at ^, J5, C, D. Hence L is a linear function of the distance X of P from A and can have no maxima or minima at any point except A, B, C, D. The stress couples at ^,D evidently vanish and at B, G are respectively equal to the moments of the reactions at A, D. The reaction at the highest point A being greater than that at D, the rod breaks at B. The given result is obtained by equating the moment of the reaction at A about B to the breaking stress. Ex. 3. A semicircular wire AB of radius a is rotating on a smooth horizontal plane about one extremity A with a constant angular velocity w. If a0 be the arc between the fixed point A and tlie point where the tendency to break is greatest, prove that tan = 7r - 0. If the extremity B be suddenly fixed and the extremity A let go, the tendency to break is greatest at a point P where ^ tan PB A =PB A. [Math. Tripos, 1886. Ex. 4. A wire in the form of the portion of the curve r = a (1 + cos d) cut off by the initial line rotates about the origin with angular velocity w. Prove that the IT 12v/2 tendency to break at the point d = - is measured by m _— lo^a^. [St John's Coll. Ex. 5. A heterogeneous rod OA is swung as a pendulum about an horizontal axis through 0. Prove that if the rod break it will be at a point P determined by the condition that the centre of gravity of PA is the centre of oscillation of the pendulum. [Math. Tripos, 1880. On Friction between Imperfectly Rough Bodies. 153. Components of a Reaction. When one body rolls on another under pressure, the two bodies yield slightly, and are therefore in contact along a small area. At every point of this area there is a mutual action between the bodies. The elements just behind the geometrical point of contact are on the point of separation and may tend to adhere to each other, those in front may tend to resist compression. The whole of the actions across the elements are equivalent to (1) a component R, normal to the ART. 155.] FRICTION BETWEEN IMPERFECTLY ROUGH BODIES. 127 common tangent plane, and usually called the reaction ; (2) a component F in the tangent plane usually called the frictioyi ; (3) a couple L about an axis lying in the tangent plane, which we shall call the couple of rolling friction ; (4) if the bodies have any relative angular velocity about their common normal, a couple N about this normal as axis which may be called the couple of tiuisting friction. The two couples are found by experiment to be in most cases very small and are generally neglected. But when the friction ■^orces are also small it may be necessary to take account of them. We shall therefore consider first the laws which relate to the friction forces, as being the most important, and afterwards those which relate to the couples. 154. Laws of Friction. In order to determine the laws of friction forces we must make experiments on some simple cases of equilibrium and motion. Suppose then a symmetrical body to be placed on a rough horizontal table and acted on by a force so placed that every point of the body is urged to move or does move parallel to its direction. It is found that if the force be less than a certain amount the body does not move. The first law of friction is therefore that the friction acts in such a direction and has such a magnitude as to be just sufficient to prevent sliding. Next, let the force be gradually increased, it is found by experiment that no more than a certain amount of friction can be called into play, and that when more is required to keep the body from sliding, sliding begins. The second law of friction asserts the existence of this limit to the amount of friction which can be called into play. Its value is called the limiting friction. The third law of friction found by experiment is that the magnitude of the limiting friction bears a ratio to the normal pressure which is very nearly constant for the same two bodies in contact, but is changed when either body is replaced by another of different material. This ratio is called the coefficient of friction of the materials of the two bodies. Its constancy is generally assumed by mathematicians. Though all experimenters have not entirely agreed as to the absolute constancy of the coefficient of friction, yet it has been found generally that, if the relative motion ot the tw^o bodies be the same at all points of the area of contact, the coefficient of friction is nearly independent of the extent of the area of contact and of the relative velocity. 155. Coulomb has pointed out a distinction which exists between statical friction and dynamical friction. The friction which must be overcome to set a body in motion relatively to another is greater than the friction between the same bodies luhen 128 MOTION IN TWO DIMENSIONS. [CHAP, IV. in motion under the same pressure. He found also that if the bodies remained in contact for some time under pressure in a position of equilibrium, the friction which had to be overcome was greater than if the bodies were merely placed in contact and immediately started from rest under the same pressure. In some bodies the difference between the statical and the dynamical friction was found to be very slight, in others it was considerable*. The experiments of Morin in general confirmed its existence. Ac- cording to some experiments of Fleeming Jenkin and J. A. Ewing, described in the Phil. Trans, for 1877, the transition from statical to dynamical friction is not abrupt. By means of an apparatus which differed essentially from any previously employed they were able to make definite measurements of the friction between surfaces whose relative velocity varied from about one hundredth of a foot per second to about one five-thousandth of a foot per second. Between the limits of these evanescent velocities the coefficient of friction was found to be decreasing gradually from its statical to its dynamical value as the velocity increased. The experiments of Coulomb and Morin were made with bodies moving at moderate velocities, but some experiments have been lately made by Capt. Douglas Galton on the friction between cast- iron brake blocks and the steel tyres of wheels of engines moving with great velocities. These velocities varied from seven feet to eighty-eight feet per second, i.e. from five to sixty miles per hour. Two results followed from his ex:periments : (1) the coefficient of friction was very much less for higher than for lower velocities, (2) the coefficient of friction became smaller after the wheels had been in motion for a few seconds. See the Report of the British Association for the meeting in Dublin, 1878. The reader will find an account of some experiments on rolling friction by Prof. Osborne Reynolds in the Phil. Trans, for 1876. 156. When bodies are said to be perfectly 7vugh it is usually meant that they are so rough that the amount of friction necessary to prevent sliding under the given circumstances can certainly be called into play. The coefficient of friction is therefore practically infinite. By the first law of friction, the amount which is called into play is that which is just sufficient to prevent sliding. 157. Application of the la-ws of Friction. Let us now extend the theory deduced from these experiments to the case in which a body moves or is urged to move in any manner in one plane. It is a known kinematical theorem, which will be proved at the beginning of the next chapter, that such a motion may be represented by supposing the body to be turning round some * The results of Coulomb's experiments are given in his Theorie des machines simples, Memoires des Savants etrangers, tome x. This paper gained the Prize of the Academie des Sciences in 1781 and was published separately in Paris, 1809. ART. 158.] FRICTION BETWEEN IMPERFECTLY ROUGH BODIES. 129 instantaneous centre of rotation. Let be the centre of rotation, then any point P of the body is moving or tends to move in a direction perpendicular to OP. The friction at P, by the first rule just given, must also act perpendicular to OP but in the opposite direction. If P move, the amount of friction at P is limiting friction and is equal to yLtP, where P is the pressure at P and fi the coefficient of friction. Thus in a moving body the direction and the magnitude of the friction at every sliding point are known in terms of the coordinates of and the pressure at the point. Suppose for example that it is required to find the least couple required to move a heavy disc resting by several pins on a hori- zontal table, the pressures at the pins being known. By resolving in two directions and taking moments about a vertical axis we obtain three equations. From these we can find the required couple and the two coordinates of 0. It sometimes happens that coincides with one of the points of support of the body. In this case the friction at this point of support is not limiting. It is only just sufficient in amount to prevent the point from sliding. Ex. A heavy body rests by three pins A, B, C on a rough horizontal table, the pressures at the pins being P, Q, R. If the body be acted on by a couple so that it is just on the point of moving, show that the centre of rotation is at a point O such that the sines of the angles AOB, BOG, CO A are as R, P, Q. But if the point thus determined does not lie within the triangle ABC, the centre of rotation coincides with one of the pins. These results follow immediately from the triangle of forces. 158. Discontinuity of Friction. The reader should. par- ticularly notice the discontinuity just mentioned. The friction at any point of support which slides is fjuR, where R is the normal pressure. But if the point of support does not slide, the friction is some quantity F, which is unknown, but must be less than fiR. Its magnitude must be found from the equations of motion. Let a moving body be placed with one point A in contact with a fixed rough plane and let the initial velocity of A be zero. The point A may either begin to slide on the plane or the body may only roll. To determine which of these motions occurs, we may adopt either of two methods. In the first method, we investigate the friction required to keep A at rest. Assuming then that the body rolls, we write down the equations of motion. The friction F is unknown, but we have a geometrical equation to express the condition that the tangential velocity of A is zero. Solving these equations we find the ratio F/P. If this ratio is less than the coefficient of friction fi, enough friction can be called into play to keep A at rest. The body therefore will begin to roll and will continue to roll as long as R. D. 9 130 MOTION IN TWO DIMENSIONS. [CHAP. IV. the ratio F/R continues to be less than fi. If the ratio F/R is greater than fjb the body slides at A. When this happens the equations written down do not represent the true motion, and we adopt the second method. In the second method, we form the equations of motion on the supposition that the point A slides on the plane. The friction is then fiR instead of F and the geometrical equation which expresses the fact that there is no slipping at A is absent. Solving these equations we find the tangential velocity of the point A of the body. If this velocity is not zero and is opposite to the direction in which the friction fiR acts when fi has a proper sign given to it, the true motion has been found. The body will slide at A and will continue to slide as long as the velocity at A does not vanish. When this occurs we again use the first method. 159. Discontinuity may also arise in other ways. When, for example, one body is sliding over another, the friction is opposite to the direction of relative motion, and numerically equal to the normal reaction multiplied by the coefficient of friction. If then, during the course of the motion the direction of the normal reaction should change sign, while the direction of motion remains unaltered, or if the direction of motion should change sign while the normal reaction remains unaltered, the sign of the coefficient of friction must be changed. This may modify the dynamical equations and alter the subsequent solution. The same cause of discontinuity operates when a body moves in a resisting Tnedium, the law of resistance being an even fuTwtion of the velocity, i.e. any function which does not change sign when the direction of motion is changed. 160. Indeterminate Motion. In some cases the motion may be rendered indeterminate by the introduction of friction. Thus we have seen in Art. 112 that, when a body swings on two hinges, the pressures on the hinges resolved in the direction of the straight line joining them cannot be found. The sum of these components can be found, but not either of them. But there is no indeterminateness in the motion. If however the hinges were imperfectly rough, there would be two friction couples, one at each hinge, acting on the body, their common axis being the straight line joining the hinges. The magnitude of each would be equal to the pressure multiplied by a constant depending on the roughness of the hinge. If the hinges were unequally rough, the magnitude of the resultant couple would depend on the distribution of the pressure on the two hinges. In such a case the motion of the body would be indeterminate. 161. Examples of Friction. Ex. 1. A homogeneous sphere is placed at rest on a rough inclined plane, the coefficient of friction being [jl, determine whether the sphere will slide or roll. Let F be the friction required to make the sphere roll. The problem then ART. 162.] FRICTION BETWEEN IMPERFECTLY ROUGH BODIES. 131 becomes the same as that discussed in Art. 144. We have, therefore, F=^R tana, where a is the inchnation of the plane to the horizon. If then ^ tan a be not greater than fi, the solution given in the article referred to is the correct one. But if fiF ^/g±5= -'L^f^^ W fg V fg a^+^^ where V is the velocity of the sphere at the epoch from which t is measured. 164. Friction couples. In order to determine by experi- ment the magnitude of rolling friction, let a cylinder of mass M and radius r be placed on a rough horizontal plane. Let two weights whose masses are P and P+_p be suspended by a fine I ART. 165.] FRICTION BETWEEN IMPERFECTLY ROUGH BODIES. 133 thread passing over the cylinder and hanging down through a slit in the horizontal plane. Let F be the force of friction, L the couple at the point of contact A of the cylinder with the horizontal plane. Imagine p to be at first zero, and to be gradually increased until the cylinder just moves. When the cylinder is on the point of motion, we have by resolving horizontally F=0, and by taking moments L =pgr. Now in the experiments of Coulomb and Morin p was found to vary as the normal pressure directly, and as r inversely. When p was great enough to set the cylinder in motion, Coulomb found that its acceleration was nearly constant, whence it followed that the rolling friction was independent of the velocity. M. Morin found that it was not independent of the length of the cylinder. The laws which govern the couple of rolling friction are similar to those which govern the force of friction. The magnitude is just sufficient to prevent rolling. But no more than a certain amount can be called into play, and this is called the limiting rolling couple. The moment of this couple bears a constant ratio to the magnitude of the normal pressure. This ratio is called the coefficient of rolling friction. It depends on the materials in contact, it is independent of the curvatures of the bodies, and, in some cases, of the angular velocity. No experiments seem to have been made on bodies which touch at one point only and have their curvatures in different directions unequal. But, since the magnitude of the couple is independent of the curvature, it seems reasonable to assume that the axis of the rolling couple, when there is no twisting couple, is the instantaneous axis of rotation. 165. In order to test the laws of friction let us compare the results of the following problem with experiment. Friction of a carriage. A cairiage on n pairs of wheels is dragged on a level horizontal plane by a horizontal force 2P with uniform motion. Find the magnitude of P. Let the radii of the wheels be respectively r^ , r^, &c., their weights w-^, w^, &c., and the radii of the axles pj, p^, &c. Let 2]F be the whole weight of the carriage, 2Qj, 2Q2. &c. the pressures on the several axles, so that W^'ZQ. Let the pressures between the wheels and axles be JRj, R^, &c. and the pressures on the ground jRj', R^, &c. Let G be the common centre of any wheel and axle, B their point of contact, and A the point of contact of the wheel with the ground. Let the angle ACB = d, supposed positive when B is behind AG. Let /jl be the coefficient of the force of sliding friction at B and / the coefficient of the couple of rolling friction at A. The equations of equilibrium for any wheel, found by resolving vertically and taking moments about A, are R' = Q + w (1), fiR (r cos d - p)-Rr sin d=fR' (2). The friction force at A does not appear because we have not resolved horizontally. The equations of equilibrium of the carriage, found by resolving vertically and horizontally, are RGoad + fiRsind^Q (3), S (-R sin^-/*iJcos^) + P=0 (4). 134 MOTION IN TWO DIMENSIONS. [CHAP. IV. The effective forces have been omitted because the carriage is supposed to move uniformly, so that the Mi) of the carriage and the wft'ci of the wheel are both zero. The first three of these equations give, by eliminating R and R', IX ( cos 6 -- cos ^ + /A sin 6 K-i) <^'- This gives the value of d. In most wheels pjr and wjQ are both small as well as/. In such a case ixco&d-B\a.d is a small quantity. If therefore ^u^tane we have d=e very nearly. The third and fourth of the equations give, by eliminating R, fismd + cosd^ (fjL smd + 008$ r^ r^ the latter by equation (5). If pjr be small, it will be sufficient to substitute for 6 in the first term its approximate value e. This gives P=s|sin.-^Q+/«±"} (6). Here we have neglected terms of the order {pjr)^ Q. If all the wheels are equal and similar we have, since ^Q=W, P = 8me- ^+/ (7). Thus the force required to drag a carriage of given weight with any constant velocity is very nearly independent of the number of wheels. In a gig the wheels are usually larger than in a four-wheel carriage, and there- fore the force of traction is usually less. In a four-wheel carriage the two fore wheels must be small in order to pass under the carriage when turning. This will cause the term sineQiPj/rj in the expression for P, depending on the radius r^ of the fore wheel, to be large. To diminish the effect of this term, the load should be so adjusted that its centre of gravity is nearly over the axle of the large wheels, when the pressure Qi in ttie numerator will be small. Numerous experiments were made by a French engineer, M. Morin, at Metz in the years 1837 and 1838, and afterwards at Courbevoie in 1839 and 1841, with a view to determine with the utmost exactness the force necessary to drag carriages of different kinds over ordinary roads. These experiments were undertaken by order of the French Minister of War, and afterwards under the direction of the Minister of Public Works. The effect of each variation was determined separately, thus the same carriage was loaded with different weights to determine the effect of pressure, and dragged on the same road in the same state of moisture. Then, the weight being the same, wheels of different radii but of the same breadth were used, and so on. The general result was that for carriages on equal wheels, the resistance varied as the pressure directly, and the diameter of the wheels inversely, whilst it was independent of the number of wheels. On wet soils the resistance increased as the breadth of the tire decreased, but on solid roads the resistance was independent of the breadth of the tire. For velocities which varied from a foot pace to a gallop, the resistance on wet soils did not increase sensibly with the velocity, but on solid roads it did increase with the velocity if there, were many inequalities on the road. As an approximate result it was found that the resistance might be expressed by a function of the form a + bV, where a and b were two constants depending on the nature of the road and the stiffness of the carriage, and V was the velocity. ART. 166.] FRICTION BETWEEN IMPERFECTLY ROUGH BODIES. 135 M. Morin's analytical determination of the value of P does not altogether agree with that given here, but it so happens that this does not materially affect the comparison between theory and observation. See his Notions Fondamentales de Mecanique, Paris, 1855. It is easy to see that M. Morin's experiments tend to confirm the laws of rolling friction stated in a previous article. 166. Problems on Friction. Ex. 1. A homogeneous sphere is projected without rotation directly up an imperfectly rough plane, the inclination of which to the horizon is a, and the coefficient of friction /x. Show that the whole time duriog which the sphere ascends the plane is the same as if the plane were smooth, and that the time during which the sphere slides is to the time during which it rolls as 2 tan a : 7//. Ex. 2. A homogeneous sphere rolls down an imperfectly rough fixed sphere, starting from rest at the highest point. If the spheres separate when the straight line joining their centres makes an angle with the vertical, prove that cos + 2;u sin = Ae^^'^, where ^ is a function of /x only. [Coll. Exam. Proceeding as in Art. 145, we show that R remains positive and that the sphere rolls until 2 sin 0//*= 17 cos 0- 10. The sphere then slides and R changes sign when satisfies the equation given in the question. Ex. 3. A rough cylinder of mass 2nm capable of motion about its horizontal axis has a particle of mass m and coefficient of friction /x placed on it vertically above the axis. The system is then slightly disturbed. Show that the particle will slip on the cylinder after it has moved through an angle 6 given by {n + 3) cos ^ - 2 = n sin dlfx.. Ex. 4. A homogeneous sphere of mass M is placed on an imperfectly rough table, the coefficient of friction of which is fi. A particle of mass m is attached to the extremity of a horizontal diameter. Show that the sphere will begin to roll or slide according as u is greater or less than „,,„ ,p^^rr^^ — ^~~o • I^ /* be equal to this value, show that the sphere will begin to roll if 5m^ <. M^ + llMm. Ex. 5. A ring of radius a is fixed on a smooth horizontal table ; a second ring is placed on the table inside the first and in contact with it, and is projected with velocity F, but without rotation, in a direction parallel to the tangent at the point of contact. Find the time that elapses before slipping ceases between the rings if the coefficient of friction between them is fx and prove that the point of contact will in this time describe an arc of length (a log 2)//*. Discuss the motion that will ensue if at the moment slipping ceases the fixed ring be released and left free to move, and prove that during the time that the inner ring rolls half round the outer one the centre of the latter will be displaced a distance -zr^ (a-b) Jlw^+4,) where m, M. are the masses of the inner and outer rings and h is the radius of the inner ring. [Math. Tripos, 1900. Ex. 6. A rod AB has two small rings at its extremities which slide on two rough horizontal rods Ox, Oy at right angles. The rod is started with an angular velocity when very nearly coincident with Ox. Show that, if the coefficient of O 2 / Q O^ friction is less than J 2, the motion of the rod is given by ^= ~ — log f 1 + n_ 2 ) 136 MOTION IN TWO DIMENSIONS. [CHAP. IV. until tan d = 2/;u,, and that when the rod reaches Oy its angular velocity is w, where and d is the angle the rod makes with Ox. What is the motion if yu,2>2 ? We deduce from the equations of motion that when /a^<2, both the reactions at the beginning of the motion act outwards from the quadrant in which the rod lies. During the motion one reaction changes sign while the corresponding friction con- tinues to act in the same direction as before : the angular velocity is found not to vanish. 167. Rigidity of Cords. After having used the apparatus with a fine cord described in Art. 164 to determine the laws of friction, Coulomb replaced the cord by a stiffer one and repeated his experiments with a view to obtain a measure of the rigidity of cords. His general result may be stated as follows. Suppose a cord ABGD to pass over a pulley of radius r, touching it at B and G, and moving in the direction ABGD. Then the rigidity may be represented by supposing the cord to be perfectly flexible, and the tension T of the portion AB of the cord which is about to be rolled on the pulley to be increased by a quantity R. The force R measures the rigidity and is equal to , where a and h are constants depending on the nature of the cord. It appears therefore that, in the equation of moments about the axis of the pulley, the rigidity of the cord which is being wound on the pulley is represented by a resisting couple of magni- tude a+bT, where T is the tension of the cord which is being bent, and a, b are two constants depending on the nature of the cord. The rigidity of the cord which is being unwound will be represented by a couple whose magnitude is a similar function of the tension of that cord. But as its magnitude is very much less than the first it is generally omitted. Besides the experiments just alluded to, Coulomb made many others on a different system. He also constructed tables of the values of a and b for ropes of different kinds. The degrees of dryness and newness and the number of independent threads forming the cord were all considered. Rules were given for com- paring the rigidities of cords of different thicknesses. On Impulsive Forces. 168. Equations of motion. In the case in which the impressed forces are impulsive the general principle enunciated in Art. 131 of this chapter requires but slight modification. Let (u, v), {v! , v) be the velocities of the centre of gravity of any body of the system resolved parallel to any rectangular axes ART. 170.] ON IMPULSIVE FORCES. 137 respectively just before and just after the action of the impulses. Let o) and «' be the angular velocities of the body about the centre of gravity at the same instants. Let Mk^ be the moment of inertia of the body about the centre of gravity. Then the effective forces on the body are equivalent to two impulsive forces measured by M{u—u) and M{v' — v) acting at the centre of gravity parallel to the axes of coordinates together with an impulsive couple measured by Mk'^ (co' — co). The resultant effective forces of all the bodies of the system may be found by the same rule. By D'Alembert's principle these are equivalent to the impressed forces. The equations of motion may then be found by resolving in such directions and taking moments about such points as may be found most con- venient. To take an example, let a single body be acted on by a blow whose components are X, Y and whose moment round the centre of gravity is L. The equations of motion are evidently M(u'-u) = X, M{v-v)=Y, Mk'{co'-(D) = L. In many cases it will be found that by using the principle of virtual work the elimination of the unknown reactions may be effected without difficulty. 169. We notice that these expressions for the effective forces depend on the differences of the momenta just before and just after the action of the impulses. We may therefore conveniently sum up the equations obtained by resolving in any direction and taking moments about any point in the two following forms : /Res. Lin. Mom.\ /Res. Lin. MomA _ /Resolved \ V after impulse / \ before impulse / V impulse / ' /Ang. Momentum\ /Ang. MomentumN _ /Moment of\ V after impulse / V before impulse / ~ V impulse / * An elementary proof of these two results is given in Art. 87. The expression for the Linear Momentum is given in Art. 74, and various expressions used for Angular Momentum are given in Art. 134. When a single blow or impulse acts on a system, we may conveniently take moments about some point in its line of action, and thus avoid introducing the impulse into the equations. We then deduce from the equation of moments that the angular momentum of a system about any point in the line of action of an impulse is unaltered by that impulse. 170. Ex. 1. A string is ivound round the circumference of a circular reel, and the free end attached to a fixed point. The reel is then lifted up and let fall so that, at the moment when the string becomes tight, it is vertical and a tangent to the reel. The whole motion being supposed to be parallel to one plane, determine the effect of the impulse. 138 MOTION IN TWO DIMENSIONS. [CHAP. IV. The reel in the first instance falls vertically without rotation. Let v be the velocity of the centre at the moment when the string becomes tight ; v', w' the velocity of the centre and the angular velocity just after the impulse. Let T be the impulsive tension, mk^ the moment of inertia of the reel about its centre of gravity, a its radius. In order to avoid introducing the unknown tension into the equations of motion, let us take moments about the point of contact of the string with the reel ; we then have m{v' -v) a + mk^u)' = (1). Just after the impact the part of the reel in contact with the string has no velocity. Hence v'-a' the angular velocity just after the change. Let {x, y) be the coordinates of the centre of gravity referred to the axes Ox, Oy at the instant of the change, and let OG = r. Since the angular momentum of the body about the point of space through which is passing is unchanged by the blow, we have, by Art. 134, M{xv' - yu + Fw') = M(xv- yu + k^oy). Let {V, V) be the resolved parts of the velocity of just after the change. Then we have by Art. 137, u = V" — yw', V =V' ■\- xw. From these three equations we easily find {¥ 4- r^) ay' = x{v-V')-y{u- U') 4- k'ay. Let L, L' be the moments round the centre of gravity G of the velocities of just before and just after the fixing, then Z = (w + yw) y — {v — xco) X, L' = U'y — V'x. The equation to find w becomes {k^ + r^){w -w)=L'-L, here L, L\ co, co' are all measured the same way round G. Another roof is given in Art. 207. If the point he suddenly fixed we have U' = 0, V = 0, and then we find {k^ + r"^) cj' =xv —yu-\- k^w. To find the blow at necessary to produce the given change. Let X, Y be the components of the blow parallel to the axes |0a;, Oy, Then by Art. 168 we have, resolving parallel to the es M{u'-u) = X, M{v'-v)=^Y. If we take the axis of x to pass through the centre of gravity, have 2/ = 0. We then find by substitution X = -M{u-U'\ Y=MjJ^^{rw-v+Vy 140 MOTION IN TWO DIMENSIONS. [CHAP. IV. 171 a. Ex. 1. A circular area is turning about a fixed point A on its circum- ference. Suddenly A is loosed and another point B on the circumference is fixed. If AB is a quadrant show that the angular velocity is reduced to one-third of its value. If AB is a third of the circumference the area is reduced to rest. Ex. 2, A disc, moving in its own plane, is reduced to rest by suddenly fixing a point O. Prove that O lies in a straight line which is parallel to the direction of motion of the centre of gravity G and is distant fc^o/w from it, where u is the velocity of G. 172. Work of an impulse. A body of mass M is acted on at a given point P by an impulsive force R. To find the change in the vis viva. An impulse is the limit of a great force acting for a very short time (Art. 84), and from this definition we may deduce the work done by the impulse and thence the loss or gain of vis viva (Art. 141). This is the course adopted in the first section of Chap. vii. We shall however here deduce the result directly from the equa- tions of impulses given in Art. 169. Let the axis of ^3? be parallel to and distant y from the line of action of the impulse. Then, the origin being at the centre of gravity, the equations of motion are by Art. 168 u'-u^RjM, v'-v = ^, a)'-co = -RylMk\ The gain of vis viva is by Art. 189 = R{2{u-y(o)-^R (y^ + k^)IMk^}. But by (1) u -co'y = u-(oy + R(y^ + k')IMk'' ; .*. gain of vis viva = i^ {{u — y(d) + {u —yco')] (2). The gain of kinetic energy is of course the half of this quantity. If V and V are the velocities of the point of applica- tion of the blow resolved in the direction of the blow just before and just after the impulse, then the gain of kinetic energy is ^{V' -\- V) R. This result is due to Kelvin. In the same way the vis viva of the relative motion is M (u -uf + M (v' - vf + Mk^ (co' - (of R^ R^y^ It follows immediately that the kinetic energy of the relative motion of the system just before and just after the action of the impulse RisUr- V)R. Both these results and the two last examples in Art. 173 b are special cases of much more general theorems, which apply to any system of bodies and any number of impulses. These, with some others equally important, are given at the end of Chapter vii., with demonstrations founded on the principle of virtual velocities. ART. 173.] ON IMPULSIVE FORCES. 141 172 a. If the impulse R make an angle ^ with the axis of a?, \et X = R cos Sf, Art. 172. \ \ 142 MOTION IN TWO DIMENSIONS. [CHAP. IV. 173 tt. If the impwging bodies are smooth with a coefficient of restitution e, we may obtain a corresponding rule for the loss of energy. We see by Art. 172 that the resolved velocity of the point of application of a blow is u' - co'y = u - coy -{- E (y^ + k^)IMIc^. Hence, when two smooth bodies impinge, the normal velocities of the points of contact (and therefore also the relative normal velocity) are, at any stage of the impact, linear functions of the reaction up to that stage (Art. 179). We write therefore where U is the normal relative velocity just before impact, U" that at any stage defined by the magnitude of R, and L is a constant which is independent of R but depends on the form of the impinging bodies. At the instant of greatest compression (Art. 179) when R^Rq, U" = 0. When the impact is concluded and R = RQ{\-\-e), let U' he the normal relative velocity. We thus have 0=U + LRo, U'=U + LR,{l^e), .-. U' = -eU. The ratio of the normal relative velocities of the points of contact just after and just before impact is therefore equal to — e. By (4) of Art. 173, the loss of kinetic energy due to an impact is ^8{U ■\- U'). Hence, if /S represent the whole blow, that is R(i{l + e), the loss of kinetic energy is \8U {\ — e), where U is the normal relative velocity just before impact. 173 h. Ex. 1. Prove that the loss of kinetic energy at the impact of two per- fectly rough inelastic uniform spheres of masses M-^, M^, is — ^ f" ^ , where u, V are the relative velocities before impact of the points of contact tangentially and normally. [Coll. Exam. 1904. Ex. 2. A disc at rest is acted on by an impulse in its own plane. Prove that the vis viva generated by the impulse is greater when the body is free than when it is constrained to turn round some fixed point. Ex. 3. Two straight lines Ox, Oy are drawn at right angles in the plane of a disc which is at rest. Suddenly the point is made to move with a given velocity in the direction Ox. Prove that the vis viva generated when the body is free is less than if it were constrained to turn about a fixed point C which lies in Oy. 174. Examples of different kinds of Impacts. Ex. 1. An inelastic sphere of radius a, sliding with a velocity F on a smooth horizontal plane, impinges on a perfectly rough fixed point or peg at a height c above the plane. Show (1) that unless the velocity V be greater than / 2gc r^ the sphere will not jump over the peg. Supposing the velocity V to have this value show (2) that the sphere c a^ + k^ will immediately leave the peg if - be greater than - -., — ^ . In this latter case ART. 174.] ON IMPULSIVE FORCES. 143 show (3) that the sphere will again hit the peg after a time t, given by the lesser root of the equation ^gH- - U ^inagt+U'^-ag co%a = 0, where U'^ = 2gc-^ — p *^^ cosa= 1 — . Show also that the roots of this quadratic are real and positive. Ex. 2. A rough inelastic sphere rolls down over the rungs of a sloping ladder without slipping or jumping, leaving each rung in turn as it impinges on the next. Show that the descent may be made without gathering or losing speed only if the slope 6 of the ladder is less than the acute angle d^ given by the equation tan (^0 + *) cota = 2-sin% / ( 1 + -^ j i and greater than the acute angle d^ given by the equation tan ^i/2 = sina (1 -cosa) (cos2a + -^j; r being the radius of the sphere, k its radius of gyration about a diameter and 2r sin a the distance between consecutive rungs of the ladder. [Math. Tripos, 1898. Let w be the angular velocity with which the sphere begins to turn round any rung just after impact, and u>' that with which it arrives at the next rung. The principle of vis viva supplies one equation connecting w, co' and 6. We have a second equation because the angular momentum is not altered by the impact, Art. 171. We obtain an inferior limit to the value of w because the vis viva must be sufficiently great to carry the centre of gravity over its highest position. We have a superior limit because the angular velocity must not be so great that the sphere leaves the rung before it arrives at the next rung. Ex, 3. A rectangular parallelepiped of mass 37n, having a square base ABCD, rests on a horizontal plane and is moveable about CD as a hinge. The height of the solid is 3a and the side of the base a. A particle m moving with a horizontal velocity v strikes directly the middle of that vertical face which stands on ^J5 and lodges there without penetrating. Show that the solid will not upset unless 9v^>5Sga. [King's Coll. Ex. 4. A vertical column in the form of a right circular cylinder rests on a perfectly rough horizontal plane. Suddenly the plane is jerked with a velocity V in a direction making an angle e with the horizon. Show that the column will not be overturned unless (1) the direction of the jerk be such that a parallel to it drawn through the centre of gravity does not cut the base, and (2) the velocity of the jerk be greater than U, where U is given by U^ = lgl {15 + cos^ d) . . COS i 1/ "t" 6) Here 21 is the length of a diagonal of the cylinder and 6 is the angle any diagonal makes with the vertical. Ex. 5. If the velocity of the jerk of the horizontal plane be exactly equal to U, find the vertical pressure of the cylinder on the plane. Show that the cylinder will not continue to touch the plane during the whole ascent of the centre of gravity unless 1 + ^ sin ^ < 3 cos 6. What is the general character of the motion if this condition is not satisfied? Let the cylinder touch the ground at the point A of the rim, and let

' = a; (3). Again, since the two rods are connected at B, the velocities of their extremities must be the same in direction and magnitude. Resolving these horizontally and vertically, we have w + aw cos a = 2aa;' cos a (4), v-au sin a = 2aw' sin a (5). These five equations are sufficient to determine the initial motion. Eliminating R between (1) and (2), and substituting for u, v, w' in terms of w from the geometrical equations, we find io = ^ . —t^ — ^ . „ , (6). 2 a (1 + 3 sm^a) In this problem we might have avoided the introduction of the unknown reaction R by the use of virtual work. Let us give the system such a displacement jt that the inclination of each rod to the vertical is increased by the same quantity 5a. The virtual work of any couple, such as rnk'^oj, is found by multiplying its moment by the angular displacement, viz. da. The work of any force, such as mu, is found by multiplying its magnitude by the linear displacement of the point of application. The principle of virtual work then gives mk^uda -m{v-V)8{Sa cos a) + mu8 (a sin a)+m {k^ + a^) u'5a + mVd (a cos a) = 0, R. D. 10 146 MOTION IN TWO DIMENSIONS. [CHAP. IV. which reduces to {2k^ + a^)w- Va sin a + 3 (v - F) a sin a + wa cos a = 0, and the solution may be continued as before. Ex. 1. Show that the direction of the impulsive action at the hinge B makes with the horizon an angle whose tangent is (3 sin^a - 2) cot a. Ex. 2. If the coefficient of restitution of the plane be e, show that the value of w given by (6) must be multiplied by 1 + e ; see Art. 404. To find the subsequent motion. This may be found very easily by the method of Vis Viva. But in order to illustrate as many modes of solution as possible, we shall proceed in a different manner. The effective forces on either of the upper rods are represented by the differential coefficients mv, nit, mk-oj, and the moment for either of the lower rods is m(k'^ + a^)u. Let 6 be the angle any rod makes with the vertical at the time t. Taking moments in the same way as before, mk^Cj + mva sin 6 - mua cos 6= -R . la cos 6 + mga sin ^ (1)', m{k^ + a'^) b\ 192. Iioss of Energy. By treating the equations (1) and (2) in exactly the same manner as those in Art. 172, we find that the gain of kinetic energy due to the impact is - FSo-RCo + h{aF^ + 2bFR + a'R^) (1), where S^, Cq, a, b, a' stand for the quantities given in equations (7) to (11). By multiplying (5) and (6) by F and R respectively we obtain aF^ + 2bFR + a'R^ = F{SQ-S)+R{Co-C). The loss of kinetic energy is therefore ^^F{So + S) + iR{Co + C) (2). Here F, R are the whole tangential and normal forces called into play, as explained in the following articles. Also Sq, Cq are the tangential and normal relative velocities of the points of contact just before impact and S, C the corresponding velocities just after impact. This result includes in a convenient form all those discussed in Art. 173. The expression (1) gives the loss of energy in terms of the relative velocities before impact and of the forces. We may eliminate the forces and express the loss of energy solely in terms of the relative velocities before and after impact. The result is 1 {aCo^ - 26^0 Cq + a'So") -{aC^- 2bSC + a'S^) 2 aa' - b^ L 193. The Representative Point. It often happens that 6 = 0, and in this case the discussion of the equations is very much simplified. But certainly in the general case, and even in the simple case when 6 = 0, it is found more easy to follow the changes in the forces if we adopt a graphical method. The point which we have to consider is this. As R proceeds from zero to its final maximum value by equal continued increments dR, F proceeds also from zero by continued increments dF, which may not always be of the same sign and which are governed by a dis- continuous law, viz. either dF = ± /judR, or dF is just sufficient to prevent relative motion at the point of contact, as explained in Art. 158. We want therefore some rule to discover the value of F. To determine the actual changes which occur in the frictional impulse as the impact proceeds, let us draw two lengths AR, AF along the normal and tangent at A in the directions NG, AN re- spectively, to represent the magnitudes of R and F at any moment of the impact. Then, if we consider AR and AF to be the co- ordinates of a point P referred to J.i^, ^i^ as axes of R and F, the changes in the position of P will indicate to the eye the changes 158 MOTION IN TWO DIMENSIONS. [CHAP. IV. that take place in the forces during the progress of the impact. At the beginning of the impact the forces R and F are zero, the representative point P is therefore situated at the origin A. As the impact proceeds the force R continually increases, hence the abscissa AR of P will also continually increase, i.e. the motion of the representative point resolved parallel to the axis of R will be always in the positive direction of the axis of R. The ordinate P of P is measured in the direction opposite to that in which the friction acts on the body M ; it follows that the motion of the representative point resolved parallel to the axis of F will indicate to the eye the direction in which the body M is sliding. This may sometimes be in one direction during the impact and sometimes in the other. It will be convenient to trace the two loci determined hy S = 0, G = 0. By reference to (5) and (6) we see that they are both straight lines. These we shall call the straight lines of no sliding and of greatest compression. To trace them, we must find their intercepts on the axes of F and R. Take AG = ^, AS = ^, AC' = ^, AS' = ~\ a a then SS\ CC will be these straight lines. Since a and a are necessarily positive, while b has any sign, we see that the inter- cepts on the axes of P and R respectively are positive, while their intercepts on the axes of R and P must have the same sign. Since aa' > ¥, the acute angle made by the line of no sliding with the axis of P is greater than that made by the line of greatest compression, i.e. the former line is steeper to the axis of P than the latter. It easily follows that the two straight lines cannot intersect in the quadrant contained by RA produced and FA produced. 194. In the beginning of the impact the bodies slide over each other, hence, as explained in Art. 158, the whole limiting friction is called into play. The point P therefore moves along a straight line AL, defined by the equation F = /jlR, where fi is the coefiicient of friction. The friction continues to be limiting until P reaches the straight line SS\ If Po be the abscissa of Sf this point we find Rq = ^ , . This gives the whole normal blow, from the beginning of the impact, until friction can change from sliding to rolling. If Rq is negative, the straight lines AL and SS' do not intersect on the positive side of the axis of P. In this case the friction is limiting throughout the impact. If Po is positive the representative point P reaches SS\ After this only so much friction is called into play as suffices to prevent sliding, provided that this amount is less than the limiting friction. If the acute angle which SS' makes with the axis of ii ART. 194.] ON IMPULSIVE FORCES. 159 is less than tan"^ ^, the friction dF necessary to prevent sliding is less than the limiting friction iidR. Hence P must travel along 88' in such a direction that the abscissa R continues to increase positively. In this case the friction does not again become limiting during the impact. But if the acute angle v^hich 88' makes with the axis of R is greater than tan~^ /x, the ratio of dF to dR is numerically greater than yLt, and more friction is necessary to prevent sliding than can be called into play. The friction therefore continues to be limiting, and P, after reaching 88', must travel along a straight line, making the same angle with the axis of R that AL does. This straight line must lie on the opposite side of 88' because the acute angle which 88' makes with AR is greater than the angle LAR. Also since the point P has crossed 88' the direction of relative sliding and therefore the direction of friction is changed. In this case it is clear that the friction continues limiting throughout the impact. An example of each of these three cases is given in the triple diagram. The figures differ in the position of the line of no sliding. In all the three figures the representative point travels from A along a straight line AL such that the angle LAR is equal to tan~^ //,. In fig. (1) the line of no sliding, viz. 88' , makes so large an angle with AR that AL does not intersect it in the positive quadrant. The friction therefore retains its limiting value throughout the impact. In the other two figures AL and 88' intersect in some point Q. In fig. (2) the angle 88' A is less than the angle LAR, the representative point therefore after reaching Q travels along Q8'. In fig. (3) the angle 88' A is greater than the angle LAR, the representative point therefore after reaching Q travels along a straight line QB on the other side of 88' such that the angle QBA is equal to the angle QAR. When P passes the straight line CC'^ compression ceases and 160 MOTION IN TWO DIMENSIONS. [chap. IV. restitution begins. But the passage is marked by no peculiarity except this. If Ri be the abscissa of the point at which P crosses Fig. 1. CC, the whole impact, for experimental reasons, is supposed to terminate when the abscissa of P is Eg = ^i (1 + ^), & being the measure of the elasticity of the two bodies. It is obvious that a great variety of cases may occur according to the relative positions of the three straight lines AL, SS' and CC. But in all cases the progress of the impact may be traced by the method just explained, which may be briefly summed up in the following rule. The representative point P travels along AL until it meets SS'. It then proceeds either along 88', or along a straight line making the same angle with the axis of P as AL does, hut lying on the opposite side of 88'. The one along which it proceeds is the steeper to the axis of F. It travels along this line in such a direction as to make the abscissa R increase, and continues to he in this straight line to the end of the impact. The complete value of R for the whole impact is found hy multiplying the ahscissa of the point at which P crosses CC hy l-\- e. The complete value of F is the corresponding ordinate of P. 8uhstituting these in the dynamical equations (1) and (2), the motion just after impact may he easily found. If /So = 0, we have 8 = — aF— hP. In this case the line of no sliding passes through the origin A. If the acute angle which this straight line makes with the axis of R is less than tan""^ /x, i.e. if hja is numerically less than fx, the representative point travels along this straight line in such a direction that its abscissa R continually increases. The friction is therefore less than its limiting value throughout the impact. If the acute angle which the line of no sliding makes with the axis of R is greater than tan~^ fju, i.e. if hja is numerically greater than fx, the representative point travels along a straight line AL making with the axis of R an acute angle LAP equal to tan~^ jx. This straight line lies on the positive or negative side of AR AHT. 196.] ON IMPULSIVE FORCES. 16X according as S is positive or negative. Since the numerical value of b is greater than a/ju, and F=± /jlR, the term — bR governs the sign of S, hence S has the opposite sign to b. It follows that the straight line AL lies within the acute angle which the line of no sliding makes with AR. Thus in fig. (1), AL is on the positive side, in fig. (2) on the negative side of AR. As AL cannot again meet the line of no sliding the friction has its limiting value throughout the impact. R S'- F Fig. 1. Fig. 2. The representative point continues its journey along either SS' or AL, as the case may be, to the end of the impact. The complete value of jR for the whole impact is found by multiplying the abscissa of the point at which P crosses GO' by 1 + e. The complete value of F is the corresponding ordinate of P. Sub- stituting these in the dynamical equations the motion just after impact may be found. 195. If the bodies are smooth, the straight line AL coincides with the axis of R. The representative point P travels along the axis of R, and the complete value of R for the whole impact is found by multiplying the abscissa of (7 by 1 + e. If the bodies are perfectly rough (Art. 156), the straight line AL coincides with the axis of F. The representative point P travels along the axis of F until it arrives at the point S. It then travels along the line of no sliding SS' until it reaches the line CC of greatest compression. If the bodies are inelastic, the coordinates Pj, F^, of this intersection are the values of R and F required. Biit if the bodies are imperfectly elastic the representa- tive point continues its journey along the line of no sliding. The complete value of R for the whole impact is then R^ = Pj (1 4- e), and the complete value of F may be found by substituting this value for R in the equation of the line of no sliding. 196. It is not necessary that the friction should keep the same direction during the impact. The friction must keep one sign when P travels along AL. But when P reaches SS\ its direction of motion changes, and the friction dF called into play in the time dt may have the same sign as before or the opposite. But it is clear that the friction can change sign only once during R. D. 11 162 MOTION IN TWO DIMENSIONS. [chap. IV. the impact. If 6 = 0, the straight line SS' is perpendicular to the axis of F, and in this case it is clear that the friction cannot change sign. It is possible that the friction may continue limiting through- out the impact, so that the bodies slide on each other throughout. The necessary conditions are that either the straight line SS' must be less steep to the axis of F than AL, or the point P must not reach the straight line SS' until its abscissa has be- come greater than i?.2. The condition for the first case is that h must be greater than /j.a. The abscissae of the intersections of AL with SS' and CC are respectively Rq = On S. and R,= afjL + b The necessary conditions for the second case are b/jb + a'' that Ri must be positive, and Rq either negative or positively greater than i^i(l + e). 197. Ex. 1. Rebound of a baU. A spherical ball, moving without rotation on a smooth horizontal plane, impinges tvith velocity V against a rough vertical wall tchose coefficient of friction is fi. The line of motion of the centre of gravity before incidence making an angle a with the normal to the wall, determine the motion just after impact. This is the general problem of the motion of a spherical ball projected without initial rotation against any rough elastic plane. Thus it applies to a billiard ball impinging against a cushion, or to a "fives" ball projected against a wall, or to a cricket ball rebounding from the ground. When the ball has any initial rotation the problem is, in general, a problem in three dimensions and will be discussed further on. In the figure the plane of the paper represents a horizontal plane drawn through the centre of the ball. The vertical plane against which the ball impinges intersects the plane of the paper in AS. Let u, V be the velocities of the centre at any time t after the commencement of the impact resolved along and perpendicular to the wall. Let w be the angular velocity at the same instant. Let jR and F be the normal and frictional blows from the beginning of the impact up to that instant. Let M be the mass and r the radius of the sphere. Then we have M{u-Vsina)= -F, M{v + Vcos a) = R, Mk^o} = Fr. ART. 197.] ON IMPULSIVE FORCES. 163 r^+k^ F The velocity of sliding of the point of contact is /Sf = M-rw= Fsina j-^— ^ . The velocity of compression of the point of contact is C= -v= Fcos « - jjv • Measure a length AS in the figure to represent -^ — pMFsina, and a length ^C to represent MFcosa, along the axes of F and R respectively. Then SB and CB drawn parallel to the directions of R and F will be the lines of no sliding and of greatest compression. Also we see that tan JS^ C = -^ — Tg tan a = ^ tan a. In the beginning of the impact the sphere slides on the wall, hence the representative point P, whose coordinates are R and F, begins to describe the straight line F=ixR. If /*> f tan a, this straight line cuts the line of no sliding SB in some point L before it cuts the line of greatest compression. Hence the representative point describes the broken line ALB. At the moment of greatest compression, F and R sxe the coordinates of B. Therefore F=^MVQina, R = MVcoQa. These results are independent of /i because we see from the figure that more "than enough friction could be called into play to destroy the sliding motion. If ju < f tan a, the straight line F=fiR cuts the line of greatest compression CB in some point H before it cuts the line of no sliding. The friction is therefore insufficient to destroy the sliding. At the moment of greatest compression F and R are the coordinates of H, F=iJ.MVcos o, R = MV cos a. If the sphere be inelastic we have only to substitute these values of F and R in the equations of motion to find the values of u, v, w just after impact. If the sphere be imperfectly elastic with a coefficient of elasticity e, the repre- sentative point P will continue its progress until its abscissa is given by R = MV cos a {I + e). Take AC to represent this value of R, and draw CB' parallel to CB. Then, as 2 tan a I)efore, we see that tan B'AC = ^ -z . 7 1+e If yu.>~ , the representative point describes some broken line like ALB', and cuts SB' before it cuts B'C. In this case F and R are the coordinates of B', F=fMVsma, R = MV cos a (l + e). U fM< ~ , the representative point describes some unbroken line like AHK, and cuts B'C before it cuts SB'. In this case F and R are the coordinates of K, F=iJ,MVcosa{l + e), R = MV cos a {l + e). Let /3 be the angle the direction of motion of the centre of the ball makes with the normal to the wall after impact, then tan^^w/i;. We see therefore that » ^ ^ 5 tan a tana- u (l + e) ,. . ^^ i ^i. 2 tan a tan p = - , or= ^--^ — , accordmg as fi is greater or less than - . Ex. 2. An imperfectly elastic cricket ball is projected so that it is rotating with an angular velocity ii about a horizontal axis perpendicular to the plane of i -.the parabola described by its centre. Just before it strikes the ground the velocity 11—2 164 MOTION IN TWO DIMENSIONS. [CHAP. IV. of the centre is V, and the direction of motion makes an angle a with the normal. Show that the angle of rebound j8 is given by either 5 2 ril e tan 6 = ^ tan a + - -^ , or = tan a- a(l + e), ^ 1 7 F cos a "^^ jaccording as fi is greater or less than - jtan a ~ ~ | . Ex. 3. A sphere of radius a rolls on the ground with velocity U and impinges normally against a vertical wall whose coefificients of friction and elasticity are ft and e. If /a (1 + e) > f the sliding will terminate before the end of the period of impact, and the sphere will therefore rebound with a horizontal velocity - TJe and a vertical velocity f U [this follows by taking moments about the point of contact]. The centre of the sphere will then describe a parabola and the sphere will after- wards impinge on the ground. If the ground be inelastic and have a coefficient of friction /*' < e + y the sliding will not terminate before the end of the impact. At the end of the impact the centre of the sphere has a velocity - U {e-^/x') and the angular velocity is (2 - Bfi') Ujla. The friction continues to act as a finite force so that the sphere finally rolls on the ground with a uniform velocity -^U {e- ^*-). Ex. 4. A thin uniform hemispherical shell of radius a with its base vertical is rotating with an angular velocity fi about a horizontal axis through its centre of gravity parallel to the base. It is placed with a point on its base in contact with a fixed rough horizontal plane. Prove that if the coefficient of elasticity is equal to e and the coefficient of friction is greater than 2, the point of contact with the plane begins after the impact to move vertically with a velocity -^aeil. 198. Ex. 1, Show that the representative point P as it travels in the manner described in the text must cross the line of greatest compression, and that the abscissa R of the point at which it crosses this straight line must be positive. Ex. 2. Show that the conic whose equation referred to the axes of R and F is. aF'^ + 2bFR + a'R^=e, where e is some constant, is an ellipse, and that the straight lines of no sliding and greatest compression are parallel to the conjugates of the axes of F and R respectively. Show also that the intersection of the straight lines of no sliding and greatest compression must lie in that angle formed by the conjugate diameters which contains or is contained by the first quadrant. Ex. 3. Two bodies, each turning about a fixed point, impinge on each other,, find the motion just after impact. Let G, G', in the figure of Art. 187, be taken as the fixed points. Taking moments about the fixed points, the results will be nearly the same as those given in the case considered in the text. Initial Motions. 199. Breakage of a support. Let a system of bodies be in equilibrium and let one of the supports suddenly give way. It is required to find the initial motions of the several bodies and the initial values of the reactions which exist between them. The problem of finding the initial motion of a dynamical system is the same as that of expanding the coordinates of the moving particles in powers of the time t. Let {x, y, 6) be the coordinates of any body of the system. For the sake of brevity ART. 199.] INITIAL MOTIONS. 165 let the suffix zero denote initial values. Thus ^o denotes the initial value of x. By Taylor's theorem we have f" ... f ^ = a-|-a7oj2 + ^o|3 + (1): the term i^o is omitted because we suppose the system to start from rest. Firstly, let only the initial values of the reactions he required. The dynamical equations contain the coordinates, their second differential coefficients with regard to t, and the unknown reactions. There are as many geometrical equations as re- actions. From these we have to eliminate the second differential coefficients and find the reactions. The process, which is really the same as the first method of solution described in Art. 135, is as follows. Write down the geometrical equations, differentiate each twice and then simplify the results by substituting for the coordinates their initial values. Thus, if we use Cartesian coordinates, let {x, y, 6) = be any geometrical relation, we have since Xq = 0, 2/0 = 0, ^0 = 0, dcl> d4> d. _ The process of differentiating the equations may sometimes be much simplified when the origin has been so chosen that the initial values of some at least of the coordinates are zero. We may then simplify the equations by neglecting the squares and products of all such coordinates. For if we have a term x^, its second differential coefficient is 2 {xx + x^), and if the initial value of X is zero, this vanishes. The geometrical equations must be obtained by supposing the bodies to have their displaced positions, because we require to differentiate them. But this is not the case with the dynamical equations. These we may write down on the supposition that each body is in its initial position. These equations may be obtained according to the rules given in Art. 135. The forms there given for the effective forces admit in this problem of some simplifications. Thus, since ro = 0, and thus by substituting in equation (1) we have found the initial motion up to terms depending on t^. 200. Secondly, let the initial motion be required. As differential coefficients of a high order sometimes present themselves in this part of the problem it will be more convenient to use accents instead of dots to represent the differential coefficients with regard to the time. Thus 35 will be written x". The number of terms of the series (1) which it may be necessary to retain depends on the nature of the problem. Suppose the radius of curvature of the path described by the centre of gravity of one of the bodies to be required. We have {x^-^y"'f ^ x'f-y'x"' Putting li — x'y" — y'x" we have after differentiation u' = x'y"'-y'x"\ "x"\ u" = xy''' - y'x'-" + a^'y" - y v:." = x'y" - y V + 2 {x'f - ?/ V^). Substituting in Taylor's Theorem and remembering that <=0, 2/o'=0, x'y" - y'x" = \ {x:'yr - x^y^') t' + i (a^oV " ^o^ VO ^' + • • - similarly (x^ + y"")^ = {x^'-' + ypf t^^ .... If then the body start from rest, the radius of curvature is zero. But if x^'y"' — x^"y^' = 0, the direction of the acceleration is stationary for a moment. We then have — u^Q yo -^0 yo •• P To find these differential coefficients we may proceed thus. Differentiate each dynamical equation twice and then reduce it to its initial form by writing for x, y, 6, &c. their initial values, and for x, y', 6' zero. Differentiate each geometrical equation four times and then reduce each to its initial form. We shall thus have sufficient equations to determine Xq\ x^", x^"', &c. ART. 201.] INITIAL MOTIONS. 