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IV^aps, piataa, charta, ate, may ba filmad at diffarant raduction ratios. Thoaa too larga to ba antiraly includad in ona axpoaura ara filmad baginning in tha uppar laft hand cornar, laft to right and top to bottom, as many framaa aa raquirad. Tha following diagrams illustrata tha mathod: Las cartas, planchas, tablaaux, ate, pauvant Atra filmte A das taux de reduction diff6rants. Lorsqua la documcint ast trop grand pour Atre raproduit an un aaul ctichA, il ast filmA A partir da I'angla supAriaur gaucha, da gaucha A droita, at da haut an baa, an pranant la nombra d'imagaa nAcaaaaira. Las diagrammas suivants illustrant la mAthoda. 1 2 3 1 2 3 4 5 6 AN ELEMENTARY TREATISE ON TBB DYNAMICS OF A SYSTEM OF RIGID BODIES. E AN ELEMENTARY TREATISE ON THE DYNAMICS OF A SYSTEM OF RIGID BODIES. /" BY EDWARD JOHN ROUTH, M.A., F.R.S., F.R.A.S., F.G.S., &c. LATE FEIiliDW AND LATE ABSIBTANX TDIOB OF 8T FETEB'S COLLEOE, CAUBBIBOE^ LATE EXAMINEB IN THE UNIVEBSITY OF LONDON. TUmH EDITION, REVISED AND ENLAB Uonljon : MACMILLAN AND CO 1877. [All Righta reserved.] PBINTBD BY a J, ci^y, a.^. AT THB 0MVBBSITY PEE88. QA HI PREFACE. In this edition I have made many additions to every part of the subject. I have been led to do this, because there are so many important applications which it did not seem proper to pass over without some notice. I have found how difficult it is not to render a book formidable to the student by its size and yet to supply some information at least on all the chief points of a great subject. I believe the reader will not find any portion treated at greater length than is necessary to render the argument intelligible. 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 will be found convenient for those who are already acquainted with the subject, as it will enable them to direct their attention to those parts in which they may feel most in- terested. It will also enable the student to select his own order of reading the subject. 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 at D'Alembert's Principle and R. D. b VI PREFACE. only read those parts of Chapter I. to which reference is made. Other readers may also wish to pass on as soon as possible to the great principles of Angular Momentum and Vis Viva. Though a different order will be found advisable for different persons, I have ventured to indicate a list of Articles to which those who are just beginning the subject 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 difHculties of Dynamics from those of Solid Geometry. I have attempted to give a slight historical notice whenever I felt it could be briefly done. This course, if not carried too far, will I believe be found to add greatly to the interest of the subject. But the success of this attempt is far from complete. In the earlier portions of the subject I had the* guidance of Montuela, and further on there was Prof. Cay ley '3 Report to the British Association. With the help of these the task became comparatively easy; but in some other portions the number of Memoirs which have been written is so vast, that anything but the slightest notice has been rendered 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. I have therefore found it necessary to correct some of the references given in the second edition, and to add references where there were none before. Throughout each chapter there will be found numerous ex- amples, many very easy and others which are intended for the more advanced student. In order to obtain as great a variety of problems as possible, I have added a further collection at the end of each chapter, taken from the Examination Papers which have been set in the University and in the Colleges. PREFACE. VII Some of these ure such excellent illustrations of dyuamical piinciples that they will certainly be of the greatest assistance to the student. I cannot conclude without expressing how much I am in- debted to Mr Webb, of St John's College, for the great assist- ance he has given me in correcting the proofs of the first eight chapters, and for the suggestions he has made to me. Most of the examples in these chapters have also been very kindly verified by him. Several others also of my friends have greatly assisted me by correcting some proof-sheeta for me, particularly Mr Edwards, of Sidney Sussex College, who has read the proofs of the last three chapters. Some portions of this edition have been written several years ago, and the printing has extended over two years. This course, though open to many objections, was rendered unavoidable by the pressure of other engagements. I have theretore found it necessary to add a few Notes, chiefly historical, at the end of the treatise. EDWARD J. ROUTH. Fetebhovse, Aptil 24, 1877. Page 77. Lino 80. For h + h read h+h'. „ 103, II 23. For in that case read in these oases. ,, 113. >» 18. For 0O8 2tf read oostf. „ 181. i> 8. For ,-«'*, tan -i?««d-«'* tan -1?. 2 -At* n 2-/[*» fi. « 141. For V read V thronghont the page. ,^ 146. II 1. Omit the word necessary. „ 255. II 2. For is read is parallel to. „ 264. M 38. For 1 read 1 . „. 287. II 14, For single read simple. „ 297. II 21. For ^h read ^2h. „ 299. II 80. For by A read by 2A. M 808. M 34. For read -0. >» >i II 35. For of three read of the axes referred to three. II II II 86. For Art. 235 read Note to Art. 235. „ 304. »l 17. For f" read <^", „ 813. II 17. For t read t, t^. „ 330. II 16. For - read +. »i »• II 17. For X read f(x). „ 331. 11 8. For - read +. CONTENTS. CHAPTER I. ON MOMENTS OF INERTIA. ABTS. 1—11. On finding Moments of Inertia by integration 12—18. Other methods of finding Moments c' Inertia 19—38. The Ellipsoids of Inertia .... 84 — 47. On Equimomental Bodies and on Inversion 48 — 66. On Principal Axes PAOES 1-9 9—15 15-23 23—32 82—46 CHAPTER II. d'alembert's principle, &c. 66 — 88. D'Alembert's Principle and the Bqnations of Motion . 84 — 87. Impulsive forces 47—60 60—64 CHAPTER III. MOTION ABOUT A FIXED AXIS. 88—91. Ecinations of Motion 65—67 92—97. The Pendulum 67—76 98 — 108. Length of the seeoads' pendulum 76 — 82 109. Oscillation of a watch-balance 82—84 110—118. Pressures on the fixed axis 84—93 119. The Centre of Percussion 93—94 120—122. The Ballistic Pendulum 94—99 CHAPTER IV. MOTION IN TWO DIMENSIONS. 123—138. General Methods and Examples 100—120 139—141. The Stress at any point of a rod 120—123 142—161. On Friction 123—131 152—168. Cn Impulsive Forces 131- -146 169—172. On Initial Motions 146—160 173—181. On Belative Motion and Moving Axes .... 151—168 Examples 159—163 CONTENTS. I< ADTS. 182—196. 197—217. 218—226. 227—238. 239—242. 243—263. 261—277. 278—283. 284—289. 290—294. 295—304. 305-316. CHAPTER V. MOTION IN THREE DIMENSIONS. Translation and Rotation . Composition of Botations . Motion referred to Fixed Axes Euler's Equations Expressions for Angular M)meutum Moving Axes and Kelative Motion Motion relative to the Earth PAORS 164—170 170—181 181—187 187—194 194-197 197—213 213-226 CHAPTER VI. ON MOMENTU.M. On Momentum, with examples 227 — 233 Sudden changes of motion 233 — 239 The Invariable Plane 239 — 242 Impulsive forces in three dimensions 242 — 247 Impact of rough elastic bodies 247—256 Examples 255—257 CHAPTER VII. VIS VIVA. 318—333. The Force-Fu ction and Work . . . . 334 — 35% Conservation of Vis Viva and energy . 353—367. Carnot's, Gauss', and Clausius' Theorems . 358—365. Newton's Principle of Similitude 366—390. Lagrange's and Sir W. K. Hamilton's Equations 391 — 398. Principles of Least Aotion and Varying Action . 399 — 409. Solution of the general equations of motion Examples . . 258—267 267—279 279—283 283—287 288—304 306—312 313—321 322—324 CONTENTS. Xi CHAPTER VIII. SMALL OSCILLATIONS. ARTS. 410—415. OsoillationB with one degree of freedom 416 426. First method of forming the equations of motion 427 — 430. Second method of forming the equations of motion 431 — 437. Oscillations with two or more degrees of freedom 43g — 443. Composition of oscillations and transference of Energy 444 — 461. Lagrange's Method of forming the equations of motion 462 — 469. The energy test of Stability, with an extension to certain cases of motion 470 — 484. Oscillations about Steady Motion with application to the Governor and Laplace's three particles, and some general theorems on Stability 485—489. Tlie Calculus of Fmite Differences 490 —495. The Cavendish Experiment 496 — 507. Oscillations of the second order Examples PAGES 325—331 831—841 341-845 845—354 854—356 356—369 369—376 375—886 886—889 389—394 395—401 401—403 CHAPTER IX. MOTION OF A BODY UNDER THE ACTION OF NO FORCES. 608—510. Solution of Euler's Equations 404—407 611 — 522. Poinsot's and Mac Cullagh's construction for the motion . 407 — 417 623 — 536. The Cones described by the invariable and instantaneous axes 417—426 637-540. Motion of the Principal Axes . . . . . 427—429 541—544. Motion when .4 ^ B 430—432 545-552. Motion when G« = J»Z'. . . . ' . . . . 432—437 653—666. Correlated and Contra-related Bodies 437 — 441 Examples 442—443 CHAPTER X. 557- 672- -571. -588. 689—698. 599. MOTION UNDER ANY FORCES. Motion of a Top 444—467 Motion of a sphere on perfectly rough surfaces of various forms and on an imperfectly rough inclined plane. Billiards 457-473 Motion of a Solid Body on a plane which is perfectly rough, imperfectly rough, or smooth .... 473 — 485 Motion of a Rod 485—487 Examples 487-489 Xll CONTENTS. CHAPTER XL PRECESSION AND NUTATION, &0. ABTB. COO— 609. On the Potential ..... 610—624. Motion of the Earth about its centre of gravity 625—634. Motion of the Moon about its centre of gravity PAOKS 490—498 499—618 614—522 CHAPTEE XII. MOTION OF A STRING OR CHAIN. 636—640. The Equations of Motion . 641-644. On Steady Motion .... 645 — 660. On Initial and Impulsive Motions 661—662. Small Oscillations of a loose chain 663 —672. Small Oscillations and energy of a tight string 623—528 628—532 632—635 636—546 647-666 NOTES. On D'Alembert's Principle 557 On Euler's Geometrical Equations 658 On the Impact of Bodies 669 On Sir W. B. Hamilton's Equations 560 On the Principle of Least Action 560 On Sphero-ConioB 662 Miscellaneous Notes 664 The student, to whom this subject is entirely new, is advised to read first the following Articles :— 1— 24, 36, 48—51, 66—68, 71, 73—93, 98—102, 110—116, 119 —120, 123—150, 152—163, 156—163, 169, 171—191, 197—208, 218—220, 227— 832, 235, 239—241, 243—246, 278—281, 284—285, 290—293, 295—298, 318-328, 834—348, 350, 366—369, 874—376, 410—412, 416—419, 424, 427—430, 444—445, 449-461, 462—464, 490—495, 608—609, 511-519, 522, 587, 641—544. I i CHAPTER I. ON FINDING MOMENTS OF INERTIA BY INTEGRA.TION. 1. In the subsequent pages of this wofk it will be found that certain integrals continually recur. It is therefore convenient to collect these into a preliminary chapter for reference. Though the bearing of these 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 jijoifi/hydxdydz, where (a, /8, 7) have particular values. In Statics two of these three exponents are usually zero, and the third is either unity or zero, according as we wish to find the numerator or denomi- nator of a coordinate of the centre of gravity. In Dynamics of the three exponents one is zero, and the sum of the other two is usually equal to 2. The integral in all its generality has not yet been fully discussed, probably because only certain cases have any real utility. In the case in which the body considered is a homogeneous ellipsoid the value of the general integral has been found in gamma functions by Lejeune Dirichlet in Vol. iv. of Liouville's 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 to restrict ourselves to the con- sideration of moments and products of inertia, as being the only cases of the integral .vhich are iseful in Dynamics. R. D. 1 i . 2 MOMENTS OF INERTIA. i. ] !M 3. If the mass of every particle of a material system bo 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 quanti^^^y that MJ(? 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 got 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 Ox, Oy be taken as axes, and if the mass of every particle of the system be multiplied by its two co- ordinates X, y, the sum of the products so formed is called the product of inertia of the system about those two axes. This might, perhaps more conveniently, be called the product of inertia of the system with reference to the two co-ordinate planes xz, yz. . 4. Let a body be referred to any rectangular axes Ox, Oy, Oz meeting in a point 0, and let x, y, z be the co-ordinates of any particle w, then according to these definitions the moments of inertia about the axes of x, y, z respectively will be A = Xm{f-\-z\ B = tm{e-\-a^), a = tm{x' + y^). The moments of inertia with regard to the planes yz, zx, xy, respectively, will be A'^^Xma?, B' = Xmy\ C' = t^.nz'. The products of inertia with regard to the axes yz, zx, xy, will be D = Xmyz, J? = %mzx, F= Xmxy. Lastly, the moment of inertia with regard to the origin will be /T = Sm (ir ' + y' + z^) = tmr\ ivhere r is the distance of the particle m from the origin. 5. The following propositions may be established without difficulty, and will serve as illustrations of the preceding defi- nitions. (1) The three moments of inertia A, B, G about three rectangular axes are such that the sum of any two of them is greater than the third. BY INTEGRATION. 3 (2) The sum of the moments of inertia about any three rectangular axes meeting at a given point is always the same ; and is equal to twice the moment of inertia with respect to that point. For A ;-5 + C=2Sm(x' + j/* + 2') = 22»ir', 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"+ G='2.m,r^, which is independent of the direction of the axes. Hence we infer that A' = \{B-\rC-A), B'==l{C^A-B), and C'=\iA+B-C). (4) Any product of inertia as D cannot numerically be so gi'eat 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^-\-2Fl + D='S.m{yl-Vxy=a. positive quantity. Hence the roots of the quadratic A0-v2Fl + B=(i are imaginary, and therefore AB>F^. (6) Prove that for any body {A + B-C){B + C-A) > ^E', {A-\-B- C){B + C-A){G+A-B) = SDEF. (7) Prove that the moment of inertia of the surface of a hemisphere of radius a and mass M about the diameter perpen- dicular to the base is Jlffa'. For, complete the ..phere, then by (2) the moment of inertia about any diameter is two-thirds of tho moment of inertia with respect to the point. 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 mor.ient of inertia of a uniform triangular plate about an axis in its plane passing through one angular pc'nt. Let ABC be the triangle, Ai/ the axis about which the moment is required. Draw Ax perpendicular to At/ and produce BC to meet Ay in D. The given triangle ABC may be regarded as the difference of the triangles ABU, AC I). Let us then first find the moment of inertia of ABD. Let PQP'Q' be an ele- mentary area whoso sides PQ, FQ' arc parallel to tho base AJJ, 1—2 MOMENTS OF INERTIA. 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 = l. I i m \ii V B — X Then the elementary area PQP'Q' is clearly r dx, and B —x its moment of inertia about Ay is fil _ dx .x\ fi being the mass per unit of area. Hence the moment of inertia of the triangle ABD Similarly if 7 be the distance of the angular point C from the axis Ay, the moment of inertia of the triangle A CD is nl~^' Hence the moment of inertia of the given triangle ABC is H'Tai^^ "y*)' Now^^)S and ^Zy are the areas of the triangles ABD, ACD. Hence if M be the mass of the triangle ABG, the moment of inertia of the triangle about the axis Ay is Ex. If each element of the mass of the triangle be multiplied by the nth 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_ /3''+l->y"^-l (w + i){n + 2) /3-7 • 7. When the body is a lamina the moment of inertia about an axis perpendicidar to its plane is equal to if>e sum of the moments BY INTEGRATION. 6 of inertia about any two rectangular axes in its plane drawn from the point where the former axis meets the plane. For let the axis of 2 be taken as the normal to the plane, then, if A, B, C be the moments of inertia about the axes, we have A = ^my\ B^Xmx", G^Xmix' + f), and therefore G=A + B, We may apply this theorem to the case of the triangle. Let fi', 7' be the distances of the points B, C 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 = f(^* + ^7 + 7^ + /3'^ + /3V + 7'^;. 8. 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 following 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- ") pendicular to the side 2a J about an axis through its centre perpendicu- ) _ lar to its plane j ~ (2) An ellipse semi-axes a and h about the major axio t = mass ^-j mass a* mass -K- , o a' + y minor axis h = mass a 4' about an axis perpendicular to its plane) _ a'+h* through the centre ] ~ "^^^^ 4 • In the particular case of a circle of radius a, the moment of 2 inertia about a diameter is mass j- , and about a perpendicular to 2 its plane through the centre mass -^ . (3) An ellipsoid semi-axes a, h, c about the axis a = mass — - — . In the particular case of a sphere of radius a the moment of , 2 inertia about a diameter = mass ■= a^ 5 »; 1 6 MOMENTS OF INERTIA. '. :! I hi 1 i f ■ i Hi ' I I' ! (4) A right solid whose sides are 2a, 26, 2c about an axis through its centre perpendicular ] _ 6' + c* to the plane containing the sides b and o j ~ ^^^^ 3" • These results may be all included in one rule, which the author has long used as an assistance to the memory. Moment of inertia ) (s"«^ 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 wanted the moment of inertia of a circle of radius a about a diameter, we notice that the perpendicular semi-axis in its plane is the radius a, and the semi-axis perpendicular to its plane is zero, the moment of inertia required is therefore M -^ , if M be the mass. If we wanted the moment about a perpendi- cular to its plane through the centre, we notice that the perpen- dicular semi-axes are each equal to a and the moment required is therefore M a" fa' = 1/ or 2* 9. As the process for determining these moments of inertia is very nearly tho same for all these cases, it will be sufficient to consider only two instances. To determine the moment of inertia of an ellipse about the minor axis. Let the equation to the eUipse hoy=- ^a'-' - «■'. Take any elementary area PQ parallel to tho axis of ift then clearly the moment of inertia is 4/t / x^i/dx = ip,- J x'' t^a'-x^dx, where n is tho mass of a unit of a 'ea. B To integrate this, put ;(;=a sin axes is ^ — . , , ,, ' , where aa', BB' are the semi-major axes of the curves. 4 (m' + »') 10. Many moments of inertia may be deduced from those given in Art. 8 by tiie method of differentiation. Thus the moment of inertia of a solid ellipsoid of uniform density p about the axis of a is known to be k trdbcp — = — • Let the ellipsoid increase indefinitely little in size, then the moment of inertia of the enclosed shell is , (4 , 6' + & a i^irabcp — ^ 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 = », - = 7. Then 4 p^ + 0" moment of inertia of solid ellipsoid = n'fppg, e ^^ .*. moment of inertia of shell = ^ "irppq (p' + ' (a) Dda. Replacing D by the variable density p, the moment of inertia required will be 1/30' (a) da. Ex. 1. Shew that the moment of inertia of a heterogeneous ellipsoid about tho 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 il/^Cft' + c'). 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 axis varymg as the distance from the centre, is 7^7 „ ■< . » — z — 5 • •' " '20 20^+0*- 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. 10 MOMENTS OP INERTIA. (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. 13. Piiop. I. Given the moments and products of inertia about all axes through the centre of grav'ty 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 moss collected at the centre of gravity about the original axes. Firstly, take the axis about which the moment of inertia is required as the axis of z. Let m be the mass of any particle of the body, which generally will be any small element. Let x, y, z be the co-ordinates of m, S, y, i those of the centre of gravity G of the whole system of bodies, x', y', z those of m referred to a system of parallel axes through the centre of gravity. Then since \m^ Xmy' Xm \mz - are the co-ordinates of the %m ' lim ' S»i centre of gravity of the system referred to the centre of gravity as the origin, it follows that Swa?' = (,„ 2my' = 0, Sw/ = 0. The moment of inertia of the system about the axis of z is V = Sw («' + 3/*), = Sm (i" -I- ^ + 2m (»" + y\ -f- 2a; . Swa/ + 2y . ^my. Now Sw (S' + p") is the moment of inertia of a mass 2m collected at the centre of gravity, and 2m (aj"+y'*) is the moment of inertia of the system about an axis through (?, also Sma;' = 0, 2my = ; 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 = 2ma;^ = tm {x + x') (^ + y'), = xy . 2m + 2m x'y + xZmy + y'Xmx' = xy%m + 'Sitnx'y. Now xy . 2m is the product of inertia of a mass 2m collected at G and Xmxy is the product of the whole system about axes through G ; whence the proposition is proved. OTHER METHODS. 11 Lot there bo two parallel axes A and 7? at tlLstancos a and b from the centre of gravity of tho body. Then, if M bo the mass of tho material system, moment of inertia] ,- , _ jmomcnt of inertia ,,., about A ) \ about B Hence when the moment of inertia of a body about one axis is known, that about any other parallol axis may be found. It is obvious that a similar proposition holds with regard to tho pro- ducts of inertia. 14. The preceding proposition m ly bo generalised as follows. Let any system be in motion, and let x, y, z be tho co-ordinates at time t of any particle of mass m, thoii -y- , -j:^ 7/7 ^"^^ *^^^ d^x cCv cPz velocities, and ;^ > j^ > j^ ^^^ accelerations of tho particle resolved parallel to the axes. Suppose tr -%} A f dx d^x dy d^y dz d*z\ V=Xm{x. j^. ^,. y, -^^, J, z, ^^, ^,j 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 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 origm plus the value of V 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, (lOS d'JT {LSR and let a; = » + x\ &c. •*• 77: = ;^ + "jT > ^^' Now since j> is an algebraic function of the second order of X, -r. , -^ ; y, &c. it is evident that on making the above sub- stitution and expanding, the process of squaring &c. Avill lead to three sets of terms, those containing only x, -7- , -1-5 , &c., those containing the products of x, x &c., and lastly those containing i, ■ ' (i^ i 12 MOMENTS OP INEBTU. dx only a?', , , &c. The first of these will on the whole make up ff> (x, r: > &c.] , and the last 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*= -r!P\ since j)r=db. Ex. 3. As an example of a different kind, let ns find the moment of inertia of an ellipsoid of mass 3/ and semiaxes (a, 6, c) with regard to a diametral j)?ane whose direc- iiou-cosines referred to the principal planes are (a, j9, 7). By Art. 8, the moments of inertia with regard to the principal axes are M — = — , M — -= — , M ■ - . 555 Hence by Art. 5, the moments of inertia with regard to the principal planes are M - , .M ■^, I/-5 . Hence the required moment of inertia is "^ (a^a? + b'^p,'^ + c'^'^^). If p ^ , M-= . Hence the required moment of inertia is ^^ 5 5 5 be the perpendicular on the parallel tangent plane, wo know by solid geometry that this IS the same sis M ■=■. Ex. 4. The moment of inertia of a rectangle whose sides are 2a, 26 about a diagonal is 2M aW , 3 ((•■!. I //.'• L Arts. 13 and 15 to the ellipse to ELLIPSOIDS OF INERTIA. 16 Ex. 6. If ki, k^ be the radii of gyration of an elliptic lamina about two conjugate diameters, then j3 + jiri= ^ ^^2 "^ pj * 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 b the radius of the 3 base, then the moment of inertia about the axis is ilf r-^'; ^bat about a straight line through the vertex perpendicular to the axis ib M^(a^ + j\, that about a slant side M ^ -^ — rj ; that about a perpendicular to the axis through the centre of 3 gravity i^ ^ ^ («" + ^S"). Ex. 8. If a be the altitude of a right cylinder, b the radius of the base, then the moment of inertia about the axis is ilf ^ and that about a straight lino through the I centre of gravity perpendicular to the axis is M «-)• Ex. 9. The moment of inertia of a body of mass At about a straight line whoso x-f _y-g _ z-h \ equation is = referred to any rectangular axes meeting at the I m n '^Im centre of gravity is I AP + Bm^ + Cn^ - 2Dmn - 2Enl - 2Flm + M{f ^+g^ + h?-(fl + gm + kn)% $ 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 + cy^ + 2dx + 2eij+l=0, M -Ha about a diameter parallel to the axis of x, is , , where M is the mass and 4 ' (ac-by II is the determinant oc - 6' + 2bed -ae^- ccP, usually called the discriminant. Ex. 11. The moment of inertia of the elliptic disc whose equation in arcal co- ordinates is

i Let ca 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 direc- tion-cosines are (a, ,8, y), we have / = -y^s- . where e is some R' constant introduced to keep the dimensions correct, and M is the mass. Hence the polar equation to the locus of Q is Me* Aoi^ + B^'+ 6V - 2Z>/3y - 2EyoL - 2Fa^. Transforming to Cartesian co-ordinates, we hav« Me* = AX^ + BY'+CZ'-2DYZ- 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 inverse square of that radius vector. The convenience of thit, 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 proper- ties of a quadric. Since a moment of inertia is essentially positive, being by definition the sum of a number of squares, it is clear that every radius vector R must be real. Hence the quadric is always an ellipsoid. It is called the momental ellipsoid, and was first used by Cauchy, Exercices de Math. Vol. ii. 20. The momental ellipsoid is defined by a geometHcal pro- perty, viz. that any radius vector is equal to some constar.c divided by the square root of the moment of inertia about that radius vector. Hence whatever co-ordinate axes are taken, we must always arrive at the same ellipsoid. If therefore the momental ellipsoid be referred to any set of rectangular axes, the coefficients of X\ Y\ Z\ -2YZ, -2ZX, -2XY in its equation will still represent the moments and products of inertia about the axes of co-ordinates. Since the discriminating cubic determines the lengths of the axes of the ellipsoid, it also follows that its coefficients are un- altered by a transf or nation of axes. But these coefficients are A + B+0, AB + BC + CA-D'-E^-F\ ABC - 2BEF - AD'' - BE' - CF\ Hence for all rectangular axes having the same origin, these are invariable and all greater than zero. ELLIPSOIDS OF IXEHTIA. 17 about the given of inertia about of the length. tor wliose direc- here € IS some ct, and M is the lis 2Fa/3. ;ry point of a which possesses ly radius vector lus vector. The ions which exist lines meeting at e known proper- )sitive, being by clear that every "ic is always an was first used geometrical pro- jonstar.c divided tout that radius iaken, we must the momental the coeflficie'^tv) lation will still •out the axes of lengths of the icients are un- fefficients are [rigin, these fire 21. It should be noticed that the constant e is arbitrary, Ithouo-h when once chosen it cannot be altered. Thus we have a series of similar and similarly situated ellipsoids, any one of [which may be used as a momental ellipsoid. When the body is a plane lamina, a section of the ellipsoid [corresponding to any point in the lamina by the plane of the [lp,mina, is called a momental ellipse of 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^=Me\ where J/ is the mass [and e* 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. T2 ^2 /I- 4. ?i2 I Taking the same notation as before, we have A = 'M j, B = M j , C = M . iPIence the ellipsoid is 4 4 a" + 6" Z^=Mt*. |Siuce 6 is any constant, this may be written If When Z=0, this becomes an ellipse similar to the boundary of given disc. Hence ;; ywe infer that the momental ellipse at the centre of an elliptic area is any similar '^d similarly situated elUpse. This also follows from Art. 18, Ex. 1. ^ Ex. 2, To find the momental ellipsoid at any point O of a material straight rod A B of mass M and length 2a. Let the straight line OAB be the axis of x, O the /•forigin, the middle point of AB, 00 =c. If the material line can be regarded as ■M ■ /a' \ ^Jipndefinitely thin, .4=0, i'=ilf( — + c') = C, henci the momental ell'psoid is 3p + Z*=e"', where e' is any constant. The momenta! ellipsoid is therefore an elongated spheroid, which becomes a right cylinder having ihe straight line for axis, vheu the rod becomes indefinitely thiiu Ek. ' The momental ellipsoid at the centre of a material ellipsoid is (62 + c") A'2 + (fl' 4 a^) P + (a2 + 6«) Z^ = e*, vhere c is any constant. It should be noticed that the longest and shortest axes of he momental ellipsoid coincide in direction with the longest and shortest axes Respectively of the material ellipsoid. 23. By a consideration of some simple properties of ellipsoids, [ho following propositions are evident : I. Of the moments of inertia of a body about axes meeting at given point, the moment of inertia about one of the principal iixes is greatest and about another least. For, in the momental ellipsoid, the moment of inertia about |iny radius vector from the centre is least when that radius vector R. D. 2 f,l '« kb' \i ■■ii . ! 18 MOMENTS OF INERTIA. 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 be 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 Xmxy = 0, %myz = 0, Sm^ic = 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 faco of the solid. The relations of these two planes to the solid are in all respects the same. Hence also the m omental 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. 24. At every point of a matenal system there ai^ 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 coefficients of — XY, — YZ, — 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 dia- meters as axes, these coefficients vanish. Hence the principal dia- meters 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 refer- ence, prove that the momental ellipsoid at the point (p, q, r) is ~2qrYZ-2 rp ZX - 2pq X !'=£*, when referred to its centre as origin. ELLIPSOIDS OF INERTIA. 