(Qocnell itmueraity Uibraty atliara, ^tta fork ^WIImSi&iiBSM..? '^^'se on the olin,anx ,3 1924 031 364 403 Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031364403 THE ELEMENTARY PART OF A TREATISE ON THE DYNAMICS OF A SYSTEM OF RIGID BODIES BEING PAET I. OF A TREATISE ON THE WHOLE SUBJECT. •»- THE ELEMENTARY PART OF A TREATISE ON THE DYNAMICS OF A SYSTEM OF KIOID BODIES. BEING PART I. OF A TREATISE ON THE WHOLE SUBJECT. Iti^ tmmixam ^^ampfl^s. EDWARD JOHN ROUTH, Sc.D., LL.D., F.R.S., &c. HON. TELLoVor PETEEHOUSE, CAMBRIDGE; FELLOW OP THE SENATE OW THE UNIVEKSITT OF LONDON. FIFTH EDITION, REVISED AND ENLARGED. UonDon: MAOMILLAN AND CO. AND NEW YOEK. 1891 [All Rights resened.] First Edition, i860. Second Edition, 1868. Third Edition, 1877. Fourth Edition, 1882. Fifth Edition, 1891. PREFACE. In this edition many improvements have been made. Though some new matter has been added, most of the changes are in the form of additional explanations and generalizations of theorems already given. In some cases also the proofs have been simplified. The numbering of the articles is the same as in the last edition, except in some cases where the additions were such as to require a rearrangement. When the last edition was printed, the book was divided into two parts in order to render it less bulky. This division has been retained. All the elements of the subject, together with some methods intended for the more advanced student, are pla'ced in the first volume. In the second part the higher applications are given. In order that the plan of the book may be understood, a short summary of the subjects treated of in the second volume has been added to the table of contents. As in the former editions, each chapter has been made as far as possible complete in itself, so that all that relates to any one part of the subject may be found in the same place. 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. The student who is just beginning dynamics may not wish to be delayed by a long chapter of preliminary analysis before he enters on the real subject of the book. He may therefore begin with DAlembert's Principle and read only those parts of chapter I. to which reference is made. Others may wish to pass on as VI PREFACE. soon as possible to the great 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. It will be observed that 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. A slight historical notice of each result has been attempted whenever it could be briefly given. Such additions, if not .carried too far, add greatly to the interest of the subject. But the suc- cess of the attempt is far from complete. In the earlier history there was the guidance of Montucla, and further on there was Prof. Cayley's Report to the British Association. With the help of these the task became comparatively easy; but in some other pdrtions the number of memoirs which have been written is so vast that anything but a slight notice is impossible. A useful theorem is many times discovered, and probably each time with some variations. It is thus often difficult to ascertain who is the first 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. It has not however been thought necessary to refer to the author's own additions to the subject except when they had already been printed elsewhere. 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 further 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, I hope that this selection will enable the student to acquire facility in solving all kinds of dynamical problems. PREFACE. Vll In constructing the examples my first care has been to follow closely the principle which each is intended to illustrate. But such instruments or applications of principles have been sought for as have been found useful in practice. Whenever some useful instrument has been found, which did not require so lengthy a description as to unfit it for an illustration, it has been preferred as an example to a merely curious and artificial construction. In the former editions differential coefficients with regard to the time have been represented by accents in the chapters after the seventh. However unsuitable such a notation may be when several independent variables are used in the same investigation, it has some advantages in such a subject as dynamics, where the differentiations are nearly always taken with regard to the time. It was not used in the earlier chapters because it was thought that it would add to the initial difficulties of the subject those of an unaccustomed notation. But now that the representation of differential coefficients by dots has been used in several standard books both on elementary and on advanced mechanics, this reason has lost much of its force. Dots and accents have therefore been used throughout this edition whenever a shortened notation has appeared to be desirable. One objection to the use of this nota- tion is that the meaning of the symbol may be 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 work- ing of examples, and care has been taken that the results should be clearly stated. I cannot conclude without expressing how much I am indebted to Mr J. M. Dodds of Peterhouse for his assistance in correcting the proof sheets. I hope that the work, having had the advantage of his revision, will be found clear of serious errors. EDWARD J. EOUTH. Peterhodse, December 8, 1890, CONTENTS. CHAPTER I. ON MOMENTS OP INERTIA. ARTS. PAGES 1 — 2. On finding Moments of Inertia by integration ... 1 3 — 9; Definitions, elementary propositions and reference table . 2 — 9 10—11. Method of Differentiation ... ... 9—10 12—14. Theorem of Parallel Axes 10—13 15 — 17. Theorem of the Six Constants of a Body . . . 13—16 18. Method of Transformation of Axes 16 — 17 19 — 32. Ellipsoids of Inertia, Invariants, &c. . . . 17 — 23 33 — 39. Equimomental Bodies, Triangle, Tetrahedron, &c. . . 24 — 28 40—44. Theory of Projections 29—31 45. Moments with higher powers . . . . 31 — 32 46. Theory of Inversion 32—33 47. Centre of Pressure, &c 33—35 48—51. Principal Axes 35—37 52—55. Foci of Inertia 38—40 56 — 59. Arrangement of Principal Axes 40 — 43 60 — 61. Condition that a Line should be a Principal Axis . . 43 — 45 62 — 65. Locus of Equal Moments, Equimomental Surface, &o. . 45 — 49 CHAPTER II. d'albmbert's peincipli?, &c. 66 — 78. D'Alembert's Principle and the Equations of Motion . 79 — 82. Independence of Translation and Rotation 83. General method of using D'Alembert's Principle 84 — 87. Impulsive Forces , . 50—61 61—63 63—65 65—69 CONTENTS. CHAPTER III. AETS. 88—91. 92—93. 94—96. 97. 98—105. 106—107. 108. 109. 110—113. 114. 115-116. 117—119. 120. 121—125. 126—129. MOTION ABOUT A FIXED AXIS. FAQES The Fundamental Theorem ... • • 70—72 The Pendulum and the Centre of Oscillation . . • 72 — 76 Effects of a change of temperature and of the buoyancy of the air . . 76-79 Moments of Inertia found by experiment .... 79 — 80 Length of the seconds pendulum with correction for resistance of the air 80—84 Construction of a Pendulum 84 — 85 The Pendulum as a Standard of Length .... 86 — 87 Oscillation of a watch balance 87 — 89 Pressures on the fixed Axis. Bodies symmetrical and not symmetrical 90 — 95 Analysis of results 95 — 97 Dynamical and Geometrical Similarity .... 97 — 98 Permanent Axes of Eotation, Initial Axes . . . 98 — 100 The Centre of Percussion 100—102 The Ballistic Pendulum 102—107 The Anemometer 107—109 CHAPTER IV. MOTION IN TWO DIMENSIONS. 130 — 133. General methods of forming the Equations of Motion 134. Angular Momentum .... 185—137. Method of Solution by Differentiation 138—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 rod 153 — 157. Laws of Friction ] 58 — 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. Bigidity of cords 168—169. Impulsive Forces, General Principles . 170 — 175. Examples of sudden changes of motion, reel, sphere, disc, column, &o. Earthquakes 176 — 178. Impact of Compound Inelastic Bodies, &c 179 — 180. Impact of Smooth Elastic Bodies 181 — 198. The general problem of the Impact of two Bodies, smooth or rough, elastic or inelastic. The representative point . 110—113 113—114 114—117 117—122 122—181 131—133 183—137 137—139 140—141 141—143 143 144—146 146—147 147—148 148—152 152—156 156—158 158—173 CONTENTS. ABTS. 199—202. Initial Motions .... 203 — 213. Eelative Motion and Moving Axes Examples PAOES 173—178 178—186 187—191 CHAPTER V. MOTION IN THREE DIMENSIONS. 214 — 228. Translation and Rotation. Base Point, Central Axis . 192 — 198 229—284. Composition of Rotations, &o 198—202 235—237. Analogy to Statics . . 202—204 238—239. The Velocity of any poiuf ... ... 204—206 240—247. Composition of Screws, &c 206—212 248—259. Euler's Equations .... ^ . . . 212—219 260. The Centrifugal Forces of a Body 219—221 261—267. Angular Momentum with Fixed or Moving Axes . 221 — 227 268—270. Examples of Top and Sphere 228—232 271 — 281. Finite Rotations. Theorems of Bodrigues and Sylvester. Screws, &c 233—238 CHAPTER VI. ON MOMENTUM. 282—287. The Fundamental Theorem, with examples . . 289—244 288—298. Sudden fixtures and changes 244—250 299—300. Gradual changes 251—254 301—305. The Invariable Plane 254—259 306 — 314. Impulsive forces in three dimensions . . . 259 — 263 315 — 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 .... 264 — 274 Examples 274—276 CHAPTER VII. VIS VIVA. 332—341. Force, Function and Work . 342. Work done by Gravity and Units of Work 843. Work of an Elastic String . 344. Work of Collecting a Body . 345. Work of a Gaseous Pressure 346. Work of an Impulse . 347. Work of a Membrane . 277—282 282 282—283 283—286 286 286—288 288 Xll COM TENTS. ARTS. 348—349. Work of Bending a Rod 350—362. Principle of Vis Viva, Potential and Kinetic Energy 363 — 364. Expressiona for Vis Viva of a Body 365—366. Examples on Vis Viva 367—372. Principle of Similitude. Models 373—374. Theory of Dimensions 375—376. Clausius' theory of stationary motion. The Virial 377—381. Carnot's theorems 382—386. The equation of Virtual Work applied to Impulses 387—388. Thomson's theorem. Bertrand'a theorem . 389. Imperfectly elastic and rough bodies . 390—394. Gauss' principle of Least Constraint . Examples PAGES 288—289 289—295 295—298 298—300 300—304 304 304—306 307—308 308—310 310—311 311—313 314—317 317—320 CHAPTER VIII. LAGEANGES EQUATIONS 395—399. Typical Equation for Finite Forces . 400. Indeterminate Multipliers . 401 — 404. Lagrange's equations for Impulsive Forces 405 — 408. Examples on Lagrange's equations 409—413. The Reciprocal Function . 414—417. Hamilton's equations .... 418—421, The modified Lagrangian Function. Its use in forming Lagrange's and Hamilton's equations 422 — 425. Co-ordinates which appear only as velocities 426 — 428. Non-conservative Forces .... 429 — 431. Geometrical equations which contain differential coefficients with regard to the time . Examples 321—325 325-326 326—328 328—331 331—334 334—337 337—339 339—342 342—344 344—347 347—348 CHAPTER IX. SMALL OSCILLATIONS. 432 — 438. Oscillations with one degree of freedom .... 349 — 353 439 — 440. Moments about the Instantaneous Axis .... 353 — 354 441 — 444. Oscillations of Cylinders, with the use of the circle of stability 355 — 358 445. Oscillations of a body guided by two curves . . . 358 — 359 446. Oscillation when the path of the Centre of Gravity is Ivuown 359 — 360 447. Oscillations deduced from Vis Viva 360 — 361 448. Moments about the Central Axis 361 — 362 449 — 452. Oscillations deduced from the ordinary equations of motion 362 — 366 453—462. Lagrange's method 366—377 463—466. Initial motions 377—379 467—469. The energy test of stability 379—381 470—476. The Cavendish Experiment 382 388 Examples 388—390 CONTENTS. CHAPTER X. ON SOME SPECIAL PK0BLEM8. ARTS. 477. Osoillatious of a rooldng body in three dimensions 478^479. Eelative indioatrix 480 — 482. Cylinder of stability and the time of oscillation . 483 — 487. Osoillatious of rough cones rolling on each other to the first order of small quantities 488 — 492. Lagrange's formula for large Tautoohronous motions . 493 — 494. Large oscillations of a particle on a rough cycloid in resisting medium 495 — 498. Large Tautochronous motions on any rough curve, with applications to epicycloid, &c. .... 499. Effect of a resisting medium on the time of oscillation 500 — 507. Conditions of stability and times of oscillations of rough cylinders to any order of small quantities 503 — 510. Conditions of stability and times of oscillation of rough cones to any order ....... PAGES 391 391—392 392—394 394—398 398—401 401—403 403—406 406 406—410 410—412 The following subjects will be treated of in the second volume. Theory of moving axes, Clairaut's theorem, and motion relative to the earth. 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. Theory of free and forced oscillations. Methods of Isolation and of Multipliers. Applications of the calculus of finite differences. Applications of the calculus of variations. Precession and Nutation. Motion of a string or chain. Motion of a membrane. 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—175, 179—186, 199. Chap. V. 214—245, 248—256, 261—269. Chap. VI. 282—285, 288—295, 299—309. Chap. Vn. 332-374. Chap. VIII. 395—409. Chap. IX. 432—463, 467—476. Chap. X. 483, 488—499. EERATUM. Page 190, line 25, /or (^-««) read (y -«»)'• CHAPTEE I. ON FINDING MOMENTS OF INERTIA BY INTEGRATION. 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 jjjoc^y^iP' dx dy dz, where (a, /S, 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 co-ordinate 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 Liouville'g 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 integiul which are useful in dynamics. R. D. 1 2 MOMENTS OF INERTIA. [CHAP. I. 3. Definitions. If the mass of every particle of a material system be 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 be 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 Osc, 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 •prodvuct 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 co-ordinate planes xz, yz. The term moment of inertia with regard to a plane seems to have been first used by M. Biuet in the Journal Folytechnique, 1813. 4. Let a body be referred to any rectangular axes Ox, Oy, 0« meeting in a point 0, and let «, y, z be the co-ordinates of any particle m, then according to these definitions the moments of inertia about the axes of x, y, z respectively will be A='2m (y^ + z% B = tm {f -I- a^), C = Sm (a;^ -I- y^). The moments of inertia with regard to the planes yz, zx, xy, respectively, will be A' = tma?, F = ^my\ C = %mz\ The products of inertia with regard to the axes yz, zx, xy, will be B = tmyz, E = tmzx, F — %mxy. Lastly, the moment of inertia with regard to the origin will be H='Zm (a^ + y + z^) = tmr\ 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 illustrations of the preceding definitions. ART. 6.] BY INTEGRATION. 3 (1) The three moments of inertia A, B, about three rectangular axes are such that the sum of any two of them is greater than the third. (2) The sum of the moments of inertia about any three rectangulai- 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 /( + B + C = 2Sm(a;2 + 2/8 + «2) = 22mr2,' 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=Sm7-^, which is independent of the directions of the axes. Hence we infer that A' = ^{B + G-A), B'=h{G + A-B). and G'=^(A+B-G). (4) Any product of inertia as D cannot numerically be so great as ^A. (5) If A, B, F be 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 = 'Z,m.('yt + x)'^=3, positive quantity. Hence the roots of the quadratic At^ + 2Ft + B = Q 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 a diameter is Jf |a^ This follows immediately from (7) by completing the sphere. 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 1—2 MOMENTS OF INERTIA. [chap. I. BG to meet Ay in D. The given triangle ABC may be regarded as the difference of the triangles ABD, AGD. Let us then tarst find the moment of inertia of ABD. Let PQP'Q' be an ele- mentary area whose sides PQ, P'Q' are parallel to the base A J), and let PQ cut Ax in M. Let /3 be the distance of the angular point B from the axis Ay, AM=x and AD = I. B — X Then the elementary area PQP'Q' is clearly I „ fe, and 3 — X its moment of inertia about Ay is fil—^dx.x% where fi is the mass per unit of area. Hence the moment of inertia of the triangle ABD = ti^j{\-^a?dx^^pl^. Similarly if 7 be the distance of the angular point G from the axis Ay, the moment of inertia of the triangle AGD is -^fil/f. Hence the moment of inertia of the given triangle ABG is J^|U,Z (/3^ - 7^). Now J?/3 and IZ7 are the areas of the triangles ABD, AGD. Hence if ilf be the mass of the triangle ABG, the moment of inertia of the triangle about the axis Ay is iilf(;8^ + jS7 + 7^). Ex. If each element of the mass of the triangle be multiplied by the «th 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 j3"+i - 7"+! (n + l)(B + 2) |3-7 /■3 /? - i^ ART. 8.] BY INTEGRATION. 5 7. When the body is a lamina the moment of inertia about an axis perpendicular to its plane is equal to the sum of the moments of inertia about any two rectangular axes in its plane drawn from tlie 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 be the moments of inertia about the axes, we have, A = %my'\ B = Sma;^ G = Sm {x' + y"), and therefore G = A + B. We may apply this theorem to the case of the triangle. Let /8', 7', be the distances of the points B, G from the axis Ax. Then the moment of inertia of the triangle about a normal to the plane of the triangle through the point A is = ^ if (,8^ + ySry + 7^ + jS'^ + ySV + 7'')- Ex. Prove that tlie moment of inertia of the perimeter of a circle of radius a and mass M 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 Ma!'. 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 memory 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^ + b'' lar to its plane J " ^^^ ^3 • {2) An ellipse semi-axes a and b about the major axis a = mass ^ , about the minor axis b = mass -r , 4 about an axis perpendicular to its plane) _ al^ + b ,2 :mass through the centre [ " """"^ 4 In the particular case of a circle of radius a, the moment of inertia about a diameter = mass -j , and that about a perpen- a" dicular to its plane through the centre = mass -^ . 6 MOMENTS OF INERTIA (3) An ellipsoid semi-axes a, h, c [chap. about the axis a = mass b'- + c^ In the particular case of a sphere of radius a the moment of b' + c' inertia about a diameter = mass -z a\ (4) A right solid whose sides are 2a, 2b, 2c about an axis through its centre perpendicular) _ ^^^^^ to the plane containing the sides b and c j a 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 [ = mass semi-axes) of symmetry j 3, 4 or 5 The denominator is to be 3, 4 or 5, according as the body is rectangular, elliptical or ellipsoidal. Thus, if we require the moment of inertia of a circle of radius a about a diameter, we notice that the perpendicular semi-axes in its plane is the radius a, and that the semi-axis perpendicular to its a' plane is zero, the moment of inertia requu'ed is therefore M-r , 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 a' + a/ 4 =^i 9. As the process for determining these moments of inertia is very nearly the same for all these cases, it will be suiBcient to consider only two instances. To determine the mortient of inertia of an ellipse about the minor axis. Let the equation to the ellipse hey = - sja^ - x'^. Take any elementary area PQ parallel to the axis of y, then clearly the moment of inertia is ra. , ra 4jn I a:^(/dx = 4/i- I x^Ja^-x^dx, Jo a J Q where /j. is the mass of a unit of area. ira* 16 ' ART. 9.] BY INTEGRATION. To integrate this, put .E = a sin 0, then the integral becomes «■' r oos^sin^^ d(l> = a' P i -COB 40 ^^^ Jo J (1 S .: the moment of inertia = uvab -r = mass -r . i 4 In the same way we may show that the product of inertia of an elliptic quadrant about its axes = mass --— . 27r To determine the moment of inertia of an ellipsoid about a principal diameter. Let the equation to the ellipsoid be -^ + jj + -i=l. Take any elementary area PNQ parallel to the plane of yz. Its area is evidently tPN . QN. Now FN is the C value of z when y — 0, and QN the value of y when 2=0, as obtained from the equa- tion to the ellipsoid ; .■. PN=- Ja"^ - x^, QN= - J a'' - x- ; .-. the area of the element = —i- (a^-x^). a^ Let n be the mass of the unit of volume, then the whole moment of inertia f'Tbc,, ^Pm + QN\ = '^j_.^(''^^> 4 "^ i , V' + c^ b"- + c^ = u TT iraoc — -- — = mass — = — . 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 -^ . 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 (a) about an axis through its centre perpendicular to its plane = Ma'', (6) about an axis through its middle point perpendicular to its plane = 2Jl/(l-«'^'')a^ 8 MOMENTS OF INERTIA. [CHAP. I. (sin 2a\ a^ 1 g — I -n • 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 j/ is the ordinate corre- sponding to X. Ex. 3. The moment of inertia of the area of the lemniscate r^=a^ cos 2$ about Q_._i_ Q a line through the origin in its plane and perpendicular to its axis is M . a'. Ex. 4. A lamina is bounded by four rectangular hyperbolas, two of them have the axes of co-ordinates 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 co-ordinate axes is J (o^ - a'^) {^ - /S'^), where oa', ^/3' are the semi-major axes of the hyperbolas. Take the equations xy=u, x^-y^=v, then the two moments of inertia are A=ffxVdudv and B=ffyVdudv, where XjJ is the Jaoobian of (u, v) with regard to (x, y). This gives at onofi A + B = ^ffdudv, where the limits are clearly ii = \a? to Ja's, i;=/32 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 co-ordinate axes. Prove that the product of inertia about the co-ordinate axes is ^ — ^[^ ^/^ , where aa', jS/3' are the semi-major axes of the curves. Ex. 6. If da- be an element of the surface of a sphere referred to any rect- /' 47r a;2"d(7 =5 ^ ,.an-2^ where r is the radius of the sphere and n is integral. Ex. 7. Taking the same axes as in the last example, prove that fx^Y'z^da=^ r^n+.L{f)L(g)L{h) ■' '' 2n + l L{n) where n=f+g + h&ndi L {/) stands for the quotient of the product of all the natural numbers up to 2/ by the product of the same numbers up to /, both included. To prove this, we notice tha-t by the last example we have f(\x + ii.y + vsf'^da= (X2 -H ;«2 + „2 m ^f!^ _ '' 2)H-1 Expand both sides and equate the coefficients of X^'ix^v^'. If we multiply the result by Dd/r we have the value of the integral for any homogeneous shell of density D and thickness dr. Regarding D as a function of r, and integrating with regard to r, we can find the value of the integral for any heterogeneous sphere in which the strata of equal density are concentric spheres. Ex. 8. If du be an element of the smfaoe of an ellipsoid referred to its principal diameters, and if jp be the perpendicular from the centre on the tangent plane, prove where a, b, c are the semi-axes and the rest of the notation is the same as before. ART. 11.] BY INTEGRATION. 9 This result follows at once from the corresponding one for a spherical shell by the method of projcctioiu, Ex. 9. Show that the volume V, the surface S, and the moment of inertia I with regard to the plane perpendicular to the co-ordinate Xj, of the sphere in space of n dimensions, whose equation is x.i^ + x^+ ...+x^' = r', are given by ™ , r2 These results follow easily from Diriohlet's theorem. See also Art. 5 (2). 10. Method of Differentiation. Many moments of inertia may be deduced from those given in Art. 8 by the method of differen- tiation. Thus the moment of inertia of a solid ellipsoid of uniform 4 h^ 4- c^ density p about the axis of a is known to be -^ Trabcp — z — . Let the ellipsoid increase indefinitely little in size, then the moment of inertia of the enclosed shell is a -{^ Trabcp — = — This differentiation can be effected as soon as the law according to which the ellipsoid alters is given. Suppose the bounding ellipsoids to be similar, and let the ratio of the axes in each be given by b =pa, c = qa. Then 4 r^ + Q^ moment of inertia of solid ellipsoid = ^ irppq ^ ^ a°; o o 4 .'. moment of inertia of shell = ^ irppq (p^ + q^) a*da. o 4t In the same way the mass of solid ellipsoid = 5 irppqa?; o .•. mass of shell = iTrppqa^da. Hence the moment of inertia of an indefinitely thin ellipsoidal jj)3 _|. ff shell of mass M bounded by similar ellipsoids is M — ^ — • o 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. 11. The moments of inertia of a heterogeneous body whose boundary is a surface of uniform density may sometimes be found by 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 ahe ^ (a) D. Then the moment of inertia of a stratum of density D will be ^'{a)Dda. Replacing D by the variable density p, the momeut of inertia required will be Jp'(a)da. 10 MOMENTS OF INERTIA. [CHAP. I. 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 ^M{b^ + €'■). 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 minor ^ . 3M ia^ + c^-5a^e^ axis varymg as the distance from the centre, is -^^ „ , + c^- 3ac^ ' Other methods of finding moments of inertia. 12. 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 integration. 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 axis 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 moments and products about all other parallel axes. The moment of inertia of a body or 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 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 ART. 14.J OTHEK METHODS. 11 the body, which generally will be any small element. Let *■, ij, z be the co-ordinates of m, ic, y, z those of the centre of gravity Q of the whole system of bodies, x', y' , / those of m referred to a system of parallel axes through the centre of gravity. Then since -= — , .^ '^ , -^^ — are the co-ordinates of the zwi im zm centre of gravity of the system referred to the centre of gravity as the origin, it follows that 2ma.' = 0, 'Zmy' = 0, 2m/ = 0. The moment of inertia of the system about the axis of z is = Sm (af + y-), = -2.m{(x + xy + (y + yy}, = Sm (^■' + f) + Xm {x^ + y"') + 2x . Xmx + 2y . Xmy'. Now Sm(oS' +y'') is the moment of inertia of a mass l-in collected at the centre of gravity, and X'm(x'^ +y'0 is the moment of inertia of the system about an axis through G, also Xmx' = 0, Smy' = ; whence the proposition is proved. Secondly, take the axes of x, y as the axes about which the product of inertia is required. The product required is = Sm xy = Sm (x + x) {y + y), = ocy . Sm + Xmx'y' + x^my' + y%mx' = xyZm + "Zmx'y'. Now ayy . Sm is the product of inertia of a mass Sm collected at G and 1,ma/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] „ ^ _ jmoment of inertia ^.^ about A ) \ about B Hence when the moment of inertia of a body about one axis 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. 14. The preceding proposition may be generalized as follows. Let any system be in motion, and let x, y, z be the co-ordinates Sj3j dii dz at time t of any particle of mass m, then -jr, -^, -rr are the dJ^x dHi d^z velocities, and ,— , -^ , j— the accelerations of the particle 12 MOMENTS OF INERTIA. [CHAP. I. resolved parallel to the axes. Suppose „ „ J / da; d?x dy d?y dz d^ to be a given function depending' on the structure and motion of the system, the summation extending throughout the system. Also let ^ be an algebraic function of the first or second order. Thus (j) may consist of such terms as aa^ + bx-^ + c (■£) + eyz +fx + where a, b, c, &c. are some constants. Then the following general principle will hold. The value of V for any system of co-ordinates is equal to the value of V obtained for a parallel system of co-ordinates with the centre of gravity for origin plus the value of \ for the whole mass collected at the centre of gravity with reference to the first system of co-ordinates. For let X, y, z be the co-ordinates of the centre of gravity, 1 , ^ _ ,0 dx dx dx' andleta; = a, + .;,&c. .-. ^^ =^ +^ , &c. Now since (/> is an algebraic function of the second order of X, -J- , -jj ; y, &c. it is evident that on making the above sub- stitution and expanding, the process of squaring &c. will lead to fine n^ir three sets of terms, those containing only x, -y- , -^ , &c., those containing the products of x, x' &c., and lastly those containing only x', T—, &c. The first of these will on the whole make up (x',-=~ , &c. ) . Hence V= "Zmtj) (x, -j- ...] + 2m(^ (x', -rr + •••) where A, B, G, &c. are some constants. dx'\ . ^. _^ dx' Now the term Sm (*-7r) is the same as xXm-^-, and this vanishes. For since Sm«' = 0, it follows that 2m -^ = 0. Simi- an larly all the other terms in the second line vanish. ART. 15.] OTHER METHODS. 13 Hence the value of V is reduced to two terms. But the first of these is the value of V for the whole mass collected at the centre of gravity, and the second of these the value of V for the whole system referred to the centre of gravity as origin. Hence the proposition is proved. The proposition would obviously be true if -^, -^-^ , -j-^, or any higher differential coefficients were also present in the function V. 15. Theorem of the six constants of a body. Oiven the moments and products of inertia about three straight lines at right angles meeting in a point, to deduce the moments and products of inertia about all other axes meeting in that point. Take these three straight lines as the axes of co-ordinates. Let A, B, G be the moments of inertia about the axes of «, y, z\ D, E, F the products of inertia about the axes of yz, zx, xy. Let a, /S, 7 be the direction-cosines of any straight line through the origin, then the moment of inertia I of the body about that line will be given by the equation I = Aa? + B^+ C7^ - 2D/37 - 2£'7a - 2i^a/3. Let P be any point of the body at which a mass m is situated, and let x, y, z be the co-ordinates of P. Let ON be the line whose direction-cosines are a, /3, 7, draw FN perpendicular to ON. Since ON is the projection of OF, it is clearly = xa. + y/3+ zy, also OP^^a^+y^+z'', and l=a?+ ^^ + y''. The moment of inertia / about ON = limFN" = tm {x" + y' + z'- (ax + ^y + yzf} = tm[(ce' + y'' + z") {0? + ^^ + 7^) -{ax+8y + yz^] = 'Zm(y'' + z') 0? -I- Sm (z'' -f x"")^ -I- 2m («'■' + f) y^ — 2,%myz . ^y — 2%mzx . ya — 2l,mxy . a^ = Aa? + B/3' + CY- 2D/S7 - 2Eyix - 2Fa^. 14 MOMENTS OF INERTIA. [CHAP. I. It may be shown in exactly the same manner that if A'B'C be the moments of inertia with regard to the planes yz, zx,_ xy, then the moment of inertia with regard to the plane whose direc- tion-cosines are a, yS, 7 is r=A'a^ + 5'/3^ + C V + 2i>/37 + 2£'7a + 2^a/3. It should be remarked that this formula differs from that giving the moment about a straight line in the signs of the three last terms. 16. When three straight lines at right angles and meeting in a given point are such that if they be taken as axes of co-ordi- nates the products %mxy, Xmyz, tmzx all vanish, these are said to be Principal Axes at the given point. The three planes which pass each through two principal axes are called the Principal Plaiies at the given point. The moments of inertia about the principal axes at any point are called the Principal moments of inertia at that point. The fundamental formula in Art. 15 may be much simplified if the axes of co-ordinates can be chosen so as to be principal axes at the origin. In this case the expression takes the simple form I = Aa^ + B^^ + Cr^K A method will presently be given by which we can always find these axes, but in some simpler cases we may determine their position by inspection. Let the body be symmetrical about the plane of xy. Then for every element m on one side of the plane whose co-ordinates are {x, y, z) there is another element of equal mass on the other side whose co-ordinates are {x, y, — z). Hence for such a body Xmxz = and l,myz = 0. If the body be a lamina in the plane of xy, then the z of every element is zero, and we have again Xmxz = 0, 'Zmyz = 0. Recurring to the table in Art. 8, we see that in every case the axes, about which the moments of inertia are given, are principal axes. Thus in the case of the ellipsoid, the three principal sections are all planes of symmetry, and therefore, by what has just been said, the principal diameters are principal axes of inertia. In applying the fundamental formula of Art. 15 to any body mentioned in the table, we may therefore always use the modified form given in this article. 17. Let us now consider how the two important propositions of Arts. 13 and 15 are to be applied in practice. Ex. 1. Suppose we want the moment of inertia of an elliptic area of mass M and semiaxes a and b about a diameter making an angle 6 with the major axis. The moments of inertia about the axes of a and b respectively are ^Mb^ and iMa^. Then by Art. 16 the moment of inertia about the diameter is iMb^eoa^e + lMa^sin^e. ART. 17.] OTHER METHODS. 15 If )• be the length of the diameter this is known from the equation to the ellipse to be the same as -j -^^ , -which is a very convenient form in practice. Ex. 2. Suppose we want the moment of inertia of the same ellipse about a tangent. Let p be the perpendicular from the centre on the tangent, then by Art. 13, the required moment is equal to the moment of inertia about a parallel axis through the centre together with Mp''=-^ —^ +Mp^ = -^ p', since pr=ab. Ex, 3. As an example of a different kind, let us find the moment of inertia of an ellipsoid of mass M and semiaxes {a, 6, c) with regard to a diametral plane whose direction-cosines referred to the principal planes are (a, j3, 7). By Art. 8, the moments of inertia with regard to the principal axes are Jil/(b^ + c^), iM{c^ + a^, lM{a^ + b^. Hence by Art. 5, the moments of inertia with regard to the principal planes axe \Ma^, iMb\ \McK Hence the required moment of inertia is iM{aV + b^fi^ + c^y\ If p be the perpendicular on the parallel tangent plane, we know by solid geometry that {his is the same as il/-^ . 5 Ex. 4. The moment of inertia of a rectangle whose sides are 2a, 26 about a , . 2M a%^ diagonalis -g- ^^^,. Ex. 5. If ftj, ^2 be the radii of gyration of an elliptic lamina about two con- jugate diameters, then o+tl=^(~2+72)- Ex. 6. The sum of the moments of inertia of an elliptic area about any two tangents at right angles is always the same. Ex. 7. If M be the mass of a right cone, a its altitude and 6 the radius of the 3 base, then the moment of inertia about the axis ia Mj^b"; that about a straight line through the vertex perpendicular to the axis iaM-={a' + jb^\, that about a slant 36^ g^2 I ^2 side M —^ „ ;,- ; that about a perpendicular to the axis through the centre of 20 a" + b^ 3 gravity is M rjr (a" + 46^). Ex. 8. If a be the altitude of a right cylinder, 6 the radius of the base, then the moment of inertia about the axis is J Jfi^ and that about a straight line through the centre of gravity perpendicular to the axis is Jilf (Ja^ + ft^). Ex. 9. The moment of inertia of a body of mass M about a straight line whose equation is — ^-^ = - — - = referred to any rectangular axes meeting at the centre of gravity is AP+Bm'+Cn'-2Dnm-2Enl-2Flm + M{f' + g^ + K'-{fl+gm + hnf}, where {I, m, n) are the direction-cosines of the straight line. Ex. 10. The moment of inertia of an elliptic disc whose equation is ax'' + 2bxy + cif + 2dx + 2ey + l = 0, 1 V f X a /3 7 y a' /3' y z o" p" 7' 16 MOMENTS OF INERTIA. [OHAP. 1. about a diameter parallel to the axis of x,isj. ^^^^, ' where M is the mass and H is the determinant ac-b^+ 2bed - ae^ - cdP-, usually called the discriminant. Ex. 11. The moment of inertia of the elliptic disc whose equation in areal co-ordinates is ij> (xyz) = (i about a diameter parallel to the side a is ,^/A\2 H (d ay where A is the area, H the discriminant and K the bordered discriminant. 18. method of transfbrmation of axes. The method used in Art. 15 to find the moment of inertia about the straight line ON is really equivalent to a change of co-ordinate axes in which this straight line is taken as a new axis, say, of f, those of 1) and f not being required. We may now generalize this into a method which is often of great practical use. Let us suppose that (fTjf) is any quadrie function, say

(a" ^'y") Smz^. In using this formula, the coefficient of Stux^ is obtained by substituting for (|?jf ) in (|i;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 sub- stituting 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 co-ordinates also, this may be done by an application of the theorem in Art. 14. Ex. 1. The co-ordinates of the centre of an elliptic area are (fgh) and the direction-cosines of its axes are (0/87) {a'^'y'), prove that Smf 2 =M{h^ + ia^y^ + i 6 V') • Ex. 2. Let Ox, Oy, Oz be the principal axes at the origin, prove that the product of inertia F' = 7im,^ti about two rectangular axes Of, Oij whose directions are (aa'o") (|8/3'/3") is given by either of the formulae Smfi; =ap^mx^+a'p'Zw.y'> + a"p"'2,rrufl = ~a^A~ a'p'B - a"|8"C. The second result follows from the first since o^ + a'|8' + a"j3" = 0. AET. 19.] ELLIPSOIDS OF INERTIA. 17 Ex. 3. Let (tvV') be the direotion-oosines of a fixed axis Of. Then as 0|, Oti turn round Of, prove that D'^ + E''^ and A'B'-F" are both constant where A', B', C, D', E', F' are the moments and products of inertia of the body referred to these moving axes. PorbyEx. 2, -D'=APy + Bp'y'+Cp"y", - E' = Aay + Ba'y' + Ca"y" ; .: D" + E"=AY{a^ + P^) + ^AByy' (oa'+/3/3') + &Oi, since a^+p'=l-y''-y'^ + y"^ and aa' + ^/3' = - 77' we have D'" + E"^ =(A-Bf (yy'f + {B-G)^ {Yy")^ + {G-A)^ (7"7)2. Similarly A'B' ~ F'^=BCy''+CAy'^+ABy"^. The Ellipsoids of Inertia. 19. The expression which has been found in Art. 15 for the moment of inertia I about a straight line whose direction-cosines are (a, ^, 7), / = ^a^ + 5/8^ + Gy^ - 2D/S7 - 2Eja - 2Fa.^, 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- cosines are (a, /8, 7), we have / = -p^ , where e is some constant introduced to keep the dimensions correct, and M is the mass. Hence the polar equation to the locus of Q is ^=Aa? + B^' + Cy' - 2D/37 - 2Eya - 2Fa.^. Transforming to Cartesian co-ordinates, we have M^ = AX' + BY' + GZ' - WYZ - 2EZX - 2FXY, which is the equation to 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 inverselsquafe of^hat" fadius^ect oS The et)nvenience of TiMs 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, Exercises de Math. Vol. 11. B. D, 2 18 MOMENTS OF INEETIA. [CHAP. T. So much has been written on the ellipBoids 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 mertia about that radius vector. Hence whatever co-ordmate axes are taken, we must always arrive at the same ellipsoid. It theretore the momental ellipsoid be referred to any set of rectangular axes, the coefficients of Z^ F^ Z\ -2YZ, -2ZX -2XY m its equa. tion will still represent the moments and products ot mertia 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 + BC+CA-D'-E-'- F\ ABC - 2DEF - AD^ - BE^ - GF\ Hence for all rectangular axes having the same origin, these are invariable and all greater than zero. 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 similarly 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. 22. If principal axes at any point of a body be taken as axes of co-ordinates, the equation to the momental ellipsoid takes the simple form AX^ + BY'' + CZ" = M^, where M is the mass and 6* any constant. Let us now apply this to some simple cases. Ex. 1. To find the momental ellipsoid at the centre of a material elliptic disc. Taking the same notation as before, we have A=iMb\ B = JJIfa«, C= Jil/(a=-)-62). Hence the ellipsoid is i Mb'' X^ + i Ma' Y'' + iM{a^ + b^) Z^ = Me*. Since e is any constant, this may be written _X2 Y2 /I IN 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 of a material straight rod AB of mass M and length 2a. Let the straight line OAB be the axis of x, the ART. 23.] ELLIPSOIDS OF INERTIA. 19 origin, G the middle point of AB, OG = c. If the material line can be regarded as indefinitely thin, ^4 = 0, B = jV(Ja^ + c'') = C, hence the momental ellipsoid is Y^ + Z''=e'', where e' ia any constant. The momental ellipsoid is therefore an elongated spheroid, which becomes a right cylinder having the straight line for axis, when the rod becomes indefinitely thin. Ex. 3. The momental ellipsoid at the centre of a material ellipsoid is where e is any constant. It should be noticed that the longest and shortest axes of the momental ellipsoid coincide in direction with the longest and shortest axes respectively of the material ellipsoid. 23. Elementary Properties of Principal Axes. By a consideration of some simple properties of ellipsoids, the following propositions are evident : I. 0/ 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 versd. 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 are 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, "Zmyz = 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. 2-2 20 MOMENTS OF INEETIA. [CHAP. I. Ex. 1. Three equal particles A, B, C axe 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 OA, GB, GC 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 =• tetrahedron. If the momental ellipsoid at their centre of gravity is a sphere prove that the tetrahedron is regular. Ex. 3. Any point in a body being given and any plane drawn through it, prove that two straight lines at right angles can be drawn in this plane through O 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 three principal axes at right angles to each other. Construct the momental ellipsoid at the given point. Then it has been shown that the products of inertia about the axes are half the coefBcients oi —XT. — TZ, — ZX in the equation to the momental ellipsoid referred to these straight lines as axes of co-ordinates. 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 (^^ + q'' + r^^X''+(^^+r^+p^y^+(^^+p^ + q^'jZ^-2qrYZ-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 -^-2 -»"^ M rp pq ~ — r'-p^ qr If {I, m, n) be proportional to the direction- cosines of the axes corresponding to any one of the values of I, their values may be found from the equations {I-{A+Mq^ + Mr^)}l+Mpqm + Mrpn=0, ■> Mpql +{I~(B + Mr^ + Mp^) }m + Mqrn = 0, i Mrpl+Mqrm+{I-{G + Mp^ + Mq'i)}n=0. ) Ex. 3. If S=0 be the equation to the momental elUpsoid at the centre of gravity referred to any rectangular axes written in the form given in Art. 19, then the momental ellipsoid at the point P whose co-ordinates are {p, q, r) is S + M {p^ + q^ + r') {X'' + Y^+Z^)-M{pX+qY+rZf=0. = 0. ART. 28.] ELLIPSOIDS OF INERTIA. 21 Hence show (1) that the conjugate planes of the straight line OP in the momental ellipsoids at and P are parallel and (2) that the sections perpendicular to OP have their axes parallel. 26. Ellipsoid of Gyration. The reciprocal surface of the momental ellipsoid is another ellipsoid, which has also been em- ployed to represent, geometrically, the positions of the principal axes and the moment of inertia about any line. We shall require the following elementary proposition. The reciprocal surface of the ellipsoid -2 + ^ + -j=listhe eUipsoid a V + ft^j/^ + c V = c*. Let ON be the perpendicular from the origin on the tangent plane at any point P of the first ellipsoid, and let I, m, n be the direction-cosines of ON, then ON^=:a^P + b'V + c^nK Produce ON to Q so that OQ^^e^jON, then Q is a point on the reciprocal surface. Let OQ = R; .: e*=t,aH^ + ljhii^ + cV) E^. Changing this to rectangular co-ordinates, we get e* = a^x^ + li^y^ + c^z'. 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 eUipsoid of gyration which maJces the moment of inertia about a perpendicular on a tangent plane equal to the product of the mass into the square of that perpendicular. If M be the mass of the body and A, B, G the principal moments, the equation to the ellipsoid of gyration is A'^ B '^ C~M- It is clear that the constant on the right-hand side must be 1/M, 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, G given in Art. 22, Ex. 1, we see that the A''' Y^ Z'^ 1 ellipsoid of gyration is -p -l- -^ H- ^q-^-, = j . 22 MOME]>fTS OF INERTIA. [CHAP. I. Ex. 2. The ellipsoid of gyration at any point of a material rod AB is ^ + _Z!- + ^' ,= 1, taking the same notation as in Art. 22, Ex. 2. It is thus ia' + c' ia' + c'' a very flat spheroid which, when the rod is indefinitely thin, becomes a circular area, whose centre is at 0, whose radius is ^K + c" and whose plane is perpendicular to the rod. Ex. 3. It may be shown that the general equation to the ellipsoid of gyration referred to any set of rectangular axes meeting at the given point of the body is A -F -E MX =0. -F B -D MY -E -D G MZ MX MY MZ M or, when expanded, {BG- D') X' + [GA - E^) Y^ + {AB - F^) Z^ + 2 {AD + EF) YZ + 2 {BE + FD) ZX + 2{GF+DE}XY=^{ABG-AD^-BE<'-CF'-WEF). The right-hand side, when multiplied by 1/, is the discriminant obtained by leaving out the last row and the last column, and the coefficients of X'\ Y', Z^, 2ZX, 2XY, 2 YZ are the minors of this discriminant. 29. The use of the ellipsoid whose equation referred to the principal axes at the centre of gravity is X^ 7" Z' 5 ^mai^ ^my^ 'S.mz^ M ' has been suggested by Legendre in his Fonctions Elliptiques. 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, j3, 7) is r = STOa;^ . a2 + 2my^ . ^2 + Smz^ . y^ + 2S>ret/2 . fiy + 22,mzx . ya + 2^mxy . a/3. Hence, as in Art. 19, we may construct the ellipsoid 2mx' . A'2 + Smi/^ .Y^ + Xmz' . Z^ + 2'Zmyz . YZ + 2J,mzx . ZX + 2Xmxy . X Y= Me*. Then the moment of inertia with regard to any plane through the centre of the elUpsoid is represented 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 ART. 32.] ELLIPSOIDS OF INERTIA. 23 from each of the coefficients of A'^, 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 Sms^ = 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 manner that the moment of inertia about the line is always the same and equal to a given quantity I. To find the equation to the cone generated by the straight line. Let the principal axes at be taken as the axes of co-ordi- nates, and let (a, /8, (Xiy^zi) + M^(j) {x^y^z^ + M^^ {x^y^z,). By properly choosing the equivalent points we may use a similar rule when ^ is any cubic or quartic function of xyz, 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 26 MOMENTS OF INERTIA. [CHAP. I. 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 co-ordinates 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 most 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 momental ellipsoid at the centre of gravity of any tri- angle may be 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 Garnet's theorem, it touches the third side GA in its middle point E. Since DF is parallel to GA 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 (o the angle between r and r. Now ON = ^r, and hence from the equation to the ellipse FN^ = f r'^, therefore moment of 1 _ o m- 3 '2 • 2 _ -^ ^'^ inertia about OE J - S^"^ • t^ si^i «- - y • ^ 5 where A' is the area of the ellipse, so that the moments of inertia of the system about OE, OF, OJD are proportional inversely to OE^, OF', OB'. 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 identical. Hence this conic is a momental ellipse at of the triangle. A normal at 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, 1/c^ = l/a" + l/b\ If the triangle be an equilateral triangle, the momental ellip- soid 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. ART. 39.] EQUlMOMENTiVL BODIES. 27 38. Ex. 1. A momental ellipse at an angular point of a triangular area touches the opposite side at its middle point and bisects the adjacent sides. Ex. 2. The principal radii of gyration at the centre of gravity of a triangle are the roots of the equation ^ 36— '"+I08 = ''' where A is the area of the triangle. Ex. 3. The direction of the principal axes at the centre of gravity of a triangle may be constructed thus. Draw at the middle point D of any side BG lengths DH= — , DH'=^— along the perpendicular, where p is the perpendicular from A on BG and k, k' are the principal radii of gyration found by the last example. Then OH, OH' are the directions of the principal axes at 0, whose moments of inertia are respectively Mk'^ and Mk'^. Ex. 4. 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=BGj2iJS. 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 M -5- , and M —^ . Ex. 5. 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 equimomental systems. Ex. 6. 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 equitnomental to the quadrilateral area. Ex. 7. 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 equimomental to the elliptic area. Ex. 8. 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 to the remainder of the mass of the sphere placed at the centre. 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 ABCD be the tetrahedron. Through one angular pomt I) draw any plane and let it be taken as the plane of xy. Let I) 28 MOMENTS OF INERTIA. [CHAP. I. be the area of the base ABG, a, ^, y 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 PQB be any section parallel to the base ABG and of thickness du, where u is the perpendicular from D on PQR. The moment of inertia of the triangle PQR with respect to the plane of scy 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 = -„Ddu. The ordinates of the middle points a + B /3 + 7 y + 0L of the sides AB, BC, CA are respectively — ^ — , — „— , ^ ■ Hence, by similar triangles, the ordinates of the middle points of The moment of inertia of the triangle PQR with regard to the plane xy is therefore sp^^'^''\[-^p) ^[-^pJ ■^[-^p)\- Integrating from m = to u= p, we have the moment of inertia of the tetrahedron with regard to the plane xy = ^{a.' + 0' + rf + ^y + ycc + a/3}, 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 xy would be 5 V 4 ; ^20 ^20 '^ ^20^' 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 mo- ments of inertia about any straight line are also equal. The two systems are therefore equimomental*. * This result was proposed as a problem in the Mathematical Tripos between the dates of the publication of the preceding and following results, thus anticipating the author by a short time. ART. 42.] EQUIMOMENTAL BODIES. 29 40._ Theory of Projections. If the distance of every point in a given figure in space from some fixed plane be increased in a 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 simplified. 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- jected figure has merely been moved through a space nD perpen- dicular to the base plane. We may therefore suppose the base plane to pass through any given point which may be convenient. 41. If two bodies are equimomental, their projections are also equimomental. Let the origin be the common centre of gravity, then the two bodies are such that '^m = 1m'; 1,mx=0, 2mV=0, &c., 'Zmaf = 'tnix'^, "^.Tuyz = '2my'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 co-ordinates are («, y, z) is transferred to {x, y, nz) and («', y', /) to {od , y\ nz). Also the elements of mass m, m become nm and 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 momental ellipse of a plane area is a momental 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, -if) where y' = ny, and any element of area m becomes w! where m' = nm. Then tmai^ = - Sm V, tmxy = — Xm'xy', tmy^ = -j ^m'y'\ The momental ellipses of the primitive and the projection are Smj/^X^ - itmxyXY + tmx'Y^ = M^, l,my"X'^ - 2Xm'xy'X'Y' + tm'afY'' = 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 as 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 30 MOMENTS OF INERTIA. [CHAP. I. 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. Ex. 3. Since any ellipsoid may be obtained by projecting a sphere, we infer by Art. 38, Ex. 8, 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 il/ -=- . If then we describe a sphere of radius p= J3r, with its centre 5 at the centre of gravity, and its mass equal to that of the tetrahedron, this sphere and the tetrahedron 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 : ,^3. 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 tetra- hedron touches each edge at its middle point and has its centre at the centre of gravity of the volume. These results may also be deduced from Art. 25, Ex. 2, without the use of projections. Ex. 1. If E^ 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 £2 p'-i yn P 2^3'^ ^2". 32'^ 2«.B~ ' ART. 45.] EQTJIMOMENTAL BODIES. 31 and that, if the roots be ±pi, ipj, ip,, the momenta of inertia with regard to the principal planes of the tetrahedron are M -^, M^^, M^ . 5 5 5 Ex. 2. If a perpendicular EF be drawn at the centre of gravity E of any faoe=4p3/j,, 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. Fmir particles of equal mass can always be found which are equimomental to any given solid body. Let 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 co-ordinate planes be Ma^, M^, and My^. By Art. 34, the mass of each particle must be IM. Let (x-^y^Zi) &b. (x^y^z^) be the required co-ordinates of these four points. . Then these twelve co-ordinates must satisfy the nine equations Sa;2=4a», I,y''=4^, 7iz^=4y^, ^xy = 0, 2j/2 = 0, Saa; = 0, Sa; = 0, Xy = 0, 2z=0. Now if we write Xi = a^j^, x^=a^^ &c. yi= Pvit y2=PV2 &o- 2i = 7fi &e. we have nine equations to find the twelve co-ordinates (fj ijj f^) &o. (f^ ijj fj) which differ from those just written down only in having a*, (3^, y' each replaced by unity. These modified equations express that the momental ellipsoid at of the four particles must be a sphere. The equations are therefore satisfied if the four points, whose co-ordinates 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 j3 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 tia'', 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^/3^. It also follows that, if one of these points, as D, 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=\OD. 45. Moments with higher powers. These moments are not often wanted in dynamics though useful in other subjects. It will therefore be sufficient to state some general results, the demonstrations of which are left to the reader. Let da be any elementary area or volume as the case may be. Let z be its ordinate referred to any plane of xy. Our object is to find the value of the integral jzMa for a triangle, quadrilateral, tetrahedron,' &c. Let the co-ordinates of the corners of the body considered be (?Oiy\Zi), (pM^^, &c. Let Hn{ZiZi &c.) represent the arithmetic mean of the different homogeneous products of ZiZ^, &c. of n dimensions, for example Hi{z^z^) = I {z^^ + z^z^ + z^zi + z.f). 32 MOMENTS OF INERTIA. [CHAP. I. Then for a triangle of area A, For a quadrilateral of area A, jz'^da- =A{Hn (Z:^Z2Z3Zi) - Z H^-i,{z^ZiZ^^\, where z' is the ordinate of the intersection of the diagonals. For a tetrahedron of volume F, ^Z'^da- = VHn (ZiZ^Z^Zi). For two tetrahedra joined together, whose united volume is V, S^ddadr= ("-X dv, and since — = '- we have x'Hv'=l - I xHv. Now dm = pdv, dm' = p'dv'. If then we take -= ( - J we have l^x'Hm'=2xHm, with similar equalities in the case of all the 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\* f k\^ in these cases respectively — = I - j or ( - J . Similar results however follow which may be all summed up in the following theorem. Theob. I. Let any body he cltanged into another by inversion with 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{ — \ . Then these two bodies have the same moments of inertia with regard to all straight lines through 0. Here n = 10, 8 or 6 according as the body is a volume, an area or an are. It also follows that the two bodies have the same principal axes at the point 0, and the same ellipsoids of gyration. We may also obtain the following theorem by the use of Sir W. Thomson's method of finding the potentials of attracting bodies by Inversion. Theor. 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 O by r, r', let p'=.p[-\ . Tlien the moment of inertia of tlie second body with regard to any point C is equal to that of the first body with regard to tlie corresponding point C multiplied by either of the equal quantities OC ■prz^ . Here n=8, 6 or 4 according as the body is a volume, area or arc. OG To prove this, consider the case in which the body is a volume. By similar triangles CP . ¥ = C'V' . OC. Hence proceeding as before, we find pdv(GPY(~^'=p'dv'(G'P'f. 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 meets the sphere. Prove that if the density had varied inversely as the sixth power of the distance from 0, 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. The theory of equimomental particles is of considerable use in finding the centre of pressure of any area vertically immersed in a homogeneous fluid under the action of gravity. It may be proved from hydrostatical principles K. D. 3 (o^)'- 34 MOMENTS OF INERTIA. [CHAP. I. that if the axis of x be in the effective surface, and the axis of y vertically downwards, the co-ordinates of the centre of pressure are ^ Product of inertia about the axes F = moment of area about Ox Moment of inertia about Ox moment of area about Ox We see therefore that two equimomental areas have the same centre of pressure. Let the given area be equimomental to particles whose masses are m„ m, &c. and let {x„ y,), (x„ y,), &c. be the co-ordinates of these particles. Then pnosy Y=^^ tmy ' tniy ' But these are the formulae to find the centre of gravity of pkrticles whose masses are proportional to Tn^yi, m^^ &c. havmg the same co-ordinates as before. Hence this rule. If any area be equimomental to a series of particles, the centre 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. 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 co-ordinates of its centre of pressure referred to the sides of the triangle itself are J (l+p/s), | (l + g/s), iil + rjs), where s=p + q + 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 malte 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 co-ordinates of the centre of pressure are B sin 9 - J" cos 9 F sin 9 - /i cos 9 ^~ 7 • ^ — Z ) where a is the area. 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. ART. 48.] POSITIONS OF THE PRINCIPAL AXES OF A SYSTEM. 