167 Rq, RJ, R^', &c., where R is any one of the unknown reactions. It is often of advantage to eliminate the unknown reactions from the equations before differentiation. We then have only the un- known coefficients x^', x^'\ &c. entering into the equations. These operations may in general be much abbreviated by some simple con- siderations. Let a dynamical equation be of the form where L, M, N, P are functions of x, y, 6 only. Differentiating twice and putting Xq=0, 2/o' = 0, ^o' = 0, we have L:ro'^ + iH/Zo'" + Nd^'" + A {Lx^" + My^" + ^^^o" + ^) = 1 . ,1 d ,, d „ ,, d If we write x = Xq + ^, y = yQ + V, <^<2- so that |, tj, &c. are small quantities it is easy to see that all the terms in L, M, &g. which contain ^'■^, r)'^, &c. disappear from the final equation. When therefore we have to find x^^", y^l^, ^^'^ by differentiating the dynamical equations, it is only necessary that the coefficients L, M, (&c. should he correct to the first poicer of the small quantities. In the same way if (.t, y, d)=Q he b. geometrical equation, we see that its fourth differential coefficient reduces to It is therefore only necessary that the geometrical equations should be correct to the second power of the small quantities. In the same way if we require the initial values of the sixth differential co- efficients we must form the dynamical equations correct to the second order and the geometrical equations to the third order. We shall afterwards see that these initial differential coefficients may be more easily deduced from Lagrange's equations. If we know the direction of motion of one of the centres of gravity under consideration, we can take the axis of i^ a tangent to its path. We then have p = ^ , where x is of the second order and y of the first order of small quantities. We may therefore neglect the squares of x and the cubes of y. This will greatly simplify the equations. If the body start from rest we have Xq = 0, and if x^' = 0, we may then use the formula p = 3 -^ . Xq The corresponding formula for p in polar coordinates may be obtained in the same way. We have when r^ {rfl'd^" - r^'" 6^") — 3(W + r;'2)t 201. Ex. 1. A circular disc is hung up by three equal strings attached to three points at equal distances on its circumference, and fastened to a peg vertically over the centre of the disc. One of these strings being cut, determine the initial tensions of the other two. 168 MOTION IN TWO DIMENSIONS. [CHAP. IV. Let be the peg, AB the circle seen by an eye in its plane. Let OA be the string which is cut, let C be the middle point of the chord joining the points of the circle to which the two other strings are attached. Then the two tensions, each equal to T, are throughout the motion equivalent to a resultant tension n R along CO. If 2a be the angle between the two strings, we have R=:2Tcosa. Let I be the length of 0(7, j3 the angle GOG, a the radius of the disc. Let {x, y) be the coordinates of the displaced position of the centre of gravity with reference to the origin 0, x being measured horizontally to the left and y vertically down- wards. Let d be the angle which the displaced position of the disc makes with AB. By drawing the disc in its displaced position it will be seen that the coordinates of the displaced position of C are a; - Z sin /3 cos d and y - Z sin /3 sin d. Hence since the length OC remains constant and equal to Z, we have X- + y^ -2lsm ^ [x COB e + y smd) = l^ cos^^. Since the initial tensions only are required, it is sufficient to differentiate this twice. Since we may neglect the squares of small quantities, we may omit x'^, and put cos^ = l, 8in^ = ^. The process of differentiation will not then be very long, for it is easy to see beforehand what terms will disappear when we equate the differential coefficients (x, i/, 6) to zero, and put for {x, y, 6) their initial values (0, I cos /3, 0). We get ?/o cos ^ = sin /3 {xq + 1 cos ^Oq). This equation may also be obtained by an artifice which is often useful. The motion of G is made up of the motion of G and the motion of G relatively to G. Since G begins to describe a circle from rest, its acceleration along GO is zero. Again, the acceleration of G relatively to G when resolved along GO is GGd cos p. The resolved acceleration of G is the sum of these two, but it is also equal to I/q cos j3 - Xq sin /3. Hence the equation follows at once. In this problem we require the dynamical equations only in their initial form. These are hixq = Rq sin ^, miJQ = mg - R^cos ^, mk^dQ = R^l sin j8 cos /3, where m is the mass of the body. Substituting in the geometrical equation we find _ fc^cosjS '>~'''^ • k'^ + l^sin^^cos'^ ' The tension of any string, before the string OA was cut, may be found by the rules of statics, and is clearly T^ = ^mg sec 7, where 7 is the angle AOG. Hence the change of tension can be found. Ex. 2. A number of uniform straight rods of the same weight and length, freely jointed end to end, are supported in a horizontal straight line, with the extreme end of the last rod fixed. If the supports are all removed at once, obtain equations to determine the initial angular accelerations of the different rods and prove that if a;„, w„+i, w„+2 are those of any three adjacent rods, w„ + 4w„+i + a>„+2 = 0- [Math. Tripos, 1903. Let a?o , Wj . . . w,^ be the angular accelerations of the m + 1 rods, Uq, u^...Uy^ the vertical accelerations of their centres, R^ , j the rods. Since k^=a^l%, we have iaw„ = i?„+i + i?„, 4aw„+i = E„+2 + ^n+i' &c (1), W„ = -Rn+l--Rn + ^» Wn+l=-Rn+2-^n+l + fl'. &e (2). ART. 202.] INITIAL MOTIONS. . 169 and by geometry m„ + aa;,^ = u^+i - aw„+i . These give R^+^_ + 4R^+^ + R^ = (3). It immediately follows by substitution from (1) that Wn+2 + ^'^n+i + '«'» = ^• To find the initial accelerations Wq ... u)^, we solve the equation of dififerences (3) by putting R^-Ac'^. This gives 0^ + 4c + 1 = 0, hence if a, j3 are the roots of the quadratic, R^ = Aa''^ + B^. To find A, B vfe examine the geometrical conditions at the ends. It is given that one end is fixed, hence UQ-~aci}Q = 0, .'. A -B = gj2j^. If the other end is free, iJ^+^^O, .'. ^a^+i + B|8^+^ = 0. These two conditions determine A and B. The problem might also have been solved by Lagrange's method. 202. Ex. 1. Two strings of equal length have each an extremity tied to a weight G and their other extremities tied to two points ^, B in the same horizontal line. If one be cut the tension of the other will be instantaneously altered in the ratio 1 : 2 cos^ ^ C. [St Pet. CoU. Ex. 2. An elliptic lamina is supported with its plane vertical and transverse axis horizontal by two weightless pins passing through the foci. If one pin be released show that, if the eccentricity of the ellipse be 4 v'lO, the pressure on the other pin is initially unaltered. [Coll. Exam. Ex. 3. Three equal particles A, B, C repelling each other with any forces, are tied together by three strings of unequal length, so as to form a triangle right- angled at ^. If the string joining B and C be cut, prove that the instantaneous changes of tension of the strings joining BA, CA will be ^TcoaB and ^TcosO respectively, where B and C are the angles opposite the strings joining CA, AB respectively, and T is the repulsive force between B and C. Ex. 4. Two uniform equal rods, each of mass m, are placed in the form of the letter Z on a smooth horizontal plane, the upper and lower extremities being connected by equal strings ; show that, whichever string be cut, the tension of the other is the same function of the inclination of the rods, and initially is f mgf sin a, where a is the initial inclination of the rods. [St Pet. Coll. Ex. 5. A horizontal rod of mass m and length 2a hangs by two parallel strings of length 2a attached to its ends : an angular velocity w being suddenly communicated to it about a vertical axis through its centre, show that the initial increase of tension of either string equals ^maw'^, and that the rod rises through a space a^u^jGg. [Coll. Exam. Ex. 6. A particle is suspended by three equal strings of length a from three points forming an equilateral triangle of side 2b in a horizontal plane. If one string be cut the tension of each of the others will be instantaneously changed in the ratio ^ 7 o"t^ • ' [Coll. Exam. 2 (a^ -b^) I Ex. 7. A sphere resting on a rough horizontal plane is divided into an infinite i number of solid lunes and tied together again with a string ; the axis through which ! the plane faces of the lunes pass being vertical. Show that if the string be cut ' the pressure on the plane will be instantaneously diminished in the ratio 457r2:2048. [Emm. Coll. 1871. Ex. 8. A smooth sphere rests on a horizontal plane and an equal sphere is supported on it, the line of centres making an angle (f> with the vertical ; prove that just after the supports are removed the ratio of the pressures on the plane and between the spheres is 2 : cos 0. [Coll. Exam. 170 ' MOTION IN TWO DIMENSIONS. [CHAP. IV. Ex. 9. A small ring of mass p is strung on a rod, of mass m and length 2a, capable of turning about one extremity as a fixed point. The system starts from rest with the rod horizontal and the ring at a distance c from the fixed point. Show that the polar coordinates of the ring referred to the fixed point are c + ro'^f'*/24 and 6>ot2/2, Find also 6^ , and prove that r^'^ = <76'o + 2c^V^, Thence find the initial radius of curvature of the path of the particle. [May Exam. 1888. Ex. 10. A solid hemisphere of mass M rests on a perfectly rough horizontal plane and a particle of mass m is gently placed on it at a distance c from the centre. Prove that the initial radius of curvature of the path described by the particle is ^mc^jMk^, where k is the radius of gyration of the hemisphere about a tangent at the vertex. [Math. Tripos, 1888. Ex. 11. A garden roller is at rest on a horizontal plane, rough enough to prevent sliding, the handle being so held that the plane through the axis of the cylinder and the centre of gravity of the handle makes an angle a with the horizon. Show that when the handle is let go the initial radius of curvature of the path described by the centre of gravity is c (sin^ a + n cos^ afjn where {n - 1) M {k"^ + aF) = ma-, c is the distance of the centre of gravity of the handle from the axis of the cylinder, m its mass, Mk^ the moment of inertia of the cylinder about its axis, and a its radius. [Math. Tripos, 1894. Ex. 12, A uniform rod of mass m and length 2a has masses equal to m attached to its ends. A string, one end of which is attached to the middle point of the rod, passes over a smooth pulley and sustains at its other end a weight 3m. The system is in equilibrium, the rod being horizontal. The particle m falls off from one end of the rod ; prove that (1) the initial acceleration of the mass 3m equals Iglll ; (2) the initial angular acceleration of the rod is IQgjVJa ; (3) the radius of curvature of the initial path of the other end of the rod is 2a (11/18)''^. [Coll. Exam. 1904. Ex. 13. A uniform cube of edge 2a and mass M rests symmetrically on two shelves, each of length 4a and mass fxM and is hinged to one shelf at the edge of the cube, and the shelves are attached to smooth hinges at a distance 8a apart, being supported in a horizontal position. If that shelf is released to which the cube is hinged, prove that the initial pressure on the edge of the fixed shelf is jfo (45 + 46ju.) -,). — ^rn—"— , and that the initial reaction at the hinge to which the cube is 45 + Ib/u attached is inclined to the horizon at an angle tan-i 5/3. [Coll. Exam. 1904. On Relative Motion or Moving Axes. 208. In many dynamical problems the relative motion of the different bodies of the system is all that is required. In such cases it will be an advantage if we can determine this without finding the absolute motion of each body in space. Let us suppose that the motion relative to some one body {A) is required. There are then two cases to be considered, (1) when the body {A) has a motion of translation only, and (2) when it has a motion of rotation only. The case in which the body {A) has a motion both of translation and rotation may be regarded as a combination of these two cases. Let us consider them in order. ART. 205.] ON RELATIVE MOTION OR MOVING AXES. 171 204. The Fundamental Theorem. Let it be required to find the motion of any dynamical system relative to some moving point C. We may clearly reduce G to rest by applying to every element of the system an acceleration equal and opposite to that of G. It is also necessary to suppose that an initial velocity equal and opposite to that of G has been applied to each element. Let f be the acceleration of G at any time t. If every particle 711 of a body be acted on by the same accelerating force / parallel to any given direction, it is clear that these are together equi- valent to a force /Sm acting at the centre of gravity. Hence to reduce any point C of a system to rest, it will be sufficient to apply to the centre of gravity of each body in a direction opposite to that of the acceleration of G a force measured by Mf^ where M is the mass of the body and / the acceleration of G. The point G may now be taken as the origin of coordinates. We may also take moments about it as if it were a point fixed in space. Let us consider the equation of moments a little more minutely. Let (r, 6) be the polar coordinates of any element of a body whose mass is m referred to G as origin. The accelerations of the particle are ~j^ -~ '^ \'Ji] ^^^ ~ ~Jtv^i' ^^^"^ ^^^ perpendicular to the radius vector r. Taking moments about G we get ( moment round G of the impressed forces y ^ f ,^dd\ _ ] plus the moment round G of the reversed dt\ dtJ ] effective forces of C supposed to act at the I centre of gravity. If the point G be fixed in the body and move with it, ddjdt will be the same for every element of the body, and, as in Art. 88, wehaveSm^^^r^^j=m^^. 205. From the general equation of moments about a moving point G we learn that we may use the equation day _ moment of forces about G dt moment of inertia about G in the following cases. Firstly. If the point G be fixed both in the body and in space ; or if the point G, being fixed in the body, move in space with uniform velocity ; for the acceleration of G is zero. Secondly. If the point G be the centre of gravity ; for in that case, though the acceleration of G is not zero, yet the moment vanishes. 172 MOTION IN TWO DIMENSIONS. [CHAP. IV. Thirdly. If the point G be the instantaneous centre of rota- tion, and the motion be a small oscillation or an initial motion which starts from rest. At the time t the body is turning about 0, and the velocity of G- is therefore zero. At the time t + dt, the body is turning about some point C" very near to G. Let GG' = da, then the velocity of G is wda. Hence in the time dt the velocity of G has increased from zero to coda, therefore its acceleration is (oda/dt. To obtain the accurate equation of moments about G we must apply the effective force Sm. coda/dt in the reversed direction at the centre of gravity. But in small oscillations co and da/dt are both small quantities whose squares and products are to be neglected, and in an initial motion w is zero. Hence the moment of this force must be neglected, and the equation of motion will be the same as if G had been a fixed point. It is to be observed that we may take moments about any point very near to the instantaneous centre of rotation, but it will usually be more convenient to take moments about the centre in its disturbed position. If there be any unknown reactions at the centre of rotation, their moments will then be zero. 206. If the accurate equation of njoments about the instan- taneous centre be required, we may proceed thus. Let L be the moment of the impressed forces about the instantaneous centre, G the centre of gravity, r the distance between the centre of gravity and the instantaneous centre G, M the mass of the body ; then the moment of the impressed forces and the reversed effective forces about (7 is L — Mw -tt . r cos GG'G. dt If k be the radius of gyration about the centre of gravity, the d(o T ,. dr writing for cos GG'G its value drjda. 207. Impulsive forces. The argument of Art. 204 may evidently be also applied to impulsive forces. We may thus obtain very simply a solution of the problem considered in Art. 171. A body is moving in any manner when suddenly a point O in the body is con- strained to move in some given manner, it is required to find the motion relative to 0. To reduce to rest, we must apply at the centre of gravity G a momentum equal to Mf, where / is the resultant of the reversed velocity of O after the change and the velocity of before the change. If w, w' be the angular velocities of the body before and after the change, and r = OG, we have by taking moments about 0, (r2 + A;2) [u)' -u}) = moment of / about 0. Now the moment about of a velocity at G is equal and opposite to the moment about G of the same velocity applied at 0. Hence if L, L' be the moments about equation of motion becomes M (k^ + r^) j, =L — Meoi ART. 209.] ON RELATIVE MOTION OR MOVING AXES. 173 G of the velocity of just before and just after the change, and k be the radius L' - L I of gyration about the centre of gravity, we have w' - w = ^^ . tC -\-T 208. Ex. Two heavy particles xohose masses are m and m' are connected by an I inextensible string, ivhich is laid over the vertex of a double inclined plane whose I mass is M\ and which is capable of moving freely on a smooth horizontal plane. i Find the force which must act on the loedge that the system may be in a state of relative equilibrium. Here it will be convenient to reduce the wedge to rest by applying to every particle an acceleration/ equal and opposite to that of the wedge. Supposing this done the whole system is in equilibrium. If F be the required force, we have by resolving horizontally {M + m-\-m')f—F. Let a, a' be the inclinations of the sides of the wedge to the horizontal. The particle m is acted on by mg vertically and mf horizontally. Hence the tension of the string is m (^ sin a+/cosa). By considering the particle m', we find the tension to be also m' [g sin a' -/cos a'). Equating these two we have ^ m' sin a' -m sin a f = -l ; q. m cos a + m cos a Hence F is found. Since /, and therefore also the horizontal and vertical accelerations of either particle, are constants, it follows that the path of either particle in space is a parabola, whose axis is parallel to the direction of the resultant acceleration of that particle. 209. A cylindrical cavity whose section is any oval curve and whose generating lines are horizontal is made in a cubical mass which can slide freely on a smooth horizontal plane. The surface of the cavity is perfectly rough and a sphere is placed in. it at rest so that the vertical plane through the centres of gravity of the mass and the sphere is perpendicular to the generating lines of the cylinder. A momentum B /.s communicated to the cube by a bloiv in this vertical plane. Find the motion of the sphere relatively to the cube and the least value of the blow that the sphere may not leave the surface of the cavity. Simultaneously with the blow B there will be an impulsive friction between the cube and the sphere. Let M, m be the masses of the cube and sphere, a the radius of the sphere, k its radius of gyration about a diameter. Let Vq be the initial velocity of the cube, v^ that of the centre of the sphere relatively to the cube, w^ the initial angular velocity. Then by resolving horizontally for the whole system, and taking moments for the sphere alone about the point of contact, we have m{v,+ V,)+MV, = B, aK+Fo)+Fa;o = (1), and since there is no sliding i?o-aa)o = (2). To find the subsequent motion, let [x, y) be the coordinates of the centre of the sphere referred to rectangular axes attached to the cubical mass, x being horizontal and y vertical, then, the equation to the cylindrical cavity being given, ?/ is a known function olx. Let \}/ be the angle which the tangent to the cavity at the point of con- tact of the sphere makes with the horizon, then tan \p = dyjdx. Let V be the velocity of the cubical mass, then, by Art. 132, m {-^+v\ + MV=B (3). If Tq be the initial vis viva and y^ the initial value of y, we have by the equation of vis viva %+A\{^\' + k^A^MV^=T,-2mg(y-y,) ...(4), 174 MOTION IN TWO DIMENSIONS. [CHAP. IV. where w is the angular velocity of the sphere at the time t. If v be the velocity of the centre of the sphere relatively to the cube, we have since there is no sliding v = au. Eliminating F and w from these equations, we have (^^;y.|,l.ta„^,)(l.^)-^-^( = 0.-2.. .(5), where Cg = ■ pr + ^^o (6). {M + m) \M+{M + m)-4 a This equation gives the motion of the sphere relatively to the cube. 210. Ex. 1. A spherical hollow of radius a is made in a cube of glass of mass M, and a particle of mass m is placed within. The cube is then set in motion on a smooth horizontal plane so that the particle just gets round the sphere, remaining in contact with it. If the velocity of projection is V, prove that V^ = 5ag + 4agm.lM. [Coll. Exam. Let us reduce the cube to rest. Let R be the normal pressure on the cube, 6 the angle the radius of the particle makes with the downward vertical. The whole horizontal effective force on the cube is Rsind. By Art. 204 we apply to every particle an acceleration R sin dIM and an initial velocity equal and opposite to V. The particle m is then acted on by a force mR sin dfM in a horizontal direction in addition to the reaction R and the weight mg. The equations of motion of the particle are mad= - ^Rsin 6 cos 6 -mg sin 6 (1), viad^ = R + ~Rsm^d-mgcosd (2). Put ^ = w and d = o}d(aldd and eliminate R, we find du 'dd 2au -^{M + m sin^ 6) + lau^m sin ^ cos ^= - 2Mg sin d [M+m), .: a -Vf/ {b-a), a being the radius of the ball, and b that of the roller. 211. Moving Axes. Next, let us consider the case in which we wish to refer the motion to two straight lines Of, Or} at right angles, turning round a fixed origin with angular velocity o). Let Ooo, Oy be any fixed axes at right angles and let the angle xO^ = 6. Let f = OM, rj = PM be the coordinates of any point P. Let u, v be the resolved velocities and X, Y the resolved accelerations of the point P in the directions Of, Or}. It is evident that the motion of P is made up of the motions of the two points M, iV by simple addition. The resolved parts of the velocity of M are d^/dt and fft) along and per- pendicular to OM. The resolved parts of the ve- locity of N are in the same way dy/dt and rjco along and perpendicular to ON. By adding these with their proper signs we have cZf dr} ^ f. Since acceleration is the rate of increase of velocity just as velocity is the rate of increase of space, we obtain the correspond- ing formulae for X, Y by writing u, v for x, y. We thus have Z = du vco, dv -\-U(0. dt ' dt In the same way by adding the accelerations of M and X we have d^x d^ii By using these formulae instead of -77^ and ^ we may refer dP ) the motion to the moving axes Of, Or}. 176 MOTION IN TWO DIMENSIONS. [CHAP. IV. 212. Ex. 1. Let the axes 0^, Orj be oblique and make an angle a with each other, prove that, if the velocity in space be represented by two components m, v parallel to the axes, w = I - w| cot a — 007) cosec a, r = 17 + ojt; cot a + u^ cosec a. In this case PM is parallel to Otj. The velocities of M and N are the same as before. Their resultant is, by the question, the same as the resultant of u and v. By resolving in any two directions and equating the components we get two equa- tions to find u and v. The best directions to resolve along are those perpendicular to 0| and Orj, for then u is absent from one of the equations and v from the other. Thus either m or v may be found separately when the other is not wanted. Ex. 2. If the acceleration be represented by the components X and Y, prove X=u- (i}U cot a- uv cosec a, Y= t + uv cot a + uu cosec a. These may be obtained in the same way by resolving velocities and accelerations perpendicular to 0| and Or). Ex. 3. If u, V be the velocities of a point P referred to rectangular moving axes rotating with an angular velocity a>, prove that the radius of curvature of the path of P in space is given by (m'-* + v^)'/p = uv -vu+ {u^ + v^) w. By taking fixed axes coincident for a moment with the moving axes the left side of this equation is seen to be xy-xy. Substituting x = u, y = v, and for x = X, y = Y their values given above the result follows at once. The ordinary expression for p in polar coordinates follows from this by writing u = r, v = rd, (i3 = d. If the independent variable is d we have ^ = 1. Ex. 4, In the case of initial motions which start from rest the formula for p in the last example becomes nugatory. Show by proceeding as in Art. 200 that p = unless iiv - ilv + 2 (w^ + 1)^) w = 0, and that in that case (?t- + b"ylp = ^ (iiv -vu) + {iiii + vv)u)+ {u^ + v where u, ii &c., v, v &c. represent their initial values, the suffix zero being omitted for the sake of brevity. 213. Ex. A particle under the action of any forces moves on a smooth curve which is constrained to turn with angular velocity w about a fixed axis. Find the motion relative to the curve. Let us suppose the motion to be in three dimensions. Take the axis of Z as the fixed axis, and let the axes of ^, rj be fixed relatively to the curve. Let the mass be the unit of mass. Then the equations of motion are .(1), where X, Y, Z are the resolved parts of the impressed accelerating forces in the directions of the axes, R is the pressure on the curve, and (Z, m, w) the direction- cosines of the direction of B,. Then since jR acts perpendicular to the curve ,df d-n dz ^ l^ + m^ + n — = 0. ds ds ds ART. 213.] ON RELATIVE MOTION OR MOVING AXES. 177 Suppose the moving curve to be projected orthogonally on the plane of |, 17, let a be the arc of the projection, and v' = di;'. Then R' also acts perpendicularly to the tangent, let {I", m", n") be the direction-cosines of its direction. The equations of motion therefore become d'^'q ,^ „ du) . _,, ,, 5r^= ^+'^''- di^+^'" \ (2). These are the equations of motion of a particle moving on a fixed curve, and acted on in addition to the impressed forces by two extra forces, viz. (1) a force wV tending directly from the axis, where r is the distance of the particle from the axis, and (2) a force rdcvldt perpendicular to the plane containing the particle and the axis, and tending opposite to the direction of rotation of the curve. In any particular problem loe may therefore treat the curve as fixed. Thus suppose the curve to be turning round the axis with uniform angular velocity. rm 1 • 1 .1 . ,1 dv ^^dx ^^dy „dz „ dr Then resolving along the tangent we have u-r=Z-^ +Y~ + Z— +u>h — , ds ds ds ds ds where r is the distance of the particle from the axis. Let V be the initial value of v, r^ that of r. Then v^- V^=2j{Xdx+Ydy + Zdz) +u}^{r^-rQ^). Let Vq be the velocity the particle would have had under the action of the same forces if the curve had been fixed. Then V(,2 - F^ = 2j {Xdx + Ydy + Zdz) . Hence i;2- ^^^^(ra-V). The pressure on the moving curve is not equal to the pressure on the fixed cwve. Since V = dr]lda; m'= -d^/da, we see that the force 2(av' acts parallel to the normal to the projected curve in the direction opposite to that due to the rotation w. Hence, reversing this force, the pressure R on the moving curve is the resultant of the pressure R' on the fixed curve and a pressure 2a)i;' acting perpendicularly both to the curve and to the axis, the last pressure being taken positively in the direction of motion of the curve. R. D. 12 178 ' MOTION IN TWO DIMENSIONS. [CHAP. IV. Thus suppose the curve to be plane and revolving uniformly about an axis perpendicular to its plane, and that there are no impressed forces. We have, resolving along the normal, - = - w^r sin + R\ where is the angle which r makes with the tangent. If p be the perpendicular drawn from the axis on the tangent, R= - + (J^p + luiv. This example might also have been advantageously solved by cylindrical co- ordinates. The fixed axis might be taken as axis of z and the projection on the plane of xy referred to polar coordinates. This method of treating the question is left to the student as an exercise. do) Ex. If w be variable, we have R = — + la^p + 2uv + -r,'>Jr^ - p^. p (it EXAMPLES*. 1. A circular hoop, whose weight is nw, is free to move on a smooth horizontal plane. It carries on its circumference a small ring, weight w, the coefficient of friction between the two being fi. Initially the hoop is at rest and the ring has an angular velocity w about the centre of the hoop. Show that the ring will be at rest on the hoop after a time {l + n)lfno. 2. A heavy circular wire has its plane vertical and its lowest point at a height h above a horizontal plane. A small ring is projected along the wire from its highest point with an angular velocity w about its centre at the instant that the wire is let go. Show that, when the wire reaches the horizontal plane, the particle will just have described n revolutions, where h(>r = 2Tr^n?g. 3. A wire in the form of a circle is capable of turning in a horizontal plane about a fixed point in its circumference, and carries a bead P which is initially projected from the opposite end A of the diameter through with a given velocity V. Supposing the mass of the wire to be double that of the bead, show that (16a'» + iah^ -r^),j>^= FV^ where r = OP, 0A=2a, ^^-gr {1 - sin (a + ^ tt)}, a being the inclination of the plane to the horizon. Show that the hoop will not remain in contact with the spike unless F2 and - + — = - + —. p p p p a a 29. A sphere whose centre of gravity is not in its centre is placed on a rough table ; the coefficient of friction being fi, determine whether it will begin to slide or to roll. 30. A circular ring is fixed in a vertical position upon a smooth horizontal plane, and a small ring is placed on the circle, and attached to the highest point by a string, which subtends an angle a at the centre ; prove that if the string be cut and the circle left free, the pressures on the ring before and after the string is cut are in the ratio M+m sin" a iMcosa, m and M being the masses of the ring and circle. [Reduce the ring to rest. Arts. 204, 210.] 31. One extremity C of a rod is made to revolve with uniform angular velocity n in the circumference of a circle of radius a, while the rod itself is made to revolve in the opposite direction with the same angular velocity about that extremity. The rod initially coincides with a diameter, and a smooth ring capable of sliding freely along the rod is placed at the centre of the circle. If r be the distance of the ring from C at the time t, prove r = -^ (e"' + e~*^^) + - cos 2nt. [Reduce C to rest, Art. 204.] 5 5 32. Two equal uniform rods of length 2a are joined together by a hinge at one extremity, their other extremities being connected by an inextensible string of length 21. The system rests upon two smooth pegs in the same horizontal line, distant 2c from each other. If the string be cut prove that the initial angular Sa'^c - l^ acceleration of either rod will be g — ^ ^ . [Take moments, for either rod alone, about the intersection of a horizontal line drawn through the hinge with a perpendicular to the rod drawn at the peg. Art. 205.] 33. A smooth horizontal disc revolves with angular velocity sJ/ul about a vertical axis, at the point of intersection of which is placed a material particle attracted to a certain point of the disc by a force whose acceleration is /x. x distance ; prove that the path on the disc is a cycloid. Art. 211. 34. A hollow cylinder of radius a rests on a rough table, and contains an insect resting within it on the lowest generator ; if the insect start off and continue to walk at a uniform velocity V relative to the cylinder in a vertical plane cutting the axis of the cylinder at right angles, then the angle 6 the axial plane containing the insect makes with the vertical is given hy a^d'^{M+ 2m sin^ ^6) = MV'^-27nagsin'^^ 6, it being understood that the cylinder is very thin. If the internal radius be b, prove ^2 [31 (A;2 + a^) + m {a^ - 2ah co&d + h^)] = G- 2mgh (1 - cos 6), where Gh"^ \M {k^ + a^) + m{a- 6)2] =V^[M {k'' + a^) + ma {a - b)f, and M, m are the masses of the cylinder and insect respectively. EXAMPLES. 183 35. A circular hoop of radius b, without mass, has a heavy particle rigidly attached to it at a point distant c from the centre, and its inner surface is con- strained to roll on the outer surface of a fixed circle of radius a {b being greater than a), under the action of a repelling force from the centre of the fixed circle equal to fji times the distance. Show that the period of small oscillations of the hoop will be 27r ( ) . Show that when c = b, all oscillations, large or small, have a \ Cfi J the same period ; and show further that in the general case the hoop may be started so that it will continue to roll with uniform angular velocity equal to / b-a\i CHAPTEE V. MOTION OF A EIGID BODY IN THREE DIMENSIONS. Translation and Rotation. 214. If the particles of a body be rigidly connected, then, whatever be the nature of the motion generated by the forces, there must be some general relations between the motions of the particles of the body. These must be such that if the motion of three points not in the same straight line be known, that of every other point may be deduced. It will then in the first place be our object to consider the general character of the motion of a rigid body apart from the forces that produce it, and to reduce the determination of the motion of every particle to as few in- dependent quantities as possible : and in the second place we shall consider how when the forces are given these independent quantities may be found. 215. One point of a moving rigid body being fixed, it is re- quired to deduce the general relations between the motions of the other points of the body. Let be the fixed point and let it be taken as the centre of a moveable sphere which we shall suppose fixed in the body. Let the radius vector to any point Q of the body cut the sphere in P, then the motion of every point Q of the body will be re- presented by that of P. If the displacements of two points A, B, on the sphere in any time be given as AA', BB', the displacement of any other point P on the sphere may clearly be found by constructing on A'B' as base a triangle A'P'B' similar and equal to APB. Then PP' will represent the displacement of P. It may be assumed as evident, or it may be proved as in Euclid, that on the same base and on the same side of it there cannot be two triangles on the same sphere, which have their sides terminated in one extremity of the base equal to one another, and likewise those terminated in the other extremity. ART. 217.] TRANSLATION AND ROTATION. 185 Let D and E be the middle points of the arcs AA', BB', and let DC, EG be arcs of great circles drawn perpendicular to AA\ BB' respectively. Then clearly CA = GA' and CB=CB\ and therefore since the bases AB, A'B' are equal, the two tri- angles AGB, A'GB' are equal and similar. Hence the dis- placement of G is zero. Also it is evident, since the dis- placements of and G are zero, that the displacement of every point in the straight line OC is also zero. Hence a body may he brought from any position, which we may call AB, into another A'B' by a rotation about OG as an axis through an angle PGP' such that any one point P is brought into coincidence with its new position P'. Then every point of the body will be brought from its first to its final position. This theorem is due to Euler. Memoires de VAcademie de Berlin 1750, and the Commentaires de Saint-Petersbourg 1775. 216. If we make the radius of the sphere infinitely great, the various circles in the figure will become straight lines. We may therefore infer that if a body be moving in one plane it may be brought from any position which we may call AB into any other A'B' by a rotation about some point G. 217. Ex. 1. A body is referred to rectangular axes x, y, z, and, the origin remaining the same, the axes are changed to x', y', z' , according to the scheme in the margin. Show that this is equivalent to turning the body round an axis whose equations are any two of the following three : (ai-l)x- -fag?/ + 032 = 0, &ia; + (62-l)2/ + &3« = 0. c^x^c^y^{c^-\)z = Q, through an angle d, where 3-4 sin^ \d — a^->r\-\-c.^. The positive directions of x' , y' being arbitrary, show that the condition that these three equations are consistent is satisfied, provided the positive direction of the axis of z' is properly chosen. See also a question in the Smith's Prize Examination for 1868. Take two points one on each of the axes of z and 2' at a distance h from the origin. Their coordinates are (0, 0, h) {a.^h, b^h, c^h), therefore their distance is h J 2 (1 - Cg). But it is also 2h sin 7 sin ^ ^ ; .'. 2 sin^ ^d sin^ 7 = 1 - Cg , where 7 is the angle zOz'. Similarly 2 sin^ ^ sin^ a = 1 - a^ and 2 sin'-^ ^ d sin^ ^ = 1 the equation to find 6 follows at once. Ex. 2. Show that the equations of the axis may also be written in the form •^ _ y _ z Ci + «3 c.^ + bs~ c^-a^-b^ + l' X'. y'. z' X «1, «2> «3 y 6i, 62, 63 z Cl, ^2, C3 186 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. 218. When a body is in motion we have to consider not merely its first and last positions, but also the intermediate posi- tions. Let us then suppose AB, A' B' to be two positions at any indefinitely small interval of time dt. We see that when a body moves about a fixed point 0, there is, at every instant of the motion, a straight line OC, such that the displacement of every point in it during an indefinitely short time dt is zero. This straight line is called the instantaneous axis. Let dO be the angle through which the body must be turned round the instantaneous axis to bring any point P from its position at the time t to its position at the time t + dt, then the ultimate ratio of dO to dt is called the angtdar velocity of the body about the instantaneous axis. The angular velocity may also be defined as the angle through which the body would turn in a unit of time if it continued to turn uniformly about the same axis throughout that unit with the angular velocity it had at the proposed instant. 219. Let us now remove the restriction that the body is moving with some one point fixed. We may establish the follow- ing proposition. Every displacement of a rigid body may be represented by a combination of the two following motions, (1) a motion of trans- lation, whereby every particle is moved parallel to the direction of motion of any assumed point P rigidly connected tvith the body and through the same space; (2) a motion of rotation of the whole body about some axis through this assumed point P. This theorem and that of the central axis are given by Chasles. Bulletin des Sciences Mathematiques par Ferussac, Vol. xiv. 1830. See also Poinsot, Theorie Nouvelle de la Rotation des Corps 1834. It is evident that the change of position may be effected by moving P from its old to its new position P' by a motion of trans- lation, and then retaining P' as a fixed point by moving any two points of the body not in one straight line with P into their final positions. This last motion has been proved to be equivalent to a rotation about some axis through P' . Since these motions are quite ijiiigpendent^ it is evident that their order may be reversed, i.e. we may first rotate the body and then translate it. We may also suppose them to take place simultaneously. It is clear that any point P of the body may be chosen as the base point of the double operation. Hence the given dis- placement may be constructed in an infinite variety of ways. 220. Change of Base. To find the relations between the axes and angles of rotation when different points P, Q are chosen as bases. ART. 222.] TRANSLATION AND ROTATION. 187 Let the displacement of the body be represented by a rota- tion 6 about an axis PR and a translation PP'. Let the same displacement be also represented by a rotation 6' about an axis Q8 and a translation QQ'. It is clear that any point has two displacements, (1) a translation equal and parallel to PP', and (2) a rotation through an arc in a plane perpendicular to the axis of rotation PR. This second displacement is zero only when the point is on the axis PR. Hence the only pomts whose displacements are the same as that of the base point lie on the axis of rotation corresponding to that base point. Through the second base point Q draw a parallel to PR. Then for all points in this parallel, the displacements due to the translation PP', and the rotation 6 round PR, are the same as the corresponding displacements for the point Q. Hence this parallel must be the axis of rotation corresponding to the base point Q. We infer that the awes of rotation corresponding to all base points are parallel. 221. ; The axes of rotation at P and Q having been proved parallel, let a be the distance between them. Let the plane q /q, of the paper intersect these .--"'"^ axes at right angles in P and PJ, „--"" Q, then Pq = a. Let PP', QQ' ^- represent the linear displace- !\ .,. -- ments of P and Q ^^spectively, ^\ ,..-"' though these need not neces- L--"" sarily be in the plane of the f paper. The rotation Q about PR will cause Q to describe an arc of a circle of radius a and angle Q, the chord Qq of this arc is 2a sin \d and is the displacement due to rotation. The whole dis- placement QQ^ of Q is the resultant of Qq and the displacement PP' of P. In the same way the rotation Q' about QB will cause P to describe an arc, whose chord Pp is equal to 2a sin \0'. The whole displacement PP' of P is the resultant of Pp and the displacement QQ' of Q. But if the displacement of Q is equal to that of P together with Qq, and the displacement of P is equal to that of Q together with Pp, we must have Pp and Qq equal and opposite. This requires that the two rotations Q, 6' about PR and QS should be equal and in the same direction. We infer that the angles of rotation corresponding to all base points are equal. 222. Since the translation QQ' is the resultant of PP' and Qq, we may by this theorem find both the translation and rotation corresponding to any proposed base point Q when those for P are given. Since Qq, the displacement due to rotation round PR, is 188 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. perpendicular to PR, the projection of QQ' on the axis of rotation is the same as that of PP'. Hence the projections on the axis of rotation of the displacements of all points of the body are equal. 223. An important case is that in which the displacement is a simple rotation 6 about an axis PR, without any translation. If any point Q distant a from PR be chosen as the base, the same displacement is represented by a translation of Q along a chord Qq — 2a sin ^6 in a direction making an angle Utt — 6) with the plane QPR, and a rotation which must be equal to about an axis which must be parallel to PR. Hence a rotation about any axis may be replaced by an equal rotation about any parallel axis together with a motion of translation. 224. When the rotation is indefinitely small, the proposition can be enunciated thus : — a motion of rotation codt about an axis PR is equivalent to an equal motion of rotation about any parallel axis QS, distant a from PR, together with a motion of translation acodt perpendicular to the plane containing the axes and in the direction in which QS moves. 225. Central axis. It is often important to choose the base point so that the direction of translation may coincide with the axis of rotation. Let us consider how this may be done. Let the given displacement of the body be represented by a rotation about PR, and a translation PP'. Draw P'N perpendicular to PR. Jf, pos sible let this same dis- placement be represented by a rotation about an axis QS, and a translation QQ' along QS. By Arts. 220 and 221 QS must be parallel to PR and the rotation about it must be 6. This translation will move P a length equal to QQ' along PR, and the rotation about QS will move P along an arc perpendicular to PR. Hence QQ' must equal PN and NP' must be the chord of the arc^ It follows that QS must lie on a plane bisecting NP at right angles and at a distance a from PR where NP=2asm^0, or, which is more convenient, at a distance y from the plane NPP' where N P' = 2y ia.ii ^6 . The rotation d round QS is to bring N to P' and is in the same direction as the rotation 6 round PR. ART. 228.] TRANSLATION AND ROTATION. 189 Hence the distance y must be measured from the middle point of NP' in the direction in which that middle point is moved by its rotation round PR. Having found the only possible position of QS, it remains to show that the displacement of Q is really along QS. The rotation 6 round PR will cause Q to describe an arc whose chord Qq is parallel to P'N and equal to 2a^m.\d. The chord Qq is therefore equal to NP\ and the translation NP' brings q back to its position at Q. Hence Q is moved only by the translation PN, i.e. Q is moved along QS. 226. It follows from this reasoning that any displacement of a body can be represented by a rotation about some straight line and a translation parallel to that straight line. This mode of constructing the displacement is called a screw. The straight line is sometimes called the central axis and sometimes the axis of the screw. The ratio of the translation to the angle of rotation is called the pitch of the screw. A rule to determine the signs is given in Art. 243. 227. The same displacement of a body cannot be constructed by two different screws. For if possible let there be two central axes AB, CD. Then AB and CD by Art. 220 are parallel. The displacement of any point Q on CD is found by turning the body round AB and moving it parallel to AB, hence Q has a displace- ment perpendicular to the plane ABQ and therefore can move only along CD. 228. When the rotations are indefinitely small, the construc- tion to find the central axis may be simply stated thus. Let the displacement be represented by a rotation wdt about an axis PR and a translation Vdt in the direction PP'. Measure a distance V sin P'PR y = ' from P perpendicular to the plane P'PR on that side of the plane towards which P' is moving. A parallel to PR through the extremity of y is the central axis. Ex. 1. Given the displacements AA', BB', CC of three points of a body in direction and magnitude, but not necessarily in position, find the direction of the axis of rotation corresponding to any base point P. Through any assumed point O draw Oa, O^, Oy parallel and equal to AA', BB\ CC. If Op be the direction of the axis of rotation, the projections of Oa, 0^, Oy on Op are all equal, each being the same as the displacement of the base point (Art. 222). Hence Op is the perpendicular drawn from O on the plane a/37. This also shows that the direction of the axis of rotation is the same for all base points. Ex. 2. If in the last example the motion be referred to the central axis, show that the translation along it is equal to Op. Ex. 3. Given the displacements AA', BB' of two points A, B oi the body and the direction of the central axis, find the position of the central axis. Draw planes through AA', BB' parallel to the central axis. Bisect AA', BB' by planes 190 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. perpendicular to these planes respectively and parallel to the direction of the central axis. These two last planes intersect in the central axis. Composition of Rotations and Screws. 229. It is often necessary to compound rotations about axes OA, OB which meet at a point 0. But, as the only case which occurs in rigid dynamics is that in which these rotations are indefinitely small, we shall first consider this case with some par- ticularity, and then indicate generally at the end of the chapter the mode of proceeding when the rotations are of finite magnitude. 230. To explain what is meant by a body having angular velocities about more than one aoois at the same time. A body in motion is said to have an angular velocity o) about a straight line, when, the body being turned round this straight line through an angle codt, every point of the body is brought from its position at the time t to its position at the time t + dt. Suppose that during three successive intervals each of time dt, the body is turned successively round three different straight lines OA, OB, 00 meeting at a point through angles coidt, coodt, w^dt. We shall first prove that the final position is the same in whatever order these rotations are effected. Let P be any point in the body, and let its distances from OA, OB, 00, respec- tively be ri, r-g, r^. First let the body be turned round OA, then P receives a displacement w^r^dt. By this motion let r^ be increased to r^ + dr^, then the displacement caused by the rotation about OB will be in magnitude Wa {'i\ + dr^ dt. But according to the principles of the differential calculus we may in the limit neglect the quantities of the second order, and the displacement becomes w^r^^dt. So also the displacement due to the remaining rotation will be co-^r^dt. And these three results will be the same in whatever order the rotations take place. In a similar manner we can prove that the directions of these displacements will be independent of the order. The final displacement is the diagonal of the parallelopiped described on these three lines as sides, and is therefore independent of the order of the rotations. Since then the three rotations are quite independent, they may be said to take place simultaneously. When a body is said to have angular velocities about three different axes it is only meant that the motion may be determined as follows. Divide the whole time into a number of small in- tervals each equal to dt. During each of these, turn the body round the three axes successivel}^ through angles coidt, (o.jdt, (Osjdt. Then when dt diminishes without limit the motion during the whole time will be accurately represented. ART. 282.] COMPOSITION OF ROTATIONS. 191 231. It is clear that a rotation about an axis OA may be represented in magnitude by a length measured along the axis. This length will also represent its direction if we follow the same rule as in statics. Let OA be the positive direction of the axis defined as in Art. 243. The rotation may be called positive or negative according as it appears to be in some standard direction or the reverse to a spectator placed with his feet at and back along OA. 232. Parallelogram of angular velocities. If tivo an- gular velocities about two axes OA, OB he represented in magnitude and direction by the two lengths OA, OB ; then the diagonal OG of the parallelogram constructed on OA, OB as sides will be the resultant axis of rotation, and its length will represent the magni- tude of the resultant angular velocity. Let P be any point in OC, and let PM, PN be drawn perpendicular to OA, OB. Since OA represents the angular velocity about OA and PM is the perpendicular distance of P from OA, the product OA . PM will represent the velocity of P due to the angular velocity about OA. Similarly OB.PN will represent the velocity of P due to the angular velocity about OB. Since P is on the left-hand side of OA and on the right-hand side of OB, as we respectively look along these directions, it is evident that these velocities are in opposite directions. Hence the velocity of any point P is represented by OA.PM-OB. PiV =OP[OA. sin CO A - OB . sin COB] = 0. Therefore the point P is at rest and OG is the resultant axis of rotation. / z:^^ Let ft) be the angular velocity about OG, then the velocity of any point A in OA is perpendicular to the plane AOB and is repre- sented by the product of co into the perpendicular distance of A from OG = q) . OA sin GO A. But since the motion is also determined by the two given angular velocities about OA, OB, the motion of the point A is also repre- sented by the product of OB into the perpendicular distance of A from OB=OB. OA sin BOA ; ,.p sin BOA ^^ . . (O = 0B. -^—7^7^-7 = OG. sm GOA Hence the angular velo.city about OG is represented in magni- tude by OG. From this proposition we may deduce as a corollary "the parallelogram of angular accelerations." For if OA, OB represent 192 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. the additional angular velocities impressed on a body at any instant, it follows that the diagonal 00 will represent the resultant additional angular velocity in direction and magnitude. 233. This proposition shows that angular velocities and angular accelerations may be compounded and resolved by the same rules and in the same way as if they were forces. Thus an angular velocity w about any given axis may be resolved into two, w cos a and o) sin a, about axes at right angles to each other and making angles a and ^tt — a with the given axis. If a body have angular velocities a)i, 0)3, 0)3 about three axes Ox, Oy, Oz at right angles, they are together equivalent to a single angular velocity o), where &> = s/ay^ + (o^ + w.^, about an axis making angles with the given axes whose cosines are respectively — , — , — . This may be proved, as in the corresponding proposition in statics, by compounding the three angular velocities, taking them two at a time. It will however be needless to recapitulate the several pro- positions proved for forces in statics with special reference to angular velocities. We may use " the triangle of angular velocities'* or the other rules for compounding several angular velocities together, without any further demonstration. 234. The Angular Velocity couple. A body has angular velocities co, w about two parallel axes OA , O'B distant a from each other, to find the residting motion. Since parallel straight lines may be regarded as the limit of two straight lines which intersect at a very great distance, it follows from the parallelogram of angular velocities that the two given angular velocities are equivalent to an angular velocity about some parallel axis 0"G lying in the plane containing OA, O'B. Let X be the distance of this axis from OA, and suppose it 0'^ , [^ B 0" r-i c ^ x\ • a y to be on the same side of OA as O'B. Let H be the angular] velocity about it. Consider any point P, distant y from OA and lying in the plane of the three axes. The velocity of P due to the rotation about OA is (oy, the velocity due to the rotation about O'B is Q)' {y — a). But these two together must be equivalent to the ! velocity due to the resultant angular velocity 12 about 0"C, and. ART. 235.] THE ANALOGY TO STATICS. 193 this is n (2/ — x), .'. (oy + (o'(y-a) = n(y-a;). This equation is true for all values of y, .*. n = (o-\-co', x=a(o'/Cl. This is the same result we should have obtained if we had been seeking the resultant of two forces o), ay' acting along OA, O'B. If co = ~co\ the resultant angular velocity vanishes, but x is in- finite. The velocity of any point P is in this case (oy-\- (o'(y—a)=a(o, which is independent of the position of P. The result is that two angular velocities, each equal to co but tending to turn the body in opposite directions about two parallel axes at a distance a from each other, are equivalent to a linear velocity represented by ao). This corresponds to the proposition in statics that " a couple " is properly measured by its moment. We may deduce as a corollary, that a motion of rotation ay about an axis OA is equivalent to an equal motion of rotation about a parallel axis O'B plus a motion of translation aw perpendicular to the plane containing OA, O'B, and in the direction in which O'B moves. See also Art. 223. 235. The analogy to Statics. To explain a certain analogy which exists between statics and dynamics. All propositions in statics relating to the composition and resolution of forces and couples are founded on these theorems : 1. The parallelogram of forces and the parallelogram of couples. 2. A force F is equivalent to any equal and parallel force together with a couple Fp, where p is the distance between the forces. Corresponding to these we have in dynamics the following theorems on the instantaneous motion of a rigid body : 1. The parallelogram of angular velocities and the parallelo- gram of linear velocities. 2. An angular velocity w is equivalent to an equal angular velocity about a parallel axis together with a linear velocity equal to cop, where p is the distance between the parallel axes. It follows that every proposition in statics relating to forces has a corresponding proposition in dynamics relating to the motion of a rigid body, and these two may be proved in the same way. To complete the analogy it may be stated (i) that an angular velocity like a force in statics requires, for its complete determina- tion, five constants, and (ii) that a velocity like a couple in statics requires but three. Four constants are required to determine the line of action . of the force or of the axis of rotation, and one to R. D. 13 194 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. determine the magnitude of either. There will also be a conven- tion in either case to determine the positive direction of the line. Two constants and a convention are required to determine the positive direction of the axis of the couple or of the velocity and one the magnitude of either. The discovery of this analogy is due to Poinsot. 236. IiJ order to show the great utility of this analogy and how easily we may transform any known theorem in statics into the corresponding one in dynamics, v/e shall place in close juxta- position the more common theorems which are in continual use both in statics and dynamics. It is proved in statics that any given system of forces and couples can be reduced to three forces X, Y, Z, which act along any rectangular axes which may be convenient and which meet at any base point we please, together with three couples which we may call L, M, N and which act round these axes. A simpler representation is then found, for it is proved that these forces and couples can be reduced to a single force which we may call R and a couple G which acts round the line of action of R. This line of action of R is called the central axis. There is but one central axis corresponding to a given system of forces. The term wrench has been applied to this representation of a given system of forces. Draw any straight line AB parallel to the central axis at a dis- tance c from it. Then we may move R from the central axis to act along AB at A, provided we introduce a new couple whose moment is Re. Combining this with the couple G, we have for the new base point A a new couple G' = s/G'^-\- R^c^, the force being the same as before. The couple G' is a minimum when c = 0, i.e. when AB coincides with the central axis. By taking moments round AB we see that the moment of the forces round every straight line parallel to the central axis is the same and equal to the minimum couple. The same train of reasoning by which these results were ob- tained will lead to the following propositions. The instantaneous motion may be reduced to a linear velocity of any base point we please and an angular velocity round some axis through the base. These are then reduced to an angular velocity which we may call O about an axis called the central axis, and a linear velocity along that axis which we may call V. The term screw has been applied to this representation of the motion. Draw any straight line AB parallel to the central axis. Then we may move n from the central axis to act round AB, provided that we intro- duce a new linear velocity represented by He. Combining this with the velocity V we have for the new base A (which is any point on AB) a new linear velocity V = ^/V^-\-c^fl^, the angular velocity being the same as before. The linear velocity V is a minimum I ART. 238.] THE VELOCITY OF ANY POINT. 195 when c = 0, i.e. when AB coincides with the central axis. We see that the linear velocity of any point A resolved in the direction AB, i.e. parallel to the central axis, is always the same and equal to the minimum velocity of translation. It will be seen that most of these results have already been obtained in Arts. 219 to 228 for finite rotations. 237. Another useful representation depends on the following proposition. Any system of forces can be replaced by some force F which acts along a straight line which we may choose at pleasure, and some other force F' which acts along some other line and does not in general cut the first force. These are called conjugate forces. The shortest distance between these is proved in statics to intersect the central axis at right angles. The directions and magnitudes of the forces F, F' are such that B, would be their resultant if they were moved parallel to them- selves, so as to intersect the central axis. Also it is known that, if 6 be the angle between the directions of the forces F, F' and a the shortest distance between them, FF'a sin 6 = GR. If the arbitrary line of action of F is such that the moment of the forces about it is zero, both F and F' act along that line in opposite directions and the magnitude of each is infinite. By help of the analogy we may obtain the corresponding propositions in the motion of a body. Any motion may be repre- sented by two angular velocities, one co about an axis which we may choose at pleasure and another o)' about some axis which does not in general cut the first axis. These are called conjugate axes. The shortest distance between these intersects the central axis at right angles. These angular velocities are such that 12 w^ould be their resultant if their axes were placed parallel to their actual positions, so as to intersect the central axis. If 6 be the angle between the axes of oj, w and a be the shortest distance between these axes, then ww'a sin S — VCi. If the arbitrary axis of ft) is such that the velocity of every point of the axis resolved along the axis is zero (Art. 137), the angular velocities o), oy' have a common axis, opposite signs and the magnitude of each is infinite. 238. The velocity of any Point. The motion of a body during the time dt may be represented, as explained in Art. 219, by a velocity of translation of a base point 0, and an angular velocity about some axis through 0. Let us choose any three rectangular axes Ox, Oy, Oz which may suit the particular pur- pose we have in view. These axes meet in and move with 0, keeping their directions fixed in space. Let u, v, w be the resolved parts along these axes of the linear velocity of 0, and (d^, (Oy, coz, the resolved parts of the angular velocity. These angular velo- cities are supposed positive when they tend the same way round the axes that positive couples tend in statics. Thus the positive 13—2 196 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. directions of cox^ (Oy, coz, are respectively from y to 2, from 2 to cc and from x to y. The whole motion during the time dt of the body is known when these six quantities u, v, w, w^, (Oy, coz are given. These six quantities may he called the components of the motion. We now propose to find the motion of any point P whose coordinates are x, y, z. Let us find the velocity of P parallel to the axis of z. Let PN be the ordinate of z and let PM be drawn perpendicular to Ox. The velocity of P due to the rotation round Ox is clearly co^PM. Resolving this along NP we get (o^PM sin NPM = co^y. Similarly that due to the rotation about Oy is — WyX and that due to the rotation about O2 is zero. Adding the linear velocity z^ of the origin, we see that the whole velo- city of P parallel to Oz is W' = W-\- (O^y — OOyX. Similarly the velocities parallel to the other axes are U =U + COyZ V =^V + COzX (Ozy, co^z. . 239. It is sometimes necessary to change our representation of a given motion from one base point to another. These formulae will enable us to do so. Thus suppose we wish our new base point to be at a point 0', the axes at 0' being parallel to those at 0. Let (^, rj, f) be the coordinates of 0' and let u\ v\ w' , wj, coy, ft)/ be the linear and angular components of motion for the base 0'. We have now two representations of the same motion, both these must give the same result for the linear velocities of any point P. Hence u + OyZ - wzy = u' + wy (z -^)- coz {y - v\ V + CO2X — COxZ = V' + COz (^ — ?) — (j^x (^ — 0> w + co^y- coyx = w'-\- coJiy - v) - «/(« - IX must be true for all values of x, y, z. These equations give coj = (Oa COz =(^z so that what- ever base is chosen the angular velocity is always the same ii direction and magnitude. See Art. 221. We also see that u', v\ are given by formulae analogous to those in Art. 238, as inde( might have been expected. ART. 241.] SCREWS. 197 The reader should compare these with the corresponding for- mulae in statics. If all the forces of any system be equivalent to three forces X, F, Z acting at a base point along three rect- angular axes together with three couples round those axes, then we know that the corresponding forces and couples for any other base point ^, t], f are Y' = Y, M' =M+Z^-X^, Z' = Z, N' =N+X'n-Yl 240. To find the equivalent Screw. The motion being given by the linear velocities (u, v, w) of some base 0, and the angular velocities, {w^, coy, coz), find the central axis, the linear velocity along it and the angular velocity round it, i.e. find the equivalent screw. Let P be any point on the central axis, then if P were chosen as base, the components of the angular velocity would be the same as at the base 0. If then Q be the resultant of the anovular velocities cox, coy, coz (Art. 233) we see that (1) The direction-cosines of the central axis are (Ox n ^V ^z cosa = ^, cos^ = J, cos7 = ^. (2) The angular velocity about the central axis is fl. (3) The velocity of every point resolved in a direction parallel to the central axis is the same and equal to that along the central axis. See Art. 222 or Art. 236. If then V be the linear velocity along the central axis we have V== u cos a + v cos /S -\-w cos 7; .'. Vfl = U(Ox + vcoy-{-W(Oz. (4) Let (x, y, z) be the coordinates of P, i.e. of any point on the central axis. Then the linear velocity of P is along the axis of rotation. Hence U -\- (OyZ — (Ozy _ '^ + 2, &c. and any linear velocities v^ , v^ , &c. is the sum of the separate invariants of the con- stituents taken two and two, or written in an algebraic form ■ 1= 2wy cos + 2wcoV sin 6, where is the angle between the direction of any linear velocity v and the axis of any angular velocity w, while d is the angle between the axes of any two angular velocities w, w' and r the shortest distance. Taking any rectangular axes each of the six components of these motions is a linear function of Wj , w^ , &c. ; v^ , v.^ , &c. The invariant I is therefore a quadratic function of the form 7= ^iiWj^ + 4 J2W1W2 + &c. + -BjjWjVi + B^^ui-^Vo + &c. + C^^v{^ + C-^^ViV,^ + &c. , where the coefficients are independent of the magnitudes of w^, Wg, &c., y^, v^, &g. Putting all the constituents equal to zero except each in turn we see that A-^-^ = 0, &c. = 0; Cii = 0, &c. = 0. Then putting all the constituents equal to zero except two in turn and comparing the results with those given in Ex. 1, we see that the other coefficients have the values given above. Ex. 3. The invariant I of two screws (w, v), (w', v') is 1= 03V + wV + {wv' + (j}'v) cos d + ww'r sin d. To prove this we add together the six invariants of the four constituents w, w', V, v' taken two and two together. 242. When the motion is equivalent to a simple rotation, it may be required to find the axis of rotation. But this is obviously only the central axis under another name, and has been found above. ART. 244] SCREWS. 