10 the greatest and ;ers. of inertia with loint, that with with regard to oint be equal very diameter is ;s are all equal, il axis at 0, and tre of gravity of the body being y = 0, '%myz = 0, tout them are by ;entre of gravity of inertia about I. Consider two parallel to a faco to the solid are mtal ellipsoid at . with regard to les parallel to all and the moment aj'e always three joint. Then it ut the axes are equation to the s as axes of co- principal dia- le principal dia- e system. But rs, hence every ig the axes of refer- "') Ex. 2. Show that the cubic equation to find the three principal moments of inertia at any point {p, q, r) may be written in the form of a determinant I-A M n rp ri J-R M ■ r' - r" rp qr qr I-C M ri^ - o' = 0. If (I, m, n) bo proportional to the direction-cosines of the axis corresponding to ' any one of the values of I, their values may be found from the ec^uationa \I-{A + Mq'> + Mr'^)]l^Mpqm-i-Mrpn=iO, j Mpql + { / - (Z? + A/r" + Mp'^) ]m + Mqrn= 0, Mrpl + Mqi-m+ {I- (C + Mp^ + M>f)ln:=0. Ex. 3. If 5-0 be the equation to the momental ellipsoid at the centre of [gravity referred to any rectangular axes written in the form given in Art. li), I then the momental ellipsoid at the point P whose co-ordinates are (p, q, r) is S+3I {p^ + 2" + !•«) (Z* +Y^ + Z^)-M(pX+qY + rZf = 0. I Hence show (1) that the conjugate planes of the straight line OP in the momental I ellipsoids at and P are parallel and (2) that the sections perpendicular to OP ^ have their axes paralleL 26. The reciprocal surface of the momental ellipsoid is ^nother ellipsoid, which has also been employed to represent, geo- iinetrically, the positions of the principal axes and the moment of "linertia about any line. We shall requue the following elementary proposition. The reciprocal surface of the ellipsoid -j + |j + ij = 1 is the ellipsoid a^x^ + IV + c'^" = e*. 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 \0N'=aH'' + b^m'^+cH^ Produce OiVto Q so that 0Q=^, then Q is a point on t e* I the reciprocal surface. Let 0Q=R; .: =a*l^ + h''m^ + chi^. Changing this to [rectangular co-ordinates, we get e*=a'x'^ + b^y^ + ch\ To each point of a material body there corresponds a series of [similar momental ellipsoids. If we reciprocate these we got lanother series of similar ellipsoids coaxial with the first, and [such that the moment of inertia of the body about the perpen- jdiculars on the tangent planes to any one ellipsoid are propor- [tional to the squares of those perpendiculars. It is, however, con- Ivenient to call that particular ellipsoid the ellipsoid of gyration I which makes the moment of inertia about a perpendicular on a I tangent plane equal to the product of the mass into the square so MOMENTS OF INERTIA. lljf f of that perpendicular. If Mho the mass of the body and A, B, the principal moments, the equation to the ellipsoid of gyration is A"^ B'^ G~ M' It is clear that the constant on the right-hand side must be -jTj., for when Y and Z are put equal to zero, ^Y' must by A definition be -r>. M 27. Conversely, the series of momontal ellipsoids at any point of a body may be regarded as the reciprocals, with different constants, 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 gy- ration. 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 com- pressed where the body is compressed. The exact reverse of this is the case in the ellipsoid of gyration. See Art. 22, Ex. 3. 28. Ex. 1. To find the ellipsoid of gyration at the centre of a material elliptic disc. Taking the values of A, B, C given in Art. 22, Ex. 1, we see that the Z2 1 ~~V ellipsoid of gyration is -— + -^ + Ex. 2. The ellipsoid of gyration at any point of a material rod AB is ■jp + r~2T— -J + r"a — i - ^> tfl'^ing tl'e same notation as in Art. 22, Ex. 2. This is a very flat ellipsoid which when the rod is indefinitely thin becomes a circular area whose centre is at 0, whose radius is /^^a^+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 = 0, A -F -E MX -F B -D MY -E -D C MZ MX MY MZ M or when expanded (UC - D^)X^ + {CA -E^)Y^-\-{AB - F-^)Z'i + 2{AB ■¥EF)YZ + 2(BE+FI))ZX+2{CF+DE)XY =^(AB0-AD^-BE^-CF'-2DEF). The right-hand side, when multiplied by M, is the discriminant obtained by leaving out the last row and the last column, and the coefficients of X\ Y'^, Z", 2ZX, 2XY, 2YZ are the minors of this discriminant. ELLIPSOIDS OF INERTIA. 31 n 29. The use of the ellipsoid -whose equation referred to the principal axes at the centre of gravity is 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. Wo may call this ellipsoid the equi- momental 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 tBa plane, then this plane envelopes an ellipsoid similtir to Lbgendre's ellipsoid. 30. There is another ellipsoid which is sometimes useJ^ By Art. 15 the moment of inertia with reference to a plane whose direction-cosines are (a, ^, 7) is /' = 2;Ha;'. a« + i,ni7/'>./3» + 2wi32. 7" + 22771^/2. j37-l-22»i2a5. 70 + 22ma!y. o^. Hence, as in Art. 19, we may construct the ellipsoid Smx'. Z4+2mj/». r»+2mz'. Z=' + 22ni?/2. YZ + 21.mzx . ZX+2,l.mxy .XY=Mi*. Then the moment of inertia with regard to any plane through the centre of the ellipsoid is represented by the inverse 9quar3 of the radius vector porpeudfoular te that plane. If we compare the equation of the momenta! ellipsoid with that of this ellipsoid^ we see that one may be obtained from the other by subtracting the same quantity from each of the coefficients of X^, 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. 5 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 *he inverse squares of the semi-axes. This ellipsoid is a reciprocal of Legendrp'd 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 2m3'=0*, The section of the fourth ellipsoid is then clearly the same as a momenta! ellipse at [ the point. If any momenta! ellipse be turned round its centre through a right angle it evidently becomes similar and similarly situated to the ellipse of gyration. So that, in the case of a lamina, any ons of these ellipses may be easily ohanr,ed I into the others. 32. A straight line passes tJirough a fixed point O a^id 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. 2 find the equation to the cone generated by the straight line. 22 MOMENTS OF INERTIA. fl Itimii m I Let the principal axes at bo taken as the axes of co-ordi- nates, and let (a, ff, 7) be the direction-cosines of the straight lino in any position. Then by Art. 17 we have Aa' + B^ + G^f = I. Hence the equation to the locus is (^-/)a«-f-(5-/)/3'+(a-/)7' = 0, or, transforming to Cartesian co-ordinates, {A-I)x' + {B-I)y'+{C-I)z''=^0. It appears from this equation that the principal diameters of the cone are the principal axes of the body at the given point. The given quantity I must be less than the greatest and greater than the least of the moments A, B, C. Let A, B, C be arranged in descending order of magnitude ; then if / be less than B, the cone has its concavity turned towards the axis C, if / be greater than B the concavity is turned towards the axis A, if 7= B the cone becomes two planes which are coincident with the central circular sections of the momental ellipsoid at the point 0. The geometrical peculiarity of this cone is that its circular sections in all cases are coincident in direction with the circular sections of the momental ellipsoid at the vertex. This cone is called an equimomental cone at the point at which its vertex is situated. 83. The properties of products of inertia of a body about different sets of axa are not so useful as to require a complete discussion. The following theorems will serve as exercises. Ex. 1. If any point be given and any plane drawn through it, then two straight lines at right angles Ox, Oy can always be found such that the product of inertia about these lines is zero. These are the axes of the section of the momental ellipsoid at the point formed by the given plane. Ex. 2. If two other straight lines at right angles Oac', O1J be taken in the same plane making an angle measured in the positive direction with Ox, Oy rc^ectively, then the product of inertia F about Ox', Oy' is given by the equation * F'=\6va2e(A-Ji), where A , B are the moments of inertia about Ox, Oy. Ex. 3. If I be the moment of inertia about any line in this plane making an angle 9 with Ox, then I=Acoi?e + Bim-9. For the section of the momental ellipsoid by the plane is the ellipse whose equation is Ax- + By- = M(*, whence the property follows at once. tL EQUIMOMENTIAL BODIES. 23 point at which i at the point aken in the same Ex. 4. Let (Nm") (^V"') be the direction-cosinca of two straight linos Ox', 0;/ I at right angles passing through the origin and referred to the principal axes at I us axes of oo-ordinates. Then the product of inertia ahout these linos is F = XX'Smx' + nix' ^my' ■{ w'Sjms'. For let (x'yV) be the co-ordinates of any point {xi/z) referred to Ox', Oij' and a I third line Oz' as nevr axes of co-ordinates. Then «'=Xa!+/ty-t-M, and y'=:\'x + n'ij + v'z. Hence, since F'='S.nu!y', the theorem follows by simple multiplication. Since XV + /*/»' + ""' = 0, we have -f" =4 XX' 4 5/*/*' + (?"»''. Ex. 5. If (X/i»') he the dircction-cosinos of an axis Ox', then the direction- cosines (XVi/') of another axis Oy' at right angles such that the product of inertia I about Ox', Oy' is zero, are given by the equations . X' ^ m' _ ^ "' _ (B-C)iu> {C-A)v\ {A-Ji)\fi' For by (4) the equations to find X'/tV are A\\'+BniJif + Ci'i>'=0,) \\'+Hfx' + i>y' = 0,) whence the theorem follows by cross multiplication. By (1) this is equivalent to the geometrical theorem. Given a radius vector Ox' of an ellipsoid, find another radius vector Oy' such that Ox', Oy' are principal diameters of the section xfOy'. Ex. 6. Let (Imn) be the direction-cosines of any given straight line Oz', and let jy, E' be the products of inertia about Oz', Oy'; 0/, Ox', where Ox', Oy' are any two straight lines at right angles. Then as Ox', Oy' turn round Oz', Z)''^ + E''^ ia constant, and D'a + E'^ ={A- B)^ (Im)' + {B- C)' {mn}^ + {C-A)^ (nl,^. For by (4), - U=Al\^-Bmit->r Cnv, - E'=Al\'-i- Bm/j! + Cnv' ; .'. jy^ + E"=AH^(X' + \''')+2ABlmQi^i.+\'iJi.') + &c. But V + X'« = l-Zi'=m2+n»,) \fi+\'ix'=-lm, ) whence by substitution the theorem follows at once. Ex. 7. If A', B' be the moments of inertia about Ox', Oy', then as Ox', Oy' turn round Oz', the value of A'B' -F'^ is constant, and A'B'-F'^=^BCl^-[- CA ;u2 + A £n\ plane making an le ellipse whose On Equimomental Bodies. 84. Two bodies or systems of bodies are said to be equi- momental when their moments of inertia about all straight lines arc C(iual each to each. I' r ii > K ' R IT B- ^ I ( ! 2t MOMENTS OF IVKRTIA. 35. If two systcmH have tho same centre of gravity, the same mass, the same principal axes and principal moments at tho centre of gi'avity, it follows from tho two fimdamcntal propositions of Arts. 13 and 15 that their moments of inertia about all straijjlit lines are equal, each to each. That the converse theorem is also true may bo shown thus. We know by Art. 13 that of all straight lines having a given direction in a body, tiiat straight line has the least moment of inertia which passes through the centre of gravity. It is clear that these least moments of inertia could not be equal in two bodies for all directions unless they had a common centre of gravity. Of all straight lines through the centre of gravity those which have the greatest and least moments of inertia are two of tho principal axes, hence these and therefore also the third principal axis must be coincident in direction if the two bodies are equi- momental. The principal moments of inertia must then be equal, because all moments arc equal. Lastly, by Art. 1.3, tho two systems could not have equal moments about two parallel axes, each to each, unless their masses were ccpial. It is easy to see that two equimomental systems must have the same momental ellipsoid, and therefore the same principal axes at every point. 3C. To find the moments and products of inertia of a triangle about an]/ axes ivhatever. If /3 and 7 be the distances of the angular points B, C, of a triangle ABC from any straight line AX through the angle A, in the plane of the triangle, it is known that tho moment of inertia M of the triangle about AX\» y (/3* + ^7 + y^, where M is the mass of the triangle. Let three equal particles, the mass of each being -^ , be placed o at the middle points of the three sides. Then it is eu ../ seen, that the moment of inertia of the three particles about AX is •which is the same as that of the triangle. The three particles treated as one system, and the triangle, have the same centre of gravity. Let this point be called 0. Draw any straight line OX' through the common centre of gravity parallel to AX, then it is evident that the moments of inertia of the two systems about OX' are also equal. Since this equality exists for all straight lines through in the plane of the triangle, it will be true for two straight lines 0X\ + (I)'- & H EQUIMOMRNTAL BODIES. 25 be shown thus. M is the mass irough in y at right angles, ami therefore also for a straight lino OZ* 1 perpendicular to the plane of the triangle. One of the principal axes at of the triangle, and of the [syHtem of three particles, is normal to the plane, and therefore the same for the two systems. The principal axes at in the plane, arc those two straight lines about which the moments of inertia J are greatest and least, and therefore by what precedes these axes [are the same for the two systems. If at any point two systems jliave the same principal axes and principal moments, they have jalso the same moments of inertia about all axes through that f)()int, and the same products of inertia about any two straight iiies meeting in that point. And if this point be the centre of Igravity of both systems, the same thing will also be true for any jother point. If then a particle whose mass is one-third that of the triangle [be placed at the middle point of each side, the moment of inertia 3f the triangle about any straight line, is the same as that of the Isystem of particles, and the product of inertia about any two ietraight lines meeting one another, is the same as that of tho Isystem of particles about tho same straight lines. § 37. Three points D, E, F can always be found such that the products and moments of inertia of three equal particles placed lit D, E, F, may be the same as the products and moments of Inertia of any plane area. For let be the centre of gravity of the area, Ox, Oy the principal axes at in the plane of the area, and il7a' and M^^ be the moments of inertia about these axes. Let {xy)y {xy), {x"y") be the co-ordinates of D, E, F, m the Imass of a particle, so that M= Sm. Then we must have m (a;*+ x' + x"^) = Jl/yS*, xy + xy + ic'y = 0. Also, since the two systems must have the same centre of jravity, ic + u;' + a;" = 0, y + y + y" = 0. Eliminating x'y, x"y" from these equations, we get diich is the equation to a momental ellipse. It easily follows, i,hat D may be taken any where on this ellipse, and E and F are it the opposite extremities of that chord which is bisected in some )oint iV by the produced radius DO, so that 0N= \0D. 38. A momental ellipsoid at the centre of gravity of any triangle may be found as follows. Vit-"^ 2G MOMENTS OF INERTIA. m MJ! -i , \' i> t Let an ellipse be inscribed in the triangle touching two of the bides AB, BO in their middle points F, I). Then, by Garnet's Theorem, it touches the third side CA in its middle point E. Since DF is parallel to CA the tangent at F, the straight line joining F to the middle point iV of DF passes through the centre, and therefore the centre of the conic is the centre of gravity of the triangle. This conic may be shown to be a momenta! ellipse of the triangle at 0. To prove this, let us find the moment of inertia of the triangle about OF. Let OE=r, and let the semi-conjugate diameter be r, and w the angle between r and r'. Now ON=^r, and hence from the equation to the ellipse FN^ = ^r'^, therefore moment ofi inertia about OF = P/. 1 '! sm 0), = M A'" irV ' where A' is the area of the ellipse, so that the moments of inertia of the system about OF, OF, OD are proportional inversely to OF^, OF^, OD^. If we take a momental ellipse of the right dimensions, it will cut the inscribed conic in F, F, and D, and therefore also at the opposite ends of these diameters. 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. 17). Hence a momental ellipsoid of the triangle has the inscribed conic for one principal section. If a and b be the lengths of the axes of thi;^ conic, c that of the axis of the ellipsoid which is perpendicular to the plane of the lamina, we have by Arts. 7 and 19 1-11 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 might take the ellipse circumscribing the triangle, and having its centre at the centre of gravity, as the momental ellipse of the triangle. 39. Ex. 1. A momental ellipse at an angular point of a triangular area touches the opposite side at its middle point and bisects tlie adjacent sides. Ex. 2. The principal radii of gyration at the centre of gravity of a triangle are i;u roots of the equation where A is the area of the triangle. llci EQUIMOMENTAL BODIES. 27 liing two of the en, by Carnot's liddle point E. he straight lino ough the centre, re of gravity of 1 ellipse of the ent of inertia of serai-conjugate Now ON=y, lr\ M ^ 2 • ttV* ' nents of inertia lal inversely to ie of the right F, and D, and eters. But two ey are identical, triangle. a principal axis 3llipsoid of the section. If a hat of the axis of the lamina, nomental ellip- the centre of CIS. pse is also a imscribing the gravity, as the gular area touches s. dty of a triangle Ex. 3. The direction of the principal axes at the centre of graviiy of a tri- [ angle may be constructed thus. Draw at the middle point D of any side BQ I lengths DII= — , BH'— — along the perpendicular, where p is the perpendicular ! P P [from A on BC and P, k"^ are the principal radii of gyration found by the last ex- f ample. Then OH, OH' are the directions of the principal axes at 0, whose I moments of inertia are respectively il/A" and Mk'-. Ex. 4. The directions of the principal axes and the principal moments at the I centre of gravity may also be constructed thus. Draw at the middle point D of BO I any side BC a perpendicular DK = 2J3' Describe a circle on OK as diameter i and join D to the middle point of OK 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 DS"' 2 and M DB^ Ex. 6. Let four particles each one-sixth of the mass of the area of a parallelo- I gi-am be placed at the middle points of the sides and a fifth particle one-third of the I same mass be placed at the centre of gravity, then these five particles and the area I of the parallelogram are equimomental systems. Ex. 6. Let four particles each one-twelfth of the mass of the area of a quadri- : lateral be placed at each corner and let a negative mass also one-twelfth be placed at the intersection of the diagonals and a sixth particle three-quarters of the same mass be placed at the centre of gravity, then these six particles and the area of the quadrilateral are equimomental systems. Ex. 7. Let three particles each one-sixth of the mass of an elliptic area be placed one at one extremity of the major axis and the other two at the extremities of the ordinate which bisects the semi-axis major, and let a fourth particle whose mass is one-half that of the area be placed at the centre of gravity. Then the moments and products of inertia of the system of four particles and of the elliptic area are the same for all axes whatever. Ex. 8. Any sphere of radius a and mass M is equimomental to a system of m four particles each of mass ^ ( - ) placed so that their distances from the centre make equal angles with each other and are each eqiial to r and a fifth particle equal to the remainder of the mass of the sphere placed at the centre. 40. To find the moments and products of inertia of a tetra- hedron about any axes whatever. Let ABGD be the tetrahedron. Through one angular point D draw any plane and let it be taken as the plane of xy. Let D I be the area of the base ABC', a, /8, 7 the distances of its angular I points from the plane of xy, and p the length of the pcrpendiculor ^" from D on the base ABC. Let PQR be any section parallel to the base ABC and of thickness du, where u is the perpendicular from D on PQR. The moment of inertia of the triangle PQR with respect to the plane 28 MOMENTS OF INERTIA. of xy is the same as that of three equal particles, each one-third its mass, placed at the middle points of its sides. The vohime of W the element PQR = -^ pdu. The ordinates of the middle points of the sides AB, BG, CA are respectively ^—^^ 9~^> '^~Y^ ' Hence, by similar triangles, the ordinates of the middle points of PQ, e;j.iJP are also ^i', ''-±^''-, ^^ The moment of inertia of the triangle PQR with regard to the plane xi/ is therefore Integrating from u = to u=p, we have the moment of inertia of the tetrahedron vfith. regard to the plane xi/ where Fis 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 xi/ would be which is the same as that of the tetrahedron. The centre of gTavity of these five particles is the centre of gi'avity of the tetrahedron, and they together 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 thir, 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.* 41, 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 pro- " This result was proposed as a Problem in the Mathematical Tripos in an interval of the publication of the preceding and following results, thus anticipating the author by a few days. <8 EQUIMOMENTAL BODIES. 29 3, each one-third The volume of B middle points ^+7 7+a » 2 ' 2 • middle points of th regard to the I)}- the moment of the tetrahedron the rest of the of gravity, the ird to the plane the centre of :e up the mass its of inertia of h the centre of . equality will , that the mo- ual. The t-vo figure in space tio, the figure fure. By pro- ical Tripos in an tlius anticipating jecting a figure from three planes as base planes at right angles in succession, the figure may be often much simplified. Thus an ellipsoid can always be projected into a sphere, and any tetra- hedron 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 nl) perpen- dicular to the base plane. We may therefore suppose the base plane to pass through any given point which may be convenient. 42. 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 2w = 2m' ; Xmx = 0, Xm'x' = 0, &c., Xmx^ = ^m'x'^, ^myz = 'liVii/'z', &c., unaccented letters referring to one body and accented letters to the other. Let both the [bodies be projected from the plane of xi/ in the fixed ratio 1 : n. Then any point whose co-ordinates are (x, y, z) is transferred to \{x, y, nz) and {x', y', z) to {x, y, nz). Also the elements of mass wi, vn 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 moinental ellipse of the projection. Let the figure be projected from the axis of x as base line, pso that any point {x, y) is transferred to {x, y') where y' = ny, ^nd any element of area m becomes m' where m' =■ nm. Then ■''?' Ill I l,mx^ = - Xm'af, tmxy = -g Xm'xy\ Xmf/ = -3 Smy". ,/si '* n n lie momental ellipses of the primitive and the projection are l^mfX' - 2XmxyXY+ Xmx^Y^^Me\ XmyX" - 2Xm'xy'X' F + Xm'x' Y" = M'e\ [■o project the former we put X' = X, Y'=nY, its equation then fbecomes identical with the latter by virtue of the above equalities Jif we put e = en. I 43. 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 these first with one Ride aa -J base line, and secondly with a diagonal as base, the square becomes successively a rectangle and then a parallelogram. Hence a momental ellipse at the centre of -gravity of a parallelogram is the inscribed conic touching at the middle points of • the sides. 30 MOriEXTS OF INERTIA. w .\ i Ex. 2. By jjrojectiug an equilateral triangle into any triangle, we may infer the results of some of the previous articles, but the method will be best exx)lained by its application to a tetrahedron. Ex. 3. Since any ellipsoid may be obtained by projecting a sphere, we infer by Ai't. 39, Ex. 8, that any solid ellipsoid of mass M is equimomental to a system of four particles each of mass -y~ ^ 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 sphere 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 of mass M is equimo- mental to a system of three particles each ^ - 21/ placed on the circumfeience of the 3 base so that the differences of their eccentric angles are equal, a fourth particle — M placed at the middle point of the straight line joining the vertex to the centre of gi'avity of the base, and a fifth particle to make up the mass of the cone placed at the centre of gravity of the volume. 44. To find the equimomental ellipsoid of 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. 40 3^.2 " _ to be M -^ . If then we describe a sphere of radius p = ^S r, o with its centre 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 ir.tr. 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 : J3. It may also be easily seen that the sphere whose radius is p = ,^3^ touches each edge of the regular tetrahedron at its middle point. Hence we infer that the equimomental ellipsoid of any tetrahedron 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, with- out the use of projections. 1 EQUIMOMEXTAL BODIES. 31 .0, we may infer the best explained by its sphere, we infer by ental to a system of ivhose linear dimeu- 10 that the eccentric fth particle equal to gravity. my given body, we ! particles placed as lid of the body. mass M is equimo- ircumfeience of the g ourth particle — M rtex to the centre of : the cone placed at 'etrahedron. :on with regard ual by Art. 23. moment with een bv Art. 40 adiu3 p = JSr, mass equal to hedron will be y face projects we infer that ntal, is similar trahedron and has its linear also be easily hes each edge ence we infer 1 touches each ntre of gravity >, Ex. 2, with- in 45, Ex 1. If E^! be the sum of the squares of the edges o' a tetrahedron, 2?" the Buia of the squares of the areas of the faces and V the volume, show that the semi- Hxesof the ellipsoid inscribed in t} J tetrahedron, touching each face in the centre of Wavity and having its centre at the centre of gravity of the tetrahedron, are the %oots of ITS P2 72 2^3 P^+2^3.P^ %nd if the roots be i-p^:kp.^J=p.^, then the moments of inertia with regard to the 'W ^F,^ W rincipal iJlanes of the tetrahedron are M -^ , M -g" , M I 4p i Ex. 2. If a perpendicular EP be di-awn at the centre of gravity E of any face = — Sphere p is the perpendicular from the opposite corner of the tetrahedron on that ,|acp, then /» is a point on the principal plane con-espondiug to the root p of the jpubic. I 46. To explain hoiu the theory of invers.n can he applied to Mnd moments of inertia. f Let a radius vector drawn from some fixed origin to any point P of a fguro be produced to P' where the rectangle OP .OP' = k^ where k is some given uantity. Then as P travels all over the given figure, P' traces out another Vhich is called the inverse of the given figure. " Let {x, y, z) be the coordinates of P, (as', ij z') those of P'; r, r' the radii vectores, dv, dv' corresponding polar elements of volume; /), p', dm, dm' their respective densities and masses. Let du be the solid angle subtended at by either dv or dv'. Then dv'=r'^dudi a/ -67 r^du "i^ov! dm=pdv, dm'= p'dv'. If then we and sincp — = - wo have sd'^ dv'= ( " ) a;^ dv. take p'—l-j p we have Ix'- dm'='^x^ dm, with similar equalities in the case of all |he other moments and products of inertia. Hence we infer, that if a homogeneous body be inverted with regard to a point ?, and the density of the new body vary inversely as the tenth power of the distance ^rom 0, then these two bodies have the same moments of inertia about all straight iiues through 0. Ex. The density of a solid sphere varies inversely as the tenth power of the listance from an external point 0. Prove that its moments of inertia about any i^traight line through is the same as if the sphere were homogeneous and equal in iensity to that of the heterogeneous sphere at a point where the tangent from icets the sphere. Prove that if the density had varied inversely as the sixth power sf the distance from 0, the masses of the two spheres would have been equal. What |s the condition thev should have a common centre of gravity ? 47. The theory of equimomental particles is of considerable ;, use in finding the centre of pressure of any area vertically im- linicrscd in a homogeneous fluid under the action of gravity. It |may be proved fi-om hydrostatical principles that if the axis of 1 S2 MOMENTS OF INERTIA. '!i ; V I I 1 4 ' tc be in the effective surface, and the axis of y vertically down- wards, the co-ordinates of the centre of pressure are Product of inertia about the axes X = Y= moment of area about Ox Moment of inertia about Ox moment of area about Ow We see therefore that two equimomental areas have the same centre of pressure. The preceding proposition may be used with considerable effect. Ex, Prove that the centre of pressure of any triangle wholly immersed is tlio 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. On the positions of the Principal Axes of a system. 48. Prop. A straight line "being given it is required to find at what point in its length it is a principal aods of the system, and if any such point eocist to find the other two principal axes at that point. Take the straight line as axis of z, and any point in it as origin. Let C be the point at which it is a principal axis, and let Cx', Gy' be the other two principal axes. Let C0 = h, ^ = angle between Ca;' and Ox. Then x' = X cos 6-\-y sin ^ 2/' = — a; sin ^ + ^ cos 6 z' = z — h Hence 1.mxz = cos OXmxz + sin OXmyz ) _ ^ — h (cos OXmx + sin d'^my) ) ~ %rtiyz = — sin 6%mxz + cos OXmyz ]_ ^ — A (— sin 6%mx + cos &%my) J " sin 20 0) (2) Imx'y = T.m {y^ — a?) ^— + 'tmxy cos 20 = (3) The last equation shows that tan2g = /fry.. (4) 2,m [x —y) ^ ' 2F B~A' accordi?ig to the previous notation. PRINCIPAL AXES. ss y vertically down- 3 are xes las have the same may be used with wholly immersed is the s of the sides and each of a system. required to find at ' the system, and if xcipal axes at that ny point in it as ncipal axis, and let Then = (1) = (2) = (3) (4) The equations (1) and (2) must be satisfied by the same value h. Eliminating h we get y.mxz Sm?