3-5 Ex. 4. In a heterogeneous fluid the pressure at any point P referred to a unit of area is given by ^ = r<\ t, regard to a tangent plane} ~ ^^ ' ==\{B+G-A) + a\ Thus the moments of inertia 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 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 be drawn, and the normals to these confocals are the principal axes at /-". By Art. 5, Ex. 3, the principal axis of least moment is normal to the confocal ellipsoid and that of greatest moment normal to the confocal hyperboloid of two sheets. * Some of the following theorems were given by Sir William Thomson 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. 42 MOMENTS OF INERTIA. [CHAP. I. 57. The moment of inertia with regard to the point P is, by Art. 14, ^(A + B+G)+OF'. Hence, by Art. 5, Ex. 3, the moments of inertia about the normals to the three confocals through P whose parameters are Aj, Xj, X3 are respectively 0P'-\, 0P'-\„ 0P'-\. 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 the three confocals through P. This gives another construction for the principal axes at P. If the confocal diminish without limit, until it becomes a focal conic, we see that the principal diameters 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. 59. If we wish to use only one quadric, we may consider the confocal ellipsoid through P. We know* that the normals to * These propositions are to be found in books on solid geometry, they may also be proved as follows. Let the confocal ellipsoid pass near P and approach it indefinitely. The base of the enveloping cone is ultimately the Indicatrix ; and as the cone becomes ulti- mately a tangent plane, one of its axes is ultimately a perpendicular to the plane of the Indicatrix. Now in any cone two of its axes are parallel to the principal diame- ters of any section perpendicular to the third axis. Hence the axes of the envelop- ing cone are the normal to the surface and parallels to the principal diameters of the Indicatrix. But all parallel sections of an ellipsoid are similar and similarly situated, hence the principal diameters of the Indicatrix are parallel to the principal diameters of the diametral section parallel to the tangent plane at P. To find the principal moments, we may reason as follows. Let a tangent plane to the ellipsoid be drawn perpendicular to any radius vector OQ of the diametral AET. 60.] POSITIONS OF THE PRINCIPAL AXES OF A SYSTEM. 43 the other two confocals arc 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. A be these principal semi-diameters, we know that Hence, if through any point P we describe the quadric «- , y"' ^^ _ 1 A+\ ^ B + \^ GT\ ~ ' the axes of co-ordinates 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 be 2A and 2A, the principal moments at P are OP' - \, OP' -X + D;\ 0P'-\ + D,\ Ex. If two bodies have the same centre of gravity, the same principal axes at the centre of gravity and the differences of their principal moments equal, each to each, then these bodies have the same principal axes at all points. GO. Condition that a line should be a principal axis. The axes of co-ordinates being tlie principal axes at the centre of gravity it is required to express the condition that any given straight line may he a principal axis at some point in its length and to find that point. Let the equations to the given straight line be «'~.f^.y-9^^-^>' nx I m n section of OP, then the point of contact T, OQ and OP will lie in one plane when OQ is an axis of the section. For draw through T a section parallel to the diame- tral section, and let 0' be its centre, and let O'Y' be a perpendicular from 0' on the tangent plane which touches at T, Then OQ, O'Y' and OP are in one plane. Now consider the section whose centre is 0' ; O'Y' is the perpendicular on the tan- gent to an ellipse whose point of contact is T. Hence O'T, 0"T do not coincide unless O'Y' be the direction of the axis of the ellipse. But this section is similar to the diametral section to which it was drawn parallel. Hence OQ is an axis of the diametral section. Let PR be a straight hne drawn through P parallel to OQ to meet in P the tangent plane which touches in T. Then RP, RT are two tangents at right angles to the ellipse PQT. Hence OiJ''' = sum of the squares of th« semi-axes of the ellipse = OP* + 0Q'^ because OP, OQ are conjugate diameters. The moment of inertia about PR, a perpendicular to a tangent plane, has been proved above to be OR^ - X, hence the moment of inertia about a parallel through P to the axis OQ is Oi^-f OQ? - \. 44) MOMENTS OF INERTIA. [CHAP. I. then it must be a normal to some quadrie _^ + .^i^+ fl_ = l (2) at the point at which the straight line is a principal axis. Hence comparing the equation of the normal to (2) with (1), we have X J y z ^„s ZTx = '^^' 5Tx = '^™' (7 + x = '^" ^■^^- These six equations must be satisfied by the same values of x, y, z, X and /ti. Substituting for x, y, z from (3) in (1), we get '^ I m n Equating the values of fi given by these equations we have l-l. £_h h_f I m _m n _ n l_ ,a~. 'a^^~'b^^~g-a •■■■■'■ ''■ 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 (3) in (2), we have \ {p + m' + n^) = —-(Al^ + Bw? + Gn"), IT which gives one value only to \. The values of X and /* having been found, equations (3) will determine x, y, z the co-ordinates 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 Mr Townsend in the Mathematical Journal. The normal and tangent plane at every point of a quadrie 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 perpen- dicular 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 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 princi- pal 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 parallel to the intersection of the plane. ART. 62.] POSITIONS OF THE PEINCIPAL AXES OF A SYSTEM. 45 61. Ex. 1. Show that the siraight line a (,x-a) = l{y- i)-= c {z - c) is at eome point in its length a principal axis of an ellipsoid whose semi-axes are ahe. 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 plane -, + — +--1 = 0, there is always some point in it at which it is a principal plane. Also this point is its intersection with the straight line f.r - A = gy - B = liz - C. Ex. 4. Let two points P, Q be so situated that a principal axis at F 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. [Mr Townsend.] For let the principal axes at P, Q meet any principal plane at the centre of gravity in p, q, and let the pei-pendicular planes cut the same principal plane in LN, MN. Also let the perpendicular planes intersect each other in UN. Then RN is perpendicular to the plane containing the points P, Q, p, q. Also since the polars of p and q are LN, MN, it follows that pq is the polar of the point N. Hence the straight line EN 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. [Mr 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 oonfoeal 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 I, = OP'-X, I, = OP'-\ + Di', I.^OP'-X + Bi. If we equate /i and J^ we have A = 0> and the point P must lie on the elliptic focal conic of the ellipsoid of gyration. If we equate I^ and /^ we have Di = I>^_, 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 A is the semi- diameter of the focal conic conjugate to OP. Hence A' + OP^ = sum of squares of semi-axes =A-C + B-C. The three principal moments are therefore I,=I.,^ OP' + C, I, = A+ B- C, and the axis of unequal moment is a tangent to the focal conic. 46 MOMENTS OF INERTIA. [CHAP. I. The second case may be treated in the same way by using a confocal hyperboloid, we therefore have /a = /j = OP^ + .8, Ii = 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 priiidpal 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 + G-A+a' + a'\ Thus the moments of inertia about the intersections of perpendicular tangent planes to the sam^ 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. Hence the moments of inertia about the tangents to any line of curvature are equal to one another; and these tangents are principal axes at the point of cmitact. On the quadric a draw a tangent PT making angles ^ and ^TT - with the tangents to the lines of curvature at the point of contact P. If I^, I, be the moments about the tangents to these lines of curvature, the moment of inertia about the tangent PT = li cos^^ + /s sitf0 = B + G-A+ (a"2 -t- a^) cos^^ -|- {d' ■{- a'^) sin^f But, along a geodesic on the quadric a, a"" sm^i^ + a"^ cos^d) is constant. Hence the moments of imrtia about the tangents to any geodesic on the quadric are equal to each other. ART. 65.] POSITIONS OF THE PRINCIPAL AXES OF A SYSTEM. 47 64. Ex. 1. If a straight line touch any two oonfooals 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 co-ordinates of the point P at which the three principal moments are equal to the three given quantities I^, I.^, I.^. [JuUien's Problem.] The elliptic co-ordinates of P are evidently a''=^{l2 + I.,-Ii- B- C + A) &c.; and the co-ordinates {x, y, s) may then be found by Dr Salmon's formuloB, " -(A--B)(A-G)'^''■ 'E■s.. 3. Let two planes at right angles touch two confocals whose semi-major axes 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'^=a^-)-a'", and the product of inertia with regard to the two planes is (aV^ - a^a'^)". 65. Equimomental Surface. The locus of all those points at which one of the principal moments of inertia of the body is equal to a given quantity is called an equimomental surface. To find the equation to such a surface we have only to put 7i constant, this gives \ = r' — I. Substituting in the equation to the subsidiary quadric, the equation to the surface becomes Through any point P on an equimomental surface describe a confocal quadric such that the principal axis is a tangent to a line of curvature on the quadric. By Art. 63, one of the intersections of the equimomental surface and this quadric is the line of curvature. Hence the principal axis at P about which the moment of inertia is / is a tangent to the equimomental surface. Again, construct the confocal quadric through P such that the principal axis is a normal at P, then one of the intersections of the momental surface and this quadric is the sphero-conic through P. The normal to the quadric, being the principal axis, has just been shown to be a tangent to the surface. Hence the tangent plane to the equimomental surface is the plane which contains the noimal to the quadric and the tangent to the sphero- conic. To draw a perpendicular from the centre on this tangent plane we may follow Euclid's rule. Take PP' a tangent to the sphero-conic, drop a perpendicular from on PP', this is the radius vector OP, because PF is a tangent to the sphere. At P in the tangent plane draw a perpendicular to PP', this is the normal PQ to the quadric. From drop a perpendicular OQ on this normal, then OQ is a normal to the tangent plane. Hence this construction : 48 MOMENTS OF INERTIA. [CHAP. I. If V he ^any point on an equimomental surface whose para- meter is 1, and OQ a perpendicular from the centre on the tangent plane, then PQ is the principal axis at P about which the moment of inertia is I. The equimomental surface becomes Fresnel's wave surface when I is greater than the greatest principal moment of inertia at the centre of gravity. The general form of the surface is too well known to need a minute discussion here. It consists _ of two sheets, which become a concentric sphere and a spheroid when two of the principal moments at the centre of gravity are equal. When the principal moments are unequal, there are two singu- larities in the surface. (1) The two sheets meet at a point P in the plane of the greatest and least moments. At P there is a tangent cone to the surface. Draw any tangent plane to this cone, and let OQ be a perpendicular from the centre of gravity on this tangent plane. Then PQ is a principal axis at P. Thus there are an infinite number of principal axes at P because an infinite number of tangent planes can be drawn to the cone. But at any given point there cannot be more than three principal axes unless two of the principal axes be equal, and then the locus of the principal axes is a plane. Hence the point P is situated on a focal conic, and the locus of all the lines PQ is a normal plane to the conic. The point Q lies on a sphere whose diameter is OP, hence the locus of Q is a circle. (2) The two sheets have a common tangent plane which touches the surface along a curve. This curve is a circle whose plane is perpendicular to the plane of greatest and least moments. Let OP' be a perpendicular from on the plane of the circle, then P' is a point on the circle. If R be any other point on the circle the principal axis at R is RP'. Thus there is a circular ring of points, at each of which the principal axis passes through the same point, and the moments of inertia about these principal axes are all equal. The equation to the equimomental surface may also be used for the purpose of finding the three principal moments at any point whose co-ordinates (x, y, z) are given. If we clear the equation of fractions, we have to determine / a cubic whose roots are the three principal moments. Thus let it be required to find the locus of all those points at which any symmetrical function of the three principal moments is equal to a given quantity. We may express this symmetrical function in terms of the coefficients of the cubic by the usual rules, and the equation to the lociis is found. ART. 65.] POSITIONS OF THE PRINCIPAL AXES OF A SYSTEM. 49 Ex. 1. If an eqiiimomental surface out a quadric oonfooal with the ellipsoid of gyration at the centre of gravity, then the intersections are a sphero-oonio and a line of curvature. But, if the quadric be an ellipsoid, these cannot be both real. For if the surface cut the ellipsoid in both, let P be a point on the line of curva- ture, and P' a point on the sphero-conic, then by Art. 59, 0P^+ D^'=OP'^, which is less than A + \. But OP^ + D{' + D.2^ = A + B + C + 3X, therefore D^'>B + G + 2\, which is >A+ 2X. Hence Dj > the greates.t 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) a sphere whose radius is . / — „.. , Art. 13, (2) the surface + Ax^ + By^+Gz^ + AB + BC + GA J ' ' ' ■ (3) the surface A'B'C - A'yV - B'zV - C'xY - ^xY^^^ = P, where A' = A + y'-\- z'^, with similar expressions for B', C B. D. CHAPTER II. d'alembert's principle, &C. 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 i.s shown that if (x, y, z) be the co-ordinates of the particle at any time t referred to three rectangular axes fixed in space, m its mass, X, Y, Z the forces resolved parallel to the axes, the motion may be found by solving the simultaneous equations. 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 3« equations, and, as shown in any treatise on statics, 3n — 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 moticm 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 the equations of motion of the several particles, and without ART. 67.] d'alembert's principle. 51 making any assumption as to the nature of the mutual actions except the following, which may be regarded aa a natural conse- quence of the laws of motion : The internal actions and reactions, of any system of rigid bodies in motion are in equilibrium amongst themselves. 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 co-ordinates referred to any fixed rectangular axes at the time t. The accele- „ , ,. 1 d'^x d^y J d^z t . j- ■, ,t. 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 mf is the resultant of F and R Hence if mfhe reversed, the three F, E 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 gi-oup 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 but acting in exactly opposite directions were applied at each point of the system these would be 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 (quantities. These, when reversed, will be in equilibrium with the given impressed 4—2 52 d'alembeet's prtnciple. [chap. ir. forces. Lastly, the equations of motion for each body may be formed, as is usually done in statics, by resolving in three direc- tions and taking moments about three straight lines. 6S. 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- nation 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 of the same in his History of the Inductive Sciences. D'Alembert uses the following words :—" Soient A, B, G, &b. les corps qui oom- posent le systfime, et supposons qu'ou leur ait imprime les mouvemens a, b, c, &c. qu'Us soient forces, k cause de leur action mutuelle, de changer dans les mouvemens a, b, c, &o. II est clair qu'on pent regarder le mouvement a imprimfi au corps A comme composfi du mouvement a, qu'il a pris, et 'd'un autre mouvement a ; qu'on pent de mSme regarder les mouvemens b, a, &a. comme composes des mouvemens b, /3; c, y; &e., d'ou il s'ensuit que le mouvement des corps A, B, G, &o. entr'eux auroit iti le m6me, si au lieu de leur donner les impulsions a, b, c, on leur eut donnfi iL-la-fois les doubles impulsions a, a; b, ;8; &c. Or par la supposition les corps A, B, G, &o. ont pris d'eux-mSmes les mouvemens a, b, c, &o. done les mouve- mens a, jS, 7, &o. doivent Hre tels qu'ils ne d^rangent rien dans les mouvemens a, b, c, &c. c'est-^-dire que si les corps n'avoient repu que les mouvemens a, /3, y, &c. oes mouvemens auroient dA se detruire mutuellement, et le systfeme demeurer en repos. De la resulte le prinoipe suivant pour trouver le mouvement de plusieurs corps qui agissent les uns sur les autres. D^composez les mouvemens a, b, a, &c. imprimis k ohaque corps, ohacun en deux autres a, a ; b, ^ ; c, 7 ; etc. qui soient tels que si I'on n'elit imprim^ aux corps que les mouvemens a, b, c, &c. ils eussent pu eonserver les mouvemens sans se nuire r^ciproquement ; et que si on ne leur eflt imprim6 que les mouvemens o, p, 7, &c. le systSme iti demeur^ 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 ART. 70.] d'ALEMBERT'S PRINCIPLE. 53 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. Thexmion produces an imponderable skeleton, carrying ponderable particles of matter. 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 equilibrium. 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 mo- mentum-acceleration can be changed instantaneously.) We come thus to the conclusion that, taking for every molecule 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 ordinary 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 solely, thus x and x will be simply abbreviations for -^' and ^ . 54 d'alembert's principle. [chap. II. 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 ex- amples or whenever 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 OAB cttJi turn freely in a vertical plane ahout 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 of 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 time. Each particle must therefore modify the motion of the other. The particle with the shorter radius hastens the motion of the other and is itself retarded by the slower motion of that other. Our object is to find the resulting motion. By using D'Alembert's principle we are able to change this dynamical problem into an ordinary statical question, which when solved by the rules of statics gives the differential equations of the motion. Let OA = a, OB = b, and let the angle the rod OAB makes with the vertical Oz be 6. The particle A describes a circular arc, hence its effective forces are known by elementary dynamics to be inad and mad'^ the former being directed along a tangent to the circular arc in the direction in which d increases and the latter along the radius AO inwards. Similarly the effective forces of the particle B are m'bd and m'bd" along its tangent and radius respectively. The directions of these effective forces are represented in fig. 1 by the double headed arrows, while the single headed Fig. {DA Fig- (3) ART. 71.] D'ALEMBERT'S PRINCIPLE. 55 arrows indicate the directions of the weights mg and m'g of the pai'ticles. By D'Alembert's principle the four effective forces when revei-sed are in equilibrium with the weights of the particles. To avoid introducing the unknown reaction at and those between the particles and the rod, let us take moments for the whole system about 0. The forces ma&' and 7nb6'^ being directed along BAO have no moments. The moments of the other two are md-d and m'¥d. Reversing these and adding the moments of the weights we have {ma^ + mV) 9 + {ma + m'b) g sin^ = (1). This is the differential equation of motion. When it has been solved and the two arbitrary constants determined by any initial conditions we shall have 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 I. This motion is therefore given by le+g smd = (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 = ; — jt- from the hinge 0. As a variation on this problem, let us fiiid tlie motion when the rod OAB moves round the vertical as a conical pendulum 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 ma sin ^(^^ being the angle made by the plane zOA with any fixed plane passing through Oz. Similarly the whole effective force on .6 is directed along its radius BF and is equal to m'b am.6j>\ The directions of these efifective forces are represented by the double headed arrows in fig. 2. Reversing these and taking moments as before about 0, we have - {ma^ + m6=) sin^ cos^<^' + {ma + m'b) g sin = 0. 56 d'alembert's principle. [chap. II. Hence the angular velocity ^ of the plane zOA round the vertical is given by . ^ (ma + m'b)g .gv ^ {ma? + mV) cosd ^ " 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. 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 (3) when it revolves uniformly round the vertica,l. Ex. 2. If the two particles are attached to by two strings OA, AB as shown in fig. 3, and the system revolves round the vertical with a uniform angular velocity 0, show that (m .AE.OE+ m' . BF . OF) ^'= {m . AE + m' . BF) '-.x, 62 d'alembert's principle. [chap. II. Let x = x + x', y = y + y', 2-Z + 2', then by Art. 14 this equa- tion becomes Now the axes of co-ordinates are quite arbitrary, let them be so chosen that the centre of gravity is passing through the origin at the moment under consideration. Then ^ = 0, y = 0, but -rr , -ji are not necessarily zero. The equation then becomes This equation does not contain the co-ordinates 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 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 inay 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 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 ART. 82.] METHOD OF USE. 63 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 the 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 resultant of the impressed forces as they really act on the body before it has been collected into its centre of gravity. 82. Example of second principle. Let us next consider an example of the second principle. Suppose the earth to be 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 whatever 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 follow immediately from this result. The centre of gravity of the 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 ooneeutric spheres. Bnt 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 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 perpendicular 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 permanence 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. 64 d'alembert's principle. [chap. II. 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 sc, y, z be the co-ordinates of the centre of gravity of any one of these bodies referred to three rectangular axes fixed in space. Let three other co-ordinates 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 Jh, h., hs be these three moments of the momentum, and let M be the mass. Then the effective forces of the body are equi- is the same for every particle, and equal to 6. Hence the moment of the mo- menta of all the particles of the body about the axis is Xmr^O, 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^^ per- pendicular to, and along the direction in which r is measured, the moment of the moving forces of m about the axis is mr^'^, hence the moment of the moving forces of all the particles of the body about the axis is S (mr^^). By the same reasoning as before this is equal to Xmr'S, 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 action of any forces. By D'Alembert's principle the effective forces when reversed will be in equilibrium with the impressed forces. To avoid intro- ART. 91.] THE FUNDAMENTAL THEOREM. 71 ducing the unknown reactions at the axis, let us take moments about the axis. 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 be the angular velocities of the body just before and just after the action of the impulses. The equations then become where all the letters have the same meaning as before, except that F, G, X, Y are now impulses instead of finite forces. Ex. 1. A circular area of weight W can turn freely about a horizontal axis per- pendicular to its plane which passes through a point G on its circumference. If it start from rest with the diameter through G vertically above G, show that the resultant pressures on the axis when that diameter is horizontal and vertically below G are respectively iJlTW and V- 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. 7? F G /c^ + 3/t**' Ex. 3. Let j^ be the resultant of j^+^^l'' and j^, and let a=g ^.v.^-^. h=a , -„. Construct an ellipse with C for centre and axes equal to 2a and 26 K' + h" measured along and perpendicular to GO. Let this ellipse be fixed in the body and oscillate with it. Prove that the pressure R varies as the diameter along which it 92 MOTION ABOUT A FIXED AXIS. [CHAP. III. acts. And the direction may be found thus ; let the auxiliary circle out the vertical through in V, and let the perpendicular from V on GO cut the ellipse in B. Then GB is the direction of the pressure B. 111. In many problems we require the vertical and horizontal components of the pressure, and more particularly the positions of the body in which either of these components changes sign. If, for instance, the body is a wedge supported by its edge on a perfectly rough horizontal table it may be regarded as turning round a horizontal axis. But the table can only exert an upward vertical pressure on the body, if then the vertical pressure changes sign as the body moves, the wedge will leave the table and the whole motion will be different. Let Q be the vertical pressure on the body, measured upwards when positive, and P the horizontal pressure, measured to the left when the centre of gravity is on the right-hand side of the vertical plane through the axis. For the sake of brevity let «=? -p:^ - ^^^ ^=3 ^^-^i ■ Then we find M I --= (a - h) sine cos, e + W^h sine As the expression for the vertical pressure is not altered by changing the sign of 0, we need only consider its changes of sign as the centre of gravity moves, say, upwards. Similar changes will occur during the descent of the centre of gravity. We see also that the vertical pressure is always upwards when the centre of gravity of the body is below the horizontal plane through the axis. Consider then the two extreme positions in which the centre of gravity lies, (1) in the horizontal plane through the axis, and (2) vertically over the axis. In these two positions Q has opposite signs if U^h>a and the intervening vanishing points are given by a quadratic equation. Hence we infer that as the centre of gravity moves from one position to the other, the vertical pressure will vanish once and change sign if fi%>a. If this inequality does not hold the vertical pressure will be directed upwards in both the extreme positions. To determine if it can vanish for any intervening position of the body we must ascertain whether the minimum value of Q is positive or negative. By difterentia- tiou we find that Q is least when oos9=— ^ , and thence by substitution we find the least value of Q to be M {b-(a- 6) cos^ d]. If U^h be less than 2 (a - 6) and greater than 2 Jb{a-b), this value of cos will be possible and the minimum value of Q will be negative. The result is that, if both these conditions are satisfied as well as 12% < a, the vertical pressure will vanish and change sign ttvice as the body moves from one extreme position to the other. But if either condition fail, the vertical pressure will not vanish between the limits. To find the exact positions at which the pressure vanishes, we have to solve the quadratic equation formed by equating Q to zero. We must also remember that the conditions of the question may exclude one or both of these positions. Thus we may show from equation (5) that unless CiVi exceed | (a - b), the body cannot go all round. Or again the body may be projected upwards at such an inclination to the vertical that both the roots of the quadratic may be excluded from the arc of oscillation. In such a case the vertical pressure will of course keep one sign during the ascent. ART. 112.] PRESSURES ON THE FIXED AXIS. 93 The horizontal pressure vanishes when 9 = or ir, and when cos 8 = ^— . . If this value of cosfl be greater than unity, the horizontal pressure vanishes and changes its direction only when the centre of gravity is vertically over or under the axis. If it he numerically less than unity, the pressure vanishes and changes sign in two more positions, which both occur when the centre of gravity is above the axis and at equal heights. 112. Secondly. Suppose either the body or the forces not to be symmetrical. Let the fixed axis be taken as the axis of z wibh 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 co-ordinates of the centre of gravity at the time t. Let a be the angular velocity of the body, /the angular acceleration, so that/= ^%mxz —flmyz (5), "2,171 (xT — yX) = Xm (xy — yx) = tmr' . 0} = Mk'Kf (6). Equation (6) serves to determine f and a, and equations (1), (2), (4), (5) then determine F, G, F,G'; H and H' are indeter- minate, but their sum is given by equation (3). Looking at these equations, we see that they would be greatly simplified in two cases. Firstly, if the axis of ^^ be a principal axis at the origin, Xmxz = 0, 2my^; = 0, and the calculation of the right-hand sides of equations (4) and (5) would only be so much superfluous labour. Hence, in attempt- ing a problem of this kind, tue should, when possible, so choose the origin that the axis of revolution is a principal axis of the body at that point. Secondly, except the determination of f and a> by integrating equation (6), the whole process is merely an algebraic substitution of y and (o in the remaining equations. Hence our results will still be correct if we choose the plane of xz to contain the centre of gravity at the moment under consideration; this will make y = 0, and thus equations (1) and (2) will be simplified. 113. Impulsive forces. If the forces which act on the body be impulsive, the equations will require some alterations. Let u, V, w, u', v', w' be the velocities resolved parallel to the axes of any element m whose co-ordinates are w, y, z. Then u = — yo), u' = — ya, v = xm, v' = xm, and w, w' are both zero. The several equations of Art. 112 will then be replaced by the following : •EX + F + F' = tm {u' -u)=- Imy («' - «) = -My{w'-a>) (1), tT + G + G' = tm(v' -v) = tmx(co' -oy) = Mx(mx, G, ==fMx. It follows that the pressures of the axis on the body due to the effective forces are equivalent to a single force, which acts at the principal point 0, and is equal to the resultant of the effective forces of the mass of the body collected at its centre of gravity. In the same way the pressure due to the effective forces when the body is acted on by an impulse may be represented by a blow which acts at the principal point 0, and whose components parallel to « y, z are the expressions on the right-hand sides of equations (l)/(2), (3)of Art. 113. Ex. 1. A heavy body can turn freely about a horizontal axis Oz which is a principal axis at 0. It starts from rest with the plane GOz through the centre of gravity G horizontal. Show that the pressure due to the effective forces alone makes an angle with the plane GOz whose tangent is half the tangent of the angle which the plane GOz makes with the vertical. Ex. 2. A quadrant of a circle of radius a can turn freely about a bounding radius as a fixed axis. Show that the pressures on the axis are equivalent to two pressures, one equal to the weight of the lamina acting at a point of the fixed radius distant iajitr from the centre, and the other at a point which divides that radius in the ratio 3 : 5. Ex. 3. A lamina can turn freely about an axis Oz in its plane as a fixed axis. It is struck by a blow F at any point A of its area in a direction perpendicular to the lamina. Show that the statical pressure on the axis is equal to a blow P acting at B where AB is a perpendicular to Oz. Show also that the pressure due to the effective forces is equal to a blow Fx^jk^ acting at in a direction opposite to the blow at B. Here the origin is the principal point of the axis, x and { are the distances of the centre of gravity and of A from Oz, Mlc^ is the moment of inertia about Oz. What is the condition that the pressure on the axis should be equivalent to a couple ? Ex. 4. A door is suspended by two hinges from a fixed axis making an angle a with the vertical. Find the motion and pressures on the hinges. . Since the fixed axis is evidently a principal axis at the middle point, we shall take this point for origin. Also we shall take the plane of xz so that it contains the centre of gravity of the door at the moment under consideration. The only force acting on the door is gravity, which may be supposed to act at the centre of gravity. We must first resolve this parallel to the axes. Let (p be the angle the plane of the door makes with a vertical plane through the axis of suspension. If we draw a plane zON such that its trace ON on the plane of xOy makes an angle (p with the axis of x, this will be the vertical plane through the axis; and if we draw OV in this plane making zOV=:a, OV will be vertical. Hence the resolved parts of gravity are X=(/ sin acos 0, y=gsinosin0, Z=-gDOSa. ART. 11.5.] IMPUJiSIVE FORCES. 97 Since the resolved parts of the effective forces are the same as if the whole mass were collected at the centre of gravity, the six equations of motion are JI/(7sinooosi^ + F+J5"= ~x = k'^w'. '}• Suppose the door to be initially placed at rest, its plane making an angle /3 with the vertical plane through the axis; then when 0=/3, u = 0; hence k'V = 2gx sin a (cos ^ - cos ^) ] and Jc'^f= - 3 sin a sin . 1 By substitution in the first four equations F, F', G, G' may be found. 115. Dynamical and geometrical similarity. It should be noticed that the equations of Arts. 112 and 113 do not depend on the form of the body, but only on its moments and products of inertia. We may therefore replace the body by any equimomental body that may be convenient for our purpose. This consideration will often enable us to reduce the compli- cated forms of Art. 112 to the simpler ones given in Art. 110. For though the body may not be symmetrical abotit a plane through its centre of gravity perpendicular to the axis of sus- pension, yet if the momental ellipsoid at the centre of gravity be symmetrical about this plane we may treat the body as if it were really symmetrical. Such a body may be said to be dynamically symmetrical. If at the same time the forces be symmetrical about the same plane, and this will always be the case if the axis of suspension be horizontal and gravity be the only force acting, we know that the pressures on the axis must certainly reduce to a single pressure, which may be found by Art. 110. B. D. 7 98 MOTION ABOUT A FIXED AXIS. [CHAP. III. 116. Ex. 1. A uniform heavy lamina in the form of a sector of a circle is suspended by a horizontal axis parallel to the radius which bisects the arc, and oscillates under the action of gravity. Show that the pressures on the axis are equivalent to a single force, and find its magnitude. Ex. 2. An equilateral triangle oscillates about any horizontal axis situated in its own plane, show that the pressures are equivalent to a single force and find its magnitude. 117. Permanent axes of Rotation. Let us suppose that any point of a body under no forces is fixed in space and that it is set in rotation about some axis which we may call Oz. We may enquire what are the necessary conditions that the body should continue to rotate about that axis as if it were fixed in space. When these conditions are satisfied the axis is called a 'permanent aoois of rotation at the point 0. To determine these conditions let us suppose some other point A of the axis to be also fixed in space. Then by using the method of Arts. 112 or 113 we may determine the pressures at A which are necessary to fix the axis. If these are zero the attach- ment at A is unnecessary and may be removed. The body will then continue to rotate about Oz as if it were fixed in space. Since there are no impressed forces acting on the body, the whole pressure on the axis is that due to the effective forces. If the axis Oz is a principal axis at any point of its length the pres- sure due to the effective forces will act at that point (Art. 114). Hence the pressure at A cannot be zero unless that point coincide with 0. The conditions are therefore satisfied if the aoois of rota- tion Oz is a principal axis at the fixed point 0. If the axis Oz is not a principal axis at any point we shall prove that it cannot be a permanent axis of rotation. To prove this we must practically return to the equations (4), (5) and (6) of Art. 112. Let F, G, H, F', G', H' be the pressures at and A. Then a = 0,a'=OA. Taking moments about Oz we have Mk'f=0\ thus the angular velocity of the body about the axis Oz is constant. It easily follows that a; = - aPx, y = - tei'y, z=0. Taking moments about the axes of x and y we have (Art. 72) - G'a' = %m (yz - zy) = w^tviyz, F'a! = Sm (zx - xz) = — a^Xmxz. Thus.F' and G' cannot be zero unless 'Zmxz = and Xmyz = 0, i.e. Oz cannot be a permanent axis of rotation unless it is a principal axis at the fixed point 0. The existence of principal axes was first established by Segner in the work Specimen Theoria TurUnum. His course of investigation is the opposite to that pursued in this treatise. He defines a principal axis to be such that when a body revolves round it the forces arising from the rotation have no tendency to alter the ART. 118.] PERMANENT AXES OF ROTATION. 99 position of the axis. Prom this dynamical definition he deduces the geometrical properties of these axes. The reader may consult Prof. Cayley's report to the British Association on the special problems of dynamics, 1862, and Bossut, Histoire des MatMmatiques, Tome ii. 118. A body at rest with one point fixed in space is acted on by an impulsive couple, it is required to find the initial axis of rotation. Let Oz be the initial axis. As before we shall regard this axis as fixed at some other point A, at which the pressures are to be equated to zero. Let L, M, N be the resolved parts of the couple about the axes. The plane of the couple is therefore L^ + Mr, + N^=0 (1). Let u', v', w' be the initial velocities of an element of the body whose co-ordinates are x, y, z, and let &>' be the initial angular velocity of the body. Then m' = - yea', v = xa exactly as in Art. 113. Taking moments about the axes of x, y, z we have L — G'a! = 2m (yw' — zv') = — Xmxz . ui'\ M + F'a' = 2m {zu! — xw') — — Xmyz .(o'>. 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 - tmxz^ - tmyzT) + MIc''^ =0 .(2). Let the momental ellipsoid at the fixed point be constructed and let its equation be A^' + Brj' + G^' -2Dri^- 2E^^ - 2F^r] = MeK The diametral plane of the axis of f is -E^-Br, + C^ = (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 cowple with regard to the momental ellipsoid at the fixed 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 the fixed point. 7-2 100 MOTION ABOUT A FIXED AXIS. [CHAP. III. 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 ellipsoid of gyration at 0, the body will begin to turn about a perpendicular from 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 (J,, m, n). Prove that the direction-cosines of the initial axis of rotation are pro- portional to f^-—i , ■ , . and 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 about either of the principal diameters of this section 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 — uf'Simyz, M—- u'Stjmz, ^=0. 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 i = u'Smj/?. Ex. 4. A body having one point fixed in space is made to rotate 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 (i;, m, n) be the direction-cosines of the axis of rotation, {\, /j,, v) those of the axis of the couple Gf. Then by the last question and' the second and third examples of Art. 18, we have X II. _ V {B - qmn~ (G - A) nl~ {A - B) Im' . (J2= {A - B)n^m^+ {B - C)Vn^+{G - A)VP. We have then to make G a maximum by variation of {Imn) subject to the con- dition P + m'+'n?=l. The positions of these axes were first investigated by Mr Walton in the Quarterly Journal of Mathematics, 1865. The Centre of Percussion. 120. 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 aads 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 vertically under 0. Let it be struck by a horizontal blow T which we may suppose to act in the plane of ART. 120.] THE CENTRE OP PERCUSSION. 101 the lamina at some point A in GQ produced. Let GA = a. Let F and G be the impulsive reactions at the fixed point G. Let a be the angular velocity of the body round G just after the blow Y has been given. The equations of motion, exactly as in Art. 110, are therefore MQ& + hy M If the pressure G on the fixed point is zero, we have by eliminating F, fc^ + A^ = ah. By Ai't. 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. 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 tlie 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 \, ri, f be the co-ordinates of any point in its line of action. Let Mk'^ he the moment of inertia of the body about the fixed axis. We have nov? 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=0 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 Z=0, Y=Mx{a'-u), Z=0 (1). iX-iZ=-(u'-a)-2myz \ (2). iY-riX= (ui'-w)Mk'^ ) 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. n. Substituting from (1) in the first two equations of (2) we have ^myz = and f = , Since the origin may be taken anywhere in the axis of rotation, let it be so chosen that Sma;2 = 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 outs 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 f = ^- 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. 102 MOTION ABOUT A FIXED AXIS. [CHAP. III. 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 + h) + lmb)}l=M {ia^+[a + hf}+\mhK The Ballistic Pendulum. 121. It is a matter of considerable 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 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 Engli.sh 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. These last experiments are still regarded as some of the most trustworthy on smooth-bore guns. 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 small bullets may also be determined by the use of some rotational apparatus. Two circular discs of paper ART. 122.] THE BALLISTIC PENDULUM. 103 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 this 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. 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 fired, the recoil causes the pendulum to turn round its acds until brought to rest by the action of gravity. A piece of tape is attach^ to the pendulum, and is drawn out of a reel during the backward m,otion of the pendulum, and thus serves to measure 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 if 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 ft) = -p^ . The subsequent motion is given (Art. 92) by the 104 MOTION ABOUT A FIXED AXIS. [cHAP. III. d^d ah . a equation -ttj = ~ pi ^^'^ " > -when ^ = we have tt = », and if a be the angle of recoil, when = a, ^ = 0. Hence }r+...> (1), Mk'e=: Fp + Rq 4-...J where F is one of the impressed forces acting on the body, whose resolved parts are F cos <^, F sin 0, and whose moment about the centre of gravity is Fp, and R is any one of the re- actions. 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. ART. 136.] ON THE EQUATIONS OF MOTION. 115 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 oc, 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 + Ge = D (2), 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 + Ge = 0, obtained by differentiaiing (2), we have an equation containing only the reactions and constants. This being true for all the geometrical 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 a? + y^ = d', containing squares instead of first powers, then its second differ- ential equation will be xx + yy + a? + y'^ = Q; and, though we can substitute for x, y, we cannot in general eliminate the terms a? and jrl 136. The reactions in a djmamical problem are in many cases produced by the pressures of some smooth fixed obstacles which are touched by the toioving 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 8—2 116 MOTION IN TWO DIMENSIONS. [CHAP. IV. 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 reactions 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 be 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 therefore 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. 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 prj?position will be often useful. Let a body be turning about a point G with an angular velocity = to in a direction opposite to the hands of a watch, and let G be moving in the direction GA with a velocity V. It is required to find the velocity of any point F resolved in any direction PQ making an angle (j) 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 a> . GP . dt. Resolving parallel to PQ, the whole displacement of P = (Fcos ^ - « . (?P sin GPN) dt. If GN=p be the perpendicular from G on PQ, we see that the velocity of P parallel to PQ is V cos <^ — mp. It should be noticed that this expression is independent of the position of P on the straight line PQ. It follows that the velocities of all points in cmy straight line PQ resolved along PQ are the same. This result will be evident if we remember that all the ART. 138.] ON THE EQUATIONS OF MOTION. 117 points in the straight line PQ are rigidly connected together, so that if the resolved velocities of the points in it were unequal, the line PQ would alter in length. When therefore we require the velocity of any point P in any direction PQ we may replace P by any other point in the line PQ so situated that its resolved velocity is more easily found. Usually the point iV is the most convenient point to use, for without quoting a formula, its velocity resolved along PQ is seen by inspection to be Fcos (f> — cop. If {w, y, 6), (a/, y , 6') be the co-ordinates of the two bodies, q, q the perpendiculars from the points {x, y), {x , y') on the direc- tion of any reaction R, i|r the angle the direction of R makes with the axis of x, the required geometrical equation will be X cos i^r + y sm ■y^ + 6q = x cos -v|f -f- jr' sin ilr -)- ffcf. If the bodies be perfectly rough and roll on each other without sliding, there will be two resolved reactions at the point of contact, one normal and the other tangential to the common surface of the touching bodies. For each of these we shall have an equation similar to that just found. But if there be any sliding friction this reasoning will not apply. The latter case will be considered a little further on. 138. Second Method of Solution. Suppose that in a dynamical system two bodies of masses M, M' are pressing on each other with a reaction R. Let the equations of motion of M be those marked (1) in Art. 135, and let those of M' be obtained from these by accenting all the letters except R, yfr and t, and writing — R for R, ■yfr and t being of course unaltered. Let us multiply the equations of motion of M by '2x, 2y, 29 respectively, and those of M' by corresponding quantities. Adding all these six equations, we get 2M (xx + yy + l) are the polar co-ordinates of the centre of gravity. ART. 140.] ON THE EQUATIONS OF MOTION. 119 If p be the distance of the centre of gravity from the instanta- neous centre of rotation of the body, /> -7- is clearly the velocity of the centre of gravity, and therefore vis viva of ilf = ilf (p= + k') , , 140. Force Function and Work. The function U in the equation of vis viva is called the force function of the forces. It may always be obtained, when it exists, by writing down the virtual moment of the forces according to the rules of statics, integrating the result and adding a constant. This definition is sufficient for our present purpose ; for a more complete explanation the reader is referred to the beginning of the chapter on Vis Viva. When the forces are functions of several co-ordinates, it may be supposed that it will often happen that the virtual moment cannot be integrated until the relations between these co-ordi- nates have been found by some other means. But it will be shown in the chapter on Vis Viva that this is not so. In nearly all the cases we have to consider the virtual moment will be a perfect dififerential. In the remarks which follow in this and in the next three articles it will therefore be convenient to suppose that the function U exists, and is a known function of the co-ordinates of the system. In a subsequent chapter we shall discuss more particularly the various forms which the force function may assume. For the present we shall merely show how to find its form for a system of bodies under any constraints which are falling through the action of gravity alone. Let X, y be the horizontal and vertical co-ordinates of any particle of the system and let the latter co-ordinate be measured downwards. Let m be the mass of the particle. The virtual moment is therefore 'Zmgdy. The force function may therefore be written TJ = J^mgdy = Xmgy + G = gylm + C, where y is the depth of the centre of gravity of the whole system below the axis of x. Sometimes to avoid the constant G we take the integral be- tween limits. The force function is then called the work of the forces as the system passes from the position indicated by the lower limit to that indicated by the upper limit. The result just arrived at may therefore be stated thus. If, as a system moves from one position to another, its centre of gravity 120 MOTION IN TWO DIMENSIONS. [CHAP. IV. descends a vertical space h, the work done by gravity is Mgh, where M is the whole mass of the system. We notice that this result is independent of any changes in the arrangement of the bodies which constitute the system, and depends solely on the vertical space descended by the centre of gravity. 141. Principle of Vis Viva. Sometimes a system may move from one position to another in one of several ways. Per- haps we do not want the intermediate motion but only the motion in the later position when that in the earlier is given. In such a case we avoid the introduction of the constant in the equation of vis viva by taking the integral in Art. 138 between limits. Thus we say that the change in] _ f twice the work done the vis viva J ~ | by the forces. In this equation the change in the vis viva is found by subtracting from the vis viva in the final position the vis viva in the first. In finding the work done by the forces, the upper limit of the integi'al (as already explained) depends on the final position of the system and the lower limit depends on the initial position. The great importance of this equation is that we have a result free from all the reactions or constraints of the system. The manner in which the system moves from the first position to the last is a matter of indifference. So far as this equation is con- cerned, we may change the mode of motion in any way by intro- ducing or removing any constraints or reactions, provided only that they are such as do not appear in the equation of virtual moments as used in statics. We must notice that some reactions will not disappear from the equation of virtual velocities in statics, for example, friction between two surfaces which slide over each other. In forming the equation of vis viva in dynamics this kind of friction, when it occurs, will appear along with the other forces on the right-hand side of the equation. As the system moves from one given position to another, it is evident that the change in the vis viva produced by each force is twice the integral of the virtual moment of that force. It follows that the whole change is the sum of the changes produced by the separate forces. Taking then any one force F, we see that, when its direction makes an acute angle with the direction of the motion of the point A "of the body at which it acts, F and df have the same sign, and the integral in the equation of vis viva is positive. The effect of the force is therefore to increase the vis viva. But when the direction of the force is opposed to ART. 142.] ON THE EQUATIONS OF MOTION. 121 the direction of the motion of A, i.e. when the force makes an acute angle with the reversed direction of the motion of A, the effect of the force is to decrease the vis viva. This rule will enable us to determine the general effect of any force on the vis viva of the system. 142. Suppose, for example, a body to move or roll under the action of gravity with one point in contact with a fixed surface which is either perfectly rough or perfectly smooth, so that there can be no sliding friction. Let it be started off in any manner, so that the initial vis viva is known. The vis viva decreases or increases according as the centre of gravity rises above or falls below its original level. As the body moves the pressure on the surface will change and may possibly vanish and change sign. In this case the body will leave the surface. The centre of gravity by Art. 79 will then describe a parabola and the angular velocity of the body about its centre of gravity will be constant. Presently the body may impinge again on the surface, but until such impact occurs the equation of vis viva is in no way affected by the body leaving the surface. But the case is different when the body impinges on the surface. To make this point clearer, let F be the reaction of the surface, A the point of the body at which it acts, and Fdf its virtual moment as in Art. 138. Then as the body moves on the surface, df is zero, and when the body has left the surface, F is zero, so that during the. motion before the impact occurs the virtual moment Fdf is zero for the one reason or the other. The reaction therefore does not appear in the equation of vis viva. But when the body impinges on the surface, the point A is approaching the surface and the reaction F is resisting the advance of A so that neither F nor df is zero. Here we measure F in the same manner as in the first part of the motion, regarding it as a very great force which destroys the velocity of J. in a very short time (Art. 84). During the period of com- pression, the force F resists the advance of A, and therefore the vis viva of the body is decreased. But during the period of restitution the force assists the motion of A, and thus the vis viva is increased. We shall show further on that the vis viva is decreased by an impact except in the extreme case in which the bodies are perfectly elastic, and we shall investigate the amount lost. As a general rule we may notice that the equation of vis viva is altered by an impact. We may find a superior limit to the altitude y to which the centre of gravity can rise above its original level. The equation of vis viva may be written /vis viva in anyN _ /initial vis\ ^ _ ^j^ \ position / V viva / ^^' 122 MOTION IN TWO DIMENSIONS. [CHAP. IV. where M is the mass of the body. Now the vis viva can never be negative, hence the centre of gravity cannot rise so high that 2Mgy > 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 vi.,iv..r*=«{(|)V@%..(0)(fy=p(f)', where P is a known function of the co-ordinates of M. The equation of vis viva will therefore take the form <^^>m and thus -^ 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 be 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 equations, 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 minutely explained. Running remarks have been made ART. 144.J ON THE EQUATIONS OF MOTION. 123 which it is hoped will clear up those difficulties which generally trouble a beginner. The attention of the student is therefore particularly dii-ected to the different principles used in the follow- ing solutions. A Iwmogeneous spliere rolls directly down a perfectly rough inclined plane under the action of gravity. Find the motion. Let a be the moliuation of the plane to the horizon, a the radius of the sphere, ink- 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 E perpendicularly to ON, secondly, the friction F acting at N along NO and mg acting vertically at C the centre. The effective forces are vix, my acting at G parallel to the axes of X and y, and a couple mk'S tending to turn the sphere round G in the direction m NA. Here $ is the angle which any straight line fixed in the body makes with a straight line fixed in space. We shall take the fixed straight Une in the body to be the radins GA, and the fixed straight line in space the normal to the inclined plane. Then 9 is the angle turned through by the sphere. Besolving along and perpendicular to the inclined plane we have mx=mg sin a~F (1), my= -mg ooaa + R (2). Taking moments about N to avoid the reactions, we have max + mJc^d=:mgasia a (3). Since there are two unknown reactions F and R, we shall require two geome- trical relations. Because there is no slipping at N we have x = a0 (4). Also, because there is no jumping, !/ = a (5). Both these equations are of the form required in the first method. Differ- entiating (4) we get x = aS. Joining this to (3) we have ^ = a^TF'''^"" <^'- 2 S Since the sphere is homogeneous, k'' = -=a^, and we have i' = - 1; sm a. 124 MOTION IN TWO DIMENSIONS. [CHAP. IV. If the sphere had been sliding down a, smooth plane, the equation of motion would have been x=gsma, so that two-sevenths of gravity is used in turning the sphere, and five-sevenths in urging the sphere downwards. Supposing the sphere to start from rest we have clearly a; = 2 . ^gsma.t^, and the whole motion is determined. In the above solutions only a few 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) we have 2 F=-mg sin a. From (2) and (5) we have R=mg cos o. 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 equa- tions, for if equation (6) had been we should have expected some error to exist in the solution. For 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 /i less than f tan a, show that the angular velocity of the sphere after a time t from rest , - , 5u ff cos a would be -^ ■' t. 2 a 145. A homogeneous sphere rolls down another perfectly rough fixed sphere. Find the motion. Let a and b 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 ij>= iBOG. Let F and B be the friction and the normal reaction at N. Then, resolving tangentiaUy and normally to the path of C, we have m,{a + b)ip=mg sin -F (1), m{a + b)^^=mg cos^-R (2). Let A be that point of the moving spliere which originally coincided with B. Then if $ 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^S=:Fa (3). It should be observed that we cannot take 6 as the angle AGO because, though CA is fixed in the body, GO is not fixed in space. The geometrical equation is clearly a(e-(j)) = b^ ...: (4) No other is wanted, since iu forming equations (1) and (2) the constancy of the distance GO has been already assumed. ART. 145.] ON THE EQUATIONS OF MOTION. 125 The form of equation (4) shows that we can apply the first method. We thus obtain F= k^ + a' mg sin 0, and we are finally led to the equation 5 {a+b)c^ = = gain4>. By multiplying by 2^ and integrating we get after determining the constant, 02 = — _2-_(l-cos0), ^ 7 a + 6 ' the rolling body being supposed to start from rest at a point indefinitely near B. This result might also have been deduced from the equation of vis viva. The vis viva of the sphere is m{v^+k^^} and v = {a+b)^. The force function by Art. 140 is mgy, if y be the vertical space descended by the centre. We thus have (a + 6)2 ^2+ i;2^2_2g, („ + 6) (1 _ cos 0), which is easily seen to lead, by the help of (4), to the same result. To find where the body leaves the sphere we must put ij = 0. This gives by (2) {a + b) (3). Since tp is the relative angular velocity of the rods BG, AB, = w'- a (4). .-.ri-^z -2aHm + —g^Birfi(j>\(ia'-ui) (7). Also from the triangle ABC p^ + 2a^=2r^ (8). From these eight equations we can eliminate a, u', r, r, p, \p and i. We shall then have a differential equation of the first order to solve, containing and 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=0 and the whole system is therefore moving as if it were a rigid body. We therefore have for a single moment w, w' and 6 all equal to each other and f =0. The two first equations become, when we have substituted for F its value ^ , (5a2 + 3r2)ai=14a20 Eliminating w and substituting for r from (8) we have the cubic ^-£("-4' (3p2 + 16a2) ip-2a)= -^— {p + 2a), which has one positive root greater than 2a. It is also required to find the motion at the instant when the rods are at right 2 angles. At this moment 0=5. and hence by(3))'=a ^5, by (5) f= -—T=a{w'-u), by (7) i=l{ ^; (J2 - Vj'- 1 30 MOTION IN TWO DIMENSIONS. [CHAP. IV. 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 BC is turning round B with an angular velocity u', while at the same time B is moving perpendicularly to AB with a velocity 2au). The velocity of E is therefore the resultant of aa' perpendicular to BC and 2aia perpendicular to AB, both velocities, of course, being applied to the point E. When we wish our results to be expressed in terms of a, uf we may use these velocities to express the motion of E instead of the polar co-ordinates (r, B). Thus in applying the principle of angular momentum, we have to take the moment of the velocity of E about A. Since the velocity 2aa is perpendicular to AB, the length of the perpendicular from A on its direction is AB together with the projection of BE on AB, which is 2a + a cos ip. Since the velocity auf 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 )u + m{k^ + a^+2a^cos(t>)u' = m{2k^ + ia^)n. 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 cos = - J and w = u' = fi. In applying the principle of vis viva, we require the velocity of E. Eegarding it as the resultant of 2aw and auf we see that, if v be its value, v^= {2au)^+ {aay + 2 . 2aui . aoi' cos + 6abcoa(4,-e){i + 4,)} = 9agooBe+3bgaos