199 243. A screw motion may thus he given in two ways. We may have given the six components of motion, which we have called {u, v, tu, Wx, (Oy, w^, which also depend on the point chosen as base. Or it may be given by the equations to the central axis, the velocity V along it, and the angular velocity O round it. In this last case a convention is necessary to prevent confusion as to the directions implied by the velocities V and fl. One direction of the axis is called the positive direction, and the opposite the negative direction. Then V is taken positive when it implies a velocity in the positive direction. So also fl is positive when the rotation appears to be in some standard direction, say clockwise, when viewed by a person placed with his back along the axis, so that the positive direction is from his feet to his head. This of course is only the ordinary definition of a positive couple as given in statics. See Art. 231. The method of determining the positive direction of the axis is easy to understand, though it takes long to explain. Describe a sphere of unit radius with its centre at the origin, and let the positive directions of the axes cut this sphere in x, y, z. Let a parallel to the central axis drawn through the origin cut the sphere in L and L'. Let the direction-cosines of the axis be given say, I, m, n. Then {I, m, n) are the cosines of certain arcs drawn on the sphere which begin at xyz, and terminate say at L, while (— I, — m, —n) are the cosines of supplementary arcs which begin at the same points xyz, and terminate at L'. Then OL is the positive direction of the axis and OL' the negative direction. With this understanding the angle between two axes is the angle between their positive directions and is determined without ambiguity of sign when the actual direction-cosines of the axes are given. 244. The position of the central axis being given, together with the linear velocity along it and the angular velocity round it, it is required to find the components of the motion when the origin is taken as the base. This is of course the converse proposition to that just discussed. Let the equation to the central axis be — r^^ = - — - = , *■ I m n where {Imn) are the actual direction-cosines of the axis. Let Fbe the linear and H the angular velocity. If {fgh) were taken as the base, the components of the linear velocities would hQlV,mV,n V, and the components of the angular velocities would be /H, mil, nfl. Hence by Art. 238, writing — /, — g, — h for X, y, z, the components of the motion when the origin is the base point are u= lV — n{mh — ng), (Ox = lO., V = mV— O (nf — Ih), coy = mfl, w — nV — VL{lg —mf), Wz^nQ.. 200 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. 245. Composition and Resolution of Screws. Given two screw motions to compound them into a single screw and conversely given any screw motion to resolve it into two screws. Two screws being given, let us choose some convenient base and axes. By Art. 244 we may find the six components of motion of each screw for this base. Adding these two and two, we have the six components of the resultant screw. Then by Art. 240 the central axis together with the linear and angular velocities of the screw may be found. Conversely, we may resolve any given screw motion into two screws in an infinite number of ways. Since a screw motion is represented by six components at any base we have in the two screws twelve quantities at our disposal. Six of these are required to make the two screws equivalent to the given screw. We may therefore in general satisfy six other conditions at pleasure. Thus we may choose the axis of one screw to be any given straight line we please with any linear velocity along it and any angular velocity round it. The other screw may then be found by reversing this assumed screw and joining it thus changed to the given motion. The screw equivalent to this compound motion is the second screw, and it may be found in the manner just explained. Or again, we may represent the motion by two screws whose pitches are both chosen to be zero, the axis of one being arbitrary. These are the conjugate axes spoken of in Art. 237. 245 a. The following viethod of compounding two screws is very convenient when the shortest distance between the axes is knoivn in position and magnitude. Let (w, v), (w', v'} represent the angular and linear velocities of the two given screws, (fi, V) those of the resultant screw. Then, by equating the invariants, ^V=uv + o}'v'+ [icv' + (a'v) cos 6 + u(a'r Bin 6, Q^=u}'^ + w'- + 2ww' cos d, where 6 is the inclination of the axes and r the shortest distance. We shall next show that the axis of the resultant screw intersects at right angles the shortest distance AA' betioeen the axes of the given screws. Since the central axis is parallel to the resultant of w, w' transferred to any base, that axis must be perpendicular to AA'. Also since AA' X intersects at right angles the axes of both I V "y '^ *^6 given screws, the velocity of every point of AA' resolved along itself is zero. Hence, since AA' is perpendicular to the central axis of the resultant screw, it must also intersect that axis. Lastly we shall show that the distance ^ of the central axis of the two screws from the middle point G of the shortest distance is given by n-^ = ^r (w2 - w'2) + (wu' _ u'v) sin d, A.RT. 246.] COMPOSITION OF SCREWS. 201 iclicre ^ is measured positively towards lo. Let Ct] be a perpendicular to the plane 3oiitaining A A' and the required central axis Oz. Equating the resolved part along Ct] of the velocity of C due to the two screws to that due to the resultant screw we have —Q^ = v sin y — v' sin y' — ^rw cos y + ^roj' cos y', where y, y' are the angles the axes AF, A'F' of the given screws make with the central axis Oz. By resolution we have fi sin 7 = w' sin Q, fi cos 7 = w + w' cos Q, fi sin7' = w sin ^, f2 cos7' = w' + w cos ^. The result follows by substituting for 7, 7'. 246. Examples. Ex. 1. The locus of points in a body moving about a fixed point which at any instant have the same resultant velocity is a circular cylinder. Ex. 2. If radii vectores be drawn from a fixed point O to represent in direction and magnitude the velocities of all points of a rigid body in motion, prove that the extremities of these radii vectores at any one instant lie in a plane. [Coll. Exam. This plane is evidently perpendicular to the central axis, and its distance from measures the velocity along the axis. Art. 228, Ex. 1. Ex. 3. The locus of the tangents to the trajectories of different points^of the same straight line in the instantaneous motion of a body is a hyperbolic paraboloid. Let AB be the given straight line, CT> its conjugate. The points on AB are turning round CD, and therefore all the tangents pass through two straight lines, viz. AB and its consecutive position A'B' , and are also parallel to a plane which is perpendicular to CD. Ex. 4. Let the restraints on a body be such that it admits of two motions A and B, each of which may be represented by a screw motion, and let w, m! be the pitches of these screws. Then the body must admit of a screw motion compounded of any indefinitely small rotations ladt, ddt about the axes of these screws accom- panied of course by the translations mojdt, m'co'dt. Prove that (1) the locus of the axes of all these screws is the surface z {x"^ + y^) = 2axy. (2) If the body be screwed along any generator of this surface the pitch is c + a cos 2d, where c is a constant which is the same for all generators and 6 is the angle the generator makes with the axis of X. (3) The size and position of the surface being chosen so that the two given screws A and B lie on the surface with their appropriate pitch, show that only one surface can be drawn to contain two given screws. (4) If any three screws of the surface be taken and a body be displaced by being screwed along each of these through a small angle proportional to the sine of the angle between the other two, the body after the last displacement will occupy the same position that it did before the first. This surface has been called the cylindroid by Sir E. Ball, to whom these four theorems are due. See his Theory of Screws. Ex. 5. An instantaneous motion is given by the linear velocities {u, v, w) along, and the angular velocities (w^, w^, wj round the coordinate axes. It is required to represent this by two conjugate angular velocities, one being about the arbitrary straight line — p- = - — - = . If Q be the angular velocity about the given axis, then tfW™ + V()},j + ICU}, — ^ -r = lu + mv + nio it where {I, m, n) are the actual direction-cosines. I, m, 202 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. The equations to the conjugate axis are y, I, m, n = lu + mv + nw, X, y, : (/- x)u+{g-y)v + {h- z) w. f, 9, h I The first of these equations may be obtained as indicated in Art. 245. Reverse Q and join it to the given motion, then the invariant of this compound motion vanishes. If the angular velocity fl be thus supposed known, the conjugate axis is the central axis of the compound motion and may be found as in Art. 245. But if the conjugate axis be required independently of S2, we may use the second and third equations. The second equation follows from the fact that the direction of motion of any point on the conjugate is perpendicular to the given axis. The third follows from the fact that the direction of motion is also perpendicular to the straight line joining the point to (/, g, h). These general equations will be simplified if the circumstances of any problem permit the coordinate axes to be so chosen that some of the constants may be zero. Thus, if the central axis of the instantaneous motion is taken as the axis of z and the shortest distance between that axis and the given straight line as the axis of a;, we have u = 0, v = 0, w^j^O, Wy = 0; ^7 = 0, h=0, and Z = 0. The equations then , wu}. ^ mo zw become -—^=ivn+fu,m, x= , y=--—. Referring to the figure of Art. 245a, / is the shortest distance OA between the given axis AF and the central axis OZ, and 7i = cos y, m = sin y where 7 is the angle AF makes with OZ. There is an apparent exception to these results when the given motion and the given axis are such that U, as found from the first equation, is infinite. This is a limiting case rather than an exception. It is easy to see that both the second and third equations are, in this case, satisfied by substituting x=f+lt, y = g + vitf z = h + nt; i.e. the conjugate axis coincides with the given axis. If fi' be the angular velocity about the conjugate axis, and S2' are together equivalent to the resultant angular velocity of the given motion ; it follows that Q' is also infinite. In this limiting case, therefore, the motion is represented by two infinite opposite angular velocities about two coincident lines. Another limiting case is when the given axis is parallel to the central axis of the given motion and the invariant of the motion is not zero. In this case I, m, 11 are proportional to w^., Wj,, w^, and the second equation represents a plane at infinity. The conjugate axis is therefore at infinity and the angular velocity about it is zero. There is a third limiting case when the invariant of the given motion is zero. If the given motion is a simple rotation about some axis, say Oz, and the given axis is not parallel to Oz and does not intersect it, fi = and the conjugate axis coincides with Oz. If the given axis is parallel to Oz or intersects it, 12 may have any value and the conjugate axis is the resultant axis of the given rotation and the reversed 12. If the given motion is a simple translation parallel to some axis Oz and the given axis is not perpendicular to Oz, 12 = and the conjugate is at infinity. If the given axis is perpendicular to Oz, 12 may have any value, and the conjugate axis is found as before ; see Art. 234. ART. 247.] COMPOSITION OF SCREWS. 203 In discussing these limiting cases analytically, it will be convenient to choose the simplified form of axes described above. Ex. 6. If one conjugate of an instantaneous motion is at right angles to the central axis the other meets it, and conversely. If one conjugate is parallel to the central axis the other is at an infinite distance, and conversely. The invariant is supposed to be finite, Ex. 7. A body is moved from any position in space to any other, and every point of the body in the first position is joined to the same point in the second position. If all the straight lines thus found be taken which pass through a given point, they will form a cone of the second order. Also if the middle points of all these lines be taken, they will together form a body capable of an infinitesimal motion, each point of it along the line on which the same is situated. Cayley's Report to the British Assoc, 1862. 247. Characteristic and focus. If tbe instantaneous motion of a body be represented by two conjugate rotations about two axes at right angles, a plane can be drawn through either axis perpendicular to the other. The axis in the plane has been called the characteristic of that plane, and the axis perpendicular to the plane is said to cut the plane in its focus. These names were given by M. Chasles in the Comptes Rendus for 1843. Some of the following examples were also given by him, though without demonstrations. Ex. 1. Show that every plane has a characteristic and a focus. Let the central axis cut the plane in 0. Resolve the linear and angular velocities in two directions Ox, Oz, the first in the plane and the second perpendicular to it. The translations along Ox, Oz may be removed if we move the axes of rotation Ox, Oz parallel to themselves, by Art. 234. Thus the motion is represented by a rotation about an axis in the plane and a rotation about an axis perpendicular to it. It also follows that the characteristic of a plane is parallel to the projection of the central axis. Ex. 2. If a plane be fixed in the body and move with the body, it intersects its consecutive position in its characteristic. The velocity of any point P in the plane when resolved perpendicular to the plane is proportional to its distance from the characteristic, and when resolved in the plane is proportional to its distance from the focus and is perpendicular to that distance. Ex. 3. If two conjugate axes cut a plane in F and G, then FG passes through the focus. If two conjugate axes be projected on a plane, they meet in the characteristic of that plane. Ex. 4. If two axes CM, ON meet in a point C, their conjugates lie in a plane whose focus is C and intersect in the focus of the plane CMN. This follows from the fact that if a straight line cut an axis the direction of motion of every point on it is perpendicular to the straight line only when it also cuts the conjugate. Ex. 5. Any two axes being given and their conjugates, the four straight lines lie on the same hyperboloid. Ex. 6. If the instantaneous motion of a body be given by the linear and angular velocities {u, v, lo), (w^, w^, wg), prove that the characteristic of the plane Ax + By + Cz + D = is its intersection with A {u + ca.^^z - w^y) + B {v + u^x - u-^z) + C {w + w-^y - w^) = 0, and its focus may be found from ? ^ = ^ ~ ^ = ^ ^ . A B G \ 204 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. For the characteristic is the locus of the points whose directions of motion are perpendicular to the normal to the plane, and the focus is the point whose direction of motion is perpendicular to the plane. When the central axis of the instantaneous motion is the axis of z, the coordinates of the focus are x=pBIC, y- -pAjC, z= -DjG and the characteristic lies on the iplaine- Ay + Bx+Cp = 0, where p is the pitch tvjb}.^ of the instantaneous motion. Ex. 7. The locus of the characteristics of planes which pass through a given straight line is a hyperboloid of one sheet ; the shortest distance between the given straight line and the central axis being the direction of one principal diameter, and the other two being the internal and external bisectors of the angle between the given straight line and the central axis. Prove also that the locus of the foci of the planes is the conjugate of the given straight line. Ex. 8. Let any surface A be fixed in a body and move with it, the normal planes to the trajectories of all its points envelope a second surface B. Prove that if the surface B be fixed in the body and move with it, the normal planes to the trajectories of its points will envelope the surface A : so that the surfaces A and B have conjugate properties, each surface being the locus of the foci of the tangent planes to the other. Prove that if one surface is a quadric the other is also a quadric. Moving Axes and Eulers Equations. 248. It has been shown in Art. 230 that when a body is moving about a fixed point the displacement in the time dt may be constructed by turning the body round three straight lines OA, OB, 00 through certain angles coidt, co^dt, w-^fiLt In the same way we may construct the displacement during the next interval* dt by rotating the body round three other straight lines OA', OB', 00' through certain other angles coi'dt, w^dt, co^dt. If these two systems of axes are infinitely close and the motion of the body is continuous, the angular velocities «/, &c., will differ from Wi, &c., by infinitely small quantities. The axes of reference are then called moving axes. It should be noticed that co^dt measures the angle of rotation round Oz, not relatively to the moving plane which contains OA and OC, but with reference to some plane fixed in space passing through the instantaneous position of 00. 249. Let Ox, Oy, Oz be the rectangular axes fixed in space and let Wa,, coy, Wz be the components of the angular velocity of a body at the time t. Let OA, OB, 00 be three rectangular axes moving about the fixed point and let q)i, co^, (0.3 be the com- ponent angular velocities of the same body at the same time. If these two systems of axes coincide in position at the time t, (^i = (*^x> 0)2 = (Oy, 0)3 = 0)2, but at the time t-\-dt the two systems will have separated and we can no longer assert that 0)3 + day^ and (Oz + d(Oz are necessarily equal. We shall now show that if the moving axes are fixed in the body, then d(03 = d(0z as far as the first order of small quantities. Le I ART. 250.] euler's equations. . 205 OR, OR be the resultant axes of rotation of the body at the times t and t + dt, i.e. let a rotation D.dt about OR bring the body into the position in which 00 coincides with Oz at the time t ; and let a further rotation Vl'dt about OR' bring the body into some adjacent position at the time t-^dt while in the same interval dt, 00 moves into the position OC. Then according to the definition of a differential coefficient d(o^ ,. . n' cos R'C'-n cos RG -di = ^^"^' dt ' dcoz T . , n' cos R'z — O cos Rz ■rf^ = i""^* dt • The angles RC and Rz are equal by hypothesis. Since 00 is fixed in the body, it makes a constant angle with OR' as the body turns round OR', the angles R'G' and Rz are therefore equal. Hence these differential coefficients are also equal. 250. The preceding proposition is a particular case of a fundamental theorem in the theory of moving axes. This general- ized theorem applies not merely to angidar velocities hut to any vector or directed quantity which obeys the parallelogram-law. By Art. 215 the moving system of axes is turning round some instantaneous axis with an angular velocity which we may call 6. Let ^1, ^2> ^3 be the components of 6 about the axes OA, OB, 00. Then in ^, the figure 6^ represents the rate at y^\ --^^ which any point in the circular arc BC / \ ^V is moving along that arc, ^2 is the rate / | \ at which any point of GA is moving f^\ [ \ along GA and so on. / y'O "---AA Let F„ F„ V, and F,, V^,Y, be il^^ ^ ^^ the components of any vector with re- 2^^--^^ gard to the moving axes OA, OB, 00, and the fixed axes Ox, Oy, Oz respectively. Let a, ^, 7 be the direction angles of Oz referred to OA, OB, 00 \ then Vz = Fi cos a + F2 cos /3 4- F3 cos 7, .•. Vz= Fi cos a + F2 cos /3 + Fg C0S7 — Fisinaa— FasiuyS/S — F3sin77. Let the axis Oz coincide with 00 at the time t, then a = Jtt, 13 = ^TT, 7 = 0. Hence Vz=V,-V,d-V,$. Now a is the angular rate at which the axis OA is separating from a fixed line Oz momentarily coincident with OG, hence d = O^. Similarly /3z= — 0^. Writing the theorem at full length we have 206 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. Similarly dV, dt dV, dt -FA + VA, dVy_ dt dV, dt -VA + VA- Let the vector V be the resultant angular velocity H of a body about an instantaneous axis (Art. 233) then Fj = a)i, ^2 = 0).,, Vs = 0)3 while Vx = cox, Vy^coy, Vz= (o^. We have therefore dwz dcos ^ , n — = ^-^A + 3 = Wg. Another very simple proof is given in the chapter on moving axes at the beginning of Vol. 11. of this treatise. 252. Euler's dynamical equations. To determine the general equations of motion of a body moving about a fixed point 0. Let X, y, z be the coordinates of any particle m referred to axes Ox, Oy, Oz fixed in space. Taking moments about the axis of z we have by D'Alembert's principle 2m {xy — yx) = iV". Let (Oxy o)y, (Oz be the angular velocities of the body about the axes, then x = (OyZ — cozy, y = (OzX — Q)xZ, z = cDxy — coyX ', .'. X = ZWy — yWz + COy {Wxy ~ (^yX) — COz {cOzX — 0)xZ), y = XCOz — ZiOx + (£>z (0)yZ — Wzy) — COx (cOxy — COyX). These we shall presently substitute in the equation of moments. VRT. 254.] euler's equations. 207 Let ft)i, W2, 0)3 be the angular velocities of the body about three (Ctangular axes OA, OB, 00 fixed in the body. Let these 3oincide with the axes fixed in space at the time t; then q)i = o)x, u, = Q)y, cos = (Oz; 6)1 = a)x, 0)2 = coy, 6)3 = coz, by Art. 249. The advantage of using axes fixed in the body is that the noments and products of inertia are then constants. If we choose xs these axes of coordinates the principal axes at the fixed point, ^vc have the additional simplification that all the products of inertia are zero. In substituting for x, y in the equation of moments we may therefore omit all the terms of x which do not contain y and all the terms of y which do not contain x. We til us have ^m {x^ + y"^) 0)3 + 2m {x"^ — y^) cDitOg = N. If A, B, G he the principal moments of inertia at the fixed point 0, this becomes C^-(A-B)co,co, = K ctt limilarly A^ - (B - 0) co,co, = L, B ^^ -{C-A) co.co, = M. 'hese are called Euler's dynamical equations. 253. We know by D'Alembert's principle that the moment ■ of the effective forces about any straight line is equal to that of the impressed forces. The equations of Euler therefore indicate i that the moments of the effective forces about the principal axes : at the fixed point are expressed by the left-hand sides of the above equations. If there is no point of the body which is fixed in space, the motion of the body about its centre of gravity is the same as if that point were fixed. In this case, if A, B, be the principal moments at the centre of gravity, the left-hand sides of Euler's equations give the moments of the effective forces about the principal axes at the centre of gravity. If we want the moment about any other straight line passing through the fixed point, we may find it by simply resolving these moments by the rules of statics. Ex. 1. If 2T = Aioi' + Bu2^ + Cu}^^ and G be the moment of the impressed forces dT about the instantaneous axis, Q the resultant angular velocity, then -r-=GQ, Ex. 2. A body (say the earth) turning about a fixed point is acted on by forces (such as the attractions of the sun or moon) which tend to produce rotation about an axis at right angles to the instantaneous axis, show that the angular velocity cannot be uniform unless either two of the principal moments of inertia at the fixed point are equal or the instantaneous axis always lies in a principal plane. The axis about which the forces tend to produce rotation is that axis about which it would begin to turn if the body were placed at rest. 254. To determine the pressure on the fixed point Let X, y, z be the coordinates of the centre of gravity referred 208 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. to rectangular axes fixed in space meeting at the fixed point, and let P, Q, R be the resolved parts of the pressures on the body in these directions. Let /t be the mass of the body. Then we have /jLX = P + SmX and two similar equations. Substituting for x its value in terms of (Ox, (Oy, (Oz we have fjL {zWy — ydiz + (Dy {(Dxy — COyX) — (Oz {(OzX — (O^z)] = P + IlllX and two similar equations. If we now take the axes fixed in space to coincide with the principal axes at the fixed point at the moment under considera- tion we may substitute for (6y and (Oz from Euler's equations. Hence with similar expressions for Q and R. 255. Ex. If G be the centre of gravity of the body, show that the terms on the left-hand sides of the equations which give the pressures on the fixed point are the components of two forces, one fi^ . (jn parallel to GH which is a perpendicular on the instantaneous axis 01, ft being the resultant angular velocit}^ and the other ft'2 . GK perpendicular to the plane OGK, where GK is a perpendicular on a line OJ ;,-.■' .• 1 . -6-C C-A A-B whose direction-cosines are proportional to — v— WoWo, — ^— WoWi, — pp-WjOJo, A a Li and Q'-^ is the sum of the squares of these quantities. 256. Euler^s geometrical equations. To determine the geometrical equations connecting the motion of the body in space with the angidar velocities of the body about tJie three moving axes, OA, OB, 00. Let the fixed point be taken as the centre of a sphere of radius unity ; let X, Y, Z and ^, P, be the points in which the sphere is cut by the fixed and moving axes respec- tively. Let ZC, BA pro- duced if necessary, meet in E. Let the angle Z^C^^/r, ZC=d, EGA = (^. It is required to determine the geometri- cal relations between 6^ (p, '\jr, and (Oi, &)2, ft)3- Draw ON perpendi- cular to OZ. Then since yjr is the angle the plane COZ makes with a plane XOZ fixed in space the. 1 I ART. 257.] euler's equations. 209 velocity of G perpendicular to the plane ZOC is CN -^ , which is the same as sin^-^, the radius 00 of the sphere being unity. JAlso the velocity of along ZG is --,-. Thus the motion of is [represented by -77 and sin^-^ respectively along and perpendi- jcular to ZG. But the motion of is also expressed by the angular 'velocities (o^ and w^ respectively along BG and GA. These two representations of the same motion must therefore be equivalent. Hence resolving along and perpendicular to ZG we have — = ftjj sm

/dt into the equation, and it will generally be found more convenient to retain (03. In this way the angular momenta of a uniaxal body about any straight lines are expy^essed in terms of the direction-angles of the axis of the body and the angular velocity about it. Secondly, instead of the unsymmetrical coordinates 6, ^J/ we may use the direc- tion-cosines ^, T], ^ of the axis of the body. Following the rule of Art. 76 we shall replace the body by a system of equimomental particles. Suppose we attach to the axis OG one or more imaginary particles so that their united moment of inertia about any axis through perpendicular to OC is equal to A. Let these particles move about with the axis. The motion of the axis is given by the angular velocities Wj, 0^2 and therefore the angular momenta of these particles about the axes OA, OB are clearly Aui, Aco^. These are the same as those of the body. Tlie angular momentum of the particles about OC is zero. Hence the angular momenta of the body about OA, OB, OC are the same as those of the particles together with an angular momentum Cw^ about OC. It follows by the "parallelogram law" that the same equality holds for all axes. Hence the angular momentum of a uniaxal body about any axis through is the same as that of one or more particles arranged along its axis of figure {so that tlj^eir united moment of inertia about a perpendicular axis through is equal to A) together with the angular momentum Cwg about the axis of figure. 216 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. Let a single particle be placed on the axis of the body at a distance unity from the origin. Its mass is therefore represented by ^. Let (^Tjf) be the coordinates of this particle referred to the axes x, y, z, then {^tj^) are also the direction-cosines of the axis. The angular momenta about the axes of coordinates are therefore *>=Hvf-4l)+''-« "-^(f|-40+^-"" -^(4J- di + Gu,.t. If we wish to use 6, 0, \p instead of the direction-cosines ^, 77, ^ we write for ^, 17, f their values ^ = sin ^ cos 1/', T7=:sin ^sini/', ^=cos^. The substitution in the last equation is easily effected if we remember the rule in the differential calculus ^dri-7jd^ = r'^d\f/. See Art. 77. We then arrive at the same results for the angular momenta h-y, h^, h^ as before. If the uniaxal body is making small oscillations and the axis OC is always so nearly coincident with the axis Oz that we can reject the squares of d, we have dv ^ u ■, . d^ f^i=-^-:d+^<^s^^ h=A£ + CiOsV, These are very simple formulsG for the angular momenta about the fixed axes. If the body is moving freely in space we use the centre of gravity instead of the fixed point. In this case it is convenient to attach to the axis tioo equal particles at equal distances on each side of the centre of gravity, so that the centre of gravity of the imaginary system is the same as that of the body. The angular momentum of the free body about any straight line is then the same as that of the system of particles together with the couple Cw.^ about the axis. Ex. 1. A body not necessarily uniaxal is turning about a fixed point 0. Three particles are attached to the principal axes at such distances a, b, c from that Ma'^ = l{B + G-A), Mb'^ = ^{C + A-B), Mc^ = i {A+B - C). Prove that the angular momentum of the body about any straight line through is equal to that of these particles. This follows at once from Art. 76. Ex. 2. A rod is constrained to remain on the surface of a smooth cone of revolution having its vertex at the point of suspension of the rod. Show that the angular motion of the rod round the axis of the cone is the same as that of a simple pendulum of length fa sin a/sin j3 where a is the length of the rod, a the semiverticaj angle of the cone and /3 the angle the axis of the cone makes with the vertical. [St John's Coll. To find the moments of the effective forces, collect the mass at an equimomental point. To find the moments of the impressed forces collect the mass at the centre of gravity. Equating the moments about the axis of the cone the result follows at once. Ex, 3. A body is turning about a fixed point and has all its principal moments of inertia at equal. If 6, 0, i/' be the Eulerian coordinates of the axes OA, OB, OC, fixed in the body, show that the angular momenta about the axes ART. 267.] EXPRESSIONS FOR ANGULAR MOMENTUM. 217 fixed in space are respectively fdyp d), K-A (cos i/z^ + sin ^sirn/'^), h^ = A [~ +cos^-^j. 267. Ex. 1. The motion of a body is given by the linear velocities {u, v, id) of the centre of gravity and the angular velocities (w^., Uy, w^), prove that the angular J* ^ 21 flf Z ?l momentum about the straight line — r— = = is equal to ^ I m n Ih^ + mh^ + nh^ + M I, m, n U, V, 10 f, g. h where M is the mass of the body, h^, h^, h-^ have the values given in Art. 262, and {I, m, n) are the actual direction-cosines of the given straight line. This may be done by the use of the principle proved in Art. 75. The angular momentum about a parallel to the given axis is clearly lh-^^-\-mh^-\-nhy We must now find the angular momentum of the whole mass collected at the centre of gravity round the given straight line and add these two results together. Referring to the figure in Art. 238, let P be the point [fgh). Let us find the angular momentum about a set of axes parallel to the given coordinate axes with P for origin. It is clear that NP produced will be the new axis of z. The moment of the velocity of the origin about NP is seen to be u. MN-v . OM, which is the same as ug - vf ; this tends in the positive direction round NP. Similarly the moments of the velocities of about the parallels to x and y will be vh-wg and wf-uh. If we multiply these three by (?i, I, m) respectively, we have the moment of the velocity of the centre of gravity about the straight line. Multiplying this by 31 we have the angular momentum of the mass at the centre of gravity. The required result follows at once. Ex. 2. To find the angular momentum of a body about the instantaneous axis and also about any perpendicular axis lohich intersects the instantaneous axis. Taking the instantaneous axis for the axis of z, we may use the expressions for 7?i, h^, 7*3 given in Art. 262. In our case 0)3. = 0, a>y = 0, and w^=0, where fl is the resultant angular velocity of the body. The angular momenta about the axes of x, y, z are therefore respec- tively \= - {"Lrnxz) 12, h^=- {^myz) Q, h^^'Zm (x^ + y^) fi. It appears therefore that the angular momentum about any straight line Ox perpendicular to the instantaneous axis Oz is not zero unless the product of inertia about those two axes is zero. To understand this properly we must remember that the angular velocities w^, u}y, u}g are used merely to construct the motion of the body during the time dt. Referring to the figure of Art. 238, let Oz be the instantaneous axis, then the particle of the body at P is moving perpendicular to the plane PLO, and therefore the direction of its velocity is not parallel to Ox and does not intersect Ox. The velocity of this particle has therefore a moment about Ox, although Ox is perpen- dicular to the instantaneous axis. Let d be the angle PMN, r = PM, then ?-2^ = yz - zy = r^w^ - xziOg — xyojy , so that the angular velocity d of the particle P about Ox vanishes when Wj. = and i>}y = 0, only when the particle lies in either of the planes xy or yz. 218 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. Ex. 3. A straight line OL turns about a fixed point O so that— =A'' where h is the angular momentum of a body and N the moment of the impressed forces about OL. Prove that every point of OL is moving perpendicularly to the plane which contains it and the resultant axis of angular momentum at 0. Ex. 4. A triangular area AGB whose mass is M is turning round the side CA with an angular velocity w. Show that the angular momentum about the side GB is ^^Mab sin^ Cu, where a and 6 are the sides containing the angle C. Ex. 5. Two rods OA, AB, are hinged together at A and suspended from a fixed point 0. The system turns with angular velocity w about a vertical straight line through O so that the two rods are in a vertical plane. If 6, be the inclinations of the rods to the vertical, a, h their lengths, M, M' their masses, show that the angular momentum about the vertical axis is w [{^M+ M') a- sin2 d + M'ah sin ^ sin + ^M '62 siu^ 0]. Ex. 6. A right cone, whose vertex is fixed, has an angular velocity w com- municated to it about its axis OC, while at the same time its axis is set moving in space. The semi-angle of the cone is ^ir and its altitude is h. If d be the inclination of the axis to a fixed straight line Oz and ^j/ the angle the plane zOG makes with a fixed plane through Oz, prove that the angular momentum about Oz is |il//i2 / sin^ ^ -^ + 1 w cos ^ | , where M is the mass of the cone. Ex. 7. A rod AB is suspended by a string from a fixed point O and is moving in any manner. If {I, vi, n) {p, q, r) be the direction cosines of the string and rod referred to any rectangular axes Ox, Oy, Oz, show that the angular momentum about the axis of z is nT^ofidm dl\ ^^a?- [ dq dp\ ,^ab / dm dp ,dq dV V dt dtj 'dYdt ^dtj 2 Y dt dt dt ^dt^ where M is the mass of the rod, and a, b are the lengths of the rod and string. 268. As examples of the use of the expressions for the an'gular momentum of a body we shall apply them to the solution of two problems on the motion of a body in three dimensions. In these the axes of reference are fixed in space, the use of moving axes being reserved for the present. For further informa- tion we must refer the reader to the second volume where a whole chapter is devoted to examples and illustrations of the different methods of finding the motion of a body in three dimen- sions. Problem I. A uniaxal top spins on a perfectly rough table with its axis nearly vertical, find the small oscillations of the top *. Let be the apex, OC the axis of the top. Let C and A be the moments of inertia about the axis OC and any perpendicular to OC through 0. Since the centre of gravity G of the top is in its axis, the impressed forces have no moment about OC. Also A = B, hence by Euler's third dynamical equation Cw^ = Q. * The general motion of a top under the action of gravity will be considered in the second volume. The small oscillations of unsymmetrical and inclined tops will be found in that volume. A slight historical account will also be given. lRT. 268.] MOTION OF A TOP. 219 ?hu« the angular velocity of the top about its axis is always the same. Let = nhe this constant angular velocity. Let ^, f), ^ be the direction-cosines of OC referred to ixed axes in space, viz. Ox, Oy, Oz where Oz is vertical. ince the axis of the top is to be always very nearly ertical we have ^=1 while f, rj are small quantities hose squares will be neglected. Let l = OG, and let he mass be represented by unity. The moments of gravity acting at G round the axes }i X, y are found by the usual formulae L^yZ-zY=-lgr,, M=zX~xZ = lg^, (There X=0, Y=0, Z= -g are the components of gravity. The angular momenta >f the body about these axes are by Art. 266, 1i-^= - At} + Cn^, h2=A^ + Cnr{. 3ince these axes are fixed in space we have -A'r}+Cnl=- glr), A^ + Cnrj = gl^. Che equation obtained by using the angular momentum about the axis of z merely ihows over again that Wg is constant, a result already deduced from Euler's equations. To solve these we put ^ = P cos {fxt+f), r)=Q sin {/xt+f)', substituting we find {Afx^ + gl)Q-C7iu.P = 0, CnfMQ-{Afi^ + gl)P^O. Chese give Aijfi + gl= ^Cnfi. 'i is unnecessary to take both the signs on the right-hand side. If we choose one dgn the effect of the other sign is merely to change the sign of fi and this merely liters the as yet undetermined constants Q and /. Without loss of generality we nay choose the upper sign. This makes both the resulting values of fi positive. [t also gives P = Q. The values of fj, are 2An=Cn^ {Chi^-'igAl)^. Representing these two by /i = Mi and ^Ug we have ^ = Pj cos (;Ui< 4- /i) + P2 cos (ac2< + /o) 7} = P^sin {fi^t + f-^)+Po sin (fi^t + f^) .vhere Pj, Po, /i ,/2 are four constants to be determined by the initial values of ^, 77, |, fj. Let us represent the initial values of the coordinates by the suflBx zero. Then lo = A cos/i + Pg cos /2 , - lo = Pj^i sin /^ + P2^2 sin A , 77o = Pi sin /i + P2 sin /o , r?^ = P^fj.^ cos /i -f P^fx^ cos /g . rhese give If 6, \p be the angular coordinates of the axis we have e2^^2 + ^2^p^2 + p^2 + 2P,P2COs{(Mi-/*2)« + /l-/2} e^^ = ^ri- ky} = i^l Vl + ^2^2 + P1P2 [H + M2) COS { (/Xi - M2) « + /l -/2 } • Supposing Pj and P2 not to be equal we see that 6 can never vanish, i.e. the axis of the top can never become strictly vertical. Also \p will never vanish unless P1P2 (/x,i + /A2) is greater than P1V1 + -P2V2' i-^- ^^^ plane ZOG will revolve round OZ always in the same direction or with temporary reversions of direction according as P1/P2 does not or does lie between fi^lfJ-i and unity. 220 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. In order that P^ = P.2 it is necessary that initially This requires that \j/ should initially differ from | {/Xj + /j.^) by small quantities of the order P. In this case \j/ will keep one sign throughout the motion and the axis will become vertical at a constant interval equal to 27r j{fx^ - fx^i. We have assumed that the values of /i are both real and unequal. If the value of n be so small that the values of ix are imaginary, the values of | and t) will contain real exponentials. In this case the values of ^ and 77 do not in general remain small. This indicates that the top has not sufficient rotation about its axis to keep the axis vertical. It will begin to fall away from the vertical position, but its subsequent motion has not been investigated here. If C^n^ — AgAl the two values of n are real and equal. In this case it will be seen tha* the equations are satisfied by ^ = Pj cos (/it + /i) + P2« cos {/xf + /a) 77 = Pi sin [fit + /i) + Pgt sin {fxt + f.^) . The original disturbance of the top has been supposed to be of the first order of small quantities. As time goes on the top will deviate from the vertical until ^, rj become so large that their squares cannot be neglected, that is until they become large when compared with the original disturbance. The subsequent motion has not here been investigated and the axis of the top might afterwards return to the immediate neighbourhood of the vertical. See Vol. 11., Art. 202 g. Ex. A uniaxal body rotates about its axis with an angular velocity n. Two inextensible strings are attached to two points on the axis at distances, each equal to b, from the centre of gravity G of the body. The other extremities of the strings are attached to two points fixed in space. The length of each string is a and its tension is T. The mass of the body is unity. Prove that the period 2irjp of the linear oscillations of G is given by ap^ = 2T, while the periods 2Trlq of the angular oscillations of the axis are given hy Aq^- Cnq = 2r (a + h) bja. [See Vol. 11. , Art. 15. 269. Problem II. To find the motion of a sphere on a perfectly rough plane. Let the plane be taken as the plane of xy and let F, F' be the frictions at the point of contact resolved parallel to these axes. Let X, Y be the resolved impressed forces which we shall suppose to act through the centre. Let a be the radius of the sphere, A; its radius of gyration about a diameter and let its mass be unity. Consider the diameters parallel to the axes of x and y. The angular momenta about them are fc^Wj and k^u}.^. These directions are fixed in space, hence we have by Art. 78 or 261, k^6}e> Fa. y F' the plane does not slide u - 1 Eliminating F, F', w^ and Wo we find ]c-C}^ = F'a, If u and V be the velocities of the centre of gravity parallel to the axes u = X^F, v=Y+F'. Also since the point of contact with = 0, v + a(j3^ = 0. du _ a^ dv di~a^Tl^ ' di These are the equations of motion of a rotation on a smooth plane under the action of the same forces but reduced in the ratio a^l{a^+k'^). Since k^ = ^a^ we may enunciate this result as follows. a^ + k^ phere moving as a particle without ART. 270.] MOTION OF A SPHERE. 221 If a homogeneous sphere* roll on a rough fixed plane under the action of any forces whatever, whose resultant passes through the centre of the sphere, the motion of the centre is the same as if the plane were smooth, and all the forces were reduced to five-sevenths of their former value. Ex. 1. If the coefficient of friction is greater than ^RjZ where R is the re- sultant impressed force parallel to the plane and Z the normal force, prove that the friction will always be sufficient to prevent the sphere from sliding. Ex. 2. A sphere is placed on an inclined plane sufficiently rough to prevent sliding, and a velocity in any direction is communicated to it. Show that the path of the centre will be a parabola. If V be the initial horizontal velocity of the centre, a the inclination of the plane to the horizon, the latus rectum will be 147-/5/7 sin a. Ex. 3. A homogeneous sphere rolls on a perfectly rough plane under the action of a force varying inversely as the square of the distance from a point in the plane of motion of the centre, prove that its centre describes a conic section; and if, when the distance of its centre from the centre of force is one-quarter of the major axis of its orbit, the sphere come to a smooth part of the plane, the major axis of the orbit will be suddenly reduced to 7/13 of its former value. [Trin. Coll. Ex. 4. A homogeneous sphere moves, without rotation, on a smooth horizontal plane, under the action of a central force such that the centre of the sphere describes an ellipse with the centre of force in the focus. If the sphere arrive at a part of the plane which is perfectly rough when the distance of its centre from the centre of force is l/7ith of the major axis of its orbit, show that the major axis is diminished in the ratio 7:5 + 2n. If the sphere come again to the smooth part of the plane when the distance of its centre from the focus is the same fraction as before of the major axis, the major axis is again diminished in the same ratio. Ex. 5. Two spheres equal in volume but of different masses attract each other according to the law of nature and roll on a rough plane. Show that they each describe ellipses relatively to their common centre of gravity with that point for a focus. Ex. 6. A uniform circular disc is rotating in its own plane with very large angular velocity about its centre O which is fixed. Prove that if a tap be given to the disc in a direction perpendicular to its plane, at a point A, the diameter through A will approximately describe a plane slightly inchned to the original position of the plane of the disc, while the diameter at right angles to it will describe the same plane as before. [Math. Tripos, 1903. 270. The principal axes are generally chosen as the axes of reference because the moments of the effective forces for these are extremely simple. Thus the somewhat long equations of Art. 252 reduce to the simple Eulerian forms when referred to principal axes. But sometimes it is important to choose other axes which suit better the geometrical conditions of the problem. The discussion of such axes is reserved for the second volume of this treatise. But when the motion is steady, so that the angular velocities are constant, the unreduced equations of Art. 252 will sometimes take so simple a form that an easy solution can be found. * This theorem was given by the author as a problem in the Mathematical Tripos 1860 ; see the solutions for that year. Another demonstration is given in the second volume by which a corresponding theorem is obtained for the case in which the sphere rolls on another sphere. 222 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. Ex. A heavy body is attached by two hinges to a horizontal axis about lohich it is capable of moving freely. The axis is made to rotate ivith a uniform angular velocity w about a vertical axis intersecting it in a point O. It is required to find the conditions that the body may be inclined at a constant angle to the vertical. Let the horizontal axis which is fixed in the body be taken as the axis of z. Then the vertical lies in the plane of xy, let it make angles 6 and \ir - 6 with the axes of X and y. The whole system turns round the vertical with an angular velocity w. Hence by resolution ajar = wcos^, Wy = wsin^, Wg = Q. Remembering that these angular velocities are constant, the general equation of moments of Art. 252 becomes - 'Lynxy (w^;- - w^') + Sm [x'^ - y^) WxWy = N. To find N, we resolve the weight Mg parallel to the axes, then X= -Mg cos 0, Y= - Mg sin 6, Z = 0. If {x, y, z) be the coordinates of the centre of gravity we have N=xY-yX. The required relation between w and 6 is therefore oP {cos Id^mxy - \ sin 2^2m (x- -y'^)}= Mg [x sind-y cos 6). The integrals Zmxy and "Zm (x^-y-) can be expressed in terms of the moments and products of inertia of the body in the usual manner. Problems on steady motion may often be easily solved by a direct application of D'Alembert's principle. Thus, in the problem just discussed, each element of the body describes with uniform angular velocity a horizontal circle whose centre is in ihe vertical axis. If r be the radius of this circle the effective force on the element is mo?r and its direction is along the radius. The body may therefore be regarded as being in equilibrium under the action of its weight and a system of forces acting directly from the vertical axis and varying as the distance from that axis. The equation found above may be obtained by taking moments about Oz. Ex. 1. If the body be pushed along the axis of z and made to rotate about the vertical with the same angular velocity as before, show that no effect is produced on the inclination of the body to the vertical. Ex. 2. If the body be a heavy disc capable of turning about a horizontal axis Oz in its own plane, show that the plane of the disc will be vertical unless lt^iJ^>gh^ where h is the distance of the centre of gravity of the disc from Oz and A; the radius of gyration about Oz. Ex. 3. If the body be a circular disc capable of turning about a horizontal axis perpendicular to its plane and intersecting the disc in its circumference, show that if the tangent to the disc at the hinge make an angle d with the vertical, the angular velocity w is given by ui^a sin d = g. Ex. 4. Two equal balls A and B are attached to the extremities of two equal thin rods Aa, Bb. The ends a and b are attached by hinges to a fixed point and the whole is set in rotation about a vertical through as in the governor of the steam-engine. If the mass of the rods be neglected show that the time of rotation is equal to the time of oscillation of a pendulum ichose length is the vertical distance of the centre of either sphere below the hinges at 0. Ex. 5. If in the last example m be the mass of either thin rod and M that of either sphere, I the length of a rod, r the radius of a sphere, h the depth of either centre below the hinge, then the length of the pendulum is — M{l + rf + \mll^ l + r M{l + r) + ^ml ON FINITE ROTATIONS. 271. When the rotations to be compounded are finite in magnitude, the rule to find the resultant is somewhat complicated. As already mentioned in Art. 229 .such rotations are not very important in rigid dynamics. We shall therefore only ART. 272.] FINITE ROTATIONS. 223 briefly mention a few propositions which may throw light on those already discussed when the motion is infinitely small. We begin with the proposition corresponding to the parallelogram of angular velocities. Rodrigues' Theorem. A body has tivo rotations, (!) a rotation about an axis OA through an angle 6 ; (2) a subsequent rotation about an axis OB through an angle ■6', and both these axes are fixed in space. It is required to compound the rotations. Let lengths measured along OA, OB represent the directions of these rotations in the manner explained in Art. 231. Let the directions of the axes OA, OB cut a sphere whose centre is at in ^ and B. On this sphere measure the angle BAC equal to \d in a direction opposite to the rotation round OA and also the angle ABC equal to \d' in the C' same direction as the rotation ,^'' ""-^^ round OB, and let the arcs inter- ,.--' ^"^^^ sect in C. Lastly, measure the angles BAG', ABC respectively equal to BAC, ABC, but on the other side of AB. The rotation d round OA will then carry any point P in OC into the straight line OC, and the sub- sequent rotation d' about OB will carry the point P back into OC. Thus the points in OC are unmoved by the double rotation and OC is therefore the axis of the single rotation by which the given displacement of the body may be constructed. The straight line OC is called the resultant axis of rotation. If the order of the rotations were reversed, so that the body was rotated first about OB and then about OA, the resultant axis would be OC. If the axes OA, OB were fixed in the body, the rotation d about OA would bring OB into a position OB'. Then the body may be brought from its first into its last position by rotations d, 6' about the axes OA, OB' fixed in space: Hence the same construction will again give the position of the resultant axis and the rotation about it. 271 a. To find the magnitude d" of the rotation about the resultant axis OC we notice that if a point P be taken in OA, it is unmoved by the rotation 6 about OA, and the subsequent rotation 6' about OB will bring it into the position P', where PP' is bisected at right angles by the plane OBC. But the rotation d" about OC must give P the same displacement, hence in the standard case 6" is twice the external angle between the planes OCA, OCB. If the order of the rotations be reversed, the rotation about the resultant axis OC would be twice the external angle at C, which is the same as that at C. So that though the position of the resultant axis of rotation depends on the order of rotation the resultant angle of rotation is independent of that order. 2T2. A rotation represented by twice any internal angle of the spherical triangle ABC is equal and opposite to that represented by twice the corresponding external angle. For since the sum of the internal and external angles is tt, these two rotations only differ by 27r ; and it is evident that a rotation through an angle 27r cannot alter the position of any point of the body. This is merely another way of saying that when a body turns about a fixed axis it may be brought from one given position to another by turning the body either way round the axis. 224 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. 273. The rule for compounding finite rotations may be stated thus : If ABC he a spherical triangle, a rotation round OA from C to B through twice the internal angle at A, followed by a rotation round OB from A to C through twice the internal angle at B, is equal and opposite to a rotation round OC from B to A through twice the internal angle at C. It will be noticed that the order in which the axes are to be taken as we travel round the triangle is opposite to that of the rotations. We observe that the sine of half the angle of rotation about each axis is pro- portional to the sine of the angle between the other tico axes. As the demonstrations in Art. 271 are only modifications of those of Rodrigues, we may call this theorem after his name. Rodrigues' paper may be found in the fifth volume of Liouville's Journal. Ex. If two rotations 6, 6' about two axes OA, OB at right angles be com- pounded into a single rotation

r z Q0& GA' . From the spherical triangle lAA', sin^^^' = sinasin^^. From the two spherical triangles BIA, BIA', we have = cos a cos j3 + sin a sin /3 cos Z, cos BA' = cos a cos /3 + sin a sin /3 cos (Z + 6), where Z = BIA. Remembering that I = cos a, //i=cos/3, 71 = cos 7, the first gives ta,nZ= -njlm, and the second gives cos BA' = lm - Im (cos 6 - tan Z sin 6) = sind {-n+ Im tan ^ d). Similarly, by changing the sign of d, we have cos GA' = sin d {m + In tan ^6); . B .-. Qosec,d5x= -xiQ.n\ 6 + mz-ny+lia,n ^d{lx-\- my -^-nz) (1), with similar expressions for by, dz. If the origin have a linear displacement whose resolved parts parallel to the Ox, Oy, Oz are a, b, c, we must add these to the values of 5x, Sy, 5z. 280 a. The central axis. Supposing that the displacement is given by a translation {a, b, c) and a rotation 6 about the axis {I, m, n) the equations of the central axis follow without difiiculty. The required axis is parallel to 01 (Art. 225) and the translation along it is equal to the projection of the translation of the origin |f (Art. 222). Any point on the central axis must therefore satisfy the equations ^ = ^ = ^-! = Za + m6 + nc (A). I m n 15—2 228 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. If {/, g, h) be the coordinates of the foot of the perpendicular drawn from the origin on the central axis we have 2f=a- I {al + bm + cn) - {bn - cm) cot |^, with similar expressions for g, h. The equation of the central axis then takes the simple form x-f^y-g^z-h I m n To obtain this value of f, we write /, g, h, for x, y, z, in the expressions for 5a;, dy, Sz. Representing the right-hand side of the equation (A) by K for brevity, and remembering that fl + gm + hn=0, we obtain lK-a={-ftsin^6 + mh-ng) Bind together with two similar equations. Multiplying these three equations by - tan h 6^ n, -m respectively and adding we find (a - ZiC) tan 1 ^ - (671 - cm) =/ (1 + tan^i ^) sin ^, which leads at once to the required value of /. 280 b. Rodrigues' formula. If (^, 77, f) are the coordinates of the middle point of the whole displacement of any point P we have ^ = x + ^5x, &c. The expressions for the component displacements then take the form dx = a + 2tsin^e {m (^ - ^c) - n {v - ib)} (2). These agree with the results given by Eodrigues. To obtain these, we notice that if after turning the body round 01 through an angle 6, we rotate it back through the same angle, it will resume its former position. If therefore we write x + dx, &c. for X, y, z on the right side of equation (1) and change the sign of 6, we should get the same left-hand side with - 5x and - 6 written for 5a; and 6. We thus have cosec d5x=+{x + 5x) tan \d + m{z + 5z) - n {y -\-hy) - I tan h^d {I [x + bx) -^^ . . .] . Remembering that Ux + m5y-\-nbz = {), because there is only a rotation, Art. 222, we find, by addition 5a; = 2 (m^j - ntj^ tan \ d, where ^j^=x + ^8x, &c. are the coordinates of the middle of the displacement due to rotation alone. When the origin has a linear displacement also, represented by a, b, c, we include these in the values of 5x, by, dz. Since ^, r], f, are the coordinates of the middle point of the ivhole displacement we write ^i = ^-^a, &c. and we then immediately obtain equations (2). Since the whole displacement of any point on the central axis is along that axis, ^, rj, f, are also the coordinates of a point on the axis. The equations of the central axis may therefore also be found by substituting these values of dx, dy, 5z in equation (A). 281. By using the formulae for 5a;, dy, 8z we can find, the components of the whole displacement of any point P due to two screw motions taken in order about axes {I, m, n), {V, m\ n') draion through any points (/, g, h), (/', g', h'). Let the rotations and translations be {6, v), {d', v'). The displacement of x, y, z due to the first is 5x='yi-f sin d {-t{x -f)+m{z -h)-n{y-g) + UP], where t = t&n^d and P=l{x-f) + m{y - g) + n{z-h), with similar expressions for 8y, Sz. The displacements 5' a;, d'y, 5'z, due to the second screw are found by writing a; + 5a!;-/', &c. for x, y, z; I', m', n' for I, m, n; ART. 281.] FINITE ROTATIONS. 229 and 6', v' for 6, v. Adding the two together we have the whole displacements Ax = 5a; + 5' a:, &c. due to both screws. There is no difficulty in the process except that in the general case the result is rather long. We thus arrive at three linear expressions for the components A.r, Ay, A2; of the whole displacement due to both screws. These are of the form Ax=a + Ax + By + Cz with similar expressions for Ay, Az. To find the central axis of the two screws we notice that the locus of points whose displacements are equal and parallel is a straight line parallel to the axis of the resultant screw, Art. 220. Putting then Ax = a, Ay = b, Az = c, we have three linear equations, any two of which determine the ratios of x, y, z, and therefore give the direction-cosines of the central axis. Let these be \, fi, v. The equation of the central axis is then Ax Ay Az ^ , — - = — ^ = — = a\ + ba + cu. X fJL V CHAPTER VI. ON MOMENTUM. 282. The term Momentum has been given as the heading of this Chapter, though it only expresses a portion of its contents. The object of the Chapter may be enunciated in the following problem. The circumstances of the motion of a system at any time 4 are given. At the time t^ the system is moving under other circumstances. It is required to determine the relations which may exist between these two motioos. The manner in which these changes are effected by the forces is not the subject of enquiry. We only wish to determine what changes have been effected in the time ^i — ^o- If the time ^i-^o be very small, and the forces very great, this becomes the general problem of impulses. This also will be considered in the Chapter. Let us refer the system to any fixed axes Oos, Oy, Oz. Then the six general equations of motion may, by Art. 72, be written in the form ^ d^z ^ „ \ Integrating these from t — Utot^t-i^, we have Let an accelerating force F act on a moving particle m during any time t-^ — U, and let this time be divided into intervals each equal to dt At the middle of each of these intervals let a line be drawn from the position of m at that instant, to represent, at the same instant, the value of mFdt both in direction and magni- tude. Then the resultant of these forces, found by the rules of statics, may be called the whole force expended in the time ^i — ^o- Thus I mZdt is the whole force resolved parallel to the axis of Z. These equations then show that ART. 283.] EXAMPLE OF A CENTRAL FORCE. 231 (1) The change produced by any forces in the resolved part of the momentum of any system is equal in any time to the whole resolved force in that direction. (2) The change produced by any forces in the moment of the mornentum of the system about any straight line is, in any time, equal to the whole moment of these forces about that straight line. When the interval ti — to is very small, the " whole force " expended is the usual measure of an impulsive force, and the preceding equations are identical with those given in Art. 86. It is not necessary to deduce these two results from the equa- tions of motion. The following general theorem, which is really equivalent to the two theorems enunciated above, may be easily obtained by an application of D'Alembert's principle. 283. Fundamental Theorem. If the momentum of any particle of a system in motion be compounded and resolved, as if it were a force acting at the instantaneous position of the particle, according to the i^ules of statics, then the momenta of all the par- ticles at any time ti are together equivalent to the momenta at any previous time to together with the whole forces which have acted during the interval. The argument from D'Alembert's principle may be made clearer by being put at greater length. If %Ye multiply the mass m of any particle P by its velocity v, the product is the momentum mv of the particle. Let us represent this in direction and magnitude by a straight line PP' drawn from the particle in the direction of its motion. For the purposes of composition and resolution this representative straight line (in accordance with the rules of statics) may be moved to any position in the line of motion ; let it therefore move with the particle. If the particle be acted on at any instant by an external force mF, a new momentum equal to mFdt is generated in the time dt. This also can be represented by a straight line and compounded with the mv of the particle. If two particles act and react on each other with a force R for a time dt, two equal and opposite momenta (viz. Rdt) are communicated to the particles. Taking all the particles, we see that the change in their momenta is equal to the resultant of every mFdt which has acted on the system. This being true for each element of time is true for any finite interval t^ — tQ. Since the resultant of every mFdt has been defined to be the whole force, the theorem follows at once. In the case in which no forces act on the system, except the mutual actions of the particles, we see that the momenta of all the particles of a system at any two times are equivalent. The two principles of the Conservation of Linear Momentum and the Conservation of Areas may be enunciated as follows. If the forces luhich act on a system he such that they have no component along a certain fixed straight line, then the motion is such that the linear momentum resolved along this line is constant. If the forces be such that they have no moment about a 232 MOMENTUM. [CHAP. VI. certain fixed straight line, then the angular momentum or the area consei^ved (Art. 77) about this straight line is constant. It is evident that these principles are only particular cases of the results proved in Art. 79. 284. Example of a central force. Suppose that a single particle m describes an orbit about a centre of force 0. Let v, v' be its velocities at any two points P, P' of its course. Then mv' supposed to act along the tangent at P' if reversed would be in equilibrium with mv acting along the tangent at P together with the whole central force from P to P'. If p, p be the lengths of the perpendiculars from on the tangents at P, P', we have, by taking moments about 0, vp=^v'p', and hence vp is constant throughout the motion. Also if the tangents meet in T, the whole central force expended must act along the line TO, and may be found in terms of v, v by the rules for compounding velocities. Ex. Two particles of masses m, m' move about the same centre of force. If li, h' be the areas described by each radius vector per unit of time, prove that mh-i-m'h' is unaltered by an impact between the particles. 285. Example of three particles. Suppose three particles to start from rest attracting each other, but under the action of no external forces. Then the momenta of the three particles at any instant are together equivalent to the three initial momenta and are therefore in equilibrium. Hence at any instant the tangents to their paths must meet in some point 0, and if parallels to their directions of motion be drawn so as to form a triangle, the momenta of the several particles are proportional to the sides of that triangle. If there are n particles it may be shown in the same way that the n forces represented by rnv, mV, &c. are in equilibrium, and if parallels be drawn to the directions of motion and proportional to the momenta of the particles beginning at any point, they will form a closed polygon. If F, F', F" be the resultant attractions on the three particles, the lines of action of F, F\ F" also meet in a point. For let X, Y, Z be the actions between the particles m'm'\ m"m, mm', taken in order. Then F is the resultant of — 7 and Z ; F' oi — Z and X ; F" oi - X and F. Hence the three forces F, F', F" are in equilibrium*, and therefore their lines of action must meet in a point 0' . Also the magnitude of each is proportional to the sine of the angle between the directions of the other two. This point is not generally fixed, and does not coincide with 0. If the attraction be directly proportional to the distance, the two points 0, 0' coincide with the centre of gravity (?, and are * This proof is merely an amplification of the following. The three forces F, F', F", being the internal reactions of a system of bodies, are in equilibrium. ART. 286.] LAGRANGE'S THREE PARTICLES. 