/ = Xmyz %nx as the con- [ition that the axis of z should be a principal axis at some point its length. Substituting in (1) we have %mx , _ ^myz __ zmxz Xmy (5) The equation (5) expresses the condition that the axis of z lould be a principal axis at some point in its length ; and le value of h gives the position of this point. The positions the other two principal axes may then be found by equa- |on (4). If Sma;« = and Xm^/z — 0, the equations (1) and (2) arc )th satisfied by h = 0. These are therofore the sufficient and jcessary conditions that the axis of z should be a principal axis the origin. Tf the system be a plane lamina and the axis of z be a normal the plane at any point, we have z = 0. Hence the conditions %mxz = and Xmyz = are satisfied. Therefore one of the l^incipal axes at any point of a lamina is a normal to the plane ftt that point. In the case of a surface of revolution bounded by planes per- pendicular to the axis, the axis is a principal axis at any point of its length. Again equation (4) enables us, when one principal axis is given, to find the other two. If ^ = a be the first value of 6, all tjje others are included in = a + n ; hence all these values give ^ly the same axes over again. 49. Since (4) does not contain h, it appears that if the axis of »,be a principal axis at more than one point, the principal axes at l^ose points are parallel. Again, in that case (5) must be satis- fed by mjre than one value of h. But since h enters only in the "fst power, this cannot be unless Xmx = 0, Swy = 0, Xmxz = 0, Xmyz = ; that the axis must pass through the centre of gravity and be a Kncipal axis at the origin, and therefore (since the origin is arbi- ■iry) a principal axis at every point in its length. If the principal axes at the centre of gravity be taken as the bs of x, y, z, (1) and (2) are satisfied for all values of h. Hence. a straight line be a principal axis at the centre of gravity, it is principal axis at every point in its length. R. D. 3 34 MOMENTS OF INERTIA. h > ^ 50. Let the system be projected on a plane perpendicular to the given straight line, so that the ratios of the elements of mass to each other are unaltered. The given straight line, which has been taken as the axis of z, cuts this plane in 0, and will be a principal axis of the projection at 0, because the projected system being a plane lamina, the conditions '^mxz = 0, z.myz = are both satisfied. Since z does not appear in equation (4), it follows that if the given straight line be a principal axis at some point G in its length, the other two principal axes at C will be parallel to the principal axes of the projected system at 0. These last may often be conveniently found by the next proposition. 51. Ex. 1. The principal axes of a right-angled triangle at the right angle are, one perpendicular to the plane and two others inclined to its sides at the angles j^tan~i -J — y^, where a and 6 are the sides of the triangle adjacent to the right angle. Take the formula tan2^ = -^--— , Art. 48, then hy Axt. 8, A = M ^, B = M ^ , F = M ah 12' Ex. 2, The principal axes of a quadrant of an ellipse at the centre are, one perpendicular to the plane and two others inclined to the principal diameters at the angles ^ tan"* - ^_^ , where a and h are the semi-axes of the ellipse. Ex. 3. The principal axes of a cube at any point P are, the straight line joining P to the centre of gravity of the cube, and any two straight lines at P perpendicular to PO, and perpendicular to each other. Ex. 4. Prove that the locus of a point P at which one of the principal axes is parallel to a given straight line is a rectangular hyperbola in the plane of which the centre of gravity of the body lies, and one of the asymptotes is parallel to the given straight line. But if the given straight line be parallel to one of the principal axes at the centre of gravity, the locus of P is that principal axis or the perpendicular principal plane. Take the origin at the centre of gravity, and one axis of co-ordinates parallel to the given straight line. Ex. 5. An edge of a tetrahedron will be a principal axis at some point in its length, only when it is perpendicular to the opposite edge. [Jullien.] Conversely if this condition be satisfied, tLd edge will be a principal axis at a 2 point C such that 0C=^ ON where N is the middle point of the edge and is the foot of the perpendicular distance between it and the opposite edge. 52. Prop. Giveti the positions of the principal axes Ox, Oy, Oz at the centre of gravity O, and the momen:,s of inertia about them, to find the positions of the principal axes at any point P in the plane ofxy, and the moments of inertia about those axes. PRINCIPAL AXES. 35 it to the right iinates parallel me point in its Let the mass of the body be M, and let A, B be the moments of inertia about the axes Ox, Oy, of which we shall suppose A the greater. Take two points 8 and // in the axis of greatest moment, one on each side of the origin so that 08=^ on -w M These points may be called the foci of inertia for that principal plane. Because these points ai'e in one of the principal axes at the centre of gravity, the principal axes at ^ and fT are parallel to the axes of co-ordinates, and the moments of inertia about those in the plane of xy are respectively A and B + M . Oti^ = A, and these being equal, any straight line through S or // in the plane of xy is a principal axis at that point, and the moment of inertia about it is equal to A. If P be any point in the plane of xy, then one of the principal axes at P will be perpendicular to the plane xy. For if ^, §- be the co-ordinates of P, the conditions that this line is a principal axis are Swi (a? — w) s = I which are obviously satisfied because the centre of gravity is the origin, and the principal axes the axes of co-ordinates. The other two principal axes may be found thus. If two straight lines meeting at a point P be such that the moments of inertia about them are equal, then provided they are in p princi- pal plane the principal axes at P bisect the angles between these two straight lines. For if with centre P we describe the momental ellipse, then the axes of this ellipse bisect the angles between any two equal radii vectores. Join SP and HP; the moments of inertia about 8P, HP are each equal to A. Hence, if PG and PT are the internal and 3-2 36 MOMENTS OF INERTIA. VV i external bisectors of the angle 8PH; PO, PT are the principal axes at P. If therefore with S and H as foci we describe anjj ellipse or hyperbola, the tangent and normal at any point are the pnncipal axes at that point. 63. Take any straight line MN through the origin, making an angle 6 with the axis oi x. Draw SM, IIN perpend icular.s on MN. The moment of inertia about it is = ^ cos'^ + ^sin"^ ==A-{A-B) sin' = A-M.{08ainey = A-M.8]\P. Through P draw PT parallel to MN, and let 8Y and HZ be the perpendiculars froni S and H on it. The moment of inertia about PT is then = moment about MN+ M. MY^ = A + M{MY- 8M) {MY+ 8M) = A + M.8Y.IIZ. In the same way it may be proved that the moment of inertia about r^ line PO passing between H and 8 is less than A by the mass into the product of the perpendiculars from 8 and H on PG. If therefore with S andU as foci we describe any ellipse or hyperbola, the moments of inertia about any tangent to either of these curves is constant. It follows from this that the moments of inertia about the • • 1 . D 1.7,. ..(8P±HPV prmcipal axes at P are equal to ij + iM ( ^ J . For if a and h be the axes of the ellipse we have a^ — b^ = 08^ A-B M and hence A^M.SY.HZ=A + Mb^ = B + Ma^ = B + M f8P + HP\' \ 2 and the hyperbola may be treated in a similar manner. 54. This reasoning may be extended to points lying in any given plane passing through the centre of gravity of the body. Let Ox, Oy be the axes in the given plane such that the product of inertia about them is zero (Art. 33). Construct the points 8 and // as before, so that 08^ and OIP are each equal to the difference of the moments of inertia about Ox and Oy divided by the mass. Draw Sy' a parallel through 8 to the axis of ?/, the PRINCIPAL AXES. 37 ia about the product of inertia about Sx, Si/' is equal to that about Ox, Oy together with the product of inertia of the whole mass collected at 0. Both these are zero, hence the section of the momental ellipsoid at 8 is a circle, and the moment of inertia about every straight line through S in the plane xOy is the same and e(jual to that about Ox. We can then sliow that the moments of inertia about PH and PS are equal ; so that PG, Pl\ the internal and external bisectors of the angle SPH are the principal dia- meters of the section of the momental ellipsoid at P by the given plane. And it also follows that the moments of inertia about the tangents to a conic whoso foci are S and H are the same. 55. Ex. 1. To ilnd tho foci of inertia of an elliptic area. The moments of iuertia about the major and minor axes are M -r and 21/ -, . Hence tho minor axis 4 4 iii the axis of greatest moment. Tho foci of inertia therefore lie in the minor axis at a distance from the centre = ^ ija^ - b', i.e. half tho distance of tho geometrical foci from the centre. Ex. 2. Two particles each of mass m are placed at the extremities of the minor axis of an elliptic area of mass M, Prove that the principal axes at ai?y point of the circumference of tho ellipse will bo the tangent and normal to the ellipse, pro- ., , m 5 e'^ Ex. 3. At the points which have been called foci of inertia two of the principal moments are equal. Show that it is not in general true that a point exists such that the moments of inertia about all axes through it are the same, and find the con- ditions that there may be such a point. Eefer the body to the principal axes at tLo centre of gravity. Let P be the point required, {x, y, z) its co-ordinates. Since the momental ellipsoid at P is to be a sphere, the products of inertia about all rectangular axes meeting at P are zero. Hence, by Art. 13, xy = 0, yz=0, zx=0. It follows that two of the three x, y, z must be zero, so that the point must be on one of the principal axes at the centre of gravity. Let this be called tho axis of z. Since the moments of inertia about three axes at P parallel to the co-ordinate axes are A + 3/ z", B + Mz"^ and C, we see that these cannot be equal unless A = B and each is less than C. There are then two points on the axis of unequal moment which are equimomental for all axes. [Poisson and Binet.] 56. Given the positions of the pnmipal axes at the centre of gravity and the moments of inertia about them, to find the positions of the principal axes*, and the principal moments at a>;y other point P. Let the body be referred to its principal axes at the centre of gravity 0, let A, B, C be its principal moments, the mass of the * Some of the following theorems were given by Sir William Thomson and Mr Towusend, in two articles which appeared at the same time in the Mathematical Journal, 1846. Their demonstrations are different from those given in this treatise. i 38 MOMENTS OF INERTIA. 1! I M m i* body hemrr taken as unity. Construct a quadric confocal with the ellipsoid of gyration, and let the squares of its semi-axes be a'-s A +\, h'^ B + \, 0^= G + \. Let us find the moment of inertia with regard to any tangent plane. Let (a, )9, 7) be the dii'cction angles of the perpendicular to any tangent plane. The moment of inertia, with regard to a parallel plane through 0, is ^ (Jcos''a + i?cos'/3+ Ccos'7). The moment of inertia, with regard to the tangent pl.ano, is formed by adding the square of the perpendicular distance be- tween the planes, viz. we get {A + X) cos'a + {B + \) cos'yS + (C+X) cos" 7, moment of inertia with re-] A + B + C gard to a tangent planej 2 B+G-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, the normals to these confocals are, by Art. 16, the principal axes at P. By Art. 5, Ex. 3, the principal axis of least moment is normal to the confocal ellipsoid and of greatest moment normal to the confocal hyperboloid of two sheets. 57. The moment of inertia with regard to the point P is, by Art. 14, s,— + OP^. Hence, by Art. 5, Ex. 3, the moments PRINCIPAL AXES. 39 )cncHcular to of inertia about the normals to the tliroo confocals through P whose parameters are \p \, \ arc respectively 0P'-\, 0P'-\, 0P'-\. 58. If wo describe any other confocal and draw a tangent cone to it whoso 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 this confocal diminish without limit, imtil it becomes a focal conic, then the priiicipal diameters of the systom 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. Wo know* that the normals to the ♦ These propositions are to bo found in books on Soliil Geometry, they may also bo proved as follows. I,et 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 prim i pal diameters of the Indicatrix. But all parallel sections of an ellipsoid are similiir and similarly situated, hence the principal diameters of the Indicatrix are parallel to the princi- pal 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 section of OP, then the point of contact T, OQ and OP will lie iu one plane when ( e •::(! ^1 40 MOMENTS OF INERTIA. other two confocals are tangents to the lines of curvature on the ellipsoid, and are also parallel to the principal diameters of che diametral section made by a plane parallel to the tangent plane at P. And if D^D^ be these princijtal semi-diameters, we know that \ =\ -Z>„ Hence, if through any point P we describe the quadric X' y A + \^ B^-\ C \ 1, 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 tjuadric, 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 2Z>, and ^D^, the principal moments at Pare OF'-K OP'-\ + D,\ OP'-X + D^\ Ex, If two bodies have the same centre tf gravity, the same principal axes at the centre of gravity and the differences of i,l.3ir principal moments equal, each to each, then these bodies have the same priuclpcl axes at all points. 60. The axes of co-ordmates being tJie principal axes at the centre of gravity it is required to express the condition that any given straight line may he a principal axis o/t some point in its length and to find that point. Let the equations to the given straight line be ^-f^y-9^z-h I m n (1). 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, /+ OP^ — sum of squares of semi-axes = -k — 4- i? — C The three prin- cipal moments are therefore I^ = I^= OF' + C, I^ = A+B — C, uud the axis of unequal moment is a tangent to the focal conic. The second case may be treated in the same way by using a confocal hyperboloid, we therefore have I^= 1^= 0P'+ B, PRINCIPAL AXES. 43 is the envelope oint P must I^ = A + C—B, and the axis of unequal moment is a tangent to the focal conic. 63. To find the curves on any confocal quadric at which a principal moment of inertia is equal to a given quantity I. Firstly. The moment of inertia about a normal to a confocal quadric is 0P* — \. If this be constant, we have OP constant, and therefore the required curve is the intersection of that quadric with any 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 same confocals are the same. 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 the intersection of the con- focals a', a" is I, = B+C-A + a:''-Va"\ 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 contact. On the quadric a draw a tangent PT making any angles (f> and T: — with the tangents to the lines of curvature at the z point of contact P. If T^, /, be the moments about the tangents to these lines of curvature, the moment of inertia about the tangent PT = /j cos'^ + ^a sin" (j> = B-^ C-A+ {a"^ + a') cos" +■ (a" + a") sui" . J-nt along a geodesic on the quadric a, a'^siu'c^ + a'^cos'^ is constant. Hence the moments of inertia about the tangents to any geodesic on the quadric are equal to each other. 64. Ex. 1. If a straight line touch any two confocals v.hose semi-major axes are a, a', the moment of inertia about it is ^ + C - il + o' + a'^. 44 MOMENTS OF INERTIA. Ex. 2. When a' body is referred to its principal axes at the centre of grafity, Bhovv how to find the coordinates of the point P at which the three principal moments are equal to three given quantities IJ^ly [Jullien's Problem.] The] elliptic co-ordinates of P are evidently a" = i (Jj + 13-/1 -5- C+^) &c. ; and the co-ordinates (x, y, z) may then be[found by Dr Salmon's formula, a:-a-'a' &o. {A-B){A-0 Ex. 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', when the confocals touch the intersec- tion of the planes; then a* + a'='=a^ + a''', and the product of inertia with regard to the two planes is aV* - a^'a'^'. 65. The locus of all those points at which one of the prin- cipal moments of inertia of the body is constant is called an equi- momental surface. To find the equation to such a surface we have only to put I^ constant, this gives \ = r^ — I. Substituting in the equation to the subsidiary quadric, the equation to the surface becomes cc y + = 1. Through any point P on an e(|ui-momcntal surface describe the 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 equi-momental 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 equi-momental 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 raomental surface and this quadric is the sphero-conic through P. The normal to the quadric, being the principal axis, has just been showr to be a tangent to the surftxce. Hence the tangent plane to the equi-momental surface, is tlic plane which contains the normal 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, (h'op a perpendicnhir from on PP, this is the radius vector OP, because PP is a tangent to the sphere. At P in the tangent plane draw a perpendicular to PP, this is the normal PQ to the (]uadric. From drop a pei-pendicular OQ on this normal, then Oi} is a normal to the tangent plane. Hence this construction, If 1^ he any point on an equi-momental surface whose para- meter is I and OQ a perpendicular from the centre on the tangent PRINCIPAL AXES. 45 plane, then PQ is the jmncipal axis at P about which the moment of inertia is the constant quantity I. The equi-momental becomes Fresnel's wa,ve surfaca when / is greater than the greatest principal moment of inert;a at the centre of gravity, 'llie general form of the surface is too Avell 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. Ax 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 t^iis tangent plane. Then PQ is a principal axis at P. Thu? iheic pre 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 sitimted 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 ^ is a circle, (2) The two sheets have a common tangent plane which touches the surface along the 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 equi-momental 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 a cubic to determine I whose roots ore the three principal moments. Thus let it be required to find the locus of all those points in a body at which any symmetrical function of the three prin- cipal moments is equal to a given quantity. We may express this symmetrical function in terms of the coefficients by the usual rules, and the equation to the locus is found. Ex. 1. If au equi-momental surface cut a quadric confocal with the ellipsoid of gyration at the centre of gravity, *lien the iutersections are a spbero conic and a line of curvature. But if tlip qualric l)c an rllipsoid, both these cannot be real. 46 MOMENTS OF INERTIA. V ,',; I' *■ U n I For if the surfEce cut the ellipsoid in both, let P be a point on the line of curvature, and 1" a point on the Bphero-conic, then by Art, 59, OP" + D^^ = OP"^, which is less than ^ + X. But OP'^ + Di« + D,* = 4 -f- B + C + 3\, therefore D^^>B + C+2\, which is >A + 2\. Hence i>j>than the greatest radius vector of the ellip- soid, which is impossible. Ex. 2. Find the Ic-us of all those points in a body at which (1) the sum cl the principal moments is equi." to a given quantity I. (2) the bnm of the products of the principal moments taken two and two together, is equal to P, (3) the product of the principal moments is equal to P, The results are (1) a sphere whose radius is a/ —ir\f " > ^^' 13. (2) the surface (x? + f+zy+{A + B + C){x^ + if + z^))_ , + Ax' + By^ + Cz' + AB + BC+CA i ' (3) the surface A'B'C - A 'yV - B'z V - C'x'y^ - 2a; Vz" = P, where A'— A +y^ + z'', with similar expressions for B*, C. CHAPTER II. d'alembert's principle, &c. 66. The principles, by which the motion of a single particle iinder 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 thioe laws of motion. It is 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, dt dt dt If we regard a rigid body as a, collection of material particles connected by invariable relations, we might write down the equa- tions of tha 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 be- tween any two are equal and opposite. Suppose there are n particles, then there will be 3w equations, and, as shown in any treatise on Statics, 3?i — 6 unknown reactions. To find the motion it will be necessary to eliminate these unknown quanti- ties. We may expect to find six resulting equations, and these will be shown, a little further on, to ^e sufficien+ to determine the motion of the body. When there are several rigid bodies which mutually act and re-act 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 making any assumption as to the nature of the mutual actions except the following, which may be regarded as a natural conse- quence of the laws of motion. The internal actions and reactions of any system of rigid bodies in rdotion are in eqidlihrium amongst themselves. 48 D ALEMBERT S PRINCIPLE. I' ) ' 1 67. To explain D'Alemhert'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 considered 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 par- ticle. Let m be the mass of the particle, {x, y, z) its co-ordinates referred to any flxed rectangular axes at the time t. The accele- d'x d^y , d?z rations of the particle are ^^, ^'^ and ~ . Let / be the resul- tant 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 mf be reversed, the three F, R, and mf are in equilibrium. We may apply the same reasoning to every particle of each body of the system. We thus have a group of forces similar to R, a group similar to F and a group similar to mf these three groups will form a system of forces in equilibrium. Now by D'Alembert's principle the group R will itself form a system of forces in equili- brium. Whence it follows that the group F will be in equilibrium with the group mf Hence If forces equal to the effective forces hut acting in exactly oppo- site directions were applied at each point of the system these woxdd he in equilibrium luith the impressed forces. 68. By this principle the solution of a dynamical problem is reduced to a problem in Statics. The process would be as fol- lows. We first choose some quantities by means of which the position of the system in space may be fixed. We then express the effective forces on each element in terms of these quantities. These reversed will be in equilibrium with the given impressed forces. Lastly, the equations of motion for each body may be formed, as is usually done in Statics, by resolving in three direc- tions and taking moments about three straight lines. (19. Before the publication of D'Alembert's principle a vast number of Dynami- cal problems had been solved. These may be found scattered through the early volumes of the Momoir.« of St Pctersburf?, Berlin and Pnris, in the works of John D ALEMBERT S PRINCIPLE. 4t» Bernoulli and the Opuscules of Euler. They require for the most part the dctormi- ration of the motions of several bodies with or without weight which push or pull each other by menus of threads or levers to which they are fastened or along which they can glide, and which having a ccrtai:. 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 tliat the centre of gravity of them all shall rise liigher 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 those, and it required the exercise of ingenuity and skill to detect the most suitable in each case. Such problems were for some tune a -sort of trial of strength among mathe- maticians. The Trait6 de Di/namique 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 616, or Whewell's version of the same in his History of the Inductive Sciences. D'Alembert uses the following words :—" Soient A, /?, C, &c. les corps qui com- posent le systeme, et supposons qu'on leur ait imprime les mouvemens a, b, c, 4c. qu'ils soient forces, h cause de leur action mntuelle, de changer dans les mouvemens n, b, c, &c. II est clair qu'on peut regarder le mouvement a imprime au corps A comme compost du mouvement a, qu'il a pris, et d'un autre mouvement a ; qu'on peut de meme regarder les mouvemens 6, c, , c, on leur eftt doun6 li-la-fois les doubles impulsions a, o; b, ^; &c. Or par la supposition les corps A, B, G, &c. ont pris d'eux-mgmes les mouvemens a, b, c, &c. done les mou- vemens a, /3, 7, &c. doivent etre tels qu'ils ne d6rangent rien dans les mouvemens a, b, c, &c. c'est-3,-dire que si les corps n'avoient rei;u que les mouvemens a, «, 7, &c. ces mouvemens auroient dft se detruire mutuellement, et le systeme demeurer en repos. De Ik xesulte le principe suivant pour trouver le mouvement de plusieurs corps qui agissent les uns sur les autres. Dccomposez les mouvemens o, 6, c&c. im- primes a chaque corps, chacun en deux autres a, a; b, /3; c, 7; etc. qui soient tels que si Ton n'eftt imprimS aux corps que les mouvemens a, b, c, &c, ils eussent pu conserver les mouvemens sans se nuire rCciproquement ; et que si on ne leur efit imprim6 que les mouvemens o, ^, 7, &c. le systeme ffit 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." 70. As an e-xample of D'Alembert's principle let us consider the following problem. A heavy body is capable of motion by two hinges about a hori- zontal axis, which axis is made to rotate with a uniform angular velocity w about a vertical axis intersecting it in a point 0. It is required to find the conditions that the body may be inclined at a constant angle to the vertical. Let the horizontal axis which is fixed in the body be taken as axis of y, and let two other axes also fixed in the body be taken origin 0. Let 6 be the angle as a set of rectangular axes with R. D. 4 r :- ' I i I h.;.. ou d'alembert's rniNCiPLE. the plane of yz makes with a vertical piano through Oj. object is to find the relation between d and a>. Our By hj'pothesis each particle P describes a horizontal circle whose centre C is in the vertical through 0. If r be the radius CP of this circle, and m the mass, the effective force on the particle is mai'i and is directed along the radius. When reversed this will act in the direction CP. The impressed forces on the body are its weight which may be supposed to act at the centre of gravity and the actions at the hinges. To avoid these last, we shall take moments about the axis Oj/. Then the moment of the reversed effective forces toge- ther with the moment of the weight will be zero. Let M be the mass of the body, x, jj, z the co-ordinates of the centre of gravity, | its distance from the vertical plane through Oy. The moment of the weight is Mg^. The resolved part of the effective force parallel to O.v has no moment about Oi/. The moment of the resolved part perpendicular to the vertical plane through Oy is mal^p if p be the distance of the particle from that plane. The equation of moments gives if CO = u Mg^ -{■ Zmoi'pu = 0. By jDrojecting the co-ordinates on CO and OP we have - ti = — X m\ 6 + z cos 6, p= X cos 6 + z ?,m. d, f = X cos 6 -{-z sin 0. Substituting we get Mg [x cos ^ + i sin 6) = o)" [\ sin 26%m (x^ — z') — cos ^Otmxz], When the shape and structure of the body are known, the integrals Sm (x^ — z^) and Sm scz can be found by the methods of DALEMBERTH PUINCIPLK. 51 the preceding chapter or by direct integration. Tliis equation will then give the required relation between 6 and w. It may be noticed that the only manner in which the form of the body enters into the equation is through its moments and products of inertia. If we change the body into any equi-mo- mental one, the equation comiecting and w will be imaltered. So far as this problem is concerned, a body may be said to bo given Dynamically when its mass, centre of gravity, principal axes, and principal moments at the centre of gravity are given. This remark will be found to be of general application. Ex. 1. If the body be pnshoil along the axis of y and made to rotate abont the vertical with tlio Barae angular velocity as before, show that uo effect is produced on the inclination of the body to the vertical. Ex. 2. If the body bo a heavy disc capable of turning about a horizontal axis Oy in its own plane, show that the piano of the disc will be vertical unless w' > "— whera h is the distance of the centre of gravity of the disc from Oij and k the radius of gyration about Oy. Ex. 3. If the body bo a circular disc capable of turning about a horizontal axis perpendicular to its plane and intersecting the disc iu its cu-cumference, show that if the tangent to the disc at the hinge make an argle with the vertical, the angular velocity w must be a./ — '- sin d Ex. 4. Two equal balls A and B ore attached to the extremities of two equal thin rods Aa, Bb. The ends a and h are attached by hinges to a fixed point O 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 spliere, I the length of a rod, r the radius of a sphere, h the depth of either centre below the hinge, then the length of the pendulum is l + r M{l-\-r) + \ml 71. To apply D'Alemherfs principle to obtain the equations of motion of a system of riyid bodies. Let (.r, y, z) be the co-ordinates of the particle m at the time t referred to any set of rectangular axes fixed in space. Then — t. V^i and -TT, will be the accelerations of the particle. Let dt' ' dt' * dt" X, Y, Z be the impressed accelerating forces on the same particle resolved parallel to the axes. By D'Alembert's principle the forces -^^-T^^ '"(^'-§)- -^-% 4—2 I' / ,) ) ;(;. 52 d'alembert's principle. together with similar forces on every particle will be in equi- librium. Hence by the principles of Statics we have the equation = zmX, ',m dC and two similar equations for y and z', these are obtained by resolving parallel to the axes. Also we have ;»i (^1^-40=^""*^-^^' and two similar equations for zx and xy ; these arc obtained by taking moments about the axes. These equations may be written in the more convenient forma d ^ dx ^ -.r (A). ,(B). d ^ dz V '7 J-, 2m -J- « 2»iZ dt dt In a precisely similar manner by taking the expressions for the accelerations in polar co-ordinates we should have obtained another but equivalent set of equations of motion. 