+G. The first equation is obtained by taking angular momentum about A for both bodies in the manner explained in Art. 78. The second is the equation of vis viva. Ex. 2. A uniform rod of length 2a has a particle attached to it by a string b ■ the rod and string are placed in a straight line on a smooth table, and the particle IS projected with a velocity V perpendicular to the string, prove that the greatest angle that the strmg can make with the rod is given by sin2i0 = a (1 -i- n)/126 ART. 149.] ON THE EQUATIONS OF MOTION 131 where n is the ratio of the mass of the rod to that of the particle. Prove also that the angular velocity then is Vjia + b). [Coll. Ex.] The common centre of gravity G moves in a straight line with uniform velocity. The vis viva and the angular momentum about G are each constant. Ex. 3. Three equal uniform bars, formed of such material that any particle repels any other with intensity proportional to the product of their masses and directly as the distance between them, are loosely jointed at their ends so as to form an equilateral triangle. If one of the connexions at the angles be severed, prove that the angular velocity of either of the outer bars when all three are in a straight line is ^ (8-4) times their angular velocity when they are at right angles to the middle bar. [Math. Tripos.] 148. The boh of a heavy pendulum contains a spherical carity 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 mlv'oi, where u 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 be the angle made by CO a fixed line in the body with the vertical, the equation of motion by Art. 89 is {MK^ + mc^) e + (Mh + mc)g am0=O, where, in finding the moment of gravity, 0, G and C have been supposed to lie in a straight line. The length L' of the simple equivalent pendulum is, by Art. 92, ~ Mh+mc 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, ~ Mh + mc It appears that L'ig. [Coll. Exam.] Ex. 3. A wire in the form of the portion of the curve r=a(l + cos B) cut ofE by the initial line rotates about the origin with angular velocity w. Prove that the TT 12 /2 tendency to break at the point 9 = ^ is measured by m — '^ uV. [St John's Coll.] J ART. 154] FRICTION BETWEEN IMPERFECTLY ROUGH BODIES. 137 Ex. 4. A rod OA whose density varies in any manner is swung as a pendulum about a 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 common tangent plane, and usually called the reaction; (2) a component F in the tangent plane usually called the friction ; (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 iV about this normal as axis which may be called the couple of twisting friction. The two couples are found by experiment to be in most cases very small and are generally neglected. But in certain cases where the friction forces 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 after- wards 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 138 MOTION IN TWO DIMENSIONS. [CHAP. IV. 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 of the two 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 when 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 was 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 sur- faces 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 gi'adually 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 beenlately 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 experiments: (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 * The results of Coulomb's experiments are given in his TMorie des viachinee simples, MSmoires des Savants Strangers, tome x. This paper gained the Prize of the AcadSmie des Sciences in 1781 and was published separately in Paris, 1809. ART. 157.] FRICTION BETWEEN IMPERFECTLY ROUGH BODIES. 139 been in motion for a few seconds. See the Report of the British Association for the meeting in DMin, 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 rough 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 there- fore 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 laws 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 ))e proved at the beginning of the next chapter, that such a motion may- be represented by supposing the body to be turning round some 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 /i-R, where R 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 co- ordinates 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 co-ordinates 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, It. 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 such that the sines of the angles AOB, BOC, COA are as R, P, Q. But if the point thus determined does not lie within the triangle ABC, the centre of rotation co- incides with one of the pins. These results follow immediately from the triangle of forces. 140 MOTION IN TWO DIMENSIONS. [CHAP. IV. 158. Discontinuity of Friction. The reader should par- ticularly notice the discontinuity just mentioned. The friction at any point of support which slides is ^iR, 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 /ii2. Its magnitude must be found from the equations of motion. As this is important let us present the argument in a slightly different form by considering the case of rolling. Suppose a body to roll on a rough surface, the friction called into play just prevents sliding, and is possibly variable in magni- tude and direction. By writing down and solving the equations of motion we can find the ratio of the friction F to the normal pressure R. If this ratio be always less than the coefBcient /x, of friction, enough friction can always be called into play to make the body roll on the rough surface. In this case we have obtained the true motion. But if at any instant the ratio -„ thus found becomes greater than the coefficient of friction, the point of contact begins to slide at that instant. In this case the equations do not represent the true motion. To correct them we must replace the unknown friction F by /iii, and remove the geometrical equation which expresses the fact that there is no slipping between the bodies. The equations must now be again solved on this new supposition. It is of course possible that a second change may take place. If at any instant the velocities of the points of contact become equal to each other, all the pos- sible friction may not be called into play. At that instant the friction ceases to be equal to /iiJ and becomes again unknown in magnitude and direction. 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 re- action 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 medium, the law of resistance being an even function 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. ART. 162.] FRICTION BETWEEN IMPERFECTLY ROUGH BODIES. 141 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. Exsunples of Friction. A homogeneous sphere is placed at rest on a rough inclined plane, the coefficient of friction heing jj,, determine whether the sphere will slide or roll. Let F be the friction required to make the sphere roll. The problem then becomes the same as that discussed in Art. 144. We have, therefore, F=f jRtana, ■where a is the inclination of the plane to the horizon. If then I tan a be not greater than /i., the solution given in the article referred to is the correct one. But if ;» < f tan a the sphere begins to slide on the inclined plane. The subsequent motion is given by the equations m'x ^mgema- jiB 0= -mg cos a + R] max + mh^ff = mga sin a whence we have, remembering that k^^^a", x=g (sin a- ix, cos a), ad = ^ii.g cos a. Since the sphere starts from rest, we have by integration X =\gt''{sixi a- ij, cos a), B=ln-t^eosa. The velocity of the point of the sphere in contact with the plane is x-a6=gt (sin a - f/* cos a). But since, by hypothesis, li is less than f tan a, this velocity can never vanish. The friction therefore will never change to rolling friction. The motion has thus been completely determined. 162. A homogeneous spliere is rotating about a horizontal diameter, and is gently placed on u rough horizontal plane, the coefficient of friction being /jt. Determine the subsequent motion. Since the velocity of the point of contact with the horizontal plane is not zero, the sphere evidently begins to slide, and the motion of its centre is along a straight line perpendicular to the initial axis of rotation. Let this straight line be taken as the axis of x, and let be the angle between the vertical and that radius of the sphere which was initially vertical. Let a be the radius of the sphere, mlc^ its 142 MOTION IN TWO DIMENSIONS. [CHAP. IV. moment of inertia about a diameter, and the initial angular velocity. Let E be the normal reaction of the plane. Then the equations of motion are clearly m,'x=tt,R I 0=mg-R\ W' whence we have x=iig, aS=-^iig (2)- Integrating, and remembering that the initial value of 6 is 0, we have a; = iM3t^ fl=fif-4A'-«^ (^)- But it is evident that these equations cannot represent the whole motion, for they make x, the velocity of the centre of the sphere, increase continually, a result quite contrary to experience. The velocity of the point of the sphere in contact with the plane is x-ad= -aQ + liigt. This vanishes at a time *i = f — (^)- At this instant the friction suddenly changes its character. It now becomes of magnitude only sufficient to keep the point of contact of the sphere at rest. Let F be the friction required to effect this. The equations of motion will then be ' mx=F \ 0=mg-R\ (5), mh^S=-Fa ) and the geometrical equation will be x=a$. Differentiating this twice, and substituting from the dynamical equations, we get F(a' + k^) = 0, and therefore F=0. That is, no friction is required to keep the point of contact of the sphere at rest, and therefore none will be called into play. The sphere will therefore move uniformly with the velocity which it had at the time tj. Substituting the value of ti in the expression for x obtained from equations (3) we find that this velocity is faii. It appears therefore that the sphere will move with a uniformly increasing velocity for a time f — and will then move uniformly with a velocity ^aSi. It may be remarked that this velocity is independent of /i. If the plane be very rough, /i is very great and the time t^ is very small. Taking the limit when n is infinite we find that the sphere begins immediately to move with its uniform velocity. 163. In this investigation the couple of rolling friction has been neglected. Its effect is to diminish the angular velocity. The velocity of the lowest point of the sphere tends to be no longer zero, and thus a small sliding friction is required to keep that point at rest. Suppose the moment of the friction-conple to be measured hyfmg, where / is a constant. Introducing this into the equations (S) the third is changed into mk^S=-Fa-fmg, the others remaining unaltered. Solving these as before we find that afmg ART. 164.] FRICTION BETWEEN IMPERFECTLY ROUGH BODIES. 143 We see from this that F is negative and retards the sphere. The effect of the couple is to call into play a frictiou-force which gradually reduces the sphere to rest. As the sphere moves we may wish to determine the effect of the resistance of the air. The chief part of this resistance may be pretty accurately represented by a force m/S— acting at the centre in the direction opposite to motion, v being the velocity of the sphere and /3 a constant whose magnitude depends on the density of the air. Besides this there is also a small friction between the sphere and the air whose magnitude is not known so accurately. Let us suppose it to be represented by a couple whose moment is myv'' where 7 is a constant of small magnitude. The equations of motion can be solved without difficulty, and we find tan-..yto-tan-x.y^^=--^>:i., 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 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. 144 MOTION IN TWO DIMENSIONS. [CHAP. IV. 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 carriage on n pairs of wheels is dragged on a level hm-izontal plane by a liarieontal force 2P with uniform motion. Find the magnitude ofV. Let the radii of the wheels be respectively j-j, r^, &c., their weights w-^, w^, &c., and the radii of the axles pj, p^, &e. Let 2W be the whole weight of the carriage, 2Qi, 2Q3, &c. the pressures on the several axles, so that W='ZQ. Let the pressures between the wheels and axles be JSj, R^, &e. and the pressures on the ground iJi', R^, &e. 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 AGB = e, supposed positive when B is behind AG. Let /t be the coefiScient 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), liU (r cos e -p)-Ilr sin e=fB.' (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 iJcos^ + /tJJsinS=Q (3), S(iJsine-/iiecose) + P=0 (4). The effective forces have been omitted because the carriage is supposed to move uniformly, so that the Mv of the carriage and the mk^d of the wheel are both zero. The first three of these equations give, by eliminating B, and B', IJ.\ cos 0~-\-sia6 co&e + ixsme r\ QJ ^'' This gives the value of B. In most wheels - and -r are both small as well as /. In r Q such a case /icosff-sinfl is a small quantity. If therefore /i=tan e we have 9=6 very nearly. The third and fourth of the equations give, by eliminating E, _ _ /t cose- sin e „ „ I u n„ f,„ ) ^=^ /.sine + cos<> « = ^ isin<> + cose 7'e+i(« + -)[ • the latter by equation (5). If £ be small, it will be sufficient to substitute for 6 in the first term its approximate value e. This gives P=s|sine|Q+/«±i^[ (6). Here we have neglected terms of the order ( ^ ) Q. If all the wheels are equal and similar we have, since SQ= IF, P=s^dw+fY^ ' (,,; Thus the force required to drag a carriage of given weight with any constant velocity is very nearly independent of the number of wheels. ART. 166.] FRICTION BETWEEN IMPERFECTLY ROUGH BODIES. 145 In a gig the wheels are usually lai-ger 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 sin e ^ Qj in the expression for P, depending on the radius 1\ 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 Q^ in the 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 6 were two constants depending on the nature of the road and the stiffness of the carriage, and V was the velocity. 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 dc Mecanique, Paris, 1855. It is easy to see that M. Morin's experiments tend to con- firm 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 lyhich to the horizon is a, and the coefficient of friction ii. Show that the whole time during which the sphere ascends the plane is the same as if the plane were smoolii, and that the time during which the sphere slides is to the time during which it rolls as 2 tan a : 7/t- Ex. 2. A uniform rod is placed at rest with one end in contact with a horizontal plane whose coefficient of friction is /j.. If the inclination of the rod to the vertical is a, show that it will begin to slide if /4 (1 -I- 3 cos" a) < 3 sin a cos a. Ex. 3. 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 ? 167. Rigidity of Cords. After having used to determine the laws of friction the apparatus with a fine cord described in Art. 164, Coulomb replaced the cord by a stififer 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 a + hT R. The force R measures the rigidity and is equal to , where a and b 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. ART. 169.] ON IMPULSIVE FORCES. 147 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. Oii 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), {id, v') be the velocities of the centre of gravity of any body of the system resolved parallel to any rectangular axes respectively just before and just after the action of the impulses. Let o) and to' 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 co-ordinates together with an impulsive couple measured by Mk^ (co — a). The resultant effective forces of all the bodies of the system may be found by the same rule. By D'Alembert's principle these will be in equilibrium with 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 convenient. 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^{a>'-w)=L. In many cases it will be found that by 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 difference 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. Mom.N _ /ResolvedN V after impulse / \ before impulse/ V impulse/' /Ang. MomentumN /Ang. MomentumN _ /Moment of\ V after impulse / \ before impulse / \ impulse / ' 10—2 148 MOTION IN TWO DIMENSIONS. [CHAP. IV. 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 wound 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 Tnotion being supposed to be parallel to one plane, determine the effect of the impulse. 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', la' 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^a' = (1). Just after the impact the part of the reel in contact with the string has no velocity. Hence v'-a^!ja- [King's Coll.] Ex. 3. 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 jerk be such that a parallel to it drawn through the centre of gravity does not cut the base, and (2) the velocity of jerk be greater than U, where U is given by U^ = ^gl {IS + 005" B) — ^y- r. COS \ P T 6) Here 21 is the length of a diagonal of the cylinder and B is the angle any diagonal makes with the vertical. Ex. 4. 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 + J sin $<:S cos 8. 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 4> be the angle made by the diagonal through A with the vertical. Then by the principle of vis viva we have (/£= + H0^=C- 2(7^008 0, where k'=V(icos''8 + ism^8), by Art. 18, Ex. 8. If the angular velocity of the cylinder vanishes when the centre of gravity is at its highest we have G=2gl. Let mR be the vertical reaction at A, where m is the mass of the cyhnder. Then —^ (I cos ^)=R-g. From these equations we find R^^ = Saos^(p-2coE^ + lcoe''e + isin^8. If R vanish we have cos 0= -J (1 ±4 sin 9). In order that R may keep one sign both these values of must be excluded by the circumstances of the case, i.e. both these values of , mil, rnk'^ii, and the moment for either of the lower rods is m(k' + a^) w'. Let $ be the angle any rod makes with the vertical at the time t. Taking moments in the same way as before, we have m/c^'cJ + mva sin 6* -jmia cos 0= -R. 2a cob 9 + iiiga am 6 (1)'. m {k^ + a^)u'- iii,k^ii + mvaBm6 + ma.Saeo!iO = R.ia cos + 2mga sin 9. . . (2)'. 154 MOTION IN TWO DIMENSIONS. [CHAP. IV. The geometrical equations are the same as those given above, with ff written for a. Eliminating B and substituting for u, v, we get (27(2 + a?)^~ + a^ [^zmBj (u sin 0) + cos 6 j^ (u cos 6)1 = iga sin 6 ; then multiplying both sides hy 01 = 6 and integrating, we get {2 (fc2 + a2) + 8a2sin2«} b!^=G-8ga cos 0. Initially, when 6 = a, u has the value given by equation (6). Hence we find that the angular velocity « when the inclination of any rod to the vertical is is given by (l + 3sin=e)w2=^. BirP»' +^ (cos a -cose). ' ia^ 1 + 3 sin-* a a 177. Ex. 1. A square is moving freely about a diagonal with angular velocity w, when one of the angular points not in that diagonal becomes fixed ; determine the impulsive pressure on the fixed point, and show that the instantaneous angular velocity will be = . [Christ's Coll.] Ex. 2. Three equal rods placed in a straight line are jointed by hinges to one another; they move with a velocity v perpendicular to their lengths; if the middle point of the middle one become suddenly fixed, show that the extremities of the other two will meet in a time -jj- , a being the length of each rod. [Coll. Exam.] Ex. 3. The points ABGD are the angular points of a square; AB, CD are two equal similar rods connected by the string BG. The point A receiving an impulse in the direction AD, show that the initial velocity of A is seven times that of the point D. [Queens' CoU.] Ex. 4. A series of equal beams AB, BC, GD is connected by hinges; the beams are placed on a smooth horizontal plane, each at right angles to the two adjacent, so as to form a figure resembling a set of steps, and an impulse is given at the end A along AB: determine the impulsive action at any hinge. [Math. Tripos.] Besult. If j?„ be the impulsive action at the n* angular point, show that -^2^-1 - ^^2M-2 - 2A'2n+3 = and that X^„+^ - SA'a^+j - 2Z2„ = 0. Thence find Z„ . Ex. 5. Two uniform rods AB, BC of equal length and mass, smoothly hinged at B, lie upon a smooth horizontal table ; the end A is struck so as to begin to move with a given velocity in a direction which makes angles 6, ^ respectively with the rods, show that, if sin (20 -0) = 3 sin 0, AB will begin to move without rotation. [June Exam. 1880.] Take moments for the rod BG about B and for both rods about A according to the rule in Art. 169. 178. The kick before and behind. A free inelastic lamina of any form is turning in its own plane about an instantaneous centre of rotation S, and impinges on an obstacle at P situated in the straight line joining the centre of gravity G to S. To find the point P when the magnitude of the blmo is a maximum''. * Poinsot, Sur la percussion des corps, Liouville's Journal, 1857; translated in the Annals of Philosophy, 1858. ART. 178.] ON IMPULSIVE FORCES. 155 Firstly, let titc obstaclt: P be a fixed point. Let GP=.r, and let R be the force of the blow. Let SG = ]i., and let u, w' be the angular velocities about the centre of gravity before and after the impact. Then ftw is the linear velocity of G just before the impact ; let v' be its linear velocity just after the impact. We have the equations , -Bx , R •"-"=![¥' ^'-^"=-M (^)' and supposing the point of impact to be reduced to rest, v' + xoi' = Q (2). Substituting for a' and v' from (1) in equation (2), we get x + li This is to be made a maximum. Equating to zero its differential coefficient with respect to x, we get x'- + 'i]ix-l?=0; :. x=-h±j¥+¥ (3). One of these values of x is positive and the other negative. Both these cor- respond to points of maximum percussion, but in opposite directions. Thus there is a point P with which the body strikes in front and a point P' with which it strikes in rear of its own translation in space more forcibly than with any other point. Ex. 1. Show that the two points P, F' are equally distant from S, and if be the centre of oscillation with regard to S as a centre of suspension, SP''=SG . SO. Ex. 2. If P be made a point of suspension, P' is the corresponding centre of oscillation. Also PP' is harmonically divided in G and 0. Ex. 3. The magnitudes of the blows at P, P' are inversely proportional to their distances from G. Secondly, let tlie obstacle be a free particle of mass m. Then, besides the equations (1), we have the equation of motion of the particle m. Let V be its velocity after impact, then F'=- . The point of impact in the two bodies will have after impact the same velocity, hence instead of equation (2) we have V' = v' + xu'. 711 13/ "4" hi Eliminating w', v', V, we get R = Mo> . V . -r^ — , ,„ 7 — -„ . This is to be made a maximum. Equating to zero its differential coefficient with respect to x, we find x=-h^^ }fi + v{l + ^^^ W- This point does not coincide with that found when the obstacle was fixed, unless m is infinite. To find when it coincides with the centre of oscillation, we must put il/ X -{-li 1i'=xh. This gives — = — — , or if l = x + h be the length of the simple equivalent pendulum, - = r • Since V' = - , it is evident that when 7J is a maximum V is m h m a maximum. Hence the two points found by equation (4) might be called the centres of greatest communicated velocity. 156 MOTION IN TWO DIMENSIONS. [CHAP. IV. There are other singular points in a moving body whose positions may be found ; thus we may inquire at what points a body must impinge against a fixed obstacle, firstly that the linear velocity of the centre of gravity may be a maximum, and secondly, that the angular velocity may be a maximum. These points, respectively, have been called by Poinsot the centres of maximum Beflexion and Conversion. Beferring to equations (1) we see that when «' is a maximum R is either a maximum or a minimum, and hence it may be shown that the first point coincides with the point of greatest impact. When u' is a maximum, we have to Rx k^ make u - =^3 = a maximum. Substituting for iJ, this gives s^ - 2 y- a; - i;^ = 0. If be the centre of oscillation, we have h. GO = h''. Let GO = h'. Then this equation becomes x^ - 2h'x - fc^ = (5). The roots of this equation are the same functions of h' and k that those of equation (3) are of li and k, except that the signs are opposite. Now S and are on opposite sides of G, hence the positions of the two centres of maximum Conversion bear to O and G the same relation that the positions of the two centres of maximum Beflexion do to S and G. If the point of suspension be changed from S to 0, the positions of the centres of maximum Beflexion and Conversion are interchanged. Ex. A free lamina of any form is turning in its own plane about an instanta- neous centre of rotation S, and impinges on a flxed obstacle P situated in the straight line joining the centre of gravity G to S. Find the position of P, firstly, that the centre of gravity may be reduced to rest, secondly, that its velocity after impact may be the same as before but reversed in direction. Result. In the first case, P coincides either with G, or with the centre of oscil- lation. In the second case if SG = h, x= GP the points are found from the equation 27^x2= fc2 (x - h). [Poinsot.] 179. Elastic smooth bodies. Two bodies impinge on each other, to explain the natiore of the action which takes place between them. When two spheres of any hard material impinge on each other, they appear to separate almost immediately, and a finite change of velocity is generated in each by the mutual action. This sudden change of velocity is the characteristic of an im- pulsive force. Let the centres of gravity of the spheres be moving before impact in the same straight line with velocities u and V. Then after impact they will continue to move in the same straight line; let u', v' be their velocities. Let 7n, m' be the masses of the spheres, R the action between them, then we have by Article 168, -R , E u-u = — , v'-v=—, nv These equations are not sufficient to determine the three quan- tities u', v', R. To obtain a third equation we must consider what takes place during the impact. ART. 179.] ON IMPULSIVE FORCES. 157 Each of the balls is slightly compressed by the other, so that they are no longer perfect spheres. Each also in general tends to return to its original shape, so that there is a rebound. The period of impact may therefore be divided into two parts. Firstly, the period of compression, during which the distance between the centres of gravity of the two bodies is diminishing, and secondly the period of restitution, in which the distance between the centres of gravity is increasing. At the termination of the second period the bodies separate. The arrangement of the particles of a body being disturbed by impact, we ought in strictness to determine the relative motions of the several parts of the body. Thus we might regard each body as a collection of free particles connected by mutual actions. These particles being set in motion might continue always in motion oscillating about some mean positions in the body. It is however usual to assume that the changes of shape and structure are so small that the effect in altering the position of the centre of gravity and the moments of inertia of the body may be neglected ; also that the whole time of impact is so short that the motion of the body in that time may be neglected. If for any bodies these assumptions are not true, the effects of their impact must be deduced from the equations of the second order. We may therefore assert that at the moment of greatest compression the centres of gravity of the two spheres are moving with equal velocities. The ratio of the magnitude of the action between the bodies during the period of restitution to that during compression is found to be different for bodies of different materials. In some cases this ratio is so small that the force during the period of restitution may be neglected. The bodies are then said to be in- elastic. In this case we have just after the impact u' = v'. This „ mm' , X , , mu + m'v gives u = ; (u — v), whence u = m+m m+m If the force of restitution cannot be neglected, let R be the whole action between the balls, R„ the action up to the moment of greatest compression. The magnitude of R must be found by experiment. This may be done by determining the values of u' and v', and thus determining R by means of equations (1). Such ex- periments were made in the first instance by Newton, and led to the result that p- is a constant ratio depending on the material of the balls. Let this constant ratio be called 1 + e. The quantity e is never greater than unity ; in the limiting case when e = 1 the bodies are said to be perfectly elastic. The value of e being supposed known the velocities after impact may be easily found. The action Ro must be first calcu- 158 MOTION IN TWO DIMENSIONS. [CHAP. IV. lated as if the bodies were inelastic, when the whole value of R may be found by multiplying by 1 + e. This gives „ mm' , \/i , \ i?= -, (m-d)(1 +e), whence w' and v' may be found by equations (1). 180. As an example, let us consider how the motion of the reel discussed in Art. 170 would be affected if the string were so slightly elastic that we could apply this theory. Since the point of the reel in contact with the string has no velocity at the moment of greatest compression, the impulsive tension found in the article referred to, measures the whole momentum communicated to the reel from the beginning of the impact up to the moment of greatest compression. By what has been said in the last article, the whole momentum communicated from the beginning to the termination of the period of restitution will be found by multiplying the tension found in Art. 170 by 1 + e, if e be the measure of the elasticity of the string. This gives T=\mm (1 + c). The motion of a reel acted on by this known impulsive force is easily found. Resolving vertically we find m(t)'-D)= - Jmj;(l + e). Taking moments about the centre of gravity, m/E*(D'= Jmua (1 + e), whence «' and u may be found. Ex. A uniform beam is balanced about a horizontal axis through its centre of gravity, and a perfectly elastic ball is let fall from a height ft on one extremity; determine the motion of the beam and ball. 'Result. Let JIf, m be the masses of the beam and the ball, 2a the length of the beam, F, V the velocities of the ball at the moments just before and after impact, w' the angular velocity of the beam. Then tj=-r^ — s— ;— > V'=V .-^ . ^ ■' (M+3m)a Sm + M 181. Rough bodies. Hitherto we have only considered the impulsive action normal to the common surface of the two bodies. If the bodies are rough an impulsive friction will clearly be called into play. Since an impulse is only the integral of a very great force acting for a very short time, we might suppose that impulsive friction obeys the laws of ordinary friction. But these laws are founded on experiment, and we cannot be sure that they are correct in the extreme case in which the forces are very great. This point M. Morin undertook to determine by experiment at the express request of Poisson. He found that the frictional impulse between two bodies which strike and slide bore to the normal impulse the same ratio as in ordinary friction, and that this ratio was independent of the relative velocity of the striking bodies. M. Morin's experiment is described in the following article. 182. A box AB which can be loaded with shot so as to be of any proposed weight has two vertical beams AC, BD erected on its lid ; CD is joined by a cross piece and supports a weight ART. 183.] ON IMPULSIVE FORCES. 159 equal to mg attached to it by a string. The weight of the loaded box is Mg. A string AEF passes horizontally from the box over a smooth pulley^ and supports a weight at i'' equal to {M ->r ni) g fi. The box can slide on a horizontal plane whose coefficient of fric- tion is yii, and therefore having been once set in motion, it moves in a straight line with a uniform velocity which we will, call V. Suddenly the string supporting mg is cut, and this weight falls into the box and immediately becomes fixed to the box. There clearly is an impulsive friction called into play between the box and the horizontal plane. If the velocity of the box im- mediately after the impulse be again equal to V, the coefficient of impulsive friction is equal to that of finite friction. The argument may be made evident as follows. Let t be the time of the fall. When the weight strikes the box, it has a hori- zontal velocity equal to V and a vertical velocity equal to gt. The box itself has a horizontal velocity V -{-ft, where M + (M-{-m)fi' Let F and R be the horizontal and vertical components of the impulse between the box and the horizontal plane. There will be an impulse between the falling weight and the box and an impulsive tension in the string AEF; by means of these the momenta generated by the external blows F and R are spread over the whole system. Let V be the common velocity of the whole system just after the impulses F and R are completed. This velocity V' is found by experiment to be equal to V. Re- solving horizontally and vertically as in Art. 168, we have {M + m + (M+m)fji.]V'-lM+{M + m)iu,}(V+ft)-mV=-F, mgt = R. Putting V =V and substituting for /, we immediately find that F = fiR. Ex. Show that the resultant impulse between the box and the falling weight is vertical. 183. When two inelastic bodies impinge on each other at some point A, the points in contact at the beginning of the im- pact have a relative velocity along the common tangent plane at A and also one along the normal. Thus two reactions will be called into play, a normal force and a friction, the ratio of these two being fi, the coefficient of friction. As the impact proceeds the relative normal velocity gets destroyed, and is zero at the moment of greatest compression. Let R be the whole momentum transferred normally from one body to the other in this very short time. This force R is an unknown reaction, to determine it we have the geometrical condition that just after impact the 160 MOTION IN TWO DIMENSIONS. [CHAP. IV. normal velocities of the points in contact are equal. This condi- tion must be expressed in the manner explained in Art. 137. The relative sliding velocity at A is also diminished. If it vanishes before the moment of greatest compression, then during the rest of the impact there is called into play only so much friction and in such a direction, as is necessary (if any be necessary) to prevent the points in contact at A from sliding, provided that this amount is less than the limiting friction. Let F be the whole momentum transferred tangentially from the one body to the other. This reaction F is to be determined by the condition that just after impact the tangential velocities of the points in contact are equal. If, however, the sliding motion does not vanish before the moment of greatest compression, the whole of the fiiction is called into play in the direction opposite to that of relative sliding, and we have jF=/iii. Generally we may dis- tinguish these two cases in the following manner. In the first case it is necessary that the values of F and R found by solving the equations of motion should be such that F' will refer to the motion just after impact. We then have, by Art. 137, u — ya—u' — y'ai = v + X(o — v' — x'co' = Resolving parallel to the tangent and normal at the point of con- tact we have, by Art. 169, M(u-U) + M'(uf-U') = 0\ M{v -V) + M'{v'-V') = o]' and by taking moments for each body about the point of contact Mk'(w-Q,) + M(u-U)y-M(v-V)x = 0\ MT' {co' - il') - M' {u' -U')y- M' (v' - V) «' = Oj ' These six equations are sufficient to determine the motion just after impact. 189. If the bodies are perfectly smooth and inelastic, the first of these six equations will no longer hold, and instead of the third we have the two equations u-U=0, u'-U' = 0, obtained by resolving parallel to the tangent for each body sepa- rately. 190. If the bodies are smooth and elastic we must introduce the normal reaction into the equations. We write down the equa- tions (1) and (2) as given below in Art. 192, except that F= 0. Then equation (4) gives the velocity G of compression at any instant of the impact. Putting = 0, we have as in equation (6) the value of R up to the moment of greatest compression, viz. Q R = -4. Multiplying this by 1 -I- e we have, by Art. 179, the com- plete value of R for the whole impact. Substituting this value of R in equations (1) and (2), we find the values oiu, v, to, u', v', as'. 191. Ex. Two smooth perfectly elastic bodies impinge on each other. Let Z», D' be the normal velocities of approach, i.e. the velocities of the point of contact of each just before impact resolved along the normal towards the other. Prove 11—2 164 MOTION IN TWO DIMENSIONS. [CHAP. IV. that the vis viva lost by the body M is equal to 4 -^ ( D' -j^ — ^ M'h' ^ ) ' *^^ notation being the same as in the next proposition. Another method of finding the change in the vis viva will be given in chapter VII. 192. Next, let the bodies be imperfectly rough and elastic. In this case, as explained in Art. 158, the friction which can be called into play is limited in amount. The results obtained in Art. 188 will not apply to the case in which this limited amount of friction is insufficient to reduce the relative sliding to zero. To determine this, we must introduce the frictional and normal im- pulses into the equations. Let R be the whole momentum communicated to the body M in the time t of the impact by the normal pressure, and let F be the momentum communicated by the frictional pressure. We shall suppose these to act on the body whose mass is M in the directions NG, MA respectively. Then they must be supposed to act in the opposite directions on the body whose mass is M'. Since B, represents the whole momentum communicated to the body M in the direction of the normal, the momentum com- municated in the time dt is dR. As the bodies can only push against each other, dR must be positive, and, by Art. 136, when dR vanishes, the bodies separate. Thus the magnitude of R may be taken to measure the progress of the impact. It is zero at the beginning, gradually increases throughout, and is a maximum at the termination of the impact. It will be found more convenient to choose R rather than the time t as the independent variable. The dynamical equations are by Art. 168 M{u-U) = -F \ M{v-V) = R \ (1), Mk-'{(o-D,)==Fy + Rx J M'{u'-U') = F \ M'(v'-V') = -R [ (2). M'k"{co'-n') = Fy'-Rx'\ The relative velocity of sliding of the points in contact is by Art. 137 ■^ 8 = u-ya)-u'-y'(o' (3), and the relative velocity of compression is by the same article C = v' + x'(o'-v-xa> (4). AKT. 193.] ON IMPULSIVE FORCES. 165 Substituting in these equations from the dynamical equations wc find S = So-aF-bR (5), G = C,-bF-a'R (6), where 8, = U -ijil-V -y'D,' (7), C, = V' + a;'D,'-V-xn (8), _J^ 1 2/" 2/'^ Mk-' M'k" *- ''• These may be called the constants of the impact. The first two, So, Co represent the initial velocities of sliding and com- pression. These we shall consider to be positive; so that the body M is sliding over the body M' at the beginning of the com- pression. The other three constants a, a, h are independent of the initial motion of the striking bodies. The constants a and a are essentially positive, while h may have either sign. It will be found useful to notice that cw! > ¥. 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= ± fidR, 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, AJU re- spectively, to represent the magnitudes of R and F at any moment of the impact. Then, if we consider AR and J. i'' to be the co- ordinates of a point P referred to AR, AF as axes of R and F, the changes in the position of P will indicate to the eye the changes 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. 166 MOTION IN TWO DIMENSIONS. [chap. IV. 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 i^ 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, C = 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 a' ■ AS = ^, AC' = ^, AS' = ^, then SS', CO' 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 F and R respectively are positive, while their intercepts on the axes of R and F must have the same sign. Since aa' > b\ the acute angle made by the line of no sliding with the axis of F is greater than that made by the line of greatest compression, i.e. the former line is steeper to the axis of F 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==fiR, where fi is the ART. 194.] ON IMPULSIVE FORCES. 167 coefficient of friction. The friction continues to be limiting until P reaches the straight line ;S;S'. If ii„ be the abscissa of o this point we find iJ„ = °—r . This gives the whole normal blow, from the beginning of the impact, until friction can change from sliding to rolling. If R^ is negative, the straight lines AL and SS' do not intersect on the positive side of the axis of F. In this case the friction is limiting throughout the impact. If R(, 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 R is less than tan~'/u,, the friction dF necessary to prevent sliding is less than the limiting friction /xdR. Hence P must travel along SS' 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 which SS' makes with the axis of R is greater than tan~^/Lt, the ratio of dF to dR is numerically greater than /a, 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 SS', must travel along a straight line, making the same angle vdth the axis of R that AL does. This straight line must lie on the opposite side of SS', because the acute angle which SS' makes with J.ii is greater than the angle LAR. Also since the point P has crossed SS' 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 R ^/^"^ F S' Fig. 1. Kg. 3. 168 MOTION IN TWO DIMENSIONS. [CHAP. IV. equal to tan-^ /i. In fig. (1) the line of no sliding, viz. 88', makes so large an angle with AR that AL does not intersect itin 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 restitution begins. But the passage is marked by no peculiarity except this. If Ri be the abscissa of the point at which P crosses CO', the whole impact, for experimental reasons, is supposed to terminate when the abscissa of P is J?2 = i?i (1+ e), e 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, 88' and GC. 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 SS', or along a straight line making the same angle with the aods of ^ as AL does, but lying on the opposite side of SS'. The one along which it proceeds is the steeper to the accis of F. It travels along this line in such a direction as to make the abscissa R increase, and continues to be in this straight line to the end of the impact. The complete valueof&for the whole impact isfownd by multiplying the abscissa of the point at which P crosses CC by 1 + e. The complete value ■ of F is the corresponding ordinate of P. Substituting these in the dynamical equations (1) and (2), the motion just after impact may be easily found. If 8o = 0, we have S=-aF- bR. 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~'/A, i.e. if b/a is numerically less than fi, 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-' /j,, i.e. if b/a is numerically greater than fi, the representative point travels along a straight line AL making with the axis of R an acute angle LAR equal to tan"' /i. This straight line lies on the positive or negative side of AR according as 8 is positive or negative. Since the numerical value ART. 196.] ON IMPULSIVE FORCES. 169 of b is greater than afi, and F=± /xB., the term - bR governs the sign of S, hence 8 has the opposite sign to b. It follows that the straight Ime AL lies within the acute angle which the line of R S'' R Fig. 1. Fig. 2. 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. 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 R 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 F. Sub- stituting these in the djmamical 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 C 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 88' until it reaches the line GC of greatest compression. If the bodies are inelastic, the co-ordinates Ri, F^, of this intersection are the values of R and F required. But 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^^Ri^l+e), and the complete value of F may be found by substituting this value for R in the equation to 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 88', its 170 MOTION IN TWO DIMENSIONS. [CHAP. IV. 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 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 S8' 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 R^. The condition for the first case is that h must be greater than fi,a. The abscissae of the intersections of AL with SS' and GG' are respectively R„ = ^ and G . . a/i + b Ml = J — , . The necessary conditions for the second case are that Ri must be positive, and R^ either negative or positively greater than iJj (1 + e). 197. Ex. 1. Kebound of a ball. A spherical ball moving without rotation on u, smooth horizontal plane impinges with velocity V against a rough vertical wall whose coefficient of friction is /n. The line of motion of the centre of gravity before incidence mahing 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 witlwut 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 a be the angular velocity at the same instant. Let It 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 If (u + F cos a) = It Mk^io = Fr] The velocity of sliding of the point A of contact is S=u-ra=Vsina =-;; . ft2 M ART. 197.] ON IMPULSIVE FORCES. 171 The velocity of oompreasion of the point of contact is C= -t)=Feosa-^Tj. M k-' Measure a length AS in the figure to represent .^ SlVaina, and a length AG to represent MV cos a, along the axes of F and R respectively. Then SB and GB drawn parallel to the directions of R and F will be the lines of no sliding and 7(3 of greatest compression. Also we see that tan BAGz ■»-3+&2 tana=|tana. In the beginning of the impact the sphere slides on the wall, hence the representative point P, whose co-ordinates are R and F, begins to describe the straight line F=ii.R. 11 /i>f tan o, 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 are the co-ordinates of B. Therefore F=fMV sin a, R = MV cos a. These results are independent of /» because we see from the figure that more than enough friction could be called into play to destroy the sliding motion. If /i<^tan o, the straight line P=:ft,R cuts the line of greatest compression GB 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 co-ordinates of //, M=IJi,MVaosa, R = MVaoBa. If the sphere be inelastic we have only to substitute these values of F and iJ in the equations of motion to find the values of u, v, w just after impact. If the sphere be imperfectly elastic with a coefiicient of elasticity e, the repre- sentative point P will continue its progress until its abscissa is given by R=MVcosa{l + e). Take AC' to represent this value of R, and draw C'B' parallel to GB. Then, as before, we see that tan B'AC = = -^ — - . 7 X + e If A4>= , -— , the representative point describes some broken line like ALB', and cuts SB' before it cuts B'G'. In this case F and R are the co-ordinates of B', i?=fJU"F8ino, R = UVcoaa.(\ + e). 172 MOTION IN TWO DIMENSIONS, [CHAP. IV. If /tf the sliding will terminate before the end Of the period of impact, and the sphere will therefore rebound with a horizontal velocity - Ue and a vertical velocity f XJ [this follows by taking moments about the point of contact]. The centre of the sphere will then describe a parabola and the sphere wiU after- wards impinge on the ground. If the ground be inelastic and have a coefficient of friction /n'-ccH-f 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 - 17 (c - f yu') and the angular velocity is (2 - oy!) Ufla. The friction continues to act as a finite force so that the sphere finally rolls on the ground with a uniform velocity equal to Ex. 4. A lamina whose plane is vertical falls in its own plane and impinges on an imperfectly rough elastic horizontal plane. At the moment before impact the velocity of the centre of gravity G is vertically downwards and equal to V, and the angular velocity Q is in such a direction that the vertical velocity of the point A of impact is F — xQ, where x, y are the horizontal and vertical co-ordinates of G referred to A as oriyin. Let «, v be the horizontal and vertical velocities of Gf just after impact, v being measured upwards. Show that three oases may occur. (1) If Vxy + arjk^o = 0, the accelerations along and perpendicular to the radius vector take the simple forms r^ ... . ip and r^o- So again the acceleration - along the normal vanishes. , , P , . If, for example, we know the initial direction of motion of the centre of gravity of any one of the bodies, we may conveniently resolve along the normal to the path. This will supply an equation which contains only the impressed forces and such tensions or re- actions as may act on the body. If there be only one reaction, this equation will suffice to determine its initial value. The rule may be shortly stated thus. Write down the geome- trical equations of the system in its general position. Differentiate each twice and then simplify the results by substituting for the co- ordinates their initial values. Write down the dynamical equations of the system supposed to be in its initial position. Eliminate the second differential coefficients and we shall have sufficient equations to find the initial values of the reactions. ART. 200.] INITIAL MOTIONS. 175 We may also deduce from the equations the values of X(„%< ^'o, 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 x 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 xy —yx and by differentiating equation (1) X ^ Xf, t -\- Xq t-^ + X(f T-^ 4" . • . X ^= Xq ~t Xq t-r Xq "j-g + . . . &c. = &c. ; .-. (x'' + y")^ = (x,"^ + y:'it' + ... x'y" - y'x" = (*„ V" - ^o' V) I + (*o V" - <"yo") l + ... These results may also be obtained by a direct use of Taylor's theorem. If then the body start from rest, the radius of curvature is zero. But if x^'y^" — Xi^"yo' = 0, the direction of the acceleration is stationary for a moment. We then have 3 (V^ + yo"f II nil ""ni " 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', yf , ff zero. Differentiate each geometrical equation four times and then reduce each to its initial form. We shall thus have sufficient equations to determine x", «„'", »„"", &c., iJ„, i?„', R", Sue, where R is any one of the unknown reactions. It is often of advantage to eliminate the unknown reactions from the equations be/ore differentiation. We then have only the un- known coefficients Xo", x„"', &c. entering into the equations. 176 MOTION IN TWO DIMENSIONS. [CHAP. IV. If we know the direction of motion of one of the centres of gravity under consideration, we can take the axis of y a tangent to its path. We then have p = ~ , where a; 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 y"^ aio = 0, and if as" = 0, we may then use the formula p = 3 -7777 . Xq 201. Ex. 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. Let be the peg, AB the circle seen by an eye in its plane. Let OA be the string which is out, let G 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 R along CO. If 2a be the angle between the two strings, we have I{=2rcos a. Let I be the length of 00, p the angle GOO, a the radius of the disc. Let (x, y) be the co-ordinates 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 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 co-ofdiuates of the displaced position of are x-lsinp cos S and y-lsinp sin 6. Hence since the length OG remains constant and equal to I, we have a;2 -1- 2/2 - 2Z sin j3 (x cos fl -I- 3/ sin 9) = J2 cos2 ^. 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 6=1, sin 6=8. 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, y, 6) to zero, and put for (x, y, 6) their initial values (0, Zcos/S, 0). We get jr'o cos |S= sin /3 (X(, + 1 cos /Sff'o). This equation may also be obtained by an artifice which is often useful. The motion of Gf is made up of the motion of G and the motion of G relatively to C. Since G begins to describe a circle from rest, its acceleration along CO is zero. Again, the acceleration of G relatively to C when resolved along CO is GCe cos j3. The resolved acceleration of G is the sum of these two, but it is also equal to 2/0 cos ^ - Xo sin /3. Hence the equation follows at once. ART. 202.] INITIAL MOTIONS. 177 In this problem we require the dynamical equations only in their initial form. These are mSo=i2Q sin ;8 my „=mg - Bf, cos p mk^Sfi = R^l sin |8 cos j3 , where m is the mass of the body. Substituting in the geometrical equation we find r> OOSB ■Ro="»ff p . l + psinS/Scos^/S The tension of any string, before the string OA was cut, may be found by the rules of statics, and is clearly T. = "'^ , where 7 is the angle AOG. Hence the o cos 7 change of tension can be found. 