233 fixed in space throughout the motion. For it is a known proposi- tion in statics that, with this law of attraction, the whole attraction of a system of particles on one of the particles is the same as if the whole system were collected at its centre of gravity. Hence 0' coincides with G. Also, since each particle starts from rest, the initial velocity of the centre of gravity is zero, and therefore, by Art. 79, G is a fixed point. Again, since each particle starts from rest and is urged towards a fixed point G, it will move in the straight line joining its initial position with G. Hence coin- cides with G. When the attraction is directly proportional to the distance, it is proved in dynamics of a particle, that the time of reaching the centre of force from a position of rest is independent of the distance of that position of rest. Hence all the particles of the system will reach G at the same time, and meet there. If Sm be the sum of the masses, measured by their attractions in the usual manner, this time is known to be ^ir/^/Xm. Similar theorems for some other laws of force are given as examples at the end of Art. 286 a and the solutions are briefly indicated. 285 rt. Any three Particles. In the general problem of three mutually attracting particles we cannot obtain a sufficient number of first integrals to determine the motion. Since there are no external forces the linear momentum in the direction of each of three coordinate axes is constant and the three equations thus obtained can be again integrated. The angular momenta about these axes are also constant, and this principle supplies three more first integrals. Besides these we have the equation of vis viva. The investigations of Bruns, Poincare and Painleve have shown that no other first integrals which are algebraic and one valued can exist. Bruns, Acta Mathenuitica, Vol. XI.; VoincQxe, Act. Math. Vol. xiii. 1890, Le>i Methodes nouvelles de la Mecanique Celeste 1892 ; Painleve, Comptes Eeiidus 1894. 286. Example of Iiagrange's Three Particles. Three particles lohose masses are m, vi', vi", mutually attracting each other, are so projected that the triangle formed by joining their positions at any instant remains alioays similar to its original form. It is required to determine the conditions of projection* . The centre of gravity will be either at rest or will move uniformly in a straight line. We may therefore consider the centre of gravity at rest and may afterwards generalise the conditions of projection by impressing on each particle an additional velocity parallel to the direction in which we wish the centre of gravity to move. Let be the centre of gravity, P, P', P" the positions of the particles at any time t. Then, by the conditions of the question, the lengths OP, OP', OP" are always to be proportional, and their angular velocities about are to be equal. Since the angular momentum of the system about is always the same, we have mr'^n + m'r'-n + m"r"-ii = constant, * Lagrange was the first to obtain the two solutions of this problem mentioned above. In the essay which gained the prize of the Academy of Sciences in 1772, he speaks of it as a merely curious problem. Another discussion is given by Laplace in Vol. IV. Chap. vi. of the Mecanique Celeste. A list of writers on this subject is given in Whittaker's report to the British Association, 1899. 234 MOMENTUM. [CHAP. VI. where r, /, r" are the distances OP, OP', OP", and n is their common angular velocity. Since the ratios r : r' : r" are constant, it follows from this equation that mi'^n is constant, i.e. OP traces out equal areas in equal times. Hence by Newton, Section ii. , the resultant force on P tends towards O. Let p, p', p" be the sides P'P" , P"P, PP' of the triangle formed by the particles, and let the law of attraction be ,,. r,,. Then, since the resultant attraction of (dist.)*^ m', m" on m passes through 0, -ttt sin P'PO= -^ sin P"PO, p K pK but, since is the centre of gravity, m'p" sin P'PO = m"p sin P"PO. Hence either the three particles are in one straight line or p"'f+ir=:p'fc+^. If }c= -1 the law of attraction is "as the distance." If k be not = - 1, we have p =p", and the triangle must be equilateral. Suppose the particles to be projected in directions making equal angles with their distances from the centre of gravity with velocities proportional to those distances, and suppose also the resultant attractions towards the centre of gravity to be proportional to those distances, then in all the three cases the same con- ditions will hold at the end of a time dt, and so on continually. The three particles will therefore describe similar orbits about the centre of gravity in a similar manner. Firstly, let us suppose that the three particles are to be in one straight line. To fix our ideas, let m' lie between m and m", and between m and m\ Then since the attraction on any particle must be proportional to the distance of that particle from 0, the three attractions, measured in the direction PP", (PP>)k ' (pp")k> (^p"p')k {pp'jk^ {PF'f {P'P"Y' must be proportional to OP, OP', OP". Since 2wOP = 0, these two equations amount to but one on the whole. Let z = P'P"jPP', so that OP _ OP' _ PP' m' -\-vi" {1 + z) -m^m"z m + m' + m"' Then we have ( vi' + j- r-^ ] {- m + m"z) = (—^ -m\ {m' + m" (1 + 2)} , which agrees with the result given by Laplace. In the case in which the attraction follows the law of nature k~2, and the equation becomes W22 {(1 + 2)3 _ 1 > _ m' (1 + 2)2 (1 - z^) - m" { (1 + zf -z^\= 0. This is an equation of the fifth degree, and it has therefore always one real root. The left side of the equation has opposite signs when 2 = and 2=00 , and hence this real root is positive. It is therefore always possible to project the three masses so that they shall remain in a straight line. Laplace remarks that if m be the sun, 111' the earth, m" the moon, we have very nearly z= » / — = — — . If then, originally, the earth and moon had been placed in the same straight line with the sun, at distances from the sun proportional to 1 and 1 + -— -, and if their velocities had been initially parallel and proportional to those distances, the moon would have always been in opposition to the sun. The moon would have been too distant ART. 286 a.] LAGRANGE'S THREE PARTICLES. 235 to have been in a state of continual eclipse, and thus would have been full every night. It has however been shown by Liouville, in the Additions a la Connaissaiice des Temps, 1845, that such a motion would be unstable. Another proof is given in the author's treatise on Dynamics of a Particle, 1898, Art. 412. The paths of the particles will be similar ellipses having the centre of gravity for a common focus. Secondly. Let us suppose that the law of attraction is "as the distance." In this case the attraction on each particle is the same as if all the three particles were collected at the centre of gravity. Each particle will describe an ellipse having this point for centre in the same time. The necessary conditions of pro- jection are that the velocities of projection should be proportional to the initial distances from the centre of gravity, and that the directions of projection should make equal angles with those distances. Thirdly. Let us suppose the particles to be at the angular points of an equi- lateral triangle. The resultant accelerating force on the particle m is ^ cos P'PO + % cos P"PO. p'K pK The condition that the forces on the particles should be proportional to their distances from shows that the ratio of this force to the distance OP is the same for all the particles. Since by a property of the centre of gravity m'p" cos P'PO + m"p' cos P"PO = {m + n\! + m") OP, it is clear that the condition is initially satisfied when p = p' = p". Hence, by the same reasoning as before, if the particles be projected in directions making equal angles with OP, OP', OP" respectively, with velocities proportional to those * distances, they will always remain at the angular points of an equilateral triangle. A discussion of the stability of this motion will be given in Vol. ii. of this work. These results may be conveniently arrived at by reducing one angular point, as P, of the triangle to rest. The resolved part of all the forces which act on each particle perpendicular to the straight line joining it to P will then be zero. The process is a little shorter than that given above, but does not illustrate so well the subject of the chapter. 286 a. Examples. Ex. 1. Show that if the three particles attract each other according to the law of nature, the paths of the particles, when at the corners of an equilateral triangle, are similar ellipses having for a common focus. Find the periodic time. Ex. 2. Three unequal particles, attracting according to the inverse kth. power of the distance, are placed at rest at the corners of an equilateral triangle. Prove that they will finally meet at their common centre of gravity. The velocities, being zero, may be said to be proportional to the distances of the particles from O and to have the proper directions. Thus the Laplacian ■ conditions of projection are satisfied. The particles therefore always remain at * the corners of an equilateral triangle and these corners move directly towards the I centre of gravity 0. The three particles therefore describe straight lines and arrive ; simultaneously at O. The time t of transit is given by V(2m) ^ = p'^^^'^(l - Af j \l - y^-'y-'dy, ' where ij, = m + m' + m" and p is a side of the initial equilateral triangle. This integral can be expressed in gamma functions by putting y'^-^ = z or 1/z according 236 MOMENTUM. [CHAP. VI. as k is less or greater than unity. When k = 3 the integration can be effected without difficulty. Ex. 3. If the solar system consisted only of the sun, earth and moon moving in one plane, prove that S {E + Mf H+{S + E + 31) EMh = constant, where h is the rate at which a unit particle at the moon describes areas about the earth, and H the rate at which the centre of gravity of the earth and moon describes areas about the sun. T. . , r- . . , ^, ^ dH SEMA /I 1 \ If the sun were fixed m space prove also that -^~ = , —-, ( -; ~ — 1 , ^ ^ dt {E + 3I)^\r^ r'V ' where r, r' are the distances of M and E from S, and A is twice the area of the triangle formed by the three bodies. [St John's Coll. , 1896. Let G be the centre of gravity of the whole system, K that of E and M. Let w be the angular velocity in space of EM, Q that of SK. Now the area conserved by the whole system about G is constant, that conserved by E and M is, by Art. 75, {E . KE"^ + M . KM^) u} + {E + M) GK^Q, and that conserved by S iq S . GS^ . fi. We have given h = Ei\P.o}, H = SK'KQ; also the distances KE, KM, GK, GS, are known in terms of the distances EBI, SK and the masses S, E, M by the definition of the centre of gravity. Making these substitutions and equating the sum of the conserved areas to a constant, the first result follows at once. The second is obtained by taking moments about *S' and K. 286 1). Jacobi's theorem. Ex. 1. A free system of particles moves under their mutual attractions only, the force function U being a homogeneous function of the ?ith degree. Prove that j^^ZmR'^ = 2 {n + 2) U+2G, i where R^, R.2, t&c. are the distances of the particles m^, iiu^, &c. from the common centre of gravity and is a constant. If the law of attraction is the inverse cube prove that "LmR- =^A + Bt-]- Ct^. Vorlesungeii i'lher Dynamik, edited by A. Clebsch, supplementary volume, page 22. To prove this we have by simple differentiation d^x^) ^ fdxV ^ d^x ^ dU dt^ \dt J dt' dx Summing this for the coordinates x, y, z and for all the particles, we have since C/ is a homogeneous function —^ (Smr^) - 2'Zmv' = 2nU. By the principle of vis viva (Art. 138 or 350) Smy2 = 2C/+C\ G-. ^Smr2 = 2(;t + 2) [7+2(7. Hence Now Sww'- = 2?MiJ2 + (2 wt) (ar-2 + y^ + ^2) , but since there are no external forces dxfdt is a constant, and therefore d!^ [xy^fdt'^ is zero. Similar results being true for y and g, the theorem to be proved follows at once. Ex. 2. Three particles attracting each other according to the inverse cube of the distance are placed at rest in any positions. Deduce from Jacobi's theorem that an impact must occur before the time t given by ^ + Cf'-^ = 0. AllT. 287.] JACOBI'S THEOREM. LIVING THINGS. 237 Since the particles start from rest B=iO and C= -217q= - S — g" "^^^re p is I the side of the triangle joining the initial positions of m', m". Also A is the initial '{ moment of inertia of the three particles with regard to their common centre of !j gravity. We notice that A is positive and G negative and that the quadratic ,J + C«2 = has real roots. If two of the particles during the motion impinge on each other, we know by the equation of vis viva that their velocities will at that moment be infinite. The whole subsequent motion also will be affected by the impact. If this impact does not occur before the time given by (74^= -A, we see that at that instant Swi?2 = 0. All the particles must therefore be in contact. It also follows from Jacobi's theorem that, if the law of attraction were the inverse cube, the present arrangement of the solar system could not be stable. If the roots of the equation A+Bt + Ct^ = are real, an impact will occur at the end of a finite time. If the roots are imaginary, since ^ is a moment of inertia and therefore positive, C must be positive, and hence the radii vectores of some of the planets must be infinite when t is infinite. Does JacohVs equation hold indefinitely ? If we assume that when two particles meet they pass through each other without resistance it might be expected that the equation ^mR^ = A + Bt+Ct'^ would hold throughout all time. But if C is negative and t sufiiciently great the two sides have opposite signs, so that the equality cannot then hold indefinitely. The cause of this discrepancy is a certain discontinuity which occurs when the particles meet. When the particles m, ni' are at a very small distance x apart we have ultimately i2 = £-/x^ where E^ = m + m'. Extracting the square root we find x= dzEjx. When the particles are approaching each other, the negative sign must be given to the root because x is positive and x negative. When the particles pass through each other, their distance x changes sign through zero but the instantaneous value of the velocity is nnaltered. We must therefore give the square root the positive sign. Hence xx changes sign, or, which amounts to the same thing, the constant E is discontinuous, changing sign suddenly when the particles meet. Each meeting therefore marks a stage at which a new problem begins and at which the values of some of the arbitrary constants have to be determined afresh. There has been much difference of opinion on the true interpretation of the equations of motion at the singular points where either the velocity or the force is infinite. We have no space for the discussion here and must refer the reader to the author's treatise on Dynamics of a Particle where also various references are given, Art. 465. 287. Examples of living things. Ex. 1. A man is fastened to a vertical axis which can turn without friction and only the man's arms are free. The system being initially at rest, explain how the man by moving his arms in space can turn his body round and face the other way. If the man move his arms in any way the whole area conserved about the axis is zero. Art. 283. Having placed his right hand close to his side, let the man push it out sideways, and then move it forward so as to describe a quarter of a horizon- tal circle. Let him next draw the hand inwards close to his body, thus bringing it back into its initial position. It is evident that each point of the arm and hand has described an area round the axis from right to left. The man's body must therefore turn round the vertical axis from left to right through such an angle that the whole area described is zero. Eepeating this process he can turn his body through any angle. 238 MOMENTUM. [CHAP. VI. In this way a person standing erect on a perfectly smooth table can turn round a vertical axis passing through his centre of gravity and face any direction he may desire. Ex. 2. A person lies down on his back on a perfectly smooth table, explain how he can turn round and face the table. Extending one arm he hits the table with it and thus acquires angular momen- tum about his axis. When he has turned through two right angles, his extended arm or arms again strike the table, and can be used to gradually stop the motion. The same effect would be produced by throwing away sideways some portion of his dress. He might also use the method described in the last example. Ex. 3. Explain how it is that a cat held with its feet upwards and let go is found, after falling through a sufficient height, to alight on its feet. During the first stage of the fall the cat stretches out its hind legs almost per- pendicularly to the axis of the body and pulls the fore legs close to the neck. In this position it twists the fore part of the body through as large an angle as it can, the hinder part turning through a smaller angle in the opposite direction, so that the whole area conserved about the axis is zero, as in Ex. 1, In the second phase of the fall the attitude of the feet is reversed, the hind legs being close to the body and the fore legs pushed out. The cat now turns the hind part of the body through the large angle, the fore part rotating through the small angle. The result is that both parts of the cat are turned round the axis through nearly equal angles. See a series of photographs of a falling cat in Nature, Nov. 22, 1894, reproduced from M. Marey's paper, Comptes Rendus, cxix. 1894. The true explanation is due to M. Guyon. M. Maurice Levy in the same volume puts the argument into a mathematical form and shows how a man placed in empty space can turn on his axis without initial velocity or the assistance of any external force. Also M. Lecornu shows how a serpent by internal motions continually repeated could rotate its body about its axis of length ivithout changing its external form or position in space. Ex, 4. A person is enclosed in a light box which is placed on a rough floor. Show by what motions he can take advantage of the friction to move the box and himself any distance along the floor. Starting from one end, he runs along the box, but not so quickly that the friction is insufficient to hold the box at rest. He thus moves his own centre of gravity and acquires momentum. Then jumping up he lifts the box off the floor and carries it with him. When gravity brings the box again to the floor, he repeats the operation. Another method is indicated in Chap. ii. Ex. 3. Certain Mexican seed vessels, called jumping beans, have been observed to move about by a series of jumps. Each bean is found to contain a grub con- siderably smaller than the cavity within which it is confined. The manner in which the grub makes the bean jump a distance equal to two or three times the length of the bean has not been properly explained. See the Royal Botanical Society, Nov. 1894, and Chambers's Journal, 1896. There is also a brief account in Nature, Nov. 19, 1896, of some recent experiments on African specimens made by Dr D. Sharp with the view of discovering the cause and object of these movements. Ex. 5. Two buckets of given weights are suspended by a fine inelastic string placed over a fixed pulley, and at the centre of the base of one of the buckets a frog of given weight is sitting. At an instant of instantaneous rest of the buckets, the frog leaps vertically upwards so as just to arrive at the level of the rim of its bucket. Prove that the ratio of the absolute length h' of the frog's vertical ascent ART. 288.] LIVING THINGS. SUDDEN CHANGES OF MOTION. 239 in space to the length h of its bucket and the time t which elapses before the frog again arrives at the base of its bucket are given by [m + m' + fif h' = 2m' {m + m') h, m'cft^ = 4 {m + m') h, the last result being independent of the frog's weight. [Walton's problem. Math. Tripos, 1864. Ex. 6. Show that a person when swinging can increase the angle of vibration by alternately crouching at the highest point and straightening himself along the rope when at the lowest point. Let 2a, 2b be the heights of the man when crouching and standing erect ; M, m the masses of the swing and man, I the moment of inertia of the swing, and c the distance of its centre of gravity from the point of support. First the system, with the man crouched, descends from rest through an angle a and has an angular velocity w at the lowest point. Suddenly when the man stands erect, the angular velocity w is changed to w'. Lastly the system ascends through an angle j3. We therefore have where A = I+7ii{l-ay^ + ^ma^, A' = Mc + m {I- a), and B, B' are obtained from A, A' by writing 6 for a. The first and third of these equations follow from the principle of vis viva, and the second from that of angular momentum. Hence 8in^ ^^Isin^ ^a = A A' IBB' . Now A'>B' since b>a ; also A>B since in swings the length I of the rope is usually longer than the height of the man. Hence /3 is greater than a. Consider the equation A(a = Bo}'; each time the man straightens himself he decreases the moment of inertia and therefore increases the angular velocity. At the highest point, when the system is instantaneously at rest, no change in the angular velocity is made by crouching, but the moment of inertia is increased. By the continued repetition of these two processes the angular velocity at each passage through the lowest point is increased. Again, the moment of gravity is greater on the descending than on the ascending arc, hence from both causes the amplitude of the swing is increased. 288. Sudden Fixtures. A rigid body is moving freely in space in a known manner. Suddenly a straight line in the body becomes fixed, or has its motion changed in some given manner. It is required to find the changes which occur in the motion of the rest of the body. Such problems as these are all solved by one mechanical prin- ciple. The change in the motion is produced by impulsive forces ; acting at points situated in this straight line. Hence, by Art. 283, ; the angular momentum of the body about the axis is the same after as before the change takes place. This dynamical principle supplies one equation which is sufficient to determine the subsequent motion of the body round the straight line. We may also use this principle in a more general case. Suppose we have any system of moving bodies which suddenly become rigidly connected together and are constrained to turn round some I axis. Then the subsequent angular velocity about this axis may \ be found by equating the angular momentum of the system about this axis after the change to that before the change. 240 MOMENTUM. [CHAP. VI. In applying this principle to various bodies it is convenient to use different methods of finding the angular momentum. The following list will assist the reader in choosing the method best adapted to each particular case. 289. Case 1. Suppose the body to be a disc moving in any manner in its own plane, and let the axis whose motion is changed be perpendicular to its plane. This case has been already solved in Art. 171. 290. Case 2. Suppose the body to be a disc turning about an instantaneous axis Ooc in its own plane with an angular velocity co. Let an axis Ox' also in its own plane be suddenly fixed. In this case the calculation of the angular momentum is so simple that we may with ad- vantage recur to first principles. Let da be any element of the area of the disc; y, y' its dis- tances from Ow, Ox', Then yto, y'w are the velocities of c?cr just before and just after the impact. The moments of the momentum about Ox just before and just after are therefore yy'a>dcr and y'-w'da-. Summing these for the whole area of the disc, we have o)"ly'^d(T==(olyy'dcT (1). Firstly, let Ox, Ox be parallel, so that the point is at infinity. Let h be the distance between the axes, then y' — y — )i. Hence we have ay'^y'^do- = (o {^y-do- — Kl^yda-]. Let ^, ^' be the moments of inertia of the disc about Ox, Ox respectively, y the distance of the centre of gravity from Ox, ilf the mass of the disc. Then we have A' on' = ft) (^ — Mhy). f Secondly, let Ox, Ox' not be parallel. Let be the origin and let the angle xOx = a, then y' = y cos a — x sin a. Let F be the product of inertia of the disc about Ox, Oy where Oy is perpen- dicular to Ox. Then by substitution in (1) we have A' (a' = w{A cos OL— F sin a). Ex. 1. An elliptic area of eccentricity e is turning about one latus rectum. Suddenly this latus rectum is loosed and the other fixed. Show that the angular velocity is - — ^-s of its former value. Ex. 2. A right-angled triangular area ACB is turning about the side AC. Suddenly ^C is loosed and BC fixed. If C be the right angle, the angular velocity is \BCIAC of its former value. ART. 293.] SUDDEN CHANGES OF MOTION. 241 Ex. 3. A rectangle ABGD has its plane vertical and its lower edge AB horizon- tal and fixed in space. A slight disturbance being given the rectangle turns round AB, but when its plane becomes horizontal the side JD is fixed and AB released. It then begins to turn round AD and when the plane is again vertical AB is fixed and AD released. Show that the final angular velocity about AB is given by the equation b}^ = 21g{l6a + 9b)l512b^, where AB = 2a and AD = 2b. Ex. 4. A point is suddenly fixed in a lamina which is instantaneously rotating about any given axis in its own plane. Show that if the new instantaneous axis is at right angles to the former the point must lie on a hyperbola one of whose asymptotes is perpendicular to the given axis and the other is its conjugate with regard to the momental ellipse at the centre of gravity. 291. Case 3. Let the body be turning round an instantaneous axis 01 with a known angular velocity co, and let some axis 0/' which intersects the former in a point be suddenly fixed. Let I, m, n be the direction-cosines of 01 referred to the principal axes at 0, and /', m', n' the direction-cosines of 0I\ Then by Art. 264, the angular momenta about these principal axes just before the change are Acol, Bcoin, C(on. The angular momentum about 01' just before the change is therefore (by Art. 265) {AW -\- Bmm! -\- Gnn')(o. If w be the angular velocity of the body about Oi' just after 01' becomes fixed in space the angular momentum is {Al"^ + Bm"^ -h Cn^) w'. Equating these we have w. Ex. 1. A solid right cone of semi-vertical angle a is rotating about a generating line. Suddenly another generating line is fixed, the axial planes through the generating lines being inclined at an angle 0. Show that the ratio of the angular velocities is equal to (2 + (4 + w) co8 0) : (6 + n), where n = iQ.rx^a. Ex. 2. When a body turns about a fixed point the product of the moment of inertia about the instantaneous axis into the square of the angular velocity is called the vis viva. Let 2T be the vis viva of the body when it is turning freely about the axis 01, and 2T' its vis viva when the axis 01' is suddenly fixed. Construct the momental ellipsoid at the point 0, and let 6 be the angle between the eccentric lines of the two axes 01, 01'. Prove that T' = T cos^ d. It follows that the vis viva is always lessened by fixing a new axis. 292. Case 4. Let the motion of the body be given by its components of motion u, v, w, (Ox, Wy, (Oz, the centre of gravity being the base point. Let the equation to the straight line whose motion is suddenly changed be — j^ = - — - = , where I, m, n are the actual direction-cosines. Suppose this straight line to be suddenly fixed in space. The angular momentum before the " fixing " is given in Arts. 263, 265, 266. If o)' be the angular velocity about this straight line after the " fixing," the angular momentum is Iw, where / is given in Art. 17, Ex. 9. Equating the two values we have w. 293. Suppose the sudden motion forced on the straight line to be represented by the velocities £/", F, W of some point P on R. D. 16 242 MOMENTUM. [CHAP. VI. the straight line, and the angular velocities 6, (f), yjr. Then the motion of the body may be represented by the linear velocities U, V, W of the same base P and the angular velocities 6 + D^l, if) + Urn, o/r + Q.n, where H is the only unknown quantity. The angular velocities d, (p, xp may be chosen in an infinite variety of ways to represent the given motion of the straight line, because an angular velocity about the straight line does not move the line itself. If 0, %» ^2) *c- ^^^^^ coordinates referred to the centre of the sun. The equation (2) then becomes /i(S + SM) = 6f.SiH(|^-,,^) + SS.¥iM2{(f2-?i)(^2-^i)-{'72-'7i)&-li)}-(3). If a be the semi-major axis of any planetary orbit, e = sin the eccentricity, n the mean motion in a Julian year, x ^^^ angle the plane of the orbit makes with the plane of xy, say, the ecliptic, we have ^7] - 7}^ = na^ cos (j) cosx (4). The terms of the second order depending on the products of the masses of the planets are omitted. The two greatest planets are Jupiter and Saturn, their masses are respectively only 1/1047 and 1/3500 of that of the sun. These terms are therefore 254 MOMENTUM. [CHAP. VT. less than the uncertainty attaching to the terms of the first order. The satellites, asteroids and comets are too small and too symmetrically distributed to exert a sensible influence on the position of the plane. Since the ratios of h^ , h.2, h^ are all that we want, we write unity for the mass of the sun and express the masses of the several planets as fractions of that of the sun. The formulae used to find h^ , h^ , h^ now become h^ = 'Z{Mna'^ cos (p cos x), /ii = S (Mwa'-^cos sin xcos;/'), /t2 = 2 {Mna^ cos (f) sin x sin xp), where xj/ is the longitude of the ascending node of the planet's orbit on the fixed plane of reference at a particular epoch. 305 a. Ex. 1. Show that the invariable plane at any point of space in the straight line described by the centre of gravity of the solar system is parallel to that at the centre of gravity. Ex. 2. If the invariable planes at all points in a certain straight line are parallel, then that straight line is parallel to the straight line described by the centre of gravity. Impulsive Forces in Three Dimensions. 306. Constrained single body. To determine the general equations of motion of a body about a fixed point under the action of given impulses. Let the fixed point be taken as the origin, and let the axes of coordinates be rectangular. Let {fix, ^y, ^z), (^a;> (^y, «»z) be the angular velocities of the body just before and just after the impulse, and let the differences cox — fl^, (Oy — £ly, Wz — ^z be called Wx, ©/, o)/. Then (Ox, Wy, «/ are the angular velocities generated by the impulse. By D'Alembert's Principle, see Art. 87, the difference between the angular momenta of the system just before and just after the action of the impulses is equal to the moment of the impulses. Hence by Art. 262 AcDx — (Emooy) coy — (Smxz) o)/ = Z "j Bcoy — (%myz) Wz — (^myx) (Ox =" M\ (1) Oft)/ — (Xmzoc) Q)x' — i^mzy) coy = N) where L, M, N are the moments of the impulsive forces about the axes. These three equations will suffice to determine the values of cox, coy, co/. By adding these to the angular velocities before the impulse, the initial motion of the body after the impulse is found. 307. Ex. 1. Show that these equations are independent of each other, and that none of the angular velocities w^., coy, Wg is infinite. This follows from Art. 20, where it is shown that the eliminant of the equations cannot vanish. Ex. 2. Show that, if the body be acted on by a finite number of given impulses following each other at infinitely short intervals, the final motion is independent of their order. I I ART. 809.] IMPULSIVE FORCES. 255 308. It is to be observed that in these equations the axes of reference are any whatever. They should be so chosen that the values of A, Xmxy, &c., may be most easily found. If the positions of the principal axes at the fixed point are known, these will in general be found the most suitable. In that case the equations reduce to the simple forms A(o^' = L, Bcoy' = M, CcD.'^N (2). The values of Wx, (Oy, co/ being known, we can find the pressures on the fixed point. For by D'Alembert's Principle the change in the linear momentum of the body in any direction is equal to the resolved part of the impulsive forces. Hence if F, G, H be the pressures of the fixed point on the body SX + i^=if.^by Art. 86 (3). = M{a)yZ - (o^'y) by Art. 238 tY+G = M{(o,'x-(o^z) XZ + H=M {(o^y - (Dy'x) 308 a. If X, Y, Z are the components of any blow, jp, q, r the coordinates of its point of application, the equations (2) may be written A{o}^-n-^) = '2{qZ-rY), &c., &c. The gain of vis viva due to the impulses is by Art. 363, A (0,^2 _ i2^2) + B (V _ fi/) + G (a,/ - O/) = (w^ + O J i:{qZ-rY) + {cjy + Qy) S {rX-pZ) + &c. Separate the terms with X, Y, Z and this becomes by Art. 238 S {X{ui + U2) + Y{v^+V2) + Z (wj^ + w^)}, where {u^, v^, Wj) {u^, v^, w^) are the resolved velocities of the point of application just before and just after the blow. See Arts. 171, 346, 384. 309. Ex. 1. A uniform disc hounded by an arc OP of a parabola, the axis ON, and the ordinate PN, has its _ vertex fixed. A blow B is given to it perpendicular to its plane at the extremity P of the curved boundary. Sup- jaosing the disc to be at rest before the application of the blow, find the initial motion. Let the equation to the parabola be y^=4oax, and let the axis of z be perpen- •dicular to its plane. Then 2mxz = 0, 1imyz = 0. Let fi be the mass of a unit of area And let ON=c. Also i:mxy = fi j lxydxdy = ju. I x^dx = 2fji. I ax^dx = A = ^lj.j y^dx=^fji,a^c^, B = fi j x^ydx=j- = ^fia^c^, ixac^ and C=A + B, by Art. 7. 256 MOMENTUM. [CHAP. VI. The moments of the blow B about the axes are L = BsJAac, M= -Be, N=0. The equations of Art. 306 will become after substitution of these values ^^fjLa^c^(ay.-^fjiac^u}j, = 2Ba^c^, j^fia^c^u}y-^fiac^u3^=-Bc, a>^=0. These equations determine the initial motion. By eliminating B we find the ratio of Wy to w^. It easily follows that if NQ is taken equal to ^V-^-f*> the disc begins to rotate about OQ. Ex. 2. One end of an inelastic string is attached to a fixed point and the other to a point in the surface of a body of mass 31. The body is allowed to fall freely under gravity without rotation. Show that just before the string becomes tight / / 1 X2 u2 ,;2\ the loss of kinetic energy due to the impact is ^V^ ( Tr + "7 "^ r^ "^ ?i" ) ' '^^®^^ ^ ^^ the resolved velocity of the body in the direction of the string just before impact^ the string only touching the body at the point of attachment, I, m, n, X, ix, v are the coordinates of the string at the instant it becomes tight and A, B, C the principal moments of inertia of the body with respect to its principal axes at its centre of inertia. [Math. Tripos, 1899. The result in the question follows from the equations of Arts. 308 and 238. Here \, fi, v are the moments about the axes of a unit force acting along the string, and I, m, n are its resolved parts. See the author's Statics, 1896, Art. 260 for references. 310. New statement of the Problem. When a body free to turn about a fixed point is acted on by any number of impulses, each impulse is equivalent to an equal and parallel impulse acting at the fixed point together with an impulsive couple. The impulse at the fixed point can have no effect on the motion of the body, and may therefore be left out of consideration if only the motion is wanted. Compounding all the couples, we see that the general problem may be stated thus : — A body moving about a fixed point is acted on by a given impulsive couple, find the change produced in the motion. The analytical solution is comprised in the equations which have bee^i written down in Art 306. The following examples express the result in a geometrical form. Ex. 1. Show from these equations that the resultant axis of the angular velocity generated by the couple is the diametral line of the plane of the couple with regard to the momental ellipsoid. See also Art. 118. Ex. 2. Let G be the magnitude of the couple, p the perpendicular from the fixed point on the tangent plane to the momental ellipsoid parallel to the plane of the couple G. Let 12 be the angular velocity generated, r the radius vector of the ellipsoid which is the axis of i^. Let K be the parameter of the ellipsoid, as in Art. 19. Prove that Kil=prG. Ex. 3. If fia;' ^2/' ^e ^^ angular velocities about three conjugate diameters of the momental ellipsoid at the fixed point, such that their resultant is the angular velocity generated by an impulsive couple G, A', B', G' the moments of inertia about these conjugate diameters, prove that A'^^= G cosa, B'Qy — G cos ^, C'Qg=GGOBy, where a, /3, y are the angles the axis of G makes with the conjugate diameters. Ex. 4. If a body free to turn about a fixed point O be acted on by an impulsive ART. 311.] IMPULSIVE FORCES. 267 couple G, whose axis is the radius vector r of the ellipsoid of gyration at 0, and if p be the perpendicular from on the tangent plane at the extremity of r, then the axis of the angular velocity generated by the blow will be the perpendicular p, and the magnitude ii is given by G = Mpr^. Ex. 5. Show that, if a body at rest be acted on by any impulses, we may take moments about the initial axis of rotation, according to the rule given in Art. 89, as if it were a fixed axis. Ex. 6. When a body turns about a fixed point, the product of the moment of inertia about the instantaneous axis and the square of the angular velocity is called the Vis Viva. Let the vis viva generated from rest by any impulse be 2T, and let the vis viva generated by the same impulse when the body is constrained to turn about a fixed axis passing through the fixed point be 2T'. Then prove that T' = Tcoa^d, where 6 is the angle between the eccentric lines of the two axes of rotation with regard to the momental ellipsoid at the fixed point. Ex. 7. Hence deduce Euler's theorem, that the vis viva generated from rest by an impulse is greater when the body is free to turn about the fixed point than when constrained to turn about any axis through the fixed point. This theorem was riterwards generalized by Lagrange and Bertrand in the second part of the first volume of the Mecanique Analytique. 311. Free single body. To determine the motion of a free body acted on by any given impulse. Since the body is free, the motion round the centre of gravity is the same as if that point were fixed. Hence, the axes being any three straight lines at right angles meeting at the centre of gravity, the angular velocities of the body may still be found by equations (1) and (2) of Art. 306. To find the motion of the centre of gravity, let (U, V, TT), (w, V, w) be the resolved velocities of the centre of gravity just before and just after the impulse. Let X, Y, Z be the components of the blow, and let M be the whole mass. Then by resolving parallel to the axes we have M{u-U) = X, M(v-V) = Y, M{w-W) = Z. If we follow the same notation as in Art. 306, the differences \u — U,v — V,w — W may be called u, v\ w . 312. Ex. 1. A body at rest is acted on by an impulse whose components [parallel to the principal axes at the centre of gravity are (Z, y, Z) and the co- [ordinates of whose point of application referred to these axes are (p, g, r). Prove I that if the resulting motion be one of rotation only about some axis, A{B - G)pYZ ^B {G - A) qZX^G {A -B)rXY=Q). Is this condition sufficient as well as necessary? See Art. 241. Ex. 2. A homogeneous cricket-ball is set rotating about a horizontal axis in [the vertical plane of projection with an angular velocity fl. When it strikes the )und, supposed perfectly rough and inelastic, the centre is moving with velocity [F in a direction making an angle a with the horizon, prove that the direction of Ithe motion of the ball after impact will make with the plane of projection an angle |tan~i - ,^ , where a is the radius of the ball. 5Fcosa R. D. 17 258 MOMENTUM. [CHAP. VI. Ex. 3. A rough lamina, turning with angular velocity fi about a fixed axis perpendicular to its plane, is impulsively gripped by a solid cone of semi-vertical angle a, whose vertex is fixed at the point where the axis meets the lamina, turning about its own axis with angular velocity w. The moments of inertia of the cone being denoted by ^, ^, G and that of the lamina by I, prove that the loss of kinetic energy is |(0 - w sin a)^ (j + ^°^ " + -77^) • [Math. Tripos, 1902. The cone will begin to roll on the lamina which can only turn about its axis, say the axis of z. Let G be the couple of reaction between the cone and lamina, its axis being that of z. Let the cone touch the lamina along the axis of x. Take moments for the cone about its principal axes DC, GA and for the lamina about its principal axis GZ ; we find C (wg' - w) = (r sin a, Ao}/= - G cos a, I (S2' - fi) = - G. Since the cone rolls on the lamina Wg' sin a - w/ cos a = 0'. Solving these we find G and thence Wg', w/, and Q\ The loss of energy follows at once. 313. Motion of any point of the body. To prove that the components of the change of velocity of any point of the body are linear functions of the components of the blow. The equations of Art. 311 completely determine the motion of a free body acted on by a given impulse, and from these by Art. 238 we may determine the initial motion of any point of the body. Let (p, q, r) be the coordinates of the point of application of the blow, then the moments of the blow round the axes are respectively qZ — rY, rX—pZ, pY—qX. These must be written on the right-hand sides of the equations of Art. 306. Let {p', q, r') be the co- ordinates of the point whose initial velocities parallel to the axes are required. Let {u^, Vi, w^, {u^, v^, w^) be its velocities just before and just after the impulse. Let the rest of the notation be the same as that used in Art. 306. Then U2 — Uj = u' + coy'r — (o^q', with similar equations for v^ — v^, w^ — w^. Substituting in these equations the value of ii, v\ w\ ©/, «/, 0)^' given by Art. 311 we see that u^ — u^,v^ — v^,w<^ — w^ are linear functions of X, Y , Z, of the form u^-u^^FX -^-GY ^EZ, where F, G, H depend on the structure of the body and the coordinates of the two points. 314. When the point whose initial motion is required is the point of application of the blow, and the axes of reference are the principal axes at the centre of gravity, these expressions take the simple forms The right-hand sides of these equations are the differential coefficients of a quadratic function of X, F, Z, which we may call E. It follows that for all blows at the same point P of the same body the resultant change in the velocity of the point ART. 315.] IMPACT OF INELASTIC BODIES. 259 P of application is perpendicular to the diametral plane of the direction of the blow with regard to a certain ellipsoid, whose centre is at P, and whose equation is E = constant. The expression for E may be written in either of the equivalent forms : 2E = ^X!±^±^' + -^{{Ap^ + Bq^+Cr^) (AX^ + BY^+CZ^) - {ApX+BqY+ GrZf) - M ^^{^Z-rYf^-{rX-pZf^^-{pY-qX)\ In this latter form we see that 1E = M{u"'-^-v"'"^w"^)^Aia^'^^B(>}^^ + Ci^;^, which is the vis viva of the motion generated by the impulse. Impact of any two bodies. 315. Two bodies moving in any manner impinge on each other. To find the motion after impact. Inelastic Bodies. If the bodies are inelastic and either perfectly smooth or so rough that the sliding must be destroyed before the termination of the impact, it is unnecessary to introduce the reactions into the equations. In either case we take the point of contact as the origin. Let the axes of x and y be in the tangent plane, and that of z be normal. Let U, V, W be the resolved velocities of the centre of gravity of one body just before the impact, and u, v, lu the resolved velocities just after the impact. Let Ha;, Hj,, fl^, a)a;, Wj,, 0)2 be the angular velocities just before and just after. Let A, B, C, D, E, F be the moments and products of inertia at the centre of gravity. Let M be the mass of the body, and X, 2/, z the coordinates of its centre of gravity. Let accented letters denote the same quantities for the other body. Then taking moments about the axes for one body we have, by Arts. 306 and 78, A{ay^-n^)-F{iOy-ay)-E{(o^-n,)-{v-V)z-^{w-W)y=^0, -F{w^-^^) + B{ayy-^y)-D{(c,-n,)-{w-W)x+{u-U)z^O, -E{co^^n^)-j){(Oy-ny)+C(co,-n,)-(u-U)y+(v-V)x = o. Three similar equations apply for the other body, differing from these only in having all the letters accented. Resolving along the axis of z for both bodies, we have M(w-W)-{-M'{w'-W') = 0. The relative velocity of compression is zero at the moment of greatest compression, we have therefore w — (o^y -\- coyX = w' — cox'y' + (Oyx\ We thus have eight equations between the twelve unknown resolved velocities and angular velocities. 17—2 260 MOMENTUM. [CHAP. VI. 316. If the bodies he smooth we obtain four more equations by resolving for each body parallel to the axes of x and y. For the one body we have u—U=0, v — V = 0, with similar equations for the other body. 317. If the bodies be rough we obtain two of the four equations by resolving the linear momenta parallel to the axes of a; and y, viz. M{u-U)-{-M'(u-U') = 0\ M{v-V) + M'{v' -V') = 0\ We have also two geometrical equations obtained by equating to zero the resolved relative velocity of sliding, viz. U — 0)yZ + (Ozy =U — COyZ' + ftj/t/' V — (jDzOC + QJa;^ =V' — (OzOC + W^Z 318. Smooth Elastic Bodies. If the bodies be smooth and imperfectly elastic, we must introduce the normal reaction into the equations. In this case we proceed exactly as in the general case when the bodies are rough and elastic, which we shall consider in the following articles. The process is of course simplified by putting both the frictions P and Q equal to zero in the twelve equations of motion (1), (2), (3) and (4). We also have the velocity G of compression equal to zero at the moment of greatest compression. Thus we have one more equation from which the normal reaction R may be found. Multiplying this value oi R hy 1 ■\- e, where e has the meaning given to it in Art. 179, we have the complete value of R for the whole impact. Substituting this last value of R in the twelve equations of motion (1) and (2), (3) and (4), the motion of both bodies just after impact is found. 319. Rough Elastic Bodies. The problem of determining the motion of any two rough bodies after a collision involves some rather long analysis and yet in some points it differs essentially from the corresponding problem in two dimensions. We shall, therefore, first consider a special problem which admits of being treated briefly, and will then apply the same principles to the general problem in three dimensions. 320. Two rough ellipsoids moving in any manner impinge on each other so that the extremity of a principal diameter of one strikes the extremity of a principal diameter of the other, at an instant when the three principal diameters of one are parallel to those of the other. Find the motion just after impact. Let us refer the motion to coordinate axes parallel to the principal diameters of either ellipsoid at the beginning of the impact. Then since the duration of the impact is indefinitely small and the velocities are finite, the bodies will not have time to change their position, and therefore the principal diameters will be parallel to the coordinate axes throughout the impact. ART. 320.] IMPACT OF ROUGH ELASTIC ELLIPSOIDS. 261 Let U, V, W be the resolved velocities of the centre of gravity of one body just before impact; u, v, tu the resolved velocities at any time t after the beginning of the impact, but before its termination. Let fl^j ^y^ ^z be the angular velocities of the body just before impact about its principal diameters at the centre of gravity; cox, cOy, odz the angular velocities at the time t. Let a, b, c be the semiaxes of the ellipsoid, and A, B, C the moments of inertia at the centre of gravity about these axes respectively. Let M be the mass of the body. Let accented letters denote the same quantities for the other body. Let the bodies impinge at the extremities of the axes c, d . Let P, Q, U be the resolved parts parallel to the axes of the momentum generated in the body M by the blow during the time t. Then —P, — Q, — R are the resolved parts of the momentum generated in the other body in the same time. The equations of motion of the body M are B(cOy-ny)=-Pc\ (1), O(a,,-a) = J M(u-U) = P\ Miv-V)=Q\ (2). M(w-W) = r] There are six corresponding equations for the other body which may be derived from these by accenting all the letters on the left-hand side and writing — P, —Q, — P, — c for P, Q, R and c on the right-hand side. Let us call these new equations respectively (3) and (4). Let 8 be the velocity with which one ellipsoid slides along the other, and the angle which the direction of sliding makes with the axis of x, then, as in Art. 192, S C0?> = U -\- C (Oy — U -{■ CWy (5), 8^1X10 = V' — c'Wx — V - C(Ox (6). Let C be the relative velocity of compression, then C = w' -w (7). Substituting in these equations from the dynamical equations we have 8 cos 6 = 8q cos 6q — pP (8), 8 sin e = SoSm do -qQ (9), G=Co-rR (10), where 8o cos 6o= V -\- c^ly -U ■\- c^y\ >Sfosin(9o=r-c'n!,'-F-cni (ii), G, = W'-W j 262 MOMENTUM. [CHAP. VI. j lie c'\ 1 1 r= h — M^ M' These are the constants of the impact. >S^0) C^o '^^^ ^be initial velocities of sliding and compression, and Oq the angle which the direction of initial sliding makes with the axis of x. Let us take as the standard case that in which the body M' is sliding along and compressing the body M, so that 8q and Go are both positive. The other three constants p, q, r are independent of the initial motion and are essentially positive quantities. 321. Exactly as in the corresponding problem in two dimen- sions, we shall adopt a graphical method of tracing the changes which occur in the frictions. Let us measure along the axes of cc, y, z three lengths OF, OQ, OR to represent the three re- actions P, Q, B. Then, if these be regarded as the coordinates of a point T, the motion of T will represent the changes in the forces. It will be convenient to trace the loci given by >S^ = 0, (7 = 0. The locus given by 8 = is a straight line parallel to the axis of R ; this we may call the line of no sliding. The locus given by C=0 is a plane parallel to the plane POQ; this we may call the plane of greatest compression. At the beginning of the impact one ellipsoid is sliding along the other, so that according to Art. 154 the friction called into play is limiting. Since P, Q, R are the whole resolved momenta generated in the time t, dP, dQ, dR are the resolved momenta generated m the time dt, the two former being due to the frictional, and the latter to the normal blow. Then the direction of the resultant of dP, dQ must be opposite to the direction in which one point of contact slides over the other, and the magnitude of the resultant must be equal to fidR, where fi is the coefficient of friction. We have therefore -— = cott^=^, — , — - — ^ (13), dQ SoSmOo-qQ {dPy + (dQy = fjL'(dRy..... (i4). The solution of these equations will indicate the manner in which the representative point T approaches the line of no sliding. The equation (13) can be solved by separating the variables. We ffet 1 I (So cos (9o - pPy = OL (So sin Oo - qQY , where a is an arbitrary constant. At the beginning of the motion ART, 324.] IMPACT OF ROUGH ELASTIC ELLIPSOIDS. 263 P and Q are zero, hence we have 1 1 fSo Coseo- pPy ^ f So s in Op- qQ y [ 'Socose, J \ Sosiudo J ^ ^^' which may also be written Scosd y _f Ssiue y . . Socoseo) "Usin6'o/ ^ ^' /sin 6 Y^-P /cos Ooy-P s=s,r44v-^r^»r^ (i7). Vsm 6o^ Vcos 6 J This equation gives the relation between the direction and the velocity of sliding. 322. If the direction of sliding does not change during the impact, 6 must be constant and equal to 6o. We see from (16) that, if p = q, then 6 = 6^', and that conversely if = 6q, S is constant unless p = q. Also, if sin 0^ or cos 6o be zero, 8 must be zero or infinite unless 6 = 6o. The necessary and sufficient condition that the direction of friction should not change during the impact is therefore p — q or sin 2^o = 0. The former of these two conditions, by (12), leads to i-i)+'='=(i-i)=« <^«>- If this condition holds, we have by (13) P = Qcot^o and therefore by (14) P = fMRcosdo, Q = fMRsmOo (19). It follows from these equations that, when the friction is limiting, the representative point T moves along a straight line making an angle tan~^ /jl with the axis of R, in such a direction as to meet the straight line of no sliding. 323. If the condition p = q does not hold, we may, by dif- ferentiating (8) and (9) and eliminating P. Q, and Sy reduce the determination of R in terms of 6 to an integral. By substituting for S from (17) in (8) and (9), we then have P, Q, R expressed as functions of 0. Thus we have the equations to the curve along which the representative point T travels. The curve along which T travels may more conveniently be defined by the property that its tangent, by (14), makes a constant angle tan-^yLt with the axis of R and its projection on the plane of PQ is given by (15). And it follows that this curve must meet the straight line of no sliding, for the equation (15) is satisfied by pP = 80^0^60, qQ = 80^1x160. 324. The whole progress of the impact may now be traced exactly as in the corresponding problem in two dimensions. The representative point T travels along a certain known curve, until 264 MOMENTUM. [CHAP. VI. it reaches the line of no sliding. It then proceeds along the line of no sliding, in such a direction that the abscissa R increases. The complete value R^ of R for the whole impact is found by multiplying the abscissa R^ of the point at which T crosses the plane of greatest compression byl-\-e, so that R^ = jRi (1 + e), if e be the measure of the elasticity of the two bodies. The complete values of the frictions called into play are the ordinates of the positions of T corresponding to the abscissa R^R,^. Substituting these in the dynamical equations (1), (2), (3), (4), the motion of the two bodies just after impact may be found. 325. Since the line of no sliding is perpendicular to the plane of PQ, P and Q are constant when T travels along this line. So that, when once the sliding friction has ceased, no more friction is called into play. If therefore sliding ceases a't any instant before the termination of the impact, as when the bodies are either very rough or perfectly rough, the whole frictional impulses are given by P^ Sq cos 00 ^ ^ /Sfp sin 6^ ~ p ' q ' If a be the arc of the curve in the plane of PQ whose equation is (15) measured from the origin to the point where it meets the line of no sliding, then the representative point T cuts the line of no slidinsf at a point whose abscissa is R = - . If the bodies are /A so rough that - < — , the point T will not cross the plane of greatest compression until after it has reached the line of no sliding. The whole normal impulse is therefore given by C R = — {l-he). Substituting these values of P, Q, R in the dynamical equations, the motion just after impact may be found. 326. Ex. 1. If ^ be the angle which the direction of sliding of one ellipsoid over the other makes with the axis of x, prove that d continually increases or continually decreases throughout the impact. And if the initial value of 6 lie between and ^tt, then 6 approaches ^tt or zero according as p> or ^ — r" — - . CoP{l + e) Ex. 3. If two rough spheres impinge on each other, prove that the direction of sliding is the same throughout the impact. This proposition was given by J. A. Euler, and by Coriolis, Jeu de billard, 1835. See Art. 322. Ex. 4. If two inelastic solids of revolution impinge on each other, the vertex of each being the point of contact, prove that the direction of sliding is the same throughout the impact. This and the next proposition have been given by M. Phillips in the fourteenth volume of Liouville's Journal. ART. 327.] IMPACT OF ROUGH ELASTIC BODIES. 265 Ex. 5. If two bodies having the principal axes at their centres of gravity parallel impinge, so that these centres of gravity are in the common normal at the point of contact, and if the initial direction of sliding be parallel to a principal axis at either centre of gravity, then the direction of sliding will be the same throughout the impact. Ex. 6. If two ellipsoids of equal mass impinge on each other at the ex- tremities of their axes c, c', and if aa' = bb' and ca' = bc', prove that the direction of friction is constant throughout the impact. Ex. 7. A billiard ball rolls without sliding on the table and impinges against a cushion, find the subsequent motion. See also Vol. ii. x\rt, 239. Ed. 1905. Let the planes of the cushion and table be called the planes of xy and xz respectively. Let the initial velocity of the centre of gravity resolved parallel to x and z he -u and - w and let the angular velocity about the vertical be n. After rebounding the ball will describe a series of very small parabolic jumps which are hardly perceptible. Finally the ball may be regarded as rolling on the table. This final motion is given by U' = - u + ^y {u + aji), W = -w + f {l + y + e)w, where y is the smaller of the two quantities ^ and /a (1 + e) wl{w^ +{u + an)"Y. 327. Two rough bodies moving in any manner impinge on each other. Find the motion just after impact. Let us refer the motion to coordinate axes, the axes of x, y being in the tangent plane at the point of impact and the axis of z along the normal. Let U, V, W be the resolved velocities of the centre of gravity of one body just before impact, u, v, w the resolved velocities at any time t after the beginning, but before the termination of the impact. Let D^x, ^yy ^z be the angular velocities of the same body just before impact about axes parallel to the coordinate axes, meeting at the centre of gravity; Wx,coyy (Oz the angular velocities at the time t. Let A, B, G, D, E, F be the moments and products of inertia about axes parallel to the coordinate axes meeting at the centre of gravity. Let M be the mass of the body. Let accented letters denote the same quantities for the other body. Let P, Q, R be the resolved parts parallel to the axes of the momentum generated in the body M from the beginning of the impact, up to the time t. Then — P,-Q,—R are the resolved parts of the momentum generated in the other body in the same time. Let (x, y, z), (oo\ y', z') be the coordinates of the centres of gravity of the two bodies referred to the point of contact as origin. The equations of motion are therefore A{(Dx-^cc)-F{(^y-^y)-E{(Dz-nz)==-yR + zQ\ -F(cox-nx) + B(coy-ny)-Diwz-nz) = -zP-\-xR[ (i), -E{cDx-nx)-D(cOy-ny) + C(aiz-nz) = -xQ + yP\ M{u-U) = P] M(v-V) = q\... (2). M(w-W) = R] 266 MOMENTUM. [CHAP. VI. We have six similar equations for the other body, which differ from these in having all the letters, except P, Q, R, accented, and in having the signs of P, Q, R changed. These we shall call equations (3) and (4). Let S be the velocity with which one body slides along the other and 6 the angle which the direction of sliding makes with the axis of x. Also let G be the relative velocity of compression, then 8 cos = u' — coyz' + (Ozy — u + coyZ - cozy) S sin $ = V — cozw' + a)x2' — v -\- cozX — w^z \ (5). C =W' — (Oxy + 0)yX —W-\- Wxy — (OyOC) If we substitute from (1) (2) (3) (4) in (5) we find, (Art. 314) ^0 cos e- S cos d = aP-]-fQ+ eR) Sosm6-Ssme=fP-hbQ + dR[ (6), Co- C=:^eP +dQ-\-cR) where So, 6o, Go are the initial values of S, 0, G and are found from (5) by writing for the letters their initial values. The expressions for a, h, c, d, e, f are rather complicated, but it is unnecessary to calculate them. 328. We may now trace the whole progress of the impact by the use of a graphical method. Let us measure from the point of contact 0, along the axes of coordinates, three lengths OP, OQ, OR to represent the three reactions, P, Q, R. Then if, as before, these are regarded as the coordinates of a point T, the motion of T will represent the changes in the forces. The equations of the line of no sliding are found by putting 8 = in the first two of equations (6). We see that it is a straight line. The equation of the plane of greatest compression is found by putting (7= in the third of equations (6). At the beginning of the impact one body is sliding along the other, so that the friction called into play is limiting. The path of the representative point as it travels from is given, as in Art. 321, by dP dQ .^ /^x •^ ^ = -^ = fidR (7), cos ^ sm ^ '^ When the representative point T reaches the line of no sliding, the sliding of one body along the other ceases for the instant. After this, only so much friction is called into play as will suffice to prevent sliding, provided that this amount is less than the limiting friction. If therefore the angle which the line of no sliding makes with the axis of R be less than tan"^ /x, the point T travels along it. But if the angle be greater than tan~^ /m, more friction is necessary to prevent sliding than can be called into play. Accordingly the friction continues to be limiting, but its direction is changed if 8 changes sign. The point T then travels along a curve given by equation (7) with 6 increased by tt. See Art. 194. ART. 330.] IMPACT OF ROUGH ELASTIC BODIES. 267 The complete value R^ of R for the whole impact is found by multiplying the abscissa R^ of the point at which T crosses the plane of greatest compression by 1 + e, where e is the measure of el isticity, so that i^2 = -^i (1 + ^)- The complete values of P and Q Ml represented by the ordinates corresponding to the abscissa R^. Substituting in the dynamical equations, the motion just after impact may be found. 329. The path of the representative point before it reaches the line of no sliding must be found by integrating (7). By differentiating (6) we have d (S cos 6) _ adP +fdQ + edR __ a^i cos 6 -\- f^i s\n 6 -\- e .„. d{S sin 6) ~ fdP + hdQ + ddR" fyucos 6 + bfism 6 + d'"^ ^' ►which reduces to ^cos2l9 + /'sin2l9 + .(9). - ,^ -^r— + ^s— cos2^ + /sm2^+-cos^4--sm^ 1 dS _2 2 "^ /x /JL TT— sm 26 -\- f cos 26 + - cos 6 sm ^ From this equation we may find >Sf as a function of 6 in the form S = Af(6), the constant A being determined from the j condition that S — So when 6 = 6^. Differentiating the first of equations (6) and substituting from (7) we get -Ad{cos6f (6)}= (fia cos + fif sin + e)dR (10), ►whence we find R = AF(0) + B, the constant B being determined from the condition that R vanishes when 6 = 6q. By substituting sthese values of S and R in the first two equations of (6) we find P and Q in terms of 6. The three equations giving P, Q, R as functions of 6 are the equations to the path of the representative ipoint. It should be noticed that the tangent to the path at any (point makes with the axis of R an angle equal to tan~^ /x. 330. If the direction of friction does not change during the impact, 6 is constant and equal to 6^, so that 6 cannot be chosen as the independent variable. In this case P = fjuR cos 6o,Q= jjlR sin 6^ and the representative point moves along a straight line making with the axis of R an angle tan~^ /i. Substituting these values of P and Q in the first two of equations (6) we have - ^^ sin 2l9o +/cos 26^ + - cos 6>o - - sin (9o = . . .(11) Z '^ /JL fJb as a necessary condition that the direction of friction should not change. Conversely, if this condition is satisfied, the equations [Q) and (7) may all be satisfied by making 6 constant. In this sase it is also easy to see that the path of the representative point intersects the line of no sliding. If Sq be zero, the representative point is situated on the line of 268 MOMENTUM. [CHAP. VI. no sliding. If the angle made by this straight line with the axis of R be less than tan~^ ^, the representative point travels along it. But if the angle be greater than tan~^yu,, more friction is necessary to prevent sliding than can be called into play. Since S^^ is zero, the initial value of 6 is unknown. In this case, differentiating the first two equations of (6) and putting >S^= 0, we see by division that the initial value of 6 must satisfy equation (11). The condition that the direction of friction does not change is therefore satisfied. This value of 6 makes the subject of integration in (9) infinite, so that the reasoning there given must be modified. But, by what has just been said, we see that the path of the representative point is a straight line, which makes with the axis of R an angle equal to tan~^ fjL, and has the proper initial value of 6. 