72. Let us consider the meaning of these equations without reference to axes of co-ordinates. The effective forces are to be equivalent to the impressed forces. But as shown in Statics any system of forces and therefore each of these is equivalent to a single force and a single couple at some point taken as origin. These resultaiic forces and couples must therefore be equivalent, each to each. If we multiply the mass m of any particle P by its velocity v we have the momentum mv of the particle. Let us represent this in direction and magnitude by a straight line PP'. Then, just as in Statics, this momentum is equivalent to an equal and parallel linear momentum at which we may represent by OM, and a couple whose moment is mvp, where p is the perpendicular dis- tance between OM and PP'. The plane of this couple is the D ALEMBKRT S rUINCIPLfc:. 53 plane containing 0^F and PP", and it may as usual be represented in direction and magnitude by an axis ON perpendicular to its plane. This couple is sometimes called an angular momentum. Let 0^f', ON' bo the positions of these two lines after an interval of time dt. Then MM', NN' will represent in direction and magnitude the linear momentum and the angidar or couple momentum added on in the time dt. Hence the elective force on any particle vi is equivalent to a single linear eti'ective force acting at represented by - ,.-, fvnd a single effective couple NN' represented by — '^ . Let OV, on be two straight linos drawn through the origit. to represent in direction and magnitude the resultant linear momentum and resultant couple momentum of the whole system at any time t. Let OV, OH' be the positions of these lines at the time t-{-dt. Then OF is the resultant of the group 03f cor- responding to all the particles of the system, and V the resultant VV of the group OM'. Hence - ,.— represents the whole linear ef- fective force of the system at the time t. By similar reasoning HH' --rr represents the resultant effective couple of the system. Thus it appears that the points Faud //trace out two curves in space whose properties are analogous to those of the hodograph in Dynamics of a particle. From this reasoning it follows, that if Vx be the resolved part of the momentum of a system in the direction of any straight line Ox, and H„ the moment of the momentum about that straight line, then — , * and —jf are re- spectively the resolved part along, and the moment about that straight line, of the effective force of the whole system. Let us now refer the whole system to Cartesian co-ordinates flOC Cm ?J fl 2 as in Art, 71. We see that m -i- , m Sr . ''^ r are the resolved dt dt dt Hence OF is the parts of the momentum of the particle m. resultant of Sm dx %m dx\ dy , ^ dz ., f dy J . ^„^ -;- , and im-rr. Also m\x-'i —y . dt dt dt \ dt "^ dt J is the moment of the momentum of the particle m about the axis of z. Hence OH is the resultant of H^'ir dx'\ diJ' y — ^ \,m ( dz di/\ VTt-'dtJ' \,m dx 'di X dz\ dtl' Now D'Alembert's principle asserts that the whole effective forces of a system are together equivalent to the impressed forces. fc ! l-MSli: 54 d'alembert's principle. ri I il Hence whatever co-ordinates may be lised, if X and L be the resolved parts and moment of the impressed moving forces re- spectively along and about any fixed straight line which we shall call the axis of x, the equations of motion are dK dt ~ = X, dJI^ dt = L. The first of these corresponds to equations (A), the second to equations (B) of Art. 71. We may notice the following cases. (1) If no impressed forces act on the system, the two lines OV, OH are absolutely fixed in direction and magnitude through- out the motion. (2) If all the impressed forces pass through a fixed point, let this point be chosen as the origin, then though OF may be variable, OZ/is fixed in position and magnitude. (3) If all the impressed forces be equivalent to a system of couples, then though OH may be variable, V is fixed in position and magnitude*. 73. The equations of motion of Art. 71 are the general equa- tions of motion of any dynamical system. They are, however, extremely inconvenient in their present form. When the system considered is a rigid body and not merely a finite number of separate particles, the 2's are all '^icfinite integrals. There are also an infinite number of xb, ?/'s and ^'s all connected together by an infinite number of geometrical equations. It will be neces- sary, as suggested in Art. 68, to find some quantities which may determine the position of the body in space and express the effective forces in terms of these quantities. These are called the co-ordinates of the bodi/f. It is most important in theoretical dynamics to choose these co-ordinates properly. They should be (1) such that a knowledge of them in terms of the time determines the motion of the body in a convenient manner, and (2) such that the dynamical equations when expressed in terms of them may be as little complicated as possible. 74. Let us first enquire how many co-ordinates are necessary to fix the position of a body. The position of a body in spa?e is given when we know the co-ordinates of some point in it and the angles which two straight lines fixed in the body make with the axes of co-ordinates. There • In a memoir on the differential coefficients and determinants of lines, Mr Cohen lias discussed some of the properties of those resultant lines. rhU. Trans. 1862. t Sir W. Hamilton uses the phrase "marks of position," but subsequent writers have adopted the term co-ordinates. Sec Caylry's licport to the Brit. Assoc, 1857. D ALEMBERTS PRINCIPLE. 55 re necessary are three geometrical relations existing between these six angles, so that the position of a body may be made to depend on sia; independent variables, viz. three co-ordinates and three angles. These might be taken as the co-ordinates of the body. By the term "co-ordinates of a body" is meant any quantities which de- termine the position of the body in space. It is evident that we may express the co-ordinates (x, i/, z) of any particle m of a body in terms of the co-ordinates of that body and quantities which are known and remain constant during the motion. First, let us suppose the system to consist only of a single body, then if we substitute those expressions for x, y, z in the equations (A) and (B) of Art. 71, we shall have six equations to determine the six co-ordinates of the body in terms of the time. Thus the motion will be found. If the system consist of several bodies, we shall, by considering each separately, have six equations for each body. If there be any unknown reactions be- tween the bodies, these will ])e included in A'^ F, Z. For each reaction there will bo a corresponding geometrical relation con- necting the motion of those bodies. Thus on the Avhole we shall have sufficient equations to determine the motion of the system. When the motion is in two dimensions these six co-ordinates become three. These ai-e the tv/o co-ordinates of the fixed point in the body, and the angle some straight line fixed in the body *makes with a straight line fixed in space. 75. Let us next consider how the equations of motion formed by resolution can be simplified by a proper choice of co-ordinates. We must find tlio resolved part of the momentum and the re- solved part of the effective forces of a system in any direction. Let the given direction be taken as the axis of x. Let {x, y, z) he the co-ordinates of any particle whose mass is m. The re- dx solved part of its momentum in the given direction is tn -jr . Hence the resolved part of the momentum of the whole system is dx - 2)H -y- • Let (x, y, z) be the co-ordinates of the centre of gravity of the system and M the whole mass. Then Mx == ^mx ; •• '^^di=^"'dt' Hence the resolved pai't of the momentum of a system in any direction is equal to the whole mass multiplied into the resolved part of the velocity of the centre of gravity. That is, the linear momentum of a system is the same as if the whole mass tuere collected into its centre of gravity. 'nf ,'m.m s 56 d'alembert's principle. !.|; fM m 1^ In the same way, the resolved part of the effective forces of a system in any direction is eqwal to the whole mass multiplied into the resolved part of the acceleration of the centre of gravity. It appears from this proposition that it will be convenient to take the co-ordinates of the centre of gravity of each rigid body in the system as three of the co-ord'inates of that body. We can then express in a simple form the resolved part of the effective forces in any direction. 76. Lastly, let us consider how the equations of motion formdti by taking moments can be simplified by a proper choice of the three remaiining co-ordinates. We must find the moment of the momentum and the moment of the effective forces about any btraifjht line. Let the given straight line be taken as the axis of x, then following the same notation as before, the moment of the mo- mentum about the axis of x is S"'(4:-i) If, then, wo Now this is an expression of the second degree, substitute y = y-Vy, ^ = s -f- s', we get by Art. 14 wlrere M is the mass of the system or body under consideration. The second term of this expression is the moment about the axis of X of the momentum of a mass M moving witli the centre of gravity. The first term' is the moment about a straight line para'llel to the axis of x, not of the actual momenta of all the several parti- cles but of their momenta relatively to that of the centre of gravity. In the case of any particular body it therefore depends only on the motion of the body relatively to its centre of gravity. In finding its value we shall suppose the centre of gravity reduced to rest by applying to every particle of the system a velocity equal and oppo- site to that of the centre of gravity. Hence Ave infer that The moment of the momentum of a system about any straight line is equal to the moment of the- momentum of the ivhole mass supposed collected at its centre of gravity and moving with it, together with the moment of the momentum of the system relaVive to Us centre of gravity about a straight line drawn parallel to the given straight line through the centre of gravity. In the same way, this proposition will bo also true if for the "momentum" of the svstom avc substitute " efteetive force." j! I Sssstfs D ALEMBERT S PRINCIPLE. B7 By taking the axis Ox through the centre of gravity, we see that the moment of the relative momenta about any straight line through the centre of gravity is equal to that of the actual momenta. 77. It appears from the preceding article that it will be con- venient to refer the angular motion of a body to a system of co-ordinate axes meeting at the centre of gravity. A general expression for the moment of the effective forces about any straight line through the centre of gravity cannot be conveniently investi- gated at this stage. Different expressions will be found advanta- geous under different circumstances. There are three cases to which attention should be particularly directed: (1) when the body is turning about an axis fixed in the body and fixed in space ; (2) when the motion is in two dimensions, and (3) Euler's expression when the body is turning about a fixed point. These will be found at the beginnings of the third and fourth chapters and in the fifth chapter respectively. 78. The quantity Sm [^-^ dy dx i.—y-r.j expresses the moment of the momentum about the axis of z. It is then called the angtilar momentum of the system about the axis of z. There is anothei' interpretation which can be given t© it. If we transform to polar co-ordinates, we have dy dx dd dt ^ dt dt Now \r^d6 is the elementary area described round the origin in the time dt by the projection of the particle on the plane of xy. If twice this polar area be multiplied by the mass of the particle, it is called the area conserved by the particle in the time dt round the axis of z. Hence .^ / dy div\ ^'''V'dt-yit) is called the area conserved by the system' in a unit of time, or more simply the area conserved. 79. We may now enunciate two important propositions, which follow at once from' the preceding results. It will, however, be more useful to deduce them' from first principles. (1) The motion of the centre of gravity of a system acted on hy any forces is the same as if all the mass were collected at the centre of gravity and all' the forces were applied at that 'point parallel to their former directions. (2) The motion of a body, acted on by any forces, about its centre of gravity is the same as if the centre of gravity ivere fixed and the same forces acted on the body. 1 !^ Its '^/ ^ ! 1 15 f H flp^ '! n I ! 58 DALEMBERTS PRINCIPLE. Tak' '^g any one of the equations (A) we have at If X, y, z bo the co-ordinates of the centre of gravity, then xXtn = Hmx ; dt S?/i = ^mX, and the other equations may be treated in a similar manner. But these are the equations which give the motion of a mass 2?^ acterl on by forces XmX, &c. Hence the first principle is proved. Taking any one of equations (B) we have tm [x-^r -y -j^^j = Sm {xY - yX). Let x=x + x' , y =y jf.y^ 2=2 + z', then by Art. 14 this equa- tion becomes "tm (. 2,/ „' A' ^'^-i/-:u^\ + , d'x It X d^y - d^x df y -^) tm = Xm{x Y- yX). 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, ^ = 0, but ~ , ~ 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 equa- tions 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 th'^ 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. 81. By the first principle the problem of finding the motion of the centre of gravity of a system, however complex the system d'alembeut's principle. 59 may be, is reduced to the problem of finding the motion of a single particle. By the second the problem of finding the angular motion of a free body in space is reduced to that of determining the motion of that body about a fixed point. In using the first principle it should be noticed that the im- pressed 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 col- lected at the centre of gravity and it were then acted on by the same centr^-l 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. From this proposition we may infer 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 mo-tion ot that point. 83. We may now collect together for reference the results of the preceding articles. Let u, V, w be the velocities of the centre of gravity of any rigid body of mass M resolved parallel to any tliree fixed rect- angular axes, let h^, h^, k^ be the three moments of tlic momentum relative to the centre of gravity about three recLangular axes fixed in direction and meeting at the centre of gravity. Then the effective forces of the body are equivalent to the three effective forces M-r- y M -j-., ^^-r acting at the centre of gravity parallel to the directions into which the velocities have been resolved, ajid to the three effective couples —i^ , ~ , -~ about the axes- meet- ing at the centre of gravity about which the moments were taken. The effective forces of all the other bodies of the system may be expressed in a similar manner. M ! f i»: I 60 d'alembert's principle. Then all these effective forces and couples, being reversed, will V-^ in equilibrium with the impressed forces. The equations of equilibrium may then be found by resolving in such directions and taking moments cbout such straight lines as may be most con- venient. Instead of reversing the effective forces it is usually found more con\euient to write the impressed and effective forces on opposite sides of the equations. Application of UAlemhert's Principle to impulsive forces. 84. If a force F act on a particle of mass m always in the same direction,, the equation of motion is where v is the velocity of the particle at the time t. Let T be the interval during which the force acts, and let v, v' be the velocities at the beginning and end of that interval. Then m (u'-u)=J Fdt. Now suppose the force F to increase without limit while the interval T decreases without limit. Then the integral may have a finite limit. Let this limit be P. Then the equation becomes m {v —v) = P.- The velocity in the interval T has increased or decreased from V to V. Supposing the velocity to have remained finite, let V be its greatest value during this interval. Then the space described is less than VT. But in the limit this vanishes. Hence the particle has not moved during the action of the force F. It has not had time to move but its velocity is suddenly changed from V to v. "We may consider that a proper measure has been found for a force when from that measure we can deduce all the effects of the force. In the case of finite forces we have to determine both the change of place and the change in the velocity of the particle. It is therefore necessary to divide the whole time of action into elementary times and determine the effect of the force during each of these. But in the case of infinite forces which act for an indefinitely short time, the change of place is zero, and the change , of velocity is the only element to be determined. It is therefore more convenient to collect the whole force expended into one measure. Such a force is called an impulse. It may be defined as the limit of a force which is infinitely great, but acts only during an infinitely short time. There are of course no such ' '^haa^M^sti^'^'-it d'alembert's principle. 61 forces in nature, but there are forces which are very great, and act only during a very short time. The blow of a hammer is a force of this kind. They may be treated as if they were im- pulses, and the results will be more or less correct according to the magnitude of the force and the shortness of the time of action. They may also be treated as if they were finite forces, and the displacement of the body during the time of action of the force may be found. The quantity P may be taken as the measure of the force. An impulsive force is measured by the whole momentum gener- ated by the impulse. 85. In deter. idning the effect of an impulse on a "body, the effect of all finite forces which act on the body at the same time may he omitted. For let a finite force / act on a body at the same time as an impulsive force F. Then as before we have rX rT Fdt fdt . m m m m But in the limit fT vanishes. Similarly the force / may be omitted in the equation of moments. 86. To obtain the general equations of motion of a system acted on by any number of impulses at once. Let u, V, w, u\ V, vo be the velocities of a particle of mass m parallel to the axes just before and just after the action of the impulses. Let X\ Y', Z' be the resolved parts of the impulse on m parallel to the axes. Taking the same notation as before, we have the equation or integrating tm{u'-'u) = tm\ Xdt = XX' (1). Jo Similarly we have the equations 2w (v' - v) =XY' (2), Xm{w'-w) = tZ' (3). Again the equation 2m (x -^ - y~j = tm (x Y- yX) I I t)2 d'alemrert's principle. becomes on integration or taken between limits, Xm[x{v'-v)-i/{u'-u)] = %{xY'-yX') (4), and the other two equations become Xm\i/{iv'—iv) — z {v —v)\ =S iyZ' — zY') (5), Xm[z{u'-u)-x{w-iv)]='Z{zX'-xZ') (6). In all the followins: investijjations it will be found convenient to use accen+'^d letters to denote the tates ' mocion after impact ■which correspond to those denoted Ir < m- . IjC letters unaccented before the action of the impulse. Sii. ; !;•. Ganges in direction and magnitude of the velocities of tl - ixt^l particles of the bodies are the only objects of investigati-m, it v i ' be more conve- nient to express the equations of motion in terms of these veloci- ties, and to avoid the introduction of such symbols ^^ -jii ~^> -f- 87. In applying D'Alembert's Principle to impulsive forces the only change which must be made is in the mode of measuring the effective foices. If (u, v, w), {u\ v, w') be the resolved parts of the velocity of any particle, just before and just after the impulse, and if 7nbe its mass, the effective forces will be measured by m{u'—^l), m (v — v), and m [lo' — V)). The quantity mf ai Art. 67 is to be regarded as the measure of the impulsive force which, if the parti- cle were separated from the rest of the body, would produce these changes of momentum. In this caae, if we follow the notation of Arts. 75 and 76, the resolved part of the effective force in the direction of the axis of z dz is the difference of the values of Sm -r just before and just after the action of the impulses, and this is the same as the difference dz of the values of M -j- at the same instants. In the same way the moment of the effective forces about the axis of z will be the difference of the values of .1^ / dii dx^ just before and just , ♦'ter the action of the impulses. We may therefore extend the general proposition of Art. 83 to impulsive forces in the following manner. Let (u, V, tu), {u', v', w) be the velocities of the centre of gravity of any rigid body of mans il/ just before and just after the action D ALEMBERT S PRINCIPLE. 03 W. (5), (6). ;onvctiient tcr impact inaccented I direction ;les of the lore conve- liese veloci- x dy dz^ Tt'tt' dt' le forces the jasuring the parts of the impulse, and ^y m{it—u), 67 is to be if the parti- •oduce these and 76, the the axis of z id just after be difference lame way the will be the of Art. 83 to Ure of gravity ter the action of the impulses resolved parallel to any three fixed rectangular axes. Let (/i,, h,^, h^, {h', //./, h^') be the three moments of the momentum relative to tlic centre of gravity about three rect- angular axes fixed in direction and meeting at the centre of gravity, the moments being taken just before and just after the impulses. Then the effective forces of the body are equivalent to the three effective forces M{u —%C), M(v' — v), M{w' — w) acting at the centre of gravity parallel to the rectangular axes together with the three effective couples (/*,'— A,), {k^ — Ik^), {hj — / . dt dd dt every particle, and equal to ^ . Hence the moment of the mo- d0 menta of all the particles of the body about the axis is "Zmr^ ->- , i.e. the moment of inertia of the body about the axis multiplied into the angular velocity, d?<^ The accelerations of the particle m are r de and-r(^ perpendicular to, and along the directions in which r is measured, J'lA the moment of the moving forces of m about the axis is mr^ -X , hence the moment of the moving forces of all the particles of the body about the axis is 2 [mr^ d^ dt' By the same reasoning as i ! ii R. D. i! w- i GO MOTION ABOUT A FIXED AXIS. before this is ('(jual to Xiiir'* , .^ , 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- ducing the unknown reactions at the axis, let us take moments about the axis. First, let the forces be 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, G)'. 2wr'— ft) . S;«r' = L, where L is the moment of the impressed forces about the axis ; moment of forces about axis O) — ft) moment of inertia about axis * This equation will determine the change in the angular velo- city produced by the action of the forces. Secondly, let the forces be finite. Then taking moments about the axis, we have d^d ^ J J d'd de moment of forces about axis de (It moment of inertia about axis ' This equation when integrated will give the values of and at any time. Two undetermined constants will make their appearance in the course of the solution. These are to be deter- dff mined from the given initial values of 6 and -j- . Thus the whole motion can be found. 90. It appears from this proposition that the motion of a rigid body about a fixed axis depends on ( 1) the moment of the forces about that axis and (2) the moment of inertia of the body about the axis. Let Mk^ be this moment of inertia, so that k is the radius of gyration of the body. Then if the whole mass of the body were collected into a particle and attached to the fixed axis by a rod without inertia, whose length is the radius of gyra- tion k, and if this system be acted on by forces having the same moment as before, and be set in motion with the same initial i' ) mmm0amm» tia of the fixed axis a reversed void intro- ) moments he angular ;tion of the the axis ; mgular velo- )ments about ues of 6 and make their to be deter- lus the whole motion of a .loment of the a of the body ,, so that k is 'hole mass of . to the fixed adius of gyra- ting the same same GENERAL PRINCIPLES. e7 dt values of and '-^ , then the whole subsequent angular or gyra- tory motion of the rod will be the same as that of the body. We may say briefly, that a body turning about a fixed axis is dyna- mically given, when we know its mass and radius of gyration. 91. Ex. A prrfecthj rough circular horizontal hoard iit capable, of revolving freely round a vertical axis through it» centre. A vian ichose weight is equal to that of the hoard walks on and round it at the edgt : when he has completed the circuit what will be his position in space f Let a bo the radius of the board, Mk' its moment of inertia about the vertical axis. Let w be the angular velocity of tlip board, u' that of tlie man about the vortical axis at any time. And let F be the action between the feet of the man and tlie board. The equation of motion of the board is by Art. 89, Mk^'^=-Fa at (1). The equation of motion of the man is by Art. 79, du' „ Ma dt .(2). Eliminating F and integrating, we get the constant being zero, because the man and the board start from rest. Let 0, e' be the angles described by the board and man round tlie vertical axis. Then w=— , w'=— , and h'e + a^' = 0. Hence, when e'-tf = 2ir, we have ^' = 7^^— -2t. at at k^ + a^ Tills gives the angle in space described by the man. It k^=- we have «'=„ t. Let V be the mean relative velocity with which the man walks along the board. Then w'-w=- of the board. «= - Va k'+a;' 2 V o a This gives the mean angular velocity On the Pendulum. 92. A body moves about a fixed horizontal axis acted on by rjravity only, to determine the motion. Take the vertical plane through the xis as the plane of refer- ence, and the plane through the axis and the centre of gravity as the plane fixed in the body. Then the equation of motion is d^d _ moment of forces , . df moment of inertia Mgh sin « r ii ' ii 08 MOTION ABOUT A FIXED AXIS. wliere A is the distance of the centre of gravity from the axis and Mk^ is the moment of inertia of the body about an axis through the centre of gravity parallel to the fixed axis. Hence (2). The equation (2) cannot be integrated in finite terms, but if the oscillations be small, we may reject the cubes and higher powers of 6 and the equation will become Hence the time of a complete oscillation is Itt sJ — r— • If h and k be measured in feet and g = 3218, this formula gives the time in seconds. The equation of motion of a particle of any mass suspended by a string / is 'J^f+f.sin^ = (3), which may be deduced from equation (2) by putting k = and // = I. Hence the angular motions of the string and the body under the same initial conditions will be identical if 1 = J^ + h' h (4). This length is called the lenr/th of the simple equivaleiH pendulum. Through G, the centre ot gravity of the body, draw a perpen- dicular to the axis of revolution cutting it in C Then C is called the centre of suspension. Produce GG to so that CO = l. Then is called the centre of oscillation. If the whole mass of the body were collected at the centre of oscillation and suspended by a thread to the centre of suspension, its angular motion and time of oscillation vvould be the same as that of the body under the same initial .cumstances. The equation (4) may be put under another form. Since CG = h and OG = 1 —h, we have (r (7. 6rO= (rad.)" of gyration about (7, " CG . CO = (rad.)^ of gyration about C, OG. 0C== (rad.)' of gyration about 0. Any of these equations show that if be made the centre of suspension, the axis being parallel to the axis about which k was taken, then C will be the centre of oscillation. Thus the centres THE PENDULUM. 69 axis and s through (2). ms, but, if nd higher 1}^. If gh a gives the suspended (3). ^ fc = and 'd the body (4). equivalent w a perpen- [1 is called = 1 Then mass of the uspended by :>n and time y under the nrm. Since [he centre of (which k was Is the centres of oscillation and suspension are convertible and the time of oscilla- tion about each is the same. If the time of oscillation be given, I is given and the equa- tion (4) will give two values of h. Let these values be h^, h^. Let two cylinders bo described \\\i\\ that straight line as axis about which the radius of gyration k was taken, and let the radii of these cylinders be /«,, K^ Then the times of oscillation of the body about any generating lines of these cylinders are the //* same, and are approximately equal to 27rA/ - . With the same axis describe a third cylinder whoso radius IS k. T\\Qnl=1k + {h -kY h , hence I is always greater than 2k, and decreases continually as h decreases and approaches the value h. Thus the length of the equivalent pendulum continually de- creases as the axis of suspension approaches from without to the circumference of this third cylinder. When the axis of suspension is a generating line of the cylinder the length of the equivalent pendulum is 2,k. When the axis of suspension is within the cylinder and approaching the centre of gravity the length of the equivalent pendulum continually increases and becomes infinite when the axis passes through the centre of gravity. The time of oscillation is therefore least when the axis is a generating line of the circular cylinder whose radius is k. But the time about the axis thus found is not an absolute minimum. It is a minimum for all axes drawn parallel to a given straight line in ihe body. To find the axis about which the time is absolutely a minimum we must find the axis about which A; is a minimum. Now it is proved in Art. 23, that of all axes through G tlie * Til 9 position of the centre of oscillation of a body was first correctly deter- mined by Huygbens in his Ilorologiim OscillatoHum jiiiblished at Paris in 1073. The most important of the theorems given in the text were discovered by him. As D'Alembert's principle was not known at that time, Hnygliens had to discover some principle for himself. The liypothesis was, that when several weights are put in motion by the force of gravity, in whatever manner they act on each other tlu^r centre of gravity cannot be made to mount to a height greater than that from which it had descended. Huygbens considers that he assumes here only that a heavy body cannot of itself move upwards. The next step in the argument was, that at any instant the velocities of the particles are such that, if thoy were separated from each other and properly guided, the centre of gravity could be made to mount to a second position as high as its first position. For if not, consider the imrticlos to start from their last positions, to describe the same paths reversed, and then again to be Jbined together into a pendulum ; the centre of gravity would rise to its first position ; but if this be higher than the second position, the liypothesis would bo contradicted. This principle gives the same equation which the modern principltj of Via Viva would give, and the rest of the solution is not of much intercut. i! 1 70 MOTION ABOUT A FIXED AXIS. axis about which the moment of inertia is least or greatest is one of the principal axes. Hence the axis about which the time of oscillation is a minimum is parallel to that principal axis through O about which the moment of inertia is least. And if Mk^ be the moment of inertia about that axis, the axis of suspension is at a distance k measured in any direction from the principal axis. ; .