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 A, B in the same horizontal line. If one be cut the tension of the other will be instantaneously altered in the ratio 1 : 2 cos2 ^ . [St Pet. Coll.] 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 -J ^10, the pressure on the other pin is initially unaltered. [Coll. Exam.] Ex. 3. Three equal particles A, B, G repelling each other with any forces, are tied together by three strings of unequal length, so as to form a. triangle right- angled at A. If the string joining B and G be cut, prove that the instantaneous changes of tension of the strings joining BA, GA will be JTcosJB and JTeosC respectively, where B and G are the angles opposite the strings joining GA, AB respectively, and T is the repulsive force between B and G. Ex. 4. Two uniform equal rods, each of mass m, are placed in the form of the letter X 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 %mg 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 a 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 aVI6g. [Coll. Exam.] Ex. 6. A particle is suspended by three equal strings of length a from three points forming an equilateral triangle of side 26 in a horizontal plane. If one string be cut the tension of each of the others will be instantaneously changed in Ex. 7. A sphere resting on a rough horizontal plane is divided into an infinite 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 iSir" : 2048. [Emm. Coll.] B. D. 12 178 MOTION IN TWO DIMENSIONS. [CHAP. IV. Ex. 8. Three equal and similar rods moveable about one common extremity are held at right angles to each other so that the three other extremities are in a horizontal plane with the common extremity either above or below. Show that if they are dropped on a smooth inelastic horizontal plane, the velocity of their centre of gravity is diminished by one-half. 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 systeni starts from rest with the rod horizontal and the ring at a distance c from the fixed point. Show that the polar co-ordinates of the ring referred to the fixed point are c-hVV/24 and 6>„t2/2. Find also 6*„, and prove that r„"" = gS^ + icS^. Thence find the initial radius of curvature of the path of the particle. [May Exam. 1888.] On Relative Motion or Moving Axes. 203. 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. 204. The Fundamental Theorem. Let it be required to find the motion of any dynamical system relative to some moving point G. 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 /be the acceleration of G at any time t. If every particle m 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 (7 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 (7 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 co-ordinafces. 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 co-ordinates of any element of a body whose mass is m referred to C as origin. The accelerations of the ART. 20o.] ON RELATIVE MOTION OR MOVING AXES. 179 ^. , d?r /dey , 1 d /d0\ , , ,. , particle *''® '^ "'^ ( j7 ) ^^^ ~di[Tf)' '^^ perpendicular to the radius vector r. Taking moments about G we get moment round C of the impressed forces plus the moment round C of the reversed effective forces of supposed to act at the centre of gravity. dd If the point G be fixed in the body and move with it, -=- iXt will be the same for every element of the body, and, as in Art. 88, d f .de\ ,„„d2(9 ~dt^' we have Sm jt (^^ jt ) = ^^' 205. From the general equation of moments about a moving point G we learn that we may use the equation d(o _ 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 C is not zero, yet the moment vanishes. 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 G, and the velocity of G is therefore zero. At the time t + dt, the body is turning about some point G' very near to G. Let GG' = da; then the velocity of G is mda. Hence in the time dt the velocity of G has increased from zero to wda, therefore its acceleration is ft) -J- . To obtain the accurate equation of moments about G we must apply the effective force 2m . <» -^ in the reversed direction drr at the centre of gravity. But in small oscillations w and -^ are both small quantities whose squares and products are to be neglected, and in an initial motion m is zero. Hence the moment of this force must be neglected, and the equation of motion will be the same as if C 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 12—2 180 MOTION IN TWO DIMENSIONS. [CHAP IV. 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 moments 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 G is L-M(o^.rcosGC'G. at If k be the radius of gyration about the centre of gravity, the equation of motion becomes dr writing for cosOG'G its value -v- . 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 rmyving in any manner when suddenly a point 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 after the change and the velocity of before the change. If u, w' be the angular velocities of the body before and after the change, and r = OG, we have by taking moments about 0, (r^+ie) (m'-u) = moment of /about 0. Now the moment about O of a velocity at G is equal and opposite to the moment about G of the same velocity applied at 0. Hence if L, 11 be the moments about G of the velocity of O just before and just after the change, and ft be the radius U — Ju of gyration about the centre of gravity, we have w' - fci= rj — ij • 208. Ex. 1. Two heavy particles whose masses are m and m' are connected by an inextensible string, which is laid over the vertex of a double inclined plane whose mass is M, and which is capable of moving freely on a smooth horizontal plane. Find the force which must act on the wedge 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 ART. 210.] ON EELA.TIVE MOTION OR MOVING AXES. 181 of the string is hi (j; sin a +/ cos a). By oonsidering the particle m', we find the tension to be also m' {g sin a' -/cos a'). Equating these two we have j_ m' sin g' - m sin g "" m' cos a' + m cos a Hence F is found. 209. Ex. 2. A cylindrical cavity whose section is any oval cnrvc and wlwse 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 tlie centres of gravity of the mass and the sphere is perpendicular to the generating lines of the cylinder. A tnomentum B is communicated to the cube by a blow in this vertical plane. Find the motion of tlie sphere relativehj to the cube and the least value of the blow that the sphere may not leave tlie 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, fc its radius of gyration about a diameter. Let V^ be the initial velocity of the cube, w,, that of the centre of the sphere relatively to the cube, u„ 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 aK + Fo) + &H = Oj ^^'• and since there is no sliding Vf,-aw^=(i (2). To find the subsequent motion, let (re, y) be the co-ordinates 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, y is a known function of x. 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 \j/= y- . Let V be the velocity ofthecubicalmass, then, byArt. 132, mr^-|-Fj-l-ikrF=B (3). If T„ be the initial vis viva and y,, the initial value of y, we have by the equation of vis viva "' {(J+ ^)'+ (f )'+ '''"i '^^^^'= '^"'^^ ^'^'^ ^*'' where u 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 = aio. Eliminating V and u from these equations, we have (|y.|(I.tan^,)(l.g)-^S=C.-2,. (5), B^ where Cg= j ■^ + 2gy„ (6). (M+m) lM + (M+m)-\ This equation gives the motion of the sphere relatively to the cube. 210. To find the pressure on the cube, let us reduce the cube to rest. Let B, be the normal pressure of the sphere on the cube, F the friction measured positively in the direction in which the arc is measured. The whole effective force on the cube is Z=Bsiti ^-^Poos \j/. By Art. 204 we must apply to every particle an acceleration 182 MOTION IN TWO DIMENSIONS. [CHAP. IV. ^opposite to this force. The sphere will therefore be acted on by a force -^ Jf in a horizontal direction in addition to the reaction R, the friction F and its own weight. Taking moments about the centre, we haye mJr' -^ = Fa (7). Besolving along a tangent to the path, m — = - i*" - — i cos ^ - mj sin ^ (8). But since there is no sliding, we have v = aa (9). Differentiating this and substituting from (7) and (8), we find ysin^cos^ k^ sin ^ . 1+7C0S2^ '"^a2+A2I+7CosV * '' where 7= 2 .3 t?- Eesolving the forces on the centre of the sphere along a normal to the path, we have -—=li + ^Xsin^-mgooB^ (II), where p is the radius of curvature of the path. Substituting for v^ its value given by (5), which may be conveniently written in the form v^l-pcos^^)=-^,{G-2y)g (12), where j3= ^ „ .- , we have two equations to find the reactions F and B. Eliminating F, we get C-2y + poos^\:^I^,P-±y = ^pP (13), where P is rather a complicated function of ^ which is not generally wanted. We have (l-^cos^2 _^+7_ I + 700s2vt /3(l-/3) ^ '- We notice that, since ^ is necessarily less than unity, P cannot vanish and is always finite and positive. If the sphere is to go all round the cavity, it is necessary that the value of v as given by (12) should be real for all values of y and cos ^. Hence the value of C as found by (6) must be greater than the greatest value of 2y. It is also necessary that li should be always positive, so that the values of cos \f/ given by the equation (13) when E = must be all imaginary or numerically greater than unity. We observe that, it C >2y and p be always positive, iS cannot vanish for any positive value of cos ^. If the equation (13), when R = 0, have two equal roots which are less than unity, the pressure on the cavity vanishes but does not change sign. In this case the sphere does not leave the cavity at the point indicated by this value of cos ^. The condition for equal roots gives us d i ,1-ScosV) 2S where p is given as a function of ^ from the equation to the cylinder. Writing |=cos ^ for brevity, this reduces to ^^f(l-;8a(I + 7r)(^ + v) = sin^{8^ + 7-(3,32 + 7»)|2 + ;S7(7-;8){^} (16). ART. 211.J ON RELATIVE MOTION OR MOVING AXES. 183 If no other real value of oos i/* makes B vanish and change sign in (13), and if also C>2y the sphere is said just to go round. We may put this reasoning in another way. If the sphere is just to go round, then R must be positive throughout and must vanish at the point virhere it is least. In this case we have R and — simultaneously zero. Differentiating (13) we notice that the differential coefficient of the right-hand side is zero, except at some singular points where p or — is dyp infinite. We notice also that the constant G which depends on the initial con- ditions disappears. In this way we again obtain equation (15). It should be observed that the point where the pressure vanishes and is a minimum cannot be the highest point of the cavity unless the radius of curvature p is a maximum or minimum at that point. This follows at once from equation (16). If we wish to find the blow B that the sphere may just go round we must examine the roots of the equations (13) and (16). To effect this we trace the curve whose abscissa is f and ordinate -q, where i; is the left-hand side of (13), from 1=0 to J= -1. The curve may undulate and the maxima and minima ordinates are given by (16). If the sphere goes round, the value of G must be such that every ordinate between f=0 and J= -1 must be positive. We therefore examine the roots of (16) and select that root which makes i; least. The value of G is found by equating this value of ?; to zero. The value of G having been found, that of B is known from (6). The result of course is subject to the limitations mentioned above. 211. Moving Axes. Next, let us consider the case in which we wish to refer the motion to two straight lines Of, Otj at right angles, turning round a fixed origin with angular velocity m. Let Ox, Oy be any fixed axes at right angles and let the angle xO^ = 6. Let ^ = OM, rj = PM be the co-ordinates 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 -^ and fw along and perpendicular to OM. A ^ y p \ y.- ■■■■ ,.■''' ^\ ,*'' \ + (v? + v^) ij where il, il &o. 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 ore a smooth curve which is constrained to turn with angular velocity a 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 |, r/ be fixed relatively to the curve. Let the mass be the unit of mass. Then the equations of motion are g_,„._l|(,M = X + ie.^ ? = ^ + ^« ) where X, Y, Z are the resolved parts of the impressed accelerating forces in the directions of the axes, E is the pressure on the curve, and (/, m, n) the direction- cosines of the direction of R. Then since R acts perpendicular to the curve ,df dv dz „ l-^+m^ + n-^ = 0. ds ds as Suppose the moving curve to be projected orthogonally on the plane of |, i;, let is the angle which r makes with the tangent. If p be the perpendicular drawn from the axis on the tangent, we have, therefore, if =— + < = lFOA. 4. Two equal uniform rods of length 2a, loosely jointed at one extremity, are placed symmetrically upon a fixed smooth sphere of radius Ja ^2, and raised into a horizontal position so that the hinge is in contact with the sphere. If they be allowed to descend under the action of gravity, show that, when they are first at rest, they are inclined at an angle oos~^ J to the horizon, that the points of contact with the sphere are the centres of oscillation of the rods relatively to the hinge, that the pressure on the sphere at each point of contact equals one-fourth the weight of either rod, and that there is no strain on the hinge. 5. A heavy uniform circular hoop of radius a and mass 2wam, which is com- pletely broken at one point, rolls with its plane vertical with uniform angular velocity u on a horizontal plane. Find the maximum and minimum values of the bending moment at any point Q of the hoop, and prove that if w be so large that the bending moment never vanishes, the greatest of these values will be ima? iin? {au^ + g), 26 being the angular distance of Q from the point of fracture. 6. Two straight equal and uniform rods are connected at their ends by two strings of equal length u, so as to form a parallelogram. One rod is supported at its centre by a fixed axis about which it can turn freely, this axis being perpen- dicular to the plane of motion which is vertical. Show that the middle point of the lower rod will oscillate in the same way as a simple pendulum of length a, and that the angular motion of the rods is independent of this oscillation. 7. A fine string is attached to two points A, B in the same horizontal plane, and carries a weight W at its middle point. A rod whose length is AB and weight W, has a ring at either end, through which the string passes, and is let fall from the position AB. Show that the string must be at least ^AB, in order that the weight may ever reach the rod. Also if the system be in equilibrium, and the weight be slightly and vertically displaced, the time of its small oscillations is 27r {ABjSgiJS)^. * These examples are taken from the Examination Papers which have been set in the University and in the Colleges, 188 MOTION IN TWO DIMENSIONS. [CHAP. IV. 8. A fine thread is enclosed in a smooth circular tube which rotates freely about a vertical diameter ; prove that, in the position of relative equilibrium, the inclination {8) to the vertical of the diameter through the centre of gravity of the thread will be given by the equation cos 9= — ^^—„, where o> is the angular velocity of the tube, a its radius, and 2a;8 the length of the thread. Explain the case in which the value of aw^ oos p lies between g and - g. 9. A smooth wire without inertia is bent into the form of a helix which is capable of revolving about a vertical axis coinciding with a generating line of the cylinder on which it is traced. A small heavy ring slides down the helix, starting from a point in which this vertical axis meets the helix : prove that the angular velocity of the helix will be a maximum when it has turned through an angle 6 given by the equation oos^9 + tan^o + e sin29=0, a. being the inclination of the helix to the horizon! 10. A spherical hollow of radius u. 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 be V, prove that V^=Sag + iag — . 11. A perfectly rough ball is placed within a hollow cylindrical garden-roller at its lowest point, and the roUer is then drawn along a level walk with a uniform velocity 7. Show that the ball will roll quite round the interior of the roUer, if V be >-^-g (b-a), a being the radius of the ball, and 6 of the roller. 12. AB, BC are two equal uniform rods loosely jointed at B, and moving with the same velocity in a direction perpendicular to their length ; if the end A be suddenly fixed, show that the initial angular velocity of AB is three times that of BG. Also show that in the subsequent motion of the rods, the greatest angle between them equals cos~'§ ; and that when they are next in a straight line, the angular velocity of BC is nine times that of AB. 13. Three equal heavy uniform beams jointed together are laid in the same right line on a smooth table, and a given horizontal impulse is applied at the middle point of the centre beam in a direction perpendicular to its length ; show that the instantaneous impulse on each of the other beams is one-sixth of the given impulse. 14. Three beams of like substance, joined together so as to form one beam, are laid on a smooth horizontal table. The two extreme beams are equal in length, and one of them receives a blow at its free extremity in a direction perpendicular to its length. Determine the length of the middle beam in order that the greatest possible angular velocity may be given to the other extreme beam. BesuU. If m be the mass of either of the outer rods, /Sm that of the inner rod, P the momentum of the blow, w the angular velocity communicated to the third rod, then maw (0+3+ 's)~^^^ ^^'"■"^ ^hen w is a maximum /3= J ^3. 15. Two rough rods ^, B are placed parallel to each other and in the same horizontal plane. Another rough rod G is laid across them at right angles, its centre of gravity being half way between them. If C be raised through any angle o and let fall, determine the conditions that it may oscillate, and show that if its ART. 213.] EXAMPLES. 189 length be equal to twice the distance between A and B, the angle through which = ) .sin a. 16. The corners A, B ot a, heavy rectangular lamina ABGD are moveable on two smooth fixed wires OA, OB, at right angles to each other in a vertical plane, and equally inclined to the vertical. The lamina being in » position of equilibrium with AB horizontal, find the velocity of the centre of gravity and the angular velocity produced by an impulse applied along the lowest edge CD. Having given that AB = 2a, BC=ia, prove that AB will just rise to coincidence with Hi wire, if the impulse is such as would impart to a mass equal to that of the lamina the velocity whose square is ^ga{2-,J2). Also find the impulsive stresses at A and B. 17. A ball spinning about a vertical axis moves on a smooth table and impinges directly on a perfectly rough vertical cushion ; show that the vis viva of the ball is diminished in the ratio lO + litau^^ : -y + 49 tan" ff, where e is the elasticity of the ball and B the angle of reflexion. 18. A rhombus is formed of four rigid uniform rods, each of length 2a, freely jointed at their extremities. If the rhombus be laid on a smooth horizontal table and a blow be applied at right angles to any one of the rods, the rhombus will begin to move as a rigid body if the blow be applied at a point distant a (1 - cos a) from an acute angle, where a is the acute angle. 19. A rectangle is formed of four uniform rods of lengths 2a and 26 respectively, which are connected by hinges at their ends. The rectangle is revolving about its centre on a smooth horizontal plane with an angular velocity n, when a point in one of the sides of length 2a suddenly becomes fixed. Show that the angular velocity of the sides of length 26 immediately becomes ^ tt »• Knd also the change in the angular velocity of the other sides and the impulsive action at the point which becomes fixed. 20. Three equal uniform inelastic rods loosely jointed together are laid in a straight line on a smooth horizontal table, and the two outer ones are set in motion about the ends of the middle one with equal angular velocities (1) in the same direction, and (2) in opposite directions. Prove that in the first case, when the outer rods make the greatest angle with the direction of the middle one pro- duced on each side, the common angular velocity of the three is ^u, and that in the second case after the impact of the two outer rods the triangle formed by them will move with uniform velocity gau, 2a being the length of each rod. 21. An equilateral triangle formed of three equal heavy uniform rods of length a hinged at their extremities is held in a vertical plane with one side horizontal and the vertex downwards. If after falling through any height, the middle point of the upper rod be suddenly stopped, the impulsive strains on the upper and lower hinges will be in the ratio of iJIS to 1. If the lower hinge would just break if the system fell through a height -p , prove that if the system fell through a height -t_ the lower rods would just swing through two right angles. 190 MOTION IN TWO DIMENSIONS. [CHAP. IV. 22. A perfectly rough and rigid hoop rolling down an inclined plane comes in contact with an obstacle in the shape of a spike. Show that if the radius of the hoop = r, height of spike above the plane = Jr and velocity just before impact = V, then the condition that the hoop will surmount the spike is V^>^gr {1- Bin. (a + It)}, a being the inclination of the plane to the horizon. Show that the hoop will not remain in contact with the spike unless V^<.i^gr . sin (a + Jir), and if it does, the hoop will leave the spike when the diameter through the point of contact makes an angle with the horizon = sin-i \^ hjsin (" + s)[ • 23. A fiat circular disc of radius a is projected on a rough horizontal table, which is such that the friction upon an element a is c Vhna, where V is the velocity of the element, m the mass of a unit of area : find the path of the centre of the disc. If the initial velocity of the centre of gravity and the angular velocity of the disc be u„, a„, prove that the velocity u and angular velocity w at any subsequent time satisfy the relation ( j;— = = — s 1 = -j. — . \3V-aH/ "oH 24. A heavy circular lamina of radius a and mass M rolls on the inside of a rough circular arc of twice its radius fixed in a vertical plane. Find the motion. If the lamina be placed at rest in contact with the lowest point, the impulse which must be applied horizontally that it may rise as high as possible (not going all round), without falling off, is M ^3ag. 25. A string without weight is coiled round a rough horizontal cylinder, of which the mass is 31 and the radius a, and which is capable of turning round its axis. To the free extremity of the string is attached a chain of which the mass is m and the length I ; if the chain be gathered close up and then let go, prove that the angle 8 through which the cylinder has turned after a time t before the chain is fuUy stretched is given by Ma0=^ (^ - oS ) . 26. Two equal rods AC, BG are freely connected at C, and hooked to A and B, two points in the same horizontal line, each rod being inclined at an angle a to the horizon. The hook B suddenly giving way, prove that the direction of the strain at C is instantaneously shifted through an angle tan~i ( = — ^ r-^ . — ^i^ \ . * \I + 6cos2a Ssmacosoy 27. Two particles A, B axe connected by a fine string ; A rests on a rough horizontal table and B hangs vertically at a distance I below the edge of the table. If .4 be on the point of motion and B be projected horizontally with a velocity u, show that A will begin to move with acceleration -Ar -r , and that the initial radius li + l I of curvature of JS's path will be (n + 1) I, where ix. is the coefficient of friction. 28. Two particles (m, m') are connected by a string passing through a small fixed ring and are held so that the string is horizontal ; their distances from the ring being a and a'. If p, p' be the initial radii of curvature of their paths when they are let go, prove that - = ^ , and - + i = - + i . 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 n, determine whether it will begin to slide or to roll. ART. 213.] EXAMPLES. 191 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 out are in the ratio Jl/+)resin^a : ilf cosa, m and M being the masses of the ring and circle. 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 G at the time t, prove '"=-£- {e'^ + e-"') + -= cos 2nt. 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 — P acceleration of either rod wiU be g m * '^ ' -±- + -±—Sa?cl 33. A smooth horizontal disc revolves with angular velocity JJi, 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 distance ; prove that the path on the disc is a cycloid. 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 the axial plane containing the insect makes with the vertical is given by o2(J2 (M+ 2m sin2 JS) = Mr^- 2mag sin^ JS, it being understood that the cylinder is very thin. If the internal radius be 6, prove ^2 \m (&2 + a?)+m (a? - 2a6 cos fl + 6=)] = C - 2mgh (1 - cos fl), where Ch^ [M(k'^ + a?) + m. (a - hf]= V'[M {k^+a')+ma [a - 1>)f. CHAPTER 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. 193 Let D and E be the middle points of the arcs AA', BB', and let DG, EG be ares of great circles drawn perpendicular to AA', BE respectively. Then clearly GA = GA' and GB = GB', and therefore since the bases AB, A'B' are equal, the two triangles AGB, A'GB' are equal and similar. Hence the displacement of G is zero. Also it is evident since the displacements of and G are zero, that the displacement of every point in the straight line OC is also zero. Hence a body may be brought from any position, which we may call AB, into another A'B' by a rotation about OC 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 I'Academie de Berlin 1750, and the Commentaires de Saint-Petersbcrwrg 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 + a0 + a^=O, through an angle 0, where 3-4 sin^ ^S = a^ + b^ + c,,. R. D. X', y', z' X «!, "a. "3 y K b„ 63 z "v •-'2. •-'3 13 194 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. The positive directions of al, 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 ExaminatUm for 1868. Take two points one on each of the axes of z and z' at a distance h from the origin. Their co-ordinates are (0, 0, h)(a3h, bgh, c^h), therefore their distance is h J2 (1 - C3). But it is also 2ft sin 7 sin JS; .-. 2 sin^p sin27=l-C3, where y is the angle 20«'. Similarly 2sm^ie sm^a = l-aj and 2 sin^ je sin2(3=l-62, whence the equation to find 6 follows at once. Ex. 2. Show that the equations to the axis may also be written in the form Cj + ttg Cg + 63 Cj - Oj - 61 + 1 ' 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'R 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 00, 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 d6 be the angle through which the body must be turned round the instantaneous axis to bring any point P from its posi- tion at the time t to its position at the time t + dt, then the ultimate ratio of dd to dt is called the angular 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 with the body and through the same space ; (2) a motion of rotation of the whole body about some aods through this assumed point P. This theorem and that of the central axis are given by Chasles. Bulletin des Sciences MatMmatiques par Femssac, vol. xiv. 1830. See also Poinsot, TMorie 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 ART. 221.J TRANSLATION AND ROTATION. 195 final positions. This last motion has been proved to be equivalent to a rotation about some axis through P'. Since these motions are quite independent, it is evident that their order may be reversed, i.e. we may first rotate the body and then trsinslate 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 hase point of the double operation. Hence the given dis- placement may be constructed in ah 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. Let the displacement of the body be represented by a rota- tion about an axis PR and a translation PP'. Let the same displacement be also represented by a rotation 6" about an axis QS 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 points 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 axes 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 of the paper intersect these axes at right angles in P and Q, then PQ=a. Let PP', QQ' represent the linear displacements of P and Q respectively, though these need not necessarily be in the plane of the paper. 13—2 196 MOTION OP A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. The rotation 6 about PR will cause Q to describe an arc of a circle of radius a and angle 6; the chord Qq of this arc is 6 2a sin ^ and is the displacement due to rotation. The whole dis- placement QQ' of Q is the resultant of Qq and the displacement PF of P. In the same way the rotation 6' about Q8 will cause 6' P to describe an arc, whose chord Pp is equal to 2a sin -^ . 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 6, 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 perpendicular to PR, the projection of QQ' on the axis of rotation is the same as that of PP'. Hence the projections cm the acds 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 9 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 a /3 Qq = 2a sin ^ in a direction making an angle — ^r— with the plane QPR, and a rotation which must be equal to 6 about an axis which must be parallel to PR. Hence a rotation about any aids may be replaced by an equal rotation about any parallel aons together with a motion of translation. 224. When the rotation is indefinitely small, the proposition can be enunciated thus : — a motion of rotation adt 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 Q8 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. ART. 226.] TRANSLATION AND ROTATION. 197 Let the given displacement of the body be represented by a rotation 6 about PR, and a translation PP'. Draw P'N perpendi- culai- to PR. If possible let this same displacement be represented by a rotation about an axis QS, and a translation QQ' along QS. By Ai-ts. 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 Q(^ 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 a NP' = 2a sin -r , or, which is more convenient, at a distance y from 9 the plane NPP' where NP' = 2y tan ^ . The rotation 6 round QS is to bring N to P' and is in the same direction as the rotation 9 round PR. 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 displapement of Q is really along QS. The rotation 6 round PR will cause Q to describe an arc whose chord Qq is a parallel to P'N and equal to 2a sin ^ . 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 198 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. 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 cmtral 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. 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 GB 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 cannot move only along GB. 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 (udt about an aads PR and- a translation Vdt in the direction PP' Measure a distance 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 draw Oa, Op, Oy parallel and equal to AA', BB', CC. If Op be the direction of the axis of rotation, the projections of Oa, Op, Oy on Op are all equal. Hence Op is the perpendicular drawn from on the plane o/Sy. 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 of 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 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. ART. 232.] COMPOSITION OF ROTATIONS. 199 230. To eocplain what is meant by a body having angula/r velocities about more than one axis at the same time. A body in motion is said to have an angular velocity &> about a straight line, when, the body being turned round this straight line through an angle a>dt, 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, OG meeting at a point through angles widt, a^dt, to^t. Then 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, respectively be ri,r.i,r3. First let the body be turned round OA, then P receives a displacement (OiVidt. By this motion let r^ be increased to r-j + dr^, then the displacement caused by the rotation about OB will be in magnitud'e m^ {r^ + 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^t. So also the displacement due to the remaining rotation will be w^^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 successively, through angles a^dt, m^dt, w^dt. Then when dt diminishes without limit the motion during the whole time will be accurately represented. 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, viz. the rotation shall appear to be in some standard direction to a spectator placed along the axis so that OA is measured from his feet at towards his head. This direction of OA is called the positive direction of the axis. 232. Parallelogram of angular velocities. If two an- gular velocities about two axes OA, OB be represented in magnitude and direction by the two lertgths OA, OB ; then the diagonal 00 of the parallelogram, constructed on OA, OB as sides will be the 200 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. resultant axis of rotation, and its length will represent the magni- tude of the resultant angular velocity. Let P be any point in 00, 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.PN = OP {OA . sinCOA - OB . sin (705} = 0. Therefore the point P is at rest and 00 is the resultant axis of rotation. Let to be the angular velocity about 00, then the velocity of any point A in OA is perpendicular to the plane AOB and is represented by the product of « into the perpendicular distance of A from OU = (o . 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 represented by the product of OB into the perpendicular distance of A from 0B= OB. OA sinBOA ; ,.o, = OB.'^^ = OG. smGOA Hence the angular velocity 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 the additional angular velocities impressed on a body at any in- stant, it follows that the diagonal OG 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 ART. 234.] COMPOSITION OF ROTATIONS. 201 and in the same way as if they were forces. Thus an angular velocity to about any given axis may be resolved into two, a cos a and ft) 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 m^, lo.^, w^ about three axes Ox, Oy, Oz at right angles, they are together equivalent to a single angular velocity w, where o) = V&Ji^ + m^ + wi, 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 a), to' about two parallel axes OA, O'B distant a from each other, to find the resulting 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"C lying in the plane containing OA, O'B. Let X be the distance of this axis from OA, and suppose it 0'- 0"- ■B -G to be on the same side of OA as O'B. Let fl 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 ay, the velocity due to the rotation about O'B is ft)' (y — a). But these two together must be equivalent to the velocity due to the resultant angular velocity fl about 0"G, and this is ii (y — OS), ■■• «2/ + ft>' (2/ - a) = fi (2/ - «). This equation is true for all values of y, .•. H = &) + w', a? = 7^- • 202 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. This is the same result we should have obtained if we had been seeking the resultant of two forces w, (o' acting along OA, O'B. If o) = — ft)', the resultant angular velocity vanishes, but x is in- finite. The velocity of any point P is in this case my + a)' {y — a) = am, which is independent of the position of P. The result is that two angular velocities, each equal to m 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 am. 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 m about an aocis OA is equivalent to an equal motion of rotation about a parallel axis O'B plus a motion of translation am 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 m is equivalent to an equal angular velocity about a parallel axis together with a linear velocity equal to mp, 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 (Jetermine the magnitude of either. There will also be a conven- tion in either case to determine the positive direction of the line. ART. 236.] THE ANALOGY TO STATICS. 203 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. In 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 d)mamics, we 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 ii 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 J. a new couple G' = '^G'^ + RV, the force being the same as before. The couple G' is a minimum when c = 0, Le. 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 fl from the central axis to act round AB, provided that we intro- duce a new linear velocity represented by flc. 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'D,\ the angular velocity being the same as before. The linear velocity V is a minimum when c = 0, i.e. when AB coincides with the central axis. We see 204 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. 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 ion: 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 R 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 = OR. 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 c* about an axis which we may choose at pleasure and another a 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 fl would 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 w, m' and a be the shortest distance between these axes, then cow'a sin0= FO. 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 cox, my, tuj, 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 directions of &)»,, ay, cog, are respectively from y to z, from ^ to a; 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, (o^ are given. These six quantities may be called the components of the motion. We now propose to find the motion of any point P whose co-ordinates are x, y, z. ART. 239.] THE VELOCITY OF ANY POINT. 205 Let us find the velocity of P parallel to the axis of z. Let PN be the ordinate of z and let PN: be drawn perpendicular to Ox. The velocity of P due to the rotation round Ox is clearly coxPM. Resolving this along iVP we get co^PM smNPM = a^y. Similarly that due to the rotation about Oy is — ooyX and that due to the rotation about Oz is zero. Adding the linear velocity w of the origin, we see that the whole velocity of P parallel to Oz is w ='w + Wxy — a>yX. Similarly the velocities parallel to the other axes are m' = M + coyZ — a>zy, v' = V -\- togoo — (Os^. 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 (^, 77, 5") be the co-ordinates of 0' and let u', v' , w', <»a;', o>y', o>z 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 + coyZ - (o-iy = w' + wy (^ - f ) - &)/ {y - ■??), V + (OgX - a^z = t)' + 6)/ (« — ^) - fOx {z — 0> w + asxy — a>yX = w' + coj {y — v) — «/ i"" — ?)> must be true for all values of x, y, z. These equations give Wx = a>x, a>y' = Oy, «/ = Wz ; so that what- ever base is chosen the angular velocity is always the same in direction and magnitude. See Art. 221. We also see that »', v, w' 206 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. are given by formulae analogous to those in Art. 238, as indeed might have been expected. 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, Y, 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 f , i), ^ are X' = X, L' = L +7t;-Z7], Y'=Y, M' = M + Z^-X^, Z' = Z, N' = N' + Xv-Yl 240. To find the equivalent Screw. The motion being given by the linear velocities {a, v, w) of some base 0, and the angular velocities (ma;, o)y, w^), find the central aads, 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 £1 be the resultant of the angular velocities Wx, (Oy, m^ we see that (1) The direction-cosines of the central axis are cosa = -^, cos^ = ^^ cos7=^. (2) The angular velocity about the central axis is Xl. (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 cosa-t-« cos/S + w cos 7; .■. VQ, = UMx + v' _ V sin 7' ~ sin 7 ~ sin a ' cos y ~ cos 7 sin a ' The first set follows from Art. 237. The second expresses the fact that the direction of the Unear motion of the point where the axis outs the shortest astance is along the axis of the screw. Ex. 10. An instantaneous motion is given by the linear velocities («, v, w) along, and the angular velocities {w^, wy, z be the angular velocities about the axes. Then x = coyZ — co-iy, y = eoja; — eoxZ, 2 = cxy — OyX ; :. X = zioy — yioz 4- Wy (w^y — o>yx) — oig {(o^ — (o^z), y = xioz — zwa 4- (»2 {tOyZ — m^y) — cox (a>xy — Wyx). Substituting in equation (1) we get %m (a^ + y^) (Og — Xnhyz . (by — ^rrta^ . a^ + ^rrhipz . a)y(o^ _ „ — Irhtrnj . {(Ox — &>/) -I- 2m {x^ — y") (OxOOy — Xnhyz . cu-ctBa J The other two equations may be treated in the same manner. The coefficients in this equation are the moments and products of inertia of the body with regard to axes fixed in space and are therefore variable as the body moves about. Let us then take a second set of rectangular axes OA, OB, OG fixed in the body, and let (Bi, 0)2, wj be the angular velocities about these axes. Since the axes Ox, Oy, Oz are perfectly arbitrary, let them be so chosen that the axes OA, OB, 00 are passing through them at the moment under consideration. 249. The axes of reference OA, OB, OG move in space. We suppose the motion determined by the three angular velocities 6>i, .(»2, (03 in the same manner as if the axes were fixed for an instant in space. The position of the body at the time t + dt may be constructed from that at the time t by turning the body through the angles (Oidt, (o^dt, (o^dt successively round the in- stantaneous positions of the axes. 214 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. Let fl be the angular velocity of the body and let OR be the instantaneous axis of rotation. Then since the two sets of axes coincide at this instant, we have the resolved angular velocities also equal, i.e. m^^a)^, Wy — as^, a^ = Wi. But at the time t-\-dt the two sets of axes have separated, so that we can no longer assert that any two components such as Wi + dwz and Wj + cZwj are necessarily equal. We shall now show that if the moving axes are fixed in the body, then dws = dco^ as far as the first order of small quantities. Let OR, OR' be the resultant axes of rotation of the body at the times t and t + dt, i. e. let a rotation ndt about OR bring the body into the position in which OG coincides with Oz at the time t; and let a further rotation D,'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 OG'. Then according to the definition of a differential coefficient dcos ,. .M' cosR'G' -n cos RG W = ^^^^* It • dcog , . .^ O' cos R'z — CI cos Rz _=U^,t g^ . The angles RG 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', hence the angles R'G' and R'z are also equal. Hence these differential coefficients are also equal. 250. The following demonstration of this equality has been given by the late Professor Slesser, and is instructive as founded on a different principle. Let A, B, C be the points in which the principal axes cut a sphere whose centre is at the fixed point. Let OL be any other axis, and let be the angular velocity about it. Let the angles LOA, LOB, LOG he called respectively o, ;8, y. Then by Art. 233 ii = ii)i cos a + Wj cos /3 + (1)3 cos 7 ; dii . , . •■■ ^ = Wi COS o + W2 cos p + W3 cos 7 - Uj sin aa - (1)3 sin ;3jS - £1)3 sin yy. Now let the line OL be fixed in spaee and coincide with 00 at the moment under consideration. Then a=|,/3=|,7=0; therefore d = a^-u^a- a^. Also a is the angular rate at which A separates from a. fixed point at C, this is clearly w^. Similarly /3= - w^. Hence fizrcig. Thus Wj.=«i, Uy=ii^, u^=Us. 251. Euler's dynamical equations. We have now proved that we may substitute in the equations of motion for co,;, &c. the angular velocities Wi, &c. about a set of axes OA, OB, OG fixed in the body and moving with it. The advantage of this transformation is that all the moments and products of inertia which occur in the equation are now constants. ART. 254.J euler's equations. 215 We can make a further simplification by properly choosing these axes in the body. Let us choose as the axes fixed in the body the principal axes at the fixed point 0. In this case the products of inertia are all zero. li A, B, be the principal moments the equations take the simple form Similai-ly A -^ — (B — G) co^co-i = L, B^-{G-A)w,w, = M. These ai'e called Euler's equations. 252. 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 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, ii A, B, G he 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. 253. Ex. 1. If 2T=Aui^-k-Bio^+Gui^ and G be the moment of the impressed forces about the instantaneous axis, Q the resultant angular velocity, prove that Ex. 2. A body turning about a fixed point is acted on by forces which tend to •prod/uce rotation about an axis at right angles to the instantaneous axis, show that the angular velocity cannot be uniform unless two of the principal moments at the fixed point are equal. 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 co-ordinates of the centre of gravity referred to rectangular axes fixed in space meeting at the fixed point, and let F, Q, R be the resolved parts of the pressures on the body in these directions. Let fi be the mass of the body. Then we have fia! = P + XmX 216 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. and two similar equations. Substituting for x its value in terms cnx, y, fflj we have /J, {zioy — ywg + Wy {(Oxy — (Oyx) — m^ (a^ — (OxZ)] = P + %mX 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 Wy and w^ from Euler's equations. We then have 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 iP . OH parallel to GH which is a perpendicular on the instantaneous axis 01, (2 being the resultant angular velocity, and the other 0'^ . GK perpendicular to the plane OGK, where GK is a perpendicular on a straight S — G C — A line OJ whose direction-cosines are proportional to — ^ — wj^s' — r~ "s^h A~B — -^— (»i(D2, and 0'* 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 angular velocities of the body about the three moving axes, OA, OB, OC. Let the fixed point be taken as the centre of a sphere of radius unity; let X, Y, Z and A, B, G be the points in which the sphere is cut by the fixed and moving axes respectively. Let ZG, BA produced if necessary, meet in E. Let the angle XZG = ■\^, ART. 256.] euler's equations. 217 ZG = 6, EGA = + -77- sm V sin 9 a* '^ at ^ ^ These two sets of equations are precisely equivalent to each other and one may be deduced from the other by an algebraic transformation. In the same way by drawing a perpendicular from E on OZ we may show that the velocity of E perpendicular to ZE is -^ sin^^, d- to E along EA is in the same way -^sinO^, and this is the same as -~ . Hence the whole velocity of A in space along AB is represented by -^ cos 6 + -^ . But this motion is also ex- pressed by «3. As before these two representations of the same motion must be equivalent. Hence we have If in a similar manner we had expressed the motion of any other point of the body as B, both in terms of coi, (03, Wg and 0, (j>, ■\fr, we should have obtained other equations. But as we 218 MOTION OF A EIGID BODY IN THREE DIMENSIONS. [CHAP. V. cannot have more than three independent relations, we should only arrive at equations which are algebraic transformations of those already obtained. 257. It is sometimes necessary to express the angular velocities of the body about the fixed axes OX, OY, OZ in terms of 0, . . Wj.= - -}- Bin ^+ -^ sin fl cos ^, OiZ Cut de , dd> . „ . , uy= — cos^ + -^ smflsm^, dt Qit dtp ^ d\f/ Sometimes it will be more convenient to measure the angular co-ordinates 6, 0, ^ in a different manner. Suppose, for example, we wish to refer the axes fixed in space to the axes fixed in the body as co-ordinate axes. To obtain the standard figure corresponding to this case, we must in the figure of Art. 256 inter- change the letters X, Y, Z with A, B, C, each with each. The angles 8, 0, ^ being measured as indicated in the figure after this change, the relations connecting them with the angular velocities about the axes fixed in space, are obtained from those in Art. 256 by simply changing wj, wj, Wj into - Wj., - Wj,, - u^ If we choose to measure B in the opposite direction to that indicated in the figure, the expressions for Wj., uy, become identical with those for Wj, w^, in Art. 256. 258. Ex. 1. If ^, g, ?• be the direction cosines of OZ with regard to the axes OA, OB, OG, show that two of Euler's geometrical equations may be put into the symmetrical form ^-gwa-H-W2=0, ^-'•ui+i"<'3=0. jf-P<^2+i'^i=0- Any one of these may be obtained by differentiating one of the expressions p= - sine cos 0, 2 = sinflsin0, r=oose. The others may be inferred by the rule of symmetry. Ex. 2. Prove that the direction cosines of either set of Euler's axes with regard to the other are given by the formulsB cos XA = - sin ^ sin + cos \j/ cos cos fi) BOB YA= cos ^ sin + sin \j/ cos cos $ ( , cos ZA = - sin e cos ) cosXB= -sin ^cos0-cos^sin0cosS) cos YB = cos ^ cos - sin xf/ sin cos S ; cos ZB = sin S sin oobXC= sin 5 cos ^^ cos YC = sin S sin ^ J cos ZC = cos 6 ART. 260.] euler's equations. '219 To prove the first three, produce XY to out AB in M, then the angle XMA = 8, MY=f, MX=W+^, MA = ^Q-3 are all small quantities, and we may neglect their products and squares. The general equation of Art. 248 reduces in this case to (7ft)3 — Dwa — Ew^ = N, where the coeflScients have the usual meanings given to them in Chap. I. We have thus three linear equations which may be written thus : Aioi — Fa^ — Ed>3 = L, - F(i)j, + Bcb^ - Dd)s = M, - Ed)^ - Dw^ + Gws = N. 260. The centrifugal forces. It appears from Kuler's Equations that the whole changes of u^, Wj, Wj are not due merely to the direct action of the forces, but are in part due to the centrifugal forces of the particles tending to carry them away from the axis about which they are revolving. For consider the equation da, N A-B Z3 — 7; H — 7r~ "i^s' dt ' N Of the increase da^ in the time dt, the part -^ dt is due to the direct action of A—B the forces whose moment is N, and the part — ^^ — wiWjdt is due to the centrifugal forces. This may be proved as follows. If a body be rotating about an axis 01 with an angular velocity a, then the moment of the centrifugal forces of the whole body about the axis Oz is (A - B) Uju^. Let P be the position of any particle m, and let x, y, z be its co-ordinates. Then x=OR, y = RQ, z=QP. Let PS be a perpendicular on 01, let OS=u, and PS=r. Then the centrifugal force of the particle m is uh-m tending from 01. The force ur'rm is evidently equivalent to the four forces or'xm, uhjm, uHm, and - uNm acting at P parallel to x,y,z, and u respectively. 220 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. The moment of ii^xm round Oz= - dfixym ahjm =(,r'xym M^jm =0 these three therefore produce no effect. The force - w^m parallel to 01 ia equivalent to the three, - aujum, - aw^um, -uwgum, acting at P parallel to the axes, and their moment round Oz is evidently (jum («i2/ - w^ic). Now the direction cosines of 01 being -^ , we get by projecting the broken line x,y,z on OI,u=-i x + -^y-] — ^z; therefore substituting )i, N' = (A - B) UiO>^, and let G be their resultant couple. The couple G is usually called the centrifugal couple. Since L'ai + M'u2 + N'ta^=0, it follows that the axis of the centrifugal couple is at right angles to the instantaneous axis. Describe the momental ellipsoid at the fixed point and let the instantaneous axis cut its surface in I. Let OH be a perpendicular from on the tangent plane at I. The direction cosines of OH are proportional to Aa^, Bia^, Cug. Since Aii)iL' + Bu^M'+CugN' = 0, it follows that the axis of the centrifugal couple is at right angles to the perpendicular OH. The plane of the centrifugal couple ia therefore the plane lOH. ART. 262.] EXPRESSIONS FOR ANGULAR MOMENTUM. 221 If yM^ be the moment of inertia of the body about the instantaneous axis of rotation, we have k'^=-^^, and T=ij.kV is the Vis Viva of the body. We may then easily show that the magnitude Q of the centrifugal couple is G = Tts,n(p, where ^ is the angle lOH. This couple will generate an angular velocity of known magnitude about the diametral line of its plane. By compounding this with the existing angular velocity, the change in the position of the instantaneous axis may be found. Expressions for Angular Momentum,. 261. We may now investigate convenient formulae for the angular momentum of a body about any axis. The importance of these has been already pointed out in Art. 75. In fact, the general equations of a motion of a rigid body as given in Art. 78, cannot be completely expressed until these formulae have been found. When the body is moving in space of two dimensions about either a fixed point, or its centre of gravity regarded as a fixed point, the angular momentum about that point has been proved in Art. 88 to be Jf^w where MJtf is the moment of inertia, and o) the angular velocity about that point. Our object is to find corresponding formulae when the body is moving in space of three dimensions. We shall show first how to find the angular momentum about a straight line which is such that one axis of reference (say, the axis of z) can be chosen parallel to it. We shall then find an expression for the angular momentum when the straight line is inclined to all three axes of reference. The former result has of course the advantage of simplicity and is therefore more generally useful. It is particularly important to find a simple form for the angular momentum of the moving body about a straight line fixed in space, for then we may use the general principle proved in Art. 78, viz. d /Angular momentum aboutN _ /Moment of im-\ dt\ a fixed straight line / \ pressed forces / " 262. Ang;ular Momentum about the axis of z. The instantaneous motion of a body about a fixed point is given by the angular velocities Wx, Oy, (o^ about three axes which meet at the 'point, find the angular momentum about the axis of z. Let X, y, z be the co-ordinates of any particle m of the body, and m', ?/', w' the resolved velocities of that particle parallel to the axes. Then by Art. 77 the moment of the momentum about the axis of z is A3 = Sm (xv' — yu!\ 222 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. Substituting u' = WyZ — a^y, v' = a^x - m^z from Art. 238, we have hs = Sot {x^ + 2/0 &»« - (Zmxz) tox — (Zmyz) Wy. Similarly the angular momenta about the axes of x and y are Ai = Sm (jf + z'^) (Ox - (Zmxy) Wy - {^mxz) a^, Aa = 2m (^^ +a^)(0y- (Zmyz) w^ - (tmyx) o)„. Here the coefficients of tox, Wy, m^ axe the moments and products of inertia about the axes which meet at the fixed point. 263. If there be no fixed point in the body we must use all the six components of motion. The fi)rm of the result depends on the point which is chosen as the base. The form is much simplified by choosing the centre of gravity as the base point, and for the reasons given in Arts. 74, 75 this is generally the most convenient point. Let Oz be the axis about which the angular momentum is required, and let Ox, Oy be two other axes, thus forming a set of rectangular axes. Let x, y, z be the co-ordinates of the centre of gravity. Let the instantaneous motion of the body be con- structed (as in Art. 238) by the linear velocities u, v, w of the centre of gravity parallel to the axes of reference and the angular velocities Wx, (Oy, cog round three parallel axes meeting at the centre of gravity. By Art. 75 the angular momentum about Oz is equal to that about a parallel axis through the centre of gravity regarded as a fixed point together with the angular momentum of the whole mass collected at the centre of gravity. The former of these has been found in the last Article and the latter is obviously M(xv — yu). The required angular momentum is therefore M (xv — yu) + Sm (a? -I- y') Wg — (Zinxz) Wx — {'S.myz) y, Wz in the expression for h^ will generally be variable and their changes may be governed by complicated laws. In such a case it is more convenient to choose axes fixed in the body, and this is the choice made by Euler in his equations of motion, Art. 251. Suppose a body to be moving about a fixed point 0, and let its instantaneous motion be given by the angular velocities ft)i, Wo, 6)3 about axes Ox', Oy', Oz fixed in the body. Then the angular momentum about the axis of / is A3' = (7ft)3 — Ewi — DtOi, where C, E and D are absolute constants, viz. G = l,m{x'^-Vy'% E = 1mafz', I) = X'm'if^. If the axes fixed in the body be principal axes, then the products of inertia vanish. These expressions for the moments of the momentum will then take the simple form hi = -4a)i, ^2' = -Bft)2, A3' = (7t»s, where A, B, G are the principal moments of the body at the origin supposed to be fixed in space. 265. From these results we may deduce a rule to find the angular momentum about an aoois fixed in space in a form which is often more convenient than that given in Art. 262. Supposing a body to be turning about a fixed point 0, we look for a set of axes Ox', Oy', Oz' such that we may easily find the angular momenta of the body about them. These will generally be some axes fixed in the body. Let the axes fixed in space about which the angular momenta are required be Ox, Oy, Oz. Let the direction cosines of either with regard to the other be given by the diagram; where for example 63 is the co- sine of the angle between the axes of z and y' (see Art. 217). Let the momenta of all the par- ticles of the body be equivalent to the three "couples" hi', h^', h, about the axes Oxf, Oy', Oni. Then the moment of the momentum about the axis Oz may be written in the form A3 = Ai'Ms + ^63 + As'Ca (1). In the same way we have A3' = AiCi + A2C2 + A3C3 (2). The simplicity of this process depends on the proper choice of the subsidiary set of axes Od, Oy', Oz'. Generally the most convenient axes to choose are the principal axes of the body at the point 0. In this case the equation (1) takes the form A3 = J.6)ia8 + Bw^i + OwjCs. X y z a! Oi a^ «3 y h h &3 z! Ci Ci Cs 224 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. We may now substitute for Wi, w^, cos their values given by Euler's geometrical equations (Art. 256). The values of the direction cosines as, 63, c^ are written at length in Ex. 2, Art. 258. When the body is uniaxal, so that the two principal moments of inertia A and B are equal, these results take a very simple form. The substitutions are rather long though not difficult, but we may greatly shorten them by taking the axes Ox, Oz' to co- incide with OE, 00 in the figure of Art. 256. Oy' will then be perpendicular to both OE and OG. These also are principal axes since the body is uniaxal, thus V> ^2', V have the simple forms J.ft)i, AfOi, Ca)3 while the direction cosines are formed by very simple trigonometrical formulae. In fact the direction cosines are those found by putting (^ = in the general forms given in Ex. 2, Art. 258. In this way we find h^=A ]-sin-v/r T- -sin6 cos0 cosi^-^I + Cms sin0 cosi/r, A2 = .4 I cos i/r -5- - sin cos sin i/r -^ • + Gws sin 6 sin i/r, h, = A sitf^^ + (7«sCOS(9. We might substitute for 0)3 its value given by Euler's third geometrical equation, but this would introduce d(\>ldt into the equation, and it will generally be found more convenient to retain 0)3. In this way the angular momenta of a uniaxal body about any straight lines are expressed in terms of the direction angles of the axis of the body and the angular velocity about it. We may find a geometrical meaning for these results which will at once supply us with an easy proof of them and enable us to write them down when wanted. The angular momenta of the body about the principal axes at the fixed point are Au^, Au^, Cioj. Suppose we attach to the axis OG one or more imaginary particles so that their united moment of inertia about any axis through perpen- dicular to 00 is equal to A. Let these particles move about with the axis. The motion of the axis is given by the angular velocities wi, wj and therefore the angular momenta of these particles about the axes OA, OB are clearly Auy, Aw^ These are the same as those of the body. The 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 Cuj about OC It follows by the "parallelogram law" that the same equality holds for all axes. Hence the angular momentum 0/ a uniaxal body about any axis through is the same as that of one or more particles arranged along its axis (so that their united moment of inertia about is equal to A) together with the angular momentum Cwj about the axis. 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 A. Let (fijf) be the oo-ordina*<«' ART. 265.] EXPRESSIONS FOR ANGULAR MOMENTUM. 225 of this particle referred to the axes x, ij, z, then (fT/f) are also the direction cosines of the axis. The angular momenta about the axes are therefore -f-4:)-^"»^- We have now to write for {, r/, f their values {=sin9cos^, )7=sinflsin^, f = cos ff. The substitution in the last equation is easily effected if we remember the rule in the differential calculus fdi; - ijdf =rt^. See Art. 77. In this way we arrive at the same results for the angular momenta hi, h^, h^ as before. If the uniaxal body is making small oscillations and the axis OG is always so nearly coincident with the axis On that we can reject the squares of 6, we have 1 = 003^ 17=9 sin ^ {■=!, hi=-A^ + Gu,s^] hi= A-^^ + Cu^tj These are very simple formulse 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 two 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 Cu, 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^=i(B + C-A), Mb''=i(G + A-B), Mc'=i{A + B-G). 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 § a sin a/sin j3 where a is the length of the rod, » the semivertical 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 the centre of gyration. 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. R.D, • 15 226 MOTION. OP A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. Ex. 3. A body is turning about a fixed point and has all its principal moments of inertia at equal. If 0, (p, ^ be the Eulerian coordinates of the axes OA, OB, OG, fixed in the body, show that the angular momenta about the axes fixed in space are respectively (de . dd>\ -sm^^+smecos^^j, ft2=.4 ( cosi/'^ + sinflsmi/'^l, 266. Angular momentum about any axis. The motion of a body is given by the linear velocities (u, v, w) of the centre of gravity and the angular velocities (wx, o>y, »z), prove that the angular momentv/m about the straight line — ^ = - — — = is equal to m n Ihi + mhj + nhj + M 1 m n I V w ■ g li where M is the mass of the body, hj, hj, hj have the values given in Art. 262, and (1, m, n) are the actual direction cosines of the given straight lime. 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 Ih^ + mh^ + nh^. 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 co-ordinate 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 (n, I, m) respectively, we have the moment of the velocity of the centre of gravity about the straight line. Multiplying this by M we have the angular momentum of the mass at the centre of gravity. The required result follows at once. 267. To find the angular momentum of a body about the instantaneous axis and also about any perpendicular axis which intersects the instantaneous axis. Taking the instantaneous axis for the axis of ^, we may use the expressions for hi,hi, h^ given in Art. 262. ART. 267.] EXPRESSIONS FOR ANGULAR MOMENTUM. 227 In our case m^— 0, Wy- 0, and m^ = O, where O is the resultant angular velocity of the body. The angular momenta about the axes of X, y, z are therefore respectively ^ = - (l,mxz) Xi, h^ = - (Zmyz) 12, A3 = Sm (a^ + y^ fl. 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 <0x, <0y, m^ 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 there- fore a moment about Ox, although Ox is perpendicular to the instantaneous axis. Let 6 be the angle PMN, r = PM, then .dO dz dy do so that the angular velocity t- of the particle P about Ox vanishes when toa, = and ay=Q only when the particle lies in either of the planes xy or yz. Examples. A triangular area AGB whose mass is M is turning round the side CA with an angular velocity u. Show that the angular momentum about the side CB is rf^Mdb sin'' Cu, where a and 6 are the sides containing the angle C. Ex. 2. 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 so that the two rods are in a vertical plane. If B,