331. Ex. 1. Let G = A -F -E yR-zQ -F B -D zP-xR -E -D G xQ-yP yR-zQ zP-xR xQ-yP and let A be the determinant obtained by leaving out the last row and the last column. Let G', A' be corresponding expressions for the other body. Then a, b, c, d, e, /are the coefficients of P-, Q^, R-, 2QR, 2RP, 2PQ in the quadric where 2E is a constant, which may be shown to be the sum of the vires vivae of the motions generated in the two bodies, as explained in Art. 314. This quadric may be shown to be an ellipsoid by comparing its equation with that given in Art. 28, Ex. 3. Show also that a, b, c are necessarily positive, and that ab>f^, bod^, ca>e^. Show that, by turning the axes of reference round the axis of R through the proper angle, we can make / zero. Ex. 2. Prove that the line of no sliding is parallel to the conjugate diameter of the plane containing the frictions P, Q. Prove also that the plane of greatest compression is the diametral plane of the reaction R. Ex. 3, The line of no sliding is the intersection of the polar planes of two points situated on the axes of P and Q, at distances from the origin respectively op 0,V and :^ — -. — — . The plane of greatest compression is the polar plane of >Sft sin Q, 2E a point on the axis of R, distant — - from the origin. C, Ex. 4. The plane of PQ cuts the ellipsoid of Ex. 1 in an ellipse, whose axer divide the plane into four quadrants; the line of no sliding cuts the plane of PQ in that quadrant in which the initial sliding Sq occurs. Ex. 5. A parallel to the line of no sliding through the origin cuts the plane of greatest compression in a point whose abscissa R has the same sign as Cq. Hence show, from geometrical considerations, that the representative point T must cross the plane of greatest compression. EXAMPLES. * 269 EXAMPLES*. 1. A cone revolves round its axis with a known angular velocity. The altitude begins to diminish and the angle to increase, the volume being constant. Show that the angular velocity is proportional to the altitude. Art. 299. 2. A circular disc is revolving in its own plane about its centre ; if a point in the circumference becomes fixed, find the new angular velocity. Art. 171 a. 3. A uniform rod of length 2a lying on a smooth horizontal plane passes through a ring which permits the rod to rotate freely in the horizontal plane. The middle point of the rod being indefinitely near the ring, any angular velocity is impressed on it, show that when it leaves the ring the radius vector of the middle point has swept out an area equal to }a^. 4. An elliptic lamina is rotating about its centre on a smooth horizontal table. If Wi, W2, W3 be its angular velocities when the extremity of its major axis, its focus, and the extremity of its minor axis respectively become fixed, prove that 5. A rigid body moveable about a fixed point at which the principal moments are A, B, C is struck by a blow of given magnitude at a given point. If the angular velocity thus impressed on the body be the greatest possible, prove that, [a, b, c) being the coordinates of the given point referred to the principal axes at 0, and {I, m, n) the direction cosines of the blow. a/1 1\6/1 l\c/l 1 al + bm + cn = 0, - (^_ - _ j + _ (^_ - _ j + - (^_ - _ :0. 6. Any triangular lamina ABC has the angular point C fixed and is capable of free motion about it. A blow is struck at B perpendicularly to the plane of the triangle. Show that the initial axis of rotation is that trisector of the side AB which is furthest from B. Eeplacing the lamina by its three equivalent particles and equating to zero the angular momentum about BC, Art. 149, it is evident that the particles at E and F (bisecting AC, AB) have equal and opposite initial velocities. It follows that the instantaneous axis bisects EF and passes through C. Considering this axis as a transversal of the triangle AEF, we deduce the result given. 7. A cone of mass vi and vertical angle 2a can move freely about its axis, which is vertical and has a fine smooth groove cut along its surface so as to make a constant angle /3 with the generating lines of the cone. A heavy particle of mass P moves along the groove under the action of gravity, the system being initially at itst with the particle at a distance c from the vertex. Show that, if 6 be the angle through which the cone has turned when the particle is at any distance r from the vertex, mk^ + Pr^sin^a _ .q ^^^ „ ^ot/S mk^ + Pc^sin^a k being the radius of gyration of the cone about its axis. 8. A body is turning about an axis through its centre of gravity when a point P in it becomes suddenly fixed. If the new instantaneous axis be a principal axis at P, show that the locus of P is a rectangular hyperbola. * These examples are taken from Examination Papers which have been set in the University or in the Colleges. 270 MOMENTUM. [CHAP. VI. Just before P is fixed the whole momentum is equivalent to a couple G acting in the diametral plane of the instantaneous axis with regard to the momental ellipsoid at 0, Art. 118 or 310. When P is fixed we may suppose the body to be at rest and acted on by the couple G ; it therefore begins to turn about the diame- tral line of the plane of G with r^ard to the momental ellipsoid at P ; see Art 297. By the question this is to be a principal axis, and it is therefore perpendicular to its diametral plane. The locus of P is therefore such that one principal axis at P is parallel to a fixed straight line, viz. the perpendicular to the plane of G. The locus is a rectangular hyperbola by Art. 51, Ex. 4. 9- A cube is rotating with angular velocity u about a diagonal when one of its edges which does not meet the diagonal suddenly becomes fixed. Show that the angular velocity u' about this edge is given by 4 ^/3a>' = w. 10. Two masses m, m' are connected by a fine smooth string which passes rotmd a right circidar cylinder of radius a. The two particles are in motion in one plane under no impressed forces, show that, if J be the sum of the absolute areas swept out in a time t by the two unwrapped portions of the string, T being the tension of the string at any time. 11. A piece of wire in the form of a circle lies at rest with its plane in contact with a smooth horizontal table, when an insect on it suddenly starts walking along the arc with uniform relative velocity. Show that the wire revolves round its centre with tmiform angular velocity, while that centre describes a circle in space with uniform angular velocity. 12. A uniform circular wire of radius a, moveable about a fixed point in its circumference, lies on a smooth horizontal plane. An insect of mass equal to that of the wire crawls along it, starting from the extremity of the diameter opposite to the fixed point, its velocity relative to the wire being uniform and equal to V. Prove that after a time t the wire will have turned through an angle 73*^"(;r3*^9- 2a V3 13. A small insect moves along a uniform bar, of mass equal to itself, and oi length 2a, the extremities of which are constrained to remain on the circumference of a fixed circle, whose radius is 2a/^3. Supposing the insect to start from the middle point of the bar, and its velocity relatively to the bar to be uniform and equal to F; prove that the bar in time t will turn through an angle 6 where a tan {d^fS) = Vt 14. A circular disc can revolve freely in a horizontal plane about a vertical asi through its centre. An equiangular spiral is traced on the disc, having the cent: for pole. An insect whose mass is n times that of the disc crawls along the curve starting from the point at which it cuts the edge. Show that, when the insect reache the centre, the disc has revolved through an angle ^ tan a log (1 + 2n), where a i the angle between the tangent and the radius vector at any point of the spiral. 15. A uniform circular disc moveable about its centre in its own plane (whic is horizontal) has a fine groove in it cut along a radius, and is set rotating wit an angular velocity w. A small rocket whose weight is an n^^ of the weight of th disc is placed at the inner extremity of the groove and discharged ; when it ha left the groove the same is done with another equal rocket, and so on. Find th angular velocity after n of these operations, and, if n be indefinitely increased, she that the limiting value of the same is we"^. EXAMPLES. 271 16. A rigid body is rotating about an axis through its centre of gravity, when a ain point of the body becomes suddenly fixed, the axis being simultaneously set ; find the equations of the new instantaneous axis ; and prove that, if it be allel to the originally fixed axis, the point must lie in the line represented by equations a^ lx + h^my + c^nz = 0, (62 -c^)j + (c^ - a^) | + (a^ -b^)^- = 0; where the Lucipal axes through the centre of gravity are taken as axes of coordinates, a, b, c p> the radii of gyration about these lines, and I, m, n the direction-cosines of originally fixed axis referred to them. Art. 296. 17. A solid body rotating with uniform velocity w about a fixed axis contains osed tubular channel of small uniform section, filled with an incompressible fluid 1 relative equilibrium ; if the rotation of the solid body were suddenly destroyed t; fluid would move in the tube with a velocity v given by vl = 2A(a, where A is the a of the projection of the axis of the tube on a plane perpendicular to the axis otatiqn, and I is the length of the tube. Any element of mass mds is moving with velocity a;r in a direction normally to plane containing the element and the axis of rotation. The normal pressures cthe tube destroy all motion perpendicular to the tube, so that we need only cisider the component wr.rddlds, Art. 307. Each element impinges on those aacent, but the hnear momentum is unaltered by this impact. Integrating the omentum along the whole tube, we have mlv=jin is called the work done by the force. If for we write the angle made by the direction AB oi the force with the direction A' A, opposite to the displacement, the product is called the work done against the force. If we drop a perpendicular A'M on AB, the work done by the force is also equal to the product F .AM, where AM is to be estimated as positive when in the direction of the force. If F' be the resolved part of F in the direction of the displacement, the work is also equal to F' . AA'. If several forces act, we can in the same way find the work done by each. The sum of all these is the work done by the whole system of forces. Thus defined, the work done by a force, corresponding to any indefinitely small displacement, is the same as the virtual moment of the force. In statics we are only concerned with the small I hypothetical displacements given to the system in applying the principle of virtual work, and this definition is therefore ^sufficient. But in dynamics the bodies are in motion, and we i simply use it as an abridged mode of expressing a fact. D'Alembert then points out that there are different kinds of obstacles and examines how their different kinds of resistances may be used as measures. It will perhaps be sufficient to observe, that the resistance may in some cases be more conveniently measured by a space-integral and in others by a time- integral. See Montucla's Historyy '"Vol. III. and Whewell's History, Vol. ii. R. D. 18 274 VIS VIVA. [chap. vii. must extend our definition of work to include the case of a displacement of any magnitude. When the points of application of the forces receive finite displacements we must divide the path of each into elements. The work done in each element may be found by the definition given above. The sum of all these is the whole work. It should be noticed that the work done by given forces, as the body moves from one given position to another, is independent of the time of transit. As stated in Art. 332, the work is a space- integral and not a time-integral. 335. If two systems of forgoes be equivalent, the work done by one in any small displacement is equal to that done by the other. This follows at once from the principle of virtual work in statics. For if every force in one system be reversed in direction without altering its point of application or its magnitude, the two systems will be in equilibrium, and the sum of their virtual moments will therefore be zero. Restoring the system of forces to its original state, we see that the virtual moments of the two systems are equal. If the displacements are finite the same remark applies to each successive element of the displacement, and therefore to the whole displacement. 336. We may now find an analytical expression for the work done by a system of forces. Let {x, y, z) be the rectangular coordinates of a particle of the system and let the mass ot this particle be m. Let (X, F, Z) be the accelerating forces acting on the particle resolved parallel to the axes of coordinates. Then mX, mY, mZ are the dynamical measures of the acting forces. Let us suppose the particle to move into the position x + dx, y + dy,jz + dz; then, according to the definition, the work done by the forces will be %{mXdx + mYdy + mZdz) (1), the summation extending to all the forces of the system. If the bodies receive any finite displacements, the whole work will be l.mj{Xdx + Ydy + Zdz) (2), the limits of the integral being determined by the extreme positions of the system. 337. Force-flinction. When the forces are such as gener- ally occur in nature, it will be proved that the summation (1) of the last Article is a complete differential, i.e. it can be integrated independently of any relation between the coordinates x^ y, z. The summation (2) can therefore be expressed as a function of the coordinates of the system. When this is the case the indefinite integral of the summation (2) is called the force-function. This name was given to the function by Sir W. R. Hamilton and Jacobi independently of each other. ART. 839.] FORCE-FUNCTION AND WORK. 276 If the force-function be called U, the work done by the forces when the bodies move from one given position to another is the definite integral U^ — U^, where U^ and U^ are the values of U corresponding to the two given positions of the bodies. It follows that the work is independent of the mode in which the system I moves from the first given position to the second. In other words, \the work depends on the coordinates of the two given extreme .positions, and not on the coordinates of any intermediate position. When the forces are such as to possess this property, i.e. when : they possess a force-function, they have been called a conservative system of forces. This name was given by Sir W. Thomson, now iLord Kelvin. :^38. There will he a forcefunction, firstly, when the external forces tend to fi^ed centres at finite distances and are functions of tlie distances from those centres ; and, secondly, when the forces due to the mutual attractions or repulsions of the particles of the system are functions of the distances between the attracting or repelling particles. i Let m{r) be the action of any fixed centre of force on a [particle m distant r, estimated positive in the direction in which r is measured, i.e. from the centre of force. Then the summation (1) in Art. 336 is clearly %m(f){r)dr. This is a complete differential. Thus the force-function exists and is equal to ^mjcf) (r) dr. Let mm'(f) (r) be the action between two particles m, m' whose distance apart is r, and as before let this force be considered positive when repulsive. Then the summation (1) becomes \1mm'(f>(r)dr. The force-function therefore exists, and is equal to Xmm^ j (j) (r) dr. If the law of attraction be the inverse square of the distance, ^ (r) = and the integral is - . Thus the force-function differs from the Potential by a constant quantity. 339. It is clear that there is nothing in the definition of the force-function to compel us to use Cartesian coordinates. If P, Q, kc. be forces acting on a particle, Pdp, Qdq, &c. their virtual moments, m the mass of the particle, then the force-function is U= ^mf(Pdp + Qdq -}- &c.), the summation extending to all the forces of the system. Ex. 1. If {p, ) be the polar coordinates of the particle m; P, Q, R the cesolved parts of the forces respectively along the radius vector, perpendicular to it n the plane of 6, and perpendicular to that plane, prove that dU = i:m{Pdr + Qrdd + Rr sin dd(p). 18—2 276 VIS VIVA. [chap. vii. Ex. 3. If {x, y, z) be the oblique Cartesian coordinates of m; X, Y, Z the components along the axes, prove that dU= I>ni { X {dx + vdy + ixdz) + Y (udx + dy + Xdz) + Z {[xdx + \dy + dz)], where (\, /*, v) are the cosines of the angles between the axes yz, zx, xy respectively. This result is due to Poinsot. 340. If a system receive any small displacement ds parallel to a given straight line and an angular displacement dO round the line, then the partial differential coefficients dll/ds and dU/dO represent respectively the resolved part of all the forces along the line and the moment of the forces about it. Since dU is the sum of the virtual moments of all the forces due to any displacement, it is independent of any particular co- ordinate axes. Let the straight line along which ds is measured be taken as the axis of 2. Taking the same notation as before, dU=lm {Xdx + Ydy + Zdz). But dx = 0, dy = 0, and dz = ds, hence we have dU^ds.XmZ', .\dUlds = lmZ. Here dU means the change produced in Uhy the single dis- placement of the system, taken as one body, parallel to the given straight line, through a space ds. Again, the moment of all the forces about the axis of z is 'Zm(a)Y—yX), but dx = — yd6, dy = ocdd , 'a,nd dz = 0. Hence the above moment _^ Ydy + Xdx + Zdz dJJ =^™-^^^ =d0- Here dll is the change produced in U by the single rotation of the system, taken as one body, round the given axis, through an angle dO. 341. As considerable use will be made of the force-function, the student will find it advantageous to acquire a facility in writing down its form. The following examples have therefore been chosen as likely to be most useful. 342. Work done by gravity. A system of bodies falls under the action of gravity. If M be the whole mass, h the space descended by the centre of gravity of the whole system, the work done by gravity is Mgh. See Art. 140. Let the axis of z be vertical and let the positive direction be downwards. Then in the summation (1) of Art. 336, X=0, F=0 and Z = g. Hence dU=I,mgdz. If i be the depth of the centre of gravity below the plane of xy, and C be any constant, we find V = Mgz + C. Taking this between limits we easily obtain the result given. Units of work. The theoretical unit of work is the work done by a dynamical unit of force acting through a unit of space. We may use the result of this example to supply a practical unit. 1 The work required to raise the centre of gravity of a given massj ART. .343.] VARIOUS KINDS OF WORK. 277 a given height at a given place may be taken as the unit of work. English engineers use a pound for the mass and a foot for the height, and the unit is then called d^ foot-pound. The term Horse- power is used to express the work done per unit of time. The unit of horse-power is usually taken to be 33000 foot-pounds per minute. The duty of a steam-engine is the actual work done by the consumption of a unit quantity, usually a bushel, of coal. A more complete account of the various units used in dynamics is given in the author's treatise on Dynamics of a Particle. Ex. 1. A force communicates to a particle whose mass is equal to that of a cubic foot of water a velocity of one foot per minute. Find the work in foot-pounds. Ex. 2. Determine the resistance of a steamer in tons when 8000 effective horse- power is required to drive it at 17^ knots (of 6080 feet) per hour. [Univ. of London, 1886. Ex. 3. Supposing a tricycle and rider weighing together 200 lbs. to run uniformly at 8 miles an hour down an incline of one in 100 against the resistances of the air and of the road, without working the pedals ; prove that to go up an incline of one in 200 at the same speed the rider must be working at the rate of -064 of a horse-power ; and that the mean pressure on each pedal will then be about 12*672 lbs., supposing the cranks to be 5 inches long and to make 100 revolutions a minute. [Univ. of London, 1886. Ex. 4. Prove that the amount of work required to raise to the surface of the earth the homogeneous contents of a very small conical cavity, whose vertex is at the centre of the earth, is equal to that which would be expended in raising the whole mass of the contents through a space from the surface equal to one-fifth of the earth's radius, supposing the force of gravity to remain constant. [Coll. Exam. 343. Work of an elastic string. Ex. If the length of an elastic string or rod which is uniformly stretched be altered, the work done by the tension is the product of the compression of the length and the arithmetic mean of the initial and final tensions. Let the length be altered from r to r'. Let p be any length between these two, let I be the unstretched length, and let E be the constant of elasticity. The tension is T = E ^-r— and acts opposite to the direction in which p is measured. The work done while p becomes p-\-dp\^ therefore equal to - Tdp. If we integrate this 2i from p = r to p = r' we find that the work required is -^^ {(^' - 0^~ (^~ ^)^}- This leads at once to the result given. If a string becomes slack, the tension is supposed to vanish, and no work is done until the string again becomes tight. In applying the rule, the compression is the difference between the two terminal lengths if the string he tight in both, whether it has been slack or not during the various changes of length which may have occurred during the process. If the string be slack in either terminal state we must in calculating the compression suppose the string to have its unstretched length in that terminal state. In the case of a rod the tension becomes negative when the rod is compressed, and the rule applies so long as the rod remains straight, and we can suppose Hooke's law to be true. 278 VIS VIVA. [chap. vii. If the string is not straight but is uniformly stretched over a surface or in a fine tube, the same rule to find the work is still true. To prove this, we divide the string into elements, each of which may be considered as straight. When the whole string is now uniformly stretched the work done is the mean of the tensions into the sum of the contractions of all the elements. This last is clearly the con- traction of the whole string. If the surface be fixed the string cannot contract without one, at least, of the extremities moving, and in this case the work is done at that extremity. If the surface move, and the extremities of the string be fixed in space, the work is transferred to the surface by means of the reactions. If the string have no effective forces, these reactions are in equilibrium with the tensions at the points A, B where the string leaves the surface. Now let the surface receive any small displacement. By the principle of virtual work the work done by the reactions on the surface is equal to that done by the two equal tensions at the points A, B. But this work is the instantaneous tension into the contraction of the string, i.e. it is - Tdp. If the surface receive a finite displacement, the work done is the integral of this expression, and the rule is of course the same as before. Whether the string have mass or not, we may consider each separate element of it as one of the moving bodies whose motion enters into the equation of vis viva. The work done by the contraction of all the elements is to be regarded as distributed over all the bodies. The work done by the equal and opposite reactions between the string and surface will then be zero. 344. Work of collecting a body. Ex. 1. If m, m' be the TYlTt}! masses of two particles attracting each other with a force — —^ where r is the distance between them, show that the work done by the mutual force when they have moved from an infinite distance apart to a distance r is . This follows from Art. 338. If the particles repel each other we regard either m or m' as negative. Ex. 2. Let two finite masses M, M' attract each other and occupy given positions. Prove that the work of bringing the par- ticles of one from infinite distances apart into their given positions Tinder the attraction of the second, supposed fixed in its given position, is the same as that of bringing the particles of the second from infinity into their positions under the attraction of the first. Prove also that this work may he found by taking both bodies in their final positions and multiplying the mass of each element of one body by the potential of the other at that element, then inte- grating throughout the volume of the former body. This integral is sometimes called the mutual work or the inutual potential of the two bodies. Let there be two sets of attracting particles which we may represent by wij, mg, &c., TOj', Wg', &c., and let the particles of each set attract the particles of the other set, but not the particles of its own set. Suppose the particles m^, m^, &g. to occupy any given positions, and let one particle mf of the second set be brought from an infinite distance to any given position, say to a position at distances ART. 344.] VARIOUS KINDS OF WORK. 279 Vj, u, &c. from the particles m^ , m^, &c. The work done is w' ( — i + — +&c..) =m'V, where V is the potential of the attracting masses at the given position of m'. Let us now bring in succession all the particles r/i/, m.2, &c. from infinite distances to their given final positions under the attraction solely of the masses ■; m^, ruo, &c. The whole work is 1,m'V, which may also be written in the sym- \ metrical form S , where r is the distance between the particles m, mf, and the [ S implies summation for every combination of each particle of one set with each particle of the other. This symmetrical form proves the first part of the pro- position. The particles may be elementary, and in that case we see that the work of collecting any mass M' into a given position under the attraction of a mass M placed in a given position is equal to jVdm', where V is the potential of the mass M at the final position of dm' and the integration extends over the whole mass of M\ Ex. 3. If the particles composing any given mass were sepa- rated from each other, work might be obtained from their mutual attractions by allowing the particles to approach each other. The work thus obtained is greatest when the particles are collected together from infinite distances. If dv be an element of volume of a solid mass attracting according to the law of nature, p the density of the element, V the potential of the solid mass at the element dv, prove that the work performed in collecting the particles composing the mass from infinite distances is ^fVpdv. The problem of determining how much work can be obtained from the bodies forming the solar system by allowing them to consolidate into a solid mass has been considered by several philo- sophers. Sir W. Thomson has calculated that the potential energy or the work which can be obtained from the existing solar system is 38 X W foot-pounds. Edin. Trans. 1854. As we bring the particles in succession into their proper places we find the whole work by multiplying the mass of each by the potential at it of the mass already collected and summing the products. We shall prove that the same result is obtained by mnltiplying the mass of each by the potential at it of the whole mass finally collected together, provided we take only half the sum. Let wij, wig, &c. be the masses of the particles ; let (1, 2); (2, 3) ; &c. be the distances between the masses m^, wig ; wig, m^ ; \ • ^^^ work of collecting M is (12) (1 2 ) rn^Tn^ m^m^' m^m^ m^m^ m^m^ m{m^ The mutual potential of M, M' is ""i V(ll') (21') (r2') j ^^ V(12') (22') (2'!')/ ' where, as explained above, the term m^ is omitted in the first bracket and m^ in the second. The theorem follows by an obvious substitution. Ex. 7. A quantity of homogeneous matter is bounded by two spheres which do not intersect, one sphere being wholly within the other. The radii of the spheres are a and h, and the distance between the centres is c. Show that the work of collectmg this matter from infinite distances is <— — + ttt + -a~\ • 345. Work of a gaseous pressure. Ex. 1. An envelope lof any shape, whose volume is v, contains gas at a uniform pressure p. Assuming that the pressure of the gas per unit of area is some function of the volume occupied by it, prove that the work done hy the pressure when the volume increases from v = a to v = b is fpdv, the limits being v = a to v = b. Divide the surface into elementary areas each equal to rfcr, then pdcr is the pressure on dcr. When the volume has increased to v + dv, let any element da- take the position dcr', and let dn be the length of the perpendicular drawn from the central point of da' on the plane of da, then pdcxdn is the work done by the pressure on da and pjdadn is the work done over the whole area. But dadn is the volume of the oblique cylinder whose base is da and opposite face da' ; so that jdadn is the whole increment of volume. The whole work done when the volume increases by 4v is therefore pdv. Ex. 2. A spherical envelope of radius a contains gas at pressure P. Assuming that the pressure of the gas per unit of area is inversely proportional to the volume occupied by it, prove that the work required to compress the envelope into a sphere of radius b is Aira'^P log a/b. Ex. 3. An envelope of any shape contains gas and the shape is altered without altering the volume. Show that the work done over the whole surface is zero. I Ex. 4. A hollow cylinder contains equal masses of two different elastic fluids ) at the same pressure P separated by a piston without weight. Show that the work done in moving the piston till the densities of the two fluids are inter- ^ changed is PA {a - b) log a/b, where A is the area of the piston, and a, b are the lengths of the portions of the cylinder occupied by the fluid. [Pembroke College, 1868. Ex. 5. A mass of air of uniform density p (1 + s) is enclosed in an envelope and I 282 VIS VIVA. [chap. vii. surrounded by air of atmospheric density p. If the mass expand until its density is equal to that of the atmosphere, prove that the work done is k ( log (1 + s) 1 + s where k is the product of the pressure and the volume. If s be small the work is very nearly ^ks^. This result is useful in the theory of sound. 346. Work of an Impulse , Ex. 1. An impulsive force acts on a body in a fixed direction in space. Show that, if F be the whole momentum communicated by the force, Uq, u^ the velo- cities of the point of application, resolved in the direction of the force, just before and just after the impulse, the work done by the impulse is \F{Uq + Wj). This result is given in Thomson and Tait's Natural Philosophy. When a force is measured in the usual way by the momentum generated per unit of time, the work is measured by the product of the force into the resolved displace- ment. But impulses are not so measured, we cannot therefore directly apply this rule to find the work of an impulse. Let us regard the impulse as the limit of a finite force acting in the fixed direc- tion for a very short time T. Let the direction of the axis of x be taken parallel to the fixed direction and let X be the whole momentum communicated during a time t measured from the commencement of the impulse. Here t is any time less than T, and X varies from zero to i^ as f varies from to T. Also, since X is the whole momentum up to the time t, X is the acting force on the body at the time t. Let M he the resolved velocity of the point of application at the time t, then m,, and Mj are the values of u when t = and t = T. Since udt is the space described in the time dt by the point of application of the force X, the work done in the time T is jtidX, from Z = to F. To integrate this we must know what function u is of X. If the body be a particle of mass m, we know that, when the time of action is very small, m{u- Uq) = X, hence, substituting for u, we find after integrating U(,F + ^F^jm. When X = F we have by definition u = Ui, :. m [u^ - u^ = F. Eliminating m, we find the work is \F{Uq-\-u^). If the body be moving in two dimensions, let u be the velocity of the centre of gravity at the time t resolved parallel to the direction of the impulse, and w the angular velocity ; we then have by Arts. 168 and 137 m {u - Uq) — X, mW (w - Wq) = Xp, u = u+ wp. Hence u = Uq + LX where i, is a quantity independent of X and therefore constant during the integration. Substituting for u, the integral takes the form F {u^ + \LF). But as before u-^^ = Uq-\-LF. Eliminating L the result follows at once. If the body be moving in three dimensions, the velocity u is known by Art. 313 to be a linear function of X, so that we may write w = m<)-|- LX, where L is a constant depending on the nature of the body. Substituting this value of u, we have the work equal to \[uq + LX) dX= UqF+L — , the limits being to F. But w^ = w^ -f LF. Eliminating L we find that the work = |(wo + Mi) F. Ex. 2. If one blow Fj be followed immediately by a second blow Fg ^-t the same point in the same straight Ime, and if u^, u-^^^u^ be the resolved velocities of the point of application before and after the blows, verify that the work ^ ("0 + ^*2) (^i + -^2) ^^ the whole blow is the sum of the works of the separate blows, viz. ^ {uq + u-^) F^ and \{u^ + u<^F<^. This follows at once, since u^ = Hq + LF-^ and u^ = u^-{-LF^. The results of Ex. 3 may be deduced from Ex. 1 in this manner. ART. 348.] VARIOUS KINDS OF WORK. 283 Ex. 3. Find the wftrk done by an impulse whose direction is not necessarily the same during the indefinitely short duration of the force. Let X, Y, Z be the components of the whole momentum given to the body in any time t measured from the commencement of the impulse. Let u, v, w be the resolved velocities of the point of application at the time t. Then, by the same fT . reasoning as before, the work done = / {Xu +Yv + Zw) dt. But by Art. 314 when T Jo . . , ^ ., , „ dE dE dE . ^ . . IS mdefinitely small u = Uq + ——, ^ = ^o + :rv-» ^ = ^o + :r^» where E is a known dX d 1 dZ quadratic function of {X, Y, Z) depending on the nature of the body. Substituting we have the work = u^X^ + v^Y^ + WQZ^+\(^dX + -^dY-\--^dz\=u^X^ + v^Y^ + io^Z^+E^, where X^, Y^, Z^, E^ are the values of X, 7, Z, E when t = T. We may eliminate the form of the body and express the work in terms of the resolved velocities of the point of application just after the termination of the impulse. Since jEj is a homogeneous quadratic function of X-^, Y^, Z-^, we have Substituting we find the work = ^^4^i X, + "^4^ Y, + "^^^^ Z^. £i 2i £ 347. "Work of a membrane equally stretched in all directions. Consider a rectangle whose sides are a and &, which may be considered as an element. Let T be the tension across any line referred as usual to a unit of length. The tension across the side a is Ta, and when the side b has increased to &' the work done by these will be Ta{b'-b). Supposing the tension across the side b' to be still T, (which is true when the rectangle is an element) the tension across the whole length will be Tb', and, when the side a becomes a', the work will be Tb' {a' - a). The whole work is therefore T {a'b' -ab), i.e. the work is the product of the tension and the change of area. If the membrane is spherical, the area is 47rr2. The increase of area is therefore 8irrdr. Hence the work done by the tensions when the radius is increased from r=a to r = b is SirjTrdr, the limits being r=a to b. If the membrane be such that we may apply Hooke's law to the tension T, we have T=E , where a is the natural radius of the membrane and E is the a coefficient of elasticity. Substituting this value of T we find that the work done A -pi by the tensions, when the radius increases from a to b, is — {b-a)^{2b + a). o a If we assume that for a soap-bubble T is constant, we find that the work done when the radius increases from a to 6 is AttT {b'^-a^). If we suppose the spherical membrane to be slowly stretched by filling it with gas at a pressure^, we have by a theorem in hydrostatics |>r= 2 T. In this case the work required has been shown to he jpdv, and, since v = ^Trr^, this leads to the same result as before. 348. Work of a couple. Ex. A given couple is moved in its own plane from one position to another ; show that the work is the product of its moment by the angle turned through. 284 VIS VIVA. [chap. vii. Any displacement of a couple is equivalent to a rotation round one extremity of its arm and a transference of the whole couple parallel to itself. The work done by the two forces during the transference is clearly zero. We need therefore only consider the work done during the rotation. Let F be the force, a the length of the arm, and let the couple be turned round one extremity A of its arm through an angle dd. The force at A does no work, and the work done by the other force is F . add. Integrating this we have the work done by the couple when it turns through any finite angle. 349. Work of bending a rod. Ex. 1. A rod originally straight is bent in one plane. If L be the stress couple at any point, p the radius of curvature, it is known both bv experiment E and by theory that Z = — , where ^ is a constant depending on the nature of the material, and the form of a section of the rod. Assuming this, prove (1) when the rod is bent into a given form, so that /9 is a known function of s (whether the forces are known or {E not) the work is ^ j-^ds, (2) when the rod is bent by known forces so that X is a known function of s (whether the form of the rod is known or not) the work is \ I ^ ds. The limits of integration are from one end of the rod to the other. Let PQ be any element of the rod and let its length be ds. As PQ is being bent, let yp be the indefinitely small angle between the tangents at its extremities, then the stress couple is E^. As \^ increases from to — the work done is i- / i^dxp, which is the same as — -^ . The work done on the whole rod is therefore Ex. 2. A uniform heavy rod of length I and weight w is supported at its two extremities so as to be horizontal. Show that the work done by gravity in bending '' '' mE- Ex. 3. A uniform light rod is supported at its extremities A and B, and supports a weight iv at any point C. If AC=a, BG = b and l = a + b, the work done by gravity in bendmg the rod is ^ c^; • Conservation of Vis Viva and Energy. 350. Def. The Vis Viva of a particle is the product of its mass and the square of its velocity. The principle of vis viva. If a system be in motion under the action of finite forces, and if the geometrical i^elations of the parts of the system be expressed by equations which do not con- tain the time explicitly, the change in the vis viva of the system in passing from any one position to any other is equal to twice the corresponding work done by the forces. 1 [^^ ART. 351.] PRINCIPLE OF VIS VIVA. 285 In determining the force-function all forces may be omitted which do not appear in the equation of virtual work. Let X, y, z be the coordinates of any particle m, and let X, F, Z be the resolved parts in the directions of the axes of the impressed accelerating forces acting on the particle. The effective forces acting on the particle m at any time t are S?x d?y (Pz ^^' ^^^' ^^- If the effective forces on all the particles be reversed, they will be in equilibrium with the whole group of impressed forces, by Art. 67. Hence, by the principle of virtual work, .„j(._S),.,(._g)a,.(.-£)4=o, where hx. By, Sz are any small arbitrary displacements of the par- ticle m consistent with the geometrical relations at the time t. Now if the geometrical relations are expressed by equations which do not contain the time explicitly, the geometrical relations which hold at the time t will hold throughout the time 8t ; and, therefore, we can take the arbitrary displacements Bx, By, Bz to be respectively equal to the actual displacements -7- Bt, ~ Bt, -r. Bt, of the particle in the time Bt Making this substitution, the equation becomes V /(PoG dx d^y dy d^z dz\ ^ ( v dx ^ ^^dy ^ r, dz\ Integrating, we get ^™ {sy + (ST + (SI = ^+ ^^^/^^'^^ + ^'^^ +^'^^>' where (7 is a constant to be determined by the initial conditions of motion. Let V and v' be the velocities of the particle m at the times t and t'. Also let C/i, C/g be the values of the force -function for the system in the two positions which it has at the times t and t'. Then l^mv' - Smi;^ = 2{U'^- U^). 351. The following illustration, taken from Poisson, may show more clearly why it is necessary that the geometrical relations should not contain the time explicitly. Let, for example, {x,y,z,t) = (1) be any geometrical relation connecting the coordinates of the particle m. This may be regarded as the equation to a moving surface on which the particle is constrained to rest. The quantities 286 VIS VIVA. [chap. vii. hx, By, Bb are the projections on the axes of any arbitrary displacement of the particle m consistent with the geometrical relations which hold at the time t. They must therefore satisfy the equation ^^^Md^^^'^^_ ax ay ^ dz The quantities -^fBt, ~^Bt, -rrBt are the projections on the axes of the displacement of the particle due to its motion in the time Bt. They must therefore satisfy the equation ax at ay at dz dt dt Hence, unless ~ is zero throughout the whole motion, we cannot take Bx, By, Bz to be respectively equal to -^ Bt, -^ Bt, -^ Bt. The equation -^ = expresses the condition that the geometrical az equation (1) should not contain the time explicitly. 352. The great advantage of this principle is that it gives at once a relation between the velocities of the bodies considered and the variables or coordinates which determine their positions in space, so that when, from the nature of the problem, the positions of all the bodies may be made to depend on one variable, the equation of vis viva is sufficient to determine the motion. In general the principle of vis viva will give a first integral of the equations of motion of the second order. If, at the same time, some of the other principles enunciated in Art. 282 can be applied to the bodies under consideration, so that the whole number of equations thus obtained is equal to the number of independent coordinates of the system, it becomes unnecessary to write down any equations of motion of the second order. See Art. 143. The principle of vis viva was first used by Huyghens in his determination of the centre of oscillation of a body, but in a form different from that now used. See the note to Art. 92. The principle was extended by John Bernoulli and applied by his son, Daniel Bernoulli, to the solution of a great variety of problems, such as the motion of fluids in vases, and the motion of rigid bodies under certain given con- ditions. See Montucla, Histoire des Mathematiques, Tome iii. 353. Initial motion. Suppose the system to begin to move from rest under the action of the forces X, Y, Z, &c. After a time dt the vis viva is given by Swv'2 = 2Sm [Xdx + Ydy + Zdz). The left-hand side of this equation is necessarily positive. We therefore infer that if a system start from rest, the initial motion must be such that the virtual work of the forces for that motion must be positive. There may be several different ways (geometrically considered) in which the system could begin to move from its initial state of rest. Let the system be com- I ART. 357.] POTENTIAL AND KINETIC ENERGY. 287 pelled to take any one of these ways of motion by obliging a sufficient number of its points to describe certain smooth curves, or by introducing any forces which have no virtual work for that particular mode of displacement. The system can now move only in one way, or as we often express it, the system has only one path open. There are two directions in which it can travel along this path. The question arises — in which direction will it begin to move ? Since the virtual work of the forces is in general positive for one of these directions and negative for the other, the system must begin to move along the former. 354. Examples of the principle. If a system be under the action of no external forces, we have X=0, Y=0, Z = 0, and hence the vis viva of the system is constant. If, however, the mutual reactions between the particles of the system are such as do appear in the equation of virtual work, then the vis viva of the system will not be constant. Thus, even if the solar system were not acted on by any external forces, its vis viva would not be constant. For the mutual attractions between the several planets are reactions between particles whose distances do not remain the same, and hence the sum of the virtual works is not zero. Again, if the earth be regarded as a body rotating about an axis and in course of time slowly contracting from loss of heat, the vis viva will not be constant, for the same reason as before. The increase of angular velocity produced by this con- traction can be easily found by the principle of angular momentum. See Art. 299. 355. Let gravity be the only force acting on the system. Let the axis of z be vertical, then we have X= 0, Y= 0, Z= -g. Hence the equation of vis viva becomes Sm?;'2 - Smv2 = _ 2Mg {z' - z). Thus the vis viva of the system depends only on the altitude of the centre of gravity. If any horizontal plane be drawn, the vis viva of the system is the same whenever the centre of gravity passes through the plane. See Art. 142. 356. Ex. If a system in motion pass through a position of equilibrium, i.e. a position in which, if placed at rest, it would remain in equilibrium under the action of the forces, prove that the vis viva of the system is either a maximum or a minimum. De Courtivron's Theorem, Mem. de VAcad. 1748 and 1749. 357. The equation of virtual work in statics is known to contain in one formula all the conditions of equilibrium. In the same way the general equation Sm (J S^ + g Sy + J hz^ = %m {Xhx + Yhy + Z^z\ may be made to give all the equations of motion by properly choosing the arbitrary displacements hx, By, 8z. In Article 350 we made one choice of these displacements and thus obtained an equation in an integrable form. If we give the whole system a displacement parallel to the axis of z we have Boo = 0, By = 0, and Bz is arbitrary. The equation . d^ ^^ drj ' d^ ^ 19—2 292 VIS VIVA. [chap. vii. Substituting these values, we get, since A = Sm (rj'^ + f^), 5 = 2m (?2 + p), (7 = Sm (f ^ + V'), Sm^i^ = -4a>a;^ 4- Bwy- + (7ct)/ - 2 (2mf?7) toajojj, - 2 (SmT/f ) oj^/W^ - 2 (Xmf^) w^cd^;. We may find the vis viva of the motion about G in another manner. Let 12 be the angular velocity about the instantaneous axis, I the moment of inertia about it. The vis viva is then clearly 10,^, Now I is found in Art. 15, and in our case Wj = fia, a;2 = fi/3, o).^ = Uy, following the notation of that article. Eliminating a, ^, y we get the same result as before. If the axes of coordinates be the principal axes at G, this reduces to %mvi^ = Aay^^ -{■ Bcoy^ + Oo)/. If the body be turning about a point 0, whose position is fixed for the moment, the vis viva may be proved in the same way to be l.mv'' = A'dii + jB'ft)/ + C. where A\ B\ C are the principal moments of inertia at the point 0, and Wx, ft>t/, ft>z are the angular velocities of the body about the principal axes at 0. 365. Examples of vis viva. Ex. 1. A rigid body of mass M is moving in space in any manner, and its position is determined by the coordinates of its centre of gravity and the angles 6, 0, i/' which the principal axes at the centre of gravity make with some fixed axes, in the manner explained in Art. 256. Show that its vis viva is given by + sin2 e {A cos2 + B sin2 V^ ti ^5 2/' •2;, t by K^, Krj, K^, KX, Kg, kz, Kt, where k is any constant, provided that we alter the accelerating forces in the ratio «: to 1. Hence if the accelerating forces are zero, it is sufficient to increase the dimensions of the elastic body and the initial values of the displacements in the ratio 1 to k, in order that the general values of ^, 77, ^ and the durations of the vibrations may vary in the same ratio. Hence we deduce Cauchy 's extension of Savart's law, viz., if ice measure the pitch of the note given by a body, a plate or an elastic rod, by the number of vibrations produced in a unit of time, the pitch icill vary inversely as the linear dimensions of the body, plate, or rod, supposing all its dimensions altered in a given ratio. 373. Theory of Dimensions. These results may be also deduced from the theory of dimensions. Following the notation of Art. 332, a force F is measured by md^xjdP. We may then state the general principle, that all dynamical equations must be such that the dimensions of the terms added together a7^e the same in space, time and mass, the dimensions of force being taken to be mass . space ' (timey To show how the principle may be used let us apply it to the case of a simple pendulum of length I, oscillating through a given angle a, under the action of gravity. Let m be the mass of the particle, F the moving force of gravity, then the time r of oscilla- tion can be a function of F, I, m and a only. Let this function be expanded in a series of powers of F, I and 7n. Thus T = ^AFH^m% ART. 374 a.] PRINCIPLE OF SIMILITUDE. • 299 where A, being a function of a only, is a number. Since r is of no dimensions in space, we have p + q = 0. Also r is of one dimension in time ; .. -2p = l. Finally t is of no dimensions in mass ; . '. p-{-r = 0. Hence p = — i, q = r = i, and since p, q, r have each only one value, there is but one term in the series. We infer that in any simple pendulum r = A \/ -^ where A is an undetermined number. See also Art. 370. Ex. 1. A particle moves from rest towards a centre of force, whose attraction varies as the distance, in a medium resisting as the velocity, show by the theory of dimensions that the time of reaching the centre of force is independent of the initial position of the particle. Ex. 2. A particle moves from rest in vacuo towards a centre of force whose attraction varies inversely as the n*^ power of the distance, show that the time of reaching the centre of force varies as the ^ {n + 1)^^ power of the initial distance of the particle. 374. Imaginary Time. The equations of motion of a system are changed into those of a similar system by multiplying the forces, lengths, masses and times by the constants F, I, /n, r, where fd^FT^. The systems however may present only an analytical similarity, for if F were negative, and /x, I positive, the ratio of the corresponding times would be imaginary. The change of sign of F is, of course, equivalent to reversing the directions of all the impressed forces. Let us suppose that the two similar systems are such that Z = l, /x = l and only so far differ that the impressed forces X, X' &c. are equal and opposite and in con- sequence t'lt= ±^/(-l), Art. 368. It follows that the same system can have two conjugate motions with opposite forces such that in one {x, y, &c.) are the same functions of t that [x', y', &c.) in the other are of t'. The initial positions are the same in the two cases, and if v=dxldt, v' = dx'ldt' are any corresponding velocities, their ratio iflv= T^/( - 1). We also evidently have dx'jdt = dxldt. Hence we arrive at the following theorem. A system of material points, subject to constraints which are independent of the time and under the action of forces ivhich depend only on the position of the several points, being given ; the integrals of the differential equations remain real ifioe replace t by tsj{-l) and the resolved initial velocities v^,Vy,Vg, of any particle by -v^sj{-l), -Vy J{-1), -Vg^/{-l), the initial positions being the same. The equations thus obtained are those of a new movement ivhich the same material particles would take if acted on by forces equal and opposite to those ivhich produced the first motion, the initial values of the coordinates and their velocities being the same in the two cases. Appell, On an interpretation of the imaginary values of the time] Comptes Rendus, Vol. 87, 1878, page 1074. Painleve, Legons siir Vinte- gration des equations differentielles de la Mecanique, 1895, page 226. 374 a. These considerations will sometimes enable us to find an interpretation for an analytical result which gives an imaginary value for the time. This will be made clear by an elementary example. Let d be the angle a simple pendulum, suspended from 0, makes with the downward vertical BOA, and let the pendulum start from rest at an angle 6 — a. Since ddfdt is initially negative the time of moving from 6 = a to 6 = 6 is given by -du Ig _ fo -dd fu V I ~ ja2v'(8inHa-sinH^)~ jl J{l-U^)^{l-Khl') 300 VIS VIVA. [chap. VII. where sin^^ = M8in|a, K: = sin|a and all the radicals are positive. Put dn K- [' ^'' K' - i^'' ^(U2 -1)^(1 -K'U2)' The times of arrival at the lowest and highest points of the circular path (starting in each case from rest) are found by writing ^ = 0, 6 = Tr, that is u = 0, u^Ijk respectively. These times are therefore given by sJ-l=K, h^^- = K'^[-l). The latter time is imaginary, showing that under the given circumstances the bob of the pendulum does not reach the highest point. We find an interpretation of the value of t.2 by reversing the impressed forces. When we have written - g for g let f^', t^ be the corresponding values of fj, t^, then i/ is now imaginary and ^2' becomes real. Thus the real path of the pendulum in either motion corresponds to the imaginary path in the conjugate motion. If the time t is counted from the instant at which the bob passes the lowest point (gravity acting downwards), the motion is given by m = sn ( t x/% ) • The two periods of this elliptic function are 4/f and 2A''^(-1), and these respectively determine the two times i^ and t^ . This example is discussed by both Appell and Painlev6 but in different ways. 375. Clausius' theory of stationary motion. To determine the mean vis viva of a system of material points in stationary motion. Clausius, Phil. Mag., August, 1870. By stationary motion is meant any motion in which the points do not continually move further and further from their original position, and the velocities do not alter continuously in the same direction, but the points move within a limited space and the velocities only fluctuate within certain limits. Of this nature are all periodic motions, such as those of the planets about the sun, and the vibrations of elastic bodies, and further, such irregular motions as are attributed to the atoms and molecules of a body in order to explain its heat. Let X, y, z be the coordinates of any particle in the system and let its mass be m. Let A', Y, Z be the components of the forces on this particle. Then m -jY = X- We have by simple differentiation (Arts. 286, 286 b), {f^ = < W-='dtV'dt) = ''\di)-^'^-d^^ and therefore 7^ ^ I = - k^-^ + dxy _ 1 m d^jx'^) 2\dtJ --2*'"'^ 4 df' Let this equation be integrated with regard to the time from to t and let the integral be divided by t, we thereby obtain [^ /dxy-^ 1 ft ^^^ VI rd{x^) fd{x^-)\-\ in which the application of the sutfix zero to any quantity implies that the initial value of that quantity is to be taken. The left-hand side of this equation and the first term on the right-hand side are till f d or\ 1 evidently the mean values of o { w7 ) ^°^ ~ ^^ ^^^'^^^ ^^^ *"^e t. For a periodic motion the duration of a period may be taken for the time t ; but for irregular motions (and if we please for periodic ones also) we have only to consider that the ART. 876.] CLAUSIUS' THEORY OF STATIONARY MOTION. 301 time t, in proportion to the times during which the point moves in the same direc- tion in respect of any one of the directions of coordinates, is very great, so that in tlie course of the time t many changes of motion have taken place, and the above expressions of the mean values have become sufficiently constant. The last term of the equation, which has its factor included in square brackets, becomes, when the time is periodic, equal to zero at the end of each period. "When the motion is not periodic, but irregularly varying, the factor in brackets does not so regularly become zero, yet its value cannot continually increase with the time, but can only fluctuate within certain limits ; and the divisor t, by which the term is affected, must accordingly cause the term to become vanishingly small for very great values of t. The same reasoning will apply to the motions parallel to the other coordinates. Hence adding together our results for each particle, we have, if v be the velocity of the particle m, mean 1 2mv'^ = - mean ^'Z{Xx+Yy + Zz). The mean value of the expressions - ^ S {Xx +Yy + Zz) has been called by Clausius the virial of the system. His theorem may therefore be stated thus, the mean semi- vis viva of the system is equal to its virial. 376. To apply this theorem to the kinetic theory of heat we premise that every body is to be regarded as a system of particles in motion. So far as this proposition is concerned, the particles may describe paths of any kind, and any particle may pass as close as we please to another. But, as no account of impacts has here been considered, we must either suppose the particles to be restrained from actual contact by strong repulsive forces at close quarters, or (which amounts to the same thing) suppose the particles to be perfectly elastic, so that the total vis viva is unaltered by the impacts. The forces which act on the system consist in general of two parts. In the first place, the elements of the body exert on each other attractive or repulsive forces, and, secondly, forces may act on the system from without. The virial will therefore consist of two parts, which are called the internal and external virials. It has just been shown that the mean semi-vis viva is equal to the sum of these tico parts. If (r) be the law of repulsion between two particles whose masses are m and m', we have Xx + X'.v' =- ^R\2^ 2i^o2 (Art. 383) we have •"01 "• -"12 ^^ -^02* It immediately follows that R^^ is greater than i^oi- Hence, if any impulsive forces act on a system in motion, the displacements of the points of application in the time dt being prescribed, the actual motion is such that the vis viva of the relative motion, before and after, is less than if the system took any other course. 387. Kelvin's theorem. If the system start from rest, the velocities represented by u, v, w are zero. We therefore have, as in the last article, ^mu'^ -f . . . = l^mu'u" + . . . ; .-. ^mu''-^-... + Sm {u"-uy+... = l.mu'^ + ... . If (as in Art. 383) we represent the vis viva of the actual ART. 388.] GENERAL THEOREMS ON IMPULSES. 307 motion after impact by 2T\ that of the hypothetical motion by 2T" and that of the relative motion by 2i2i2, this equation becomes r + R^, = T". Let a system he at rest and he set in motion hy jerks or impulses at given points, so that the motions of these points are prescribed, then the vis viva of the subsequent motion is less than that of any other hypothetical motion of the system in which these points have the prescribed motions. Natural Philosophy by Thomson and Tait, Art. 312. 388. Let the impulses be given. Consider two geometrically possible motions of the system. Let one of these be the actual motion in which ii\ v, w' are the resolved velocities of the particle m, and let the second be any other motion, such that we can compel the system to take that motion by introducing the proper friction- less constraints. For instance each particle may be constrained to move in any direction (geometrically possible) by attaching it, like a bead, to a smooth wire. Let u' , v" , w" represent the resolved velocities of the particle m in this motion. Supposing the system to have the first motion, let us give it a virtual displacement along the second, then Sm {u' - u) u" + ... = %Xu" + ... . Supposing the system to have the second motion, and that the work of the constraining reactions is zero. Art. 362, we have 2m {u" -u)u''-\- ... = lXu"-\- ...; .'. 2m (u — u)u''+ ... = 2m (it'— u)u'' + ... ; .-. 2m(w' - uy + ... + Xmu"'+ ... = Xmu'^ + ... . Representing the vis viva of the actual motion after impact by 2T', that of the hypothetical motion by 2T" and that of the relative motion by 211^2, this equation gives It follows that T' is greater than T". Suppose a system in motion to he acted on hy any impulses, the vis viva of the subsequent motion is greater than if the system were subjected to any additional constraints and acted on by the same impulses. We thus arrive at a theorem of Lagrange generalized first by Delaunay in Liouville's Journal, Vol. v., and afterwards by Bertrand in his notes to the Mecanique Analytique. See Art. 379. Comparing Kelvin's and Bertrand's theorems we perceive that, when the motions of the points of application of the impulses are t given, the subsequent motion may be found by making the vis viva a minimum, but, when the impulses are given, the subsequent motion may be found by introducing some constraints and making the vis viva a maooimum. 20—2 308 VIS VIVA. [chap. VII. 388 a. Sxamples. To understand these two principles properly we should examine their application to some simple cases of motion. Ex. 1. A body at rest having one point fixed is struck by a given impulse, find the resulting motion. See Art. 308 and Art. 310. Let L, M, N be the given components of the impulse about the principal axes at 0. Then, if the body begin to turn about an axis fixed in space whose direction cosines are {I, m, n), the angular velocity w is found by Art. 89 from (A Z2 + £m2 + Cn^) (a=Ll + Mm + Nn. To find the axis about which the body begins to turn when free, we must by Lagrange's Theorem make the vis viva a maximum. That is, we have {AV^ + Bm^ + Cn^) w^ = maximum. We have also the condition l- + m^ + n^=l. Treating these three equations in the usual manner indicated in the differential , , „ - At Bm Cn calculus, we find , -v = -rr = -^ • L M N These equations determine the direction cosines of the axis about which the body begins to turn. Ex. 2. Four equal rods at rest are joined together by smooth hinges so as to form a rhombus ABCD, the angle at A being 60°. Apply Kelvin's theorem to show that if the corner A is suddenly moved with velocity V along the diagonal CA, the initial angular velocity of any rod is 3F/7a where 2a is the length of any rod. If the angle at ^ is 2^, and w the angular velocity of any rod, the vis viva is F2 + 8Vau} sin + lOa^oj^ sin2^ + 2a^-u)^ cos^^ + 2kW. Equating to zero the differential coefficient with regard to w, we obtain the initial value of w, which reduces to the given result when 20 = 60°. Ex. 3. A body is in motion with a point fixed in space. Suddenly a straight line OG fixed in the body is made to move round in a given manner ; find the motion, Art. 293. Let the instantaneous position of OG be the axis of z. Let the previous motion of the body be given by the angular velocities Wj , w^, W3 and the prescribed motion of OG by the angular velocities 0, 0, about the axes of x and y. Let Q be the required angular velocity of the body about Oz. The vis viva of the relative motion, before and after, is A{0- (a^f + B (0 - W2)^+ C (fi - ^3)2 - 2D (0 - Wg) (O - w^) -2E{0- w^) (O - Wg) -2i^(<9-wi){0-wo). This is to be made a minimum by Art. 386. Differentiating with regard to fi, (ft - wg) - D (0 - W2) -E{0- wi) = 0. This equation expresses the fact that the angular momentum about OG is unaltered. Ex. 4. A rod AB at rest is acted on by an impulse F perpendicularly to its length at the extremity A, and that extremity begins to move with a velocity/. Find the point in AB about which the rod will begin to turn (1) when F is given and (2) when / is given. If AO = x, show that both Kelvin's theorem and Bertrand's theorem require the same function of x to be made a minimum. Ex. 5. A system is moving in any manner. A blow is given at any point per- pendicular to the direction of motion of that point. Prove that the vis viva is increased. A ART. 389.] GENERAL THEOREMS ON IMPULSES. 309 This follows from the first of the equations in Art. 383 ; for the virtual work of this force (there called A) vanishes in the initial motion. Hence T'=T + Rq^. Ex. 6. A system at rest, if acted on by two different sets of impulses called A and B, will take two different motions. Prove that the sum of the virtual works of the forces A for displacements represented by the velocities in the motion B is equal to the sum of virtual works of the forces B for displacements represented by the velocities in the motion A . See Art. 383. Ex. 7. Two equal uniform rods AB, BC, smoothly jointed at B, and each of mass m, lie making an angle a with one another on a horizontal table, and pass at their middle points through smooth fixed rings. To the free end A is given a velocity ?; in a direction towards and perpendicular to BC. Prove that the kinetic energy of the motion is ^mv^ (2 - cos a cos 3a) and that it is greater by -^viv^ cos2 2a than it would be if there were no ring on BC. [Use Kelvin's theorem.] [Math. Tripos, 1904. 389. Imperfectly elastic and rough bodies. When two bodies of an imperfectly elastic and rough system impinge on each other, we may deduce from the equations of Art. 382 some extensions of Carnot's theorems. Let (uvw) {u'v'w') {u"v"w") be the resolved velocities of a particle m just before the impact begins, at the moment of greatest compression, and just after the con- clusion of the impact. Let the vis viva of the system at these epochs be represented by the symbols 2T, IT', 2T'\ Let the vis viva of the relative motion at any two of these epochs be represented by 2Rq^, 2jRi2, 2iJo2. If the bodies impinging are perfectly smooth we have by the same reasoning as in Arts. 378 and 380 Swi{K -w)m' + &c.}=0 (1), Sm{(tt"-'u)w' + &c.} = (2). Since the whole impulse between the two bodies bears to the impulse up, to the moment of greatest compression the ratio 1 + e : 1 we may deduce from Art. 382 the two following equations i:m{{u" -u)u + &G.\ = {l + e)'Zm{{u'-u)u + &c.} (3), I,m{{u" -u) u" + &c.} = {l + e) Sm {{u'-u) u" + &g.