■ : 93. Ex. 1. Find the time of tlie small oscillations of a cube (1) when one side is fixed, (2) when a diagonal of one of its faces is fixed; the axis in both cases being horizontal. Result. If 2a be a side of the cube, the length of the simple ecjuivalent pendu- lum is in the first case ^ - a, and in the second case - a. Ex. 2. An elliptic lamina i" such that when it swings about one latus rectum as a horizontal axis, the other latus rectum passes thi'ough the centre of oscillation, prove that the eccentricity is J. Ex. 3. A circular arc oscillates about an axis through its middle point perpen- diculai" to the plane of the arc. Prove that the length of the simple equivalent pendulum is independent of the length of the arc, and is equal to twice the radius. Ex. 4. The density of a rod varies as the distance from one end, find the axis perjoendicular to it about which tlie time of oscillation is a minimum. Jiesult. The axis passes through either of the two points whose distance from the centre of gravity is -^r a, where a is the length of the rod. Ex. 5. Find what axis in the area of an ellipse must be fixed that the time of a small oscillation may be a minimum. Result, The axis must be parallel to the major axis, and bisect the semi-minor axis. Ex. 6. A uniform stick hangs freely by one end, tlie other end being close to the ground. An angular velocity in a vertical plane is then communicated to the stick, and when it has risen through an angle of 90", the end by which it was hanging is loosed. What must be the initial angular velocity so that on falling to the ground it may pitch in an upright position ? Jiesult. The reriuired angular velocity w is given by 2a ^ 2a j(2n + l) 2\ (2^^+1)^+1 where n is any integer, and 2a is the length of the rod. Ex. 7. Two bodies can move freely and independently under the action of gravity about the same horizontal axis ; their masses are m, m', and the distances of their centres of gravity from the axis are h, h'. If the lengths of their simple equi- valent pendrJums be L, L', prove that when fastened together the length of tli o . , . J , ... , mhL + m'h'L' equivalent pendulum will be - ,- .--- • mn + mh.' THE PENDULUM. 71 Ex. 8. Wlien it is required to regulate a clock without stopping the pendulum, it is usual to add or subtract some small weight from a platform Attached to the pendulum. Show that in order to make a given alteration in the going of the clock by tlie addition of the least possible weight, the platform must be placed at a dis- tance from the point of suspension equal to half the sir. pie equivalent pendulum. Show also that a sliglit error in the position of the platform will not affect tho weight required to be added. Ex. 9. A circular table oentre is supported by three legs AA\ BB', C(7' which rest on a perfeunly rough horizontal floor, and a heavy particle P is placed on tho table. Suddenly one leg CC gives way, show that tho tabic and the particle will immediately separate if pc be greater than k^ ; where p and c arc the distances of P and respectively from the MweAB joining the tops of the legs, and k is the radius of gyration of the table and legs about the line A'B' joining the points where tho legs rest on the floor. The condition of separation is that the initial normal acceleration of the point of the table at P should be greater than the normal acceleration of the particle itself. Ex. 10. A string without weight is placed round a fixed ellipse whose plane ia vertical, and the two ends are fastened together. The length of the string is greater than the perimeter of the ellipse. A heavy particle can slide freely on the string and performs small oscillations under the action of gravity. Prove that the simple equivalent pendulum is the radius of curvature of the confocal ellipse passing through the position of equilibrium of the particle. 94. In a clock whicli is regulated by a pendulum, it is neces- sary that tlie time of oscillation should be invariable. As all substances expand and contract with every alteration of tempera- ture, it is clear that the distance of the centre of gravity of the pendulum from the axis and the moment of inertia about that axis will be continually altering. The length of the simple equi- valent pendulum does not however depend on either of these elements simply, but on their ratio. If then we can construct a pendulum such that the expansion or contraction of its different parts does not alter this ratio, the time of oscillation will be un- affected by any changes of temperature. For an account of the various methods of accomplishing this which have been suggested, we refer the reader to any treatise* on clocks. We shall here only notice for the sake of illustration one simple construction, whicli has been generally used. It was invented by George Graham about the year 1715. Some heavy fluid, such as mercury, is enclosed in a cast-iron cylindrical jar into the top of whicli an iron rod is screwed. This rod is then suspended in tho usual manner from a fixed point. The downward expansion of the iron on any increase of temperature tends to lower the centre of oscillation, but tho upward ex- pansion of the mercui'y tends on the contrary to raise it. It is re(iuirod to doter- * Rcid oil Clocks; Denison's treatise on Clocls and Clockmnldnn in Wcalc's Si'rics, 1W)7; Caiitain Kator's treatise on .l/rr/jft;;/''? in Laidnor's Ciiflopiritiuy 1880. i I t ■; 72 MOTION ABOUT A FIXED AXIS. mine the condition that the position of the centre of oscillation may on tlie whole be unaltered. Let Mk'^ be the moment o^ inci-tia of the iron jar and rod about the axis of sus- pen iioE, c the distiUiofl ii; tlaalr common centre of gravity from that axis. Let I be the len<3th of the iKniinam from the point of suspension to the bottom of the jar, a the internal ratlius of the jar. Let nM be the mass of the mercury, k the height it occupies in the jar. The moment of inertia of the cylinder of mercury about a straight line through its centre of gravity perpendicular to its axis is by Art. 18, Ex. 8, nM ( To + t ) • Hence the moment of inertia of the whole body about the axis of suspension is Mn\^ + aud the moment of the whole mass collected at its centre of gravity is Mn(l-^+Mc. The length L of the simple equivalent pendulum is the ratio of these two, aud on reduction we have X= n (^-lh + P + :->" ^¥T) (!)• + c Let the linear expansion of the substance which forms the rod and jar be denoted by a and that of mercury by /3 for each degree of the thermometer. If the thermo- meter used be Fahrenheit's, we have a =-0000065668, /3 = -00003336, accordiug to some experiments of Dulong and Petit. Thus we see that o and /3 are so small that their squares may be neglected. In calculating the height of the mercury it must be remembered that the jar expands laterally, and thus the relative vertical expan- sion of the mercury is 3j8 - 2a, which we shall represent by y. If then the temperature of every part be increased t", we have a, I, Jc, c, all increased in the ratio l + a< : 1, while h is increased in the ratio 1-H7i; : 1. Sir>(.j L is to be unaltered, we have rdL dL \da dL , dL, dL dk dc )' But Lis & homogeneous function of one dimension, hence dL dL, dL, dL dL ^ , da, dl dk dc dh The condition becomes therefore by substitution a 0-7 hdL Ldh' Let A, Bhe the n" "aerator and denominator of the expression for L given by equation (1). Then 'ikng the logarithmic differential I dh ~ .i' ^ B B V""T~ + 2'^* THE pry;jULUM. 7Z he whole il'-Dii the required condii . . is da of 8UH- Let I be )f the jar, iry, h the lie through ion is two, aud on (1). ai be denoted ; the thermo- accordiug to 80 small that •cnry it must jrtical expau- |«, I, h c, all 1. Si^HJ L Lr L given by 3(/3-«) > This calculation is of more theoretical than practical importance, for the nume- rical values of a and /3 depend a good deal on the purity of the metals and on the mode in which they have been worked. The adjustment must therefore be finally made by experimt 'it. In the iuvestiijation we have supposed a and /3 to be absolutely constant, but this is only a itv near approximation. Thus a change of 80" Fah. would alter /3 by less than a t ftieth of its value. When the adjustment is made the compensation is not strictly correct, for the iron jar and mercury have been suppo.ed to be of uniform temperatiu'e. Now the different materials of which the pendulum is composed absorb heat at different rates and therefore while the temperature is changing there will be some slight error in the clock. 95. Another cause of error in a clock pendulum is the buoy- ancy of the air. This produces an upward force acting at the centre of gravity of the volume of the pendulum equal to the weight of the air displaced. A very slight modification of the fundamental investigation in Art. 92 will enable us to take this into nccoiint. Let V be the volume of the pendulum, D the density of the air ; I\, h^, the distances of the centres of gravity of the ma;-s and volume respectively from the axis of suspension, Mkj' the moment of inertia of the mass about the axis of suspen- sion. Let us also suppose the pendulum to be symmetrical about a plane through the axis and either centre of gravity, equation of motion is then J. lie iric' -^"^ = - 3Ii/h, sin 6 + VDg\ sin 6 (!)• By the same reasoning as before we infer that if I be the length of the equivalent pendulum M' , , VD (2). The density of the air is continually changing, the changes being indicated by variations in the height of the barometer. Let h be VD the value of h^ — h^ —j.j for any standard density D. Suppose the actual density to be Z> + 8D and let l+ol be the corresponding length of the seconds pendulum, then we have by differentiation Ic'Bl , V8D , ,, , -rp = Aj "Tf" » ^^^ therefore I " h M D ' :i /! ; 11 ■ t ' ; t ! li i \ l\ 7^ MOTION ABOUT A FIXED AXIS. If T be the time of oscillation, we have 8T im V (J and .• T 2 1' 96. Ex. 1. II the centres of gravity of the mass and vohime were very nearly coincident and th., weight of the air displaced were T-^^nr <*^ ^^ weight of the penduhira, show that a rise of one inch in the barometer would canae «n error in the seconds pendulum of nearly 2 sec. per day. Ex. 2. If we affix to the peudiUum rod produced upwards a body of the same volume as the pendulum bob but of very small weight, so that the centre of gravity of she volume lies in the axis of suspension, shovv that the correction for buoyancy vanishes. This method was suggested in 1871 by the Astronomer Eoyal, but ho remarks that this construction woiUd probably be inconvenient in practice. Ex. 3. If a barometer be attached to the pendulum show that the rise or fall of the mercury as the density of the air changed could be so arranged as to keep the time of vibration unaltered. This method was suggested first by Dr Robinson of Armagh in 1831 in the fifth volume of the memoirs of the Astronomical Society, and afterwards by Mr Deiiison in the Astronomical Notices for Jan. 1873. In the Armagh Places of Stars published in 1859, Dr Robinson describes the difficulties he found in practice before he was satisfied with the working of the clock. The theory of this construction is that in differentiating equation (2) we are to suppose k^ and h^ variable and I constant. This gives — ^ — = 8 (^^^i) - 8 {h^VD). V Let r be the rise of the barometer in the glass tube, ?•' the fall in the cistern, then r' — nvr, whore jftis a known fraction depending on the dimensions of the barorriter. Let a and h be the depths of the mercmy in the tube and cistern below the axis of suspension, 1c the diameter of the tube, p the density of the mercury. Since irc'^pr is the quantity of mercury added to the top of tne mercury in the tube and taken away from tiie istern, we have 8(iWi=) = Tcv|(a-|y-^6 + 0'|, These are acrurate if the barometer be merely ". bent tube so that the cylinders transferred are similar as well as equal; in this case m — l. If the area of the cistern be grt it ;r than that of the tube wo have here neglected the difl'erence of the moments of inertia of the two cylindp .r about asf - through their centre of gravity. As r is seldom more than one inch, wo may write tln.'e d{Mk^) = 7,v'',>r{a;^--b-^), d(Mhi) = Trc^pr{a-h). Since D is very small, we may neglect the variations of Vh.^ when multiplied by D. Thus we have ST) ..r^IIpa + h-l D " rm., " r ''' THE PENDULUM. 75 where H=b-a is the height of the barometer. If the temperature of the au- bo luialtercil wo have 5/) SIf -'j ami )• (1 + m) = dU. The required condition is therefore irc-Ifp JI a + b~ I , It in clearly necessary that n + b>l. The jar of merenry in Graham's mercurial l)eudiiliim might be used as the cistern of the barometer, as Mr Denisou remarks. The height of the barometer being 30 inches this would hardly be effective unless the pendulum was longer than the seconds pendulum, which is about 39 inches. Prof. Bankine read a paper to the British Association in 18/33 in which ho proposed to use a clock with a ccnti-ifugal or revolving pendulum, part of whicli should consist of a siphon barometer. The rising and falling of the barometer would affect the rate of going of the clock and thence the mean height of the mercurial column dm'ing any long period would register itself. Ex. 4. If the pendulum be supposed to drag a quantity of air with it which bears a constant ratio to the density D of the surrounding air and adds yD to the moment of inertia of the pendulum without increasing the moving power, show that the change produced in the simple equivalent pendulum by a change of density SD is given by 51=7 ,..- . Show that this might be included in Dr Eobinson's mode of correcting for buoyancy. 97. In many experimental investigations it is necessary to determine the moment of inertia of the body experimented on about some axis. If the body be of regular shape and be so far homogeneous that the errors thus produced are of the order to be neglected, we can determine the moment of inertia by calculation. But sometimes this cannot be done. If we can make the body oscillate under gravity about any axis parallel to the given axis placed in a horizontal position, we can determine by equation (4) of Art. 92 the radius of gyration about a parallel axis through the centre of gravity. This requires however that the distances of the centre of gravity from the axes should be very accurately found. Sometimes it is more convenient to attach the body to a pendulum of known mass whose radius of gyration about a fixed horizontal axis has been previously found by observing the time of oscilla- tion. Then by a new determination of the time of oscillation, the moment of inertia of the compound body, and therefore of the given body, may be found, the m.asses being known. If the body be a lamina, Vire may thus find the radii of gyra- tion about three axes passing through the centre of gravity. By measuring three lengths along these axes inversely proportional to these radii of gyration, we have three points on a momental ellipse at the centre of gravity. The ellipse may then be easily con- structed. The directions of its principal diameters are the princi- pal axes, and the reciprocals of their lengths '.epresent on the same scale as before the principal radii of gyration. n ! II h 70 JIOTION ABOUT A FIXED AXIS. If the body be a solid, six observed radii of gyration will deter- mine the principal axes and moments at the centre of gravity. But in most eases some of the other circumstances of the par- ticular problem under consideration will simplify the process. On the length of the Seconds Pendulum. 98. The oscillations of a rigid body may be used to determine the numerical value of the accelerating force of gravity. Let t be the half time of a small oscillation of a body made in vacuo about a horizontal axis, h the distance of the centre of gravity from the axis, k the radius of gyration about a parallel axis through the centre of gravity, j ' . n we have by Art. 92, k'' + h' = \hT' (1), where \ = '-^ so that X is the length of the simple pendulum TT whose complete time of oscillation is two seconds. We might apply this formula to any regular body for which A;* and h could be found by calculation. Experiments have thus been m^'^^e with a rectangular bar, drawn as a wire and suspended from one end. In this case , which is- the length of the h simple equivalent pendulum is easily seen to be two-thirds of the length of the rod. The preceding formula then gives X or g as soon as the time of oscillation has been observed. By inverting the rod and taking the mean of the results in each position any error arising from want of un'formity in density oi figure may be partially obviated. It has, lowever, been found impracticable to obtain a rod sufficiently uniform to give results in accordance with each other. 99. If we make a body oscillate about two parallel axes in succession not at the same distance from the centre of gravity, we get two equations similar to (1), viz. k'+h" = U'T"] ^*'^- Between these two we may now eliminate k"^, thus ''-J^^hr'-hW' (3). This equation gives X. Since k"^ has disappeared, the form and structure of the body is now a matter of no importance. Let a body be constructed with two apertures into which knife edges I.ENOTII OF THE SECONDS PENDULUM. 77 ieter- avity. ! par- ?rmiue at T be ) about Din tbe gh the .•(1). Qdulum r which „ve thus spended I of the s of the or g as Qvorting ion any re may icticable ordance axes in Lvity, we (2). ,.(3). )rm and Let a Ife edges can be fixed. By means of these resting either on a horizontal phine or in two triangular apertures to prevent shpping, the body can be made to oscillate through small arcs. The perpendicular distances h, h' of the centre of gravity from the axes must then be measured with great care. The formula will then give \. 100. In Capt. Kater's method the body has a .sliding weight in the form of a ring which can be moved up and down by means of a screw. The body itself has the form of a bar and the apertures are so placed that the centre of gravity lies between them. The ring weight is then moved until the two times of oscillation are exactly equal. The equation (3) then becomes /<+// = T .(4). which determines \. The advantage of this construction is that the position of the centre of gravity, which is very difficult to find by experiment, is not required. AH we want is ^ + h , the exact distance between the knife edges. The disadvantage is that the ring weight has to be moved until two times of oscillation, each of which it is difficult to observe, are made equah 101. The equation (3) can be written in the form We now see that if the body be so constructed that the times of oscillation about the two axes of suspension are very nearly equal r^ — r'^ will be small, and therefore it will be sufficient in the last term to substitute for h and li their approximate values. The position of the centre of gravity is of course to be found as accu- rately as possible, but any small error in its position is of no very great consequence, for these errors are multiplied by the small quantity t'^ — t '^ The advantage of this construction over Kater's is that the ring weight may be dispensed with and yet the only element which must be measured with extreme accuracy is h -\ h , the distance between the knife edges. 102. Tn order to measure the distance between the knife edges, Captain Kater first compared the different standards of length then in use, in terms of each of which he expressed the length of his pendulum. Since then a much more complete com- parison of these and other standards has been made under the direction of the Commission appointed for that purpose in 1843. FML Trans. 1857. Having settled his unit of length, Captain Kater proceeded to measure the distance between the knife edges by means of micro- 78 MOTION ABOUT A FIXED AXIS. j I: r ; i! scopes. Two different metliods were used, which liowever cannot be ()*Bcribed here. As an illustration of the extreme care neces- sary in these measurements, the following fact may be mentioned. Though the images of the knife edges were always perfectly sharp and well defined, their distance when seen on a black ground was •000572 of an inch less than when seen on a white ground. This difference appeared to be the same whatever the relative illumi- nation of the object and ground might be so long as the difference of character was pr^erved. Three sets of measurements were taken, two at the beginuing of the experiments, and the third after some time. The object of these last was to ascertain if the knife edges had suffered from u^k?. The mean results of these three dif- fered by less than a ten-thousandth of an inch from each other, the distance to be measured being 3944085 inches. 103. The time of a single vibration cannot be observed di- rectly, because this would requii*© the fraction of a second of time as shown by the clock to be estinoiated either by the eye or ear. The difficulty may be overcome by observing the time, say of a thousand vibrations, and thus the error of the time of a single vi- bration is divided by a thousand. The labour of so much counting may however be avoided by the use of "the method of coinci- dences." The pendulum is placed in front of a clock pendulum whose time of vibration is slightly differept. Certain marks made on the two pendulums are observed by % telescope at the lowest point of their arcs of vibration. The field of view is limited by a diaphragm to a narrow aperture across which the marks are seen to pass. At each succeeding vibration one pendulum follows the other more closely, and at last its mark is completely covered by the other during their passage across the field of view of the telescope. After a few vibrations it appears again preceding the other. In the interval from one disappearance to the next, one pendulum has made, as nearly as possible, one complete oscillation more than the other. In this manner 530 half-vibrations of a clock pendulum, each equal to a second, Avere found to orrespond to 532 of Captain Kater's pendulum. The advantage of this method of observation is such, that an error of one second in noting the interval between two coincidences would occasion an error of only 0*63 in the number of vibrations in 24 hours. The ratio of the times of vibration of the pendulum and the clock pendulum may thus be calculated with extreme accuracy. The rate of going of the clock must then be found by astronomical means. 104. The time of vibration thus obtained will require several corrections which are called "reductions." For instance, if the oscillation be not so small that we can put sin ^ = ^ in Art. 92, we must make a reduction to infinitely small arcs. The general method of effecting this will be considered in the chapter on Small Pi LENGTH OP THE SECONDS PENDULUM. cannot neces- tioned. ' sharp nd was ThiH illumi- ference s were rd after le knife ree dif- 1 other, :ved di- of time ? or ear. say of a ingle vi- 3ounting f coinci- sndulum •ks made le lowest nited by rks are follows covered iw of the ling the ext, one cillation Ions of a espond of this n noting error of ratio of sndulum of going Irr le several if the \i. 92, Ave general )n Small Oscillations. Another reduction is necessary if wo wish to reduce the result to what it would have been at the level of the sea. The attrnction of the intervening laud may be allowed for by Dr Young's rule {Phil. Trans, 1819). We may thus obtain the force of gravity at the level of the sea, supposint^- all the land above this level were cut off and the sea constrained to keep its present level. As the level of the sea is altereil by the attraction of the land, further corrections are still necessary if we wish to re- duce the result to the surface of that spheroid which most nearly represents the earth. See Cainh. Phil. Trans. Vol. X. M. Baily gives as the length of the pendulum vibrating in half time a mean solar second in the open air in this latitude 39'13.S inches, and the length of a similar pendulum vibrating sidereal seconds 38'919 inches. 105. The obsen'ations must be made in the air. To correct for this we have to make a reduction to a vacuum. This reduction consists of throe parts: (1) The correction for buoyancy, (2) Du Buat's correction for the air dragged along by the pendulum, (3) The resistanc' of the air. Let V be the volume of the pendulum which may be found by measuring the dimensions of the body. As the "rtiductiou to a vacuum " is only a correction, any small unavoidable errors in calculating the dimensions will produce an effect only of the second order on the value of X. Let p be the density of the air when tho body is oscillating about one knife edge, p' the density when oscillating about tho other. If the observation be made within an hour or two hours, we may put p = p'. The effect of buoyancy is allowed for by supposing a force Vp ^ 23 WIST MAIN STRUT WltSTIR.N.Y. I45M ( 71* ) 372-4503 ,.<*• ^ ^ 80 MOTION ABOUT A FIXED AXIS. ii edges succeed each other r.t a short interval we may put p=p', and then Dn Buat's correction will disappear. This is of course a very great advantage. We then have* h + h' ,n-%,j-;,._,,,(,.^). the last term heing very small because t and r' are nearly equal. de The resistance of the air will be some function of the angular velocity ^ of the pendulum. Since -^ is very small we may expand this function and take only the first power. Supposing Maclaurin's theorem not to fail, and that no coefficient of a higher power than the first is very great, this gives a resistance proportional to (10 di' The equation of motion will therefore take the form iir . where — is the time of a complete oscillation in a vacuum and the term on the n right-hand side in that due to the resistance of the air. The discussion of this equation will be found in the chapter on Small Oscillations. 106. In constructing a reversible pendulum to measure the force of gravity, the following are points of importance. 1. The axes of suspension, or knife edges, must not be at the same distance from the centre of gravity of the mass. They should be parallel to each other. 2. The times of oscillation about the two knife edges should be nearly equal. 3. The external form of the body must be symmetrical, and the same about the two axes of suspension. 4. The pendulum must be of such a regular shape that the dimensions of all the parts can be readily calculated. These conditions are satisfied if the pendulum be of a rect- angular shape with two cylinders placed one at each end. The external forms of these cylinders are to be equal and similar, but one is to be solid and the other hollow, and such that by calcula- tion of moments of inertia the distance between the knife edges is to be as nearly as possible equal to the length of the simple equi- valent pendulum. 5. The pendulum should be made, as far as possible, of one metal, so that as the temperature changes it may be always similar to itself. In this case since the times of oscillations of similar bodies vary as the square root of their linear dimensions, it is easy to reduce the observed time of oscillation to a standard tem- * This formula was mentioned to the author as the one used in the late experi- ments by Capt. Heaviside to determine the length of tlie seconds pendulum. LENGTH OF THE SECONDS PENDULUM. 81 =.hr^-h-, perature. The knife edges however must be made of some strong substance not likely to be easily injured. 107. Ex. 1. If the knife edges be not perfectly sharp, let r be the difference of their radii of curvature, show that h?-h'^ + {h^h')r \ very nearly when the pendulum vibrates in vacuo. It appears that the correction vanishes if the knife edges be only equally sharp. By interchanging the knife edges we have the same equation with the sign of r changed. By making a few observa- tions we may thus determine r A proposition similar to this has been ascribed to Laplace by Dr Young. Ex. 2. A heavy spherical ball is suspended successively by a very fine wire from two points of support A and B whose vertical distance b has been carefully measured, thus forming two pendulums. The lowest point of the ball is, on each suspension, made to be as exactly as possible on the same level, which level is approximately at depths a and a' below A and B respectively. If r be the radius of the ball, wliich is small compared with a or a', and I, V the lengths of the simple ,.a very nearly. By coimt- l — V 2 equivalent pendulum, prove that —7— = 1 - - , , , , b 5(a-r)(a-r) ing the number of oscillations performed in a given time by each pendulum, show I how to find ratio -, . V Thence show how to find g and point out which lengths must be most carefully measured and which need only be approximately found, so as to render this method effective. This method is mentioned in Cirant's iirtory of Physical Astronomy, page 155, as having been used by Bessel. 