?+gl)Q-CmP=0\ Cn^iQ-(A^l?+gl)P=Of■ These give An^+gl= ±Gnft.. It is unnecessary to take both the signs on the right hand side. If we choose one sign the effect of the other sign is merely to change the sign of /i and this merely alters the as yet undetermined constants Q and /. Without loss of generality we may choose the uj^er sign. This makes both the resulting values of /i positive. It also gives P= Q. The values of /t, are _Cn^ {ay-igAl)^ '^~2A 2A Bepresenting these two by /;t=/^ and /j^ we have |=Pi cos {fijt +/i) +P^ cos (iitat 4-/2) i7=Pi sin (ftt+Zi) +P2 si" (m2*+/2) where P„ P^tf^fi are four constants to be determined by the initial values of |, ij, |, ij. Let us represent the initial values of the co-ordinates by the suffix zero. Then fo=PlOOS/i-t-P2COS/2 7;o=PiSin/i+P2Sin/2 - io = Pilh. sin/i +-P2/*2 siii/2 % =-Pi/*i oos/i+Pa^ C0S/2. These give -Pi' (Ih - M2)"= (% - Ihiof + do + Wo)') Ps'(ft-A«2)'=('7o-ftlo) + (io+Mo)' I ■ If fl, ^ be the angular co-ordinates of the axis we have fl2=|i!.(- V''=PiHP2H2PiP2Cos {((iJi-;«2) t-J-A-Za)} 92^=^77 - kn=Pih + Pi^l^i + PiPi (h+IH) COS {(Ml -fti) «+/i -fi)- Supposing Pi and Pj not to be equal we see that d can never vanish, i. e. the axis of the top can never become strictly vertical. Also ^ will never vanish unless ■PA K+A4) « greater than P^ii^+P^jx^ i.e. the plane ZOO 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 /Uj//*! and unity. In order that Pi=P2 it is necessary that initially 2 (ft - Aij) (f4 - H = W - IH^) (f ' + ')')• This requires that ^ should initially differ from 4 (ft + Ma) ^y small quantities of . the order P. In this case yji will keep one sign throughout the motion and the axis will become vertical at a constant interval equal to 2tI{ii^- /i^). We have assumed that the values of /it are both real and unequal. If the value of n be so small that the values of 11 are imaginary, the values of f and ij will contain real exponentials. In this case the values of | and j; do not in general remain smaU. This indicates that the top has not sufficient rotation about its axis to keep the axis vertical. It will fall away from its initial position. 230 MOTION OF A EIGID BODY IN THREE DIMENSIONS. [CHAP. V. If CH?=igAl the two values of /t are real and equal. In this case it will be seen that the equations are satisfied by I = Pj cos (/*« +/i) + Pat cos (nt +/a) t; = Pj sin (nt +/i) + Pjf sin (/it +fji, so that the motion is in general unstable. The axis of the top cannot remain nearly vertical unless the initial conditions are such that Py=asaie, <<)j=0. Remembering that these angular velocities are constant, the general equation of moments of Art. 248 becomes ~2mxy (u^-<,iy^) + 'Sm(x^-y^ u^y=N. To find N, we resolve the weight Mg parallel to the axes, then X= -MgaoaO, Y= - Mg sin 9, 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 w2 {cos 2ei>mxy - J sin 2flSm {x^ -y^)}=Mg {xiiaff-y cos S). The integrals Xmxy and 2m (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^th uniform angular velocity a horizontal circle whose centre is in the vertical axis. If r be the radius of this circle the effective force on the element is BiwV and its direction is along the radius. The body may therefore be regarded as being in eqmlibriimi 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 ab'out 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 ft^u^ > gh, where h is the distance of the centre of gravity of the disc from Oz and k 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 8 with the vertical, the angular velocity w must be . / — ? — . V o sm e 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 whose length is the vertical distance 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, i the radius of a sphere, h the depth of either centre below the hinge, then the length of the pendulum is —-^(^ + '')'' +4'"^" . l + r M{l + r) + iml ART. 271.] FINITE ROTATIONS. 233 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 yery important in rigid dynamics. We shall therefore only 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. Bodrlgues' Tbeorem, A body lias two rotations, (1) a rotation about an axis OA through an angle 8; (2) a subsequent rotation about an axis OB through an angle $', and both these axes are fixed in space. It is required to compound the rotations. Let lengths measured along OA, OB represent 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 A and B. On this sphere measure the angle BA G equal to ^8 in a direction opposite to the rotation round OA and also the angle ABC equal to ^0' in the same direction as the rotation round OB, and let the arcs intersect in G. Lastly, measure the angles BAG', ABG' respectively equal to BAG, ABG, but on the other side of AB. The rotation 8 round OA mil then carry any point P in OC into the straight line OC", and the subsequent rotation 8' about OB will carry the point P back into 00, Thus the points in 00 are unmoved by the double rotation and OG is there- fore the axis of the single rotation by which the given displacernent of the body may be constructed. The straight line 00 is called the resultant axis of rotation. If the order of the rotatious 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 8 about OA would bring OB into a position OB'. Then the body may be brought from its first into its last position by rotations $, ff 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. To find the magnitude 8" of the rotation about the resultant axis 00 we notice that if a point P be taken in OA, it is unmoved by the rotation 8 about OA, and the subsequent rotation 8' about OB will bring it into the position P', where PP' is bisected at right angles by the plane OBG. But the rotation 8" about OC must give P the same displacement, hence in the standard case 8" is twice the external angle between the planes OCA, 0GB. 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 234 MOTION OF A RIGID BODY IN THREE DIMENSIONS, [CHAP. V. of rotation depends on the order of rotation the resultant angle of rotation is independent of that order. 272. 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 ir, these two rotations only differ by 2ir ; and it is evident that a rotation through an angle 2ir cannot alter the position of any point of the body. This is merely another way of Baying 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. 273. The rule for compounding finite rotations may be stated thus : If ABC be 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 DC from B to A through twice the internal angle at 0. 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. 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. 1. If two rotations 9, 6' about two axes OA, OB at right angles be com- pounded into a single rotation (p about an axis OC, then tan (704 = tan — ooseo g , tan COB = tan ^ cosec jj , and cos^=cos j;Cos-=-. 274. Sjrlvester's Tbeorem. From Rodrigues' theorem we may deduce Sylves- ter's theorem by drawing the polar triangle A'B'C Since a side B'C is the supplement of the angle A, a rotation represented in direction and magnitude by 2B'C' differs from that represented by 2A in the opposite direction by a rotation through an angle 27r. But a rotation through 27r cannot alter the position of the body, hence the two rotations 2B'C' and 2 A are equivalent in magnitude but opposite in direction. If therefore A'B'C he any spherical triangle, a rotation represented by twice B'C followed by a rotation twice C'A' produces the same displacement of the body a/s a rotation twice B'A'. By a rotation B'C is meant a rotation about au axis perpendicular to the plane of B'C which will bring the point B' to C. 275. The following proof of the preceding theorem was given by Prof. Donkin in the Fhil. Mag. for 1851. Let ABG be any triangle on a sphere fixed in space, ART. 278.] FINITE ROTATIONS. 235 a/37 " triangle on an equal and conoentrio sphere moveable about its oentte. The sides and angles of 0/87 are equal to those of ABC, but differently arranged, one triangle being the inverse or reflection of the other. If the triangle 0187 be placed in the position I, so that the sides containing the angle a may be in the same great circles with those containing A, it is obvious that it may slide along AB into the position n, and then along BC into the position III; into which last position it might also be brought by sliding along AC. To sMde 0187 along AB is equivalent to moving jS and a each through an arc twice the arc AB about an axis perpendicular to the plane of AB. A similar remark applies when the triangle slides along BC or AC. Hence, twice the rotation AB followed by twice the rotation BG produces the same displacement as twice the rotation AG. 276. Rotation Couples. If it be required to compound the rotations about two parallel axes, the construction of Bodrigues requires only a slight modification. Instead of arcs drawn on a sphere, let planes be drawn through the axes making with the plane containing the axes the same angles as before; their intersection will be the resultant axis. One case deserves notice. If 6=-S', the resultant axis is at infinity. A rotation about an axis at infinity is evidently equivalent to a translation. Hence a rotation 8 about any axis OA followed by an equal and opposite rotation about a parallel axis O'B distant a from OA is equivalent to a translation 2a sin ^d perpendicular to a plane through 0^ making an angle ^d with the plane containing the axes and in the direction of the chord of the arc described by any point in OA. These results also follow easily from Art. 223. 277. Conjugate Rotations. Any given displacement of a body may be repre- sented by two finite rotations, one about any given straight line and the other about some other straight line which does not necessarily intersect the first. When a dis- placement is thus represented, the axes are called conjugate axes and the rotations are called conjugate rotations. Let OA be the given straight line, and let the given displacement be represented by a rotation ^ about a straight line OR and a translation OT. We wish to resolve this rotation about OR into two rotations, one about OA to be followed by a rotation about OB, where OB is some straight line perpendicular to OT. To do this we follow the rule in Art. 271, we describe a sphere whose centre is and radius unity and let it intersect OA, OR, OT in A, B and T. Make the angle ARB equal to the supplement of 5 , and produce BB to B so that TB = -=, and join AB. By the triangle of rotations the rotation tp is now represented by a rotation about OA which we may call B, followed by a rotation about OB which we may call 0'. By Art. 276 the rotation B' is equivalent to an equal rotation B' about a parallel axis GD, together with a translation, which may be made to destroy the translation OT. This will be the case if the angle OT makes with the plane of OB, CD be i{ir-B') on the one side or the other of OT according to the direction of the rotation, and if the distance r between OB, CD be such that 2r sin4e'= OT. The whole displacement has thus been reduced to a rotation 6 about OA followed by a rotation B' about CD. 278. Composition of Screws. Any two successive displacements of a body may be represented by two successive screw motions. It is required to compound these. Let the body be screwed first along the axis OA with linear displacement a and angle of rotation 0, and secondly along the axis CD with displacement a' and angle 236 MOTION OF A RIGID BODY IN THREE DIMENSIONS. [CHAP. V. e'. Let OG be the shortest distance between OA and CD, and for the sake of the perspective let it be called the axis of ]). Let be the origin and let the axis of x be parallel to CD, so that OA lies in the plane of xz. Let OC=r, and the angle AOx=a. Draw a plane xOT making with the plane of xz an angle Jfl', and let it cut yz in OT. Draw another plane AOE making with xz an angle ^ff , and cutting the plane xOT in OR. Produce ^10 to a point P, not marked in the figure, so that PO=a, and let us choose P as a base point to which the whole displacement of the body may be referred. The rotation 6' is equivalent to a rotation ff about Ox together with a translation along OT = a- sin J 9' by Art. 223. By Art. 271 the rotation 8 about OA followed by d' about Ox is equivalent to a rotation O about OR where is twice the 1 iT,m ii_ i - i^ . e smAx angle ART, so that sm -=smT;.-; — =- . 2 2 smija; The whole displacement is now repre- sented by (1). a translation of the base point P to 0, (2) the rotation £2, (3) a further linear translation which is the resultant of the translations 2r sin J9' along OT and a' along Ox. By Art. 219 these displacements may be made in any order, being all connected with the same base point. They may therefore be compounded into a single screw by the Tule given in Art. 225. This is called the resultant screw. A screw equal and opposite to the resultant screw will bring the body back to its original position. The angle of rotation of the resultant screw is fi and its axis is parallel to OR by Art. 220. It follows by Art. 272 that the sine of half the angle of rotation of each screw is proportional to the sine of the angle between the axes of the other two screws. To find the linear displacement along the axis of the resultant screw, we must by Art. 222 add together the projections on OR of the three displacements OT, a, a'. The projection of OT =2r am ie' cos TR = 2r cos Ty .coa TR, which is twice the projection of the shortest distance r on the axis of rotation. If T be the linear displacement, we have T=2r cos Ry + a cos RA + a' ooaRx. 279. If the component screws be simple rotations, we have a=0, a'=0, and it Q ~ff ft' may be shown without difficulty that T Bin;j=2rsin5sin -sino. It has been A 2 2 shown in Art. 277 that any displacement may be represented by two conjugate rotations in an infinite number of ways, but it now follows that in all these ABT. 280.] FINITE ROTATIONS. 237 a at r sin 2 sin - sin a is the same. When the rotations are indefinitely small, and equal to udt, u'dt respeotively, this becomes Jruu' {dt)^ sin a ; that is, the product of an angular velocity into the moment of its conjugate angular velocity about its axis is the same for all conjugates representing the same motion. Ex. 1. If the component screws be simple finite rotations, show that the equations to the axis of the resultant screw are -a:tan0+)/sin^+«oos-=rsm-, ?/cos - -2sin- = rsin^oos0'cot-, where iff is the angle xOB and is the resultant rotation. The first equation expresses the fact that the central axis lies in a plane which bisects at right angles a straight line drawn from perpendicular to OB, in the plane xOR to represent the linear translation in that direction. The second expresses the fact that the central axis lies in a plane parallel to TOR at a distance from it determined by Art. 225. These equations may also be deduced from those of Eodrigues given in Art. 281. To effect this we must write for (a, b, e) the resolved parts of the translation along OT. Since however the positive direction of the rotation in Eodrigues' formula has been taken opposite to that chosen in the preceding article, we must write for {I, m, n) the direction cosines of OR with their signs changed. The equations to the central axis of any two screws may be found by either of these methods. Ex. 2. Let the motion be constructed by two finite rotations B, 6' taken in order round axes OA, CD at right angles to each other, and let CO be the shortest distance between the axes. Let the two straight lines OP, CP be drawn in the a ff R plane DCO such that the angle FOG=^ and tan PC0=sin2- cot-. Then if P i 6 6 be moved backwards by the rotation B (yc forwards by the rotation B\ in either case its new position is a point on the central axis. Ex. 3. If OA, OB be the axes of two screws at right angles, with linear dis- placements a and b, the point P is the intersection of two parallels to the straight lines described in the last example; these parallels being drawn respectively at distances ^ tan ^ and s ( 1- + "ot^ (p' sin^ o ) > where ^, f '• Then we have, as before, i'=x + -^&B., but since Sx, Sy, Sz, are increased by a, b, c we must write f ' - 5 > ''' ~ 2 ' ^' ~ 2 for f , ij, f. We thus obtain Sa;=a + 2tan- |™ U'- |) -» (V- gjl • with similar expressions for Sy and Sz. 281. The equations to the central axis follow from these expressions without difficulty. The whole displacement of any point in the central axis is along the axis, so that (|', V. D the co-ordinates of the middle point of the displacement are co-ordinates of a point in the axis, and Sx, Sy, Sz are proportional to {I, m, n) the direction-cosines of the axis. Hence .■H2tangf(r-|)-.(y-|)} ^^2 ta.| |. (,> - |) - Z (r-|)} I m . + 2tanHi(V-|)-m(|'-g} n Each of these is evidently equal to la + mb + nc, which is the linear displacement along the central axis. The results of this and the preceding Article are due to Rodrigues. 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 ^0 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 motions. 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 ti — to. If the time t^ — to 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 Ox, Oy, Oz. Then the six general equations of motion may, by Art. 72, be written in the form Integrating these from ^ = i„ to i = ^i, we have %m -YT = Sm Zdt, [^-('l-^S)],;-S'»/I'(«>'-»^)'"- Let' an accelerating force F act on a moving particle m during any time ti — to, 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 ti — to. Thus I mZdt is the whole force resolved parallel to the axis of Z. J tft These equations then show that 240 MOMENTUM. [CHAP. VI. (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 amy forces in the moment of the momentivm, 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 — t^ 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 rules of statics, then the mmnenta of all the par- ticles at any time tj 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 we multiply the mass ro 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. It may 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 compotmded with the viv of the particle. If two particles act and re-act on each other with a force R for a time dt, two equal and opposite momenta (viz. Bdt) 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^-tf,. 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 Conservation of Areas may be enunciated as follows. If the forces which act on a system be 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. ART. 285.] EXAMPLE OF A CENTRAL FORCE. 241 If the forces be such that they have no moment about a certain fixed straight line, then the moment of the momentum or the area conserved 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 mi/ 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 = i/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, ml move about the same centre of force. If ft, ft' be the double areas described by each per unit of time, prove that tiih + 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 mv, m'v', &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 attraction 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", ml'm., mm', taken in order. Then F is the resultant of — F 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. * This proof is merely an amplification of the following. The three forces F, F', P", being the internal reactions of a system of three bodies, are in equili- brium by D'Alembert's Principle. R. D. 16 242 MOMENTUM, [CHAP. VI. If the attraction be directly proportional to the distance, the two points 0, 0' coiacide with the centre of gravity 0, and are fixed in space throughout the motion. Eor 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 0. 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 Q. 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 -7 , . 286. Example of Iiaplace's Three Particles. Three particles whose masses are m, m', m", mutually attracting each other, are so projected that the triangle formed by joining their positions at any instant remains always similar to its original form. It is required to detei-mine the conditions of prelection. 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 moment of the momenta of the system about is always the same, we have mrhi + m'r'^n + mV'% = constant, where r, 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 mrhn is constant, i.e. OP traces out equal areas in equal times. Hence by Newton, Section ii, the resultant force on P tends towards 0. 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 ,-. ^ ,. . Then, since the resultant attraction of (dist.)* m', m" on m passes through 0, m' m" -^jSinP'PO = ^sinP"PO, but, since is the centre of gravity, m'p" sin P:P0 = m"p' sin P"PO. Hence either the three particles are in one straight line or p"*+'=p'*+i. K h=-l the law of attraction is "as the distance." If A; be not= -1, we have p'=p", and the triangle must be equilateral. ART. 286.] LAPLACE'S THKEE PARTICLES. 243 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 m' m" w!' TO m m' (PP7 {FF'f (P"P')* (PP')*' (VF")" {P'P") • must be proportional to OP, OP', OP". Since SmOP=0, these two equations amount to but one on the whole. Let 2 = ^^, so that Q£ _ '»' + '»" (1 + g ) OP' _ -m + m"z PP'~m + m'+m"' Then we have PP' PP' m+m' + m" which agrees with the result given by Laplace, by whom this problem was first considered. In the case in which the attraction follows the law of nature ife = 2 and the equation becomes mz^ {(1+2)3-1} -to' (1 + 2)2(1 -2:3) -m"{(l + «)3-z3}=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, m' the earth, to" the moon, we have very nearly 2= a/ — s = fj^- ^^ then, V oTO J.UU 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 + r^ , 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 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 Connaissanee dee Temps, 1845, that such a motion would be unstable. 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 th» 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. 16—2 244 MOMENTITM. [CHAP. VI. Thirdly. Let us suppose the particles to be at the angular points of an equi- lateral triangle. The resultant force on the particle to is ^ cos P'PO + ^ cos P"PO. p '* p * The condition that the forces on the particles should be proportional to their distances from O shows that the ratio of this force to the distance OP is the same for all the particles. Since m'p" cos P'PO + m"p' eoa P"PO = (m + to' + 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 with equal velocities in directions making equal angles with OP, OP', OP" respectively, they wiU always remain at the angular points of an equilateral triangle. A discussion of the stability of this motion will be given in a later part of this work. 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 equal ellipses having for a common focus. Find the periodic time. Ex. 2. If four particles be placed at the corners of a quadrilateral whose sides taken in order are a, h, c, d and diagonals p, p', then the particles cannot move under their mutual attractions so as to remain always at the corners of a similar quadrilateral unless (pV - fd") (c" + a") + (a"c» - pV") (*" + <*") + (^"d" - a"c») (p» + /») = 0, where the law of attraction is the inverse (n - !)"■ power of the distance. Show also that the mass at the intersection of b, c divided by the mass at the intersection of c, d is equal to the product of the area formed by a, p', d divided by the area formed by a, b, p and the difference s^- -^ divided by -j - t^ . These results may be conveniently arrived at by reducing one angular point, as A, of the quadrilateral to rest. The resolved part of all the forces which act on each particle perpendicular to the straight line joining it to A will then be zero. The case of three particles may be treated in the same manner. The process is a little shorter than that given in the text, but does not illustrate so weU the subject of the chapter. 287. When the system under consideration consists of rigid bodies we must use the results of Art. 74 to find the resolved part of the momentum in any direction. The moment of the momentum about any straight line may also be found by Art. 75 in Chap, ii., combined with Art. 134 in Chap, iv., if the motion be in two dimensions, or with Art. 262 in Chap, v., if the motion be in three dimensions. 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 reqxiired to find the changes which occur in the motion of the rest of the body. ART. 290.] SUDDEN CHANGES OF MOTION. 245 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 will supply 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 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. 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 Ox in its own plane with an angular axis o>. 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 advantage recur to first principles. Let da- be any element of the area of the disc ; y, y' its dis- tances from Ox, Ox'. Then yco, y'a' are the velocities of der just before and just after the impact. The moments of the momentum about Oaf just before and just after are therefore yy'wda and l/^m'da. Summing these for the whole area of the disc, we have a>'ty"'d(T = (otyy'da (1). 246 MOMENTUM. [CHAP. VI. Firstly, let Ox, Ox' be parallel, so that the point is at infinity. Let h be the distance between the axes, then y' = y-h. Hence we have w'ty'^da = w {l, \j/ have been chosen to make the component l8+m4> + ni + M(vh- wg) = A'H^ - F'D.y - E'n^, - F(o^ + Bmy - Dm, + M(wf- uh) = - F'D,^ + B'D.y - D'D.^, - Ea^ - B(Oy +G hence we have h = Sif/c^o) COS7 + tM (x^^ -y^\ . The values of ^1, h^ may be found in a similar manner. The position of the invariable plane is then known. 303. The Invariable Plane may be used in Astronomy as a standard of reference. We may observe the positions of the heavenly bodies with the greatest care, determining the co-ordi- ART. 304.] THE INVARIABLE PLANE. 257 nates of each with regard to any axes we please. It is, however, clear that, unless these axes are fixed in space, or if in motion unless their motion is known, we have no means of transmitting our knowledge to posterity. The planes of the ecliptic and the equator have been generally made the chief planes of reference. Both these are in motion, and their motions are, known to a near degree of approximation, and will hereafter probably be known more accurately. It might, therefore, be possible to calculate at some future time what their positions in space were when any set of valuable observations were made. But in a very long time some error may accumulate from year to year and finally become considerable. The present positions of these planes in space may also be transmitted to posterity by making observations on the fixed stars. These bodies, however, are not absolutely fixed, and, as time goes on, the positions of the planes of reference can be determined from these observations with less and less accuracy. A third method, which has been suggested by Laplace, is to make use of the Invariable Plane. If we suppose the bodies forming our system, viz. the sun, planets, satellites, comets, &c., to be subject only to their mutual attractions, it follows from the preceding articles that the direction in space of the Invariable Plane at the centre of gravity is absolutely fixed. It also follows from Art. 79 that the centre of gravity either is at rest or moves uniformly in a straight line. We have here neglected the attractions of the stars. These, however, are too small to be taken account of in the present state of our astronomical knowledge. We may, therefore, determine to some extent the positions of our co-ordinate planes in space, by referring them to the Invariable Plane, as being a plane which is more nearly fixed than any other known plane in the solar system. The position of this plane may be calculated at the present time from the present state of the solar system, and at any future time a similar calculation may be made founded on the then state of the system. Thus a knowledge of its position cannot be lost. A knowledge of the co-ordinates of the Invariable Plane is not, however, sufficient to determine conversely the position of our planes of reference. We must also know the co-ordinates of some straight line in the Invariable Plane whose direction is fixed in space. Such a line, as Poisson has suggested, is supplied by projecting on the Invariable Plane the direction of motion of the centre of gravity of the system. If the centre of gravity of the solar system is at rest or moves perpendicularly to the Invariable Plane, this method fails. In any case our knowledge of the motion of the centre of gravity is not at present sufficient to enable us to make much use of this fixed direction in space. 304. If the planets and bodies forming the solar system can be regarded as spheres whose strata of equal density are con- centric spheres, their mutual attractions act along the straight R. D. 17 258 MOMENTUM. [CHAP. VI. lines joining their centres. In this case the motion of their centres is the same as if each mass were collected into its centre of gravity, while the motion of each about its centre of gravity would continue unchanged for ever. Thus we may obtain another fixed plane by omitting these latter motions altogether. This is what Laplace has done, and in his formulae the terms depending on the rotations of the bodies in the preceding values of /h, Aa, h^ are omitted. This plane may be called the Astronomical Invariable Plane to distinguish it fi-om the true Dynamical Invariable Plane. The former is perpendicular to the axis of the momentum couple due to the motions of translation of the several bodies, the latter is perpendicular to the axis of the momentum couple due to the motions of translation and rotation. The Astronomical Invariable Plane is not strictly fixed in space, because the mutual attractions of the bodies do not strictly act along the straight lines joining their centres of gravity, so that the terms omitted in the expressions for hi, h.^, h^ are not absolutely constant. The effect of precession is to make the axis of rotation of each body describe a cone in space, so that, even though the angular velocity is unaltered, the position in space of the Astronomical Invariable Plane must be slightly altered. A collision between two bodies of the system, if such a thing were possible, or an explosion of a planet similar to that by which Olbers supposed the planets Pallas, Ceres, Juno and Vesta, &c., to have been produced, might make a considerable change in the sum of the terms omitted. In this case there would be a change in the position of the Astronomical Invariable Plane, but the Dynamical Invariable Plane would be altogether unaffected. It might be supposed that it would be preferable to use in Astronomy the true Invariable Plane. But this is not necessarily the case, for the angular velocities and moments of inertia of the bodies forming our system are not all known, so that the position of the Dynamical Invariable Plane cannot be calculated to any near degree of approximation, while we do know that the terms into which these unknown quantities enter are all very small or nearly constant. All the terms rejected being small compared with those retained, the Astronomical Invariable Plane must make only a small angle with the Dynamical Invariable Plane. Although the plane is- very nearly fixed in space, yet its intersection with the Dynamical Invariable Plane, owing to the smallness of the inclination, may undergo considerable changes of position. In the MScaniqiie Celeste, Laplace calculated the position of the Astronomical Invariable Plane at the two epochs, 1750 and 1950, assuming the correctness for this period of his formulae for the variations of the eccentricities, inclinations and nodes of the planetary orbits. At the first epoch the inclination of this plane to the ecliptic was l°-7689, and the longitude of the ascending ART. 308.] IMPULSIVE FORCES. 259 node 114°"3979; at the second epoch the inclination will be the same as before, and the longitude of the node 114°'3934. 305. Ex. 1. Show that the invariable plane at any point of space in the straight line deaoribed 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 co-ordinates be rectangular. Let (D,,,, Hj,, D,g), (cox, (Oy, cog) be the angular velocities of the body just before and just after the impulse, and let the differences a^ — O,^, (Oy — fly, m^ — fl^ be called 0)^', ojy, ft)/. Then lOx, a>y', eo/ are the angular velocities generated by the impulse. By D'Alembert's Principle, see Art. 87, the difference between the moments of the momenta of the particles 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 Am^ — (Zmxy) eoy — (2ma;z) «/ = i "j Ba)y — (%myz) &>/ — (^myx) Wx = M\ (1), Oft)/ — (tmzx) ft)/ — (S.m2y) ft)j,' = iVj where L, M, N are the moments of the impulsive forces about the axes. These three equations will suffice to determine the values of ft)/, Wy, ft)/. By adding these to the angular velocities before the impulse, the initial motion of the body after the impulse is found. 307. Ex. 1. Shovsr that these equations are independent of each other, and that none of the angular velocities a^ Wy, w^ 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. 308. It is to be observed that these equations leave the axes of reference undetermined. They should be so chosen that the values of A, %mxy, Ike, may be most easily found. If the 17—2 260 MOMENTUM. [chap. VI. 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 Awx = L \ GmJ=N) .(2). The values of ca^, aty, w/ being known, we can find the pres- sures 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, Q, H be the pressures of the fixed point on the body .(3). .SX+F=Jlf. J by Art. 86 = M{a>y'z - mly) by Art. 238 l.Z + H= M{(oJy - (Oy'x) 309. Ex. A wnifwm 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. Supposing the disc to be at rest before the application of the blow, find the initial motion. Let the equation to the parabola be y^=iax, and let the axis of z be perpen- dicular to its plane. TheTiSmxii=0,'Smyz=0. Let ju be the mass of a unit of area aai let ON =c. Also 2mxy=fi jfxydxdy^n i x^dai=2ii I ax''dx=-/juic^, Arz-fi r''y^dx=^lia^ci, li=iij° x^ydx = -im^c^, and C=^ + B, by Art. 7. The moments of the blow B about the axes are L = B tjiac, M= - Be, N=0. The equations of Art. 306 will become after substitution of these values ART. 310.] IMPULSIVE FORCES. 261 j^ /i.a'i ct (I), - 3 /i oeS a)„ = 2Ba^ c4 4 1 » 2 = /JUV^C' W|, - ; lioc^a^ — -Be t o From these u^, mj, may be found. By eliminating B we have J = — -J^L^. 7 Hence, if NQ be taken equal to ^r iVP, the disc will begin to rotate about OQ. The 75 Ji resultant angular velocity will be ttt; — 7 00. 26 ij.ac^ 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 been 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 be the angular velocity generated, r the radius vector of the ellipsoid which is the axis of 0. Let M^ be the parameter of the ellipsoid. Prove that r- = — . 12 pr Ex. 3. If Oj., Qy, Oj be 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 AV^ = GcoBa, B'fij,= Gco3/3, C'fi,= 6o037, where a, /3, 7 are the angles the axis of Gf makes with the conjugate diameters. Ex. 4. If a body free to turn about a fixed point be acted on by an impulsive couple G, whose axis is the radius vector /■ 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 is given by G = MprSi. 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. 262 MOMENTUM. [CHAP. VI. 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 oaUed 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'=T cob's, 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 afterwards generalized by Lagrange and Bertrand in the second part of the first volume of the Micanique 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, W), (u, V, w) be the resolved velocities of the centre of gravity just before and just after the impulse. Let X, T, 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 {X, Y, Z) and the co- ordinates of whose point of application referred to these axes are (p, q, r). Prove that if the resulting motion be one of rotation only about some axis, A{B-C)pYZ + B{G- A) qZX+ G(A-B) rXY=(l. Is this condition suificient 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 fi. When it strikes the ground, 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 the motion of the ball after impact will make with the plane of projection an angle tan-ij = , where a is the radius of the ball. 5 V cos a 313. Motion of any point of the body. The components of the change of velocity of amy point of the body are linear funclions of the components of the blow. The equations of Art. 311 com- pletely determine the motion of a free body acted on by a given ART. 314] IMPULSIVE FORCES. 263 impulse, and from these by Art. 238 we may determiuc the initial motion of any point of the body. Let {p, q, r) be the co-ordi- nates of the point of application of the blow, then the moments of the blow round the axes are respectively qZ — rY, rX — pZ, pT — qX. These must be written on the right-hand sides of the equations of Art. 306. Let {p, g', r') be the co-ordinates of the point whose initial velocities parallel to the axes are required. Let {ui, Vi, Wi), {u.2, «2, w-i) 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 Un — tOi = u' + (Uy'r' — (Oi'q', ■with similar equations for v^ — Vi, w^ — Wi. Substituting in these equations the value of a', v, w', cox, eoy, Wz given by Art. 311 we see that u^^ — Mi, v-i — v-^, w^ — tVx are all linear functions of X, T, Z of the first degree of the form xo,-tH=:FX+GT+HZ, where F, G, H depend on the structure of the body and the co- ordinates 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 «3- -1*1 = \M* B -S)- Ply- c -f^. "2- -«! = = -?« ■Qi4 ^3 z-f^. W2- -'»! = .-f.v- ^MtA +§)^- The right-hand sides of these equations are the differential coefEoients of a quadratic function of X, Y, 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 P of application is perpendicular to the diametral plane of the direction of the blow with regard to a certain ellipsoid, wlwse centre is at P, and whose equation is E= constant. The expression for E may be written in either of the equivalent forms : 2E = ^"+^l + ^\ ^A mp-2 + Sq-->+Cr^](AX^ + BY^ + GZ-')-(ApX+BqY+CrZf\ = \i +^(qZ-rYf + t(rX-pZ) + -(pY-qXf. In this latter form we see that it is = M (u'2 + v"' + w'-) + Au/' + Bun'- + Co.;-, which is the vis viva of the motion generated by the impulse. 264 MOMENTUM. [CHAP. VI. Impact of any two bodies. 315. Two bodies moving in any ma/nner impinge on each other. To find the motion after impact. Inelastic Bodies. If the bodies be inelastic and either perfectly smooth or perfectly rough, it is unnecessary to introduce the reactions into the equations. In such a case we take the point of contact as the origin. Let the axes of x and y be in the tangent plane, and tha/t 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, w the resolved velocities just after the impact. Let fix, Oy, D,i, cox, coy, lo^ be the angular velocities just before and just after. Let A, B, 0, 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, y, z the co-ordinates 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 76, ^ (a^-ft^) - i^ K-Oy) - ^(<»^-n,) - (?) - F) ^ +(w- F)2/=0, - ^ (oj^-fla,) + £ («j,-Oj,) - Z)(«2-02) - (w- Tf ) a; + (m - 17)2 =0, -^K-Ila,)-i)(«»j,-f2y) + (7(w2-Ii^)-(w-C/')y+(D- F)a;=0. Three similar equations apply for the other body, differing from these only in having all the letters accented. Kesolving along the axis of z for both bodies, we have M{w-W) + M' (w' - TF') = 0. The relative velocity of compression is zero at the moment of greatest compression, we have therefore w — a^y + Wjr" = wf — (o^y' + tOy'x'. We thus have eight equations between the twelve unknown re- solved velocities and angular velocities. 316. If the bodies be 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 perfectly rough we obtain two of the four equations by resolving the linear momenta parallel to the axes of X and y, viz. M(u-U) + M'(u'-U') = 0' M(v -V) + M' (v' - V) = 0,' ■ ART. 320.] IMPACT OF ROUGH ELASTIC ELLIPSOIDS. 265 We have also two geometrical equations obtained by equating to zero the resolved relative velocity of sliding, viz. u — WyZ + w^y = m' — ■'"'■"J "* .J»iV*iiig, . AtJ. f = m' - Wj,'/ + w/y'l : =V' — tOjV + WajVj ' 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 the fractions P and Q both equal to zero in the twelve equations of motion (1), (2), (3) and (4). We also have the velo- city 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 of R by 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 essen- tially 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. Tivo 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 co-ordinate 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 co-ordinate axes throughout the impact. 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 of the impact, but before its termination. Let il^, ^y, ^« be the angular velocities of the body just before impact about its principal diameters at the centre of gravity ; w^, o)y, o)i the angular velocities at the time t. Let a, b, c be the semiaxes of the ellipsoid, and A, B, G 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 266 MOMENTUM. [CHAP. VI. same quantities for the other body. Let the bodies impinge at the extremities of the axes c, c'. Let P, Q, R 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 A (cox - n^) = Qc •> B(a,y-ny) = -Pc\ (1) G («, -a,)=o J M{u-U) = P\ M(v-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,—R,—c' for P, Q, R and c on the right-hand side. Let us call these new equations respectively (3) and (4). Let S be the velocity with which one ellipsoid slides along the other, and 6 the angle which the direction of sliding makes with the axis of x, then S COS0 = U'+C'tOy —U+ CtOy (5), ;Si sin ^ = «^' — c'lOx' — V — ccog; (6). Let C be the relative velocity of compression, then G = w'-w (7). Substituting in these equations from the dynamical equations we have S cos^ = S„ cos0,-pP (8), S sine = So sine,- qQ (9), C=C,-rR (10), where So cos 00=11' + c'ily' -■U+ cD,y\ /S„sin^„==F'-c'a/-F-cfti (11), Go = W'-W J p-- ■m + M' ^B + B' ? = 4- 1 M' ^i ^ A' r = 1 1 M' .(12). ART. 321. j IMPACT OF ROUGH ELASTIC ELLIPSOIDS. 267 These are the constants of the impact. S,,, G^ are the initial velocities of sliding, and 60 the angle which the direction of initial sliding makes with the axis of w. Let us take as the standard case that in which the body M' is sliding along and compressing the body M, so that 8^ and G^ 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 two dimensions, we shall adopt a graphical method of tracing the changes which occur in the frictions. Let lis measure along the axes of x, y, z three lengths OI', OQ, OR to represent the three reactions P, Q, R. Then, if these be regarded as the co-ordinates of a point T, the motion of T will represent the changes in the forces. It will be convenient to trace the loci given hy S = 0, C = 0. The locus given by 8=0 is a straight line parallel to the axis of R, which we may call the Zme of no sliding. The locus given by (7 = is a plane parallel to the plane POQ, which 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 in 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 ftdR, where fi is the coefficient of friction. We have therefore ^ = cot^=|^=^;? (13), dQ 80 sm 00 -qQ {dPf-\-{dQy = ^'{dKf (14). 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 get (/So cos e, -pPy = a {8, sin 6, - qQY, where a is an arbitrary constant. At the beginning of the motion P and Q are zero, hence we have / 80 C0& 00 -pP \^ _ ( 8, sin 0, - qQ \q , . V S,ooB0, ) \ 8osm0, ) ^ '^^' which may also be written 8 cos0 \p _ / 8 8in0 \g ,^„ 8oC08do) ~\8,sin0j ^ '' 268 MOMENTUM. [CHAP. VI. sm e \-zzz /COS 0o\w:^ or 8 This equation gives the relation between the direction and the velocity of sliding. 322. If the direction of slidiog does not change during the impact, 6 must be constant and equal to 6o. We see from (16) that, ii p = q, then 6 = d„; and that conversely if = ^o, iS is constant unless p = q. Also, if sin d^ or cos ^o be zero, S must be zero or infinite unless 6 = 6o. The necessary and sufficient condition that the direction of friction should not change dwring the impact is therefore p = q or sin 20„ = 0. The former of these two conditions, by (12), leads to i^-^)^'i^'-i)-' (!«)• If this condition holds, we have by (13) P = Q cot 0„ and therefore by (14) P = fiR cos ^o) ,, Qv Q=fiRsme4 It follows from these equations that, when the friction is limiting, the representative point T moves along a straight line making an angle tan~^/t4 with the axis of iJ, 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 8, reduce the determination of R in terms of 6 to an integral. By substituting for 8 from (17) in (8) and (9), we then have P, Q, R expressed as functions of 6. 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~^yu. 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 = /So cos 00, qQ = 8^ sin ff^. 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 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 Ra of R for the luhole impact is found by multiplying the abscissa Rj of the point at which T crosses the plane of greatest compression by 1 + e, so that Ra = Ri (1 + e), if e be ART. 326.] IMPACT OF ROUGH ELASTIC ELLIPSOIDS. 269 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 = K^. 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 at 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 _ S„ cos 00 g ^ So sin 0^ P <1 ' If o- be the arc of the curve whose equation is (15) from the origin to the point where it meets the line of no sliding, then the representative point T cuts the line of no sliding at a point whose _ (T C abscissa is R = - . If the bodies be so rough that - < — , the /i ° fx, r 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 i2 = — (1 + e). Substituting these values of P, Q, Rin. the dynamical equations, the motion just after impact may be found. 326. Ex. 1. If B be the angle which the direction of sliding of one ellipsoid over the other makes with the axis of x, prove that continually increases or continually decreases throughout the impact. And if the initial value of 6 lie between and - , then approaches „ or zero according asp > ox < q. Show also that the repre- sentative point reaches the line of no sliding when 6 has either of these values. Ex. 2. If the bodies be such that the direction of sliding continues unchanged during the impact and the sliding ceases before the termination of the impact, the S r roughness must be such that u > ;; — = ; . C!oPO- + 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 first given by Coriolis. Jeu de hillard, 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. PhiUips in the fourteenth volume of Liouville's Journal. 270 MOMENTUM. [CHAP. VI. Bk. 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' = W 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. 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 2 be - It and - w and let the angular velocity about the vertical be n. After rebounding the ball wUl 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 + an) W'= -w+^ (l+y+e)w where 7 is the smaller of the two quantities f and fi{l + e)wl{v}^ + {u + an)^]*. 327. Two rough bodies moving in any manner impinge on each other. Find the motion just after impact. Let us refer the motion to co-ordinate 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.^, 0,y, O^ be the angular velocities of the same body just before impact about axes parallel to the co-ordinate axes, meeting at the centre of gravity; a>x,(Oy, w^ the angular velocities at the time t. Let A, B, C, D,E, F be the moments and products of inertia about axes parallel to the co-ordinate 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){x', y', z) be the co-ordinates of the centres of gravity of the two bodies referred to the point of contact as origin. The equations of motion are therefore A{(o^-Q,^)-F{my-n,y)-E{w,-il,) = -yR + zQ\ -F{a>„-il^) + B{my-D,y)-D{m,-a,) = -zP + xR\...{l), -E{a>^-n^)-i)(o^y-ny)+G(w,-n,) = -xQ+yP] ART. 328.] IMPACT OF ROUGH ELASTIC BODIES. 271 M(v-V) = q\ (2). M(w-W) = b] 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 the angle which the direction of sliding makes with the axis of x. Also let C be the relative velocity of compression, then S cos = 11 — asy'z' + w/y' — u+ (OyZ — (i>zy\ S sin = v' — w/x' + ft)// —v+ a^ — (Oxz\ (5). G = w' — wjy' + Wy'x —w-\- (Oxy — (Oyx) If we substitute from (1) (2) (3) (4) in (5) we find S, COS0-S cos = aP +fQ + eR) S, sm0 - S sine =fP + bQ + dpi (6), C,-G=eP + dQ + cR) where So, 0o, G^ are the initial values of *S', 0, 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 these. 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 co-ordinates, three lengths OP, OQ, OR to represent the three reactions, P, Q, R. Then if, as before, these be regarded as the co-ordinates of a point T, the motion of T will represent the changes in the forces. The equations to the line of no sliding are found by putting ;S = in the first two of equations (6). We see that it is a straight line. The equation to the plane of greatest compression is found by putting G = 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 — = '^-h, = imR (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 sufiice to prevent sliding, provided that this amount is less than the limiting 272 MOMENTUM. [CHAP. VI. friction. If therefore the angle which the line of no sliding makes with the axis of B be less than tan~'/A, the point T travels along it. But if the angle be greater than tan~'ytt, 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 equations (7) with d increased by tt. 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 elasticity, so that iia = -Ri (1 + e). The complete values of P and Q are represented by the ordinates corresponding to the abscissa -R2- 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 (8 cos 6) _ adP +fdQ + edR _ a/j, cos +ffi sin ^ + e , , dls sme)~ fdP + bdQ + ddR~ f,x,cQsd + b(j.sme + d'"^ ^' which reduces to 1 JO — s — I- —s- cos 20 +f sin 20+- cos6 + - sin 1 af^, bocP, 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 2P 2F and = — -. — J . The plane of greatest compression is the polar plane of So cos 0„ Sq sin $„ t on the axis of R R. D. 18 2E a point on the axis of R^ distant -^ from the origin. 274 MOMENTUM. [CHAP. VI. Ex. 4. The plane of PQ outa the ellipsoid of Ex. 1 in an ellipse, whose axes divide the plane into four quadrants; the line of no sliding cuts the plane of PQ in that quadrant in which the initial sliding S^ 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 B has the same sign as C^. Hence show, from geometrical considerations, that the representative point T must cross the plane of greatest compression. .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. 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. 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 -^ . 4. An elliptic lamina is rotating about its centre on a smooth horizontal table. If (i)j, (oj, Wj 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 7^65 Wj "" Ua (1)3 ■ 5. A rigid body moveable about a fixed point at which the principal moments are A, B, G 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, 6, c) being the co-ordinates of the given point referred to the principal axes at 0, and (l, m, n) the direction cosines of the blow, al + bm + cn=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. 