} (4). The left-hand side of either of these equations, after multiplication by dt, is equal to the virtual work of the whole impulse, and the summation on the right-hand side, after multiplication by dt, is equal to the virtual work of the impulse of compression. These are taken for the same displacement and are therefore in the ratio 1-f-e : 1. In the first equation the displacement chosen is the actual displacement just before impact. In the second equation the displacement chosen is that just after impact. These are both consistent with the geometrical conditions. The above four equations may be conveniently expressed in the forms T'-T=-R,, (5), T"-T'=R^^ (6), T" -r {l-\-e)-veT=R^-{l + e)B^^ (7), T"-T'{l^e)^eT=eR^-{l + e)R^^ (8). If we eliminate the E's from these equations, we find T"-T'=-e^{T'-T) (9), thus the gain of vis viva due to restitution or explosion is ^ into the loss of vis viva due to compression. If we eliminate the T's, we find ^oi=7rT^!\2= 2 (^O)- 1-e If we eliminate T',',J?oi, i?i2, we find r"-T=-r Rq^ (11), which may be regarded as an extension of Carnot's third theorem in Art. 380. 310 VIS VIVA. [chap. VII. Suppose next that the bodies impinging are rough, and slide on each other during the whole impact, the friction acting always in the same direction. The friction now bears a constant ratio to the normal pressure throughout the impact. The equations (.3) and (4) hold as before. The separate equations (1) and (2) no longer hold, but instead we may form the single equation Sw {{u" -u)u' + &c.} = {l + e) 2m {{u' - ti) u' + &c.} (12), by the same reasoning as in equations (3) and (4). The equation (12) may be expressed in the form T" - T' {l + e) + eT=R-^^ + eB^-^ (13). Joining (13) to (7) and (8) we have three equations connecting the six quantities T, T', T", i?oi, JRo2» -^12- We easily find Rq^_R^_T"-T' {l-\-e) + eT (H-e)2~^~ e(l + e) We may deduce from these equations the following theorem. When one body of a system impinges on another^ the three states of motion {viz. that just before, that just after, and that at the moment of greatest comprest^ion) are so related that the vis viva of the relative motion of any tivo bears to the vis viva of the relative motion of any other two a ratio which depends only on the coefficient of elasticity. Let us suppose a system to be acted on by an impulsive force whose direction in space remains unchanged during its time of action. A theorem similar to that just enunciated applies to any three epochs in the time of action of this impulse, provided these epochs are such that the whole impulse exerted in the interval from the first epoch to the second bears a known ratio (say 1 : e) to the whole impulse exerted in the interval from the second to the third. Eepresenting the vires vivae of the system at the three epochs by 2T, 2T', 2T" as before, and the vires vivae of the relative motions by 27^0^, 2Rq2, 272^2 > we notice that the equations (3), (4) and (12) apply to the motions of the system at the three epochs. The equation (14) will therefore give the same relations as before between the six quantities T, T', T", R^^, R^^, R^^. We may obtain an easy proof of this theorem by combining the results of Arts. 385, 386 with Art. 313. Let X be an impulse, and let the axis of x be taken parallel to its direction. By Art. 385 the vis viva of the relative motion before and after the impulse is proportional to X{u' -u). But, by Art. 313, u' -u is a linear function of X, and vanishes with X. It is therefore proportional to X. The vis viva of the relative motion is therefore proportional to X^. It immediately follows that iJoi, -Ro2' ^12 ^^^ proportional to 1, (1 + e)^ e^. The remaining part of the theorem follows from Art. 386. Letting X now represent the impulse from the first to the second epoch, we have T'-T = lX{u' + u), T"-T' = iXe{u" + u'). It easily follows that T" -T' -e{T' -T) = ^Xe {u" - u). Since the right-hand side of this equation is RQ^el{l-\-e), by Art. 385, the remaining part of equation (14) has been proved. When two elastic systems impinge on each other, the theorems contained in equation (14) are true for the impulse on each system. They therefore follow by simple addition for the two impinging systems regarded as one. 390. Oauss' measure of the " constraint." The expression, called 2R in the previous articles, which represents the vis viva of the relative motion, has been interpreted by Gauss in another manner. Let the particles m^, 7»2, &c. of a system just before the action of any impulses occupy positions which we shall call Pi , 2^2 . &c. Let us suppose that the particles if free would under the action of these impulses and their previous momenta acquire such velocities that in the ART. 392.] GENERAL THEOREMS ON IMPULSES. 311 time dt subsequent to the impulses they would describe the small spaces PiQi, p^q.^^ &c. But if the particles were constrained in any manner consistent with the geometrical conditions which hold just before the action of the impulses, let us suppose that they would under the same impulses and their previous momenta describe in the time dt subsequent to the impulses the small spaces Pxr^, p^r.^, &c. Then the spaces q^^r^^, q^r^, &c. may be called the deviations from free motion due to the constraints. The sum Sm (qr)^ is called the " constraint." 391. We may also measure the constraint by the ratio of this sum to {dt)^. We then take Piqi, &c. Pir^^, &c. to represent, not the displacements in the time dt, but the velocities of the particles just after the action of the forces in the two cases in which the particles are free or constrained. Referring to D'Alembert's principle in Art. 67, we see that pq represents the resultant of the previous velocity and of the velocity generated by the impressed force on the typical particle m, while qr represents the velocity generated by the molecular forces*. If we suppose that the lengths pq, qr, &c. represent velocities and not displace- ments, let (u, V, xo) be the components of pq in any motion, and (w', v', w') the components of pr in any other motion ; then Sm {qrY='Lm {{u' - uf+{v' - vf + («?' - xof] measures the " constraint" from one motion to the other. This is precisely what we have represented by the symbol 2E, with suffixes to define the two motions compared. 392. Gauss' principle of least constraint. Suppose a system of particles in motion and constrained in any given manner to be acted on by any given set of impulses. Let 2T" be the vis viva of the subsequent motion. This is the actual motion taken by the system. Let us now suppose that the particles were forced to take some hypothetical motion consistent with the geometrical conditions by introducing some further constraints. Let 2T" be the subsequent vis viva in this hypothetical motion. Thirdly, let us suppose that all constraints were removed so that the particles were acted on solely by the given set of impulses. Let 22"" be the subsequent vis viva in this free motion. Let 2T be the initial vis viva common to all the motions. Let 222^2 > ^iJjg, 2^23 be the vires vivae of the relative motions of the first, second and third subsequent motions as denoted by the suffixes. By Bertrand's theorem, since the hypothetical motion is more constrained than the actual motion, we have T' = T" + R^^ • * Gauss' proof of the principle is nearly as follows. By D'Alembert's principle the particles m^, m.^, &c., if placed in the positions r^, rg, &c., would be in equilibrium under the action of these molecular forces alone. Let us apply the principle of virtual work, and displace the system so that the typical particle m describes a space rp, making an angle with the direction rq of the molecular force on m. Then since the product m [rq] measures the molecular force on m, we have 2wi {rq) {rp cos 0) = 0. But qp^ = qr^ + rp^-2qr .rp cos . Hence we easily find Sm {qp}^ = Sm {qr)^ + Sm (?-p)2. In the actual motion the particles move from p^, &c. to r^, &c. and the "con- straint " is Sm {qr)^. If the particles had been forced to take any other hypothetical courses, by which they were brought into the positions p^, &c., the "constraint" would be 2wt {qp)"^. Gauss' Principle asserts that the former is always less than the latter. 312 VIS VIVA. [chap. VII. Also, since each of these is more constrained than the free motion, Hence we have R^s = iJjg + -R12 • Therefore JR23 is always greater than Ry^. It follows that the motion which the system actually takes when subject to any impulses is such that the "constraint" from the free motion is less than if the system took any other motion consistent with the geometrical conditions. This result is true whichever way the ' ' constraint " is measured. 393. If we suppose the system to be acted on by a series of indefinitely small impulses, these impulses may be regarded as finite forces. We therefore infer the following theorem, which is usually called Gauss'' principle of least constraint. The motion of a system of material points connected by any geometrical relations is always as nearly as possible in accordance with free motion; i.e. if the constraint during any time dt is measured by the sum of the products of the mass of each particle into the square of its distance at the end of that time from the position it would have taken if it had been free, then the actual motion during the time dt is such that the constraint is less than if the particles had taken any other position. Gauss remarks that the free motions of the particles when they are incom- patible with the geometrical conditions of the system are modified in exactly the same way. as geometers modify results which have been obtained by observation, i.e. by applying the method of Least Squares so as to render them compatible with the geometrical conditions of the question. 394. Ex. Any number of particles m^, m^, dx. are acted on by any forces whose components are m^X^, %J^ij %-2'i, d;c. Their coordinates x^, y^, z-^; X2, 2/2* ^2> ^^' ^^^ connected together by some relation such as (t>{x-^, c^-c.) = 0. {For instance the particles may be beads slung on a string of given length ichose extremities are tied together.) It is required to form the equations of motion. Let V, V, W be the resolved velocities of the typical particle r;t at the time t ; w, V, w its resolved velocities just after the action of the impulse whose resolved parts are mXdt, mYdt, mZdt, on the supposition that the particle is perfectly free. But as the typical particle is not perfectly free, let u\ v', w' be its actual resolved velocities at the same instant. Then to find u', v\ lo' we make 2Ry^='2.m{_{u' -uf+{v' -vf + {w' -wY^ = m\mm\xm (1), where the S implies summation for all the particles. This quantity is to be a minimum for all variations of u', v', 10' subject to the condition 2 {yv' + (l>,w') = (2), where the S here also implies summation for all suffixes. To make R^^ a minimum we take the total differential of each of these quantities with regard to all the accented letters, multiply the second by some indeterminate multiplier \, and add the results together. Equating to zeto the coefficients of du' &c. we obtain the three typical equations m{u' -u)+\(f>y.=0, m{v' -v) + \(py=0, m{w'-iv) + \ , where ,&Lc.) .....(1), with similar equations for y and z. It should be noticed that these equations are not to contain dd/dt, d(f)/dt, &c. The independent variables in terms of which the motion is to be found may be any we please, with this restriction, that the coordinates of every particle of the body can, if required, be expressed in terms of them by means of equations which do not contain any differential coefficients with regard to the time. When the system admits of such a choice of independent coordinates, it is said to be holonomous. This name is due to Hertz, Die Principien der Mechanik* , 1894. * The following is taken from the translation by Jones and Wallay, 1896. A material system between whose possible positions all conceivable continuous motions are also possible motions is called a holonomous system. The term means that such a system obeys integral laws whereas material systems in general obey only differential conditions. 318 Lagrange's equations. [chap. viii. The number of independent coordinates to which the position of a system is reduced by its geometrical relations is sometimes spoken of as the number of degrees of freedom of that system. Sometimes it is referred to as being the iiiimher of independent motions of which the system admits. In this chapter total differential coefficients with regard to t will in general be denoted by accents. Occasionally dots will be used as before, and sometimes the differential coefficients will be dec d'^x written at length. Thus -r- and -r- will in general be written x' and x". If 2T be the vis viva of the system, we have 2T=tm{x'^ + y'^ + z'^) (2); also since the geometrical equations do not contain 0\ ', &c., ^'=1-1^'-^^*'-^^^ («)■ with similar equations for y' and z. In these the differential coefficients of/ &c. are partial. Substituting in the expression (2) we see that 2T takes the form ^T:=A,,e'^' + 2A,,d'4>'-\-...+B,e'-\-B,'+...+C ...(4), where the coefficients A^^, &c., ^i, &c., and C are functions of t, 6, (/), &c. The quadratic terms, i.e. those containing the squares and products of 6\ (f/, &c., come from the substitution of all the terms of x\ except dfjdt and those in y', z corresponding to it. If the geometrical equations do not contain the time explicitly, t is absent from the equations (1), the term df\dt is also absent in (3), and the expression for 2T is reduced to the quadratic terms alone. We may briefly write (4) in the form ^T = F{t, e, , &c., d\ (i>\ &c.) (5). When the system of bodies is given, the form of F is known. It will appear presently that it is only through the form of F that the effective forces depend on the nature of the bodies considered ; so that two dynamical systems which have the same F are dynamically equivalent. It should be noticed that no powers of 6', (j>\ &c. above the second enter into this function, and that, when the geonietmcal equations do not contain the time explicitly, it is a homogeneous function of 6', <^', <^c. of the second order. 397. Virtual work of the eflfective forces. To find the virtual moment of the momenta of a system, and also that of the effective forces, corresponding to a displacement produced by varying one coordinate only. ART. 398.] VIRTUAL WORK. 319 Let this coordinate be 6, and let us follow the notation already explained. Let all differential coefficients be partial, unless it be otherwise stated, excepting those denoted by accents. Since X, y\ z' are the components of the velocity, the virtual moment of the momenta is Sm {x'hx + y'hy + z'hz), where hx, hy, hz are the small changes produced in the coordinates of the particle m by a variation hO of 6. This is the same as H'%-y'%^''>- If 2 r be the vis viva given by (2) of the last article dT ^ f ,dx dT ^ ( ,dx \ But, differentiating (3) partially with regard to 6', we see dx' dx dT thafc TTT, =-T7i- Hence ttv ^^ is equal to the virtual moment of dd dd dd ^ the momenta. 398. The virtual work of the effective forces is Omitting the factor hO, this may be written in the form where the -77 represents a total differential coefficient with regard to t. We have already proved that the first of these terms is 7 irp -J- -^ . It remains to express the second term also as a differ- ential coefficient of T. Differentiating the expression for 2T partially wdth regard to 6, dT ^ ( ,dx' , . But, differentiating the expression for x^ with regard to 6, ax ax ax ^^ ax §/ , dS^dddt'^W^ '^dM4>'^^ ' d dx and this is the same as ^7- -y^ . Hence the second term may be dT dtdd' written -v^ , and the virtual work of the effective forces is du therefore I ^ 7^, — tt^ 1 hd, [dtdd' dd) 320 LAGRANGE'S EQUATIONS. [CHAP. VIII. The following explanation will make the argument clearer. The virtual work of the effective forces is clearly the ratio to dt of the difference between the virtual moments of the momenta of the particles of the system at the times t + dt and t, the displacements being the same at each time. The virtual moment of the momenta at the time t is first shown to be -—, dd. Hence i ^n:, + -r -ttt, dt ] 80 dd \dd dtdd J is the virtual moment of the momenta at the time t + dt corresponding to a dis- placement 5^ consistent with the positions of the particles at that time. To make the displacements the same, we must subtract from this the virtual moment of the momenta for a displacement which is the difference between the two displacements dx at the times t and t + dt. Since bx= -rrbd, this difference for the variable x is ciu — i ^ni\ dtdd. We therefore subtract on the whole Sm \x' — { ^ ) dt + &c.[ 86, and dt\ddj { dt\ddj \ dT this is shown to be -j^ dt86. ciu 399. Lagrange^s equations for finite forces. To deduce the general equations of motion referred to any coordinates. Let U be the force-function, then t/" is a function of 6, cf), &c. and t. The virtual work of the impressed forces corresponding to a displacement produced by varying 6 only is -^ SO. But by D'Alembert's principle this must be the same as the virtual work of the effective forces. Hence ddT_dT_dU dt dd' dd ~ dd' ^. ., , , d dT dT dU . . Similarly we have -^, -ytt — tt = t , , &c. = &c. ^ dt d<^ d^ d(^ It may be remarked that if Y be the potential energy we must write — Y for JJ. We then have d^d^_dT dV_ dt dd' dd'^ dd~ ' with similar equations for c^, >/r, &c. In using these equations, it should be remembered that all the differential coefficients are partial except that with regard to t. Let us write L = T + U, so that L is the difference of the kinetic and potential energies. Then, since U is not a function of d', , i^, &c., 6\ ', -^jr', &c. and that there are two geometrical equations connecting these co- ordinates, viz. f(d, (f), &c.) = 0, F(e, (f), &c.)=0 (1). To simplify the explanation, we suppose that there are only twc such geometrical equations, but it will be seen that the process is quite general and will apply to any number of conditions. By the principle of virtual work we have d dL dL\ ^^ fd dL dL\^^ Also %^^'^%^'^^^'^-^^ (^)^ and ^^^ + ^f ^*+^^'=^^ (^)- Since the coordinates 6, ' d(l>'^ d^'^^dcp' i M &c. = j ' There are here as many equations as coordinates. Joining these to the equations (1) we have sufficient equations to find all the coordinates and the two multipliers X and /j,. ART. 401.] IMPULSIVE FORCES. 323 These equations may be put into a simpler form. We notice that the geometrical functions f and F do not contain Q' , ', &c. (see also Art. 396). Let us then write L^^L + \f+,jLF (6), and treat Zj as if it were the Lagrangian function. If we substitute this value of L^ in the typical equation dt dO' de ^ ^' where 6 stands for any one of the coordinates, and simplify the results by remembering that/= 0, i^= 0, we obtain in turn all the equations (5). The same process will also supply the geometrical equations (1), if we include \ and /^ among the coordinates. Thus, since Zj contains no \', we have dL^jdX' = ; hence, writing X for 6, the equation (7) gives /= 0. If the geometrical equations (1) contain t, the argument and the result are the same, for the arbitrary variations hd, h^, must (as in Art. 351) be consistent with the geometrical equations which hold at the time t. Ex. A particle under the action of no impressed forces is constrained to remain on the curve x^ + y'^=1axt. Show that x = at\l + coBiB +—\\ , y = atQm(B + — \. 401. Lag^range's equations for impulsive forces. Let the system, defined by the arbitrary coordinates 6, , t^c, be subjected at the time t to impulsive forces which act at definite points. It is required to deduce the changes produced in the motion. Let hU \>Q the virtual moment of the impulsive forces pro- duced by a general displacement of the system. Then from the geometry of the system, we can express d>U in the form BU = PSe + QScj, + (1). The virtual moment of the momenta given to the particles is Sm [{oc/ - Xo) Bx + (y/ - yo) By + (z,' - z^) Bz] (2), where {x^, y^, z^'), (d?/, 3//, z-^) are the values of {x', y\ z') just ! before and just after the action of the impulsive forces. Let us suppose that every possible motion of the point {x, y, z) is given by x' = a^6' + a^(\>' + ,..+a (3), iwith similar expressions for y and z when 61, b^, &c.; Ci, c^, &;c. are written for a^, a^, &c. Here ai, aa, &c. are known functions of the coordinates 6, (j>, (Sec. and t. They would be df/dd, df/dcf), &c. las given by equations (3) of Art. 396 if the system were holono- Jmous, but this restriction is not necessary* for our present purpose. * See a memoir by MM. Beghin and Rousseau in the Journal de Mathematiques^ Liouville, Tome ix. 1903. m 21—2 324 Lagrange's equations. [chap. viii. Since the virtual displacement must be consistent with the geometrical conditions which hold at any instant, we have, when only the coordinate 6 is varied (as in Art. 397), hx = a-^hOy hy — bi80, Bz = CiBO. . •. 2m (x'Bx + y'hy + z'hz) — Xm {a-^x' + 61/ + Ci/) W. But since 2T = 2m {x"^ + y'^ + z"^) and the partial differential coefficients dxjdO'^a^, dy/dd' = biy dz'/dd' = Ci by equations (3), ., ^ f ,dx' ,dy' ,dz\^^ dT ^^ these are = 2m j ^ ^, + y j^ + ^ ^, j 8^ = _ 8ft Let Of!, (fid, &c., ^/, ^i', &c. be the values of 6', , &c. just before and just after the impulses, and let To, 7\ be the values of T when these are substituted for 6', ', &c. The virtual moment (dT^ dT\ of the momenta is then f j^\ — -^, J hO. The Lagrangian equa- tions of impulses may therefore be written dT,_dT,^ do; ddd ' with similar equations for <^, a/t, &c. These equations are some- times written in the convenient forms ©:-• ©>*-• ■ where the brackets enclosing any quantity imply that that quantity is to be taken between the limits mentioned. Sometimes when no mistake can arise as to the particular limits meant, these are omitted, and only the brackets, with perhaps some distinguishing marks, retained. When the quantity in brackets (as in our case) is a linear function of the variables 6\ ', &c. of the first order, another meaning can be given to the expressions. The brackets may then he said to indicate that 6^ — 6^, q, &c. are to be written for 6\ , (Ssc. after all other operations indicated within the brackets have been performed. I 402. If we interpret our equations by the general principles' of Art. 283, viz., that the momenta of the particles just after an impulse compounded with the reversed momenta just before are equivalent to the impulse, we see that it will be convenient to dT call -Tw, the generalized component of the momenta with regard to 6, a name suggested in Thomson and Tait's Natural Philosophy. More briefly we may say that this is the ^-component of the mo- mentum. In the same way we may define the ^-component of the effective forces to be t; -ta, — tt, , when the system is holonomous. dt dO dd -^ ART. 403.] OBLIGATORY MOTIONS, &C. 825 Suppose for example that a variation 80 of any coordinate has the effect of turning the system as a whole about some straight line through an angle 86, then dT/dd' is equal to the angular momentum about that straight line. But, if the variation 80 move the system as a whole parallel to some straight line through a space 80, then dTjdO' is the linear momentum parallel to that straight line. These results also follow immediately from the general ex- pression dT ^ ( ,U ,5y ,8z f ,8x ,hy ,8z\ dd' given above. Let the given straight line be the axis of z. In the first case 5x= - ydd, 8ij = x5e, 8z = 0, hence the expression reduces to 2m (- x'y + y'x), which is the angular momentum. In the second case 8x = 0, 8y = 0, 8z = Sd, hence the expression becomes 2wj2;', which is the linear momentum. The equations for impulsive forces were not given by Lagrange. They seem to have been first deduced by Prof. C. Niven from the Lagrangian equation d dT_dT _dU dtdd' 'dd~'dd' We may regard an impulse as the limit of a very large force acting for a very short time. Let t^, t-^ be the times at which the force begins and ceases to act. Let us integrate this equation between the limits 1 = 1^ and t = t-^. The integral of the first v~dT~\ti dT term is ^j-r which is the difference between the initial and final values of — r: . Ldd'X dd' The integral of the second term is zero. For dTjdd is a function of 6, , &c., d', (}>', &c. which, though variable, remains finite during the time t^-t^. li A be its greatest value during this time, the integral is less than A [t-^ - t^, which ultimately vanishes. Hence the Lagrangian equation becomes -j^, ' = — - . See a paper in the \_dd Jta dd Mathematical Messenger for May, 1867, Vol. iv. page 82. 403. Obligatory Motions and Sudden Fixtures. A system of bodies is moving in a given manner. Suddenly certain points are seized and constrained to move under neio conditions. Find the subsequent motion. To simplify matters let the system have four coordinates 6, 0, \p, x» ^^^ 1®^ *wo points A, Bhe suddenly constrained to remain on two planes which move parallel to themselves with given velocities a, j8, the motions of the points along the planes being perfectly free and unrestricted. If, for example, A and B coincide and the motion is in two dimensions, this is equivalent to saying that the point A is suddenly made to move in a given direction with a given velocity. Art. 171. Let p, q be the distances (or any convenient functions of the distances) of A and B from two fixed planes parallel to the moving planes ; then p, q are known functions of 6, 0, xp, x> and two geometrical equations of the form P=f{e, 0, i^, x) = a + <^t, q = F{d, 0, ^, x) = b + ^t (1). have been introduced into the system. By the introduction of these constraints the variables p, q have become determined, and the system has then only two degrees of freedom. We shall however still consider the system to have four degrees of freedom and to be acted on by two impulses such that the subsequent motion satisfies the equations (1). The solution would be much simplified if the coordinates were originally so chosen that p, q are two of them, the other two (say 6, 0) being any independent 326 LAGRANGE'S EQUATIONS. [CHAP. VIII. quantities. If this choice has not been made we can analytically effect the change of coordinates from 6, the coordinates of the relative motion because their arbitrary variations {p, q having the values given by (1) in terms of t) move the system into all positions consistent with the constraints, while p, q may be called the coordinates of the constraint because their arbitrary variations would contradict the conditions of the constraint. This choice of coordinates is exactly the same as that made in Art. 293. Since the impulses act normally to the moving planes we have dU=Pdp+Qdq, where P and Q may be taken as measures of the impulses. The Lagrangian equations therefore become (i)>o. (^.);-. m>- m> The two first only are required to find the change of motion and these may be summed up in the following rule ; the generalized components of momenta with regard to the coordinates of the relative motion are unchanged by the impulses. This is really the generalized form of the rule already given in Art. 288. We see also that when the subsequent motion only is required it is unnecessary to calculate the force function U, it is sufficient to know the form of T. When it is important to use coordinates 6, (f>, \p, x which are not those of the constraints and relative motion we slightly alter the arrangement. We now write 8U=P{fQ5e+f^54>-V&c.) + Q{FQ5d + F^5 + &c.), where as usual sufl&xes denote partial differential coefficients. The Lagrangian equations then become Joining these four to the given relations (1) we have sufficient equations to find the subsequent values of d', 0', f, x ^^^y i^ required, the two quantities P, Q. Ex. A point in a moving disc is suddenly made to move with given com- ponent velocities a, /3 parallel to the axes. Find the subsequent motion. This is the problem already solved in Art. 171. Let p, q be the distances of O from the axes ; the equations of constraint are p = at, q = pt. Let 6 be the angle OG makes with the axis of x, OG = r. Then 2T = (p' - r B'm 66')'^+ {q' + r coa eey + k^d"^. Here the relative motion has only one coordinate, viz. d, dT z±= -(p'-r sin dd') r sin d + {q' + r cos 60') rcosd + k'^d'. dd If (as in Art. 171) u, v are the resolved velocities of G before impact, w the angular velocity, we see that p/ = m + r sin ^w, g-o' = y - r cos ^w, 6^' = ^, just before the impact, while just after Pi=a, qi=p. Substituting these values in the expression for dTjdd' and equating the results we find the value of 6' just after impact. This value agrees with that given for w' in Art. 171. 404. When two smooth elastic systems impinge on each other at one point we divide the duration of the impact into the two periods of compression and restitu- tion, Arts. 179, 185, &c. Let d^;, q', &c.; d^;, &c.; 6^', &c.; be the values of the velocities of the coordinates just before impact begins, at the moment of greatest compression and at the moment of separation respectively. Let U-^ be the work of the blow of compression, then Z/j has the measure of that blow as one factor, the ART. 405.] IMPACT OF ELASTIC SYSTEMS. 327 other factors being known from the geometry of the figure. It follows that U^e is the work function of the blow of restitution. We thus have two sets of equations with similar equations for «> «i9li«Md in lit. 398, ira OMUMt «M it M « t*w*fa*M>^ «|WlMa to ntew Iht nwAcr of ^m Ml^ «M Ok «fWtioii off tis TiT% ^^lieki c^^nes 1V» fiaa tiM a«ttli«i7 ooMftwte « oaa fi ^«o amsl kvvo noowno lo tto imtiol offonif. Ik*l^,^dMyul^4#U<<^^1tlMimtiat«lo»Boffi^f,4liidtt^ aokoi, givo # OMi ^ in tanM of t, ond tOno of Hko 6m OG, TW fmntwyimiiii^ ovniMns Cor tte motmi off Hm I OL on tonril ^ ■»!»« C^Q^ A^MI?, and Jk^l, I is 1£ho kMEttt off Aia mm " .(2^ -(tyH*y=--^(ty-(ty-f'~-H £1 orihr OnA liM aM«MM off IIm two liBW 0« o»a OL M(ir bt lito M»^ llM t^ OfMliiHW (1) oni (S) wm bo Ow auM. Has anU %e Om omk it olAv Cii^a, ixr •«i^« £a tiw first «M^ no woat ka^to tt^.>tft» of eqaatioiui are iJentk*! if Mhl=A, TM» ki Hie iwne fonmila. _. , „_- .„.;i44 in Art. 92. Ex. Two eqioal !!«»▼/ imiiiarm rods, C^ <7J?, ItmIj hingiBd «4 C, «re moirfiif in any manner under the aetion of graTitj. Eiq^#M Uie laa^tU eaergj of ibe ijflem in terms of tlie coordinates of C and tiie dsreetion eosisies (f, ot, «) (V A** ^) >vsk Lagrange** ■^uatiatu. Taking as axes of referenes the prindpal axes at thie fixed point. We cannot talEC (<^, m,, m'^ as tiie indepeadeiU ratialiles, beeanse the eoordinales of ererjpartide of the bo^ cannot be caressed in terms of them withont liifoiodtieing difSetential coeflieients into the geometrical eqpiatioiM. (See Art. 996.) Let as therefore express <^, m,, <^ in terms of ^, ^, ^. By Art. 2$6, we hare )4>"' + &c., U=F^{e) + F.,{^) + &c. The Lagrangian equation to find d is Integrate this by inspection If^id) e'^=Fi{e)+a (1). Similarly 1/3 (0) '^=F^ (0) + i3 and so on. These results may also he obtained by a change of variables. Put The Lagrangian equations are #x' = %i^\ ^y'=^, &c.; dt dx dt^ dy ^ j dx J dd dx .: if,{e)e'-^ = F^{d) + a; and the other integrals follow by symmetry. Another solution. The following solution is instructive. It is evident that, with these forms of T and U, the several Lagrangian equations are really inde- pendent, the 6 equation contains only 6, the equation only and so on. Each equation taken alone represents a possible motion in which one coordinate only varies. For each there will be a separate equation of energy, and these are the first integrals just found. Ex. 4. Iiiouville's Integrals. Let a dynamical system be such that T=^M{A^r^ + A2'^ + &c.} (1), U+C={F^{d}+F^{'^=i/% &c. "We then have by the principle of vis viva ^M{x''- + y"' + &c.)=U+C (4). The X Lagrange's equation is d ,_^ ,. , d3I , ,„ ,^ . . dU Substitute from (4), and we find I i,^ , d ,-r ,, rdM ,^^ _,. ^^dU~] dx I ^^di<«^>=U(^+'^)+*srJd« <«>• Now ~ {M {TJ+ C)] = — ~~ and ^ is a function of x only, hence by integrating (5) we obtain the first of the integrals (3). The final integrals are given in Liouville^s Journal, 1846, Vols, xi., xii., 1849, Vol. xiv., p. 291. It will be proved in Art. 431 that this process is equivalent to a change of the independent variable from t to t, where dt = MdT. Ex. 5. A dynamical system is such that T=M {iA^^e'^ + A^^e'' + iA^^(t>'^, U+C = BjM, where M is a function of 6 and 0, but A^^, A-^^, A.^2 ^^^ ^ ^^^ functions of (f> only. Prove that a first integral of Lagrange's equations is M{A,^d' + A,2cp'}=a, where a is an arbitrary constant. To prove this we combine the 6 Lagrangian equation with the equation of energy. It may also be deduced by inspection, if the independent variable is • changed as explained in Art. 431. Ex. 6. The elliptic coordinates of a particle are \, /x, v, and the particle is constrained to move on a fixed ellipsoid X. The force function U being given by , and the kinetic energy by Art. 365, Ex. 4, deduce from Liouville's integrals that ( ' (»2 _ ^2\2 U^ _ ^2) „'2 /„2 _ ^2\2 /^2 _ ^2) ^'2 ' where h, k are the semi-axes of the focal conies. By division the discovery of the path is reduced to integration. This solution applies in the following cases or any combinations of them. (1) When the force tends from the centre and varies as the distance, we have 2U=r^ = \'^ + fx^ + p'^-h^-k\ .: U{iJ?-v^=F^{ii) + F^{v). (2) When the force acts perpendicularly to the plane of yz and varies inversely as the cube of the distance from that plane, we have \ixv= ±hkx, hence U=AlfjPv^ and {/jfi - j/2) U has the required form. (3) When the force is central and such that U=AI{p^-m?)i where p is the distance of the particle from either of the fixed 332 LAGRANGE'S EQUATIONS. [CHAP. VIII. points x=±hkl\, y = 0, z = and m'^\^ = {X^ -h'^){\'^-k'^). We notice that m = when \=h or k, that is when the ellipsoid becomes a plane. Solutions of this and other problems of the same kind are given in the author's treatise on Dynamics of a Particle under the heading Motion on an ellipsoid, Chap. VII. 408. Examples on, impulses. Ex.1. A rhomhus, formed of four jointed rods, falling from rest tcith a diagonal vertical impinges with velocity V on a fixed horizon- tal inelastic plane at the corner A. Find the suhsequent initial motion. This is the problem solved in Art. 176 ; for the sake of comparison we shall here give two solutions both founded on the Lagrangian equations. Let the mass of each rod be unity. Let x be the altitude of the centre of gravity of the rhombus, 6 the inclination of any rod to the vertical. If we take X and 6 as the coordinates of the system, we have T = 2 {a;'2+ (it^ + a^) 6'^]. If P be the impulse at ^, we have W=Pb{x-2acoQd) = P5x + 2aP&Yn.eZd. The Lagrangian equations are by Art. 401 4 {x^ - Xq') = P, 4 (/c2 + a2) (0/ - do') = 2aP sin d. The initial and final values of x' are Xq = -V, x^ = - 2aw sin 6 ; those of d' are e' = 0, dy = w. Hence putting k'^ = la^ and eliminating P we have w = s - t , o • 2^ 3 ' " ^ ci J. + o sm^ u which is the same result as in Art. 176. Remark on the choice of coordinates. The objection to the solution given above is that we have to use all the Lagrangian equations though the impulse is not required. If loe wish to avoid introducing the impulse into the equations, tee must use such coordinates that the variation of one alone while the other is constant does not alter the point of application of the hloxo. When the coordinates chosen are X and e a variation of either alone alters the position of A. But if we take as coordinates 6 and the ordinate y of the point A which strikes the plane, a varia- tion of 6 alone does not alter the position of ^, so that the virtual moment of any force acting at A does not enter into the equation thus formed. In the same way if the magnitude of the blow at A were wanted we should use an equation formed by the variation of some coordinate, such as y, which does alter the position in space of A. The coordinates y and d have been called in Art. 403 the coordinates of the constraint and of the relative motion respectively. Taking as coordinates y and 6, we find r=2 {?/'2 - ^ay'e' sin d + {k'^ + a^ + 4.a^ sin^ 6) d'^]. The single equation now required is ( ;t77 ) =0, so that it is unnecessary to calculate U. The limits of y' are y^'^ -V, y^' = 0; those of 6' are ^0' — ^' ^1=^- The value of w follows without difficulty. If the ground is elastic we follow the rule given in Art. 404. Since ^0' = ^ ^^® angular velocity of each rod after the rebound is found by multiplying the value of w given above when the ground is inelastic by (1 + e). Ex. 2. Six equal uniform rods form a regular hexagon loosely jointed at the angular points ; a blow is given perpendicularly to one of them at its middle point, show that the opposite rod begins to move with one-tenth of the velocity of the rod struck. [Math. Tripos, 1882. We take as one coordinate the distance of the point of application of the blow from the axis of x (supposed to be parallel to the rod struck) and as the other coordinate the angle 6 which either of the adjacent rods makes with the axis of x. ART. 410.] EXAMPLES ON IMPULSES. 333 This choice is made because a change of d alone does not alter the point of application of the blow. Since cos^ = ^ we have 1T= 6y'2 + llay'd' + 4 (Sa^ + r-) 6'^-, where 2a is the length of any rod. The single Lagrangian equation required is that dTjdd' is unaltered and therefore is equal to zero. Since the velocities of the two rods to be compared are y' and y' + 2ad' the result follows at once. Ex. 3. A beam, placed on a smooth horizontal plane, has one extremity fixed; and a ball A of mass m is placed in contact with it at a given distance a from the fixed extremity. Determine at what distance b another ball B of mass /x must impinge directly on the beam that the greatest possible velocity may be communi- cated to the ball A by the impact. The beam and balls are inelastic. [Math. Tripos, 1844. Let 6' be the angular velocity of the beam, y' the velocity of the ball B, the relative velocity of approach of the ball and beam is then z' = y'-hd' and dU= -Pdz. If we take d and z as coordinates the one Lagrangian fact required is that dTjdd' is unaltered by the impact. We have 2T= {ma^ + Mk^) d"" + ix{z' + hd')\ since the limits are 6^=0, ^/ = w, Zq'=V, 2i' = 0, we find {ma' + M A;2 + fxb'') o} = fxbV, .: fxb^ = ma^ + Mk^ when w is a maximum. 409. Sir W. R. Hamilton has put the general equations of motion into another form, which is sometimes more convenient for investigating the general properties of a dynamical system. This transformation may be made to depend on the lemma given in the following article. In what follows we confine ourselves to the elementary properties of re- ciprocation. The subject will be resumed and treated more fully in the second volume. Sir W. Hamilton's demonstration of his equations requires that T should be a homogeneous quadratic function of the velocities, and this is generally true in dynamics. The extension to the case in which the geometrical equations contain the time explicitly is due to Donkin, Fhil. Trans. 1854. 410. The Reciprocal Function*. Let T^ he a function * We may deduce from this lemma the method of solving partial differential equations by reciprocation, sometimes called Legendre's method and sometimes De Morgan's method. Let the partial differential equation be {x, y, z-^,p, q) — Q, where p and q are the partial differential coefficients of z-^ with regard to x and y. If we write z^:= - z-^+ px -\- qy , we have by the lemma x^dz^jdp, y = dzjdq. Hence ■ this rule ; substitute for x, y, z-^ from the auxiliary equations dz^ dz,) dz» dz^ dp ^ dq^ ^ ^ ^ dp ^ dq and treat p, q as the independent variables. Thus we have a new differential equation which may be more easily solved than the former. Let the solution be z^=f{p, q), then, by the auxiliary equations, x, y and z^ have all been found in terms of two auxiliary quantities p and q, and further these quantities have a geometrical meaning. This method may be extended to any number of variables and orders. Also as in Art. 418 we may if we please modify the equation for some only of these variables. Ex. If the equation be xp^ + yq^ = z^, show that z.2 = ^^F ( ^(i_ \ )' whence X, y, z can be found in terms of the auxiliary quantities by differentiation. 334< Lagrange's equations. [chap. viii. of any quantities ivhich it luill he presently found convenient to call e\ ^', &c. Let dT^ _ dT^ _ then 6', (j)', &c. may he found in terms of u, v, (&c., from these equa- tions. Let T^=-T^ + ud' + v' + &c., and let T^ he expressed in terms of u, v, <&c., the quantities 6\ ~ defy' '^^' To prove this, let as take the total differential of T^, we have dT,= -^dd-h(-^ + v}jde'+e'du + 8zc. By the conditions of the lemma the quantity in brackets vanishes. Now if T^ be expressed as a function of 6, u, , v, &c, only, and not 6', ', &c., we have •''>*'i- dT, = ^de-{-^du + Szc. Comparing these two expressions for dl\ we have dT^ dTi , dT^ ^, Thus we have a reciprocal relation between the functions T^ and T^. We find T^ from T^ by eliminating 0', cf)', &c. by the help of certain equations, we now see that we could deduce Ti from Tz by eliminating u, v, &c. by the help of similar equations. We shall therefore call T^ the reciprocal function of Ti with regard to the accented letters 6', (/>', &c. 411. It should be noticed that, if T^ be a homogeneous quadratic function of the accented letters 6', , &c., then u6' -\-v4) + &>c. = '2.Ti, and therefore T2 = T^, but is differently expressed. Thus Tj is a function of 6', ^\ &c. and not of u, v, &c., while T^ is a function of u, v, &c. and not of 6', cf)', &c. We notice that in this case T^ is a homogeneous quadratic function of u, v, &c. 412. If Ti be the semi-vis viva of a dynamical system, this process is really equivalent to changing from the use of component velocities to the use of the corresponding component momenta. Either may be used to determine the motion of the system, some- times the one set being the more convenient and sometimes the other. ART. 413 a.] THE RECIPROCAL FUNCTION. 335 413. Examples on the Reciprocal Function. Ex. 1. The position in space of a body of mass M is given by x, y, z, the rectangular coordinates of its centre of gravity, and d, , \f/ the angular coordinates of its principal axes at the centre of gravity, as used in Chap. v. Art. 256. If two of the principal moments of inertia are equal, and if ^, 77, ^, u, v, w, be the components of momentum corresponding respectively to x, y, z, 6, (p, \p, the vis viva 2Ti is given in Art. 365, Ex. 1. Show that the reciprocal function is 2ro |2 + ^2+^2 y;. 2 ^2 (10 + -7; + r C0S^)2 M -AC' Asin'd As a useful case we notice that the reciprocal function of where the terms containing the products are absent, is T -^ — 1 v^ 2 A-^i 2 A22 We observe that if Tj is a one-signed positive function, T^ must also be a one- signed positive function. If the vis viva 2T^ be given by the general homogeneous expression Ex. 2. 2T^ = A^^d'^ + 2A^^d'(p' + show that the reciprocal function of 1\ may be written in the form To=- 2A u -^12 V A12 Ann where A is the discriminant of Tj. Thus the coefficients of u^, v^, 2uv, dc. in T^ are the minors after division by 2A of A^^, An^, ^12 » ^c. See also Chap. i. Art. 28, Ex. 3. Ex. 3. If ^, T], &c. be partial differential coefficients of a function P of x, y, &c. with regard to those variables respectively, prove that x, y, &c. are also partial differential coefficients of a function Q of ^, 77, &c. with regard to these variables respectively. If P be homogeneous and of n dimensions prove also that Q={n-1) P, For instance P may be the potential function in Attractions, or the velocity potential in Hydrodynamics. Ex. 4. Regarding Tj as a function of d', ', &c. in terms of du, dv, &c. Substituting in the second set the theorem follows at once. 413 a. We notice that for any given position of the system, the coordinates, if independent, may have any given velocities; so that 0, 0, &c. being given, 6', ', &c.), and therefore u6^ + V(j>^ + &c. = 2T. Hence H = - L + ud' + v(l>' + &c. = T -U. Thus H is the sum of the kinetic and potential energies, and is therefore the whole energy of the system. 415. To express the Lagrangian equations of impulses in the Hamiltonian form. Referring to Art. 401, we see that the Lagrangian equations of motion may be written in the typical form dO'l, ^• Let H be the reciprocal function of T, and let us replace u, v, &c. by P, Q, &c. Then these equations take the typical form ^ , ^ , dH ART. 416.] THE HAMILTONIAN TRANSFORMATION. 337 Thus the changes in the velocities of the generalized coordi- nates are immediately determined by simple differentiation when the reciprocal function of T has been written down. 416. Examples on tlie Hamiltonian Equations. Ex. 1. To deduce the equation of Vis Viva from the Hamiltonian equations. Since H is a function of {d, ^, &c.), {u, v, &c.) we have, if accents denote total differential coeflficients with regard to the time, ,,, dH dH ,, dH , ^ dH dt dd du dt Thus the total differential coefficient of H with regard to t is always equal to the •partial differential coefficient. If the geometrical equations do not contain the time explicitly, this latter vanishes and we have H=h, where 7i is a constant. Ex. 2. To deduce Euler's equations of motion from the Hamiltonian equations. Taking the same notation as in the corresponding proposition for Lagrange's equations. Art. 406, we have dT , . ^ ^ , dT ^ dd' i ^ ^ ^' d(f>' ^ dT w = -fr, = { - -^^i cos + Bo}2 sin (p) sin 6 + Cwg cos 6. Before we can use the Hamiltonian equations we must by Art. 411 express T in terms of {u, v, w). To do this we solve these equations to find Wj, Wg, Wg in terms otu.v.w. We find Aio^=usmd) + (vco8 6-tv)—. — ?, Bcoo = u cos -{v cos d-w) ~^—\ . ^ ^ ^ ' sin ^ AJso by Art. 414 H=l {A coja + Bui^^ + Gcj^^) - U. As we only require one of Euler's equations, let us use ^— = -v', -^ — dt and this leads at once to the third Euler's equation in Art. 252. The latter of the two Hamiltonian equations leads to one of the geometrical equations of Art. 256. Thus the six Hamiltonian equations are equivalent to all the three dynamical and the three geometrical Eulerian equations. Ex. 3. A sphere rolls down a rough inclined plane as described in Art. 144. We have T==^^maW^ and U=mgad sin a. Is it correct to equate IZ^ to the difference of these functions ? Verify the answer by obtaining the equations of motion given |in Art. 144. See Art. 411. Ex. 4. A system being referred to coordinates d, ', \//' the point P will lie somewhere on the quadric Tj^= U yfhere Uis the instantaneous value of the force function. Then since u= ^-/, v= -r^., w= -—j, we see that Q will also lie on a quadric, which is dd dtp d^ the polar reciprocal of the quadric Tj with regard to a sphere whose centre is at the origin, and whose radius is equal to JiU. Let this reciprocal quadric be T^= U. Then, since these quadrics possess recipro- cal properties, we see that d = -~, = -^, ^ = -y-^. I Ex. 1. If the coefficients of the two quadrics T^ and T^ be functions of any dT dT quantity 6, show geometrically that -—^ = — ^. Thence deduce the remaining du da r xu TT -1. • X- • r d^ , dll , dH ^ three of the Hamiltonian equations, viz. -«=——, -v =^— , -w =-^--, where dO d(p dy ," p. 62. Ex. 2. Show that the form of Tg ^^s used in Geometry is the same as that given in Art. 413, Ex. 2. 417. Reciprocal Theorems*. Let us suppose that two sets or arrangements of impulses are applied to the same system * The reciprocal theorem is primarily due to Eayleigh who has given many illustrations of it. See Phil. Mag. 1874 and the Theory of Sound. There is also a memoir by Helmholtz in Crelle's Journal, 1886. Many examples are given by Prof. Lamb in the Proc. London Math. Soc. 1888. The one at the end of Art. 417 is taken from the article Dynamics in the Encyc. Brit. ART. 418.] THE MODIFIED LAGRANGIAN FUNCTION. 339 of bodies at different times, the system being in each case previously placed in a given state of motion. Let be respectively the work function due to each set as explained in Art. 401. Let ^Z, /, &c. d^, 4>^, &c. be the velocities generated by each set. If the vis viva be represented by the general ex- pression for 2T given in Art. 413, Ex. 2, we have p,=A,,e; + A,,cf>^+..., Q, = A,,e^+ko (i), while P2, Q25 &c. are represented by similar expressions with 6^, &c. written for 0^, &c. It immediately follows by substitution from (1) that PA'+Qii>.' + ... = PA'+Q2i + (2), each being equal to a symmetrical expression. Using the language of the principle of virtual velocities, it follows that the sums of the virtual moments of either set of impulses for the actual dis- placement produced by the other set are equal. Let each of these systems consist of a single blow and let A, B he the points of application. To trace their effects let two of the coordinates, say 6, (j), be the ordinates of A, B measured in the direction of the blows. Then if Pj, Q^ are the blows the work will respectively be PiBd, Q^Bcj). The reciprocal equation (2) then becomes Pi^/=Q.0i' (3), all the other terms being zero. The blow at A affects all the co- ordinates and causes velocities 6^, 0/, &c. That at B also affects all the coordinates, but this proposition shows that the velocity of B due to the blow at A and that at A due to the blow at B are in the same ratio as the blows to which they are respectively due. The equation (3) also follows very easily from Art. 41.5. As an example of this theorem consider the case of a straight chain of rods hinged each to the next. A blow at any point A will produce a certain velocity at any point B ; the theorem asserts that an equal blow at B will produce an equal velocity at A. An impulsive couple acting on any rod will produce a certain angular velocity in a rod B, an equal couple acting on the rod B will produce an equal angular velocity in the rod A. If a blow F acting at a point A produce an angular velocity w in a rod B, then a couple Fa on the rod B will produce a linear velocity ua at the point A. 418. The Modified Lagrangian Function. Sir W. Hamilton transforms all the accented letters 6\ (f>\ &c. into the corresponding letters u, v, &c. But we may also apply the Lemma to change some only of the Lagrangian coordinates into the corresponding Hamiltonian coordinates, leaving the others unchanged. We may thus use a mixture of the two kinds of equations. With one and the same function we can use Lagrange's equations for those coordinates for which they are best adapted, and the 22—2 340 lagkange's equations. [chap. viii. Hamiltonian equations with the remaining coordinates, if we think their forms preferable. The substance of this theory, as given in Arts. 418 to 425, is taken from the author's essay on " Stability of Motion," 1876. 419. To explain this more clearly let us consider a system depending on four coordinates, 6, 4*,^,i]- Let Li be the Lagrangian function. Let us now suppose that we wish to use Lagrange's equations for the .coordinates ^, rj and the Hamiltonian equations for the coordinates 0, (p. To do this we use the two formulae of transformation -T7i = u, -rn =v, and we put dd d(p ^ ■ Zg = — Xj + ud' + V(j)\ We have as in Art. 414 the two sets of Hamiltonian equations, ~d^' ""'"dd' We must now include f ', rj' among the unaccented letters spoken of in the Lemma of Art. 410, so that we have c^Zg _ dLi dL^ _ dLi d^' ~~W dl ~~~~d^' with two similar equations for r]. Thus the two Lagrangian equations for f, 77 are still true if we replace L^ by L^ ; so that we have the two sets of Lagrangian equations, d dL^ _ dL^ d dL^ _ dL^ dt d^' d^ ' dt di] drj 420. The function L^ might be called the modified function^ but it is more convenient to give this name to the function with its sign changed. The definition may be repeated thus : — If the Lagrangian function X be a function of 6, 6\ will be L = L — u6' — vcj)', where u = -jw, , v = -.-., , and we suppose 6\ , 6', <^' and all the other letters, L' is a function of 6, cf), u, v and all the other letters. These two functions L, L' possess the property (by Art. 410) that their partial differential coefficients are the same with respect to all letters except d\ <^\ u, v. As regards these four we have dL dL , dL' ^, dL' ,, We may form the dynamical equations, for the coordinates with ART. 421.] THE MODIFIED LAGRANGIAN FUNCTION. 341 regard to which the function has been modified by the Hamiltonian rule, as if L^ — — L' were the Hamiltonian function, and for the remaining coordinates by the Lagrangian rule, as if either L^ or L' were the Lagrangian function. The function L^ may be also called the reciprocal function of the Lagrangian function L^ with regard to the coordinates 0, , &c., because it is obtained from L^ just as T^ is obtained from T^ in Art. 410, except that we operate only on such of the coordinates as we please. It is however convenient to distinguish the two operations by different words. We shall use the word Reciproca- tion when we change all the coordinates, and Modification when we change only some. 421. To find a general expression for the modified Lagrangian function after the necessary eliminations have been performed. Let the vis viva IT be given by the homogeneous quadratic expression T=ir,,^'2+ r,^^V + ... + ir^^r+ r,^^r + ... , so that the Lagrangian function is L = T+U, where iiJ is a function of the co- ordinates 6, 0, ^, &c. We intend to modify L with regard to d, **^'+ rrf.^^'-^rf>y+- - ... = hT^^^"'+T^/v'+... + U-id'{u-X)-^', &g. in terms of ^', t)', &c. by the help of determinants. Substituting their values in the expression (3), we find hT^^^-+T^/v' + &c. + U+^ 0, u-X, v-Y, u-X, V-Y, T.J., where A is the discriminant of the terms in T which contain only d', 0', &c. It may also be derived from the determinant just written down by omitting the first row and the first column. We may expand this determinant, and write the modified function in the form 1 0, V u, T, V, T, 2A 0, X, Y, 0<^ ' -^(jxf)^ X To. T, e4>' ■^it,> 342 LAGRANGE'S EQUATIONS. [CHAP. VIII. dT dT where u, v, &c. as usual stand for -—-. , -^—, , &c., and X, Y, &c. are given by da d x= r,^r + T,^v' +•:, Y= T^^r + T^n-n' + ■■; &c. = &c., so that X, Y, &c. may be obtained from u, v, &c. by omitting the terms which contain d', , &c. become -^ -^^ = 0, &;c. Integrating, we have dL dL where u, v, &c. are absolute constants whose values are known from the initial conditions. By the help of these equations we may find 6\ ', &c. in terms of ^', rj', &c., so that the problem is really reduced to that of finding f, 77, &c. The names kinosthenic and speed coordinates have both been suggested by Prof J. J. Thomson for coordinates which enter into the Lagrangian function only through their differential coefficients (Phil. Trans. 1885, and Applications of Dynamics to Physics and Chemistry, 1888). We may now simplify the process of finding these remaining coordinates ^, rj, &c. by modifying the Lagrangian function so as to eliminate the variables 6', <^', Szc, and introducing in their place the constant quantities u, v, &c. We write and eliminate 6\ (j)\ <&;c. by help of the integrals just found. The equations to find f, rj, <&:c. may he deduced by treating ± L' as the Lagrangian function. , 423. When the system starts from rest the modified function takes a simple form. Suppose the Lagrangian function L to be a homogeneous quadratic function of 0', \ &c. Then, referring to the first integrals found above, and remembering that the initial values of 6\ ', &c. are all zero, we have u = 0, v = 0, &c. = 0. ART. 424.] ABSENT COORDINATES. 345 TJius the modified function L is equal to the original function, but is differently expressed. The function Z is a function of 0\ i^' , &c.; the function L' is the value of L after we have eliminated the differential coefficients Q\ ^', &dc. by help of the first integrals. The result of the elimination can be deduced from Art. 421. The first and third determinants are here zero. We have therefore 0, X, Y, ., ^'=^«T+^s/'''+*"- + ^+^ We may deduce this expression from the Lagrangian function Zi by a simple rule, viz., omit all the terms ivhich contain the differential coefficients 6\ 0', d'c. to be eliminated, and add the determinantal term loritten doion above. 424, Example of the Solar System. As an example let us consider the case of three particles whose masses are m^, m^, m^ mutually attracting each other according to the Newtonian law and moving in any manner in one plane. Referring these to any rectangular axes, their vis viva and force-function will be functions of the six Cartesian coordinates and their differential coefficients. But we may move the origin and turn the axes round the origin without altering the vis viva or the force-function. It follows that each of these functions is independent of three of the coordinates, though it may depend on their differential coefficients with regard to the time. We may therefore modify the Lagrangian function and make it depend only on the three other coordinates. The vis viva of the system is equal to the vis viva of the whole mass collected at the centre of gravity together with the vis viva relative to the centre of gravity. The former is easily written down and is in our case a constant ; let us turn our attention to the latter. Let G be the centre of gravity, draw Ga, (r/3, Gy to represent in direction and magnitude the velocities of the three particles, i.e. let a, ^, y trace out their hodographs. Then the sides of the triangle a^y represent the relative velocities of the particles, and the vis viva of the system is represented by m-^Ga^-\-m^G^'^-\-m^Gy^. Since the momentum of the system relative to its centre of gravity resolved in any direction is zero, it follows that G is the centre of gravity of three particles m^, m^, m.^ placed at a, ^, y. By a well-known property of the centre of gravity we have m-^m^{apf+ =/a{wJi {Gaf +...}, where jx is the sum of the masses. It immediately follows that the vis viva of any system relative to its centre of gravity = — L, ^ •'^ , where v-^2 is the relative velocity of the particles m^, Wg. This formula for the relative vis viva is evidently true for any number of particles. It was obtained by Sir E. Ball by a different method in the Astronomical Notices for 1877. Let a, b, c, A, B, C be, as usual, the sides and angles of the triangle formed by joining the particles. Let 6 be the angle made by the side c with any straight line fixed in space. Let accents as usual denote differential coefficients with regard to the time. Then we have 2mi?n2V = 7Mi7W2 {c'^ + c^d'^}+m^m^ {b'^ + b^ {$' + A^} + m^m^ {a'^ + a^ {d'-B')-}. Thus, if 2r be the vis viva relative to the centre of gravity, we have 2T=Pd'^ + 2Qd' + R, 344 Lagrange's equations. [chap. viii. where P, Q, R are functions only of the triangle, and not of 6. We have fMP = m-^m2C^ + fxR: How we shall express these must depend on the coordinates we wish to use. Thus we may choose any three parts of the triangle, except the three angles, as co- ordinates. Ex. Supposing it to be convenient to choose the distances b and c of two of the particles from the third, and the angle A subtended by those two at that third particle, as the coordinates of the triangle, show that P, Q, R may be expressed in terms solely of 6, c, A and their differential coefficients by the help of the following results a^ = b'^ + c^-2bcGosA, ^ {be sin A) = b^ A' + aW + 2bc' sin A, a'-'^ + a^B'^ = b'-^ + c'^-2b'c' co9A+bW^ + 2bA'c' ainA. These admit of easy geometrical demonstrations. 425. We may also modify the Lagrangian function with regard to d. To do this we put u = dTldd' = Pd' + Q. We notice that, since the force-function U is not a function of 6, u is by Art. 422 an absolute constant. We now form the modified function L'=L-ud = ^ h (/. This function may now be used as if it were the Lagrangian function to find any changes in the triangle joining the three particles. We may also notice that the angular velocity in space, viz. 6', of the side of the triangle joining ?%, Wg is given by the equation Pd'+Q = u, where w is a constant. Ex. 1. Show that P is equal to the moment of inertia of the three particles about the centre of gravity. Ex. 2. Show that fi^ {PR - Q'^) may be written in the symmetrical form {m^m2C^ + in^m^b^ + m.^m^a^} {m^m2c'^ + m^mjb''^ + m^m^a'^] + m^m2m^ {wi {bcA')^ + m2 {caBy + ms{abC'f}. Ex. 3. Show that the quantity u is equal to the angular momentum of the system about the centre of gravity. See Arts. 397 and 402. Ex. 4. Show that we may take for fiQ either of the forms m-^^ {m2C^B' - vi.^b^C), or m2{m^a^C' -vi-i^c^A'), the effect of the change being to add to the Lagrangian function L' a quantity equal to B' or C respectively. See Art. 399, Ex. 2. 426. Non-Conservative Forces. To explain how Lagrange's equations are to he used when some of the forces are non-conservative. Lagrange's equations in the form given in Art. 399 can be used only when the forces which act on the system have a force-function. If however P5d be the virtual work of the impressed forces obtained by varying 6 only, Q50 the vir- tual work obtained by varying only, and so on, it is clear from Art. 399 that Lagrange's equations may be written in the typical form — -7^ - -—-z=P. at ad do 427. It is often convenient to separate the forces which act on the system into two sets. Firstly those which are conservative. The parts of P, Q, &c. due to these forces may be found by differentiating the force-function with regard to 6, , &c. Secondly those which are non-conservative, such as friction, some kinds of ART. 427.] NON-CONSERVATIVE FORCES. 345 resistances, &c. The parts of P, Q, &g. due to these must be found by the usual methods given in statics for writing down virtual work. Though the non-conservative forces do not admit of a force-function, yet sometimes their virtual works may be represented by a differential coefficient of another kind. Thus suppose some of the forces acting on a particle of a body to be such that their resolved parts parallel to three rectangular axes fixed in space are proportional to the velocities of the particle in those directions. The virtual work of these forces is S {fi-^x'bx + fi^y'dy + /JL^^'dz) , where fx.^, ix^, fi^ are three constants which are negative if the forces are resistances. Eor example, if the particles are moving in a medium whose resistance is equal to the velocity multiplied by a constant k, then A^i » /^ > A^a are each equal to - k. Put Since [x, y, z) are functions of 6, 0, &c. given by the geometry of the system we dx dx have, as in Art. 396, x' = -zr- + t-^ 6' -i- . . . at ad with similar expressions for the other coordinates. Substituting we have F expressed as a function of 6, 0, &c., d\ ^', &c. We also notice that, as in Art. 397, -r^ = :t^- Differentiating F partially we have uo da ^.^(..^^^\ ^,. \_v/.. ..«^^ dd' -^. = ^ifH-'% + &c.y^(^f.,x'^^ + &c.y dF ^^ dF ^ „ ^ f , fdx ^„ dx ^ \ ) In this case, therefore, if U be the force-function of the conservative forces, F the function just defined, 05^, 4»50, &c. the virtual works of the remaining forces, Lagrange's equations may be written d dT _dT _dU _dF dtdd'~ dd ~dd ~ dd'^ ' with similar equations for 0, xj/, &c. We may notice that, if the geometrical equations do not contain the time explicitly, the function i^ is a quadratic homogeneous function of 6', 0', &c. If the forces whose effects are included in F be resistances, then fi^, fx^, fx.^, &c. are all negative. In this case F is essentially a positive function of the velocities, and in this respect it resembles the function T representing half the vis viva. If we treat the equations written down above exactly as Lagrange's equations are treated in Art. 407 to obtain the principle of vis viva we find ^(T-U) = e'e + &c.-^,d'-&G., dt^ ' dd but in this case F also is a homogeneous function of 6', &c. Hence we find ^^{T-U) = e'e + &c.-2F. We therefore conclude that, if the geometrical equations do not contain the time explicitly, and if there be no forces present but those which may be included in the potential function U and in the function F, then F represents half the rate at which energy is leaving the system, i.e. is dissipated. The use of this function was suggested by Lord Eayleigh in the Proceedings of the London Mathematical Society, June, 1873. The function F has been called by him the Dissipation function. 