108. The length of the seconds pendulum has been used as a national standard of length. By an Act of Parliament passed in 1824, it was declared that the distance between the centres of the two points in the gold studs in the straight brass rod then in the custody of the clerk of the House of Commons, whereon the words and figures "standard yard, 1760" were engraved, shall be the original and genuine standard of length called a yard, the brass being at the temperature of 62° Fah. And as it was expedient that the said standard yard if injured should be restored of the same length by reference to some invariable natural standard, it was enacted, that the new standard yard should be of such length that the pendulum, vibrating seconds of mean time in the latitude of London in a vacuum at the level of the sea, should be 39"1393 inches. On Oct. 16, 1834, occurred the fire at the Houses of Parlia- ment, in which the standards were destroyed. The bar of 1760 was recovered, but one of its gold pins bearing a point was melted out and the bar was otherwise injured. In 1838 a commission was appointed to report to the govei-n- ment on the course best to be pursued under the peculiar circum- stances of the case. R. D. 6 y I Hill f 82 MOTION ABOUT A FIXED AXIS. In 1841 the commission reported that they .were of opinion that the definition by which the standard yard is declared to be a certain brass rod is the best which it is possible to adopt. With resp'ict to the provision for restoration they did not recommend a reference to the length of the seconds pendulum. " Since the passing of the act of 1824 it has been ascertained that several elements of reduction of the pendulum experiments therein re- ferred to are doubtful or erroneous: thus it was shown by Dr Young, Phil. Trans. 1819, that the reduction to the level of the sea was doubtful ; by Bessel, Astron. Nachr. No. 128, and by Sabine, Phil. Trans. 1829, that the reduction for the weight of air was erroneous ; by Baily, Phil. Trans. 1832, that the specific gravity of the pendulum was erroneously estimated and that the faults of the agate planes introduced some elements of doubt ; by Kater, Phil. Trans. 1830, and by Baily, Astron. Soc. Memoirs, Vol. IX., that very sensible errors were introduced in the operation of comparing the length of the pendulum with Shuckburgh's scale used as a representative of the legal standard. It is evident, therefore, that the course prescribed by the act would not neces- sarily reproduce the length of the original yard." The commission stated that there were several measures which had been formerly accurately compared with the original standard yard, and by the use of these the length of the original yard could be determined without sensible error. In 1843 another commission was appointed to compare all the existing measures and construct from them a new Parliamentary standard. Unexpected difticulties occurred in the course of the comparison, which cannot be described here. A full account of th.e proceedings of the commission will be found in a paper contributed by Sir G. Airy to the Royal Society in 1857. Oscillation of a Watch Balance. 109. A rod B'CB can turn freely about its centre of gravity C which is fixed, and is acted on by a very fine spiral spring CPB. The spring has one end fixed in position in such a manner that the tangent at C is also fixed, and has the other end B attached to the rod so that the tangent at B makes a constant angle with the rod. The rod being turned through any angle, it is required to find the time of oscillation. This is the construction used in watches, just as the pendulum is used in clocks, to regulate the motion. Let Cx be the position of the rod when in equilibrium, and let be the angle the rod makes with Cx at any time t, MF the moment of inertia of the rod about G. Let p be the radius of /I OSCILLATION OF A WATCE BALANCE. 83 curvature at any point P of the spring, p^ the value of p when in equilibrium. Let (a?, i/) be the co-ordinates of P referred to G as origin and Cx as axis of x. Let us consider the forces which act on the rod and the portion BP of the spring. The forces on the rod are X, Y the resolved parts of the reaction at C parallel to the axes of co-ordinates, and the reversed eflfective forces which are J2i3 equivalent to a couple 3Ik^ -j^ . The forces on the sj»ring are, the reversed effective forces which are so small that they may be neglected, and the resultant action across the section of the spring at P. This resultant action is produced by the tensions of the innumerable fibres which make up the spring, and these are equivalent to a force at P and a couple. When an elastic spring is bent so that its curvature is changed, it is proved both by experiment and theory that this couple is proportional to the change of curvature at P. We may therefore represent it by eI ], where E depends only on the material of which the spring is made and on the form of its section. Taking moments about P to avoid introducing the unknown force at P, we have ^^^f= E Cv.)--^ + Yx. This equation is true whatever point P may be chosen. Con- sidering the left side constant at any moment and (a;, y) variable, this becomes the intrinsic equation to the form of the spring. Let BP = s, multiply this equation by ds and integrate along the whole length I of the spiral spring, we have ds i Now — is the angle between two consecutive normals, hence P ds is the angle between the extreme normals. Now at A the normal to the spring is fixed throughout the motion, therefore 6—2 J I lA i'^' 84 MOTION ABOUT A FIXED AXIS. K 1 is the angle between the normals at B in the t'tvo positions in which 6 - 6 and ^ = 0. But since the normal at B makes a constant angle with the rod, this angle is the angle 6 which the rod makes with its position of equilibrium. Also if x, y be the co-ordinates of the centre of gravity of the spring at the time t, we have \xds = xl, \yds = yl. Hence the equation of motion becomes Let us suppose that in the position of equilibrium there is no pressure on the axis (7, then X and Y will, throughout the motion, be small ^^aantities of the order 6. Let us also suppose that the fulcrum is placed over the centre of gravity of the spring when at rest. Then if the number of spiral turns of the spring be numerous and if each turn be nearly circular, the centre of gravity will never deviate far from C. So that the terms Yx and Xy are each the product of two small quantities, and are therefore at least of the second order. Neglecting these terms we have ^^ de " I ^• Hence the time of oscillation is 27r /Mm It appears that to a first approximation the time of oscillation is independent of the form of the spring in equilibrium, and depends only on its length and on the form of its section. This brief discussion of the motion of a watch balance is taken from a memoir presented to the Academy of Sciences. The reader is referred to an article in Liouville's Journal, 1860, for a further investigation of the conditions necessary for isochronism and for a determination of the best forms for the spring. Pressures on the fixed cuds. 110. A body moves about a fi,xed axis under the action of any forces, to find the pressures on the axis. First. Suppose the body and the forces to be symmetrical about the plane through the centre of gravity perpendicular to the axis. Then it is evident that the pressures on the axis are reducible to a single force at G the centre of suspension. Let F, G be the actions of the point of support on the body resolved along and perpendicular to CO, where is the centre oJ tl re li fo h( pi b( tb ID bi V{ PRESSUKES ON THE FIXED AXIS. 85 II of gravity. Let X, Y be the sum of the resolved parts of the impressed forces in the same directions, and L their moment round C. Let CO = h and d = angle which CO makes with any straight line fixed in space. Taking moments about C, we have d?d _ L de~M{k^^K') ^ ^• The motion of the centre of gravity is the same as if all the forces acted at that point. Now it describes a circle round C\ hence, taking the tangential and normal resolutions, we have ^de = -w- (2)' -^Kdt) — M" (^)- Equation (1) gives the values of -^ and -j-, and then the pressures may be found by equations (2) and (3). If the only force acting on the body be that of gravity, let be measured from the vertical. If the body start from rest in that position which makes CO horizontal, we have X=Mg cos 6, Y^ — Mg smd, L = — 3fghsiud; d'e_ gh . ^ •*• dt' — F+T"'''' integrating, we have but when G = — , -j: vanishes, therefore (7=0; substituting these values (2) and (3), we get y M 86 MOTION ABOUT A FIXED AXIS. -F=Mg COS 0.-j^j^, where B is the angle which CO makes with the vertical. Let "^ be the angle the direction of the pressure at G makes with the line GO, the angle being measured from GO downwards to the left, then cot -^ = ( 1 + 3 p) cot 0, which is a convenient formula to determine the direction of the pressure*. 111. Secondly. Suppose either the body or the forces not to be symmetrical. Let the fixed axis be taken as the axis of z with any origin and plane of xz. These we shall afterwards so choose as to sim- plify our process as much as possible. Let x, y, i be the co-ordi- nates of the centre of gravity at the time t. Let 0) be the angular velocity of th«. acceleration, so that /= -— . y, f the angular Now every element m of the body describes a circle about the axis, hence its accelerations along and perpendicular to the radius vector r from the axis are — wV and fr. Let be the angle * Let il/.iZ be the resultant of F and G, and let a=g~^ and &=g /' g, cos' i^ sin'^iA' 1 then - J^ - + — pX= _. Construct an ellipse with C for centre and axes equal to a and h measured along and perpendicular to CO. Then the resultant pressure varies as the diameter along which it acts. And the direction may be found thus ; let the auxiliary circle cut the vertical in F, and let the perpendicular from F on CO cut the ellipse in R. Then CR is the direction of the pressure. PRESSURES ON THE FIXED AXIS. 87 which r makes with the plane of xz at any time, then from the resolution of forces it is clear that — = - ft)V cos 6 —fr sin ^ = — tJ'x —fiji similarly -^ = — m^y +fx. These equations may also be obtained by differentiating the equations a? = r cos ^, y = r aind twice, remembering that r is constant. Conceive the body to be fixed to the axis at two points, distant a and a from the origin, and let the reactions of the points on the body resolved parallel to the axes be respectively F, Q, H\ The equations of motion of Art. 71 then give tmX+ F+F'=^tm^ = Xm{- o>'x -ft/) = -cB'J/5-/ify (1), tmY+G+G'='Zmj^,=Xm{-a>'y+fx) ^-tJ'My+fMx (2), 2mZ+ir+ir' = Sw^ = (3). Taking moments about the axes, we have tm{2/^-z7)-Ga-G'a' = 'Zm(y^-zj^^ = (o^Xmyz —fXmxz (4): by merely introducing z into the results in (2), • tm{zX-xZ) + Fa + F'a==Xm(z~-x^^ = — a>*Xmxz —fXmyz (5), %m{xY-.yX) =tm{x^^-y^) = Mk'\f (6). f 1 i i. 'I III % 88 MOTION ABOUT A FIXED AXIS. Equation (6) serves to determine / and w, and equations (1), (2), (4), (5) then determine F, G, F', 0'\ //and W are indeter- minate, but their sum is given by equation (3). Looking at these equations, we see that they n'ould be greatly simplified in two cases. First, if the axis of ^ be a principal axis at the origin, "^inxz = 0, "Zmyz = 0, and the calculation of the right-hand sides of equations (4) and (5) would only be so much superfluous labour. Hence, in at- tempting a problem of this kind, we 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 / and to by integrating equation (G), the whole process is merely an algebraic substitution of / and to in the remaining equations. Hence our results will btill be correct if we choose the plane of wz to contain the centre of gravity at the moment under consideration ; this will make ^ = 0, and thus equations (1) and (2) will be simplified. 112. If the forces which act on the body be impulsive, the equations will require some alterations. Let to, to' be the angular velocities of the body just before and just after the action of the impulses. In the case in which the body and forces are symmetrical, the equations (1), (2), (3) of Art. 110 become respectively ^^'-^^ilJl/ZTF) "^^' Y-\- G " "^ " 7, («.•-<.) = ^^ (2), = ^-^ • (3). where all the letters have the same meaning as before, except that F, G, X, Y are now impulsive instead of finite forces. Let us next consider the case in which the forces on the body are not symmetrical. Let u, v, w, u', v , w' be the velocities resolved parallel to the axes of any element m whose co-ordinates are x, y, z. Then u = — yto, u' = — yto', v = xto, v' = xto', and w, V) are both zero. The several equations of Art. Ill will then be replaced by the following: 2 A^ ^-FaF = Sw {\i -u) = - tmy {to' - to) = -3/^(0,' -a,) (1), PRESSURES ON THE FIXED AXIS. 89 2 r + O' + 6^' = 2m (v - v) = Swa; (o)' - 0)) = Mu:((o'-(o) (2), 2Z+//+//' = (3), XQ/Z-zY) -Ga- G'a' = Xm [y (w'-w)-z{v -?;)} = — Xmxz . (o>' — w) (4), 2 {zX- xZ) +fli + Fd = tm {z (u' -u)-x{w'-w)} = — Imi/z . («()' — ', F, F\ O, G' and the sum H+ H' oi the two pressures along the axis. These equations admit of simplification when the origin can be so chosen that the axis of rotation is a principal axis at that point. In this case the right-hand sides of equations (4) and (5) vanish. Also if the plane of xz be chosen to pass through the centre of gravity of the body, we have ^ = 0, and the right-hand side of equation (1) vanishes. 113. Ex. A door is suspended by tieo 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, -Tfe 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 imder consideration. The only force acting on the door is gravity, which may he supposed to act at the centre of gravity. We must first resolve thie parallel to the axes. Let if> be ■/ 1 fj 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 ^ with the axis of x, this will be the vortical plane through the '/■■ ii 90 MOTION ABOUT A FIXED AXIS. nxis; and if wo draw V in tiiis plane making ZO V=a, OV will be vertical. Hence the resolved parts of gravity are X =<7 sin a cos 0, r=(7HinoHin0, Z=:-*7C0flo. 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 3/j/ sin a cos + /•+/"= -u^Mx (1), Mfj Bina Bin + + 0'=fMx (2), -iMircoso + //+//'=0 (S), -Oa+0'a=0 (4), Afg COB ax + Fa -F'a-0 (5), booause the fixed axis is a principal axis at the origin, - Mfj Bin a Bin if,. x = Mk''.-^^ (6). Integrating the last equation, we have C + 2r/ sin o cos ^ = fc '*w'. Suppose the door to be initially placed at rest, with its plane making an angle /3 with the vertical plane through the axis; then when 0=/3, u=0; hence k'^w^ = 2r/i sin o (cos - cos /3) ) and k'^/= ~u sinasin.x )' By substitution in the first four equations F, F', 0, 0', may bo found. 114. It should be noticed that these equations 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 equi- momental body that may be convenient for our purpose. This consideration will often enable us to reduce the compli- cated forms of Art. Ill to the simpler ones given in Art. 110. For though the body may not be symmetrical about 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 fouod by Art. 110. 115. 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 gi'avity. 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. PRESSURES ON THE FIXED AXIS. 91 116. If a body be set in rotation about any axis which is a principal axis at some point in its length, and if there be no impressed forces acting on the body, it follows at once from these conditions that the pressures on the axis are equivalent to a single resultant force acting at 0. Hence if be fixed in space, the body will continue to rotate about that axis as if it also were fixed in space. Such an axis is called a permanent axis of rotation at the point 0. If the body be entirely free and yet turning about an axis of rotation which does not alter its position in space, we may suppose any point we please ii* the axis to be fixed. In this case the axis must be a principal axis at every point of its length. It must therefore by Art. 49 pass through the centre of gravity. The existence of principal axes was first established by Scgner in the work Specimen Theorim Turhinum. His course of in- vestigation is the opposite of 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 position of the axis. From 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, 18U2, and Bossut, Histoire de MatMmatiqiie, Tome ii. 117. Suppose the body to start from rest and to be acted on by a couple, let us discover the necessary conditions that the pressures on the fixed axis may be reduced to a single resultant pressure. Supposing such a single resultant pressure to exist, we can take as origin that point of the axis at which it is intersected by the single resultant. Then the moments of the two pressures on the axis of rotation about the co-ordinate axes will vanish. Hence since © = the equations (4), (5), and (6) of Art. 112 become L = -fXmxz, M=-fXmi/z, N=Mky, where we have written L, My iVfor the three moments 'Zm{yZ—z Y), &c. of the impressed forces about the co-ordinate axes. The plane of the couple whose resolved parts about the axes are L, M, N, is known by Statics to be LX + 3fY + NZ=0, or in our case, -tmxzX-XmyzY+Mk"Z=0 (1). Let the momental ellipsoid at the fixed point be constructed, and let its equation be AX' + BY'+CZ'-2DYZ- 2EZX- 2FXY= e*. The equation to the diametral plane of the axis of Z is ^EX-DY+ CZ^O (2). i ii n u I 92 MOTION ABOUT A FIXED AXIS. Comparing (1) and (2) we see that the plane of the resultant couple must be the diametral plane of the axis of revolution. Since the pressures on the axis are equivalent to a single resultant force acting at some point of the axis, we may suppose this point alone to be fixed and the axis of rotation to be other- wise free. If then a body at rest with one point fixed be acted on by any couple, it will begin to rotate about the diametral line of the plane of the couple with regard to the momental ellipsoid at the fixed point. Thus the 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. If the acting couple be an impulsive couple, the equations of motion, by Art. 112, will be the same as those obtained above when (o is put zero and '- we have V = -Tr- . 2 sin ^ \gh. But the chord of the arc of the recoil is 6 = 2c sin 2 ; nhk' i—r :. v= — T. va/i. cf The magnitude of k' may be foimd experimentally by ob- serving the time of a small oscillation ^ the pendulum and rifle. If '£ be a half-time we have r= tt a/ -r- (^^t. 92.) This is the formula given by Poisson in the second volume of his Mecaniqiie. The reader will find in the Philosophical Maga- zine for June 18.54, an account of some experiments conducted by Dr S. Haughton from which, by the use of this formula, the initial velocities of rifle bullets were calculated. Tlio formula must however be regarded as only a first approximation, for the recoil of the pendulum when the gan is fired without a baU has been altogether neglected. In Dr Haughton's experiments the (barge of powder was compijiratively small, and this assumption was nearly correct. But in some of Dr Button's experi- ments, where comparatively large charges of powder were used, the recoil without a ball was found to be very considerable. To allow for this Dr Button, following Mr Eobins, assumed that the effect of the charge of powder on the recoil of the gun is the same either with or without a ball. If p be the momentum generated by the powder, the whole momentum gene- rated in the pendulum will be mv+p instead of mv. Proceeding as before, we find If we now repeat the experiment, with an equal charge without a ball, we have p _ „ "') be the polar co-ordinates of any par- ticle m referred to the centre of gravity of the body as origin. Then r' is constant throughout the motion, and -—- is the same rid for every particle of the body and equal to t- . Thus the an- gular momentum h, exactly as in Art. 88, is THE EQUATIONS OF MOTION. 101 where M1^ is the moment of inertia of the body about its centre of gravity. The angle is the angle some straight line fixed in the body makes with a straight line fixed in space. Whatever straight dd lines are chosen -^ is the same. If this be not obvious, it may be shown thus. Let QA, O'A' be any two straight lines fixed in the body inclined at an angle a to each other. Let OB, OB be two straight lines fixed in space inclined at an angle /9 to each other. Let AOB=0, A'0'B' = ff, then ^ + /3 = l^+a. Since a and y9 are independent of the time, -j1 = -ji • -By this propo- sition we learn that the angular velocities of a body in two di- mensions are the same about all points. The general method of proceeding will be as follows. Let {x, y) be the co-ordinates of the centre of gravity of any body of the system referred to rectangular axes fixed in space, M the mass of the body. Then the effective forces of the body are together equivalent to two forces measured by JZ-rr, M-z^ at dv acting at the centre of gravity and parallel to the axes of co- ordinates, together with a couple measured by Mh^ -^ tending to turn the body about its centre of gravity in the direction in which 6 is measured. By D'Alembert's principle the effective forces of all the bodies, if reversed, will be in equilibrium with the impressed forces. The dynamical equations may then be formed according to the ordinary rules of Statics. For example, if we took moments about a point T"hose co- ordinates are (p, q) we should have an equation of the form M ((-.)f-(y-.)'i^}+3f^g=A where L is the moment of the impressed forces and the other letters have the same meaning as before. In this equation (p, q) may be the co-ordinates of any point whatever, whether fixed or moving. Just as in a statical problem, the solution of the equations may frequently be much simplified by a proper choice of the point about which to take moments. Thus if we wished to avoid the introduction into our equations of some unknown reaction, we might take moments about the point of application or use the principle of virtual velocities. So again in resolving !' ii ;i 102 MOTION IN TWO DIMENSIONS. d'x \l I our forces wo might replace the Cartesian expressions M -rp , M -j^ by the polar forms dt* M ff-'(f)'} „„a./14(4*) r dt\ dtt lor the resolved parts parallel and perpendicular to the radius vector. If V be the velocity ol the centre of gravity, p the radius of curvature of its path, we may sometimes also use with advantage the forms M-yr and M— for the resolved parts of the effective dt p ^ forces along the tangent and radius of curvature of the path of the centre of gravity. 124. As we shall have so frequently to use the equation formed by taking moments, it is important to consider other forms into which it may be put. Let the point about which wj are to take moments be fixed in space, so that it may be chosen as the origin of co-ordinates. Then the moment of the effective forces on the body M is ih(4r-4:)+^4:}= The attention of the reader is directed to the meaning of the several parts of this expression. We see that, as explained in Art. 72, the moment of the effective forces is the differential coefficient of the moment of the momentum about the same point. The moment of the momentum by Art. 76 is the same as the moment about the centre of gravity together with the moment of the whole mass collected at the centre of gravity, and moving with the velocity of the centre of gravity. The moment round the centre of gravity is by the first Article either of Chap. iii. or Chap. IV. equal to Mk^ -r: and the moment of the collected mass is Jf [a; -^ — y -^ J , where {x, y) are the co-ordinates of the centre of gravity. Hence in space of two dimensions we have for any body of mass M angular momentum round the origin ^(4?-4)+^^l- If we prefer to use polar co-ordinates, we can put this into another form. Let (r, ^) be the polar co-ordinates of the centre of gravity, then, angular momentum round ") _ , , , d^ ^, g dd the origin ) dt dt' If V be the velocity of the centre of gravity, and p the per- pendicular from the origin on the tangent to its direction of THE EQUATIONS OF MOTION. 103 B radius le radius Ivantage eflfective :h of the squation er forms , wj are losen as effective ig of the ained in ferential ae point, moment le whole velocity gravity equal to -yi:& Hence M d0 dt' his into jentre of the per- ction of motion, the moment of momentum of the mass collected at tho centre of gravity is Mvp, so that wo also have angular momentum round) _ , - . -, , d$ the origin [ ~ r ^t' ] dt It is clear from Art. 7G that ihis is the instantaneous angular momentum of the body about the origin, whether it is fixed or moveable, though in the latter case its dift'erential coefficient with regard to t is not the moment of the effective forces. Since the instantaneous centre of rotation may be regarded as a fixed point, when we have to deal only with the coordinates and with their first differential coefficients with regard to the time, we . have angular momentum round the instantaneous centre = i/(r' + A;') dt' If Mk'^ be the moment of inertia about the instantaneous centre, this last moment may be written MTc* -^ . In taking moments about any point whether it be the centre of gravity or not, it should be noticed that the Mk* in all these formulae is the moment of inertia with regard to the centre of gravity, and not with regard to the point about which ^ "^ arc taking moments. It is only when we are taking moments ubout the instantaneous centre or about a fixed point that we can use the moment of inertia about that point instead of the moment of inertia about the centre of gravity, and in that case our expres- sion for the angular momentum includes the angular momentum of the mass collected at the centre of gravity. 125. Suppose we form the equations of motion of each body by resolving parallel to the axes of co-ordinates and by taking moments about the centre of gravity. Wc shall get three equations for each body of the form M -jw = i^cos ^ + jR cos -Jr -H . . . M d^ df = JP sin ^ + JB sin -^^ + . . . Mk !^d'e _ de = Fp. +Mq iV, where F is any one of the impressed forces acting on the body, whose resolved parts are J^cos (f>, Fsin ^, and whose moment about the centre of gravity is Fp, and B is any one of the re- actions. These we shall call the Dynamical equations of t!ie body. : I L i:n i 104 MOTION IN TWO DIMENSIONS. 1 t! , -; I \ Bc'widt'8 these there will be certain geometrical cquatiuuM expressing the connections of the system. As every such forced connection is accompanied by a reaction and every reaction bv some forced connection, the number of geometrical equations will be the same as the number of unknown reactions in the system. Having obtained the proper number of equations of motion we proceed to their solution. Two general methods have been proposed. First Method. Ditferuntiate the geometrical equations twice with respect to t, and substitute for v , , -jtj , ^ , , from the dynamical ecjuations. We sliall then have a sufficient number of equations to determine the reactions. This method will oe of great advantage whenever the geometrical equations are of the form Ax + Bi/+ Ce = D (2), where A, B, C, D ave 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 obtained by differentiating (1), 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. For suppose any geo- metrical equation took the form x'-¥f = c\ containing squares instead of first powers, then its second dif- ferential equation will be d'x . d^y /dx^ X df'^'^df ^ + (ly-. 72 _ 7 2 and though we can substitute t'or-~ , -^, we cannot, in general, eliminate the terms dxY di ™^ ('fX THE EQUATIONS OF MOTION. 105 12C. The reactions iu a ilynamical problem are in many cases producetl by tho pressures of some smooth fixed obstacles which are touched by tho moving bodies. Such obstacles can only push, and therefore if the equation showed 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 eciuations. At first tho system moves under certain constraints, and our equations are found on that suppo- sition. At some instant which may be determined by the vanish- ing of some reaction, ore of the bodies leaves its constraints and the equations of motion have to be changed by the omission of this 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 this kind of discontinuity can never occur. If a moving body be in contact with another, they will either separate at the beginning of the motion or will always continue in contact. 127. Suppose that in a dynamical system we have two bodies which press on each other with a reaction R; let us consider how we should form the corresponding geometrical equation. We have clearly to express the fact that the velocities of tiie points of contact of the two bodies resolved along the dire^v tion of R are equal. The following proposition will be oftei; useful. Let a body be turning about a point G with an angular velocity -j7 = f^ 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 P resolved in any direction PQ, making an angle (f) 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 pei-pendicular to CP through a space w . GP . dt Resolving parallel to PQ, the whole displacement of P = {Vco!i^(o.GPsmGPN)dt. ill : , 1 • i ' 1( i .M -, "I W \M f 106 MOTION IN TWO DIMENSIONS. If ON'=p be the perpendicular from G on PQ, we see that the velocity of P parallel to PQ is = F cos — top. It should be noticed that this is independent of the position of P on the straight line PQ. It follows that the velocities of all points in any straight line PQ resolved along PQ are the same. In practice, therefore, we only use that point in the direction of PQ which is most convenient, and this is generally the foot of the perpendicular from the centre of gravity. If (x, y, 6), {x, y\ 0) be the co-ordinates of the two bodies, q, q' the perpendiculars from the points (ar, y), [x, y') on the direc- tion of any reaction B, y^ the angle the direction of R makes with the axis of x, the required geometrical equation will be dx di , + ^ d0 di X cosylr + -f sinylr + -j-q= -^cosf dt dt^ dt j^lyl ^3 . , d& , If the bodies be perfectly rough and roll on each other without sliding, there will be two 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. This case will be considered a little further on. 128. Second Method of Solution. Suppose 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. 125, and let those of M' be obtained from these by accenting all the letters except R, i/r and t, and writing — R for Ry ifr and t being of course unaltered. Let us multiply the equation of JW by 2 -3- , 2 ^ , 2 -5- respectively, and those of M' by corresponding quantities. Adding all these six equations, we get ^^ \dt de ^ dt dt^" dt dej ^ **'• cir<( J ^-^ ' , du d& + &C. + 2B(cost| . , dy d& -2«H^^'+»°^f +?'f)- The coefficient of R will vanish by virtue of the geometrical equation obtained in the last Article. And this reasoning will apply to all the reactions between each two of the moving bodies. / THE EQUATIONS OF MOTION. 