7. A cone of mass m and vertical angle 2a can move freely about its axis, and has a fine smooth groove cut along its surface so as to make a constant angle /3 vsrith 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 rest with the * These examples are taken from Examination Papers which have been set in the University or in the Colleges. ART. 331.] EXAMPLES. 275 particle at a distance c from the vertex. Show that, if S he the angle through which the cone has turned when the particle is at any distance r from the vertex, ?rtfc2 + P»fdn% ^ 29 sin „ . oot /3 vik' + Pc^sin^a k heing the radius of gyration of the cone ahout its axis. 8. A body is turning about an axis through its centre of gravity when a point in it becomes suddenly fixed. If the new instantaneous axis be a principal axis with respect to the point, show that the locus of the point is a rectangular hyperbola. 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 about this edge as axis = — p . 10. Two masses m, ml are connected by a fine smooth string which passes round a right circular cylinder of radius a. The two particles are in motion in one plane under no impressed forces, show that, if A 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 uniform angular velocity, whUe that centre describes a circle in space with uniform angular velocity. 12. A uniform circular wire of radius a, movable 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 5 =tan-M -=ti 2a Vs Va/3 13. A small insect moves along a uniform bar, of mass equal to itself, and of length 2a, the extremities of which are constrained to remain on the circumference of a fixed circle, whose radius is —p . 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 V\ prove that the bar in time t will turn through an angle -p tan-' — . 14. A rough circular disc can revolve freely in a horizontal plane about a vertical axis through its centre. An equiangular spiral is traced on the disc, having the centre for pole. An insect whose mass is an «th that of the disc crawls along the curve, starting from the point at which it cuts the edge. Show that, when the insect reaches the centre, the disc has revolved through an angle — ^ log ( 1 + - 1 , where o is the angle between the tangent and the radius vector at any point of the spiral. 18—2 276 MOMENTUM. [CHAP. VI. 15. A uniform circular disc moveable about its centre in its own plane (which is horizontal) has a fine groove in it cut along a radius, and is set rotating with an angular velocity u. A small rocket whose weight is an n"' of the weight of the disc is placed at the inner extremity of the groove and discharged; when it has left the groove the same is done with another equal rocket, and so on. Find the angular velocity after n of these operations, and, if n be indefinitely increased, show that the limiting value of the same is ue~'. 16. A rigid body is rotating about an axis through its centre of gravity, when a certain point of the body becomes suddenly fixed, the axis being simultaneously set free; find the equations of the new instantaneous axis; and prove that, if it be parallel to the originally fixed axis, the point must lie in the line represented by the equations w'lx + hhny + cHz = 0, (6* -c^)j+ (c" -a?)^+ (a^-ft^) - =0 ; where the principal axes through the centre of gravity are taken as axes of co-ordinates, a, 6, u are the radii of gyration about these lines, and {, m, n the direction-cosines of the originally fixed axis referred to them. 17. A solid body rotating with uniform velocity u about a fixed axis contains a closed tubular channel of small uniform section, filled with an incompressible fluid in relative equilibrium ; if the rotation of the sohd body were suddenly destroyed the fluid would move in the tube with a velocity ^- , where A is the area of the projection of the axis of the tube on a plane perpendicular to the axis of rotation, and I is the length of the tube. 18. A gate without a latch, in the form of a rectangular lamina, is fitted with a universal joint at the upper corner, and at the lower corner there is a short bar, normal to the plane of the gate and projecting equally on both sides of it. As the gate swings to either side from its stable position of rest, one or other end of the bar becomes a fixed point. If h be the height of the gate, h tan a its length, and 2/3 the angle which the bar subtends at the upper corner, show that the angular velocity of the gate as it passes through the position of rest is impulsively dimin- ished in the ratio -r-^ — -j — 5^ , and that the time between successive impacts when the oscillations become small decreases in the same ratio, the weights of the bar and joint being neglected. CHAPTER VII. VIS VIVA. The Force-function and Work. 332. Time and space integrals. If a particle of mass m is projected along the axis of x with an initial velocity V and is acted on by a force F in the same direction, the motion is given by the equation m -,— = F. Integrating this with regard to t, if v be the velocity after a time t, we have m(v-V)=\' Fdt. ,(v-V)=f Jo If we multiply both sides of the differential equation of the doc second order by -tt and integrate, we get* ^m(v^-V')=rFdx. * It is seldom that Mathematicians can be found engaged in a controversy such as that which raged for forty years in the last century. The object of the dispute was to determine how the force of a body in motion was to be measured. Up to the year 1686, the measure taken was the product of the mass of the body and its velocity. IJeibnitz, however, thought he perceived an error in the common opinion, and undertook to show that the proper measure should be the product of the mass and the square of the velocity. Shortly all Europe was divided between the rival theories. Germany took part with Leibnitz and Bernoulli ; while England, true to the old measure, combated their arguments with great success. France was divided, an illustrious lady, the Marquise du Chatelet, being first a warm supporter and then an opponent of Leibnitzian opinions. Holland and Italy were in general favourable to the German philosopher. But what was most strange in this great dispute was, that the same problem, solved by geometers of opposite opinions, bad the same solution. However the force was measured, whether by the first or by the second power of the velocity, the result was the same. The arguments and replies advanced on both sides are briefly given in Montucla's History, and are most interesting. For these however we have no space. The controversy was at last closed by D'Alembert, who showed in his treatise on Dynamics that the whole dispute was a mere question of words. When we speak, he says, of the force of a moving body, we either attach no clear meaning to the word or we understand 278 VIS VIVA. [CHAP. VII. The first of these integrals shows that the change of the momentum is equal to the time-integral of the force. By applying similar reasoning to the motion of a dynamical system we have been led in the last chapter to the general principle enunciated in Art. 283, and afterwards to its application in determining the changes produced by very great forces acting for a very short time. The second integral shows that half the change of the vis viva is equal to the space-integral of the force. It is our object in this chapter to extend this result also, and to apply it to the general motion of a system of bodies. 333. Vis viva. For purposes of description it is convenient to give names to the two sides of this equation. Twice the left- hand side is usually called the vis viva of the particle, a term introduced by Leibnitz about the year 1695. Half the vis viva is also called the kinetic energy of the particle. Many names have been given to the right-hand side at various times. It is now commonly called the work of the force F. When the force does not act in the direction of the motion of its point of application the term ''work" requires a more extended definition. This we shall discuss in the next article. 334. Work. Let a force F act at a point J. of a body in the direction AB, and let us suppose the point A to move into any other position A' very near A. If (/> be the angle made by the direc- tion AB of the force with the direction AA' of the displacement of the point of application, then the product F . AA' . q,os(I} 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. only the property that certain resistances can be overcome by the moving body. It is not then by any simple considerations of merely the mass and the velocity of the body that we must estimate this force, but by the nature of the obstacles overcome. The greater the resistance overcome, the greater we may say is the force ; provided we do not understand by this word a pretended existence inherent in the body, but 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 oases be more conveniently measured by a space-integral and in others by a time-integral. See Montucla's History, Vol. in. and Whewell's History, Vol. ii. ART. 336.] FORCE-FUNCTION AND WORK. 279 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 hypothetical displacements given to the system in applying the principle of vu-tual work, and this definition is therefore sufficient. But in dynamics the bodies are in motion, and we must extend our definition of work to include the case of a dis- placement 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 fi-om 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 forces be equivalent, the work done by one in any small displacement is equal to thai 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 co-ordinates of a particle of the system and let the mass of this particle be m. Let (X, Y, Z)he the accelerating forces acting on the particle resolved parallel to the axes of co-ordinates. 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, z + dz; then according to the definition the work done by the forces will be 2 (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 Sm j {Xdx + Ydy + Zdz) (2), the limits of the integral being determined by the extreme positions of the system. 280 VIS VIVA. [chap. VII. 337. Force-function. 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 co-ordinates x, y, z. The summation (2) can therefore be expressed as a function of the co- ordinates of the system. When this is the case the indefinite integral of the summation (2) is called the force-fwnction. This name was given to the function by Sir W. R Hamilton and Jacobi indepen- dently of each other. 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 Ui—Ui, where U^ and CTj, 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 moves from the first given position to the second. In other words, the work depends on the co-ordinates of the two given extreme positions, and not on the co-ordinates 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 to the system by Sir W. Thomson. 338. There will be a force-fwnction, firstly, when the esdemal forces tend to fixed centres at finite distances and are functions of the 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. 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 2m<^(r-)dn This is a complete differ- ential. Thus the force-function exists and is equal to 1m\^{r)dr. Let mm'(r)dr. If the law of attraction be the inverse square of the distance, (fi(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 co-ordinates. If ART. 340.] FORCE-FUNCTION AND WORK. 281 P, Q, &c. be forces acting on a particle, Pdp, Qdq, &c. their virtual moments, m the mass of the particle, then the force-function is U= tmf{Pdp + Qdq + &c.), the summation extending to all the forces of the system. Ex. 1. If (p, If), z) be the cylindrical or semi-polar co-ordinates of the particle m; P, Q, Z the resolved parts of the forces respectively along and perpendicular to p and along 2, prove that dI7=Sm(P(ip-(- Qpd^ + ^dz). Ex. 2. If (r, 6,. . 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 If. Also let U^, U^ be the values of the force-function for the system in the two positions which it has at the times t and if. Then 351 The following illustration, taken from Poisson, may show more clearly why it is necessary that the geometrical relations ART. 352.] PRINCIPLE OF VIS VIVA. 291 should not contain the time explicitly. Let, for example, z = z+^. Then, by a property of the centre of gravity, zm^ = 0, "Zmr) = 0, 'tm^= 0. Hence Sm^ = 0, Sm^^ = 0, Sm^=0. Now the vis viva of at at at a body is Substituting for x, y, z, this becomes ^•»{(S)'-(l)"-©V^{(S)'-©'-(S)"i All the terms in the last line vanish, as they should, by Art. 14. The first term in the first line is the vis viva of the whole mass 2m, collected at the centre of gravity. The second term is the vis viva due to rotation round the centre of gravity. This expression for the vis viva may be put into a more con- venient shape. 364. Firstly. Let the motion he in two dimensions- See Art. 139. Let V be the velocity of the centre of gravity, r, 9 its polar co- ordinates referred to any origin in the plane of motion. Let n be the distance from the centre of gravity of any particle whose mass is m, and let Vi be its velocity relatively to the centre of gravity. Let m be the angular velocity of the whole body about the centre of gravity, and Mil? its moment of inertia about the same point. The vis viva of the whole mass collected at G is Mv^, which may be put into either of the forms *-^{(S)'-(f)"l=^l©"--(S)"}- The vis viva about is XmVi^. But since the body is turning about G, we have Vi = ryto. Hence %mv^ = m' . Xmr^ = <»" . Mh?. The whole vis viva of the body is therefore %mv^ = MtP + M¥m\ ART. 364.] EXPRESSIONS FOR VIS VIVA. 297 If the body be turning about an instantaneous axis, whose distance from the centre of gravity is r, we have v = rco. Hence where Mk"^ is the moment of inertia about the instantaneous axis. Secondly. Let the body he in motion in space of three dimen- sions. Let V be the velocity oi G ; r, 6, "^ its polar co-ordinates referred to any origin. Let m^, (Oy, m^ be the angular velocities of the body about any three axes at right angles meeting in O, and let A, B, C be the moments of inertia of the body about the axes. Let f, rj, f be the co-ordinates of a particle m referred to these axes. The vis viva of the whole mass collected at G is Mv^, which may be put equal to "{(sy- (i)'- ©1 « *{(S)"-«(Sy-(l)'} . according as we wish to use Cartesian or polar co-ordinates. The vis viva due to the motion about G is ....=s-{(§)V(|)V(D^ x, , d? J. drj ^ dt 5, Substituting these values, we get, since A = %m {rf + ^), B = tm (?^ + ^), (7 = 2m (f + 7)% Xmvi' = ^tB/ 4- 5ft)/ 4- Cm/ - 2 (Xm^r)) coxtoy — 2 (Zmt]^) coytoz — 2 (Xm^^) cogcox. We may find the vis viva of the motion about G in another manner. Let fi be the angnlar velocity about the instantaneous axis, I the moment of inertia about it. The vis viva is then clearly IQ". Now I is found in Art. 15, and in our case (i)j=na, w^ — ap, U3=(}7, following the notation of that article. Eliminating a, /3, y we get the same result as before. If the axes of co-ordinates be the principal axes at G, this reduces to tmvi' = Aa>x' H- 5ft)j,2 + (7ft)/. 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 tmi^ = AW + B'toy' + G'mi, where A', B', G' are the principal moments of inertia at the point ■■, ;, 298 VIS VIVA. [chap. vn. 0, and mx, cay, w^ 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 co-ordinates of its centre of gravity and the angles d, (p, ip 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 + sin^ e {A oos^ (l>+B sin" 0) f ''H- 2 (B - A) sin S sin cos 0^^. Show also that, when two of the principal moments A and B are equal, this expression takes the simpler form This result will be often found useful. Ex. 2. A body moving freely about a fixed point is expanding under the in- fluence of heat, so that in structure and form it remains always similar to itself. If the law of expansion be that the distance between any two particles at the temperature 6 is equal to their distance at temperature zero multiplied by f{6), show that the vis viva of the \>odiy=Ao,J'+BiOy'+Cu/+^{A+B + q (^^^^^ , where A, B, C are the principal moments at the fixed point. Ex. 3. A body is moving about a fixed point and its vis viva is given by the equation 2T = .4 Wj.2 + Bw/ + CuJ' - iDwyia, - 2Ea^x - 2Fa^y . Show that the angular momenta about the axes are ^ — , -= — , ^ — . owj. duty auj Let the body be moving freely and let 2T^ be the vis viva of translation. Prove that, a x,y,z\ye the co-ordinates of the centre of gravity referred to any rectangular axes fixed or moving about a fixed point, and if accents denote differential coefficients with regard to the time, the linear momenta parallel to the axes will be dTo dTf, dT„ dud ' dy' ' dz' ' Thus the vis viva, like the force-function, is a scalar function whose differential coefficients are the components of vectors. See Arts. 262 and 840. In the case of the semi-vis viva, these are the resultant linear momentum and the resultant angular momentum round the centre of gravity. 366. Froblems on tbe Principle of vie viva. Ex. 1. A circular wire can turn freely about a vertical diameter as u fixed axis, and a bead can slide freely along it under the action of gravity. The whole system being set in rotation about the vertical axis, find the subsequent motion. Let M and m be the masses of the wire and bead, a their common angular velocity about the vertical. Let a be the radius of the wire, Mk^ its moment of inertia about the diameter. Let the centre of the wire be the origin, and let the axis of y be measured vertically downwards. Let 9 be the angle which the axis of y makes with the radius drawn from the centre of the wire to the bead. ART. 366.] PROBLEMS ON THE PRINCIPLE. 299 It is evident, since gravity acts vertically and since all the reactions at the fixed axis must pass through the axis, that the moment of all the forces about the vertical diameter is zero. Hence, taking moments about the vertical, we have Myia + ma?ui sin^ e = h. And by the principle of vis viva, ilf ft V + m { a¥8 + o= sin" eu2 } = C + ^ga cos 6*. These two equations will suffice for the determination of 6 and u. Solving them, we get -rrr—^ — ' . ., - +ma'( -r,] =C + 2mga cos $. This equation cannot be integrated, and hence 6 cannot be found in terms of t. To determine the constants h and G we must recur to the initial conditions of motion. Supposing that initially $ = ir, and ^=0 and u = a, then h=Mk^a. and C=imga+Mk'^a?. See Art. 352. Ex. 2. A lamina of any form rolls on a perfectly rough straight line under the action of no forces ; prove that the velocity v of the centre of gravity G is given by ,.2 t)'=c'' -J — Tj, where r is the distance of G from the point of contact, k the radius of gyration of the lamina about an axis through G perpendicular to its plane, and c some constant. Ex. 3. Two equal beams connected by a hinge at their centres of gravity so as to form an X are placed symmetrically on two smooth pegs in the same horizontal line, the distance between which is b. Show that, if the beams be perpendicular to each other at the commencement of the motion, the velocity of their centre of / Wg^ gravity, when in the line joining the pegs, is equal to . / ^ . ,g , where k is the radius of gyration of either beam about a line perpendicular to it through its centre of gravity. Ex. 4. A uniform rod is moving on a horizontal table about one extremity, and driving before it a, particle of mass equal to its own, which starts from rest indefinitely near to the fixed extremity; show that, when the particle has described ■dL distance r along the rod, its direction of motion makes with the rod an angle tan-i ^ - [Christ's Coll.] Ex. 5. A thin uniform smooth tube is balancing horizontally about its middle point, which is fixed; a uniform rod such as just to fit the base of the tube is placed end to end in a line with the tube, and then shot into it with such a horizontal velocity that its middle point shall only just reach that of the tube ; supposing the velocity of projection to be known, find the angular velocity of the tube and rod at the moment of the coincidence of their middle points. [Math. Tripos.] Result, li m be the mass of the rod, m' that of the tube, and 2a, 2a' their respective lengths, v the velocity of the rod's projection, w the required angular velocity, then O'^-J^^. Ex. 6. If an elastic string, whose natural length is that of a uniform rod, be attached to the rod at both ends and suspended by the middle point, prove by means of vis viva that the rod will sink until the strings are inclined to the horizon at an 300 VIS VIVA. [CHAP. VII. angle 6, which satisfies the equation cot' - - cot 5 - 2m= 0, where the tension of the string, when stretched to douhle its length, is n times the weight. [Math. Tripos.] Ex. 7. The centre C of a circular wheel is fixed and the rim is constrained to roll in a uniform manner on a perfectly rough horizontal plane so that the plane of the wheel makes a constant angle a with the vertical. Bound the circumference there is a uniform smooth canal of very small section, and a heavy particle which just fits the canal can slide freely along it under the action of gravity. If m be the particle, B the point where the wheel touches the plane, and B = LBGm, and if m be the angular rate at which B describes the circular trace on the horizontal plane, prove that (■-=-) = -coso cose-m'cos^o cos^S + oonst., where a is the radius of the wheel. AnnaUs de Gergonne, Tome xix. Ex. 8. A regular homogeneous prism, whose normal section is a regular polygon of n sides, the radius of the circumscribing circle being o, roUs down a perfectly rough inclined plane whose inclination to the horiison is «.. If «„ be the angular velocity just before the m"" edge becomes the instantaneous axis, then 27r /„ „ 2ir\2 / „ 27r v 8+ COS — /2 + 7COS — \ / . 8+cos — \ 2 gsmg 2L-I — I I a gam a n \ "" ~^7"!i 27^1 T ^^Jl""-^ '~~^k,A 27r /• osm-5 + 4cos — \ 8 + oos — / \ asm-5 + 4cos — / The Principle of Similitude. 367. What are the conditions necessary that two systems of particles which are initially geometrically similar should also be mechanically similar, i.e. that the relative positions of the particles in one system after a time t should always be similar to the relative positions in the other system after another time f, such that i bears to < a constant ratio ? In other words, a model is made of a machine, and is found to work satisfactorily, what are the conditions that a machine made according to the model should work as satisfactorily ? The principle of similitude was first enunciated by Newton in Prop. 32, Sect. vii. of the second book of the Principia. Biit the demonstration has been very much improved by M. Bertrand in Gahier osxodi. of the Journal de I'dcole Poly technique. He derives the theorem from the principle of virtual work so as to avoid that necessity of considering the unknown reactions which enters into some other modes of proof. Since all the equations of motion may be deduced from the general principle of virtual work, that principle seems to afford the simplest method of investigating any general theorem in Dynamics. 368. Let {x, y, z) be the co-ordinates of any particle of mass m in one system referred to any rectangular axes fixed in space, and let (X, Y, Z) be the resolved part of the impressed moving ART. 369.] PRINCIPLE OF SIMILITUDE. 301 forces on that particle. Let accented letters refer to corresponding quantities in the other system. Then the principle of virtual work supplies the two following equations : t {(X -mx)Sx + &c.} = 0, t{(X'-m'x')Sa)' + &c.}=0. It is evident that one of these equations will be changed into the other if we put X' = FX, V = FY, &c., x' = Ix, y' = ly, &c., m' = \xin, &c., Il = Tt, &c., where F, I, fi, t are all constants, pro- vided that id = F-T^. In two geometrically similar systems we have but one ratio of similarity, viz. that of the linear dimensions, but in two mechanically similar systems we have three other ratios, viz. that of the masses of the particles, that of the forces which act on them, and that of the times at which the systems are to be compared. It is clear that, if the relation just established hold between these four ratios of similitude, the motions of the two systems will be similar. Suppose then that the two systems are initially geometrically similar, that the masses of corresponding particles are propor- tional each to each, and that they begin to move in parallel directions with like motions and in proportional times, then they will continue to move with like motions and in proportional times provided the external moving forces in either system are propor- , , mass X Hnear dimensions „. ,, , , , .,. tional to 771 — rj . bmce the resolved velocities (timef of any particle are t- , &c., it is clear that in two similar systems the velocities of corresponding points at corresponding times are , ^ linear dimensions ,./> t ■ j^ ., proportional to -. . It we eliminate the time ^ ^ time between these two relations, we may state, briefly, that the con- dition of similitude between two systems is that the moving forces , , , . , , mass X (velocity)" must be proportional to j-. 3^ ■r^-'- . ^ '^ linear dimensions 369. On Models. M. Bertrand remarks that, in comparing the working of a model with that of a large machine, we must •take care that all the forces bear their proper ratios. The weights of the several parts will vary as their masses. Hence we infer that the velocity of working the model must be made to be pro- portional to the square root of its linear dimensions. The times of describing corresponding arcs will also be in the same ratio. When the speeds of working the model and the large machine are thus related it is convenient to apply to them the terms " corresponding velocities." 302 VIS VIVA. [chap. VII. If there be any forces besides gravity which act on the model, these must bear the same ratio to the corresponding forces in the machine, if the model is to be similar to the machine. If the model be made of the same material as the machine, the weights of the several parts will vary as the cubes of the linear dimensions. Hence the impressed forces must be made to vary as the cubes of the linear dimensions. For example, in the case of a model of a steam-engine, the pressure of the steam on the piston varies as the product of the area of the piston into the elastic force. Hence, the elastic force of the steam used must be proportional to the linear dimensions of the model. Supposing the impressed forces in the two systems to have, each to each, the proper ratio, the mutual reactions between the parts of the systems will, of themselves, assume the same ratio. For if, by giving proper displacements according to the principle of virtual work, we form equations of motion to find these reactions, it is easy to see that they will be, each to each, in the same ratio as the forces. Since sliding fidction varies as the normal pressure, and is independent of the areas in contact, these fiictions will bear their proper ratio in the model and machine. This, however, is not the case with rolling friction. Recurring to Art. 164, we see that the rolling friction varies inversely as the diameter of the wheel, and therefore bears a greater ratio to the other forces in the model than it does in the machine. If the resistance of the air be proportional to the product of the area exposed and the square of the velocity, the resistances will bear the proper ratio in the model and the machine. 370. Examples. As an example, let us apply the principle to the case of a rigid body oscillating about a fixed axis under the action of gravity. That the motions of two pendulums may be similar they must describe equal angles, corresponding times are therefore proportional to the times of oscillation. Since the forces vary as the mass into gravity, we see that when a pendulum oscillates through a given angle, the square of the time of oscillation must vary as the ratio of the linear dimensions to gravity. As a second example consider the case of a particle describing an orbit round a centre of attraction whose force is equal to the product of the inverse square of the distance and some constant n. The principle at once shows that the square of the periodic time must vary as the cube of the distance directly, and as /* inversely. This is Kepler's third law. In Mr Froude's experiments to determine the resistance to ships he employed small models. The following rule used by him will be a third example. If the linear dimensions of a ship be n times those of the model, and if at a speed V the measured resistance to the model be B, then at the corresponding speed, viz., mi V, the resistance to the ship will be n^M. Ex. Experiments are to be made on the deflection of a bridge 50 feet long and weighing 100 tons, when an engine weighing 20 tons passes with a velocity of 40 miles per hour, by means of a model bridge 5 feet long and weighing 100 oz. ART. 372.] PRINCIPLE OF SIMILITUDE. 303 Find the weight of the model engine, and if the model bridge be of such stiffness that its statical central deflection under the model engine be one-tenth of the statical central deflection of the bridge due to the engine, show that the velocity of the model engine must be 18-55 feet per second. [Coll. Exam.] 371. Savart's Tbeorem. In the twenty-ninth volume of the Annales de Chimie (Paris, 1825) Savart describes numerous experiments which he made on the notes sounded by similar vessels containing air. He says that if we construct cubical boxes and set the air in motion, as is ordinarily done in organ pipes, we find that the number of vibrations in a given time is proportional to the reciprocals of the linear dimensions of the masses of air. This law was verified between extreme limits, and its truth tested over many notes. He says that he frequently used the law during his researches, and never once found that it led him wrong. This result having been obtained for cubes, it was natural to examine whether the same law held for prismatic tubes on square bases. After numerous experiments he found the same law to be true. He then tested the law with corneal pipes in which the opening was always of the same solid angle, then with cylindrical pipes, then with pipes whose bases were equilateral triangles. These he made to sound in different ways, putting the mouth-piece for instance at different points of the length of the tube. In all cases the same law was found to hold, for tubes whose diameters were very small compared with their lengths as well as for those whose diameters were very great. This law he again found applicable to masses of air set in motion by com- munication from other vibrating bodies. Hence he inferred the following general law, which he enunciated as an experimental fact. When masses of air are contained in two similar vessels, the number of vibra- tions in a given time [i.e. the pitch of the note sounded] is inversely proportional to the linear dimensions of the vessel. This theorem of Savart's follows at once from the principle of similarity. Divide the similar vessels into corresponding elements, then the motions of these elements will be similar each to each if the forces vary as —■ — ^ • But by Marriotte's (time)2 law the force between two elements varies as the product of the area of contact into the density. Hence the times of oscillation of corresponding particles of air must vary as the linear dimensions of the vessel. 372. The first person who gave a theoretical explanation of Savart's law was Cauchy, who showed, in a Menwire presented to the Academy of Sciences in 1829, that it followed from the linearity of the equations of motion. He refers to the general equations of motion of an elastic body whose particles are but slightly dis- placed even though the elasticity is different in different directions. These equa- tions, which serve to determine the displacements ({, rj, f) of a particle in terms of the time t and the co-ordinates (x, y, z), are of two kinds. One applies to all points of the interior of the elastic body and the other to all points on its surface. These are to be found in all treatises on elasticity. An inspection of the equations shows that they will continue to exist if we replace f , ij, f, x, y, z, t by icf , kti, icf, kx, Ky, kz, Kt, where k is any constant, provided that we alter the accelerating forces in the ratio k 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 {, ri, ^ and the durations of the vibrations may vary in the same ratio. Hence we deduce Cauchy's extension of 304 VIS VIVA. [CHAP. VII. Savart's law, viz., if we 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 will 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 d?as of Art. 332, a force F is measured by m -^ . We may then state the general principle, that all dynamical equations must be such that the dimensions of terms added together are the same in space, time and mass, the dimensions of force being taken to be mass . space (timef ' Let us apply this 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 t of oscillation 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 m. Thus T = tAFH^mr, where A, being a function of a. only, is a number. Since t is of no dimensions in space, we have p + q = 0. Also t is of one dimension in time ; .". — 2p = 1. Finally t is of no dimensions in mass ; :. p-\-r = 0. Hence p = — \, q = r = \, 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 t = A a/ -^ where A is an undetermined number. See also Art. 370. 374. Ex. 1. A particle moves from rest towards a centre of force, whose attrac- tion 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 77, + 1 reaching the centre of force varies as the — h— th power of the initial distance of the particle. Clausing theory of stationary motimi. 375. To determine the mean vis viva of a system of material points in stationary motion. Olausius, 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 ART. 376.] CLAUSIUS' THEORY OF STATIONARY MOTION. 305 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 .r, y, z be the co-ordinates of any particle in the system and let its mass be m. Let X, Y, Z be the components of the forces on this particle. Then m^ = X. We have by simple differentiation, d'ix') „d f dx\ „fdxy „ d^x and therefore _y=__.x+^^^ Let this equation be integrated with regard to the time from to t and let the integral be divided by t, we thereby obtain in which the application of the suffix 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 m fdxy 1 evidently the mean values of ■=^[--^\ and -^xX during the time 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 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 co-ordinates, is very great, so that in the 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 co-ordinates. Hence adding together our results for each particle, we have, if v be the velocity of the particle m, mean 5 ^mv^ = - mean g S {Xx +Yy + Zz). The mean value of the expression - 5 S (Xa: -I- F?/ -t- Z«) has been called by Clansius the virial of the system. His theorem may therefore be stated thus, f fte mean semi- vig 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 liere 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 R. D. 20 306 VIS VIVA. [chap. VII. thing) suppose the particles to be perfectly elastic, so that the total vis viva is unaltered hy 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 viriah. It has just been shown that the mean semi-vis viva is equal to the sum of these two parts. If (r) be the law of repulsion between two particles whose masses are m and m', we have Xx + X'x'=z -0(r) — ^-x-^{r) — — x' = 4i{r)^ . And, since forthe two other co-ordinates corresponding equations may be formed, we have for the internal virial - JS (Xx +Yy + Zz)= - SJj-^ (r). Let the volume be increased, the system remaining similar to itself. Every r is now increased so that dr=pr, where j3 is an infinitely small quantity. If W be the work of the internal repulsions, we have dW= ^ (r) /3r. If V be the volume of the dW body, dV=3pV. Hence -SJr0()-)= -fF^. This supplies another expression for the internal virial, if we understand W to represent the mean work. As to the external forces, in the case most frequently to be considered the body is acted on by a uniform pressure normal to the surface. If p be this pres- sure, da- an element of the surface, I the cosine of the angle the normal makes with the axis of x, -^2Xx = ^ Cxpld -^12 ^^^ proportional to 1, (1 + e)", eK 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=iX(u'+u), T"-T' = iXe{u"+u'). It easily follows that T"-T'-e{T'-T) = lXe{u"-u). Since the right-hand side of this equation is iJ„2e/(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. Examples. 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 m^ition. 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 axie fixed in space whose direction cosines are {I, m, n), the angular velocity w is found by Art. 89 from (AP + Bm'' + Cn') umLl + Mm + Nn. 314 VIS VIVA. [CHAP. VII. 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 (A P + Brrfi + Cn^)i^= maximum. We have also the condition P + m?+ri'=l. Treating these three equations in the usual manner indicated in the differential , , „ , Al Bm On calculus, we find y = jtr ==~Kr • These equations determine the direction cosines of the axis about which the body begins to turn. Ex. 2. A body is at rest with one point fixed in space. Suddenly a straight line OGfiaed in the body begins to move round in a known manner, find the motion of tlie body. See Art. 293. Take the instantaneous position of OG as the axis of z, and let be the origin. Let the motion of OC be given by the angular velocities $, ^ about the axes Ox, Oy, and let u be the required angular velocity of the body about Oz. Then, by Sir W. Thomson's theorem, we make the vis viva of the body a minimum. We .482 + 502 + Cu2 - 2D0U - 2E8a - 2Fe=mm., where A, B, &o. are the moments and products of inertia at 0. Differentiating we have Cw-D0-£6» = O. Thus u has been found. This last equation expresses the fact that the angular momentum about the axis of z is unaltered by the blow. Ex. 3. A rod AB at rest is acted on by an impulse ii' perpendicular 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 Sir W. Thomson's theorem and Lagrange's or Bertrand's theorem require the same function of x to be made a minimum. Ex. 4. 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. 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+B„i. Ex. 5. 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. Since T=0, T'=E|,„ and r" = iJ|,2, the result follows by comparing the third equations in Art. 383 and Art. 388. 390. Gauss' measure of tbe "constraint." The expression, called 12 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^, &c. of a system just before the action of any impulses occupy positions which we shall call Pi,Pi, &o. Let us suppose that the particles if free would under the action ART. 392.] gauss' principle of least constraint. 315 of these impulses and their previous momenta acquire such velocities that in the time dt subsequent to the impulses they would describe the small spaces piq^, p^q^i &o. 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 jpi^i, pjj'j, &c. Then the spaces q^r^, q^r.^, &c. may be called the deviations from free motion due to the constraints. The sum Sni {qry is called the " constraint." 391. We may also measure the constraint by the ratio of this sum to {dt)". We then take p^q^, &o. pjr,, &o. 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, w) be the components of pq in any motion, and («', v', w') the components of pr in any other motion ; then Sm {qrf=Zm {(«' - uf + {v' - vf + (w' - wf} measures the "constraint" from one motion to the other. This is precisely what we have represented by the symbol 2B, 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 21" 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 21"' 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 2T'" be the subsequent vis viva in this free motion. Let 2T be the initial vis viva common * Gauss' proof of the principle is nearly as follows. By D'Alembert's principle the particles mj, m^, &c., if placed in the positions j-j, r^, efcc, 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 vi describes a space rp, making an angle

y^' ^ai &C. are connected together by some relation such as (xj, ^=0,&o., where fidt has been written for \. The equations in this form might have been derived directly from the principle of virtual work. By that principle we have Sm r(^ -x\sx + &o.1=0 with the condition S [^^.to + &c.] = 0. Multiplying the second by an indeterminate multiplier /n, adding the results together, and equating to zero the coefficients of 5a;, &o. we obtain the same results as before. EXAMPLES*. 1. A screw of Archimedes is capable of turning freely about its axis, which is fixed in a vertical position : a heavy particle is placed at the top of the tube and runs down through it ; determine the whole angular velocity communicated to the screw. Remit. Let n be the ratio of the mass of the screw to that of the particle, a the angle which the tangent to the screw makes with the horizon, h the height descended by the particle. If w be the angular velocity generated, prove that iir'a'(n+l) {n+ siiv' a) = 2gh ooa' a. 2. A fine circular tube, carrying within it a heavy particle, is set revolving about a vertical diameter. Show that the difference of the squares of the absolute velocities of the particle at any two given points of the tube equidistant from the axis is the same for all initial velocities of the particle and tube. 3. A circular wire ring, carrying a small bead, lies on a smooth horizontal table; an elastic thread, the natural length of which is less than the diameter of the ring, has one end attached to the bead and the other to a point in the wire ; the bead is placed initially so that the thread coincides very nearly with a diameter of the ring; find the vis viva of the system when the string has contracted to its original length. * These examples, except the last two, are taken from the Examination Papers which have been set in the University and in the Colleges. 318 VIS VIVA. [chap. VII. 4. A straight tube of given length is capable of turning freely in a horizontal plane about one extremity, two equal particles are placed at different points within it at rest; an angular velocity being given to the system, determine the velocity of each particle on leaving the tube. 5. A smooth circular tube of mass M has placed within it two equal particles of mass m, which are connected by an elastic string whose natural length is f of the circumference. The string is stretched until the particles are in contact, when the tube is placed flat on a smooth horizontal table and left to itself. Show that, when the string arrives at its natural length, the actual energy of the two particles is to the work done in stretching the string as 2 (M^ + Mm+m?) : {M+2m) {2M+m). 6. An endless flexible and inextensible chain, in which the mass per unit of length is jn through one continuous half, and ft' through the other half, is stretched over two equal perfectly rough uniform circular discs (radius a, mass M) which can turn freely about their centres at a distance b in the same vertical line. Prove that the time of a small oscillation of the chain under the action of gravity is / M+{Tra + b)(fi. + pi') '"' V 2(^-/)sf ■ 7. Two particles of masses m, m' are connected by an inelastic string of length a. The former is placed in a, smooth straight groove, and the latter is projected in a direction perpendicular to the groove with a velocity V. Prove that the particle m will osciUate through a space -, , and that, if m be large compared with m', the m+m time of oscillation is nearly -rp ( 1 - 7— ) • 8. A rough plane rotates with uniform angular velocity n about a horizontal axis which is parallel to it but not in it. A heavy sphere of radius a, being placed on the plane when in a horizontal position, rolls down it under the action of gravity. If the centre of the sphere be originally in the plane containing the moving axis and perpendicular to the moving plane, and if x be its distance from this plane at a subsequent time t, before the sphere leaves the plane, then a;=— — =i — I - 84a - 60c e^ ' -e ^' )--;-^smnJ, e being the distance from the axis to the plane measured upwards. 9. The extremities of a uniform heavy beam of length 2a slide on a smooth wire in the form of the curve whose equation is r=a (1 -cos 6), the prime radius being vertical and the vertex of the curve downwards. Prove that, if the beam be placed in a vertical position and displaced with a velocity just sufficient to bring it into a horizontal position, tanfl = - -je v&' _g'' Vj^tl , where 9 is the angle through which the rod has turned during a time t. 10. A rigid body, whose radius of gyration about G the centre of gravity is k, is attached to a fixed point C by a string fastened to a point A on its surface. CJ ( = 6) and AG{ = a) are initially in one line, and to G is given a velocity V at right angles to that line. No impressed forces are supposed to act, and the string is attached so as always to remain in one right line. If e be the angle between AG and AC ART. 394] EXAMPLES. 319 at time t, show that I ^j = -p ) i .a , J sm^ 9 ' ""^ '^ "^® amplitude of «, i.e. 2 sin-' — 7^ , be very small, the period is , . 2 Jab " Vja{a+b) 11. A fine weightless string having a particle at one extremity is partially coiled round a hoop, which is placed on a smooth horizontal plane, and is capable of motion about a fixed vertical axis through its centre. If the hoop be initially at rest and the particle be projected in a direction perpendicular to the length of the string, and if s be the portion of the string unwound at any time t, then s2_62=_*L vV + 2Vat, m+iJ. where b is the initial value of s, m and /i the masses of the hoop and particle, a the radius of the hoop and V the velocity of projection. 12. A square, formed of four similar uniform rods jointed freely at their ex- tremities, is laid upon a smooth horizontal table, one of its angular points being fixed : if angular velocities u, u' in the plane of the table be communicated to the two sides containing this angle, show that the greatest value of the angle (2a) between them is given by the equation cos 2a= - - — '- 6 m2+w'2" 13. Two particles of masses m, m' lying on a smooth horizontal table are con- nected by an inelastic string extended to its full length and passing through a small ring on the table. The particles are at distances o, a' from the ring and are pro- jected with velocities v, v' at right angles to the string. Prove that, if mv^a^=m'v'^a'^, their second apsidal distances from the ring will be a', a respectively. 14. If a uniform thin rod PQ move, in consequence of a primitive impulse, between two smooth curves in the same plane, prove that the square of the angular velocity varies inversely as the difference between the sum of the squares of the normals OP, OQ to the curves at the extremities of the rods and ^^ of the square of the whole length of the rod. 15. Assuming that the muscular power or moving force of an animal varies as the sectional area of its limbs, and that its weight varies as its volume, prove that two animals of similar forms, but of different dimensions, can make jumps of exactly the same height, the height being measured by the vertical distance described by the centre of gravity after the animal has left the ground. 16. The extremities of a uniform beam of length 2a, slide on two slender rods without inertia, the plane of the rods being vertical, their point of intersection fixed, and the rods inclined at angles j and - 7 to the horizon. The system is set rotating about the vertical line through the point of intersection of the rods with an angular velocity a, prove that if 9 be the inclination of the beam to the vertical at the time t and a the initial value of 6 . rdey (Scos^a-t-sin^a)" „ ,„ » , • , , . , 6^ / • ■ a> 4( — ) -I-*- =-;; ^~ la' = (i aoa^ a + Bm^ a) 0^ + —(sm a- sm 6). \dtj 3cos^ff + 8m''$ * ' a ^ 17. A perfectly rough sphere of radius a is placed close to the intersection of the highest generating lines of two fixed equal horizontal cylinders of radius c, the axes being inclined at an angle 2a to each other, and is allowed to roll down be- 320 VIS VIVA. [chap. VII. tween them. Prove that the vertical velocity of its centre in any position will be sin a cos A ^<^9 (<^ + c)(l-sm^) l J ^ .^ ^^^ inclination to the horizon of the ^ \ 7 - 5 oos^ cos^ a ) ^ radius to either point of contact. d^x dT 18. Let a complete integral of the equation -3-3 = ^ , in virhich T is a function of X, be x=X, X being a known function of a and b, two arbitrary constants, and t. Then the solution of ^ = ;j- + -=- , i! being a function of as, may also be repre- sented by a; = JL provided that a and 6 are variable quantities determined by the equations -^=k-^T- , -^ = - k ^r- , where ft is a function of a and 6 which does not at do at da contain the time explicitly. 19. A satellite, considered as a particle, revolves about its primary with an angular velocity Q, and the primary rotates about an axis which is perpendicular to the plane of the satellite's orbit with an angular velocity n. Show that the angular momentum h of the system about its centre of gravity and the energy E are given by h=Cn+Da~i, 2E = Cifi-DSi% where C is the moment of inertia of the primary about the axis of rotation and D is a quantity depending on the masses of the bodies. Trace the curves whose ordinates are h and E and abscissa is x=DCl~*. Show that the latter curve belongs to one or other of two species according as a maximum and a minimum ordinate do or do not exist, i.e., according as the biquadratic h=x+CD^x-» has two real roots or none. Show also that the real roots correspond to the case in which the primary always turns the same face to the satellite. 20. Assuming the results of the last example, determine the effect on the motion of a continual loss of energy (due to tidal friction or any other cause), the angular momentum h being constant. Show that, when the circumstances of the system are such that the energy curve is of the second species, the satellite must ultimately fall into the planet. If the energy curve is of the first species, show that, according to the initial value of Q, the satellite will either fall into the planet or will approach the planet until it reaches a certain distance, when the two will revolve as a rigid body. To obtain these results imagine two points to be placed with the same abscissa, one on the momentum line and the other on the energy curve, and suppose the one on the energy curve to guide that on the momentum line. Since the energy decreases, it is clear that, however the two points are set initially, the point on the energy curve must always slide down a slope, carrying with it ithe other point. The final positions of the points will thus depend on the existence or absence of a mini, mum ordinate in the energy curve. See a paper by G. H. Darwin on the secular effects of tidal friction in the Proceedings 0/ tlie Royal Society, June 1879, or Thomson and Tait's Treatise mi Natural Philosophy, Vol. i, Part 11. App. Gb. CHAPTER VIII. Lagrange's Equations. 395. Two advantages of Lagrange's equations. Our object in this section is to form the general equations of motion of a djmamical system freed from all the unknown reactions and expressed, so far as is possible, in terms of any kind of co-ordinates which may be convenient in the problem under consideration. In order to eliminate the reactions we shall use the principle of virtual work. This principle has already been applied to obtain the equation of vis viva, by giving the system that par- ticular displacement which it would have taken if it had been left to itself. But since every dynamical problem can, by D'AMmbert's principle, be reduced to oHe in statics, it is clear that, by giving the system proper displacements, we must be able to deduce, as in Art. 357, not the vis yiva equation only, but all the equations of motion. 396. Let the co-ordinates of any particle m of the system referred to any fixed rectangular axes be {x, y, z). These are not independent of eaclr other, being connected by the geometrical relations of the system. But they may be expressed in terms of a certain number of independent variables whose values will determine the position of the system at any time. Extending the definition given in Art. 73, we shall call these the co-ordinates of the system. Let them be called 6, if), yjr, &c. Then x, y, z, &c. are functions of 6, 0, &c. Let x=f{t,e,^,&c.).:. (1), with similar equations for y and z. It should be noticed that these equations are not to contain -jr , -j- , &c. The independent variables in terms of which the motion is to be found may be any we please, with this restriction, that the co-ordinates 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. The number of independent co-ordinates to which the position of a system is reduced by its geometrical relations is sometimes B. D. 21 322 LAGRANGE'S EQUATIONS. [CHAP. VIII. spoken of as the number of degrees of freedom of that system. Sometimes it is referred to as being the number 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 written at length. Thus -^ and -5-^ will in general be written a/ and x". If 2T be the vis viva of the system, we have 2?'=2m(a;'2 + y'= + /'') (2); we also have, since the geometrical equations do not contain 6', ,&.c.e',4>',8c(i). 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', (f>', &c. ahove the second enter into this function, and that, when the geometrical equations do not contain the time explicitly, it is a homogeneous function of &, ^ , &c. of the second order. 897. Virtual work of the effective 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 co-ordinate only. Let this co-ordinate 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 2m (i»'Sa; + i/'Sj/ + ^'S^), where hx, Sy, Bz are the small changes produced in the co-ordinates of the particle m by a variation 80 of 0. This is the same as H''B*'^7^^%>- ART. 398.] VIRTUAL WORK. 323 If 2 r be the vis viva given by (2) of the last article dT „ ( ,dx' dT „ f ,dx' . \ But, differentiating (3) partially with regard to 6', we see that -j^ = ^ . Hence the virtual moment of the momenta is equal to -j^ SO. 398. The virtual work of the effective forces is Omitting the factor 86 for the moment, this may be written in the form d ^ f , dx _ g \ „ I , d dx dt % m (^/_^+&c.)-Sm(^'^g+&c.)^ where the -^ represents a total differential coefficient with regard to t. We have already proved that the first of these terms is T- j^ . It remains to express the second term also as a differ- ential coefficient of T. Dififerentiating the expression for 2T partially with regard to 6, dT ^ ( ,dx' dd = tm[x'-^+&c). But, differentiating the expression for x with regard to ff, dx' _ d^a> d^x ., d^x , W-dWdt + de^^ +Wd4'^ +'^°-' and this is the same as -y- -ttt . Hence the second term may be dT ^* ^^ written -j^ , and the virtual work of the effective forces is d dT dT\ therefore (^^-^)S^, The following explanation will make the argument clearer. The virtual work of the effective forces is clearly the ratio to dt of tlije 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 -rr/ SB. Hence (^^i+ ^^tt, dt\58 d$ \de dtdd' J is the virtual moment of the momenta at the time t + dt corresponding to a dis- placement S6 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 21—2 324 LAGRANGE'S EQUATIONS. [CHAP. VIII. momenta for a displacement which is the difference between the two displacements at the times t and t + dt. Since Sx=^iB, tMs difference for the variable x is du ^ ( ^^ d/ SB. We therefore subtract on the whole Sm | x' ^^ (^) <*« + *«•} ^^< and dT this is shown to be ^7; dt SB. dB 399. Lagrange's equations for finite forces. To deduce the general equations of motion referred to any co-ordinates. Let U be the force-function, then Z7 is a function of 6, , &c. and t. The virtual work of the impressed forces corresponding to a displacement produced by varying 6 only is -5^ SO. But by D'Alembert's principle this must be the same as the virtual work of the effective forces. Hence ±dT_dT^dU dt dd' de~ dd' „. ., , , d dT dT dU „ J. Similarly we have ^ ^, - ^ = ^ , &c. = &c. It may be remarked that if V be the potential energy we must write — V for U. We then have ±dT_dT dV^ dt dd' de "^ de ' with similar equations for , yfr, &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 + JT, so that L is the difference of the kinetic and potential energies. Then, since U is not a function of 6', (f)', &c., the Lagrangian equations of motion may be written in the typical form d dL dL _^ diW'dd^ Thus it appears that, when the one function L is known, all the differential equations of motion may be deduced by simple partial differentiations. The function L is called the Lagrangian function. These are called Lagrange's general equations of motion. Lagrange only considers the case in which the geometrical equations do not contain the time explicitly, but it has been shown by Vieille, in Liouville's Journal, 1849, that the equations are still true when this restriction is removed. In the proof given above we have included Vieille's extension, and adopted in part Sir W. Hamilton's mode of proof, Phil. Trans., 1834. It differs from Lagrange's in two respects; firstly, he makes the arbitrary displacement such that only one co-ordinate varies at a time, and secondly, he operates directly on T instead of ^mx'^. ART. 400.] INDETERMINATE MULTIPLIERS. 