346 LAGRANGE'S EQUATIONS. [CHAP. VIII. 428. Ex. 1. If any two particles of a dynamical system act and react on each other with a force whose resolved parts in three fixed directions at right angles are proportional to the relative velocities of the particles in those directions, show that these may be included in the dissipation function F. If F^. , Vy, V^ be the com- ponents of the velocities, fiiV^, ix^Vy, fi^V^ the components of the force of repulsion, the part of F due to these is -|S (fJ'-iV'x^ + f^z^y^ + f^s^z^)' This example is taken from the paper just referred to. Ex, 2. A solid body moves in a medium which acts on every element of the surface with resisting forces partly frictional and partly normal to the surface. Each of these when referred to a unit of area is equal to the velocity resolved in its own direction multiplied by the same constant k. Show that these resistances may be included in a dissipation function F, where F='^{ff{u^ + v^ + 10^) +Ao}^^ + 5aj/ + Cw/ - 2D(ayb}, - 2^w,w^ - 2Foj^(Oy} , . where a is the area, A, B, &c. the moments and products of inertia of the surface- of the body, and {u, v, w) the resolved velocities of the centre of gravity of cr. 429. Systems not holonomous. To explain how Lagrange's equations can be used in some cases when the geometrical equations contain differential coefficients with regard to the time. It has been pointed out in Art. 396 that the independent variables 0, , &c. used in Lagrange's equations must be so chosen that all the coordinates of the bodies in the system can be ex- pressed in terms of them without introducing 6', (f>, &c. But when we have to discuss a motion like that of a body rolling on a perfectly rough surface, the condition that the relative velocity of the points in contact is zero may sometimes be expressed by an equation which, like that given in Art. 137, necessarily involves differential coefficients of the coordinates. In some cases the equation expressing this condition is integrable. For example : when a sphere rolls on a rough plane, as in Art. 144, the condition is x — aO' = 0, which by integration becomes x — a6 = b, where h is some constant. In such cases we may use the condition as one of the geometrical relations of the motion, thus reducing by one the number of independent variables. But when the conditions cannot easily be cleared of differential coefficients, it is often convenient to introduce the reactions and frictions into the equations among the non -conservative forces in the manner explained in Art. 426. Each reaction has an accom- panying equation of condition, and thus we always have sufficient equations to eliminate the reactions and determine the coordinates of the system. The elimination of the reactions may generally be most easily effected by recurring to the general equation of virtual work and giving only such displacements to the system as make the virtual work of these forces disappear. Suppose, to fix our ideas, that a body is rolling on a perfectly rough surface. Let 6, , &c. be ART. 429.] SYSTEMS NOT HOLONOMOUS. 347 the six coordinates of the body, then by Art. 137 there will be three equations of the form A = ^i^' + 5af +...=0 (1), the other two being derived from this by writing 2 and 3 for the suffix. These three equations express the fact that the resolved velocities in three directions of the point of contact are zero. The equation of virtual work may be written (Art. 398) [dtde'-der^^'''^W^^^^' ('>' where U is the force-function of the impressed forces. Since the virtual works of the reactions at the point of contact have been omitted, this equation is not true for all variations of 6, (j>, &c., but only for such as make the body roll on the rough surface. But the geometrical equations L^, Xo, L^ express the fact that the body rolls in some manner, hence B6, B + ...= (3). If we use the method of indeterminate multipliers (see Art. 400), the equations of virtual work are transformed in the usual manner into d dT dT _dU dL, dL, dL, ,,. dtdO'~dd-dd'^^W^-^d¥'^''W ^*^' with similar equations for the other coordinates , yfr, &c. These joined to the three equations Xj, L^, L^ are sufficient to determine the coordinates of the body and X, //., v. This process will be very much simplified, if we prepare the geometrical equations Xi, L^, L^ by elimination, so that one dif- ferential coefficient, as 6', is absent from all but the first equation, another, as \ absent from all but the second, and so on. When this has been done, the equation for becomes ±dT_dT_dU dX, dtdO' dd~ dO^ dO' ^ ^• Thus \ is found at once. The values of //, and v may be found from the corresponding equations for ^, -^/r. We may then sub- stitute their values in the remaining equations. It is here supposed that some of the equations of condition represented by equation (1) do not admit of exact integration. The systems here considered are therefore not necessarily holono- mous, see also Art. 396. In Art. 232 of the second volume of this treatise this method is applied to find the oscillations of a heavy sphere set rotating about a vertical axis and placed on the summit of a fixed rough surface. 348 LAGRANGE'S EQUATIONS. [CHAP. VIII. 429 a. The method of indeterminate multipliers is really an introduction of the unknown reactions into Lagrange's equations. Thus let Ri, jRg) -^3 he the resolved parts of the reaction at the point of contact in the directions of the three straight lines used in forming the equations L^, L^, L^. Then L^, L^, L^ are proportional to the resolved relative velocities of the points of contact. Let these velocities be k^L^, k^^L^, k-^L^. Then if 6 only be varied the virtual velocity of R^ is k^A^W, which may be written k^ -y^ hO. Similarly the virtual velocities of Ro and R^ do are k^ ,^ W and k^ -twf 80. Hence, by Art. 426, Lagrange's equations are of the form d_dT_dT _dU dL^ dL, dL, dt dO' dd ~dO^ "'^^ dO' "^ "'^^ dO' "^ "' ' dd' ' Comparing this with the equations obtained by the method of indeterminate multipliers we see that \, /jl, v are proportional to the resolved parts of the reactions. The advantage of using the method of indeterminate multipliers is that the reactions are introduced with the least amount of algebraic calculation, and in the manner which is most convenient for the solution of the problem. 430. Appell's Equations. There is another method of forming the general 'equations of motion besides that of Lagrange which has at least the advantage of not being restricted to holonomic systems*. To simplify the discussion let us however first suppose that the geometrical equations do not contain any differential * The first writer who extended Lagrange's equations to systems in which the equations of condition are not expressible in an integrable form was Ferrers, Quarterly Journal of Mathematics, No. 45, Vol. xii. 1872. He replaces Lagrange's d equation by another of the form d dT ^ ( ,dQ^\ dU I dtde' -r-ryr ^^ j ^^ > where 6, 0, &c. are the generalized coordinates, x^. is a Cartesian coordinate con- nected with a mass m^ subject to the condition which may be integrable or not integrable. The method explained in Art. 429 of applying Lagrange's equations to systems not holonomous by using indeterminate multipliers was first given in the third edition of this treatise, 1877. It requires no new function. The equations of Appell are briefly explained in the Comptes Rendus, tome cxxix. 1899, and more fully developed in the Journal de Mathemdtiques (formerly known as Liouville's Journal), tome vi. 1900. The theory given in Arts. 430 b, &c. is chiefly founded on the latter account. There are also some memoirs on The Equations of Mechanics by P. Jourdain in the Quarterly Journal of Mathematics, 1904, 1905. It appears that he had inde- pendently arrived at the equations given by Appell. Appell in Art. 462 of his Traite de Mecanique has given a list of foreign writings on this subject, the earliest being dated 1888. ART. 430 b.] appell's equations. 349 coefficients with regard to the time. We have seen in Arts. 398, 399 that the equations of motion are included in the form ,dx „dy „dz\ dU ^''\''"re*y"f0'-'"de) = M W' where the dxjdd, &c. are the partial differential coefficients of x^ &c. when any one coordinate as d is alone varied. We then have by (3) of Art. 396 It follows from the latter equation that the partial differential coefficients dx" _df _dx W'~dd~dd' we may therefore write (1) in the form ^ / „dx" „dy" „dz"\ dU If then we introduce a new function the equation (2) becomes dS__dU dS _dU dd"~ dd' dip"~dct>' ^"^ ^^^' since d stands for any one of the coordinates. Here the differential coefficients with regard to 6", ", &c. on the left-hand side and d, (p, &c. on the right-hand side are partial. The function S has been called the energy of the accelerations. When we have constructed a method of expressing the function S in terms of the coordinates d, 0, &c. including their first and second differential coefficients with regard to t, the equations (3) give the differential equations of motion of the system. The right-hand sides are deduced from the force-function U exactly as in Lagrange's equations. 430 a. In calculating the function S we may obviously omit all terms which do not contain the second differential coefficients 6", 0", &c. for all such terms disappear in the partial differential coefficients which occur in equations (3). We also notice that the function S in general contains quadratic and first powers of 6", cji", &c. 430 h. Let us noio apply similar arguments to systems ivhich are not holonomous. Let us suppose that the displacements have been made to depend on K-\-p coordi- nates having p relations between them, so that the variations of k of these are arbitrary. Let these be q-^, q.-^, ... q^. Let x, y, z be the coordinates of any point of the system referred to axes fixed in space, then every possible motion of that point, consistent with the geometrical conditions, are given by dx = a-^dq^-\-a<^dq^-\- ... •\-a^dq^-radt\ dy = bj^dq^ + b^dq2+ ... +b^dq^ + bdt\ W' dz = Cidq.j^ + c^dq2+ ... +c^dq^+cdt) Let q^, . . . q^,p be the p other variables which have been introduced into our 'equations to assist in expressing the geometrical conditions. Let these be related to the former variables by equations of the form ^?K+i-"i^3'i + a2^9'2+ ••• +oi^dq^ + adt\ \ (2). 350 LAGRANGE'S EQUATIONS. [CHAP. VIII. Here the coefficients «! ... c^ and aj ... \^ as well as a, b, c, a .., \ may be functions of all the variables q^ ... q^^^ and t. The right-hand sides of these equations are not necessarily exact differentials, but may express geometrical conditions in the same way that the equations (1) of Art. 429 expressed the conditions that the body there mentioned was rolling on a surface. 430 c. To form the equations of motion we use the principle of virtual work as in Arts. 398, 399. This equation is 'Zm{x"dx + y"dy + z"8z) = 'E{Xdx+Y8y + Z8z) (3). Since the virtual displacements 5a;, 5y, 5z are to be consistent with the geometrical equations which hold at the time t, we use the equations (1) and (2) without their last terms, hence dx = a^dq^+ ... +a^8q^ (4), with similar expressions for dy, 5z. Since dq^... dq^ are arbitrary the equation (3) decomposes into the k following equations 2??i {x"a-^ + y"bj^ + z"Ci) = dUldqi j = } (5). Xm{x"a^ + y"b^ + z"c^) = dUldq^^ These correspond to equations (1) of Art. 430. By dividing the equations (1) of Art. 430 b by dt and differentiating the quotient with regard to t, we obtain ^" = %(?l" + «2Q'2"+ ••• +^k?k" + *° (^)' with similar expressions for y" and z" obtained by writing 6 and c for a. The terms which do not contain q^" ... q^" are included in the &c. It is evident that a^ = dx"ldqi", b-^ = dy"ldqf^' and so on. Hence the equations (5) become ^'»(^"^^^"l7^^"l?')=''^"'«' <"• with similar equations for the other coordinates. If we now construct the function S=^^m{:x"^^ + y"-^ + z"^) (8), the equations of motion are dS^ _dU _dS__dU dS _dU dqi'~dqj^' dq^'~dq^^ '' dq^" ~ dq^ ^ '' To form the equations of motion of a system whether holonomous or not it is sufficient to express the function *S' so that it contains no other second differential coefficients than those of the coordinates q^.-.q whose variations are regarded as arbitrary. If in constructing the function S any second differential coefficients of the remaining coordinates made their appearance they should be eliminated by using the conditions (2). After division by dt, these conditions take the linear form ?',+! = cti2i'+ ...+%q,'-^a, i with similar expressions for q'^_^_^ , &c. By differentiation we obtain q"^^-^ . • ■ Q^^+p in terms of q^" . . . q^", and these should be substituted in the function S. 430 d. If the forces do not admit of a force-function U, we proceed as explained in Art. 426. Let P^dq^ be the virtual work of the forces produced by varying q^ only, P2dq<2. that obtained by varying g-g only and so on. We then replace dUjdq^, dUldq^, &c. by P^, P^, &c. 430 e. Another proof. We may also deduce Appell's equations (as he has also done) from Gauss' principle of least constraint by translating the formula of Art. 394 a into generalized coordinates. This principle applies to systems not holonomous because it has not been assumed in the proof that x, y, z are integral functions of the coordinates q-^^ 32 ••• S'* • ART. 430^.] appell's equations. 351 By Gauss' principle the accelerations assumed by the system are such as to make 2jRj3 = Sm {{x" - X)'^ + {y" - Y)^+{z" - Z')^} =minimum subject to the geometri- cal conditions of the problem. By differentiating (1) of Art. 430 b we have x" = a^q^' + a^qc^" + ... +%q^", where all terms are omitted which do not contain the second differential coefficients ^i"> (I2" ••• Q " (A-rt. 430a). There are similar expressions for y" and z" . Again if P^, Pg, &c. are the generalized equivalents of Z, F, Z we have •Lm {Xbx + Yhy + Zhz) = P^ dq^ + P^5q^+... as in Arts. 426 and 430 d. Hence by (4) of Art. 430 c, Xa^+Yb^ + Zc^ = P^, where n has any value from n=l to n = K. It follows that Sm {Xx" + Yy" + Zz") = P^ q{' + P^ q^' + cfec. Putting 2/S'=Sw (a;"-+7/"2 + 2"2) we have to make E,3 = S'-(Pig'/' + P2g2" + &C.)+lSm(Z2+ 72 + ^2)^ a minimum with regard to q-[' , q^' ... q^'. Since these second differential coefficients do not appear in the last term of Pjg, we find by differentiation dSldq" = P^, dSjdq^ = P^, &c., and these are the equations to be proved. 430/. To find the function S for any given system of bodies we follow the analogy of Lagrange's function T. Since S* is a quadratic function of x'\ y", z'\ we first deduce from the general theorem of parallel axes (Art. 14) that the value of 8 for a system of Cartesian axes is equal to that for a parallel system of axes with the centre of gravity for origin plus the value of 8 for the whole mass collected at the centre of gravity with reference to the first system. We also notice that since x"'^ + y"''^^-z"'^ is the resultant acceleration of the particle m the value of »S' must be the same for the same bodies, however the coordinates may be transformed. 430 g. To investigate the form of the function S for a body free to turn about a fixed point lohen referred to the principal axes at 0. These axes are either fixed in the body or (if two or more of the principal moments of inertia at are equal) may move in an arbitrary manner, yet so that they remain principal axes. The space- velocities of any point [x, y, z) are by Art. 238 u^w.^z-w^y, v = o}.^x- (jo-^z, w = u^y - ca^x. The x component of acceleration is X=duldt-vd.^ + wd2 = - a;(a>2^ + Wg^) +?/ {wg (wj - d^) + Wj^g " ^3'} +2 {^3 (wj - 6^) + (^id-^ + u^'}, by using the formulae of Art. 251. ••• Z2=-.^2[a,3'2-2a;3'{a,2(a,i-^i) + a;i^2}] + ^2|-^^/2 + 2a;2'{w3(wi-^i)+a;i^3}], where only terms which contain Wj', Wg', co-/ have been retained (Art. 430 a). Terms depending on the products xy, yz, zx have also been rejected as they will presently disappear when the summation S is effected. The expressions for Y^ and Z^ can be written down by symmetry. We now form 2,Sf=Sm(Z2+ 72 + ^2) and substitute 2Xmx'^ = B+C-A, 2Sm?/2=C + 4-P, 2:Emz^ = A+B- C, .-. 2S = ^wi'2 + Bu}^'^ + Ca;3'2 -2iOj'{{B-C) «2W3 + 4 ('^2^3-<*'3^2)} - 2W2' {{C -1) W3W1+ P (Wg^i - Wj^g) } - 2W3' {(^ - P) Wj W2+ C (Wi^2 - ««'2^l)}- 352 Lagrange's equations. [chap. viii. If the axes are not principal axes, we must add to these three sets of terms con- taining D, E, F respectively as factors. The first set is 2D [ - W2' W3' - W/ (W2^ - W32) + Wg' { W2 (Wi - ^1) + Wi 62} - W3' {W3 (wi - d^) + Wi ^3}] , those with E and F follow by symmetry. 430 h. To deduce Euler^s equations. Let the moving axes be fixed in the body, then ^1 = wi , d^ = W2, ^3 = ci'3 . We then have 2S=Ao}^^-+ Boj^'^+Cw^"^ -2{B-C) wawgoj/ -2{C-A) u^^w^w^ -2{A-B) u^u^u)^ ; since the body has three degrees of freedom, we have to choose three variables Q'lj 9'2» Ss which are to be arbitrary and whose variations should express every possible small displacement of the body. These conditions are satisfied if we put dq-^ = (a^dt, dq^ = w^dt, dq^=u}^dt. We then have 2S = Aq^"^ + Bq2"''+Cq,"'^-2{B-C)q2'qs'qi"-2{G-A)qs'q^'q^"-2{A-B)q{q,'q,". The equation dSjdq-^' = dUldq^ then gives Aq-^' -{B-G)q^q^ = dTJldq^. This becomes Euler's first equation (Art. 252) when we write qi=w^, &g. and dUldq^ = L (Art. 340). 430 i. An elliptic disc rolls in a vertical plane on a rough ground. To form the equation of motion. Let ^, 7) be the coordinates of the centre C, ^ being measured along the ground and 77 vertically upwards. Let 6 be the angle the major axis makes with the vertical. Let P be the point of contact and let CN=7] be the perpendicular from G, let also PN=u. Since the point P of the body is at rest we have ^ If the mass be unity, we have 2^ = r'2 + V" + Ce"^ U= - gtt. Hence by eliminating ^", tj" 2S={G + 'n^ + u^)d"'' + 2L^^ + u^\e'^d'\ where only terms which contain d" are retained. The equation dSldd" = dUldd gives Since the boundary is an ellipse, both 77 and u are known functions of 6. * We may easily verify this result by using the ordinary equations obtained by resolving and taking moments. 430^'. To determine the motion of a circular disc or hoop rolling on a rough ground but not necessarily in a vertical plane. In the figure the disc GP is drawn with its plane perpendicular to the paper, GM is a perpendicular from the centre G on the . ground and P is the point of contact. Let GP, GB, \ 'i GC be the moving axes of reference; since GG is \ [ ^ y^C fixed in the body its motion whether deduced from \r>^ the angular velocities (w^, W2, Wg) of the body or the ^'"Q^. angular velocities (^j, Q^, 6^) of the axes must be the \^ n same, hence ^i = a>i, d^^w^. \ Let u, V, w be the components of the velocity of G J- ' N ^ — along the axes of reference, then since P is instanta- ^ neously at rest, M=0, v + aw3 = 0, w-a(>}2 = 0. ART. 431.] CHANGE OF THE INDEPENDENT VARIABLE. 353 The component accelerations of G are by Art. 251 X = dujdt -vd^ + wd2 = awg ^3 + acog^, Y= dvjdt -w6i + ud^ = - awg' - ao)^ w^ , Z = dw/dt - ud^ + vd-^ = awg' - aoj^ Wj . If Sq be the part of S which depends on the motion of G we have 2^0 = 2m {Z2 + 72 + Z2) = a^ {u,^'^ + u,.p + 2wi ( Wg Wg' - Wg Wg') } , where all terms which do not contain w^', w./ or w^ are omitted. Let S^ be that part of S which depends on the motion relative to G. By writing A — B, ^i = a>j, ^2 = '^2 i'^ -^^^- 430 p we deduce 2;Sfi = ^ (Wi'2 + Wg'^) + CW3'2+ 2 (^6'3 - CW3) (WiW/ - W2W1'). The complete value of S is found by adding together S(, and S^ (Art. 430/). We now introduce the variables qi, q^, q^ where as before dq^ = u}^dt, dq2 = o}2dt, dq^=oi).^dt (Art. 430 h) and deduce 2S = Aq,"-' +{A + a2) q^'"' + (C + rt^) q^"2 + 2 {^^3 - Cq,') {q^q^' - q^q^') + 2a2^i' {q^q^' - q^'q^'). We notice that since 6^ is an angular velocity (not an acceleration) we are not obliged to eliminate it before differentiating the function S (Art. 480 a). We have yet to consider the differential coefficients dUjdq^, &c. When the body receives the angular displacements dq^^, dq^, dq^ the centre G moves and the body turns round P as an instantaneous centre. Hence dUjdq^, &c. are the moments of the forces about axes parallel to GA, GB, GC but having their origin at P (Art. 340). These moments are dUldq-^ = 0, dU I dq2= - ga COB 0, dUldq^ = Q, where 6 is the angle the plane of the disc makes with the horizontal ground. The equations of motion of the disc are therefore Aq^' - {Ad^- Gq^) q^ =0, {A + a?) q^' + {Ad^- Cq^) q^ - o?q{q^= - ga cos 6, {G + a^)qs" +a^^'q^' = 0. The problem of the motion of a disc or hoop rolling on a rough ground is also discussed by another method in Art. 244 of Vol. 11. of this treatise. The variables there used are the two angles 6, xp of Euler and Wg. If we write in the equations (5) of this article g-/ = Wi =-;/'' sin ^, q2=u}o=6', qs=<^3, d^ = \f/'coB6 we arrive at equations equivalent to those in Vol. 11, The interpretation of these equations will be found in that volume. 431. Change of the independent variable. A system of n degrees of freedom IS defined by the Lagrangian function L = T+U+G, T = lA^^d'- + A^2^'' + &c (1). Let us now change the independent variable i to r and put P=dTldt. For the ake of distinctness let suffixes applied to the coordinates 6, 0, &c. mean differentia- ions with regard to r just as accents denote differentiations with regard to t ; then = Pdi, 0' = P0i, &c. We now write T,=iA,,e,'+A,2e,,+ (2), that Tj differs from T only in having r written for t. The equation of vis viva therefore T=P^T^=U+G (3). e shall now prove that we may take as a Lagrangian function either of the forms L, = PT,+ ^-±^=2{iU+C)T,}i (4), le second being derived from the first by using (3). Here P is an arbitrary function )f the coordinates 6, ^, &c. R. D. 23 354 LAGRANGE'S EQUATIONS. [CHAP. VIII. The new Lagrangian equations will then be d dL^ dL^ d dL^ dLi dT~ddl~~dd' dTd^^~~d^ &c (5). TJ+G\dP P2 ) dd To prove this we form the partial differential coefficients dLJddi, dLj^jdd, since d' = Pd^, j^, &c., ■ 4{§^'^h{-^-'-P)§ <')• Substituting in the Lagrangian equation ±dT_dT dU dt dd' ~ de'^ dd ' and using the equation (3) because d/dt is a total differential coefficient we arrive at the new typical equation -j- —■ = -~ , dT du^ do where dr has been written for Pdt. The first of the two forms for Lj given by the equation (4) should be used when we desire to simplify the original form of the Lagrangian function L by a proper choice of the arbitrary factor P. Thus in the example solved in Art. 431a, we transfer a factor 31 from the expression for T to that for U. The second form may be used when we wish that the new independent variable r should have any special vahie, while the form of P is a matter of indifference. For example, in Art. 431 b we replace t by one of the coordinates 6 and thus eliminate the time from the Lagrangian equations. ! 431 a. As an example consider Liouville's integral, Art. 407, Ex. 4. We have T=^M{A^d'^ + A^((>'^ + &c.}, U+C={F^{d) + F^{<(>) + &c.}IM, where A^ is a function of 6 only, A^oi (p only, &g., while M may be a function of all the coordinates. Taking P=l/ilf we form the Lagrangian function L, = h{A,ei^ + A2cf>^^ + &G.} + {F^{e)+F,{) + &G.}. The Lagrangian 6 equation then becomes d^:^'^'^ 2~dd^'--d~d ' ■■ 2d^^^^^^'-^dd~^'' Hence by an easy integration we have ^A^d^^ = F,{e) + a, .-. iA^M^d'^ = F^{e) + a. This is the integral already arrived at in Art. 407. 431 6. When the paths of the particles are alone required, we may eliminate the time from the Lagrangian equations by using a new function instead of the Lagrangian function. In this method we choose some one coordinate 6 to be the independent variable and regard the others '^ + A^' d0i ' d~ d(t> ^^' where the differential coefficients of T and T' are partial. The equation of energy gives T'd'^=U+C; :. d'={^^\ (4). ^, T . ^. d dT dT dU ^ The Lagrangian equation — :t-; - 3— = ^— becomes at aq> acp a

T' where all the differential coefficients are partial except the djdd. Since XJ is not a function of 0i , this becomes I4^'''^^*^''*==^**^+^'^'J* <'»• If then we use Q={{U+G) T']^ as if it were the Lagrangian function and regard • 6 as the independent variable, we have the equations d^dQ__dQ d^dQ__dQ . dd d^ d(t>' dd dxjy^~ dxl^ ' ^'' from which the paths viay he found. This result also follows from the theorem of Art. 431 by putting dr=dd, and we have here reproduced in another form so much of that article as is required for our present purpose. Since dT = Pdt we have P = ddldt, and 6-^ = 1, ldd and so on. It immediately follows from (2) and (3) of Art. 431 that T^=T', P=fE±^\ . The Lagrangian function given by (4) of that article becomes L^ = 2{{U+G)T'}^ = 2Q. 431 c. We notice that however the expressions for the vis viva and the work function maybe altered, yet so long as the product {U+G)T' remains unchanged the general equations of the paths are determined by the same relations between the coordinates 6, (f>, &c. The times of describing the paths may however be altered. 431 d. Since in the Lagrangian equations, the letters 6, <{>, &c. represent prbitrary functions of the quantities or coordinates which determine the position of the system, it is evident that we have here taken as the independent variable any arbitrary function of the coordinates. 431 e. If some one coordinate, say 0, is absent from the product {U+G)T' (though 2" contains the differential coefficient of 0) we have dQld

unless T' and U are separately independent of 0. But when G is given by the initial conditions this limitation is not necessary. If we substitute for dT'ldtp^ and 6' the values given by (3) and (4) of Art. 431 &, Pthis integral becomes dTjd^' = 2a which is the same as that obtained in Art. 407, Ex. 5» 23—2 356 LAGKANGE's equations. [chap. VIII. 431/. To make a comparison of methods, let us use the function Q to investi- gate the paths when T and U have the forms given in Art. 431 a. We have T'=\M{A^ + A^^^ + &Q.}, U+C={F^{d)+F^{,^ + &G.)G, G = Fi{d) + F^{4>)+&G. r^^ ^- d dQ dQ . The equation -jt: tt = ^rr becomes do d-^A^ _ Gj^ dA^ Aj^ + A2(f>i^ + &Q. dF^ dd Q ~ 2Q d0" "^ 2Q d^" ' . Gi d (G^ \_1 /G,ydA.dF^ ■ Q dd\ Q '^y-2\ Q J W^ d^^^' .1 d fG,y lfG<(>,ydA,_dF. •'2'^^dd\ Q J '^2\~Q~) d^-d0^i' Substitute for G and Q and we find A, + A^ 4>i'+:. ^ ^201^ ^ ^sj^i" ^Slc the third and other fractions follow by symmetry. Since ^j^ = d' + &c.) + A,+ U+C, where ^n, &c., A^, &c., and Aq are functions of the coordinates but not of t. The equation of vis viva is then (by Art. 407, Ex. 2), ^A^^d'^ + &c.-Af^=U+C. Proceeding exactly as before we change dt into dr by taking as a new Lagrangian A A-TJ A-G function L^ = P {\A^.^ 6^^ + &c. ) + {^i ^i + Ac.) + "-"p^ , where as before d^ = ddjdr and P = drfdl. The equation of vis viva gives 431 h. The elimination of the time from the Lagrangian equations is given by Painleve in his Lemons sur V integration des equations differ entielles de la MecaniquSy 1895, page 237. By an application of the principle of least action he obtains the function here called Q and writes the equations in the typical form -v— -^ = — . dq^dq,: dq^ From these he deduces (page 239) that the Lagrangian equations may be written in the two forms ddT_dT_dU d dT^ ^^' -o dt dq' dq dq ' dr dq' dq ~ ' where T' = T{U+C) and dT = {U+C)dt. This special result follows from that given here by putting P=U+C. Its importance lies in the fact that by this change the motion is made to depend on that of a system moving under no forces. The elimination of the time from Lagrange's equations is also given by Darboux in his Legons sur la theorie generals des surfaces, Art. 571, 1889. He expresses the result in the same form as Painlev6. EXAMPLES. 357 EXAMPLES 1. Two weights of masses m and 2m respectively are connected by a string which passes over a smooth pulley of mass m. This pulley is suspended by a string passing over a smooth fixed pulley, and carrying a mass 4m at the other end. Prove that the mass Am moves with an acceleration which is one twenty-third part of gravity. 2. A uniform rod of mass 3m and length 2Z has its middle point fixed, and a mass m attached at one extremity. The rod when in a horizontal position is set rotating about a vertical axis through its centre, with an angular velocity equal to tj{2ngll). Show that the heavy end of the rod will fall till the inclination of the rod to the vertical is cos"^ {isJn'^+1 - n), and will then rise again. 3. A rod of length 2Z is constrained to move on the surface of a hyperboloid of revolution of one sheet with its axis of symmetry vertical, so that the rod always lies along a generator. If the rod start from rest, show that r'2 - 2ar'd' sin a + a^ d'^ + sin^ (i{r"-^\ P) d"^ + 1g cos a{r- r^) = 0, {a2 + sin2 ^ (^2+ 1 p)| 0' _ ar' sin a = 0, where r is the distance measured along a generator from the centre of gravity to tlie principal circular section, 6 is the excentric angle of the point in which the generator meets this circular section, a is the radius of the circular section, and a is the inclination of the rod to the vertical. 4. A ring of mass m and radius b rolls inside a perfectly rough ring of mass M and radius a, which is moveable about its centre in a vertical plane. If 6, (p be the angles turned through by the rings from their position of equilibrium, prove that ad + h(t) = {a-h)\p, Mad" = mh) sin d respectively. Let 6 be the angle turned round by the body in moving from i the position of equilibrium into the position B'A'P. Then, since before disturb- jance A'C and AO were in the same straight line, we have e = l.GDE = (f> + (j>', where GA' meets OAE in D. Also, since one body rolls on the other, the arc AP = arc A'P, .-. p0 = pep', .-. = -,e. 7 cos a e. p + p Again, in order to take moments about P, we re- quire the horizontal dis- tance of G from P ; this may be found by projecting the broken line PA' + A'G on the horizontal. The projection of PA' = PA' cos (a + 0) = p(f) cos a when we neglect the squares of small quantities. The projection of A'G is rd. Thus the hori- zontal distance required is (— ^ If h be the radius of gyration about the centre of gravity, the ecjuation of motion is If L be the length of the simple equivalent pendulum, we have k'^ + r^ pp — r — = , cos a — r. L p + p 442. Circle of Stability. Along the common normal at the point of contact A of the two cylindrical surfaces measure a length AS = s, where - = - + — , and de- . ^ P P scribe a circle on AS as diameter. Let AG, produced if necessary, cut this circle in JSf. Then GN= s cos a — r, the positive direction being from N towards A. The length L of the simple equivalent pendulum is given 366 SMALL OSCILLATIONS. [CHAP. IX. by the formula L . GN = sq. of rad. of gyration about A. It is clear from this formula, that if (r* lie without the circle and above the tangent at ^, L is negative and the equilibrium is unstable, if within, L is positive and the equilibrium is stable. This circle is called the circle of stability. This rule will be found very convenient to determine not only the condition of stability of a heavy cylinder resting in equilibrium on one side of a rough fixed cylinder, but also to determine the time of oscillation when the equilibrium is disturbed. An ex- tension of the rule to cases of rough cones and other surfaces will be given further on. 44.3. It may be noticed that the preceding result is per- fectly general and may be used in all cases in which the locus of the instantaneous axis is known. Thus p is the radius of curva- ture of the locus in the body, p that of the locus in space, and a the inclination of its tangent to the horizon. If dx be the horizontal displacement of the instantaneous centre produced by a rotation dO of the body, the equation to find the length of the simple equivalent pendulum of a body oscillating under gravity may be written ^_+r^_dx__ ^^ dd ^' This follows at once from the reasoning in Art. 441. It may also be easily seen that the diameter of the circle of stability is equal to the ratio of the velocity in space of the instantaneous axis to the angular velocity of the body. Ex. 1. A homogeneous sphere makes small oscillations inside a fixed sphere so that its centre moves in a vertical plane. If the roughness be sufficient to prevent all sliding, prove that the length of the equivalent pendulum is seven-fifths of the difference of the radii. If the spheres were smooth the length of the equivalent pendulum would be equal to the difference of the radii. Ex. 2. A homogeneous hemisphere being placed on a rough fixed plane, which is inclined to the horizon at an angle sin-^|^2, makes small oscillations in a vertical plane. Show that, if a is the radius of the hemisphere, the length of the equivalent pendulum is ^^ (92 - 5 x/14) a. * Let R be the radius of curvature of the path traced out by G as the one cylinder rolls on the other, then we know that B = -tt-^ , and that all points with- out the circle described on AS as diameter are describing curves whose concavity is turned towards A, while those within the circle are describing curves whose convexity is turned towards A. It is then clear that the equilibrium is stable, unstable, or neutral, according as the centre of gravity lies within, without, or on the circumference of the circle. ART. 445.] A BODY GUIDED BY TWO CURVES. 367 444. If the body be acted on by any force which passes through the centre of gravity, the results must be slightly modified. Just as before, the force in equi- librium must act along the straight line joining the centre of gravity G to the instantaneous centre A. When the body is displaced, the force cuts its former line of action in some point F, which we shall assume to be known. Let AF=f, taking / positive when G and F are on opposite sides of the locus of the instan- taneous centre. Then it may be shown by similar reasoning, that the length L of the simple equivalent pendulum under this force, supposed constant and 'f+r direction of the force makes with the normal to the path of the instantaneous centre. If we measure along the line AG a. length AG' so that -— -, = --^ + -Te, . tlien the equal to gravity, is given by — , - = , cos a L. p + p where a is the angle the k^ + r^ expression for L takes the form — = — = G'N. AG' AG AF' The equilibrium is therefore stable or unstable according as G' lies within or without the circle of stability. 445. Oscillations of a body resting on two curves. Two points A, B of a body are constrained to describe given curves, and the body is in equilibrium under the action of gravity. A small disturbance being given, find the time of an oscillation. Let C, D be the centres of curvature of the given curves at the two points A, B. Let AG, BD meet in 0. Let G be the centre of gravity of the body, GE a perpen- dicular on AB. Then in the position of equilibrium OG is vertical. Let i, j be the angles which CA, BD make with the vertical, and let a be the angle AOB. T>(-t A', B', G', E' denote the positions into which A, B, G, E are moved when the body is turned through an angle 6, and let 0' be the point of intersection of the normals at A', B'. Let ACA' = (P, BDB' = (f>'. Since the body may be brought from the position AB into the position A'B' by turn- ing it about through an angle 6, we have ■0. Also GG' is CA.(f> _ BD.' OA ~ OB ultimately perpendicular to OG, and we have GG'=OG .6. Also let x, y be the projections of 00' on the horizontal and vertical through O. Then by projections a: cosJ + t/8inj = distance of 0' from OD = OD . sin3i) Ex. 2. The extremities of a uniform heavy rod of length 2c slide on a smooth wire in the form of a parabola, whose axis is vertical, and whose latus rectum is equal to 4a. If the rod be slightly displaced from its position of stable equilibrium, 2ac 2(il2a^+c^ prove that the length of the equivalent pendulum is 7-^ —. , or — 2_ . 2" > according as the length of the rod is greater or less than the latus rectum of the parabola. In the first case the rod in its stable position of equilibrium passes through the focus and is inclined to the horizon. In the second case the rod is horizontal. When the length of the rod is equal to the latus rectum the oscillation is not tauto- chronous, see Art. 450. If the rod start from rest at a small inclination a to the horizon, it will become horizontal after a time - ( ^5- ) / {1 - (f)^)'^ d'^=C - gyo"^"^, where Xq is the value of dxld0 vfhen = a; dif- ferentiating we get {xQ^ + k^)(p= -gyo"- E. D. 24 370 SMALL OSCILLATIONS. [CHAP. IX. If L be the length of the simple equivalent pendulum, we have where for 6 we are to write its value a after the differentiations have been eifected. It is not difficult to see that the geometrical meaning of this result is the same as that given in the last article. This analytical result was given by Mr Holditch, in the eighth volume of the Cambridge Transactions. It is a convenient formula when the motion of the oscillating body is known with reference to its centre of gravity. Ex. 1. The lower extremity of a heavy uniform beam of length a slides on a weightless inextensible string of length 2a, whose extremities are attached to two fixed points in the same horizontal line, and the upper extremity slides on a vertical rod which bisects the line joining the two fixed points. Prove that the only position of equilibrium is vertical, and that the time of a small oscillation about this position is »<» ,^, ) where 2J(a^ - b^) is the distance between the two fixed points. ^{Sg{2b-a)} [Math. Tripos. The lower extremity of the rod may be regarded as moving in a circle of radius a^jb. Express the coordinates (x, y) of the middle point in terms of the angle 6 which the rod makes with the vertical. The result follows by the principle of vis viva. Ex. 2. The extremities of a rod slide on the circumference of a three-cusped hypocycloid whose plane is vertical. The radius of the circumscribing circle is 3a, and one of the cusps is at the highest point of the circle. Prove that the length of the equivalent pendulum is ^a. [Math. Tripos, 1872. First prove that in this hypocycloid the rod as it slides with its two ends on the side branches BE, BE always touches the lowest branch BD. Its middle point R describes a circle with centre 0, and radius a where O is the centre of the circum- scribing circle. If BOR = (p, the angle which the rod makes with the tangent at the cusp B is 10. The result then follows by using the principle of vis viva. 448. Moments about the Instantaneous Axis. When a body moves in space with one independent motion there is not in general an instantaneous axis. It has, however, been proved in Art. 225 that the motion may always be reduced to a rotation about some central axis and a translation along that axis. Let / be the moment of inertia of the body about the instan- taneous central axis, O the angular velocity about it, Fthe velocity of translation along it, M the mass of the body, then by the prin- ciple of vis viva ^Ifl^ + ^MV^= U + G, where U is the force- function, and G some constant. Differentiating we get da 1 di j^ydv_du dt^2 dt^ a dt ~ndt' In the time dt the body turns round the instantaneous axis through an angle Hdi, and advances along that axis a space Vdt; we therefore have dll = Ladt + ZVdt where L is the moment of the impressed forces about the central axis and Z the component ART. 450.] MOMENTS ABOUT THE INSTANTANEOUS AXIS. 371 along it (Art. 340). Let p be the pitch of the screw-motion of I the body, then F = j^H. The equation of motion therefore becomes 1 If the body be performing small oscillations about a position of equilibrium, we may reject the second and third terms, and the equation becomes .^ ,. o\ ^^ r . n ^ {1 4- i/pO -T7=L'\-'pZ. \ If there be an instantaneous axis, ^ = 0, and we see that we may take moments about the instantaneous axis exactly as if it were fixed in space and in the body. Ex. A rigid body moves in any manner about a fixed point. If fi is the (angular velocity, I the moment of inertia, L the moment of the impressed forces, each about the instantaneous axis, prove that ^r- — (I122) = L. [Arts 215, 252.] 2i\i dt A uniform rough heavy circular disc of radius a has its edge touching a horizontal ■table and rests against the pointed top of a peg of vertical height h fixed in the table. »tn the position of equilibrium its plane makes an angle a with the table. Show that llihe length of the simple equivalent pendulum for a small oscillation in which there US no slipping is ah sec a tan a/4 {h - a sin a). [Math. Tripos 1904. Second Method of forming the Equations of Motion. 449. Let the general equations of motion of all the bodies be brmed. If the position about which the system oscillates be mown, some of the quantities involved will be small. The squares md higher powers of these may be neglected, and all the equations vill become linear. If the unknown reactions be then eliminated ihe resulting equations may be easily solved. If the position about which the system oscillates be unknown, t is not necessary to solve the statical problem first We may by me process determine the positions of rest, ascertain whether they ire stable or not, and find the time of oscillation. The method of )roceeding will be best explained by an example. 450. Ex. The ends of a uniform heavy rod AB of length 21 we constrained to move, the one along a horizontal line Ox, and the )ther along a vertical line Oy. If the whole system, turn round Oy vith a uniform angular velocity co, it is required to find the posi- ions of equilibrium and the time of a small oscillation. Let X, y be the coordinates of G the middle point of the od, 6 the angle OAB which the rod nakes with Ox. Let R, R' be the re- ictions at A and B resolved in the )lane xOy. Let the mass of a unit »f length be taken as the unit of mass. The accelerations of any element ^; ■ Ir of the rod whose coordinates are 24—2 372 ^ SMALL OSCILLATIONS. [CHAP. IX. (f, 7)) are ^— «^f parallel to Ox, t'jA¥^) perpendicular to the plane xOy, and -j- parallel to Oy. As it will not be necessary to take moments about Ox, Oy, or to resolve perpendicular to the plane xOy, the second acceleration will not be required. The resultants of the effective forces l^dr and r}dr, taken throughout the body, are ^Ix and 2ly acting at G, and a couple 2lk^Q tending to turn the body round G. The resultants of the effective forces ay^^dr taken throughout the body are a single force acting at G = ay^{x + r cos 6) dr = co'^x . 21, and a r+l p couple * round G= \ co^ (^ + r cos 6) rsin6dr = (o^ .21 .- sin 6 cos J —I o ' the distance r being measured from G towards A. Then we have, by resolving along Ox, Oy, and by taking moments about G, the dynamical equations 2lx = -R+ay'x.2l 2ly = -R-\-g.2l 72 2lk^e = Rx- Ry-(oK2l.- sin ^cos 6 .(1). We have also the geoinetrical equations x=l cos 6, y=l sin 6 (2). Eliminating R, R', from the equations (1), we get xy—yx-v k^d = gx — ay^xy — J coH^ sin ^ cos ^ (3). To find the position of rest. We observe that if the rod were placed at rest in that position it would always remain there, and that therefore x = 0, y = 0, 6 = 0. These give f(x, y, 6) = gx— co^xy — ^coH'^ sin 6cos 6 = (4). Joining this to equations (2), we get 6 = ^y , c>r sin 6 = ~r~ and thus the positions of equilibrium are found. Let any one oi these positions be represented by ^ = a, x — a, y = h. To find the motion of oscillation. Let x = a + x', y = h -\- y 6 = a + 0', where x, y' , 6' are all small quantities, then we must substitute these values in equation (3). On the left-hand side since x, y, 6, are all small, we have simply to write a, h, a, foi * If a body in one plane be turning about an axis in its own plane with ar angular velocity w, a general expression can be found for the resultants of th< centrifugal forces on all the elements of the body. Take the centre of gravity G as origin and the axis of y parallel to the fixed axis. Let c be the distance of G from the axis of rotation. Then all the centrifugal forces are equivalent to a single resultant force at (r — Jw^ (c + x)dm = ox^ . Mc, since x = 0, and a single resultant couple =jb}^{c + x)ydm = o}^jxydm, since ^ = 0. ART. 450.] SECOND METHOD OF FORMING THE EQUATIONS, ETC. 373 X, y, 6. On the right-hand side the substitution should be made by Taylor's Theorem, thus We know that the first term f(a, 6, a) = 0, because this is the very equation (4) from which a, h, a were found. .-. ay - bx + k^B' =(g- co%) x - w^ay - ^coH^ cos 2a . 6'. But, by putting ^ = a + ^' in equations (2), we get by Taylor's Theorem w' = — lsmoL. 6', y' — l cos a . 6', also a = l cos a,b = l sin a. Hence the equation to determine the motion is (^2 4. ^2) ^ + Li sin a + I coH' cos 2a] 6' = 0. Now, if gl sin a + ^(o^P cos 2a = 71 be positive when either of the two values of a is substituted, the corresponding position of equi- librium is stable, and the time of a small oscillation is 27r a / • If n be negative the equilibrium is unstable, and there can be no oscillation. If o)^ > Sg/4fl, there are two positions of equilibrium of the rod. It will be found by substitution that the position in which the rod is inclined to the vertical is stable, and the other position unstable. If o)^ < 3^/4^ the only position in which the rod can rest is vertical, and this position is stable. If n = 0, the body is in a position of neutral equilibrium. To determine the small oscillations we must retain terms of an order higher than the first. By a known transformation we have xy-yx = j^ (I'd). Hence the left-hand side of equation (3) becomes (P + k'^)d. The right-hand side becomes by Taylor's Theorem ^^ {^gl cos a - g coH' sin 2a j j-^ -f &c. When 71 = 0, we have a = j7r and co^='SglU. Making the necessary substitutions, the terms of the second order vanish, and the equation of motion becomes ,j,^ i.2\^^^— 9^ a'z Since the lowest power of 9' on the right-hand side is odd, and its coefficient negative, the equilibrium is stable for a displace- ment on either side of the position of equilibrium. Let a be the initial value of 6\ then the time T of reaching the position of equilibrium is / 4 (Z^-f- A;^) T'* dd' put ^=a^, then T = ^ ^^^ . J^ ^^j=^^ . -. 374 SMALL OSCILLATIONS. [CHAP. IX. Hence the time of reaching the position of equilibrium varies inversely as the arc. When the initial displacement is indefi- nitely small, the time becomes infinite. This definite integral may be otherwise expressed in terms of the Gamma function. It may be easily shown that I - — -^ — =^ — -^JX- . Vut d)^ = x. 451. This problem might have been easily solved by the first method. For, if the two perpendiculars to Ox, Oy at A and B meet in iV, N is the instantaneous axis. Taking moments about N, we have the equation r+^ dr {P -{-t)e = gl cos ^ - I a>2 il + ry sin cos d ^ = gl cos 6 - ^P(o^ sin 6 cos 6. If we represent the right-hand side of this equation by f{0), the position of equilibrium can be found from the equation /(a) = and the time of oscillation from the equation doL 452. Ex. 1. If the mass of the rod AB is M, show that the magnitude of the couple which constrains the system to turn round Oy with uniform angular velocity is ikf -^ w — sin 26. Would the magnitude of this couple be altered if Ox or Oy had any mass ? Ex. 2. The upper extremity of a uniform beam of length 11 is constrained to slide on a smooth horizontal rod without inertia, and the lower along a smooth vertical rod, through the upper extremity of which the horizontal rod passes ; the system rotates freely about the vertical rod, prove that if a be the inclination of the beam to the vertical when in a position of relative equilibrium, the angular velocity of the system will be (3^ sec ajU)^, and, if the beam be slightly displaced from this position, show that it will make a small oscillation in the time T where m -J (sec a + 3 cos a). [Coll. Exam. In the example in the text the system is constrained to turn round the vertical with uniform angular velocity, but in this example the system rotates freely. The angular velocity about the vertical is therefore not constant, and its small variations must be found by the principle of angular momentum. Lagrange s Method of forming the Equations of Motion. 453. Advantages of the Method. We now propose to state Lagrange's method of forming the equations of motion. This method has several advantages. It gives us the equations of motion free from all reactions, and is therefore specially useful when we have to consider the motions of several bodies connected together. It also gives us a larger choice of quantities which we may ART. 454.] LAGRANGE'S METHOD. 375 take as coordinates. Again, as soon as we have written down the Lagrangian function we may deduce from this one function all the equations of motion, instead of deriving each from a separate principle. On the other hand, this function must be calculated so as to include the squares of the small quantities. Now in small oscillations we retain only the first powers of the small quantities, so that, when only a few equations are wanted, it is often more convenient to obtain these by resolving and taking moments. It will be seen, therefore, that the method is best adapted to oscillations which have more than one degree of freedom. For this reason we shall here only state the general mode of forming the equations of motion, so that we may be able to apply the method to the solution of problems. But we shall postpone the general discussion of Lagrange's determinant to the second part of this work. 454. The object of Lagrange's method is to determine the oscillations of a system about a position of equilibrium. It does not apply to oscillations about a state of steady motion. For example, if a heavy particle were suspended by a string from a fixed point, the string is vertical when the system is in equi- librium, and the oscillations about this position could be found by Lagrange's method. If however the particle were made to describe a horizontal circle, as in the conical pendulum, the oscillations about the circular steady motion could not be found by this method. In the same way when a hoop rolls on the ground in a vertical plane, it may make small oscillations from one side to the other of the plane. These oscillations cannot be found by Lagrange's method. A method of investigating the oscillations of a system about a state of steady motion will be given in the next volume. We shall assume, for the present, that the forces which act on the system have a force function. We shall also assume that the geometrical equations do not contain the time explicitly, and do not contain any differential coefficient with regard to the time. In Lagrange's method it is essential that the coordinates chosen should be such small quantities that we may reject all powers of them except the lowest which occur. They should generally be so chosen that they vanish in the position of equili- brium. But with this restriction they may be any whatever. Let us represent them by the letters 6, <^, &c. Then if the system oscillate about the position of equilibrium, these quantities will be small throughout the motion. Let n be the number of these coordinates. As before, let accents denote differential coefficients with regard to the time. 376 SMALL OSCILLATIONS. [CHAP. IX. Let 2T he the vis viva of the system when disturbed from its position of equilibrium, then as in Art. 396 we may express T as a homogeneous quadratic function of 6', cj)', &c. of the form 2T = ^n<9''+2J[i2 6''''+&c (1). Here the coefficients A^ &c. are all functions of 6, (f), &c. and we may suppose them expanded in a series of some powers of these coordinates. If the oscillations are so small that we may reject all powers of the small quantities except the lowest which occur, we may reject all except the constant terms of these series. We shall therefore regard the coefficients J.„ &c. as constants. Let U be the force-function of the system when disturbed from the position of equilibrium. Then we may also expand U in a series of powers of 6, ^, &c. Let this expansion be 2U=2Uo + 2B,d + 2B,4> + &c. + B^O' + 2B,^e(l> + &c. ...(2). Here Uo is a constant, which is evidently the value of U when 6, cf), &c. are all zero. It is necessary for the success of Lagrange's method that both these expansions should be possible. In the position of equilibrium, we must have, by the principle of virtual work, -j-r = 0, -7-7 = 0, &c. = (see also Art. 340). If the coordinates chosen are such that they vanish in the position of equilibrium, it immediately follows that B^ = 0, ^2 = 0> &c. = 0. If the coordinates have not been so chosen they must yet vanish for some position of the system close to the position of equilibrium. The differential coefficients of U, i.e. B^, B^, &c., are therefore necessarily small. The terms B^d, B^(\>, &c. are thus of the second order of small quantities and the quadratic terms of U connot be neglected in comparison with them. We may also notice that the equilibrium values of 6, (j>, &c. may be found beforehand by equating to zero the several first differential coefficients of u. But this is generally unnecessary, as these values of 6, , &c. will appear in the sequel (see also Art. 449). We have now to substitute the expanded values of T and U in the n Lagrange's equations d^dT_dT^dU I dtdO' de dO ^ ^' ■ with similar equations for (/>, a/t, &c. Since the expression for T does not contain 6, (f), &c., we have <^^ n <^^ an d0 = ^'d4 = ^'^''- ART. 455.] LAGRANGE'S METHOD. 377 The n equations (3) therefore become A,,6" + A,," + ... = B, + B,,e + B^_+ ...\ (4). &c. = &c. J [These are Lagrange's equations to determine the small oscillations of any system about a position of equilibrium. 455. Method of Solution. We have now to solve these equations. We notice that they are all linear, and that therefore 6, (j), &c. are properly represented by a series of exponentials of the form Me^*. But, as we are seeking an oscillatory motion, it is more convenient to replace these exponentials by the correspond- ing trigonometrical expressions. Since the equations do not contain any differential coefficients of the first order, it will be found possible, on making the trial, to satisfy them by means of the following assumption. ^ = a + ifi sin (p^t + €i) + M^ sin (p^t + eg) + &;c.^ ^ = /3+ iVi sin (p,t + 6i) + iVa sin (pj; + e^) + &c. I . . .(5). &c. = &c. J Taking the trigonometrical terms separately, they may be written in the typical form (9 = if sin (pt + e), (/) = iV^ sin {pt + e), &c. = &c. If we now substitute these in equations (4) we have (Aup' + ^n) M + {A,,p' + B,,) N + 8zc. = 0\ (A,,p' + B,,) M + {A,,p' + B^) i\r + &c. = 1 (6). &c. &c. = OJ Eliminating M, N, &c. we have the determinantal equation ^iijp' + ^ii, ^12P' + A2, &c. =0 (7). ^I2i)'+-5l2, ^22^' + ^22, &C. &c. &c. &c. This determinant, it will be observed, is symmetrical about the leading diagonal. If there be n coordinates, it is an equation of the n^"^ degree to find _p^ It will be shown in the second part of this work that all the values of p^ are real. Taking any root positive or negative, the equations (6) determine the ratios of iV, P, &c. to M, and we notice that these ratios also are all real. If all the roots of the determinantal equation are positive, the equations (5) give the whole motion, with 2?^ arbitrary constants, viz. M^, M^, M^... Mn and e^, eg ... e^. These have to be determined by the initial values of 0, , &c., &, ', &c. If any root of the determinantal equation is negative, the corresponding sine will resume its exponential form, the coefficient If 378 SMALL OSCILLATIONS. [CHAP. IX. being rationalized by giving the coefficient M an imaginary form. In this case there is no oscillation about the position of equili- brium. The position is then said to be unstable. It may be noticed that for every positive value of p^ given by the equation (7) there are two equal values of p with opposite signs. No attention however should be here given to the negative values of p. To prove this, we notice that the solution of the linear differential equations is properly represented by a series of exponentials. Now each sine is the sum of two ex- ponentials with indices of opposite signs. Both the values of p have therefore been included in the trigonometrical expressions assumed for 6, , &c. The constants a, /3, &c. in the trial solution (5) are evidently the coordinates of the central position about which the system oscillates. Substituting these values of 6, cj), &c. in the equations (4) we have = B, + BuOl + B,,^ + &c.^ = B,-{-B,,oL + B^l3-\-&^c\ (8). = &c. J These equations determine the values of a, /S, &c. Since the equations of motion are satisfied by these constant values of the coordinates without any terms containing the time, it follows that a, /3, &c. are the coordinates of the equilibrium position of the system. That this is so, follows also from the rules given in statics to find the position of equilibrium of a system when the function U is known. According to these rules, we find the equili- brium values of the coordinates 6, , &c. by equating to zero the first differential coefficients of U with regard to 6, (f>, &c. The equations thus obtained are evidently the same as the equations (8). When a root (say^j^^ of the determinantal equation (7) is zero, the correspond- ing terms in (5) reduce to constants. It also follows from (7) that the eliminant of the equations (8) is zero, so that either the equations (8) are not independent or the values of a, j8, Ac. are not so small that their squares can be neglected. In the former case that part of the solution (5) which depends on the root p^^ takes another form. Putting d = a + At,

' + ( A:^ + a") 0'^} , U=UQ-lmg{ld'^ + a , and negative when p^=gjl, the roots are separated by the latter value of p^; the roots, if equal, are therefore given hy p^ = gll. Since tbe determinantal equation is then not satisfied unless ap^ is also zero, the roots cannot be equal unless a = 0. If a = 0, it is easy to see that the roots are not equal. ART. 458.] laorange's method. 381 If the string is attached to the middle point of a uniform rod, we have a = and k^ finite. In this case one root of the Lagrangian determinant is zero, i.e. p.r = 0, while the other root is 2^1^ = gjl. Supposing that the position of the system is also required, we have T = ^m ( Pe'^ + A; V^), U=Uq- \mgm. The Lagrangian equations are therefore ld" + ge = 0, 0" = O, .-. ^ = Lsin(^if + e),