107 that the ►sition of es of all le same, direction e foot of bodies, he direc- kes with eh other f contact, ,ce of the equation If friction d a little ynamical Lch other be those led from writing multiply those of uations. )metrical ling will : bodies. Suppose the body M to press against some external fixed obstacle, then in this case B acts only on the body M, and its coefficient will be restricted to the part included iu the first bracket. But the velocity of the point of contact resolved along the direction of R must vanish, and therefore the coefficient of R is again zero. Let A be the point of application of the impressed force F, and let -4 be the velocity of .4 resolved along the direction of action oiF. Then we see that the coefficient of 2F is -4- . at df It also follows from the definition of -j that Fdf is what is called in Statics the virtual moment of the force F. "We have thus a general method of obtaining an equation free from the unknown reactions of perfectly smooth or perfectly rough bodies. The rule is, Multiply the equations having dx It' X d'y iw«^ M-r^, M-jY, Mk^-j^, &c. on their left-hand sides by df di df -jr, -J- , &c., and add together all the resulting equations for all cLZ dt the bodies. The coefficients of all the unknown reactions will be found to be zero by virtue of the geometrical equations. The left-hand side of the equation thus obtained is clearly a perfect differential. Integrating we get «m-$M^'h'^-'-^hf- where C is the constant of integration. In practice it is usual to omit all the intermediate steps and Avrite down the resulting equation in the following manner: where U is the integral of the virtual moment of the forces. This is called the equation of Vis Viva. Another proof will be given in the chapter under that heading. 129. The left-hand side of this equation is called the vis viva of the whole system. Taking any one body M, we may say thau ^dy\\ ,dtj visviva of Jf =ilf-|l-^j + -^m If the whole mass were collected into its centre of gravity and were to move with the velocity of the centre of gravity, k would be ) 1; M 108 MOTION IN TWO DIMENSIONS. m zero, and the vis viva would be reduced to the two first terms. These terms are therefore together called the vis viva of transla- tion, and the last term is called the vis viva of rotation. If V be the velocity of the centre of gravity, we may write this equation vis viva of M= Mv^ + il/A;' (-r-J • If we wish to use polar co-ordinates, we have v.vivaor^=..g)V.(t)V..(|)] . where [r, ^) are the polar co-ordinates of the centre of gravity. If p be the distance of the centre of gravity from the instanta- dd neous centre of rotation of the body, p -tt is clearly the velocity at of the centre of gravity, and therefore vis viva of J/ = ilf (p^ + A;") ^^y . The right-hand side of the equation of vis viva, after division by 2, is called sometimes the force function of the forces and sometimes the luork of the forces. It may always be obtained by writing down the virtual moment of the forces according to the rules of Statics, integrating the result and adding a constant. Frequently it is convenient to avoid introducing the unknown constant C by taking the integral between limits. We then subtract from the left side the initial vis viva, and from the right side the initial value of the force function. 130. 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 consideration. 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 that they should have the simplest possible forms. 131. In many cases a great simplification of the equations will be effected by a proper choice of the direction in which to resolve the forces, or of the points about which we take moments. First we should search if there be any directioi. in which the resolved part of the impressed forces vanishes. By resolving in this direction wc get an ccjuation which can bo immediately fl ! THE EQUATIONS OF MOTION. 109 integrated. Suppose the axis of x to be taken in this direction ; lot M, M', &c. be the masses of the several bodies, x, al, &c. the abscissae of their centres of gravity, then by Art. 123 we have ''w*''w^ which by integration gives ^S + ^'f + ..=0, = c, where is some constant to be found from the initial conditions. This equation may also be again integrated if required. This result might have been derived from the general princi- ples of the conservation of the translation of the centre of gravity laid down in Art. 79. For since there is no impressed force parallel to the axis of x, the velocity of the centre of gravity of the whole system resolved in that direction is constant. 132. Next we should search if there be any point about which the moment of the impressed forces vanishes. By taking moments about that point we again have an equation which admits of immediate integration. Suppose this point to be taken as origin, and the letters to have their usual meaning, then by the first article of this chapter we have w^^- yf,)^MU ^1 = tlie % referring to summation for all the bodies of the system Integi'ating as in Art. 124 we have {^(^l-/|)+^43=^- where C is some constant to be determined by the initial condi- tions of the question. This equation expresses that if the impressed forces have no moment about any point, the angular momentum about that point is constant throughout the motion. This result follows at once from the reasoning in Chap, li. 133. A homogeneous sphere rolls directly down a perfectly rough inclined plane under the action of gravity. Find the motio7i. Let a he tlie inclination of the plane to the horizon, a the radius of the sphere, mJc'^ its moment of inertia about a horizontal diameter. Let be that point of the inclined plane which was initially touched by ti.ri sphere, and N the point of contact at the time t. Then it is obviously convenient to choose for origin and OIV for the axis of ,«■. I If \\ \ :' ■, 1 i I I 110 MOTION IN TWO DIMENSIONS. The forces which act on the sphere are first the reaction R perpendicular to ON, secondly, F the friction acting at N along NO and mg acting vertically at C the centre. The effective forces ^ro m -^ , m -A acting at C parallel to the axes of x and y d'0 and a couple mk* -z-^ tending to turn the sphere round C in the direction NA, '^ Here 6 is the angle any fixed straight line in the hody makes with a fixed straight line in space. We shall take the fixed straight line in the body to be the radius CA, and the fixed straight line in space the normal to the inclined plane. Then is the angle turned through by the sphere. Resolving along and perpendicular to the inclined plane we have m-^=mgBxaa-F (1), m ■^= -mg COB a + R (2). Taking moments about N to avoid the reactions, we have ""^d^"*"^ ^~'"^**'"*" ^ '" 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=ae (4). Also because there is no jumping y=a (5). Both these equations are of the form described in the first method. Differcn- tiating (4) we get j^ =<* ^ • Joining this to (3) we have diJ = a-«T*«^'^°" (^^- 2 Since the sphere is homogeneous, Ji^=^a^, and we have 5 d^x 5 . If the sphere had been sliding down a smooth plane, the equation of motion would have been d'x EXAMPLES. Ill 80 that two-sevenths of gravity is used in tnming the sphere, and five-sevenths in urging the sphere downwards. Supposing the sphere to start from rest we have clearly 1 6 «=2 and the whole motion is determined. jflfsino.e', In the above solutions, only a few of the equations of motion have been used, and if only the motion had been required it would have been unnecessary to write down any equations except (3) and (4). If the reactions also be required, we must use the remaining equations. From (1) we have From (2) and (5) we have F-=mg Bin a. R=mg COB a. It is usual to delay the substitution of the value of k^ in the equations until the end of the investigation, for this value is often very complicated. But there is another advantage. It serves as a verification of the signs in our original eqiiations, for if equation (6) had been we should have expected some error to exist in the so! ition. 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 sin a, show that the angular velocity of the sphere after a time t from rest would be 5/1 g cos a •i l1 11 ■1l '1 ' n i t. 134. A homogeneous sphere vols 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 = / BOO. Let F and B be the friction and the normal reaction at N. Then resolving tangentially and normally to the path of C we have nm Vi »1 i ' . I! 'I 112 MOTION IN TWO DIMENSIONS. (a + 6)^=flrsm^-- (1), (« + ^)(ty=^°°«^-^ <2). Let ^ be that point of the moving sphere which originally coincided with £. Then if bo 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 dt^~mk'^ ^ '' It should be observed that we cannot take as the angle ACO because, though CA is ^xed in the body, CO is not fixed in space. The geometrical equation is clearly a(e-) = b

Y 10 o ,, the rolling body being supposed to start fron rest at a point indefinitely near B, This result might also have been deduced from the equation of vis viva. Tho vis viva of the sphere is m | v" + P ( -37 ) | and r = (a + 6) -^ . The force function is m lgdy=mffy if y be the vertical space descended by the centre. We thus have <" + ^)'' (5)' + *"©' = ^^^" + ^^ (1-COS0), which is easily seen to lead, by help of (4), to the same result. To Und whero the body leaves the sphere we must put R=0. This gives by (2) {a + h)(-^\=gcos3_ yn increatdd by M — and that the p-a radius of curvature of the curve at the point of contact. 135. A rod OA can turn about a hinge at 0, lohile the end A rests on a smooth toedge which can slide along a smooth horizontal plane through 0. Find the motion. Let a=the inclination of the wedge, il!f=its mass and «= OC. ;: ' \ have left the R. D ^njaatti ^! I ;i U 114 MOTION IN TWO DIMENSIONS. Let l=ihe length of the beam, m=itB mass aud 0=A OC. Let iZ- the reaction at 4. Then we have the dynamical equations, d'x _ iZ sin o ,, , dt^-~M~ ^^'• ,„ M . cos (a-O)- mg ^ cos 6 d£^~ mk^ _ ■ ^" and the geometrical equation, x = -. — .siu(o-e) (3). sma ' It is obvious we must apply the second method of solution. Hence ...dxd^x „ ...dedW , „de „„\ . dx . , ..d0\ ^'^didt^+^'''^dlW'=-'''^^''''^dt+^^n''dt-^^'^'^''-^^di\' The coefficient of R is seen to vanish by differentiating ec[uation (3). Inte- grating we have This result might have been written down at once by the principle of vis viva. For the vis viva of the wedge is clearly M{-r-\ and that of the rod Mk^ \dt/' The virtual moment of the forces is - mgdy where y is the altitude above OC of the centre of gravity of the rod OA, hence twice the force function is C-2mgy. Since y=^l sin 0, this reduces to the result already written down. Substituting from (3) we have \M^^^,-coBHa-e) + mh-'\(~^y=C-mglBm0 (4). If the beam start from rest when 0=p, then C=mgl sin /3. This equation cannot be integrated any further. We cannot therefore find in terms of t. But the angular velocity of the 'jeam, and therefore the velocity of the wedge, is given by the above equation. 136. Two rods A B, BC are hinged together at B and can freely slide on a amooth horizontal plane. The extremity A of the rod AB is attached by another hinge to a fixed point on the table. An elastic string AC, whose unstretched length is equal to A B or BC, joins A to the extremity C of the rod BC. Initially the two rods and the string form an equilateral triangle and the system is started with an angular velocity CI round A . Find the greatest length of the elastic string during the motion. Find also the angular velocities of the rods when they are at right angles^ and the least value of Q that this may be possible. Let the length of either rod be 2a, mP the moment of inertia of cither about its centre of gravity, so that k^=~ . Let D and E be the middle points of the rods, and let (r, 6) be the polar co-ordinates of E referred to A as origin. The only forces on the system are the reaction of the hinge at A and the tension of the elastic string A C. If we search for any direction in which the sum of the resolved parts of these vanishes, we can find none, since tho direction of the EXAMPLES. 115 (1). (2), .(3). di\' (3). Inte- of vis viva. e OC of the mgy. Since '©•• (4). )re find in locity of the / slide on a by another died length ally the two ted with an g during the •ight angles^ ler about its of the rods, the tension sum of the ion of the reaction is at present unknown. But since the lines of action of both these forces pass through A, their moments about A vanish, and therefore, by Art. 132, the angular momentum about A is constant throughout the motion aud equal to its initial value. Let u, u'A . The vis viva of AB is by the same article m (P + a") w" since it is turning round 4 as a fixed point. The initial values of these are respectively m (Sa" + k^) fl* and m (/fc" + a") QK If T be the tension of the string, p its length at time /, the force function of the tension is X{- T)dp. According to the rule given in Statics to calculate virtual moments, the minus sign is given to the tension because it acts to diminish p; and the limits are 2a to p because the string has stretched from its initial length 2a to p. By Hooke's law T=E ^ ^ "" y so that, by integration, the force .unction= -E-~ 2^ , , ^j 6- . -" -- 4„ • The reaction at A does not appear by Art. 128. The equation of vis viva is therefore m(;!^ + a')wHmj^|^y + >-2(^y + JfcV2J=wi(2i!;' + 4a'')n'-£^^'^ (2). There are only two possible independent motions of the rods. We can turn A B about A and BC about B, all other motions, not compounded of these, are incon- sistent with the geometrical conditions of the question. Two dynamical equations 8—2 116 MOTION IN TWO DIMENSIONS. are snffioient to dotemaine these, and these we have jnst obtained. All the other equations which may be wanted must be derived from geometrical considerations. We must now express the geometrical conditions of the question. Let ^ bo tho supplement of the angle ABC, then rf* r'=5o' + 4o'cos^ Since ^ is the relative angular velocity of the rods BC, AB, dt = w - w dr di' - 2a' Bin (w' - «) Let \f> be the angle EAB, then sin f = sin .(8). .(4), .(6). (C), d^p de , and smce 37 = 37 - w, we have dt dt +-^Bin^ p> ^ and -77 . (tt 0ib (8). Wo shall d then have a differential equation of the first order to solve, containing ^ and -~ Hi' It is required to find the greatest length of the elastic string during the motion. At the moment when p is a maximum, -i^=0 and tho whole system is therefore moving as if it wt.e a rigid body. We therefore have for a single moment w, u' and ■^ all equal to each other and ^7=0. The two first equations become, when we have substituted for I? its value — , (6a«+3r2)w=14««0 3J? I- (Sa" + 3r«) w2 = 14a« 0* - ^ (/) - 2af Eliminating w and substituting for r from (8) we have the cubic (3p»+16a«){p-2a) = 2i^'. (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 angles. At this moment =^ and hence by (3) r = a V^. by (5) -3^ = - -j= a{u'-u), do 1 ^y (7) -J- = H (w' + 4«). Substituting in equations (1) and (2) we get at 5 17 EXAMPLES. 117 From those two equations we may easily find u and (■/. It is easily seen that the 10 E values of u, u' will not be real unless 0'>t;^ (\/2 - 1)*. 7 ma ^ ' We moy often save ourselves the trouble of some elimination if wo form tho equations derived from the principles of angular momentum and vis viva in a slightly different manner. Tho rod BC is turning round P with an angular velocity w', while at the same time B is moving perpendicularly to AB with a velocity 2au. The velocity of E is therefore the resultant of aw' perpendicular to BC and 2au per- pendioiUar to AH, both velocities, of course, being applied to the point E. When we wish our results to be expressed in terms of u, u' we may use these velocities to express the motion of E instead of the polar co-ordinates (r, 0). Thus in applying the principle of angular momentum, we have to take tho moment of the velocity of E about A . Since tho velocity 2au is perpendicular to AB, the length of the perpendicular from A on its direction isAB together with the projection of BE on A B, v^hich is 2a + a cos if>. Since the velocity att/ is perpen- dicular to BE, the length cf tho perpendicular from A on its line of action is BE together with the projectioa of ^B on BE, which is a + 2a cos ^. Hence the angu- lar momentum of the rod BC about A is, by Art. 124, ink"- w'+ 2inau (2a + a cos ) + mau' {a + 2o cos 0). The principle of augula'" momentum for the two rods gives therefore m (P + Sa" + 2a' cos 0) w + wi (i» + a" + 2a' cos 0) w' = m (2/fca + 4a') 0. Tho 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 0= - 4 and w = w'=0. In applying the principle of vis viva, we require the velocity of E. Begarding it as the resultant of 2au and au' we see that, if v be this velocity, «' = (2aw)' -f- (aw')'' + 2 . 2aw . ow' cos . The initial value being found, as before, by putting cos 0= - J, «=«'=0, the princi- ple of vis viva gives, by Art. 129, ,(/)-2a)a m (A" + 5a') oP + m (A' + a') w" + ima^ uu' cos = m (2i' + 4a«) 0' - £ ' 2a The force function is foimd in the same manner as before. If we join to this equa- tion (4) given above, and substitute p=4acos --, we have just three equations to find u, w', and if>. If these quantities are all that are required, as in the two cases con- sidered above, this form of solution has the advantage of brevity. When p is a maximum, we put w=w', when the rods are at right angles, we put cos 0=0. The equations then lead to the results already given. 137. The boh of a heavy pendulum contains a spherical cavity tohich is filled with water. To determine the motion. Let be the point of suspension, the centre of gravity of the solid part of the pendulum, MK^ its moment of inertia about and let 00=h. Let C 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. i j I f i % 118 ' II ' i MOTION IN TWO DIMENSIONS. The ofTtictivo forccB of the water are by Art. 128 equivalent to the efTcctive force o! the whole mass collected at ItH centre of grovity together with a couple niA' - where w in the angnlar velocity of the water, and m/t' itn moment of inertia ahout a diameter. But u has jUHt been proved zero, hence thiw couple may be omitted. It followH that in all problems of this kind wliere the body does not turn, or turns with uniform angular velocity, we may collect the body into a .lingle particle placed at its centre of gravity. The pendulum and the collected fluid now form a rigid body turning about n fixed axis, hence if be the angle CO a fixed line in the body makes with the vertical, the equation of motion by Art. 88 is d^0 (MK^ + «ic«) ^ + (Mh + mc) g sin = 0, where in finding the moment of gravity, 0, and C Lave been supposed to lie in a straight line. The length L' of the simple equivalent pendulum is, by Art. 92, ~ Mh + mc Let ml' bo the moment of inert' \ of the sphere of water about a diameter. Then if the water were to become solid and to bo rigidly connected with the case, the length L of the simple equivalent pendulum would bo, by similar reasoning. L = Mh+mn It appears that L''' + c'). Tho moment of inertia of the fluid coliocttMl into its ctMitro is ^ttJ'.c*. When wo add those togothor o disappoars, so that tho whole moment of inertia is independent of tho position of the cavity. The motion of a uniform triangular area moving under tho action of gravity is another example. If we replace the area by throe wires forming its perimeter hut without weight, the geome- trical conditions of the motion will in general be inialterod, and if we also place at the middle points of these wires three weights, each one-third of the mass of the triangle, this body will have all its characteristics the same as that of the real triangle, and may replace it in any problem. When a string connecting two parts of a dynamical system passes over a rough pulley, it was formerly the custom to con- sider tho inertia of rotation of the pulley by replacing it by another pulley of the same size but without mass and loaded with a particle at its circumference. If a be the radius of the pulley, k its radius of gyration about the centre, m its mas.s, the mass of the particle is -jj^». so that in a cylindrical pulley the mass of the particle is half that of the pulley. This mass must then be added on to the other particles attached to the string. For example, if two heavy masses j\f, M' be connected by a string passing over a cylindrical pulley of mass m, which can turn freely about its axis, the ecpiation of motion is h {M^M'^f^'%^(U-M'), wliere v is the velocity. Here the inertia of tlio pulley is taken account of by simply adding - - to the mass moved. If the pulley be moveable in space as well as free to rotate, its inertia of trans- lation is as usual taken account of by collecting the whole mass into its centre of gravity. As this representation of the inertia of rotation is not often used now, the demonstration of the above remarks, if any be needed, is left to the reader. Ex. 1. A rod AB whose centre of gravity is at t'ae middle point C oi AB has its extremities A and B constrained to move along t'lro straight lines Ox, Oy inclined at right angles and is acted on hy any forces. Shew that the motion is the same as if the whole mass were collected into its centre of gravity and all the forces reduced A" in the ratio 1 + — j : 1 where 2a ia the length A B and it is the radius of gyration about the centro of gravity. ti. n 120 MOTION m TWO DIMENSIONS. I: -h r I'ij i ^ir K( Ex. 2. A cironlar disc whose centre of gravity is in its centre rolls on a perfectly rough curre under the action of any forces, she^ that the motion of the centre is the same as if tha curve were smooth and all the forces were reduced in the ratio 1 + — : 1, where a is the radius of the disc and k is the radius of gyration about the a* centre. But the normal pressures on the curve in the two cases are not the same. In any position of the disc they differ by X yzJa ^^^^^ ^ is the force on the disc resolved along the normal to the rough curve. On the stress at any point of a rod. 139. Suppose a rod OA to be in equilibrium under the action of any forces, it is required to determine the action across any section of the rod at P. This action may be conceived to be the resultant of the tensions positive or negative of the innumerable fibres which form the material of the rod. All these we know by Statics may be compounded into a single force B and a couple O acting at any point Q we may please to choose. Since each por- tion of the rod is in equilibrium, these must also be the resultants of all the external forces which act on the rod on one side of the section at P. If the section be indefinitely small it is usual to take Q in the plane of the section, and these two, the force R and the couple G, will together measure the stress''^ at the section. If the rod be bent by the action of the forces, the fibres on one side will all be stretched and on the other compressed. The rod will begin to break as soon as these fibres have been sufiici- ently stretched or compressed. Let us compare the tendencies of the force B and the couple G to break the rod. Let A be the area of the section of the rod, then a force F pulling the rod will cause a resultant force R= F, and will produce a tension in the F fibres which when referred to a unit of area is equal to -j . The same force F acting on the rod at an arm from P whose length is p, will cause a couple O = Fp, which must be balanced by the couple formed by the tensions. Let 2a be the mean breadth of the rod, then the mean tension referred to a unit of area produced F v by is of the order 7 . - . Now if the section of the rod be very small - will be large. It appears therefore that the couple, when it exists, will generally have much more effect in breaking the * Sir W. Thomson has appropriated the word strain to the alterations of volume and figme produced in an clastic body by the forces applied to it, and the word Ktress to the elastic pressures. ON STRESS. 121 a perfectly e centre is [u the ratio 1 about the t the same. on the disc he action jross any to be the umerable know by couple G jach por- ■esultants ie of the usual to force R * at the fibres on bd. The n suffici- encies of be the rod will n in the r. The e length by the eadtli of jroduced be very le, when :ing the of volume the word rod than the force. This couple is therefore often taken to measure the whole effect of the forces to break the rod. The " tendency to break" at any point P of a rod. OA of very small section is measured by the moment about P of all the forces which act on either of the sides OP or PA of the rod. The resolved part of the force B perpendicular to the rod is called the shear. This is therefore equal to all the forces which act on either of the sides OP or PA resolved perpendicular to the rod. If the rod be in motion the same reasoning will, by D'Alem- bert's principle, be applicable ; provided we include the reversed effective forces among the forces which act on the rod. In most cases the rod will be so little bent that in finding the moment of the impressed forces we may neglect the effects of curvature. If the section of the rod be not very small, this measure of the "tendency to break" becomes inapplicable. It then becomes necessary to consider both the force and the couple. This does not come within the limits of the present treatise, and the reader is referred to works on Elastic Solids. In the case of a string the couple vanishes and the force acts along a tangent to the string. The stress at any point is there- fore simply measured by the tension. 140. A rod OA, of length 2a, and mass m, which can turn freely about one extremity 0, falls under the action of gravity in a vertical plane. Find the " tendency to break" at any point P. Let du be any element of the rod distant u from P and on the side of P nearer the end A of the rod, and let OP=x. Let be the angle the rod makes with the vertical at the time t. The effective forces on du are hVi m du, ^ .d^0 , du, , ,fdey -(x+«)^ and -m^^ («: + «) (^^j du respectively perpendicular and along the rod. The impressed force is wi — g acting ^a vertically downwai'ds. The effective forces being reversed the tendency to break at P is equal to the moment about P of all the forces whiclr act on the part PA of the rod. If this be called L, we have the limits being from u^ to «= 2a - x. Also taking moments about 0, the equation of motion is W> d^d Hence we easily find m. -^ ^/a - "*5"* ^"^ ^'' innmnB ,„ ^ I 122 MOTION IN TWO DIMENSIONS. The meaning of the minus sign is that the forces tend to bend PA round P in the opposite direction to that in which has been measured. To find where the rod supposed equally strong throughout is most likely to break, we must make L a maximum. This gives -7- =0 and therefore x=-rr. The ' ax A point required is at a distance from the fixed end equal to one-third of the length of the rod. This point, it should be noticed, is independent of the initial conditions. To find the shear at P we must resolve perpendicularly to the rod. If the result be called 1", we have du . . r du , . d-$ the limits being the same as before. This gives mg sin 9 Y= lOa" (2a - x) (2a - 3x), I I which vanishes when the tendency to break is a maximum, and is a maximum at a distance from the fixed end equal to two- thirds of the length of the rod. To find the tension at P we must resolve along the rod. If the result be called X, we have ^ r du ^ r du , , (deY ^= -> 27*^"°' ^ + >2-a (^ + «) \dt) ' If the rod start from rest at an inclination o to the vertical, we find, by integrating the equation of motion, ( -jr ) = k^ (cos a - cos 6), Hence X=^^(2a-x){- 4a cos ^ -1- 3 (cos a - cos 9) (2a f a;)}. From these equations we may deduce the following results. (1) The magnitudes of the stress couple and of the shear are independent of the initial conditions. (2) The magnitude of either tho couple or the shear at any given point of the rod varies as the sine of the inclination of the rod to the vertical. (3) The ratio of the magnitudes of the stress couples at any two given points of the rod is always the same, and the same proposition is also true of the shear. (4) The tension depends on the initial conditions and unless the rod start from rest in the horizontal position, the ratio of the tensions at any two given points varies witl- the position of tho rod. 141. A rigid hoop complctehj cracked at one point rolls on a perfectly rough horizontal plane and is acted on by no forces but gravity. Prove that the wrench couple at the point of tlic hoop most remote from the crack tcill he a maximum ivhen- ever, the crack being lower than the centre, the inclination of the diameter through 2 the crack to the horizon is tan~^~ . [The Math. Tripos, 1864.] TT Let u be the angular velocity of tho hoop, a its radius. The velocity of any point P of the hoop is the resultant of a velocity aw parallel to the horizontal pliiue and an equal velocity aw along a tangent to the hoop. Tlio first is co^istant in direction and magnitude and therefore gives nothing to tho acceleration of P. The latter is constant in magnitude but variable in direction and gives aw' as the acceleration which is directed along a radius of the hoop. Lot A be tue cracked point, /i tho other end of tho diameter, V tho centre,, tho inclination of ACfi to ON STRESS. 123 tho horizon. Let PP" be any element on the upper half of the circle, BCP=(t>. Then the wrench couple, or tendency to break, at B is proportional to / [ - aw' a sin + gr {a COS ^ - a cos { = - 2a''w^ + ga^ (cos ^jr + 2 sin 0). •'0 2 * This is a maximum when tan — -, Ex. 1. A semicircular wire ^ J5 of radius a is rotating on a smooth horizontal plane about one extremity A with a constant angular velocity w. If a = ir-. If the extremity £ be suddenly fixed and the extremity A let go, prove that the tendency to break is greatest at a point P where ^ tan PBA = PBA. Ex. 2. Two of the angles of a heavy square lamina, a side of which is a, are connected with two points equally distant from the centre of a rod of length 2a, so that the square can rotate about the rod. The weight of the square is equal to the weight of the rod, and the rod when supported by its extremities in a horizontal position is on the point of breaking. The rod is then held by its extremities in a vertical position, and an angular velocity w is then impressed on the square. Shew that it will break if « > V'i- [Coll. Exam.] Ex. 3. A wire in the form of the portion of the curve r=a (1 + cos 0) cut off by the initial line rotates about the origin with angular velocity w. Prove that the TT 12 v/2 „ tendency to break at the point tf=^ is measured by m — ^ — w^a'. [St John's CoU.] I' y of any tal plane taut in P. The as the cracked AC/i to On Friction hetiveen Imperfectly Bough Bodies. 