325 Ex. 1, If we change the co-ordinates in Lagrange's equation from 6, 0, &c. to any others x, y, which are connected with 8, 0, &o. by equations which do not contain differential ooefiicients with regard to the time, show by an analytical transformation that the form of Lagrange's equations is not altered, i.e. that the transformed equations are the same as the original ones with x, y, &o. written for 0, 0, &e. This is of course evident by dynamics. Ex. 2. If two sides 6, c and the included angle A of any triangle be taken as the co-ordinates 8, ip, \j/, prove that the Lagrangian equations are satisfied by L=B'. This easily follows from the last example by a change of co-ordinates. 400. Indeterminate Multipliers. In order to use these equations it is necessary to express the Lagrangian function L in terms of the independent co-ordinates of the system. If the geo- metrical conditions are somewhat complex it may be very trouble- some to do this. It is sometimes convenient to express Z as a function of more than the necessary number of co-ordinates and to have geometrical relations connecting them. Suppose that we have L expressed as a function of the co-ordinates 6, (j), yjr, &c., 6", cj}', yjr', &c., and that there are two geometrical equations connecting these co-ordinates, viz. 7(6', (^, &c.) = 0, F (0, ^, Sic.) = (1). To simplify the explanation, we suppose that there are only two 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 f d dL dL\ ^Q , (d dL dL\ ^ , , j, „ ,„-, Also ^8^-)-^S0-f&c. = O (3), Jjp JET and ^Sd + ^S + 8zc. = (4). Since the co-ordinates 6, <^, &c. are connected by two geometrical equations, two of them are dependent variables ; let these be 6, . Following the argument explained in the differential calculus, we multiply (3) and (4) by two arbitrary quantities \ and fi, and add the products to (1). We now choose X and /jl so that the coefficients of 80, Scf) may be zero. The remaining co-ordinates yjr, &c., being independent, the coefficients of Syfr, &c., must also vanish. We thus have .(5). d dL dt d0' dL ^ df , dF „^ -d0-'''d0+f'd0-^ d dL dt d^ dL^.df^ dF _ &c. = oj 326 LAGRANGE'S EQUATIONS. [CHAP. VIII. There are here as many equations as co-ordinates. Joining these to the equations (1) we have sufficient equations to find all the co-ordinates and the two multipliers X and /*. These equations may be put into a simpler form. We notice that the geometrical functions / and F do not contain 6', 4>', &c. (see also Art. 396). Let us then write L,=L + y+fj.F. (6), and treat L^ as if it were the Lagrangian function. If we substitute this value of Zj in the typical equation dt dff dd ^ ^' where 6 stands for any one of the co-ordinates, and simplify the results by remembering that /= 0, ^=0, we obtain in turn all the equations (5). The same process will also supply the geometrical equations (1), if we include X and fi among the co-ordinates. Thus, since L^ contains no \', we have dLJdX' = ; hence, writing \ for 0, the equation (7) gives /= 0. 401. Lagrange's equations for impulsive forces. To deduce the general equations of motion for impulsive forces. Let BUi be 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 SUi in the form SU, = PSd + QB + .... The virtual moment of the momenta imparted to the particles of the system is tm {(x,' - x:) hx + (y/ - 2/„') hy + {z( - zl) Iz], where («„', yi, Zo), («i', Vi, z{) are the values of (a;', y', /) just before and just after the action of the impulsive forces. Let 6o, (po, &c., ^i', ^i', &c. be the values of 6', ', &c. just before and just after the impulses, and let ^o, T^ be the values of T when these are substituted for 6', ^', &c. Then, as in Art. 397, the virtual moment of the momenta is ( "^^-^ - ^M B9. The \a^i ddu J Lagrangian equations of impulses may therefore be written de,' do,' ~ ' with similar equations for (j>, ■^, &c. ART. 403.] COMPONENTS OF THE MOMENTUM. 327 402. These equations are sometimes written in the convenient forms m>^- ©>*-. 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 ff, <})', &c. of the first order, another meaning can be given to the expressions. The brackets may then be said to indicate that d^' — 60, i — ', &c. after all other operations indicated within the brackets have been performed. 403. 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 call --Tn, 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 the ^-component of the momentum dT is -j^ . In the same way we may define the ^-component of the „^f. „ . . d dT dT effective lorces to be -y- -^n- — rn ■ at do da Suppose for example that a variation hd of any co-ordinate has the effect of turning the system as a whole about some dT straight line through an angle hO, then -^ is equal to the angular momentum about that straight line. But, if the variation hO move the system as a whole parallel to some straight line through dT a space h6, then -r^, is the linear momentum parallel to that straight line. See Arts. 306, 308. These resiilts also follow immediately from the general expression dT ^ / ,dx' , ,dy' , ,dz'\ given in Art. 397. Let the given straight line be the axis of z. In the first ease x'=-y$', y'=xB', z'=0, hence the expression reduces to Sm( -x'j/H-i/'a;), which is the angular momentum. In the second case a!'=0, y' = 0, z' = 6', hence the expression becomes Zmz', which is the linear momentum. 328 Lagrange's equations. [chap. viii. 404. The equations for impulsive forces were not given by Lagrange. They seem to have been first deduced by Prof. Niven from the Lagrangian equation dt de' de" de ' We may regard an impulse as the limit of a very large force acting for a very short time. Let tg , t^ be the times at which the force begins and ceases to act. Let us integrate this equation between the limitsi=t|,andi=ti. The integral of the first term is -337 ^ which is the difference between the initial and final values of -=-; . \_jio J t(j ad dT The integral of the second term is zero. For -rj is a function of 6, ', &c. do which, though variable, remains finite during the time tj-fj. If 4 be its greatest value during this time, the integral is less than A (tj - {„), which ultimately vanishes. Hence the Lagrangian equation becomes 33; P= -7^. See a paper \_do _\tQ do in the Mathematical Messenger for May, 1867. 405. Examples of Lagrange's equations. Before pro- ceeding to discuss the properties of Lagrange's equations, we may illustrate their use by the following problems. A body, two of whose principal moments at the centre of gravity are equal, turns under the action of gravity about a fixed point 0, situated in the axis of unequal moment. To determine the conditions that there may be a simple equivalent pendulum. Def. If a body be suspended from a fixed point under the action of gravity, and if the angular motion of the straight line joining to the centre of gravity be the same as that of a string of length I to the extremity of which a heavy particle is attached, then I is called the length of the simple equivalent pendulum. This is an extension of the definition in Art. 92. Let 00 be the axis of unequal moment, A, A, G the principal moments at the fixed point, and let the rest of the notation be the same as in Art. 365, Ex. 1. Then 2T=A (ff'Hsin^ ef^) + G(i>'+feoa ef, U= Mgh cos d + constant, where h is the distance of the centre of gravity from the fixed point, and gravity is supposed to act in the positive direction of the axis of ii. Lagrange's equations will be found to become j^ (AS') - ^ sin e cos S^'^ + Cf {' + f cos $) sin 9= - Mgh sin 8, ^{C(0' + f cose)}=0, j{G{tt>' + \p' cos 8) coiB + A sin^ 8f}=0. Integrating the second of Lagrange's equations, we have (t>' + foos8=n, where « is a constant expressing the angular velocity about the axis of unequal moment. (See Art. 256.) Integrating the third we have Gn aosB + A siv? Bf = a, ART. 406.] EXAMPLES. 329 where a is another constant expressing the moment of the momentum ahout the vertical through 0. (See Arts. 264 and 265, also Art. 403.) There are errors, sometimes made in using Lagrange's equations, which we should here guard against. If uj be the angular velocity ahout OG, we know by Euler's equations, Art. 251, that Uj is constant. If n be this constant, the vis viva of the body may be correctly written in the form 2T=4 (9'2 + sin2ef ») + Cn2. But, if this value of T were substituted in Lagrange's equations, we should obtain results altogether erroneous. The reason is, that, in Lagrange's equationef, all the differential coefficients except those with regard to t are partial. Though u^ is constant, and therefore its total differential coefficient with regard to ( is zero, yet lis partial differential coefficients with regard to 6, ip, &a. are not zero. In writing down the value of T, preparatory to using it in Lagrange's equation, no properties of the motion are to be assumed which involve differential coefficients of the co- ordinates. This has been already indicated in Art. 396. But we must introduce into the expression any geometrical relations which exist between the co-ordinates and which therefore reduce the number of independent variables. Instead of the first equation, we may use the equation of vis viva, which gives A (sin^ e^j/'^ + e"^) =p + 2Mgh cos e. To find the arbitrary constants a and |8 we must have recourse to the initial values of 6 and ^. Let 9„, ^oj-jT' jT ^^ ^^^ initial values oi 8, f, —, -^i then the above equations become . .^dtp Gn . ... A-iif. , Gn . — (t)"+(f)'=— •(t)"HtT*'^'<""— • .(1). J These equations, when solved, give 8 and ^ in terms of f, and thus determine the motion of the line OG. The corresponding equations for the motion of the simple equivalent pendulum OL are found by making (7=0, A=:Ml^, and h=l, where I is the length of the pendulum. These changes give 8in=ef=sin»9„^'' at at .(2). •'■""(f)"+o'"'-.(f')'-^(sy*''f'«>-"«j In order that the motions of the two lines OG and OL may be the same, the two equations (1) and (2) must be the same. This will be the case if either Cn=0, or 8 = 6^. In the first case, we must have «=0, or C=0, so that either the body must have no rotation about OG, or the body must be a rod. In the second case, we must have throughout the motion 8 and f constant, so that the body must be moving in steady motion making a constant angle with the vertical. In either case, the two sets of equations are identical if Mhl=A. This is the same formula as that obtained in Art. 92. 406. Ex. 1. Show how to deduce Euler't equations, Art. 251, from Lagrange's equations. Taking as axes of reference the principal axes at the fixed point, 2T=Aui' + Boi./+Go,i,K 330 Lagrange's equations. [chap. viii. We cannot take (01^,(1)2, u^) as the independent variables, because the co-ordinates of every particle of the body cannot be expressed in terms of them without introducing differential coefficients into -the geometrical equations. (See Art. 396.) Let us therefore express Wi , Wj , Wg in terms of 0, <(>, ^. By Art. 256, we have oil— 6' 8'" 't'-'l'' sin 8 cos \ u2=S'cos0 + ^'sinesin0\. . U3=0' + ^'cosS J As it is only necessary to establish one of Euler's equations, the others follow- ing by symmetry, we need only use that one of Lagrange's equations which gives the simplest result. Since tfi' does not enter into the expressions for oi^, u>^, it is most convenient to use the equation dt d' d~ dip' be seen by differentiating the expressions for u^, w,. Also, by Art. 340, if Jf be the moment of the forces about the axis of C, -r;-=N. djp Substituting we have -rACu^) -(A-B) la^u^ = N, which is a typical form of Euler's equations. Ex. 2. A body turns about a fixed point and its vis viva is given by 2T=Auii^ + Bui2^ + Cu^ - 2Du^a^ ~ iEa^a^ - iFu^u^ . Show that, if the axes are fixed in the body, but are not necessarily principal axes, Euler"s equations of motion may be written in the form d dT^_dT , dT dt dwi do>2 with two similar equations. This result is given by Lagrange. 407. Ex. Deduce the equation of vis viva from Lagrange's equatiom. If the geometrical equations do not contain the time explicitly, Z" is a homo- geneous function ote', ', &c. of the second degree. Hence 2T =—,e' + —,d>' + ... do dtp Difierentiating this totally, we have 2~ = e'^ (^\ +i^e" + &c., dt at \mi0 J did where the &c. implies similar expressions for 0, ^, &c. If we now substitute on the right-hand side from Lagrange's equations, we have But, smce T is a function of 8, $', 0, 0', &c. , — = —0' + — n" + *» ' '^'^ ' dt de dd' ^ "•' subtracting this from the last expression we have — = !^e'+^0' + &o. dt dQ d', &c. may he found in terms of u, v, &c., from these equa- tions. Let Tj, = - Ti + u0' + v^' + &c., and let T3 he expressed in terms of u, v, t&c, the quantities &, ', &c. heing eliminated. Then will It may he that Tj is a fimction of some other quantities, which it will presently he fmmd convenient to designate hy the unxiccented letters 0, ' * We may deduce from this lemjna 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 ^+ &°- By the conditions of the lemma the quantity in brackets vanishes. Now if T« be expressed as a function of 0, u, , v, &c. only, and nob ff, (}>', &c., we have dT, = ^de+'^du + Sic. do du Comparing these two expressions for dT^ we have dT, . dT, dT, ^ = -^and^=0. Thus we have a reciprocal relation between the functions T^ and fa. We find T^ from T^ by eliminating 6', ', &c. by the help of certain equations, we now see that we could deduce T-y from Tr^ 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', tj)', &c. 411. It should be noticed that, if T^ be a homogeneous quadratic function of the accented letters 6', ^', &c , then ud' + v^' + &c. = 2Ti, and therefore T^=Tj^, but is differently expressed. Thus Ti is a function of ff, , \j/, the vis viva 23"! is given in Art. 365, Ex. 1. Show that the reciprocal function is Ex. 2. If the vis viva 2Ti be given by the general expression 2T^ = A^^0^-i-'iAi^e'', Ac. Then d^T d'^T wiU -r~ , ^7-^ , &a. be equal to the minors of the corresponding constituents of du^ dudv the determinant A, each minor having its proper sign and being divided by A. To prove this, we take the total differential of the two sets of equations, dT dT u=^, (fee, e'= 3-^. (fee. From the first set we find de', d' + &c. = y - !/■. Thus H is the sum of the kinetic and potential energies, and is therefore the whole energy of the system. 415. To eaypress the Lagrangian equations of impulses in the Hamiltonian form. Referring to Art. 402, we see that the Lagrangian equations of motion may be written in the typical form 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 416. Examples on tbe Hamiltonian Equations. Ex. 1. To deduce the equation of Vis Viva from the Hamiltonian equations. Since H is a function of (8, 0, &o.), (u, v, &o.) we have, if accents denote total differential coefficients with regard to the time, „, dH dH ^ dH , . dH dt de du dt so that 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 therefore we have H=: h, where h is a con- stant. 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 , . ^ r, ^ dT „ de' ^ i T ^0' dT w=^,={- Aui cos + 5(02 s™ 0) s^'i ff + Cwj cos 6. Before we can use the Hamiltonian equations we must by Art. 411 express T in terms of (m, v, w). To do this we solve these equations to find ta^, u^, Wj in terms COB of u.v.w. We find Aw,=UBmd) + (v eoBe-w) -^-r , 1 ^ ^ ' sin 9 , sin di Bwo=« COS -{vcob6- id) -. — J . ^ ^ ^ ' sin Also by Art. 414 H=i [Aa^^ + B «/ + Coi,^ - U. As we only require one of Euler's equations, let us use ^= -"'i 'j~—'P'- 336 LAGRANGE'S EQUATIONS. [CHAP. VIII. The former of these gives Aa-.-^+Bu«-^ - -j- = - C ^ , a a a at ,.,.,, . Bwo _ Aoi, dU ri^"s which IS the same as Aia, —t^ - Bun-=r --s-i = ~^ jI > '4 ^ B dip dt and this leads at once to the third Euler's equation ip Art. 251. 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 aU the three dynamical and the three geometrical Eulerian equations. Ex. 3. A sphere rolls down a rough inclined plane as descrihed in Art. 144. We have T=^ma?e'^ and V=mgaB sin a. Is it correct to equate H to the difference of these functions ? Verify the answer by obtaining the equations of motion given in Art. 144. Ex. 4. A system being referred to co-ordinates 6, , &o., and the corresponding momenta u, v, &u., in the Hamiltonian manner, it is desired to change the co- ordinates to X, y, &c., where 0, (p, &e. are given functions of ^, y, &o. Show that if f, 17, &c. be the corresponding momenta, then where the suflfixes as usual denote partial differentiations. Show also by a purely analytical transformation that the Hamiltonian equations with 0, u, &o. change into the corresponding ones with x, ^, &o. Ex. 5. The Lagrangian function is a function of 0, (p, &c. and 0', ', &c. into the corresponding letters u, v, &c. But we may also apply the Lemma to change some only of the Lagrangian co-ordinates into the corresponding Hamiltonian co-ordinates, 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 co-ordinates for which they are best adapted, and the Hamiltonian equations with the remaining co-ordinates, 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 co-ordinates, 0, 4i, ^, v- I^et ij be the Lagrangian function. Let us now suppose that we wish to use Lagrange's equations for the co-ordinates f, ij and the Hamiltonian equations for the co-ordinates 0, . To do this we use the two formulae of transformation -jff = u, jT7 = "". ^^^ ^^ P^* X2 = - Xi -f- ud' + V(j>'. We have in consequence the two sets of Hamiltonian equations, dL^ ,_ dLi ^ = d^' "*- de- '^-^' ^- #• We must now include f, r)' among the unaccented letters spoken of in the Lemma of Art. 410, so that we have dL^ dL, dL^ ^ __ dL, df' ~ df ' d^ d^' K.D. 22 338 Lagrange's equations. [chap. viii. with two similar equations for t]. Thus the two Lagrangian equations for f, rj are still true if we replace L^ by L^; so that we have the two sets of Lagrangian equations, d dL^ _ dig d dL^ _ dL^ JtW~'d¥' Jt~di}'~~dv' 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 i be a function of 6, & , (j), <^, &c., then the function modified for (say) the two co-ordinates d, (f> will be L' = L-ud'- v^', where u = -^ , v = j-, , and we suppose 6', <^' eliminated from the function L'. Thus X is a function of 6, ' and all the other letters, L' is a function of 0, (f>, u, v and all the other letters. These two functions L, L' possess the property (by Art. 410) that their partial differential coefiicients are the same with respect to all letters except 0', , &c., because it is obtained from Li just as T^ is obtained firom Ti in Art. 410, except that we operate only on such of the co-ordinates as we please. It is however convenient to use the two words in slightly different senses. We shall use the word Reciprocation when we change all the co-ordinates, and Modification when we change only some. 421. To find a general expression for the modified Lagrangian fwnction after the necessary eliminations have been performed. Let the vis viva 2T be given by the homogeneous quadratic expression r=Te9y+rfl^ey+... + r«^+rflf9'i'+..., so that the Lagrangian function is L=T+17, where J7 is a function of the co- ordinates e, ' + ... = «- Teii' - r^,,,' - &0, =&0. For the sake of brevity let us call the right-hand members of these equations " - -^i v~Y, &o. Since T is a homogeneous function, we have ^ .(!)• + P'(M + X) + 40'(5; + r) + &c. But by definition the modifipd function L'=-L„ is L' = L-uB'-v4>'- ... ■•(2). ■•(3)- -ie'{u-X)-i!f,'{v-Y)-&B.] Solving equations (1) we find 8', ', &c. in terms of ^', r/', &a. by the help of determinants. Substituting their values in the expression (3), we find J' L'=Til ~ +T^r,^'v' + &e. + U+^ 0, u-X, v-Y, u-X, Tee, Tej,, v-Y, Te^, T^^, where A is the discriminant of the terms in T which contaih only 8', 4>', &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 L'=Tu~+Ti.,il.W + & X=Tei^' + Te,,r,' + ..., Y=T^i^' + T^nv'+.-; &e. = &a., so that X, Y, &e. may be obtained from u, v, &c. by omitting the terms which contain 8', ip', &c. , i. e. the co-ordinates to which we intend to apply the Hamiltonian equations. It should be noticed that the first of the three determinants in the expression for L' contains only the momenta u, v, &c. and the co-ordinates. The second does not contain u, v, &c. but is a quadratic function of f', V, &a. The third contains terms of the first degree in f, t]', &b. multiplied by the momenta u, v, &c. 422. Case of absent co-ordinates. In many cases of small oscillations about a state of steady motion, and in some other problems, the Lagrangian function L does not contain some of the co-ordinates as 0, ', &c., and introducing in their place the constant quantities u, v, &c. We write L' = L-ue'-v' ..., and eliminate 6', ^', &c. hy help of the integrals just found. The equations to find ^, rj, &c. may he deduced hy 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 6', the vir- tual work obtained by varying only, and so on, it is clear that Lagrange's equations may be written in the typical form -^--t^,- -j^ = P. dt du du 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, ij), &a. Secondly those which are non-conservative, such as friction, some kinds of resistances, &c. The parts of P, Q, &c. 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 AKT. 427.] NON-CONSERVATIVE FORCES. 343 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 where /j.^, ft^, /j.^ are three constants which are negative if the forces are resistances. For example, if the particles be moving in a medium whose resistance is equal to the velocity multiplied by a constant k, then /i^, ii^, fji^ are each equal to - k. Put Since (x, y, z) are functions of 0, 0, &c. given by the geometry of the system we have, as in Art. 396, x'=-r--\ 8' + ... at d$ with similar expressions for the other co-ordinates. Substituting we have F expressed as a function of d, , &a., d', (ji, &o. We also notice that, as in Art. 397, -T2i= :n- Differentiating F partially we have da aa _dF d8' dF dF[, 'de' d' = 'Z{iJi^x'Sx + &Q.). In this case, therefore, if U be the force-function of the conservative forces, F the function just defined, QSd, $50, &c. the virtual works of the remaining forces, Lagrange's equations may be written d dT_dT_dU dF dt dB' de " de de'^ ' with similar equations for 0, ^, &c. We may notice that, if the geometrical equations do not contain the time explicitly, the function F is a quadratic homogeneous function of e', 0', &c. If the forces whose effects are included in F be resistances, then /m^, /ji^, /jl^, &e. 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+&o.-§e'-&c., but in this case F also is a homogeneous function of 8', &c. Hence we find ^(T-U) = 8'e + &o.-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 Rayleigh in the Proceedings of the London Mathematical Society, June, 1873. The function F has been called by him the Dissipation function. 344 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 V^, Vy, V^ be the com- ponents of the velocities, /j-iV^, i^V^, /ijF^ the components of the force of repulsion, the part of F due to these is - - S {f^V^^ + fji^V/ + fj^VJ'). 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 multipUed by the same constant k. Show that these resistances may be included in a dissipation function F, where F=^{(r{w' + v^ + w^)+Au^^ + Buiy' + CwJ' - 2Z»Uj,a>, - 2Ea^a^ - 2Fw^ay} , where o- is the aiea, A, B, &c. the moments and products of inertia of the surfaee of the body, and (m, v, w) the resolved velocities of the centre of gravity of a: 429. Indeterminate Multipliers, &c. To explain how Lagrange's equations can be used in some cases when the geometrical equations contain differential coefficients ivith 7'egard to the time. It has been pointed out in Art. 396, that the independent variables 6, (f), &c. used in Lagrange's equations must be so chosen that all the co-ordinates 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 in- volves differential coefficients of the co-ordinates. 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 a/ — a6' = 0, which by integration becomes x — a6 = h, 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. 427. Each reaction has an accom- panying equation of condition, and thus we always have sufficient equations to eliminate the reactions and determine the co-ordinates 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 ART. 430.] INDETERMINATE MULTIPLIERS. 345 a body is rolling on a perfectly rough surface. Let 6, (j>, &c. be the six co-ordinates of the body, then by Art. 137, there will be three equations of the form L, = A,e' + B,' + ... = (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'~d0)^^ + ^:'- = d6^^ + ^' <2), where U is the force-function of the impressed forces. Since the virtual work of the reactions at the point of contact have been omitted, this equation is not true for all variations of 6, , &c., but only for such as make the body roll on the rough surface. But the geometrical equations L^, L^, L, express the fact that the body rolls in some manner, hence hd, S^, &c. are connected by three equations of the form AS6> + £iS^+... = (8). If we use the method of indeterminate multipliers (see Art. 400), the equations of virtual work are transformed in the usual manner into ddT_dT_dUdL, dL, dL, dt dO' de~ de '^'^ dO'^f^ dd'^" dd' ^*^' with similar equations for the other co-ordinates ', absent from all but the second, and so on. When this has been done, the equation for becomes ddT_dT_dU dL, dt dO' dd'^ de '^ dd' ^ ^• Thus \ is found at once. The values of fi and v may be found from the corresponding equations for ^, ■^. We may then sub- stitute their values in the remaining equations. 430. The method of indeterminate multipliers is really an introduction of the unknown reactions into Lagrange's equations. Thus let Ri, R„ i?,, be the resolved parts of the reaction at the point of contact in the directions of the three straight lines 346 LAGRANGE'S EQUATIONS. [CHAP. VIII. 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 KiA, k^L^, kJO^. Then if only be varied j}he virtual work of -Ri is KiA^SO, which may be written K^-y^hQ. Similarly the virtual works of R^ and R^ are k^ rj ^0 and «s -yj BO- Hence, by Art. 426, Lagrange's equations are of the form d dT dT dU r, dL, t> dL, dL, dtde'~de~d0^ "'^^ W + "'^^ dff + "'^^ dff ■ Comparing this with the equations obtained by the method of indeterminate multipliers we see that, X, /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 just that manner which is most convenient for the solution of the problem. 431. Ex. Form by Lagrange's method tlie equations of motion of a homoge- neous sphere rolling on an inclined plane under the action of gravity. Let the axis of x be taken down the plane along the line of greatest slope, and let the axis of y be horizontal and that of z normal to the plane. Let (x, y, a) be the co-ordinates of the centre of gravity of the sphere, B, , \p the angular co-ordi- nates of three diameters at right angles fixed in the sphere in the manner explained in Art. 256. Then, if the mass be taken as unity, the vis viva is by Art. 365, Ex. 1, T£=x"^ + y'-^+k^{(4,' +f cosey^ + e'^ + sm^ef",. The resolved velocities parallel to the axes of x and y of the point of the sphere in contact with the plane are to be zero. These give the oouditions x' - aa,, = 0, y' + a, xji. Taking these in turn, we have x" = g + 'K, y" = ii, h^ (d" + cl>'f sin 9) = -\a cos yp- im sin \j/, K^ -J- ((fi' + f oos 8)= -\a sin S sin f + iJ.a sin B cos \j/, k^^ (,^'coa6 + f) = 0, ART. 431.] EXAMPLES. 34T The last equation shows that 0'oosfl + ^' is constant. Prom this we infer, by Art. 257, that the angular velocity u, of the sphere about a normal to the plane is constant throughout the motion. Eliminating /t from the two preceding equations, and substituting for y*" from the last we find - p = e" cos i/' + " sin e sin ^ + Y' sin 9 cos i/- - e'f sin \j/ + e'' cos B sin yp. n If But this is — . In the same way we find - ^ = ^ . Substituting these values of \ and li in the first two of Lagrange's equations we have These are the equations of motion of a particle acted on by a constant force parallel to the axis of x. The centre of gravity of the sphere therefore describes a parabola, as it it were under a constant acceleration, equal to f j, tending along the line of greatest slope. This solution is rather complicated, but the problem has been selected to show how we may use Lagrange's equations as specially illustrating the remarks made in Art. 429. So far as this particular problem is concerned a very simple and short solution may be obtained by the ordinary processes of resolving and taking moments. For this we refer the reader to Art. 269 and also to the chapter on the motion of a body under any forces in the second part of this work. 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 puUey 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 2! 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 ^2^ . Show that the heavy end of the rod will fall till the inclination of the rod to the vertical is cos~i ( y/n' + 1 - re) , and will then rise again . 3. A rod of length 21 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'^ - iar'e' sin a + a?e'^ + sin^o (r" + J V) 0"' + 2g cos a (r - r„) = 0, {a^ + sin^a (r^H- J P)} 6' - ar' sin a = 0, where r is the distance measured along a generator from the centre of gravity to the principal circular section, B is the excentric angle of the point in which the generator meets this circular section, a is the radius of the circular section, and o is the inclination of the rod to the vertical. * These examples are taken from the Examination Papers which have been set in the University and in the Colleges. 348 Lagrange's equations. [chap. viii. 4. A ring of mass vi and radius 6 rolls inside a perfectly rough ring of mass M and radius o, which is moveable about its centre in a vertical plane. If 8, tj> be the angles turned through by the rings from their position of equilibrium, prove that ae + h=(a-l>)^, Mae"=mh sin n{t-t')f {t')dt', where x„, x^ are the values of x and x when t = 0. [Math. Tripos, 1876.] 437. It will be often found advantageous to trace the motion of the system by a figure. Let equal increments of the abscissa 352 SMALL OSCILLATIONS. [CHAP. IX. of a point P represent on any scale equal increments of the time, and let the ordinate represent the deviation of the co-ordinate x from its mean value. Then the curve traced out by the repre- sentative point P will exhibit to the eye the whole motion of the system. In the case in which a and h — a^ are both positive the curve takes the form here represented. The dotted lines correspond to the ordinate ± Ae~^*. The representative point P oscillates between these, and its path alternately touches each of them. In just the same way we may trace the representative curve for other values of a and b. The most important case in dynamics is that in which a = 0. The motion is then given by w-^=Asha. {^/bt + B). The representative curve is then the curve of sines. In this case the oscillation is usually called harmonic. 438. Ex. 1. A system oscillates about a mean position, and its deviation is measured by x. If «„ and x^ be the initial values of x and x, show that the system will never deviate from its mean position by so much as \ "- " ° 2 \ ^ if b be greater than a". Ex. 2. A system oscillates about a position of equilibrium. It is required to find by observations on its motion the numerical values of a, b, c. Any three determinations of the co-ordinate x at three different times will generally supply sufficient equations to find a, b, c, but some measurements can be made more easily than others. For example, the values of x when the system comes momentarily to rest can be conveniently observed, because the system is then moving slowly, and a measurement at a time slightly wrong will cause an error only of the second order, while the values of t at such times cannot be conveniently observed, because, owing to the slowness of the motion, it is difficult to determine the precise moment at which x vanishes. If three successive values of x thus found be ajj, x^, x^, the ratio of the two successive arcs x^-x^ and ^^ - ajj is a known function of o and 6, and one equation ART. 439.] MOMENTS ABOUT THE INSTANTANEOUS CENTRE. 353 can thus be formed to find the constants. If the position of equilibrium is unknown, we may form a second equation from the fact that the three arcs c c c ^ c ^1 - T > •'"a - r > ^3 - r also form a geometrical progression. In this way we find y , which is the value of x corresponding to the position of equilibrium. The position of equilibrium being known, the interval between two successive passages of the system through it is also a known function of a and 6, and thus a third equation may be formed. Ex. 3. A body performs rectilinear vibrations in a medium whose resistance is proportional to the velocity, under the action of an attractive force tending towards a fixed centre and proportional to the distance therefrom. If the observed period of vibration is T, and the co-ordinates of the extremities of three consecutive semi- vibrations are p, q, r, prove that the co-ordinate of the position of equilibrium, and the time of vibration if there were no resistance are respectively -a2 pr — q p + r~2q and T k + A ^ log^— 'Vl *. [Math. Tripos, 1870.] First Method of forming the Equations of Motion. 439. When the system under consideration is a single body there is a simple method of forming the equation of motion which is sometimes of great use. Let the motion be in two dimensions. It has been shown in Art. 205, that if we neglect the squares of small quantities we may take moments about the instantaneous centre as a fixed centre. Usually the unknown reactions will be such that their lines of action will pass through this point, their moments will then be zero, and thus we shall have an equation containing only known quantities. Since the body is supposed to be turning about the instan- taneous centre as a point fixed for the moment, the direction of motion of any point of the body is perpendicular to the straight line joining it to the centre. Conversely, when the directions of motion of two points of the body are known, the position of the instantaneous centre can be found. For if we draw perpendiculars at these points to their directions of motion, the perpendiculars must meet in the instantaneous centre of rotation. The equation may, in general, be reduced to the form ifj.a ^"^ — /i^oment of impressed forces aboutN df~\ the instantaneous centre / ' where is the angle some straight line fixed in the body makes with a fixed line in space. In this formula Mk' is the moment of inertia of the body about the instantaneous centre, and since d'd the left-hand side of the equation contains the small factor -^ R. D. 23 354 SMALL OSCILLATIONS. [CHAP. IX. we may here suppose the instantaneous centre to have its mean or undisturbed position. On the right-hand side there is no small factor, and we must therefore be careful either to take the moment of the forces about the instantaneous centre in its disturbed position, or to include the moment of any unknown reaction which passes through the instantaneous centre. Ex. If a, body with only one independent motion can be in equilibrium in the same position under two different systems of forces, and if ij, L^ are the lengths of the simple equivalent pendulums for these systems acting separately, then the length L of the equivalent pendulum when they act together is given by L Lj L2 440. Ex. 1. A homogeneous hemisphere performs small oscillations on a perfectly rough horizontal plane : find the motion. Let C be the centre, G the centre of gravity of the hemisphere, N the point of contact with the rough plane. Let the radius = a, CG=c, 6= A. NOG. Here the point N is the centre of instantaneous rotation, because, the plane being perfectly rough, sufficient friction is called into play to keep N at rest. Hence taking moments about N {k''+GN^)e=-gc. sme. Since we can put GN=a-c in the small terms, this reduces to {k^+{a-cf}e+ge.e=0. Therefore the time of a small oscillation is 2ir / k' + (a- («-«)" 2 3 It is clear that k''+c^=Bq. of rad. of gyration about G =-=a^, and that c=-^a. 8 If the plane had been smooth, M would have been on the instantaneous axis, GM being the perpendicular on ON. For the motion of N^ is in a horizontal direction, because the sphere remains in contact with the plane, and the motion of G is vertical by Art. 79. Hence the two perpendiculars GM, NM meet on the instantaneous axis. By reasoning similar to the above the time is found to V c be 2ir Ex. 2. Two circular rings, each of radius a, are firmly jointed together at one point so that their planes make an angle 2a with one another, and are placed on a perfectly rough horizontal plane. Shew that the length of the simple equivalent pendulum is Ja (1 + 3 cos^ a) cos a ooseo^ o. [Math. Tripos.] ART. 441.] OSCILLATIONS OF CYLINDERS. 355 441. Oscillations of Cylinders. A cylindrical surface of any form rests in stable equilibriimi vmder gravity on another perfectly rough cylindrical surface, the axes of the cylinders being horizontal and parallel. A small disturbance being given to the upper surface, find the time of a small oscillation. Let BAP, B'A'P be the sections of the cylinders perpendicular to their axes. Let OA, GA' be normals at those points A, A' which before disturbance were in contact, and let a be the angle made hy AO with the vertical. Let OPO be the common normal at the time t. Let G be the centre of gravity of the moving body, then before distiirbance A'G was vertical. Let A'0 = r. Now we have only to determine the time of oscillation when the motion decreases without limit. Hence the arcs AP, A'P will be ultimately zero, and therefore C and may be taken as the centres of curvature of AP, A'P. Let p = OA, p' = CA', and let the angles AOP, A'GP be denoted by (/>, <^' respectively. Let 6 be the angle turned round by the body in moving from the position of equilibrium into the position BA'P. Then, since before disturbance A'G and AO were in the same straight line, we have 6 = /: GDE= + ^', where GA' meets OAE in D. Also, since one body rolls on the other, the arc AP = arc A'P, .:p = p'') ^ p + p Again, in order to take moments about P, we require the horizontal distance of G from P ; this may be found by projecting the broken line PA' + A'O on the horizontal. The projection of PA' = PA' co%{a + 6) = p<\>cosa when we neglect the squares of 23—2 356 SMALL OSCILLATIONS. [CHAP. IX. small quantities. The projection of A'O is r0. Thus the hori- zontal distance required is f "^ , cosa — rjd. If k be the radius of gyration about the centre of gravity, the equation of motion is If L be the length of the simple equivalent pendulum, we have P + r^ op' — ^ — = -'-'—, 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 A8 = s, where - = - + -, and describe a circle on AS as diameter. Let AG, produced if necessary, cut this circle in N. Then 01^ = 8 cos a — r, the positive direction being from N towards A. The length L of the simple equivalent pendulum is given by the formula L . GN = sq. of rad. of gyration about A. It is clear from this formula, that it G* lie without the circle * 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 iJ = - j=^ , so 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. 444] THE CIRCLE OF STABILITY. 357 and above the tangent at A,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. 443. 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 d6 of the body, the equation to find the length of the simple equivalent pendulum of a body oscillating under gravity may be written 1 *^8° ^^^ expression for L takes the form — = — = G'N. The ec[uilibiium is therefore stable or unstable according as G' lies within or without the circle of stability. 445. Osoillatlons of a body resting on two curveB. 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 eijuilibrium OG is vertical. Let i, j be the angles which CA, BD make with the vertical, and let a be the angle AOB. Let A', B', G', E' denote the positions into which A, B, G, E are moved when the body is turned through an angle 8, and let 0' be the point of intersection of the normals at A', B'. Let AGA'=, BDB'=^'. Since the body may be brought from the position AB into the position A'B' by turning it about through GA.it> _ BD. and the expression becomes Ij sm a sm a If 04 and OB be at right angles, this takes the simple form '^=0G-20F, where F is the projection on OG of the middle point of AB. Ex. 1. A heavy rod ACB rests in equilibrium in a horizontal position within a surface of revolution whose axis is vertical. Let 2a be the length of the rod, p the radius of curvature of the generating curve at either extremity of the rod, i the inclination of this radius of curvature to the vertical. Prove that, if the rod be slightly disturbed, so that it makes small osoiUationa in a vertical plane, the length , . , . , . J , . opsin^icos j (1 + 3 cot^i) of the eqmvalent pendulum is — ^, . „ ., ■' . 3 (a - p sm' i) 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, prove that the length of the equivalent pendulum is . _ , or ~ ^-^ — ^ , 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 horizoiital after a time ( 5- ) I (1 - ip*)'^d^. The first case of this question was set in a Caius Coll. paper. Ex. 3. The extremities of a rod of length 2a slide upon two smooth wires, which form the upper sides of a square whose diagonal is vertical, prove that the length of the equivalent pendulum is fa. [Math. Tripos.] 446. Oscillation when patb of centre of gravity is known. A body oscillates about a position of equilibrium imder the action of gravity, the radius of curvature of the path of the centre of gravity being known, find the time of an oscillation. Let A be the position of the centre of gravity of the body when it is in its position of equilibrium, O the position of the centre of gravity at the time (. Then since in equilibrium the altitude of the centre of gravity is a maximum or mini- mum, the tangent at A to the curve AQ is horizontal. Let the normal GG to the curve at G meet the normal at A in C. Then, when the oscillation becomes iudefi- 360 SMALL OSCILLATIONS. [chap. IX. nitely small, G is the centre of curvature of the curve at A. Let AG=s, the angle ACG=\j/, and let R be the radius of curvature of the curve at A. Let 6 be the angle turned round by the body in moving from the position of equilibrium into the position in which the centre of gravity is at Cf ; then — is the angular velocity of the body. Since Gf is moving along the tangent at G, the centre of instantaneous rotation lies in the normal GO, at such a point that OG|=vel.ofG = g, GO^^. de Let MK^ be the moment of inertia of the body about its centre of gravity, then taking moments about 0, we have rPf) (h-'+OG^'^^=-g.OGsmf. li d^ OG Now ultimately when the angle 6 is indefimtely small 5 = 33 = ~p~ ! equation of motion becomes the Hence if Z. be the length of the simple equivalent pendulum we have 447. Oscillations found by Vis Viva. When the system of bodies in motion admits of only one independent motion, the time of a small oscillation may frequently be deduced from the equation of vis viva. This equation is one of the second order of small quantities, and in forming the equation it is thus necessary to take into account small quantities of that order. This sometimes involves rather troublesome considerations. On the other hand, the equation is free from all the unknown reactions, and we thus frequently save much elimination. The method of proceeding will be made clear by the following example, by which a comparison may be made with the method of the last article. The motion of a body in space of two ditnensions is given by the co-ordinates x, y of its centre of gravity, and the angle which any fixed line in the body makes with a line fixed in space. The body being in equilibrium under the action of gravity, it is required to find the time of a small oscillation. Since the body is capable of only one independent motion, we may express {x, y) as functions of 0, thus x=F(0), y=f{0). ART. 448.] MOMENTS ABOUT THE INSTANTANEOUS AXIS. 361 Let Mk^ be the moment of inertia of the body about an axis through its centre of gravity, then the equation of vis viva becomes i:^ + y^ + k''&^ = C-2gy, where G is an arbitrary constant. Let a be the value of 8 when the body is in the position of equilibrium, and suppose that, at the time t, e=o + ^. Then, by Maolaurin's theorem, 2/=Vo+2/o'0 + 2/o"|-+-. where y,/, y^' are the values o^ ^ . jg^ when 9= o. But in the position of equili- brium y is a maximum or minimum; .•. 2/0'= 0. Hence the equation of vis viva becomes (x^'^ + k^)^^=G-gy^'\ where a;/ is the value of -75 when e = a; dif- ferentiating we get {x^^ + k^)', the angle which the rod makes with the tangent at the cusp B is J 0. 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 362 SMALL OSCILLATIONS. [CHAP. IX. Art. 225 that the moment 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 -^ID? + -^MV=U +C, where U is the force- function, and G some constant. Differentiating we get dO 1 rf/ „ F dV _ dU dt'^2 dt'^ n dt ~€ldf Let L be the moment of the impressed forces about the in- jjj stantaneous central axis, then L = y^-t: by Art. 340. Let p be the pitch of the screw-motion of the body, then V = pQ>, The equation of motion therefore becomes If the body be performing small oscillations about a position of equilibrium, we may reject the second and third terms, and the equation becomes If there be an iastantaneous axis, p = 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. Second Method of forming the Equations of Motion. 449. Let the general equations of motion of all the bodies be formed. If the position about which the system oscillates be known, some of the quantities involved will be small. The squares and higher powers of these may be neglected, and all the equations will become linear. If the unknown reactions be then eliminated the resulting equations may be easily solved. If the position about which the system oscillates be unknown, it is not necessary to solve the statical problem first. We may by one process determine the positions of rest, ascertain whether they are stable or not, and find the time of oscillation. The method of proceeding will be best explained by an example. 450. Ex. The ends of a wniform heavy rod AB of length 21 are constrained to move, the one along a horizontal line Ox, and the other along a vertical line Oy. If the whole system turn round Oy ART. 450.] SECOND METHOD OF FORMING THE EQUATIONS, ETC. 363 vnth a uniform angular velocity m, it is required to find the posi- tion of equilibrium and the time of a small oscillation. Let X, y be the co-ordinates of G the middle point of the rod, 6 the angle OAB which the rod makes with Ox. Let R, R be the reactions at A and B resolved in the plane xOy. Let the mass of a unit of length be taken as the unit of mass. J^ >1 Rl £ W 3 The accelerations of any element dr of the rod whose co- d"? 1 d ordinates are (f , rj) are -^ — oj^f parallel to Ox, ^ -=- (^^to) perpen- dicular to the plane xOy, and -^ 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 \dr and ridr, taken throughout the body, are 2^05 and 2ly acting at G, and a couple 2W0 tending to turn the body round G. The resultants of the effective forces m^^dr taken throughout the body are a single force acting at G' = I (o^{x + r cos 6) dr = uy'x . 21, and a couple* round G= j (o^(x + rcoa9)rsin0dr=a)'.2l. ^aind cos 6, the distance r being measured from Q towards A. Then we have, by resolving along Ox, Oy, and by taking moments about G, the dynamical equations * If a body in one plane be turning about an axis in its own plane with an angular velocity u, a general expression can be found for the resultants of the centrifugal forces on all the elements of the body. Take the centre of gravity Gf 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 G = j(a'(c + x) dm=u^. Mc, since a; = 0, and a single resultant couple =ju^{c + x) ydm = i^ jxydm, since ^ = 0. 364 SMALL OSCILLATIONS. [CHAP. IX. (1)- 2lx = -R' + ay'x.2l 2ly = -R+g.2l n 2We = Rx-R'y-o)K2l.^ sin cos 9 We have also the geometrical equations X = I cos 0, 2/ = Z sin (2). Eliminating R, R', from the equations (1), we get ooy — yx + k^0 = gx — m^xy — <»^ k sin cos 9 (3). o 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, = 0. These give f(x, y, 0)=gx — m'ciyy — w'' ^ sin ^ cos ^ = (4). Joining this with equations (2), we get = ^ , or sin 5 = j-^ , and thus the positions of equilibrium are found. Let any one of these positions be represented by = a, x = a, y = b. To find the motion of oscillation. Let x=a + x', y = b + y', = a + 0', where x', y', ff are all small quantities, then we must substitute these values in equation (3). On the left-hand side, since x, y, 0, are all small, we have simply to write a, b, a, for X, y, 6. On the right-hand side the substitution should be made by Taylor's Theorem, thus /(. + .', 6 + ,', a + ^') = |^' + f3/' + |^'- We know that the first term f{a, b, a) will be zero, because this is the very equation (4) from which a, b, a were found. We therefore get 72 ay' - bx' + m' =(g- 10%) x' - ar'ay' - m^ 5- cos 2a . 0'. o But, by putting 6 = a-\-!9' in equations (2), we get by Taylor's Theorem a/ = — lsma.0', y' =1 cos a . 0'. Hence the equation to determine the motion is Q? + ^') 5 + (9^ sin a + 1 foH'' cos 2a) ff = 0. 4 Now, if gl sin a 4- K «>H^ cos 2a = n be positive when either of the two values of a is substituted, the corresponding position of equi- ART. 450.] SECOND METHOD OF FORMING THE EQUATIONS, ETC. 365 librium is stable, and the time of a small oscillation is 2^^'- + k^ If n be negative the equilibrium is unstable, and there can be no oscillation. If «' > ^, 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 eo'K-M 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 Hence the left-hand side of equation (3) becomes {l^ + P) S. The right-hand side becomes by Taylor's Theorem /72 / 9 \ /?'^ 1^^ (gl cos a - ^ (oH^ sin 2a j j— ^ + &c. When n = 0, we have a = |^ and «' = ^f • Making the neces- sary substitutions the equation of motion becomes Since the lowest power of 6' 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 (I' + k') /•' ggl_ . V gl JoJoL'-e'*' gl Jo Ja. put ff = a0, then a ~V gl -Jojl^*' 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. 366 SMALL OSCILLATIONS. [CHAP. IX. This definite integral may be otherwise expressed in terms of the Gamma function. It may be easily shown that I , = '. f—l . 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 N, N is the instantaneous axis. Taking moments about N, we have the equation ft)" (Z + r-y sin ^008^27 4P = glcQ)^d — -^ , Sac. Then if the system oscillate about the position of equilibrium, these quantities will he small throughout the motion. Let n be the number of these co-ordinates. 368 SMALL OSCILLATIONS. [CHAP. IX. As before, let accents denote differential coefficients with regard to the time. 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', <^', &c. of the form 2T=AJ'^ + 1A,,e',i>'^A^^'^ + &c (1). Here the coefficients A-^ &c. are all functions of 6, + &c. + B^O' + 2B,,etj> + &c....{2). Here Uo is a constant, which is evidently the value of 17 when 0, , &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 princi- ple of virtual work, -j^ = 0, tt = 0. ^-c = (see also Art. 340). If the co-ordinates chosen are such that they vanish in the position of equilibrium, it immediately follows that B^ = 0, jB^ = 0, &c. = 0. If the co-ordinates 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^O, iB^, &c. are thus of the second order of small quantities and the quadratic terms of U cannot be neglected in comparison with thern. We may also notice that the equilibrium values of 6, +..\ (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, degree to find p''. It will be shown in the second part of this work that all the values oip' are real. Taking any root positive or negative, the equations (6) determine the ratios of N, P, &c. to M, and we notice that these ratios also are all real. If all the roots of the determmantal equation are positive, the equations (5) give the whole motion, with 2n arbitrary constants, viz. Mi, M^, M^-.-Mn and e^, e2,...6„. These have to be determined by the initial values of 6, ^, &c., 6 , (/)', &c. If any root of the determinantal equation is negative, the R. D. 24 370 SMALL OSCILLATIONS. [CHAP. IX. corresponding sine will resume its exponential form, the coefficient 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 0, , &c. The constants a, /S, &c. in the trial solution (5) are evidently the co-ordinates of the central position about which the system oscillates. Substituting these values of 0, , &c. by equating to zero the first differential coefficients of U with regard to 0, ^, &c. But the equations thus obtained are evidently the same as the equations (8). 456. Periods of Oscillation. We see from (5) that each of the n co-ordinates 0, ^, &c. is expressed in a series of as many sines as there are separate values of p'. Thus, when there are several independent ways in which the system can move, there are as many periods of oscillation. These are clearly equal to — , — , &c. Generally we want only these periods of oscillation and not the particular position occupied by the system at any instant. In such a case we may in any problem omit all the steps of the argument and write down the determinantal equation at once. We then use the following rule. Expmd the force function U and the semi vis viva T in ascending powers of the co-ordinates 0, (f), Sc, and their differential coefficients 0', (j)', &c., all powers above the second being rejected. Then, omitting the accents or dots ART. 458.] LAGRANGE'S METHOD. 371 in the expression for T and retaining only the quadratic term in U, equate to zero the discriminant of p^T + U. The roots of the equa- tion thus formed will give the required values ofip. The mode of using this rule in conjunction with the method of indeterminate multipliers will be given in the second part of this treatise. 457. Position of the system. If it be also required to find the position of the system at any time, we must determine the values of the constants. Referring to equations (6) we see that the ratios of M, N, P, &c. for any particular trigonometrical term in the solution (5) are the same as the ratios of the minors of the constituents of any line we please in the Lagrangian determinant (7). In these minors we of course substitute the value oip^ which belongs to the particular trigonometrical term we are consider- ing. In this manner the coefficients of all the trigonometrical terms are found in terms of those which occur in the series for any one co-ordinate. As already explained, these remaining n coeffi- cients, and the n constants from Ci, ... 6„ must be found from the given initial values of the n co-ordinates 0, P-gl alp' I =0. I alp^ p'(lt' + a^)-ag\ 24-2 872 SMALL OSCILLATIONS. [CHAP. IX. This quadratic gives two values otp^. If these be pj^ and p^", we have e=Mi sin (pi* + El) + M2 sin (jp^f + eg) , Writing 31fi=a\ show that the ratio of the periods of the two oscillations cannot lie between 2±;^3. Ex. 2. Two heavy particles, masses M and m, are tied to a string and suspended frota a fixed point 0, the lengths OM, Mm of the string being respectively a and 6. If the particles make small transverse oscillations find the two periods of oscilla- tion, and show that they cannot be equal. Show also that one period is double the other if i{M+m){a+ b)^=2Mab. Ex. 3. A smooth thin shell of mass M and radius a rests on a smooth inclined plane by means of an elastic string, which is attached to the sphere, and to a peg at the same distance from the plane as the centre of the sphere, while a particle of mass m rests on the inner surface of the shell. In the position of iequilibrium the string is parallel to the plane, find the times of oscillation of the system when it is slightly displaced in a vertical plane, and prove that the arc traversed by the particle and the distance traversed by the centre of the shell from their positions of equilibrium can always be equal if (M+m+mcosa) gl=Ea{l + oo3a.), where £ is the coefficient of elasticity of the string, / its natural length, and a the inclination of the plane to the horizon. Caius Coll. Ex. 4. A three-legged table is made by supporting a heavy triangular lamina on three equal legs, the points of support being the angular points of the lamina; if the legs be equally compressible and their weights be neglected, then the system of co-existent oscillations of the top consist of one vertical oscillation and two angular oscillations about two axes at right angles in its plane, and the periods of the latter are equal and double that of the former. St John's Coll. Ex. 5. A bar AB of mass m and length 2a is hung by two equal elastic cords AC, BD, which have no sensible mass, and have unstretched lengths l^. G and D are fixed points in the same horizontal line, and CD = 2a. Investigate the small oscillation of the bar when it is displaced from its position of equilibrium in the vertical plane through CD, and show that the periodic times of the horizontal and vertical oscillations of the centre of gravity of the bar, and of the rotational oscilla- tion, are those of pendulums of lengths I, l-l^,,^ ('- W respectively, where I is the length of either cord when the system is in equilibrium. Math. Tripos. Ex. 6. Three equal particles mutually attracting each other according to the Newtonian law are constrained to move like beads along the smooth sides of an equilateral triangle. In equilibrium they occupy the middle points of the sides. Prove that the equilibrium is unstable unless the initial displacements and the initial velocities are equal, and in this latter case find the time of a small oscillation. Ex. 7. A heavy body whose centre of gravity is H is suspended from a fixed point 0. A second body whose centre of gravity is G is attached to the first at some point A situated in OH produced. The system oscillates freely in a vertical plane, prove that the quadratic giving the periods is {(MK'^+ma?)p^-(Mh+ma)g} {&y-6s}=ma26y, where MK'^ and m}? are the moments of inertia of the two bodies about and A respectively. Mso OH=h, OA = a, AO = h. What do these periods become when ART. 460.] lagranue's method. 