where MK^ and mk^ are the moments of inertia of the two bodies about O and A respectively. Also OH=h, OA = a, AG = b. What do these periods become when (1) the upper body, and (2) the lower, is reduced to a short pendulum of slight mass? The first case occurs when the attachment of a pendulum to its point of support is not quite rigid, so that the pendulum may be regarded as supported by a short string. The second case occurs when a small part of the mass of a pendulum is loose and swings to and fro at each oscillation. Ex. 10. A uniform circular disc of mass M and radius a is held in equilibrium on a smooth horizontal plane by three equal elastic strings of modulus X, natural length Iq and stretched length I. The strings are attached to the disc at the extremities of three radii equally inclined to one another and their other ends are attached to points of the plane lying on the radii produced. Show that the periods of vibration of the disc are 27r . / ^, , and 2ir x/ — . ,t— ; — r , where u = 2wZL/3X. [Math. Tripos, 1898. 459. Principal Coordinates. To explain what is meant by the principal coordinates of a dynamical system. When we have two homogeneous quadratic functions of any number of variables, one of which is essentially positive for all values of the variables, it is known that by a real linear trans- formation of the variables we may clear both expressions of the terms containing the products of the variables, and also make the coefficients of the squares in the positive function each equal to ART. 460.] LAGRANGE'S METHOD. 385 unity or some given positive constants. If the coordinates 6, <^, '&C. be changed into f, 77, &c. by the equations <^ = Atif + /^7; + &c.| (9), &c. = &c. J we observe that 6\ (f>', &c. are changed into f, rj', &c. by the same transformation. Also the vis viva is essentially positive. Hence we infer that by a proper choice of new coordinates, we may express the vis viva and the force-function in the forms I 2(?7-C^o) = 26,?+26,77 + &c. + 6nr+6^7;2+...r I These new coordinates f, 77. &c. are called principal coordinates of the dynamical system. A great variety of other names has been given to these coordinates ; such as harmonic, simple and normal coordinates. Usually J.„, ^22? ^c. are made unity. It is usually understood (when not otherwise stated) that prin- cipal coordinates are so chosen that they vanish in the position of equilibrium. We then have 61 = 0, 62 = 0, &c. = 0. 460. When a dynamical system is referred to principal co- ordinates which do not necessarily vanish in the position of equilibrium, Lagrange's equations take the form A-^i^" — 6uf = 61, A^T}" — h^T] = 62, &c. = &c. 30 that the whole motion is given by f = a + ^ sin (pi^ + 61), 7; = 6 -I- i^ sin {p^t + €2), &c., sphere E, F, &c., ei, e^, &c. are arbitrary constants to be deter- mined by the initial conditions, and A-^^p-^ = — bn, A^p^^ = — 63.2, &c. ind a, h, &c. are the values of f, t], &c. in equilibrium. If we substitute the trigonometrical values of f, 77, &c. in ihe formulae of transformation given above, we obviously reproduce ;he equations (5) of Art. 455, where the general coordinates 0, 0, &c. tre expressed as trigonometrical functions of t. We may therefore )btain one set of principal coordinates, viz. fi, 771, &c., which vanish in the position of equilibrium, by writing e= a + ifif, + i/2^i+...) = ^ + A^,f, + i\r^77, + ... (10), &c. = &c. J wrhere the values of a, ^, &c., M^, M^, &c., Ni, A^a, &c. may be bund by the methods explained in Art. 455. All other sets of principal coordinates may be found from these by taking ^ = a + E^i, r) = h-\-Fr)^, &c. When the initial conditions are such that throughout the motion all the principal coordinates are constant except one, the lystem is said to be performing a principal or harmonic oscilla- 384 SMALL OSCILLATIONS. [CHAP. IX. tion. It performs a compound oscillation when any two or more are variable. We may therefore say that any possible oscillation of the system about a position of equilibrium is analysed by Lagrange's method into its simple or component oscillations. From this reasoning we infer the important theorem that if the equilibrium of a system is stable for the principal oscillations it is stable for all oscillations. The theorem that the general oscillations of a system may be resolved into certain primary oscillations which can have a simultaneous existence is sometimes called the principle of the co-existence of small oscillations. 461. It is clearly important to determine the peculiarities of a principal oscillation by which it can be recognized apart from all mathematical symbols. The physical peculiarities of a principal oscillation are : 1. The motion recurs at constant intervals, i.e. after one of these intervals the system occupies the same position in space as before, and is moving in exactly the same way. 2. The system passes through the position of equilibrium, twice in each complete oscillation. For, taking f as the variable co- ordinate, we see that ^ — a vanishes twice while p^t increases by Stt. 3. The velocity of every particle of the system becomes zero at the same instant, and this occurs twice in every complete oscillation. For d^/dt vanishes twice while p:^t increases by Stt. The positions of rest may be called the extreme positions of the oscillation. 4. Let the system be referred to any coordinates 6, (f>, &c. whose equilibrium values are (as before) a, 0, &c. When the system is performing a principal oscillation these are all variable, but the ratios of d — a, 4> — ^, &c. to each other are constant throughout the motion*. For, referring to the formulae of trans- formation (10), we see that, when rj^, fi, &c. are all zero and only ^i is variable, — a :=0. Substituting for djcp the ratio of the minors of the second row of Lagrange's determinant we have I . ap^ - z {Ip^ - g)=0. This determines the value of z corresponding to the two periods p^=p-^, P^=V^- The two principal oscillations are exhibited in the figure, z = AE being negative in ig. (1) and positive in fig. (2). The actual oscillation is constructed by the super- Dosition of these two kinds of motion. It is interesting to notice the way in which one principal oscillation disappears vben either the length I of the string or the linear dimensions a of the body iiminishes without limit. Referring to Lagrange's determinant we see that in both iases one value of p"^ is very great so that the period of the disappearing oscillation s very short. The visible motion is therefore reduced to a harmonic oscillation )erformed in the finite time given by the other value of p^ together with a tremulous notion. The values of Ip^ and ap^ given by the Lagrangian determinant, when I and a vanish respectively, and p^ is infinite, are ultimately (k^ + a^) gjk^ and a-g/k^. The lorresponding values of z are the positive quantity [k'^ + a'^)la and zero. The II R. D. 25 386 SMALL OSCILLATIONS. [CHAP. IX. disappearing oscillation is that represented in fig. (2) ; the points and E are fixed, and the extent of the tremulous motion is geometrically more and more limited as either OA or AE becomes evanescent. In the oscillation which does not disappear E ultimately coincides with 0. Ex. 2. In the experiments conducted by Borda, Cassini, Arago and Biot to determine the length of the seconds' pendulum by observing the time of oscillation of a sphere supported by a wire, it has always been supposed that the diameter of the sphere, which in the position of rest was vertical, continues during the whole vibration to be in the same straight line as the wire. Show that the value of the seconds' pendulum thus found is too short by k^jaP of itself ; where a is the radius of the ball, k the radius of gyration about a diameter and I the length of the string. In the experiments the ball used was so small that this correction is insensible. [Airy, Camb, Trans, vol. iii. 1829. In these experiments it is almost impossible to avoid giving the sphere a slight spin about the diameter which in equilibrium is vertical. Treating the sphere and the supporting wire as a rigid body, rotating with an angular velocity n about the wire, we see by Art. 268 that the time of oscillation of such a system is 2irl{ix-^- ix^). Substituting the values oi fx^, ix^ given in that article and writing - g tov g we easily see that the length of the pendulum as observed is too long by a'^n^l25gP of itself very nearly. This result agrees with that given by Poisson in the Connaissance des Terns 1816. This correction also is insensible. Ex. 3. If {^1, ^i), (^2' ^2) ^^® *^^ ^^o values of 6, for two principal oscillations, prove that in example (1) of Art. 458, 16^6.2= -a^i^g- ^^ *wo equal particles A, B, are suspended by a string from a fixed point 0, prove also that 2adjd2= -b(f)i(p2i where OA = a, AB = h. These relations between the principal oscillations are special cases given by the method of Multipliers, vol. ii. , Art. 398. 462. Equal Roots in Lagrange^s Determinant. When some of the roots of the equation giving p^ are equal, we know by the theory of linear differential equations that either (1) terms of the form (At -\- B) sin pt enter into the values of 6, (f), &c., or (2) there must be an indeterminateness in the coefficients M, N, &c. given by Art. 455. Referring the system to principal coordinates, which vanish in the position of equilibrium, w^e see by Art. 460, that the first alternative is in general excluded. If two values of p^ are equal, say b^ and 622, the trigonometrical expressions for ^ and 7) have equal periods, but terms which contain ^ as a factor do not make their appearance. The physical peculiarity of this case is that the system has more than one set of principal or harmonic oscillations. For it is clear that, without introducing any terms containing the products of the coordinates into the expressions for T or U, we may change f, 77 into any other coordinates ^1, t]-^, which make p+ 77^= ^1^ + ^^ the other coordinates f, &c. remain- ing unchanged. For example we may put ^ = fi cos a — rji sin a and r) = Si sin « + '^1 cos a, where a has any value we please. These new quantities fi, rji, f, &c., are evidently principal coordinates, according to the definition of Art. 459. One important case must however be noticed, viz., when one or more of the values of p are zero. If, for example, bn = 0, we have ^ = At + B, where A and B are two undetermined con- ART. 463.] LAGRANGE'S METHOD. 387 stants. The physical peculiarity of this case is that the position of equilibrium from which the system is disturbed is not solitary. To show this, we remark that the equations giving the position of equilibrium are -^^ = 0, -^ = 0, &c., where U has the value These in general require that f, 77, &c. should all vanish, but if 5ii = they are satisfied whatever ^ may be, provided that 77, f, &c. are zero. In any case however f must be very small, because the cubes of f, 77, &c. have been rejected. It follows therefore that there are other positions of equilibrium in the immediate neigh- bourhood of the given position. Unless the initial conditions of disturbance are such as to make the terms of the form At-\-B zero, it may be necessary to examine the terms of higher orders to obtain an approximation to the motion. This line of argument requires that the equations of motion should be of the Lagrangian form. In other cases the existence of equal roots in the fundamental determinant may introduce powers of the time outside the trigonometrical expressions. As the motion is greatly changed by the introduction of these terms, it is important to have a criterion to determine beforehand lohether they are present or not. The general conditions that all powers of the time are absent from the solution of a system of linear differential equations are given in vol. ii., Art. 281. Ex. 1. A heavy particle of mass m rests in equilibrium within a right circular smooth fixed cylinder whose generating lines are horizontal. If the particle be disturbed, form Lagrange's equations of motion, and show that in their solution there may be terms of the form At + B. Ex. 2. A rough thin cylinder of mass m and radius h is free to roll inside another thin cylinder of mass M and radius a. The whole system is placed in equilibrium on a smooth horizontal plane. A small disturbance being given, show ithat the three values of p'^ are p^ = 0, p^ = Q and p2__ _ y_ ^ Interpret this result. If X be the space rolled over, the angle turned through by the outer cylinder, and 6 the inclination to the vertical of the plane containing the axes, show that all three coordinates have a common periodic term, while x and each have additional independent terms of the form At + B. How would the results be altered if the horizontal plane were perfectly rough? 463. Initial Motions. We may also use Lagrange's method [to find the initial motion of any system as it starts from a position lof rest. See Art. 199. As before we must choose for our co- )rdinates some quantities whose higher powers can be rejected. It generally convenient to choose them so that they vanish in the litial position. As in Art. 454 we have 2T= And"" + 2A,,d'' + A,,'' + &c., ^here A^^, &c. are functions of 6, (f), &c. Since the system starts from rest, 6, , &c. are in the beginning of the motion all small [uantities. If we reject all powers of 6, , &c. except the 25—2 388 SMALL OSCILLATIONS. [CHAP. IX. lowest which occur, we may regard A^ &c. as constants whose values are found by substituting for 6, ''-h.., = B, I (1). &C. = &C.J From these we may deduce the initial values of 6'\ (/>", &c. If X, y, z be the Cartesian coordinates of any point P of the system, we may, by the geometry of the question, express these as functions of Q, , &c., Art. 396. Thus suppose that x—f{d, 6, &c.), then we have initially, since 6\ (j>' are zero, with similar expressions for y and z. The quantities x'\ y'\ z" are evidently proportional to the direction cosines of the initial direc- tion of motion of the point P. In this way the initial direction of motion of every point of the system may be found. 464. Initial Radius of Curvature. As explained in Art. 200, we sometimes want more than the initial direction of motion of any point P of the system. Suppose that we also want the initial radius of curvature of the path of P. We must find the values of x" , x'", &c., and then substitute in any of the formulae given in Art. 200. If, as before, x=f{d, " +..., where suffixes as usual indicate partial differential coefficients with respect to 6, , &c. If y=F{d, (f>, &c.) there are of course similar expressions for y", &c., and in three dimensions for z", &c. If the point P be so situated that for every possible motion of the system it can begin to move only in some one direction, we take the axis of x perpendicular to that direction. We then have x" = Q for all initial variations of 6, '+C'^ (1), where A, B, G are given functions of 6, "=VQ, Bd" + C^"=U^ (3), which give the initial values of 6", (p". To find the initial values of d"\ (f>"' we differentiate (2) with regard to t and put ^' = 0, 0' = O. We obviously have d"' = 0, 0'" = 0. To find d'\ 0'^ we differentiate (2) twice. Noticing that when 0' = 0, 0' = 0, |(P.') = .>^P + 3r(."| + 0"A) d«3 ^ {Pd'^ + Qd'' + R'^) = 2 {Pd"-' + Qd"" + B"\ where P, Q, R are any functions of d, ^'' = L, Bd''^+G"^ (5), we obtain L, M in the symmetrical forms dd'^^ d"^ with similar expressions for y" and y". We therefore have « r Also we have by (3) and (4), {e"^^-"d% .(9). "d-) = U^M-U^L .(10). The equations (8), (9) and (10) determine the radius of curvature p in terms of the initial values of the accelerations d'\ Fq, the equation (1) shows that T is '+ where bn, 622 » ^c. are constants, and Uo is the value of U in the position of equilibrium. Taking as a type any one of Lagrange's equations ddT_dT^dU dt dd' d6~ dO' we have e"-bnO=0, with similar equations for cj), yfr, &c. If b^ is positive, this equa- tion gives 6 in terms of real exponentials, and the equilibrium is unstable for all disturbances which affect 0, except such as make the coefficient of the term containing the positive exponent vanish. If 611 is negative, 6 is expressed by a trigonometrical term, and the equilibrium is stable for all disturbances which affect 6 only. In this demonstration the values of 6n, 622? &c. are supposed not to be zero. If in the position of equilibrium f/" is a maximum for all possible displacements of the system, we must have b^, 622 j ^c. all negative. Whatever disturbance is given to the system, it will oscillate about the position of equilibrium, and that position is then stable. If C/" is a maximum for some displacements and a minimum for others, some of the coefficients ^n, b^, &;c. will be negative and some positive. In this case if the system be dis- turbed in some directions, it will oscillate about the position of equilibrium ; if disturbed in other directions, it may deviate more and more from the position of equilibrium. The equilibrium is therefore stable for all disturbances in certain directions, and un- stable for disturbances in other directions. If [/" is a minimum in the position of equilibrium for all displacements, the coefficients 611, 622, &c. are all positive, and the equilibrium is then unstable for displacements in all directions. Briefly, we may sum up the results thus : — ART. 470.] THE CAVENDISH EXPERIMENT. The system will oscillate about the position of equilibrium for all disturbances if the potential energy is a 'minimum for ail dis- placements. It will oscillate for some disturbances and not for j others if the potential energy, though stationary, is neither a maoci- I mum nor a minimum. It will not oscillate for any disturbance if the potential energy is a maximum for all displacements. It appears from this theorem that the stability or instability of a position of equilibrium depends, not on the inertia of the system, but only on the force function. The rule is, give the system a sufficient number of small arbitrary displacements, so that all possible displacements may be compounded of these. By examining the work done by the forces in these displacements we can deter- mine whether the potential energy is a maximum or minimum or neither. We have assumed in this proof that when V is expanded in powers of 6, ^33 = -t-» • But, since (xoo Q/y ciz sum of these is zero, h^^, &22' ^33 cannot all have the same sign. Ex. 2. Hence, show that, if any number of particles mutually repelling each )ther be contained in a vessel, and be in equilibrium, the equilibrium will be instable unless they all lie on the containing surface. [Sir W. Thomson, now [Jord Kelvin, Garnb. Math. Journal, 1845. Reprint, viii., p. 100.] 470. The Cavendish Experiment. As an example of the [node in which the theory of small oscillations may be used as I means of discovery we have selected the Cavendish Experiment. The object of this experiment is to compare the mass of the arth with that of some given body. The plan of effecting this )y means of a torsion-rod was first suggested by the Rev. John Siichell. As he died before he had time to enter on the experi- nents, his plan was taken up by Mr Cavendish, who published ;he result of his labours in the Phil. Trails, for 1798. His experiments being few in number, it was thought proper to lave a new determination. Accordingly, in 1837 a grant of £500 vas obtained from the Government to defray the expenses of he experiments. The theory and the analytical formulae were upplied by Sir G. Airy, while the arrangement of the plan of operation and the task of making the experiments were under- iaken by Mr Baily. Mr Baily made upwards of two thousand . 394 SMALL OSCILLATIONS. [chap. IX. experiments with balls of different weights and sizes, and sus- pended in a variety of ways, a full account of which is given in the Memoirs of the Astronomical Society, Vol. xiv. The experiments were, in general, conducted in the following manner. 471. Two small equal balls are attached to the extremities of a fine rod called the torsion-rod, and the rod itself is sus- pended by a string fixed to its middle point G. Two large spherical masses A, B are fastened on the ends of a plank which can turn freely about its middle point 0. The point is vertically under C and so placed that the four centres of gravity of the four balls are in one horizontal plane. Firstly, suppose the plank to be placed at right angles to the torsion-rod, then the rod will take up some position of equilibrium called the neutral position, in which the string has no torsion. Let this be represented in the figure by Ca. Now let the masses A and B be moved round into some position B^A^, making a not very large angle with the neutral position of the torsion-rod. The attractions of the masses A and B on the balls will draw the torsion-rod out of its neutral position into a new position of equi- librium, in which the attraction is balanced by the torsion of the string. Let this be represented in the figure by CE^. The angle of deviation E^Ca, and the time of oscillation of the rod about this position of equilibrium are observed. Secondly, replace the plank AB at right angles to the neutral position of the rod, and move it in the opposite direction until the masses A and B come into some position A^B^ near the rod but on the side opposite to B^A^. Then the torsion-rod will perform oscillations about another position of equilibrium CE2 under the influence of the attraction of the masses and the torsion of the string. As before the time of oscillation and the deviation EoCa are observed. In order to eliminate the errors of observation, this process is repeated over and over again, and the mean results are taken. iRT. 471.] THE CAVENDISH EXPERIMENT. 395 riie positions B^Ai and A^B^, into which the masses are alternately out, are as nearly as possible the same throughout all the ex- Deriments. The neutral position Got of the rod very nearly Disects the angle between B^Ai and AzB^, but as this neutral Dosition, possibly owing to changes in the torsion of the string, s found to undergo slight changes of position, it is not to be considered in any one experiment coincident with the bisector )f the angle AfiB^. Let Cx be any line fixed in space from which the angles may )c measured. Let b be the angle xGa, which the neutral position )f the rod makes with Co) ; A and B the angles which the alter- late positions, B^A^ and A^B^, of the straight line joining the '('litres of the masses, make with Coo; and let a = ^(A + B). Also or X be the angle which the torsion-rod makes with Cx at the ,ime t. Supposing the masses to be in the position ^i, B^, the moment ihout GO of their attractions on the two balls and on the rod will )e a function only of the angle between the rod and the line ^i^ii ct this moment be represented by (J. — x). The whole apparatus s enclosed in a wooden casing to protect it from any currents )f air. The attraction of this casing cannot be neglected. As it nay be different in different positions of the rod, let the moment )f its attraction about CO be ^jr (x). Also the torsion of the string s very nearly proportional to the angle through which it has )een twisted. Let its moment about CO be E {x — h). If then / be the moment of inertia of the balls and rod about h(j axis CO, the equation of motion is Now a — X is a small quantity, let it be represented by f. Jubstituting for x and expanding by Taylor's theorem in powers f ^, we get Let ^.^4>'{A-a)-^-ia)^E^ {A-a)+^{a)-E{a-h) Then x=e + L^m{nt + L'), rhere L and L' are two arbitrary constants. We see therefore lat in the position of equilibrium the angle made by the torsion- with the axis of x is e, and the time of oscillation about 16 position of equilibrium is 27r/^. Let us now suppose the masses to be moved into their alternate SMALL OSCILLATIONS. [CHAP. IX. position A^B^] the moment of their attraction on the balls and rod is now — (x — B). The equation of motion is therefore Let a = x — ^, then, substituting for B its value 2a — A, we find by the same reasoning as before ^ = e' + iV" sin (nt + JSf'), where n has the same value as before, and ^,_^ . -cl>(A-a)+y!r(o.)-E{a-h) ^-""^ 'K^ • ^ In these expressions, the attraction sjr (a) of the casing, the coefficient of torsion E and the angle b are all unknown. But they all disappear together, if we take the difference between e and e. We then find (A-a)- fiM (mP + m'Q). If r be the radius of either ball, we have I=2m (c2 + |r2) + im' (c - rf, ^hich may be represented by J = mP' + m'Q', where P' and Q' are known functions of the linear dimensions of the rod and balls. [Hence we find by substituting in equation (A) ^ mP + rriQ _e — e /^ttV ^ • mP' + m'Q' ~ ~Y~ ' \T) Let E be the mass of the earth, R its radius and g the force !of gravity, then* g = /jbE/R'^. Substituting for /jl, we find M _ e-e' /SttV JL_ mP' + m/Q' E~ 2 '[tJ ' gR'' mP + m'Q" The ratio mjm' was taken equal to the ratio of the weights of the ball and rod weighed in vacuo, but it would clearly have been more accurate to have taken it equal to their ratio when weighed in air. For, since the masses attract the air as well as the balls, the pressure of the air on the side of a ball nearest the attracting mass is greater than that on the furthest side. The difference of these pressures is equal to the attraction of the mass on the air displaced by the ball. 474. By this theory the discovery of the mass of the earth I has been reduced to the determination of two elements, (1) the I time of oscillation of the torsion-rod, and (2) the angle e — e' between its two positions of equilibrium when under the influence of the masses in their alternate positions. To observe these, a small mirror was attached to the rod at C, with its plane nearly perpendicular to the rod. A scale was engraved on a ver- tical plate at a distance of 108 inches from the mirror, and the image of the scale formed by reflection on the mirror was viewed in a telescope placed just over the scale. The telescope was furnished with three vertical wires in its focus. As the torsion-rod turned on its axis, the image of the scale was seen in the telescope to move horizontally across the wires, and at any instant the * In Baily's experiment, a more accurate value of g was used. If e be the ellipticity of the earth, m the ratio of centrifugal force at the equator to equatorial E gravity, we have g = iJ.^{l-\-m~2e- (fm - e) cos^X}, where R is earth's polar radius and X the latitude of the place. SMALL OSCILLATIONS. [CHAP. IX. number of the scale coincident with the middle wire constituted the reading. The scale was divided by vertical lines one-thirteenth of an inch apart and numbered from 20 to 180 to avoid negative readings. The angle turned through by the rod when the image of the scale moved through a space corresponding to the interval of two divisions was therefore -^ . yj^ . J = 73'' '46. But the division lines were cut diagonally and subdivided decimally by horizontal lines; so that not only could the tenth of a division be clearly distinguished, but, after some little practice, the frac- tional parts of these tenths. The arc of oscillation of the torsion- rod was so small that the square of its circular measure could be neglected ; but as it extended over several divisions it is clear that it could be observed with accuracy. A minute description of the mode in which the observations were made would not find a fit place in a treatise on dynamics, we must therefore refer the reader to Baily's memoir. In this investigation no notice has been taken of the effect of the resistance of the air on the arc of vibration. This was, to some extent at least, eliminated by a peculiar mode of taking the means of the observations. In this way also some allowance was made for the motion of the neutral position of the torsion-rod. We have also not considered what relative dimensions should be given to the different parts of the instrument, consistent with its proper support, so as to obtain the most accurate result. Such considerations are hardly suited to a general treatise on dynamics. In the original experiments the attracting masses A and B were large, and brought near the small balls m and m. As a rapid oscillation of the rod was inadmissible, the moment of inertia I of the rod and balls was large and the torsion of the string was small. The size of the instrument was not handy. It was very important that the whole instrument should be kept at the same uniform temperature. As this could not be completely accomplished slight air currents were set up both within and without the wooden casing. Thus the oscillation of the rod was sometimes irregularly affected and the torsion of the string altered. 475. The density of water in which the weight of a cubic inch is 252'725 grains (7000 grains being equal to one pound avoirdupois) was taken as the unit of density. The final result of all the experiments was to determine for the mean density of the earth the value 5"6747. Many experiments have been made besides those by Cavendish and Baily, a full account of which is given by Poynting in his Adams Prize Essay, 1894. We may allude to the results of Cornu and Bailie (see Gomptes Bendus, 1873 and 1878). They made several improvements in the apparatus arid found the mean density to be 5'56. They considered that they had found an error in Baily's method of taking his means, and that, if this were corrected, Baily's result would become 5*55. The observations made by Jolly at Munich and Poynting at Manchester are also important; th« former gave 5*692 and the latter 5'4934 as the mean density. ART. 476.] THE CAVENDISH EXPERIMENT. 399 A great improvement in the mode of conducting the experi- munt has been made by Boys. It is clear that every diminution ill the size of the apparatus is an advantage, provided the extent )t' oscillation remains sufficiently large for accurate measurements. The apparatus is then more easily kept at one temperature, and can be made more free from currents of air. Now Boys discovered a method of making fine quartz wires, which are not only sufficiently strong to carry the beam, but are also free from some other defects of ordinary metallic wires. The result of his ex- periments gave 5"527(), which is considered to be a very near approximation to the truth. Proceedings Royal Soc. 1889. 476. Three general methods have been employed to determine the mean density. In the first a balance is used as in the Cavendish experiment. In the second the mass of the earth is compared with that of a mountain by observing the deviation produced in a plumb-line by the attraction of the latter. The mountain chosen was Schehallien, and the density of the earth was found to be a little less than five times that of water. See Fhil. Trans. 1788 and 1811. From some observations near Arthur s Seat, the mean density of the earth was given by Lieut.-Col. James of the Ordnance Survey as 5'316. See Phil. Trans. 1856. In the third method the force of gravity at the bottom of a mine is compared with that at the surface, by observing the times of vibration of a pendulum. In this way the mean density of the earth was found to be 6-566. Airy, Phil. Trans. 1856. The following summary of results is taken from Poynting's Essay. Approximate date Experimenter Method Result 1737-40 1774-6 1855 1821 1880 1854 1883 1885 1797-8 1837 1840-1 1852 1870 1889 1879-80 1878-90 1884- 1886-8 1889 Bouguer Maskelyne and Hutton James and Clarke Carlini Mendenhall Airy Von Sterneck Von Sterneck Cavendish Keich Baily Eeich Cornu and Bailie Boys Von Jolly Poynting ( Konig, Kicharz and ) I Krigar Menzel ) Wilsing Laska Plumb-line and Pendulum Plumb-line Inconclusive 4-5 to 5 5-316 4-39 to 4-95 5-77 6-565 5-77 about 7 5-448 5-49 5-674 5-583 5-56 to 5-50 in progress 5-692 5-493 in progress 5-579 in progress Mountain Pendulum Mine Pendulum Torsion Balance Common Balance Pendulum Balance 400 EXAMPLES. [chap. IX. EXAMPLES*. 1. A uniform rod of length 2c rests in stable equilibrium with its lower end at the vertex of a cycloid whose plane is vertical and vertex downwards, and passes through a small smooth fixed ring situated on the axis at a distance b from the vertex. Show that, if the equilibrium be slightly disturbed, the rod will perform small oscillations with its lower end on the arc of the cycloid in the time /a -fc^ + S (b — c)H ^'"'\/ "— V— rro-^ — ^— 5 where 2a is the length of the axis of the cycloid. V 3g{b^-4ac) 2. A small smooth ring slides on a circular wire of radius a which is con- strained to revolve about a vertical axis in its own plane, at a distance c from the centre of the wire, with a uniform angular velocity u where u}"^ (c ^2 + a)=g ^2; show that the ring will be in a position of stable relative equilibrium when the radius of the circular wire passing through it is inclined at an angle 45° to the horizon ; show also that, if the ring be slightly displaced, it will perform a small oscillation in a time T where {TI2Tr)^ g{c^8 + a)=aJ2{cj2 + a). 3. A uniform bar of length 2a, suspended by two equal parallel strings each of length b from two points in the same horizontal line, is turned through a small angle about the vertical line through the middle point, show that the time of a small oscillation is 2ir^bk^lga^. 4. Two equal heavy rods, connected by a hinge which allows them to move in a vertical plane, rotate about a vertical axis through the hinge, and a string whose length is twice that of either rod is fastened to their extremities and bears a weight at its middle point. If M, M' be the masses of a rod and the particle, and 2a the length of a rod, prove that the angular velocity about the vertical axis when the rods and string form a square is a / — ^ • - — ^ — ; prove also that, if the weight be slightly depressed in a vertical direction and the system left to itself, the time of a small oscillation is 27r . / .^ . rr-z — -^rjr, • 5. A ring of weight TT which slides on a rod inclined to the vertical at an angle a is attached by means of an elastic string to a point in the plane of the rod, so situated that its least distance from the rod is equal to the natural length of the string. Prove that, if 6 be the inclination of the string to the rod when in equilibrium, cot^-cos ^ = wcosa, where W/n is the modulus of elasticity of the string. Also if the ring be slightly displaced the time of a small oscillation will be 2irsJ{nllg(l - sin^^)}, where I is the natural length of the string. 6. A circular tube of radius a contains an elastic string fastened at its highest point equal in length to one-eighth of its circumference, and having attached to its other extremity a heavy particle which hanging vertically Would double its length. The system revolves about the vertical diameter with an angular velocity sjgja. Find the position of relative equilibrium, and prove that, if the particle be slightly 27r ,^fir la ta + ^ika t disturbed, the time of a small oscillation is t t \/ ~- l^^^- 4:ou.j \/7r + 4 V 9 7. A heavy uniform rod AB has its lower extremity A fixed to a vertical axis, and an elastic string connects B to another point C in the axis such that AG=a, AB = asJ2; the wholie is made to revolve round AC with such angular velocity that the string is double its natural length and horizontal when the system * These examples are taken from the Examination Papers which have been set in the University and in the Colleges. EXAMPLES. 401 is in relative equilibrium, and then left to itself. If the rod be slightly disturbed in a vertical plane, prove that the time of a small oscillation is 2ir,J^al21g^ the weight of the rod being sufficient to stretch the string to twice its length. Art. 452. 8. Three equal elastic strings AB, BG, GA surround a circular arc, the ends being fixed at ^. At J5 and G two equal particles of mass m are fastened. If I be the natural length of each string supposed always stretched, and \ the modulus of elasticity, show that if the equilibrium be disturbed the particles will be at equal distances from A after intervals irj^/mllX. Art. 454. 9. A particle of mass M is placed near the centre of a smooth circular horizontal table of radius a, strings are attached to the particle and pass over n smooth pulleys which are placed at equal intervals round the circumference of the circle ; to the othpr end of each of these strings a particle of mass M is attached ; show that the time of a small oscillation of the system is 27r f ) . \ n gj 10. Two discs slide in a circular tube of uniform bore containing air, exactly fitting the tube. The two discs are placed initially so that the line joining their centres passes through the centre of the tube, and the air in the tube is initially of its natural density. One disc is projected so that the initial velocity of its centre is a small quantity. If the inertia of the air be neglected, prove that the point ou the axis of the tube equidistant from the centres of the discs moves uniformly and that the time of an oscillation of each disc is 27r^iHa7r/4P, where M is the mass of each disc, a the radius of the axis of tube, and P the pressure of air on the disc in its natural state. 11. A uniform beam of mass 31 and length 2a can turn round a fixed horizontal axis ctt one end ; to the other end of the beam a string of length I is attached and at the other end of the string a particle of mass m. If, during a small oscillation of the system, the inclination of the string to the vertical is always twice that of the beam, then M {Bl -a) = 6/;i {I + a). Art. 458. 12. A conical surface of semivertical angle a is fixed with its axis inclined at an angle 8 to the vertical, and a smooth right cone of semivertical angle /3 is placed within it so that the vertices coincide. Show that time of a small oscillation = 27r;y/ (sin (a-/3)cosec ^a/^r), where a is the distance of the centre of oscillation of the cone from the vertex. 13. A number of bodies, the particles of which attract each other with forces varying as the distance, are capable of motion on certain curves and surfaces. I Prove that, if Ay B, C be the moments of inertia of the system about three axes 1 mutually at right angles through its centre of gravity, the positions of stable I equilibrium will be found by making A + B + G a minimum. Art. 469. 14. A particle is in motion within a triangle ^SC, and is attracted perpendicu- larly to the sides with forces each equal to /x times the perpendicular distance. Show that the motion is expressed by two terms of the form Psin {«^(\(u) + a}, where (\ - 1) (X - 2) + 2 cos A cos B cos G = 0. Show that the roots of this quadratic are real and positive. Examine the case of an equilateral triangle, and in that case verify the above iLSult independently. 15. The force between two small masses attracting according to the law of the inverse square of the distance is equal, at distance a, to a very small fraction n of the weight of either. They are suspended by two strings of length I from two points situated in a horizontal plane, at a distance apart equal to a, and are set to perform small vibrations in the same vertical plane; prove that the motion of each is compounded of two harmonic motions whose periods are very nearly as 1 : l + 2nlla. R. D. ' 26 CHAPTEE X. ON SOME SPECIAL PROBLEMS. Oscillations of a Rocking Body in three dimensions. 477. A heavy body oscillates in three dimensions with one degree of freedom on a fixed rough surface of any form in such a manner that there is no rotation about the common normal. Find the motion. 478. The Relative Indicatrix. Let be the point of contact when the heavy body is in equilibrium. Let the common normal be the axis of z, and let the other two axes be at right angles in the common tangent plane. The equations to the portions of the surfaces in the neighbourhood of may be written in the forms z = ^{ax- + 2bxy + cy^) + &c. z' = J(aV + 2b' xy + cY) + &c. Let an ordinate move round the origin so that the portion z — z between the surfaces is constant and equal to any indefinitely small quantity A, This ordinate traces out an evanescent conic on the plane of xy whose equation is (a - a') x^-\-2{b- b') xy-\-{c- c') 3/^ = 2 A. Any conic similar and similarly situated to this, lying in the tangent plane and having its centre at 0, is called the Relative Indicatrix of the two surfaces. Let OR be any radius vector of this indicatrix, then the difference of the curvatures of the two sections made by a normal plane zOR (or their sum, if they are measured in oppo- site directions) varies inversely as the square of OR. This of course follows from the definition of the conic by a well-known argument in solid geometry. Thus, let (r, z){r, z') be the co- ordinates of two points on the two circles of curvature at the same distance from the axis of z. We have ultimately 2pz = r'^ and 2p'z' = r\ Also z — z' = A, hence, eliminating z and /, we see that the difference of the curvatures varies inversely as r\ Let OR be a tangent to the arc of rolling determined by the geometrical conditions of the question. Let p, p be the radii of I ART. 481.] THE TIME OF OSCILLATION. 403 curvature of the normal sections through OR, taken positively when the curvatures are in opposite directions, and let - = - H — , . s P P Then s may be called the radius of relative curvature. The three following propositions are of use in dynamics. 479. Prop. The Instantaneous Axis. Let 01 and Oi/ be two conjugate diameters of the relative indicatrix, then, if Oy be a tangent to the arc of rolling, 01 is the instantaneous axis, and, if 6 be the indefinitely small angle turned round the in- stantaneous axis, the arc a of rolling is given by (t — Bs sin yOI. To prove this, measure in the plane yz along the surfaces two lengths OP and OP' each equal to ut, if D be the diameter of the cylinder of stability drawn with its axis parallel to /'/', and if PW cut the cylinder in V, we have PV .cosKPW=D cosKPz'. Substituting in the equation, the expression in brackets takes the form PV - OG, which is ultimately equal to GV. We thus obtain the second result. We might also find the periods by the method of vis viva. Oscillations of Cones in three dimensions. 488. Oscillations of Cones to the first order. A heavy cone of any form oscillates on a fixed rough conical surface, the vertices being coincident. It is required to find the time of a small oscillation. The motion of a cone about its vertex regarded as a fixed point is conveniently discussed by the help of spherical trigonometry. Let be the common vertex, G the centre of gravity of the moving cone, OG = h. With centre 0, and radius equal to OG, describe a \ /^ sphere ; it is on this sphere that lue ,-^"" '-p^ shall suppose our spherical triangles to \ © & back to its position of equilibrium, is gh sin z {GM - GG'), which on substitution becomes M=gh {a cos n sin {z-r)-d sin r sin z}. Equating this moment with the sign changed to K^d, the result to be proved follows immediately. We may obtain this equation by the analytical method given in Art. 509. We there replace the geometry here used by a process of differentiation, which may be extended to any higher degree of approximation. 486. Examples. Ex. 1. If the upper body be a right cone of semi-angle p, and if it be on the top of any conical surface, we have n=0 and r=p. The preceding expression then takes the form K^ hL sm {z + p) sin^ p sin {p + p') Ex. 2. A heavy right cone of angle 2p and altitude a, suspended by its vertex from a fixed point in a rough vertical wall, is oscillating, prove that the length of the equivalent pendulum is \a sec p (1 +5 cos^ p). Let the cone when in equilibrium touch the plane along the vertical Oz. At the time «, let the generator ON be the line of contact, where zON=(r. Let OA be the axis. Resolving gravity along and perpendicular to the line ON, and taking 408 ON SOME SPECIAL PROBLEMS. [CHAP. X. moments about the instantaneous axis ON, we have K'^d= -g sin a- . ^a sin p. Now, if the cone turn round ON through an angle Mt, the centre A of' the base advances a space a sin p . 6dt, hence, if ^ If be a perpendicular on ON, H advances an equal space. But it does advance a space OH . d and altitude a is divided by a plane through the axis. One of the halves rests in equilibrium with its axis along a generator of a fixed right cone of angle 2p', the vertices being coincident, prove that the length L of the equivalent pendulum is given by fn Q.1C* 2 ,i2atan2p . , al sin{p'-\-z) {97r2 + 16 tan^ p]i — — — ^ = Stt sm z tan p' - 4 tan p — ^-'^--J ^ '^^ 5L r r cosp' where z is the inclination of the line of contact to the vertical measured upwards. 487. Condition of Stability of Cones to the first order. To determine the condition of stability when a heavy cone rests in equilibrium on a perfectly rough cone fixed in space. It is evident that we must have the length L of the equivalent pendulum, found in Art. 483, equal to a positive quantity. This Ifeads to the following construction, which is represented in the figure of Art. 483. Measure along the common normal CI to the cones a length IS = s, such that cot s = cot p + cot p. From S draw an arc SR perpendicular to IGW, then cos 71 = cot s . tan III. Then L is positive and the equilibrium is stable if the centre of gravity of the moving cone be either below the common generator of the two cones, or above the generator at an angle r such that cot r > cot -s + cot IRj provided IR is less than a right angle. When the vertex is very distant the cones become cylinders. In this case, if the arc 2 become a quadrant, the condition of stability is reduced to r < IR. This agrees with the condition given in Art. 442. Large Tautochronous Motions. 488. When the oscillations of a system are not small, the equation of motion cannot always be reduced to a linear form, and no general rule can be given for the solution. But the oscil- lation may still be tautochronous, and it is sometimes important to ascertain whether this is the case. Various rules to determine this question are given in the following Articles. ^RT. 489.] LAKGE TAUTOCHRONOUS MOTIONS. 409 When a particle oscillates on a given smooth curve either n a vacuum or in a medium whose resistance varies as the velocity we know that the oscillation is tautochronous about she position of equilibrium if the tangential force P = m^s where s s the length of the arc measured from the position of equilibrium ind 7n is a constant, Art. 434. If therefore any rectifiable curve s given, the proper force to produce a tautochronous motion 3an at once be assigned. Thus a catenary is a tautochronous 3urve for a force acting along the ordinate equal to m^y, because :he resolved part along the tangent is obviously ni^s. The eqid- mgular spiral is tautochronous for a central force fir tending to 3he pole, because the resolved part along the tangent being m^5, vvhere iii^ = fi cos^ a, the time of arrival at the pole is the same br all arcs. In the same way the epicycloid and hypocycloid ire also tautochronous curves for a central force tending from )r to the centre of the fixed circle and varying as the dis- ance, because since r^ = As^-\-B, the resolved part along the angent, viz. jurdr/ds, varies as s. In all these cases the time )f arrival at the position of equilibrium is the least positive root )f the equation tan nt — — u/k (Art. 434) where 2kv is the re- iistance and n^ -\- k^= m^. The whole time from one position of nomentary rest to the next is Tr/n. d^x I dx ^ 489. If the equation of motion he ;7/2 ~ -^ ( 77/ ' ^ ) ' uhere F is a homogeneous function of the first degree, then, in what- ever position the system is placed at rest, the time of ai^riving at the msition determ,ined by x = is the same. (1 dx\ - -tt] ' Let X md f be the coordinates of two systems starting from rest in two lifferent positions, and let x = a, f = Ka initially. It is easy to .ec that the differential equation of one system is changed into hat of the other by writing f = kx. If therefore the motion of )iie system is given by x = (f){t. A, B), that of the other is given )y ^ = K(l>(t, A', B'). To determine the arbitrary constants A, B md A', B', we have exactly the same conditions, viz. that, when ^ = 0, b = a and dcfi/dt = 0. Since only one motion can follow from a ingle set of initial conditions, we have A' — A, and B' = B. ience throughout the motion f = kx, and therefore x and f anish together. It follows that the motions of the two systems re perfectly similar, and the times equal. This result may be obtained also by integrating the differential q nation. If we put px= dx/dt, we find, after eliminating x, that he variables p and t can be separated, showing that ^ is a unction of t + B. Hence by an easy integration x = A(f) {t + B). /Vhen t = 0, dxjdt = 0, and therefore ^' (B) = 0. Thus B is known -nd X vanishes when (^ + 5) = 0, whatever be the value of A. 410 ON SOME SPECIAL PROBLEMS. [CHAP. X. It must be noticed that if the force be a homogeneous function of the velocity and X, the motion is tautochronous only in a certain sense. It may happen that the "system arrives at the position determined by x = only after an infinite time, or the time of arrival may be imaginary. Thus, suppose the homogeneous function to be ni^x, where m^ is positive, then the system starting from rest moves continually away from the position a; = 0. The value of x is evidently represented by an exponential function of x which never ceases to increase with the time. It is therefore necessary in applying the rule to ascertain whether the time given by the equation

dy^ where (f> stands for (y) and accents as usual denote differential coefficients. Let c/)/^' =f(y\ substituting we have df~f\dtj f\dt) "^-^ \fdt^ where / has been written for f{y). The last two terms of this expression form a homogeneous function of/ and dyjdt of the first ART. 492.] LARGE TAUTOCHRONOUS MOTIONS. 411 degree, and therefore Lagrange's formula has been proved. This demonstration is due to Bertrand. Liouville's Journ. Vol. xii. 1847. The motion begins from rest with any iuitial value of x and ends when x = 0. Hence, writing x = (y), we see that in the second equation the motion begins with dy/dt = and with any initial value of y, and terminates when c^ (y) = 0. Now dx/dt does nut in general vanish when x = 0, since the system arrives with some velocity at the position of equilibrium. But since dx ,,, ^dy — (t>'-f(y), that the motion terminates when /(^z) = 0. 491. Effect of a resisting medium. If the motion is tautochronous according to Lagrange's formula in a vacuum, the motion is also tautochronous in a medium whose resistance varies as the velocity. The only effect of such a resistance is to introduce an additional term, viz. 2kv, of the first degree into the arbitrary function F. This theorem is due to Lagrange. If the resistance is 2kv + /«V, we write Lagrange's equation in the form d^x fix) , , V'' „ ,, . Putting the coefficient of v"^ equal to k, we find by integration that f{x) = Ce'''^ + A/K. If x is measured from the position of equilibrium, at which by Lagrange's theorem f{x) = 0, we must have A= — kG. The result is that for this law of resistance, the motion is tautochronous if the impressed force is P = G (e"'^ — I). vFhis result agrees with those given by Euler and Laplace. 492. We can give an easy independent proof of this theorem. For the sake of simplicity let the system be a particle moving from rest towards a point A of equi- librium on a smooth given curve under the action of a tangential force P. The equation of motion is, ii — =v. tt - k'v^ + 2kv = - P. at at jThis equation can be written in the form — (e"y) + 2k (e'*y) = - Pe'*, provided -f=- k'v, i.e. u= - k's. Put e^ds = dio, .'. -^-j + 2k -t~ + P^""'* = 0. The time of arrival at the point w = will be independent of the arc if we put !Pe-«'« = m2ip, Art. 434. Now io= - — e-'^'^ + C, and if s is measured from the Iposition at which w = 0, we have k'C = 1. We therefore have P= — r («'<'* - 1) which is the same result as before. Also the time of arriving at the position io = is given by the least positive root of the equation tan nt= -nJK where n^ = m'^- k\ If > m^ the particle arrives at the position io = after an infinite time. Art. 434. Laplace remarks that the expression for the force P is independent of the joefficient k of that part of the resistance which varies as the velocity, while the 412 ON SOME SPECIAL PROBLEMS. [CHAP. X. time of arrival at the position of equilibrium is independent of the coefl&cient k' of that part of the resistance which varies as the square of the velocity, Mecanique Cileste, Vol. i., page 38. Ex. 1. Find the smooth curve such that the motion of a heavy particle in a medium whose resistance is 2kv-\-k'v'^ may be tautochronous. Since gravity is the 7/i ft?/ Ill only force we put P—— («"'* ~^)—9^.'y ''- 9y = ~2 (^"'^ " ''''■^)' Ex. 2. Find also the curve when the impressed force tends to the origin and is equal to fir^. 493. Motion on a rough cycloid. A heavy particle slides from rest on a rough cycloid placed with its axis vertical, in a rnedium whose resistance varies as the velocity, show that the motion is tautochronous. Let be the lowest point of the cycloid, P the particle, OP — Sy. so that the arc is measured from in the direction opposite to that of the motion. Let the normal at P make an angle -xjr with the vertical, let p be the radius of curvature at P, and a the diameter of the generating circle. Then, by known properties of the cycloid, s = 2a sin sjr, p=2a cos yjr. Let yu, be the coefficient of friction, g the accelerating force of gravity, and let the mass be unity. Then, if R be the pressure on the particle measured positively inwards and v = ds/dt, we have -r. = fiR — gsin-yfr— 2kv, - = R — g cos -^^ (1). at p Eliminating R, the equation of motion becomes ^_^^,2+2/cv + -^sin(.|r-e)=0 (2), dt p cos € '^ ^ where tan e = /z. This may be written -r. (e"v) + 2k (e''v) + -^ e« sin (^/r - e) = 0, az cos € provided 37 = — /"'- > *-^- u= — fJ^'^- Put e'^'^ds = dw ; az p d^w ^ dw q , • / , \ .V Now w =je~^'^ 2a cos yjrdylr = 2a cose e"'*''' sin (\/r — e). The equation therefore reduces to d^w ^ dw a ^ The motion is therefore tautochronous, Art. 434. At what- ever point of the cycloid the particle is placed at rest, it arrives at the point A determined by w = 0, i.e. -v^ = e, in the same time. The point A, at which the tautochronous motion terminates, is clearly an extreme position of equilibrium in which the limiting friction just balances gravity. ART. 495.] LARGE TAUTOCHRONOUS MOTIONS. 413 The time of arrival at A is given by the least positive root of the equation tan nt = — n/tc, where n^ + k.^ = gj^a cos'-' e ; the whole time from one position of momentary rest to the next being tt/ti-. So long as the particle is moving in the same direction the constant fx retains the same sign, Art. 159. The motion is there- fore given by g-Mv^ sin (^/^^ - e) = Ae-"^ sin {nt + B\ where, as before, n^ + fc^ = gj2acos^ € and A, B are constants. When the particle arrives at the next position of rest, it will begin to return or will remain there at rest according as the value of -v/r at that point is greater or less than the angle of friction. We may also deduce the tautochronism of the motion from Lagrange's theorem. Proceeding as in Art. 491 and equating the coefficient of v'^ to /x//), we find a value "t f{s) which makes the Lagrangian equation become the same as that of the particle on the cycloid. 494. Historical Summary. That a smooth cycloid is tautochronous in vacuo for a heavy particle was first proved by Huygens in his Horologium Oscillatorium, 1673. Newton extended this to the case in which the resistance is 2kv, and also proved that a smooth epicycloid is tautochronous for a central force varying as the distance. That the oscillations on a cycloid are tautochronous when the curve is rough has been deduced by Bertrand from Lagrange's formula, Liouville, Vol. xiii., 1848. He ascribes the proposition to Necker, who published it in the Memoires des savants etr angers, Vol. iv., 1763. Euler practically determined the force which would make a smooth curve tautochronous when the resistance is k'v'^, Mechanica, 1736. This result was afterwards extended by Laplace to the case in which the resistance is 2kv + k'v'^, Mecanique Celeste, Tome i., page 36. Puiseux has a memoir on smooth tautochronous curves in vacuo, and also for heavy bodies when the resistance is kV, Liouville, Vol. ix., 1844. He remarks that he has avoided the use of series, such as that employed by Poisson in his Mecanique (see Art. 197). He discusses the tautochrone in vacuo when the force is central and varies as the distance and shows that the curve is an epicycloid, a hypocycloid or a certain spiral. Haton de la Goupilli^re proves that the epicycloid when rough is also tautochronous and points out briefly that this fact is not affected by a resistance 2kv, Liouville, Vol. xiii. Darboux in a note to the Mecanique de Despeyrous, 1884, shows that when friction is taken account of the only tautochronous curves are those discussed by Puiseux. 495. Motion on any rough ctirve. A particle, starting from rest, moves on a rough curve of given form in a medium whose resistance is k'v^, under the action of forces which depend only on the position of the particle. Prove that the necessary condition that the time of arriving at the position of equilibrium should be independent of the arc described is where P=G-fjiH is the excess of the tangential force G over the part fiH of the friction, and m is a "q constant. Find also the time of transit. Let A be the point at which the tautochronous motion terminates, M the position of the particle at any time t, AM=s, so that s is measured from A in the direction opposite to that of motion. Let the tangent at M make an angle ^ with the axis 414 ON SOME SPECIAL PROBLEMS. [CHAP. X. of X, and let ^ and s increase together. Let the tangential and normal components of the force be G and B. ; the tangential component G acting towards A, and the normal component H acting outwards, i.e. opposite to the direction in which p is measured. We shall suppose p to be positive throughout the arc. The equations of motion are therefore - = R-H, v~=^fjiE-G + K'v^ (1). P «s Since the particle starts from rest, we see that R and H are initially equal and thus have the same sign. We shall suppose that H is positive throughout the motion, so that the impressed force urges the particle outwards. It follows that R also is positive throughout the motion. The friction continues therefore to be represented by fxR, without any discontinuous changes in the sign of fi, such as would happen if R were to change sign xcithout a corresponding change in the direction of the friction. (See Art. 159.) Eliminating R we find vf^=f^-^+K'v^-{G-^H) (2). Let P=G-/ji,H, so that P is the whole impressed force urging the particle along the tangent towards the point A . We may prove that P must be positive throughout the motion until the particle reaches A. If P be zero at any point B, then, placing the particle at rest at B, it will remain there in equilibrium, and therefore the times of reaching A from all points will not be the same. We see also by the same reasoning that the point A must be one at which P is zero. (See Art. 489.) Writing dsjd^ for p, the equation of motion becomes ~-2(f. + K'p)v^=-2pP, .: vh-2fi^l^-2K's = c^- I 2pPe-2fi^-2K'sdxly, where a is the angle the tangent at A makes with the axis of x. As x// is greater than a throughout the motion the constant of integration, viz. c^, must be positive. We notice that the integral on the right-hand side is independent of the position of the starting point of the particle and depends only on the intrinsic equation of the curve and the point A. Let us represent this integral by z\ and take z as the coordinate of the particle. We have z = c when the particle is starting from rest, and z = when it arrives at the point A determined by \f/ = a. The intrinsic equation of the curve being given, we can imagine ^ and s to be expressed as functions of z. Putting then e-t'-'i'-K-'s ds = 4> {z) dz, the time T of transit from z = c to « = is easily seen to be [z) dz : ^{c^-z'^) To find the form of the function which makes this result independent of the arc we equate to zero its differential coefficient with regard to c. Putting 2 = c^ we have '{ck)m jov/{l-a' ■• dc jo x/(i-a • This integral cannot be zero for all values of c unless 0'(c^) = O. If it were not zero we could by taking c small enough make 0'^c^) keep one sign from ^ = to ^ = 1 ; thus every term of the integral would have the same sign and the sum could not be zero. Writing then (2) = l/m, we see that the time of transit is r=7r/2m. Putting tt= - [jLxp- k's, for the sake of brevity, we have to find P from the two dz f^ equations me"=;^, 2/ pPe^''dxp = z'^. ART. 498.] LARGE TAUTOCHRONOUS MOTIONS. 415 Integrating the first from ;^ = a to \p, i.e. 2 = to z and substituting in the second have {w|e"ds}2 = 2jPe2'*dj?. Differentiating and reducing, this leads to we -/: V#, ••. mV = ^-(M + 'c'p)P. Since P vanishes when ^=a, we verify the theorem that the point at which the tautochronous motion terminates is a position of equilibrium ; Art. 489. Ex. Show that this law of force is included in the Lagrangian expression for a tautochronous force. Comparing the Lagrangian equation as written in Art. 491 with (2) of this- article, term for term, we find an expression for /(s), i.e. - P, which agrees with that given above. By deducing the condition of tautochronism from Lagrange's expression we prove that it is sufficient, the mode of proof adopted above shows that the condition is also necessary. 496. Ex. 1. Euler's theorem. A particle moves on a smooth curve under the action of a tangential force P which is some function of the distance s of the particle from the position A of equilibrium, and the time of arrival at A from rest in any position is independent of the arc. Prove that if the motion take place in vacuo, P=Cs; and if in a medium whose resistance is k'v^, P = C (e" '* - 1) . This should be proved by the method of Art. 495, not deduced as a particular case from the general result. Ex. 2. A heavy uniform string is placed within a thin smooth cycloidal tube with its base horizontal. Prove that the time of oscillation is the same for all arcs and is independent of the length of the string. 497. Determine how the time of arrival at the position A of equilibrium in Art. 495 would be modified if the resistance were changed to 2kv + k'v^. The equation of motion (2) of Art. 495 now becomes 37 = At — + k'v^ - Ikv - P. at p As in Art. 493 this may be written in the form — {e^v) + 2k {e^v) + e^'P = 0, providied u= - fi\p- k's. Put e^'-ds = die, .: -r-^ + 2«- — - + e'^P = 0. The time of arrival at the point A , determined by iv = 0, becomes independent of the arc if the last term is equated to m~ic. We then have P = m^e-^je'^ds, which is the same value as P as before. The time of arrival at the position of equilibrium is now given by the least positive root of tannT= -w/k where n^ = m^- k^, the time from one position of rest to the next being tt/w, Art. 434. 498. Epicycloids d'c. Supposing the curve to be rough, the resistance to be 2kv, the force central and equal to Xr, and the tautochronic period to be given, prove that the differential equation of the curve is p = ip, where i{l-\-m^l\) = l + p?, and X is positive when the force is repulsive. The constant m is a function of the period whose value is given in Art. 497; when the resistance is zero the tauto- chronic period is 7r/2m. Trace also the curves included in this equation. In this case G= -\dpjd\J/, H=\p; see the figure at the beginning of Art. 495. Since k' = 0, the condition of tautochronism takes the simpler form m^p = dPjd-^ - /xP. Substituting for P its value G - /xH, we arrive at the given result. To trace the curves p = ip, we notice that- = l-^ — -ttito in the epicycloid in which a and h are the radii of the fixed and rolling circles respectively and that in. the hypocycloid h is negative. 416 ON SOME SPECIAL PROBLEMS. [CHAP. X. If we sketch the curve whose ordinate is i and abscissa b we see at once that there are two asymptotes defined by / = 1 and b= - ^a. As b increases from - 00 to -a, i varies from 1 to ; as & increases from - a to 0, i is negative being CO when b= -^a; as b increases from to oo , i varies from to 1. Thus i may have any negative value and any positive value less than unity. (1) If i has any negative value, b is negative and lies between - a and 0. Since i (1 + wi-'/X) = l + ij.^it follows that m^j\ should lie between - oo and - 1. The curve is therefore a hypocycloid and the central force is attractive. (2) If i is positive and less than unity, b may have any positive value or any negative value between - oo and - a. This requires that m^/X should lie between p.- and 00 . The curve is therefore either an epicycloid or a hypocycloid and the central force is repulsive. (3) If i is positive and greater than unity, the curve takes other forms. Putting i-l = a^, its differential equation becomes--^ =a2p. By rotating the axis of X round the origin through an appropriate angle, the integral may be reduced to one of the forms Since in any curve the projection of the radius vector on the tangent is equal to dpldxp, we have r2 =2)2 + {dpld^l/y\ cot iyp-e) = dpjpdxp. "We can therefore express the polar coordinates r, 6 in terms of \f/ as an auxiliary angle. Tracing the curves we find two kinds of spirals according as we take the upper or lower signs, together with an equiangular spiral whose angle /3 is given by sin2^ = l//. Since the two kinds of spirals do not pass through the origin (for this would require both p = and dpjd\p = 0), the point of equilibrium at which the tautochro- nous motion is to terminate is found by making tan = 1/yu where (p is the acute angle which the radius vector makes with the tangent. In the equiangular spiral the point of equilibrium is the origin for the central force vanishes at that point. In the first kind of spiral the angle 0, i.e. yjy~d, varies from ^tt when ^ = to tan~i 1/a when \f/ is infinite, and in the second kind

a; the arc to be described being on that side of the position of equilibrium in which tan0a or lx 1 we must have in the former case [x'^>vi^j\. We have therefore the following cases, (1) force attractive; if m'^j\< -1 the curve is a hypocycloid, if m^/X > - 1 but < the curve is the first spiral or the equiangular spiral according to the position of the point at which the motion is to terminate; (2) force repulsive, i.e. 7n'^j\>0, the curve is an epi- or hypocycloid if m^/X lies between fi^ and oo , and is the second spiral if m^/X lies between and jjr. 499. Ex. 1. A system having one degree of freedom is defined by 2T=3P6"^, U=f{d). Prove that the motion is tautochronous if U=C {jMdd}^. [Put Mdd = ds, and use Art. 496.] [Appell. ART. 501.] OSCILLATIONS TO THE SECOND ORDER. 417 Ex. 2. A system having two degrees of freedom is defined by where A, B, C are given functions of 6, 0. Investigate the constraint which must be introduced into the system that the motion may be tautochronous. [Assume

. Substituting dr for -r- from the subsidiary equations of Art. 501, the equation of motion is therefore dd {k^ + r^) 6 + rz sin n^^ = gr sin 0. The method of proceeding is the same as that in Art. 502. We expand each coefficient by Taylor's theorem in powers of 6, which is to be so chosen as to vanish in the position of equilibrium. To do this we require the successive differentials of these coefficients to any order expressed in terms of the initial values only of (f>, n, and r. The first differentials are given in the subsidiary equations of Art. 501. To find the others we continually differentiate these subsidiary equations, until we have obtained as many differential coefficients as we require. 506. To form the equation to the first order. Let the initial or equilibrium values of n and r be a and h. The equation is therefore {h^ + k'^) e = gr sin = z sm n sin (p + rz cos I —j, by substituting from the subsidiary equations. This by reduction becomes — (r sin 0) =r cos - 2 cos (0 - n). 27—2 -izco,a-h)ge + gz^^zcoBaf^-^ + ~—j^--—^Y-^ 420 ON SOME SPECIAL PROBLEMS. [CHAP. X. In equilibrium G lies in the vertical through the point of contact, hence the initial value of

0 &c. ^„4i>0. Let also i/j ... Vn+i ^6 the distances of the corners of the simplex from a space i^„_i. The distance of a point from E^_^ is therefore 2/i^i+ ... +2/n+i^n+i ^^^ *^® moment of inertia is i'i^ = J(yili+ ... +y,i4-i^n+i)^^^' where dv is an element of volume. In consequence of the symmetry of this expression it is equal to a (i/i2 + . . . + 2/\+i) + 26 (i/i2/2 +...+yrys+...+ yr^n+i)> where a = j^-i^dv, b = j^i^2^v. But with rectangular coordinates Xi^...x^, dv = dx^...dx^, and because each a; is a linear function of n of the variables li...|n+i, say ?i...|„, we have a=Cjj...^,H^,...dU, b=Cjj...^,^^d^,...d^^, where C depends solely on the relations between the two systems of coordinates X and ^ and the integrals are to be taken for all positive values of the variables ^i...^n s"ch that ?i+...+^„ ^ ~ — ~ \^)y A (m) A (m") - A (m'") \{F-P) _ m' {B - P) _ - A (m) ~ A (m') "" m{D-P) _m'{G-P) m" {C - P) m{F-P) _ m' [B - P) _ _ m'"{C-P) , -A(m) ~ A(m') ~ ~ A (m'") ^ '' A(m) -A(m') A (m") ^^'' As the two resolutions for the four particles taken as one system and the equation of moments give identities, these eight equations are equivalent to five independent conditions. From these we may deduce the ratios of the four masses when the form of the quadrilateral is given. They also determine P and give a relation between the sides of the quadrilateral, which must exist if the motion is possible. Eliminating the ratios of the masses we find BD-FG _ AC-FG _ BD-AC ~B + D-F-G~A + C-F-G~B+D-A-G ^ '' Any two of these values of P give FG{A + G-B-D) + {F+G){BD-AG) + AG{B + D)-BD{A + q = (6). This condition may also be written {B - F) {A - G) {G - D)=={G - F) {D - G) {B - A), or in another form obtained by interchanging B and D. 428 NOTES. Besides (6) we have the known geometrical relation which must exist between the lengths of the sides and diagonals of any quadrilateral. By adding or subtracting the equations (1) to (4) taken two together we eliminate P and deduce the ratios of the masses. We find, for the ratios of adjacent masses, m{D-F) _ m' {B~G) vi' {A - G) _ m" {G - F) m'" [G - G) _ m {A -F) A{m) ~ A{m') ' A (m') ~ A (m") ' °'' A{m'") ~ A (m) '"^'' and for the ratios of opposite masses m{A-D) _ m" {B - G) m' {A - B) _ m'" {D - G ) A(m) ~ A(m") ' A (m') ~ A (m'") ^'' If the quadrilateral is such that each of the four particles is outside the triangle formed by the other three, the areas A(m), A{m') &c. in the equations (1) to (4) are all positive. We then see by glancing at these equations that, if the masses are positive, the numerical value of P {i.e. ifij'Lm) must separate those of F, G from A,B, G, D. Since both diagonals cannot be less than every side, it follows that, if the law of attraction is an inverse power, each of the quantities A, B, G, D must be greater than both F and G. It also follows immediately from (8) that the greatest and least sides of the quadrilateral are opposite to each other and that each diagonal is longer than any side. For example, it is evident that the particles could not lie at the corners of a parallelogram unless the four sides are equal and each angle greater than 60°. Also by equations (1) to (4) the masses at opposite corners are equal. The results (6) and (7) were given as an example in the text of the third edition (1877) of this treatise (Art. 282, Ex. 2), but were omitted in the sixth edition to make room for examples then considered to be more interesting. They were obtained by reducing one of the four particles to rest. There are two memoirs dated 1895, 1897 by A. G. Wythoff on the dynamical stability of a system of four mutually attracting particles. For the results we refer the reader to the Nieuw Archief Voor Wiskunde. A reference is made to a paper by C. Krediet in the same Journal, where several interesting propositions on the equilibrium of the four attracting particles are arrived at. This paper the author of this book has not seen. In volume XXXV. of the Quarterly Journal of Mathematics (Oct. 1903) there is a memoir by Prof. E. 0. Lovett on the positions and small oscillations of an infinitesimal satellite acted on by three masses which move in steady motion with special reference to the case in which the three masses are equal. The Phil. Mag. for March 1904 contains an investigation by Prof. J. J. Thomson of the stability and periods of oscillation of a number of corpuscles arranged at equal intervals around the circumference of a circle, with an application to the structure of the Atom. If the mutually attracting particles start from rest, as described in Art, 285, the sum of the resolved parts of the momenta m-^^v^, m^v^, &c. in any direction, and the sum of the moments of the momenta about any straight line are zero. Since these are the necessary and sufficient conditions of equilibrium of a system of forces, we may apply to the system of moving particles any of the theorems proved in "Statics" for systems of four, five, &c. forces in equilibrium. Thus if four particles start from rest the invariant of the momenta of any two is equal to that of the other two, and therefore by a known theorem the ratios of the four momenta PROOF OF LAGRANGE'S EQUATIONS. 429 particles meet, the invariant of the other two vanishes and therefore the lines of motion of wzg, m^ must intersect or be parallel. If five particles start from rest their five lines of motion can at every instant be cut by two straight lines. If there are six particles their lines of motion are in involution. In this way we may obtain numerous curious and interesting, though nnf very useful, theorems. Art. 399. The Proof of I^agrange's Equations. The proof of Lagrange's equations in Arts. 397 to 399 a may be arranged somewhat differently by usmg as a lemma an extension of the theorem given in the first example of the last article named. Lemma. Let L be any function of the variables x, y, &c., x', y', &c. and t. If we express x, y, &g., as functions of any independent variables 6, , &g.) (1), with similar expressions for y, z, &c. .: x'=ff+fQd' + &c (2), where the &c. refers to the other variables v^ 14 DAY USE RETURN TO DESK FROM WHICH BORROWED ASTRONOMY, MATHEMATICS- This book is dtfJA'tJreUlltlai'JfJf^cI below, or on the date to which renewed. Renewed books are subject to immediate recall. -UEC — s^rasB — TJEfr a 7 t9fi6 cj ,\. , ; M'o^iBmQ LD 21-40m-5,'6r, (F4308sl0)476 General Library University of California Berkeley YC 102356 U.C.BERKELEY LIBRARIES Wp. CLh^(^i 0%"^ ^ ASTRONOMY UBRARf miX.^^;