373 (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 he 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. 459. Principal Co-ordinates. To explain what is meant by the principal co-ordinates 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 unity. If the co-ordinates 6, , &c. be changed into f , rj, &c. by the equations <}> = fi,^ + fi^rj + &c.[ (9), &c. = &c. J we observe that 9', ^', &c. are changed into ^', rj', &c. by the same transformation. Also the vis viva is essentially positive. Hence we infer that by a proper choice of new co-ordinates, we may express the vis viva and the force function in the forms 2{U-U,)==2b,^+2hv + &G.+bu^' + b^rf+ J- These new co-ordinates ^, rj, &c. are called principal co-ordinates of the dynamical system. A great variety of other names has been given to these co-ordinates; such as harmonic, simple and normal co-ordinates. It is usually understood (when not otherwise stated) that prin- cipal co-ordinates are so chosen that they vanish in the position of equilibrium. We then have 6i = 0, \ = 0, &c. = 0. 460. When a dynamical system is referred to principal co- ordinates which do not necessarily vanish in the position of equi- librium, Lagrange's equations take the form f _ bn^ = 6, , v" - Kv = b„ &c. = &c. so that the whole motion is given by ^ = a -I- ^ sin (pjt -)- ei), r] = b+F am {pjk + e^), &c., where E, F, &c., ei, 62, &c. are arbitrary constants to be deter- mined by the initial conditions, and p^ = — bu,pi = — bw, &c. and a, b, &c. are the values, of ^, t;, &c. in equilibrium. If we substitute the trigonometrical values of f, 77, &c. in the formulae of transformation given above, we obviously reproduce 374 SMALL OSCILLATIONS. [CHAP. IX. the equations (5) of Art. 455, where the general co-ordinates 6, , &c. are expressed as trigonometrical functions of t. We may therefore obtain one set of principal co-ordinates, viz. fi, rj-^, &c., which vanish in the position of equilibrium, by writing 4> = P + F,^, + N,'n, + ..\ (10), &c. = &c. J where the values of a, y8, &c., M-^, M^, &c., N^,, N^, &c. may be found by the methods explained in Art. 455. All other sets of principal co-ordinates may be found from these by taking ^ = re + ^|i, 'r} = b + Fri^, &c. When the initial conditions are such that throughout the motion all the principal co-ordinates are constant except one, the system is said to be performing a principal or harmonic oscilla- 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.. It is therefore important to determine the peculiarities of a principal oscillation by which it can be recognized apart from all mathematical symbols. 461. 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 ^ as the variable co-ordinate, we see that | — a vanishes twice while pit 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 -^ vanishes twice while p{t increases by 27r. The positions of rest may be called the extreme positions of the oscillation, 4. Let the system be referred to any co-ordinates 0, , &c., or (2) there must be an indeterminateness in the coefficients M, N, &c. given by Art. 455. Referring the system to principal co-ordinates, which vanish in the position of equilibrium, we see by Art. 460, that the first alternative is in general excluded. If two values of p" are equal, say bn and 622, the trigonometrical expressions for ^ and ri have equal periods, but terms which contain i 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 co-ordinates into the expressions for T or u, we may change ^, tj into any other co-ordinates ^i, rii, which make ^^ + 71" = ^^ + 7]^, the other co-ordinates ^, &c. remain- ing unchanged. For example we may put f = f 1 cos a — % sin a and ?; = ^1 sin a -I- % cos a, where a has any value we please. These new quantities fj, 171, ^, &c., are evidently principal co-ordinates, according to the definition of Art. 459. One important exception must however be noticed, viz., when one or more of the values of p are zero. If, for example, 5n = 0, we have- ^ = At + B, where A and B are two undetermined con- 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 -j^ = 0, -t— = 0, &c., where TJ has the value These in general require that f, 17, &c. should all vanish, but if 611 = they are satisfied whatever ^ may be, provided that ij, ^, &c. are zero. In any case however ^ must be very small, because the cubes of f, 7), &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. 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 6 is free to roll inside another thin cylinder of mass M and radius u,. The whole system is placed in equilibrium on a smooth horizontal plane. A small disturbance being given, show that the three values of ■f are i)''=0, ^==0 and :^-^^^ -^. Interpret this 2M a-h ^ result. If ic be the space rolled over, the angle turned through by the outer cylinder, and $ the inclination to the vertical of the plane containing the axes, show ART. 464.] INITIAL MOTIONS. 377 tbat aU three oo-ordinates 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 of rest. See Art. 199. As before we must choose for our co- ordinates some quantities whose higher powers can be rejected. It is generally convenient to choose them so that they vanish in the initial position. As in Art. 454 we have 2T=AnO" + 2A,,e'<}i' + A^tji'^ + &c., where A^, &c. are functions of 0, j>, &c. Since the system starts from rest, 0, , &c. their initial values. We require also the expansion of TJ given in the same article, viz., 2(U-Uo) = 2B,0 + 2B,<1> + &c. Since the initial position of the system is not close to a position of equilibrium, the first difi'erential coefficients of U with regard to 0, (f), &c. are not small. The terms Bi0, B^iJ3, &c. are not now small quantities of the second order and hence it is unnecessary to retain the quadratic terms of U. Proceeding exactly as in Art. 454 the equations of motion are Ane" + A,,cl>"+...=B,\ A,^0" + A^^"+...=B^ \ (1). &c. = &C.J From these equations we may determine the initial values of 0", (/>", &c. If X, y, z be the Cartesian co-ordinates of any point P of the system, we may, by the geometry of the question, express these as functions of 0, (p, &c.. Art. 396. Thus suppose that x =f(0, , &c.), then we have initially, since 0', 0' 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 Badlus 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 378 SMALL OSCILLATIONS. [CHAP. IX. must find the values of a", a;'", &c., and then substitute in any of the formulje given in Art. 200. If, as before, x=f{B, 0, &c.) we find by differentiation that initially ■ x"=feB"+f^i>" + ..., x"'=ftd"'+U"' + -, where suffixes as usual indicate partial differential coefficients with respect to e, (p, &c. If y=F(S, 0, &c.) there are of course similar expressions for y", &o., and in three dimensions for z", &b. 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" = for all initial variations of $, 4>, &o. It follows that/9=0,/^=0, &c. = 0. Hence a!"'=0, and the value of x"" depends only on 6", If,", &c., and not on &'", 0"", &e. It is therefore unnecessary to differen- tiate the dynamical equations (1) to find these higher differential coefficients. The axis of y being parallel to the initial direction of the motion of P, the value of y" is finite. Hence, taking the formula at the end of Art. 200, we find that the initial radius of curvature p of the path of P is given by (Fflr + .F^0"+-)° 465. In order to find the higher differential coefficients of S, 0, &c. when they are required, it may be necessary to form the equations of motion (1) to a higher degree of approximation. There can of course be no difficulty in retaining the first few powers of 6, 0, &c. which occur on either side of the equation. After differen- tiation we put zero for each of the quantities 0, 0, &c., 6', ipl, &o. But it is often more convenient to use Leibnitz' theorem. We have to substitute 2r=.4ufl'2 + 2^u9'0'+... in the Lagrangian equations, to differentiate the results, and to put d=0, 0'=O, &o. after the differentiations have been performed. Taking the first differential co- efficient with regard to t we find fl"'=0, 0"'=O, &c. Taking the second differential coefficient of the d-equation we find -4u«""+4i20""+... =B^6"+B^i," + ... where for the sake of brevity we have written cPA dA „„ cLA ^„ W-le^ +d0^ +••• which is evidently true initially. The other equations are formed in the same way. 466. Examples of Initial Motion. Ex. 1. A smooth plane of mass M is freely moveable about a horizontal axis lying within it and passing through its centre of gravity, the radius of gyration of the plane about the axis being k. The plane being inclined at an angle a to the horizon, a sphere of mass m is placed gently on it. If initially the centre of the sphere be in a vertical through the axis of the plane, and ART. 467.] THE ENERGY TEST OF STABILITY. 379 if h be its initial height above that axis, show that the angle which the initial direction of motion of the centre makes with the vertical is given by (Mk'> + mW') tan

Fo, the equation (1) shows that jT is < 2\ + Fi — Fj. Thus throughout the subsequent motion the vis viva lies between zero and a small positive quantity, and therefore the motion of the system can never be great. Also, since T is necessarily positive, the system can never deviate so far from the position of equilibrium as to make F greater than T^ + Fi. These two results may be stated thus : — If a system he in equilibrium i/ri a position in which the potential energy of the forces is a minimum,' or the work a maadmum,for all displacements, then the system if slightly displaced will never acquire any large arniownt of vis viva, a/nd will never deviate far from the position of equiUbriwm. The equilibrium, is then said to be stable. 380 SMALL OSCILLATIONS. [CHAP. IX. It will be shown that this reasoning may in certain oases be extended to de- termine whether a given state of motion as well as a given state of equilibrium is stable. See also the Treatise on the Stability of Motion, Chap, vi., 1877. 468. If the potential energy be an absolute maximum in the position of equilibrium, V is less than Vo for all neighbouring positions. By the same reasoning we see that T is always greater than Ti+Vi—V^, and the system cannot approach so near the position of equilibrium as to make V greater than Tj + Vj,. So far therefore as the equation of vis viva is concerned, there is nothing to prevent the system from departing widely from the position of equilibrium. To determine this point we must examine the other equations of motion*. If any principal oscillation can exist, let the system be placed at rest in an extreme position of that oscillation, then the system will describe the complete oscillation and will therefore pass through the position of equilibrium. But, if Ti be zero, V can never exceed Fj, and can therefore never become equal to Fo. Hence the system cannot pass through the position of equilibrium. It is unnecessary to pursue this line of reasoning further, for the argument will be made clearer in the next article. 469. We may also deduce the test of stability from the equa- tions which determine the small oscillations of a system about a position of equilibrium. Let the system be referred to its prin- cipal co-ordinates, and let these be 6, , i/r, &c. If &n 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 * This demonstration is twice given by Lagrange in his Mecaniqv£ Analytique. In the form in which it appears in the first part of that work, V is expanded in powers of the co-ordinates, which are supposed very small ; but in Section vi. of the second part this expansion is no longer used, and the proof appears almost exactly as it is given in this treatise up to the asterisk. The demonstration in the next article is simplified from that of Lagrange by the use of principal co- ordinates. ART. 469.] THE ENERGY TEST OF STABILITY. 381 vanish. If bn is negative, d is expressed by a trigonometrical term, and the equilibriuni is stable for all disturbances which affect & only. In this demonstration the values of bn, b^^, &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^^, b^i, &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 fT^ is a maximum for some displacements and a minimum for others, some of the coefficients b^, 622, &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 f7 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 : The system mil oscillate about the position of equilibriv/m for all disturbances if the potential energy is a minimum for all dis- placements. It will oscillate for some disturbances and not for others if the potential energy, though stationary, is neither a maxi- mum nor a minimum. It will not oscillate for any dist'wrbance if the potential energy is a maodmumfor 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. Ex. 1. A perfectly free particle is in equilibrium under the attraction of any number of fixed bodies. Show that, if the law of attraction be the inverse square, the equiUbrium is unstable. [Earnshaw'a Theorem.] Let be the position of equilibrium, Ox, Oy, Oz any three rectangular axes, d^V {A-a) + ^{a)-E{a-h) + [4>'{A-a)-y^'{a) + E]l Let ^,^nA-a)-^'{a) + E^ and ^^^_^HA-a)^-Ha)-E{a-h) Then a; = e + i sin (nf + L'), where L and L' are two arbitrary constants. We see therefore that in the position of equilibrium the angle made by the torsion- rod with the axis of x is e, and the time of oscillation about the position of equilibrium is — . Let us now suppose the masses to be moved into their alternate position A^Bs^; 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 a; = e' + JV sin {nt + N'), where n has the same value as before, and -^{A-a)+^{a)-E{a-h) ■■a + - Iv? In these expressions, the attraction ■^(a) of the casing, the coefiSeient of torsion E and the angle h are all unknown. But they all disappear together, if we take the difference between e and e'. We then find <^{A —a) _e-e' (^ir I 2 1$)' w, where T is the time of a complete oscillation of the torsion-rod about either of the disturbed positions of equilibrium. Thus the attraction <^{A — a) can be found if the angle e — e' between the ART. 473.] THE CAVENDISH EXPERIMENT. 385 two positions of equilibrium and also the time of oscillation about either can be observed. 472. It is sometimes wrongly objected to the Cavendish Ex- periment that the attractions of the balls A and B are supposed to be great enough to be measured, while the much greater attractions of surrounding objects, such as the house, &c., are neglected. But this is not the case. The attractions of all fixed bodies are included in that of the casing. These are therefore not neglected but eliminated from the result. It is to eifect this elimination that we have to observe both e' — e and the time of oscillation. We thus really form two equations, and from these we eliminate those attractions which we do not want to find. 473. The function ^(A — a) is the moment of the attractions of the masses and the plank on the balls and rod, when the rod has been placed in a position Of, bisecting the angle AiGB^ be- tween the alternate positions of the masses. Let M be the mass of either of the bodies A and B, m that of one of the small balls, m' that of the rod. Let the attraction of ilf on m be represented by /i-T^i where D is the distance between their centres. If (p, q) be the co-ordinates of the centre of A^ referred to Gf as the axis of x, the moment about of the attraction of both the masses on both the balls is 2f,Mm\ ^2 _ 91 where c is the distance of the centre of either small ball from the centre G of motion. Let this be represented by /iMmP- The. moment of the attractions of the masses on the rod may by integration be found to be /jlMiu'Q, where Q is a known function of the linear dimensions of the apparatus. The attraction of the plank may also be taken account of. Thus we find ^(A-a) = fiM (mP + m'Q). If r be the radius of either ball, we have which may be represented by / = 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 + m 'Q '^^^■mP' + m'Q' R. D. 25 386 SMALL OSCILLATIONS. [CHAP. IX. Let E be the mass of the earth, R its radius and g the force E* of gravity, then g = (i> -pj- . Substituting for /t, we find M e-e' /27rV 1 m! ^^ m B 2 KlfsB?-^ ■ m The ratio — , was taken equal to the ratio of the weights of m ^ 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 has been reduced to the determination of two elements, (1) the 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 G, 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 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 =-^ . =-jr^ . ^ = 73"'46. But the Xo 108 ^ 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 * In Baily's experiment, a more accurate value of g was used. If e be the elliptioity of the earth, m the ratio of centrifugal force at the equator to equatoreal gravity, and \ the latitude of the place, we have g=/i~h-2e+ ( - m - e ) eos^'xl . ART. 476.] THE CAVENDISH EXPERIMENT. 387 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 oseUlation 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. A plan of using a quartz fibre as the supporting string has been proposed by C. V. Boys, by which the whole apparatus can be made on so small a scale that the two difficulties of keeping the temperature uniform and of dealing with large balls as the attracting masses are very much reduced. See the Proceedings of the Royal Society, May, 1889. 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 '67 47. The most important experiments after Baily which were con- ducted on this plan were those of Cornu and Bailie. See Comptes Rendus, Tome Lxxvi., 1873 and Tome Lxxxvi., 1878. They made several improvements in the apparatus which we cannot here describe. They made the mean density to be 5'56. They con- sidered that they had found an error in Baily's method of taking his means. If this were corrected Baily's result would become 5'55. 476. Two other methods of finding the mean density have been employed. In 1772 Dr Maskelyne, then Astronomer Koyal, suggested that the mass of the earth might be 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 Phil. 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. The other method, used by Sir G. Airy, is to compare the force of gravity at the bottom of a mine with that at the surface, by observing the times of vibration of a pendulum. In this way 25—2 388 SMALL OSCILLATIONS. [CHAP. IX. the mean density of the earth was found to be 6'566. See Phil. Tram. 1856. Within the last ten years the density of the earth has been found by observing how a very delicate balance is disturbed by the near approach of large attracting masses. The experiments were conducted by Jolly in Munich and Poynting in Manchester. The result was 5'69. EXAMPLE S*. 1. A uniform rod of length 2c rests in stable ec[nilibrium 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 6 from the vertex. Show that, if the equilibrium be slightly disturbed, the rod wiU perform small oscillations vrith its lower end on the arc of the cycloid in the time 4^ / "-if+^^^- ^l , where 2a is the length of the axis of the cycloid. V 3g{V-iac) 2. A small smooth ring slides on a circular vrire 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 vrire, with a uniform angular velocity » / — 7= — ; 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 the time \ aj2 c j2 + a] i 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 2t 4. Two equpl heavy rods connected by a hinge which allows them to move in t. 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 n 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 » / — ^-pz . =r= — ; prove ^ 2aj2 ^ also that, if the weight be slightly depressed in a vertical direction and the fia /I M+3M' system left to itself, the time of a small oscillation is 2v \,/ — ^ . -^ — ^r^-^, " ' V 15^ M+2M' 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 * These examples are taken from the Examination Papers which have been set in the University and in the Colleges. ART. 476.] EXAMPLES. 389 situated that its least distance from the rod is equal to the natural length of the string. Prove that, if $ be the inclination of the string to the rod when in W equilibrium, cote-oose= — oosa, where w is the modulus of elasticity of the String. Al so if the ring be sHghtly displaced the time of a small oscillation will /wi 1 be 2jr ^ _ j^_^.^3g , 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 g 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 . /- . Find the position of relative equilibrium, and prove that, if the particle be slightly dis- turbed, the time of a small oscUlation is J! ^'^'"' /- . Jt + 4: V 9 7. A heavy uniform rod AB has its lower extremity A fixed to >.. vertical axis, and an elastic string connects B to another point G in the axis such that AB AC = —i^=a; the whole is made to revolve round AC with such angular velocity that the string is double its natural length and horizontal when the system 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 27r . / ==— , the weight of the rod being sufficient to stretch the string to twice its length. 8. Three equal elastic strings AB, BC, CA surround a circular arc, the ends being fixed at A. At B 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 /ml 9. A particle of mass M is placed near the centre of a smooth circular horizontal table of radius u., strings are attached to the particle and pass over n smooth pullies which are placed at equal intervals round the circumference of the circle; to the other end of each of these strings a particle of mass M is attached; show that the time of a small oseillation of the system is 2ir I | . 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 on 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 . / — -— - , where M is the 'V i" 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. distances from A after intervals ' 390 SMALL OSCILLATIONS. [CHAP. IX. 11. A uniform beam of mass M and length 2a can turn round a fixed horizontal axis at 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{3l-a) = 6m (l+a). 12. A conical surface of semivertical angle a is fixed with its axis inclined at an angle S to the vertical, and a smooth right cone of semivertical angle /lis placed within it so that the vertices coincide. Show that time of a small oscillation = 2ir . / . „"' , where a is the distance of the centre of oscillation of the cone V gsmB 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. Prove that, if ^, JS, C be the moments of inertia of the system about three axes mutually at right angles through its centre of gravity, the positions of stable eijuilibrium will be found by making A + B + G & minimum. 14. A particle is in motion within a triangle ABC, and is attracted perpendicu- larly to the sides with forces each equal to />, times the perpendicular distance. Show that the motion is expressed by two periodic terms of the form Psin{iv'(M + <»}. where (X - 1) (X - 2) + 2 cos A cos B cos C=0. Shew that the roots of this quadratic are real and positive. Examine the case of an equilateral triangle, and in that case verify the above result 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 - of the weight of either. They are suspended by two strings of length I from two n 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 = 1 + ^. CHAPTER 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 =i {aa? + Ibxy + cy"") + &c. / = i (aV + Wxy + cV) + &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') a^ + 2 (6 - 6') xy + ic- c') 2/^ = 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-knowu 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 2/3V = r''. Also z — z'=X hence, eliminating z and z', we see that the difference of the curvatures varies inversely as rK 392 ON SOME SPECIAL PROBLEMS. [CHAP. X. 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 curvature of the normal sections through OR, taken positively . ,. . ,,111 when the curvatures are in o'p'pomte directions, and let - = - + —, . s p p Then s may be called the radius of relative curvature. We have the three following propositions which are of use in D3Tiamics. 479. Peop. The Instantaneous Axis. Let 01 and Oy 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 be the indefinitely small angle turned round the in- stantaneous axis, the arc o- of rolling is given by o- = ^s sin yOI. To prove this, measure in the plane yz along the surfaces two lengths OP and OP' each equal to , — sm r sin z. hL ^ ' sm (p + p) The dynamical principle used in obtaining this result is that of taking moments about the instantaneous axis, Art. 448. If O' be the position of the centre of gravity at the time t, and d the angle between the planes GOI, G'OI, we have K^e = M (1), where M is the moment of g acting at 0' about the instantaneous axis at the time t. If OP be a neighbouring generator of the fixed cone and the angle POI be a; the moment M' about OP of g acting at G' is a fiinction of 6 and a. We therefore have to the first order of small quantities M' = Aa + Be (2), where A and B are two expressions which depend on the form of the cone. Finally, if OP be the instantaneous axis at the time t, we have M' = M and o- sin (p + p') = Osinp sinp' (3). Eliminating either cr or ^ from these equations the time of oscillation can be deduced. The relations (2) and (3) are established in an elementary manner in Arts. 484 and 485. The steps in the investigation correspond to those used in the oscillation of cylinders (Art. 441), the chief difference being that the straight lines used in the figure for cylinders are here replaced by spherical arcs. The proof of the relation (3) presents no difficulty, but in the general case when both the rolling and the fixed cone are of any forms the figure required to obtain the relation (2) is rather complicated. In particular cases, such as when the fixed surface is plane or the rolling cone is one of revolution, there is considerable simplifica- tion, the extent of which is pointed out in some of the examples in Art. 486. In these the proof, as adapted to the special case under consideration, is again briefly sketched. By considering the parts of M' due to 6 and a separately, we may arrive at their values without requiring any figure more 396 ON SOME SPECIAL PROBLEMS. complicated than that already drawn in this Article, is as follows. . [chap. X. The proof Suppose (1) that (r=0, then M' ia the moment round 01 of g acting at G' parallel to the vertical WO. Since the body is turned round 01 through an angle $, the are GG'=hd ainGI. Eesolving g parallel and perpendicular to 01, the latter component is g sin WI and its moment round 01 is g sin WI . GG'. Sub- stituting for the spherical arcs WI and GI their values z and r, the moment becomes - ghB sin r , sin z. Suppose (2) that 6=0, then if' is the moment round the neighbouring generator OP of g acting at G parallel to WO. Eesolving g along and perpendicular to GO, the latter component is g sin WG, and acts at G along a tangent to the spherical arc GI. To find its moment round OP we resolve it perpendicular to the plane OGP and multiply the component by h sin GP. The required moment is therefore the product of g sin WG, sin IGP and h sin GP. Since o- cos n and IGP . sin GP both express the perpendicular distance of P from the arc GI, the required moment becomes ghir sin (2 - r) cos n, where z-r has been written for WG. The complete value of M is therefore M=gh{(r cosm Bin(2-r)-9 aiar ainz}. 484. As the heavy cone rolls on the surface, the point on the sphere which is at I in equilibrium takes the position I', and P is the new point of contact. Let the arc IG assume the position I'G', and let the centre G of the osculating cone move to C. Let a-=IP be the arc rolled over, and let ff be the angle turned round by the cone. Since this angle is ultimately the same as GPC, we have CO' = 6 sin p. Also CC" cosec (pH-/)') and o-coseep' are each equal to the angle IDP. We thus „ , „ sin p sin p' find 17 = 0-^ c.. sin (p + p') 485. The vertical OW cuts the sphere in W. To find the moment of the weight about OP we must resolve gravity parallel and perpendicular to OP. The former component has no moment, and the latter is g sin WP. Let this latter act parallel ART. 486.] OSCILLATIONS OF CONES. 397 to some straight line KO. The moment required is the product of resolved gravity into the projection of OG' on a straight line OH, which is perpendicular to both OK and OP. Thus the spherical triangle HKP has all its sides right angles. In equilibrium 6 lies in the vertical plane WOI, and as the cone rolls G moves to G', so that the arc OG' is perpendicular to 71'/, and equal to $ sin r. Let this are cut TI'P in M. The projection required is h cob HG'= -h . MG' since HM is a right angle. Since FI makes with PH an angle which is ultimately equal to n, we have GM sinPTG sin(z-r) ,..,,„, ^ . , . ,, = ■ Trri- = '^^ ultimately. The moment required, urging the cone o-oosn smWI sin« _ i i o b back to its position of equilibrium, is ghsinz (QM - GG'), which on substitution becomes M=gh {(T cos n sin (z-r) -Bsinramz}. Equating this moment with the sign changed to K'ff, 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 pre- ceding expression then takes the form K^_ sin{z + p')sm^p hL~ Bin (p + p') Ex. 2. A right cone of angle 2p and altitude a, suspended by its vertex from a fixed point in a rough vertical wall, makes small oscillations, prove that the length of the equivalent pendulum is , 5 cos/) Let the cone when in equilibrium touch the plane along the vertical Oz. At the time t, let the generator ON be the line of contact, where zON = a. Let OA be the axis. Eesolving gravity along and perpendicular to the line ON, and taking moments about the instantaneous axis ON, we have K''S= - p sin (7 . ja sin p. Now, if the cone turn round ON through an angle 6dt, the centre A of the base advances a space a sinp . 6dt, hence, if AH be a perpendicular on ON, H advances an equal space. But it does advance a space OH . da i.e. a cos pda. We therefore have etan/)=ff. Substituting this value of 6 in the above equation and quoting the value of K^ from Art. 18, Ex. 7, the length of the equivalent pendulum is found without difficulty. Ex. 3. A right cone of angle 2p and altitude a oscillates on a perfectly rough plane inclined to the vertical at an angle z, the length of the equivalent pendulum is a (1 + 5 cos" p) 5 cos p cos z Besolve gravity into g cos z acting down the plane and a perpendicular compo- nent which can be neglected. Then proceed as in the last question. Ex. 4. A right cone of angle 2p 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 398 ON SOME SPECIAL PROBLEMS. [CHAP. X. fixed tight cone of angle 2p', the vertices being coincident, prove that the length L of the equivalent pendulum is given by fn d -,^1. a .1 iai&n^p „ . , , ., sin(p'+«) {97r^ + 16tan2p}4 — =r=— '^ = 37rsin2 tanp'-4tanp — , , where » 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 leads to the following construction, which is represented in the figure of Art. 483. Measure along the common normal GI to the cones a length 18 = s, such that cot s = cot p + cot p. From S draw an arc SR perpendicular to IGW, then cos n = cot s . tan IR. 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 J" > cot ^ + cot IR. When the vertex is very distant the cones become cylinders. In this case, if the arc z become a quadrant, the condition of stability is reduced to »" < IR. This agrees with the condition given in Art. 442. Large Tautochronou^ 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 ease. Various rules to determine this question are given in the following Articles. 489. Show that, if the equation of motion be d^x . dx TTj = a homogeneous fimction of the first degree of -3- and x, then, in whatever position the system is placed at rest, the time of arriving at the position determined by x = is the same. Let the homogeneous function be written "f [- -^) ■ Let a; \3S 0/0/ and ^ be the co-ordinates of two systems starting from rest in two different positions, and let x = a, ^ = «a initially. It is easy to see that the differential equation of one system is changed into ART. 491. J LARGE TAUTOCHRONOUS MOTIONS. 399 that of the other by writing f = kx. If therefore the motion of one system is given by a; = ^ {t, A, B), that of the other is given by ^ = K=a and ^ = 0. Since only one motion can follow from a single set of initial conditions, we have A' = A, and B' = B. Hence throughout the motion ^=kx, and therefore x and f vanish together. It follows that the motions of the two systems are perfectly similar. This result may be obtained also by integrating the differential equation. If we put - tt =i', we find x = A<}>(t + B). When t = 0, dx * T- = 0, and therefore (t + B) = 0, whatever be the value of A. 490. 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 a; = only after an infinite time, or the time of arrival may be imaginary. Thus, suppose the homo- geneous function to be w?x, where m^ is positive, then the system starting from rest moves continually away fi'om the position x = 0. The value of x is evidently represented by an exponential func- tion 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 stands for ^ (y) and accents as usual denote differential coefficients. Let ^=/(y), substituting we have df f\dt) f\dt)'^-'\fdt)' where / has been written for f{y). The last two terms of this •expression form a homogeneous function of / and -^ of the first degree, and therefore Lagrange's formula has been proved. This demonstration is due to Bertrand. The motion begins from rest with any initial value of x and ends when a; = 0. Hence, writing x = ^ (y), we see that in the second equation the motion begins with -^ = and with any dx initial value of y, and terminates when (y) = 0. Now -t- does not in general vanish when x = 0, since the system -arrives with dec OAJ some velocity at the position of equilibrium. But -jr = ' (y) -^ > hence 4>' iy) does not vanish when x = 0. It follows therefore, since ^ = ^' . fly), that the motion terminates when /(y) = 0. When the particle is constrained to move on a rough curve under the action of a tangential force, Lagrange's rule may be put under another form. If F be the tangential force tending to ART. 493.] LARGE TAUTOCHRONOTJS MOTIONS. 401 diminish the arc s, the equation of motion takes the form (Art. 495) dt' ~ [dtj p It is proved in Arts. 495 and 496 that the motion will be tauto- chronous if where m is some constant and -^jr is the angle the tangent makes with a straight line fixed in space. The tautochronous motion terminates at the point determined by F=0 and the constant interval is -^r— . 2m 492. Effect of a resisting medium. If the motion of any system is tautochronous according to Lagrange's formula in vacuo, it will also be tautochronous in a resisting medium, if the effect of the resistance is to add on to the differential equation of motion a term proportional to the velocity. This theorem is due to Lagrange. The proof of this is easy, for to introduce the resistance of such a medium into the equation of motion is merely to increase the homogeneous function F hy a. term of the form k-tt- This is permissible, since the only restriction on the form of F is that it must be a homogeneous function of the first degree. 493. Motion on a rough cycloid. A heavy particle slides from rest on a rough cycloid placed with its axis vertical, show that tlie motion is tautochronous. Let be the lowest point of the cycloid, P the particle, OP = s, so that the arc is measured from in the direction opposite to that of the motion. Let the normal at P make an angle •^ with the vertical, let p be the radius of curvature, and a the diameter of the generating circle. Then, by known properties of the cycloid, s = 2asini/r, p = 1a coa^y^. Let p, 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 inwards when posi- tive, and V the velocity, we have - = it — (jrcosi/r '' I (1). dh ^ . ,\ -^^=p,B-gmxt B. D. 2G 402 ON SOME SPECIAL PROBLEMS. [CHAP. X. Eliminating R the equation of motion becomes S=p(IJ-^('^°^-^'°'^^ ^^^- Substituting for p and s their values in terms of yfr, this becomes - cos i|r -^ + (sin f + iM cos -yjr) ^-^ j = ^ (sin a^ - /a cos ilr). Writing /i = tan e this is identical with t {e-* sin (t - e)} + ^^^ {e-* sin (^ - e)] = 0. Since -^ is initially zero, the solution of this equation is , cos ej ' where -d is a constant depending on the initial value of i/r. The motion is therefore tautochronous. At whatever point of the cycloid the particle is placed at rest, it arrives at a point A deter- mined by €"'*■'' sin (-«|r — e) = in the same time, and this time is -'"'' sin (■Jr-e) = A cos (, i- - IT 2a '-iS)'^^Hi)m^^^^'^' g cos 6 . / — . The point A at which the tautochronous motion terminates is clearly an extreme position of equilibrium in which the limiting friction just balances gravity. We have here given an independent proof of this result, but it might have, been deduced at once from Lagrange's rule. We may write his theorem in the form since the last two terms on the right-hand side constitute a homogeneous function of f(s) and ds/dt of the first order. Com- paring this form with the equation (2) we have f'(s) + A fj. .. . . , •' J, s = - . fis) = sm ■yir — u, cos i/r. f{s) p ■'^^ r A- r Substituting in the first of these equalities the value of f{s) given by the second, we find that the former is identically satisfied by choosing 1 + 2aA = — p.^ It follows that the equation (2) is merely an example of Lagrange's rule. The motion is therefore tautochronous for arcs terminating at the point determined by tan y}r = fjL. The same result follows immediately from the general theorem proved in Art. 495. We have F = g (sin lir — yii cos lir). The dF equation of condition m'p = —- — ixF is therefore satisfied identi- AKT. 495.] LARGE TAUTOCHKONOOS MOTIONS. 403 cally if m = sec e . V'.7/2a. The tautochronous arcs terminate at the point given hy F=0 and the period is 7r/2m. 494. That cycloidal oscillations in a medium in which the resistance varies as the velocity are tautochronous has been proved by Newton in the second book of the Principia, Prop. xxvi. That the oscillations are tautochronous when the cycloid is rough has been deduced by M. Bertrand from Lagrange's formula, given in Art. 491, see Limmlle's Journal, Vol. xiii. M. Bertrand ascribes the proposition to M. Necker, who published it in the fourth volume of the Memoires prdsent6s a I'Academie des Sciences par des savants Strangers. It follows of course from Lagrange's pro- position (Art. 492) that the cycloid is tautochronous when the medium resists as the velocity, and at the same time the cycloid is rough. 495. Motion on any rough curve. A particle starts from rest and is con- strained to move along a rough curve wider the action of any forces, find the conditioTis of tautochronous motion. Let A be the point at which the tautochronous motion terminates, P the position of the particle at any time t, AP = s, so that s is measured from A in the direction opposite to that of motion. Let the tangent at P make an angle xp with the axis of X, and let ^ and s increase together. Let the tangential and normal components of the force on P be G and H ; the tangential component G acting on P to urge the particle towards A, and the normal component H acting outwards, i.e. opposite to the direction in which p is measured. Let the letters R, v, ft, have the same meaning as before. We shall suppose p to be positive throughout the arc. The equations of motion are therefore -=B-H, v^ = !iR-G (1). p ds * ' Since the particle starts from rest, we see that JJ 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 repre- sented by liR, without any discontinuous changes in the sign of fj,, such as would happen if R were to change sign without a corresponding change in the direction of the friction. (See Art. 159.) Eliminating R we find ''S='^7-(^-''^^) ■(^)- Let F=G-/iH, so that F is the whole impressed force urging the particle along the tangent towards the point A. We may prove that F must be positive throughout the motion until the particle reaches A. If F were zero at any point B, then, placing the particle at rest at B, it will remain there in equilibrium, and there- fore 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 F is zero. (See ds Art. 490.) Writing jj- for p, equation (2) becomes 404 ON SOME SPECIAL PROBLEMS. [CHAP. X. therefore vM-"-i^*=c^-2 r pFe-^i^^d^p (4), where o is the angle made by the tangent at A with the axis of x. As ij/ is greater than o throughout the motion, the constant of integration, viz. c^, must be positive. If we put 2 I''' pFe-^'"''d^l/=x\ »-i"l'ds = ,p{z)dz, this equation reduces to the form ■/; 0(2^^ J ^gj_ The lower limit is determined by the value of z at the point where the motion begins. Referring to equation (4) we see that, since v=:0, we have z = c. The upper limit is determined by the value of z at the point where all the tautochronous motions are to end. This has been defined by 1//= a, and therefore by s=0. If the force F be such that (p {z) is constant, the integration of (5) presents no 1 TT difficulty. Writing — for (z) we then have f = „- ■ Since this result is indepen- dent of c, the motion is tautochronous. Wherever the particle be placed at rest on the curve, it will reach the position A in the same time, and this time will be -^ . The supposition we have made regarding the value of z gives I m ['''e-i"l'ds |'= 2 j* pFe-^'I'di^, differentiating and reducing we find F=:m^e'"''['''e~'"l'pd

=d^-''^ (7). 496. We shall now show that unless 0(z) be constant the motion will not he tautochronous. To prove this we must find what forms of 0(2) will make the integral (5) independent of c. Put 2 = c|, and let {z) be expanded in a series of some powers of z not necessarily integral, say let {2) = 2^2". Then we have ZNd^ Now since J is less than unity, the integrals in all the terms are less than the integral in the term defined by n=0, and therefore they are finite. Since every element in each integral is positive, none of the integrals can vanish. Hence this series of powers of c cannot be independent of c unless it contain only the one term determined by n=0. But this makes (2) a ooustant and leads to the solution we have already discussed. 497. Ex. Show that this law of force coincides with that given by Lagrange's formula. ART. 498.] LARGE TAUTOGHRONOUS MOTIONS. 405 We have here to determine when eauation (2) ooiuoides with Lagrange's formula. We therefore take as his form Comparing this with (2), term for term, we find ^ - ^ =^F 1^ , which leads to the OS as reqmred form for F. 498. Motion on a rough epicycloid. A particle acted on bij a repulsive force, varying as the distance and tending from a fixed point, is constrained to move along a rough cinve, find the curve that the motion may be tautochronous. Let r be the radius vector of the particle, \r the repulsive force acting along it. Let p he the perpendicular on the tangent from the origin, then the projection of the radius vector on the tangent is known to be ■— . Let A be the point on the curve at which the tantochrouous motion is to terminate. Besolving the radial force along the tangent towards A and along the normal outwards, the com- ponents are respectively G=-\-^ and H=\p, Art. 495. Since f =Gf-/iff, we have *'= ~ ^ jX - Mi"- Substituting in equation (7) of Art. 495 we find Tti^ d^p therefore p = ^„ « (1). The equation to an epicycloid generated by the rolling of a circle whose radius is 6 on a fixed circle whose radius is a is known to be 2 „r^-a:^ c^-a^ P'=''^rz^' -■•• P = ^P (2), where r is the radius vector measured from the centre of the fixed circle as origin and c=a+2b. We find therefore that the epicycloid is a tautochronous curve for a central repulsive force varying as the distance. The time of arriving at the -1- J. .,.. . . T , ... to" uV + a^ position of equihbnum is ^r— , where m is given by — = ^- ' , . Zm \ c^ — a^ If 4> be the angle made by the tangent with the radius vector, the position A of equihbrium is given by cotd) = |ii. Since »=rsino!i, we have by (2) r'' = — gA_JJlz_ a' + 11,'c' Let this value of r be represented by r^. Since a is less than c, it is easy to see that a circle described with centre and radius )•„ intersects the epicycloid. If A be any one of these points of intersection and G the nearest cusp, a particle start- ing from rest at any point D between A and G will describe the arc DA iu a time which is independent of the position of D. Ex. 1. Show that the equiangular spiral is not included in the formula (1). For if we write p sin^ a=^, we have (/i^H- 1) sin'' a greater than unity, which may be shown to be impossible if the particle is to move at all. Since the angle made by the tangent with the radius vector is the same at all points of the curve, we also notice that there is no point of equilibrium at which a tautochronous motion could properly terminate. 406 ON SOME SPECIAL PROBLEMS. [CHAP. X. Ex. 2. If the central force be attractive and vary as the distance, show that a hypocycloid is a tautochronous curve. 499. Effect of a Besisting Mediuni on the time. We know by Lagrange's theorem that, if the motion on a rough curve be tautochronous in a vacuum, it is also tautochronous when the motion occurs in a medium resisting as the velocity. Let us now determine how the time of arrival at the position of equilibrium is affected by the presence of such a resisting medium. Eeferring to Art. 495, the equation of motion there marked (2) now takes the form dt^~p\dt) dt ' where 2/c is the coefficient of the resistance due to the medium. This equation may be put into the form i{^-^'%)M'''''ih-^''-''- dt ' Let us put e~'^'^ds = dw and suppose that w vanishes when ^=a. Substituting for F its value given by equation (6) of Art. 495 we find d'w „ dw . . This solution of this by Art. 434 is y e-i^'^pd^ji = w=Ae-''*' aos,(^m^-K^t + B). To find the constants of integration we notice that -^, and therefore -j-, is zero when t=0. This gives tan R=: To find the time of arriving at the ijm? — K^ position of equilibrium we put ^ = a, this gives ijm^ -K^t + B=^. The time t of the tautochronous motion is therefore given by the equation Jm''-KH = ^ + taTr^ I „ „ ■ ^ 2 i^rn? - iC We notice that the time depends only on m and k and not on the form of the curve. Ex. If the resistance of the medium be so great that k is equal to or greater than m, the solution by Art. 434 takes another form. Show that in both these cases the time of arriving at the position of rest is made infinite by the resistance of the medium. Oscillations of Cylinders and Cones to the second order. 500. Condition of Stability of Cylinders to tbe bigher orders. When a heavy cylinder rests in equilibrium on one side of a fixed rough cylinder as in Art. 441, the condition of stability is that the centre of gravity should lie within a certain circle called the circle of stability. If the centre of gravity lie on the boundary of this circle the equilibrium is called neutral, but it is generally either stable or unstable, a higher degree of approximation however, being re- quired to distinguish the two. We may reach any degree of approximation by the AUT. 501.] OSCILLATIONS TO THE SECOND OEDER. 407 following easy process, which amounts to the oontmued differentiation of a certain quantity until we arrive at a result which is not zero. The sign of this result distinguishes between the stability and instability of the cylinder. The magnitude of the result, joined to some other elements, enables us to form the equation of motion. 501. In equilibrium the centre of gravity is in the vertical through the point of contact. Let the body be turned round through any angle d, so that G in the figure is the position of the centre of gravity, and I the point of contact. Let IV be vertical. Let CID be the common normal to the two cylinders, G and D being the centres of curvature of their transverse sections. Let p=GI, p'=DI, and let z = _ + - , so that 2 is the radius of relative curvature. z p p Let IG=r, the angles GIG=n, GIV=, and let IP=ds. Then we have the four following subsidiary equations (Ir . dn cosm 1 -;- = sin «, ^- = ds ds r p d(p 1 cos n ds _ ds~ z r ' d6 ' Since GI is the radius vector of the upper curve referred to an origin G fixed relatively to it, and --m is the angle made by this radius vector with the tangent at I, the first of these subsidiary equations is evident. To obtain the second we notice that G is the centre of curvature, so that the distance GC is constant as well as the radius of curvature, when / moves a short distance ds along the are. Now GG^=r'' + p^-2pr cos n, therefore 0=(r- pcoBn)dr+preinndn. Substituting for dr its value from the first subsidary equation, this immediately gives the second. To obtain the third equation we notice that ^ + m is the angle made by the normal DI to the lower curve, which is fixed in space, with a straight line also fixed in space. Hence -t- + ^ = ~< wlie°oe the third equation follows from the second. The fourth equation has been proved in Art. 441 ; the proof may be summed up as follows. If GP, DP' be the two normals which are in a straight line when the body has turned through an angle dB, then d$ = PCI+P'DI, , /l 1\ ds which gives "*l~ + ^l~T' 408 ON SOME SPECIAL PROBLEMS. [CHAP. X. 502. In equilibrium the centre of gravity of the body must be vertically over the point of support. Hence ^=0. In any other position of the body the value of ^ is given by the series de'^ds' 1.2 + *"- If in this series the first coefficient which does not vanish be positive and of an odd order, it is clear that the line IG moves to the same side of the vertical as that to which the body is moved. The equilibrium is therefore unstable for dis- placements on either side of the position of equilibrium. If the coefficient be negative the equilibrium is stable. On the other hand if the term be of an even order, it does not change sign with s, the equilibrium is therefore stable for a displacement on one side and unstable for one on the other side. The first differential coefficient is given by the third subsidiary equation. The second differential coefficient is found by differentiating this subsidiary equation and substituting for ^- and ^- from the others. The third differential coefficient as as may be found by repeating the process. In this way we may find any differential coefficient which may be required. 503. If the first differential coefficient -^ be not zero, the equilibrium is stable as or unstable according as its sign is negative or positive. This leads to the condition that r must be respectively less or greater . than z cos n, which agrees with the rule given in Art. 441. If ^ = 0, we have r=z cos n, so that the centre of gravity lies on the circumference of the circle of stability. Differentiating we have p7 z ds \pj z^ Ip "*" ?y ■ The equilibrium is stable or unstable according as this expression is negative or positive. If the tranverse section be a circle or a straight line these expressions admit of great simplification. 504. Ex. 1. A heavy body rests in neutral equilibrium on a rough plane inclined to the horizon at an angle n. Show that, unless ^=tan n, the equilibrium is stable for displacements on the one side and unstable for displacements on the other. But, if this equality hold, the equilibrium is stable or unstable according as — | is positive or negative. Here ds is measured along the arc in the direction down the plane. ART. 506.] OSCILLATIONS TO THE SECOND ORDEE. 409 Show also that these conditions imply that the equilibrium is stable or unstable according as the centre of the conic of closest contact to the upper body is without or within the circle of stability. Ex. 2. If a convex spherical surface rest on the summit of a fixed convex spherical surface in neutral equilibrium, the equilibrium is really unstable. But if the lower surface have its concavity upwards the equilibrium is stable or unstable according as its radius is greater or less than twice that of the upper surface, and is really neutral if its radius is equal to twice that of the upper surface. The moveable spherical surface in this example must of course be weighted so that its centre of gravity is at such an altitude above the point of support that the equilibrium is neutral to a first approximation. Thus, when the radius of the lower surface is twice its radius, its centre of gravity lies on its surface, i.e. at a distance twice its radius from the point of contact. The centre of gravity is outside or within the surface according as the radius of the lower surface is less or greater than twice its radius, and when the lower surface is plane the centre of gravity lies at the centre. In this last case also the equilibrium is really neutral. 505. Oscillations of Cylinders to the higher orders. To form to any degree of approximation the general equation of motion of a cylinder oscillating about a position of equilibrium. Following the same notation as before and taking the figure of Art. 501, the equation of vis viva is (k'' + r^)i^=C+2U, where U is the force function and k the radius of gyration of the body about its centre of gravity. Differentiating this with regard to $, as in Art. 448, we have The right-hand side of this equation is by Art. 340 the moment of the forces about the instantaneous axis, and is therefore in our case equal to gr sin 0. Substituting for — from the subsidiary equations of Art. 501, the equation of motion is therefore {Tc^ + r^)d + rz Binni^=grsm. The metSd*9 bTprooeedincLiis the same as that in Art. 502. We expand each coefficient by Ta^BlSgttheorem'in powers of $, which is to be so chosen as to vanish in the position of equikfei'um. To do this we require the successive differentials of these coefficients to any order expressed in terms of the initial values only of , 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 ft. The equation is therefore {h^ + k^)S=grBin((,. We have to find r sin to the first power of $. Now d . . dr . deb , . . , ^ / 1 cos n\ — (r sm0) = ;^sm0+r ^^cos0 = « smnamip + rz cos [ 1 , d0 du ao \z r j by substituting from the subsidiary equations. This by reduction becomes — (i-sin 0) = )'CO3 0-«cos (0-)j). R. D. 27 410 ON SOME SPECIAL PROBLEMS. [CHAP. X. In equilibrium G lies in the vertical through the point of contact, hence the initial value of

«■ where L is the moment of gravity about "OZ. ■* As the cone rolls, the point J moves along the intersection of the fixed cone with the sphere. Let IP=ds be the arc described in a time dt. It will be con- venient to take s as the co-ordinate by which the position of the cone is determined. By the same reasoning as in Art. 484 we fi;Dd dt sin p. sin p' '■' ^ '■ ART. 509i] OSCILLATIONS TO' THE SECOND ORDER. 411 We have now to find the moment of gravity about 07. We again use the same argument as in Art. 485. Resolving gravity along and perpendicular to 01, the S*.::: former component has no moment, and the latter is g sin z. Let this latter com- ponent act parallel to some straight line KO, then KWI is an arc in a vertical plane. The moment required is then the product of resolved gravity into the projection of OQ on OS, where H is the pole of the arc KWI. Thus the moment is gJi sin z cos SG. To find cos HG produce HG to cut KWI in M. Then, in the right-angled triangle GIM, we have sin GM= sin 01 sin GIM. The moment L is therefore ' L = .-.ghsxarsmzsm. (re- ^) (3). When th? forms of the cones are kupwiij.we can express K, r, z, n and tj/ in terms of s or any other co-ordinate we may choose. The equaliion of motion will then be known. This process may be effected by the help of the four following subsidiary equations dr . dz . .\ -7-= Sinn, as dn ds = sin^ .(4). = cot raosn-cotp ^=cot«cos jp + cot p' The proof of these is left to the reader. They may be obtained by the same reason- ing as in the case of the cylinder, with only such modifications as are made necessary by using spherical instead of -plane triangles. 509. To find to any degree of approximation the equation of motim of a right cone oscillating about a position of equilibrium. Since the cone is a right cone, we have K^ constant. The equation of motion is therefore K'^-^=L, where fi and L have the values given in equations (2) and (3) at of Art. 508. We notice that L = 0, and therefore n = ^ in the position of equilibrium. Let the co-ordinate s be so chosen that it also vanishes in this position. We have therefore now to expand fi and L in powers of s. To effect this we use Taylor's theorem, thus ^ (dL\ ^fd^L\ «2 ^ 412 ON SOME SPECIAL PEOBLEMS. [CHAP. X. where the bracket implies that s is to be made equal to zero after the difierentia- tions have been performed. Now these differentiations may all be performed without any difficulty, by using the expression for L given in (3) and continually substituting for -=- , 5- , &c. their values given in the subsidiary equations (4). We as as may treat fi in the same way. The formation of the equation of motion is thus reduced to the differentiation of a known expression and the substitution of known functions. We may use this method to obtain the equation of motion to the first power. Thus we have K^--j-= ~gh T^ {sin r sin 2 sin (ra- ^)}s. Substituting for Q and retaining on the right-hand side only those terms which do not vanish when \j/=mne obtain K^ cPs ( . , , sin fl sin p' . . ) gh dt^ I sm(/)+p') ) which gives the same result as in Art. 483. If the eone is not a right cone, we may express K^ in terms of r and n and proceed in the same way. 510. Ex. A heavy right cone rests in neutral equilibrium on another right cone which is fixed in space, the vertices being coincident. Show that the equation of motion, including the squares of smaU quantities, is E^ d^s sin p sin p' . , , . , , , „ ^ , ^ , »^ -r 33= - -= — 7 7T sm(r-2)siun{(cotz + 2ootr)cosTO-cotp} — . END OF THE FIRST VOLUME. CAMBRIDGE: PRINTED BY 0. J. CLAY, M.A. AND SONS, AT THE UNFTERSITY PRESS. ^'mVV'A!