142. 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 be- tween 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 com- pression. The whole of the actions across all the elements are equivalent to (1) a component R, normal to the common tan- gent plane, and usually called the reaction; (2) a component # in the tangent plane usually called the friction ; (3) a couple L about an axis lying in the tangent plane and which we shall call the couple of rolling friction ; (4) if the bodies have any relative angular velocity about their common normal, a couple N about this normal as axis which may be called the couple of twisting friction. 124 ^•OTION IN TWO DIMENSIONS. 143. These two coupres 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. 144. When one body presses against another over any small area, the force of friction acts in such a direction and with such a magnitude that it is just sufficient to prevent sliding. Both the magnitude and direction of friction may. therefore, be unknown beforehand, and their determination will be part of the problem under consideration. It is found by experiment that no more than a certain amount of friction can be called into play, and when more is required to keep the bodies from sliding on each other, sliding will begin. T'lis amount is called limiting friction. The magnitude of this limit is found to bear a ratio to the normal pressure which is rery nearly constant for the same two bodies. Though all experimenters have not entirely agreed with each other as to the accuracy of this result, 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, this ratio is nearly independent of the extent of the area and of the relative velocity. If, however, the bodies have remained in contact for some time under pressure in a position of equilibrium, it is found that, for the more compressible bodies, the ratio is a little greater than after motion has begun. This ratio has been called the coefficient of friction of the materials of the two bodies. Its constancy is generally assumed by mathema- ticians. When the friction which can be called into play is insuf- ficient to prevent sliding, the bodies slide on each other. In this c-,ise the magnitude of the friction is equal to its limiting value, and the direction of the friction is opposite to that of relative motion. 145. If the bodies be perfectly rough, the coefficient of friction is infinite, and there is no limit to the amount of friction which' can be called into play. There can, therefore, be no sliding be- tween the bodies. 146. Discontinuity of motion will often occur when a body moves under the action of friction. Suppose the body rolls on a rough surface, the friction called into play just prevents sliding, and is possibly variable in magnitude 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 coefficient fi 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 should be greater than the co- to be in a certain lecessary Qy small h such a Both the inknown problem no more lay, and on each friction. e normal bodies. ich other ally that, points of extent of lies have position e bodies, n. This terials of athema- is insuf- In this ig value, relative friction n which ing be- a body lis on a sliding, writing ratio of } always always surface. at any the co- IMPERFECT FRICTION. 121 i efficient of friction, the point of contact will begin to slide at that moment. Jn this case the equations do not represent the true motion. To correct them we must replace the unknown friction F by fiR, and remove the geometrical equation which expresses the fact that there is no slipping between the bodies. The ecjua- tions must now be again solved on this new supposition. It is of course possible that another change may take place. If at any instant the velocities of the points of contact become equal to each other, all the possible friction may not be called into play. At that instant the friction ceases to be equal to fiR and becomes again unknown in magnitude and direction. 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 .he normal reaction multiplied by the c efficient of friction. If then, during the course of the motion the direction of the normal reaction should change sign, while the direction of motion remains un- altered ; or if the direction of motion should change sign Awhile the normal reaction should remain unaltered, the sign of the coefficient of friction must be changed. This may modify the dynamical equations and alter the subsequent motion. The same cause of discontinuity operates when a body moves in a resisting medium, when the law of resistance is an even function of the velocity, or any function which does not change sign when the direction of motion is changed. In some cases the motion may be rendered indeterminate by the introduction of friction. Thus, we have seen in Art. Ill, 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 was no indeterminateness in the motion. If however these hinges were imperfectly rough, there would be two friction couples, one at each hinge, acting on the body. The common axis of these couples would be the straight line joining the hinges. The magnitude of each would be- equal to the pressure resolved along its axis 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. 147. A homogeneous sphere is placed at rest on a rough inclivnd plane, the copffieient offrictio.t being /i, determine whether the sphere tvill slide or roll. Let F be the friction required to make the sphere roll. The problem then F becomes the same as that discussed in Art. 133. We have, therefore, - =^ tan a, whore a is the inclination of the plane to the horizon. 126 MOTION IN T'VO DIMENSIONS. m If then I tan a be not greater than /tc, the solution given in the article referred to is the correct one. But it /k^ tan a the sphere will begin to slide on the inclineJ plane. The subsequent motion will be given by the equations m j-j =nig sina-fxR 0= -mg cos a + R vw -, ji- + vik^ ^ = mga f in o dfi dt^ whence we have, rememberiug that k^ = i a', d-x ^.i= 9 (sin a -n cos a) d^0 . g ^, = |M-cosa Since the snhere starts from rest, we have by integration aj^Jf^t" (sino-/xcosa) \ e=^fji.^t'^ cos a j The Velocity of the point of the sphere in contact with the plane is dx 10 ^ , . . . But since, by hypothesis, n 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. 148. A homogeneous sphere is rotating about a horizontal diameter, aiid is gently placed on a rough horizontal plane, the coefficient of friction being /i. Deter- mine the subsequent motion. Since the velociiy of the point of contact with the horizontal plane is not zero, the sphere will evidently begin to slide, and the motion of its centre will be along a straight line perpendicular to the initial axis of rotation. Let this straight line be taken as the axis of x, and let 6 be the angle between the vertical and that radiiis of the sphere which was initially vertical. Let a be the radius of the sphere, mJc^ its moment of inertia about a diameter, and Q the initial angular velocity. Let R be the normal reaction of the plane. Then the equations of motion are clearly d^x m^^ = y.R Q=mg-R \ m: (1), whence we have d^ dt^'' ■M d^e , g (2). ,d9 Integrating, and remembering that the initial value of — is 0, we have x = \ixgt'^ ) * a .(S). referred ) on the ir vanish, has thus ', and is . Deter- not zero, along a line be radius of mk^ its Jet R be ... (1). (2). IMPERFECT FRICTION. 127 But it is evident that these equations cannot represent the whole motion, for they would make j- , the velocity of the centre of the sphere, increase continually. This is quite contrary to experience. The velocity of the point of the sphere in contact with the plane is dx do .-. , This vanishes at a time ti=; .(4). ■(6). (8). At this instant the friction suddenly changes its character. It now becomes only of sufficient magnitude 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 dH _ -\ = m(/-R and the geometrical equation will be a; = a9. 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 noi:e will be called into play. The sphere will therefore move uniformly with the velocity which it had at the UX time t^. Substituting the value of t^ in the expression for ^ obtained from equa- (It tions (3) we find that this velocity is f aO. 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 f aQ. It may be remarked that this velocity is independ- ent of /x. If the plane be perfectly rough, n is infinite, and the time t^ vanishes. The sphere therefore immediately begins to move with a uniform velocity =f aO. 149. In this investigation the couple of rolling friction has been neglected. Its effect would be to diminish the angular velocity. The velocity of the lowest point of the sphere would then tend to be no longer zero, and thus a small sliding friction will be required to keep that point at rest. Suppose the moment of the friction- couple to be measured by Jmg, where / is a constant. Introducing this into the equations (5) the third is changed into mP - , = -Fa -frng, the others re:aainiug imaltered. Solving these as before we find We see from this that F is negative and retards the sphere. The effect of the couple is to call into play a friction-force which gradually reduces the sphere to rest. As tho sphere moves in the air we may wish to determine the effect of its resist- ances. The chief part of this resistance may be pretty accurately represented by a t V li I? ■J i \m mm 128 MOTION IN TWO DIMENSIONS. force m/3 - 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 will be also a small friction between the sphere and 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-x..v/to_tan- 7^^= .^^^f^,, \' fg V fg a' + k* where V is the velocity of the sphere at the epoch from which t is measured. 150. In order to determine by experiment 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 + phe 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= aud by taking moments L =pgr. Now in the experiments of Coulomb and Morin p was found to vary as the novmal pressure directly, and as r inversely. When p was great enough to set the cylinder in motion. Coulomb found that the acceleration of the cylinder was nearly constant, and thence we may conclude 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 con- tact, 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 bodi-,;^ which touch at one point only and have their curvatures in all direc- tions unequal. But since the magnitude of the couple is indepen- dent 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. In order to test these laws of friction let us compare the results of the following problem with experiment. > being the ) density of ere and air represented tude. The ired. nitude of placed on ses are P ! cylinder !. Let F tact A of )e at first st moves. resolving )w in the ry as the was great that the hence we it of the at of the re similar fnitude is a certain 3 limiting tant ratio called the Is in con- 3, and, in i-^ which all direc- I indepen- that the pie, is the ipare the IMPERFECT FRICTION. 129 151. A carnage on n pairs of wheels is dragged on a level horizontal plane by a horizontal force 2P tcith uniform motion. Find the magnitude of P. Let the rndii of the wheels be respectively r,, r^, &c., their weights w^, u\, &o., and the radii of the axles p^, p^, &o. Let 2Wbe the whole weight of the carriage, 2Qp 2Qjj, &c. the pressures on the several axles, so that W= ZQ. Let the pressures between the wheels and axles be Ri, i?a, &c. and the pressures on the ground Ii\, R\, &c. Let C be the common centre of any wheel and axlo, P their point of con- tact, and A the point of contact of the wheel with the ground. Let the angle ACP = 6 supposed positive when P is behind AC. Let p. be the coeflacient of the force of sliding friction at P and / the coefficient of the couple of rolling friction at A. The equations of equilibrium for any wheel, foimd by resolving vertically and taking moments about A, are R'=q + ic (1), p.R{rQos0-p)-Rr&iue=fR' (2). The friction force at A does not appear because we have not resolved horizontally. The equations of equilibrium of the carriage, foimd by resolving vertically and hori- zontally, are Rco^0 + p.Rmne = Q (3), "L (R sin. e-iiR COS d)-\-P = Q (4). The effective forces have been omitted because the carriage is supposed to move uniformly, so that the M-r.ol the carriage and the mh"^ — of the wheel are both Up (It zero. The first three of these equations give by eliminating R and R' u(costf--)-sintf ,, . costf+/isintf r\ QJ ^ This gives the value of 9. In most wheels - and ^ are both small as well as /. In ° r Q such a case /* cos tf - sin is a small quantity. If therefore ja=tane we have d=e very nearly. The third and fourth of these equations give by eliminating R p^^ HCOBO-Bine p. sin + cos (/tsinff + cos^r r ) by equation (5). If - be small, it will be sufficient to substitute for 6 in the first term its approximate value e. This give"? i.=ZJ,m.5«^/«ll"i (6). Here we have neglected terms of the order y-jQ' If all the wheels are equal and similar we have, since 2Q= TV, P=sinejt7+/^ (7). Thus the force required to drag a carriage of given weight with any constant velocity is very nearly independent of the number of wheels. R. D. 9 ! P\ W: \m 130 MOTION IN TWO DIMENSIONS. In a gig tho wheels are usually larger than in a four-wheel carriage, and there- fore the force of traction is usually less. In a four-wheel carriage the two fore wheels must be small in order to pass under the carriage when turning. This will cause the term sin e - Qj in the expression for P containing the radius r^ of the fore wheel to be large. To diminish the effect of this term, tho load should bo eo adjusted that its centre of gravity is nearly over tho axle of the large wheels, tho pressure Qj in tho numerator of this term will then be small. A variety of experiments were made by a French engineer, M. Morin, at Metz in the years 1837 and 1838, and afterwards at Coiu-bevoie in 1839 and 1841, with a view to determine with the utmost exactness the force necessary to drag carriages of different kinds over the ordinary roads. These experiments were undertaken by order of the French Minister of War, and afterwards under the directions of the Minister of Public Works. The eiiect of each element 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 tho same state of moistvire. Then the weight being the same, wheels of different radii but the same breadth were used, and 80 on. The general results were that for carriages on equal wheels, the resistance varied as the pressure directly and the diameter of the wheels inversely, and was independ- ent of the number of v jieels. On wet soils the resistance increased as the breadth of the tire was 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 wore many inequalities on the road. As an approximate result it was found that the resistance might be expressed by a formula of the kind a + bV, where a and b are two constants depending on the nature of the road and the stiffness of the carriage, and V is 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 de 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. Ex. 1. A homogeneous sphere is projected without rotation directly up an imperfectly rough plane, the inclination of which to the horizon is a, and the coefficient of friction fi. Show that the whole time during which the sphere ascends the plane is the same as if the plane were smooth, and that tLe iiuie 2 tan a during which the sphere slides is to the time during which it rolls as 1 : 7 fi Ex, 2. A homogeneous sphere of mass M is placed on an imperfectly rough table, the coefficient of friction of which is fi. A particle of mass m is attached to the extremity of a horizontal diameter. Show that the sphere will begin to roll or slide according as u is greater or less than „ .,„ ,„., s . this value, show that the sphere will begin to roll. If fjL be equal to Ex. 3. A rod AB has two small rings at its extremities which slide on two rough horizontal rods Ox, Oy at right angles. The rod is started with an angular velocity when very nearly coincident with Ox, show that if the coefficient of fric- IMPULSIVE FORCES. 131 and there. le two foro This will B Ti of the ihoultl be Fo wheels, the , at Metz in 1841, with n rag carriages clertaken by itions of the 1 separately, the effect of ). Then the ere used, and stance varied ras independ- i the breadth odent of the a gallop, the on solid roads he road. As pressed by a on the nature ogether agree illy affect the amentales de tend to con- rectly up an a, and the the sphere hat iliii vlme 2 tan a rfectly rough s attached to !gin to roll or be equal to slide on two h an angular icient of fric- tion is less than ^2, the motion of the rod is given by fl = ■ ^ log f 1 + ' ^ , J until 2 tan 6=-, and that when the rod reaches Oy, its angular velocity is w, where ill What is the motion if /u' > 2 ? (2-/i«)(4-/x'0* On Impulsive Forces. 152. In the case in which the impres.sed forces are impulsive the general principle emuiciated in Art. 123 of this chapter re- quires but slight modification. Let {u, v), {ii, 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 CO and &>' be the angular velocities of the body about the centre of gravity at the same instants. And let MJc' be the moment of inertia of the body about the centre of gravity. Then the effective forces on the body are equivalent io two 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 a couple measured by M¥{w'-(o). 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. In many cases it will be found that by the use of Virtual Velocities the elimination of the unknown reactions may be effected without difficulty. 153. A string is wound round the circumference of a circular reel, and the free end is 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 vwtion being supposed to take place in one plnne, 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', u 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 radias. In order to avoid introducing the imknown 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 + mKW=0 (1). y— 2 ^1 132 MOTION IS TWO DIMENSrONfl. JuHt after the impact tlio part of the reel in contact with the string has no velocity. Henco v'-aw'^O (2). av Solving these we have w' = a-+k-' If the reel be a homogeneous cylinder 2v k^ = -x , and in this case wo have w'= ^ , f = r, impulsive tension, we have resolving vertically 7/i(v'~v)= - T. If it bo required to find the (3). Hence To find the subsequent motion. The centre of the reel bcyins to descend verti- cally, and there is no horizontal force on it. Hence it will continue to descend in a vertical straight lino, and throughout all the subsequent motion the string is vertical. The motion may therefore be easily investigated as in Art. 18.S. If we put o = 7i I and let F=tho finite tension of the string, it may be shown that F=one- 2 third of tlie weight, and that the reel descends with a uniform acceleration = 5 f;. o The initial velocity of the reel has been found in this article =v', so that the space 1 2 descended in a time t after the impact is =v't + - . -yt^. Ex. 1. An inelastic sphere of radius a sliding on a smooth horizontal plane impinges on a fixed rough point at a height c above the plane, show that if the velocity of the sphere bo a/ — / --^j— > it will just roll over the point. Ex. 2. A rectangular parallelepiped of mass 3j», having a square base ABCD, rests on a horizontal plane and is moveable about CD as a hinge. The height of the solid is 3a and the side of the base a. A particle m moving with a horizontal velocity v strikes du'ectly the middle of that vertical face which stands on ^ S and lodges there without penetrating. Show that the solid will not upset unless ^^ fja. [King's Coll.] v'> 9 154. Four equal rods each of length 2a and mass m are freely jointed so as to form a rhombus. The system falls from rest with a diagonal vertical under tlie action of gravity and strikes against a fixed horizontal inelastic plane. Find the subse- quent motion. Let AB, BC, CD, DA be the rods and let .4C be the vertical diagonal impinging on the horizontal plane at A. Let V be the velocity of every point of the rhombus just before impact and let a bo the angle any rod makes with the vertical. Let u, V be the horizontal and vertical velocities of the centre of gravity and u the angular velocity of either of the upper rods just after impact. Then the effective forces on either rod are equivalent to the force m [v - V) acting vertically and mu horizontally at the centre of gravity and a couple mk^u tending to increase the angle a. Let R be tho impulse at C, the direction of which by the rule of symmetry is horizontal. To avoid introducing the reaction at £ into onr equations, let us take moments for the rod BC about B and we have mJc^u + m (y - V)a sin a - mua cos a = - ]i.2a cos a (1). ug has no (2). UB cyliniler to find the .(3). jscend verti- deHcend in he string ia IbH. If we that F=one- 2 eration = Kf/' liat the space izontal plane ow that if the at. hase ABCD, lie height of a horizontal on ^B and upset unless inted so as to der tlie action ind the subse- al impinging the rhomhus cal. gravity and w it. Then the ing vertically g to increase ly the rule of owe eqxiations, (!)• IMPULSIVE FORCES. 133 Each of the iower rods will begin to tarn round its extremity A aa a fixed point. If w' bo its angular velocity juHt after impact, the moment of tlio momentum about A just after impact will bo m(ifc- + a')w' and just before will be mrasino. The difference of these two is tlio moment about A of the effective forces on either of the lower rods. Wo may now tiiiio moments about A for the two rods AB, BO together and wo have m (P + a") w' - m Va sin a - mk^u + m (u - V)a sin a + mw . 3a cos a = TJ . 4a cos o . . . (2), The geometrical equations may be found thus. Since the two rods must make equal angles with the vertical during the whole motion we have w' = w (3). Again, since the two rods are connected at // the velocities of the extremities of the two rods must bo the same in direction and magnitude, liosolviug those hori- zontally and vertically, we have « + aw cos o = 2aw' cos o (4), t>-acosina = 2aw'8iuo (5). These five equations are sufficient to determine the initial motion. Eliminating R between (1) and (2), substituting for m, v, u' in terms of u from the geometrical equations, we find _3 Tsino "~2 ' all + aBiu^ia' ^"'' In this problem we might have avoidoil ! he intvoductiou of the unknown reaction R by the use of Virtual Velocities. Sui^oso wo give tho system such a displace- ment that tho incliuatiou of each rod to the vortical is increased by the same quantity Sa. Then tho principle of Virtual Velocities gives »iA'«5o - m [v - V)S (3a cos a) + mu5 (a sin o) + m (k" + a^) w'5a + m Vd {a cos a) = 0, which reduces to (2^-3 + rt^) w - Va sin a + 3 (v - F) a sin a + ua cos c = 0, and the solution may be continued as before. Ex. 1. Prove that the rhombus loses by the impact , — „ . ■- of its 1 + 3 sm* a momentura. Ex. 2. Show that the direction of the impulsive action at tho hinges B or D makes with the horizon an angle whose tangent is — . tan a To find the subsequent motion. This may be found very easily by the method of Vis Viva. But in order to illustrate as many modes of solution as iiossible, we shall proceed in a different manner. The effective forces on either of the upper rods will be represented by tho differential coefficients m-j-, m . , mk^^, and the moment for either of tho lower rods will be m (P + a*) -z- . Let d be the angle any rod makes with tho vertical at the time t. Taking moments in the same way as before, we have • mP -,+ ni -rr a sin ^ - ?» -7- a cos 6= - B .2a cos 6 + t)wa sin (1)', dt at at " ^ " m (k^ + a^)-, — mk^ -r -t- wi-p a sin ^ + m -.- . 3a cos = 7? . 4a cos + Imna sin ^. . . (2)'. dt dt dt dt o \ / i \-\ 134 MOTION IN TWO DIMENSIONS. The geometrical equations are the same as those given above, with d written for a. Eliminating R and substituting for u, v, we get do multiplying both sides by u=-7- and integrating, we get {2 (i« + a- ; i- 8a« sin2 6] w^ = C - Sffa cos $. Initially when 0=a, u has the value given by equation (6). Hence we find that the angular velocity w when the inclination of any rod to the vertical is is given by {l + 3sbxH)i^=^l . - ^^f.°-. +^(coBa-cosg). ' 4a^ 1 + d sm* a a Ex, 1. A square is moving freely about a diagonal with angular velocity u, when one of the angular po'nts not in that diagonal becomes fixed; determine the impulsive pressure on the f-.cd 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 -^r— , a being the length of each rod. [Coll. Exam.] yv Ex. 3. The points ABCD are the angular points of a square; AB, CD are two equal similar rods connected by the string BO. The point A receives an impulse in the direction AD, show that the initial velocity of A is seven times that of the poin*. D. [.Queens' CollJ Ex, 4. A series of equal beams AB, BC, CD is connected by hinges; the beams arc 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 A B : determine the impulsive action at any hinge. [Math. Tripos.] Result. If Xn be the impulsive action at the »"■ angular point, show that ^ir^i - 5^sn+a - 2X2,1+3 - and Zjn+j - 5Z8„+i - iX^^ = 0. Thence find Z„. 155. A free lamina of any form is turning in its own plane abotit an imtanta- neous 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 wlien tlie magnitude of the blow is a maximum*. First, let the obstacle P be a fixed point. Let GP=x, and let It be the force of the blow, l^st SO=h, and let w, 0/ be the angular velocities about the centre of gravity before and after the impact. Then hu * Poinsot, Surla percussion des corps, Liouvilles Journal, 1857; translated in the Annals of Philosophy, 185*^. IMPULSIVE FORCES. 135 written e find that al is is velocity w, srmine the as angular nges to one the middle ities of the 11. Exam.] CD are two an impulse that of the linges; the to the two se is given ge. [Math. show that m imtanta- the straight magnitude w, (■/ he the Then ha ranslated in is the linear velocity of Q just before the impact; let v' be its linear velocity just after the impact. We have the equations -Rx (i). (2). w — w = Ml and supposing the point of impact to he reduced to rest, v' + xu)'=0 Substituting for u/ and v' from (1) in equation (2), we get »= + &■' This is to be made a maximum. Equating to zero its differential coefficient with respect to x, we get x^ + 2hx-l?=0 •. (3); One of these values of x is positive and the other negative. Both these corre- spond to viaximum points of percussion, hut opposite in direction. 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, P' are equally distant from S, and if be the centre of oscillation with regard to /S as a centre of suspension, SP^=SO . SO. Ex. 2. If P be made a point of suspension, P' is the corresponding centre of oscillation. Also PP' is harmonically divided in and 0. Ex. 3. The magnitudes of the blows at P, P' are inversely proportional to their distances from G. Secondly, let the 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, • • r — — • m 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'. Eliminating w', v', V, we get (M + m) ^-i + mx" This is to be made a maximum. Equating to zero its differential coefficient with respect to a;, we find ;= - 7t± ^h^ + k^ {^'^m) <*^' 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 k^=xh. This gives M _x + h ~h~ m , ov \il=x + h\iQ the length of the simple equivalent pendulum,— ■=-, Since F'= — , it is evident that when i? is a maximum V m h m is a maximum. Hence the two points found by equation (4) might bo called the centres of greatest communicated velocity. ^i 138 MOTION IN TWO DIMENSIONS. There are otber singiilar points in a moving body whose positions may be found; thus Vie might inquire at what point a body must impinge against a fixed obstacle, that first the linear velocity of the centre of gravity might be a maximum, or secondly, that the angular velocity might be a maximum. These points, respec- tively, have been called by Poinsot the centres of maximum Reflexion and Conver- sion. Beferriug to equations (1), we see that when v' 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 gieatest impact. When u' is a maximum, we have to make a - ryr« = maximum. Mir Substituting for H, this gives «* - 2 t- oj - /;*= 0. If be the centre of oscillation, we have GO_=j . Let this length be represented by h'. Then this equation becomes x^-2h'x-k^=0 (5). The roots of this equation are the same functions of h' and I- that those of equation (3) are of k and k, except that the signs are opposite. Now /u, the sphere will separate slightly from the wall before sufficient friction has been called into play to reduce the tangential velocity of the point of contact to zero. In this ease we must replace F by fxR in the equations. At the momei ^. of greatest compression we have as before v' = 0. This gives R = mv cos a. By substituting in the equations the motion of the sphere may be found. The initial velocity of the point of contact is easily seen to be n' -au' = v(Hm o-/«5coso). If this were negative, the friction at the end of the impact would be acting in the direction of relative motion, which is impossible. This solution is therefore correct only if ^ tan a> fi. If the sphere be imperfectly elastic, a normal force of restitution is called into play equal to emv cos a. If then ^ mv sin ahe ix(l + e)mv cos o, we must put R=:(l + e)mveoaa, F-ix{l+e)mvcosa, and the same equations will now give u', v' and w'. 164. Two rough bodies of any form impinge on each other in a given manner. It is required to find the motion just after impact. Let O, 0' be the centres of gravity of the two bodies, A the 142 MOTION IN TWO DIMENSIONS. point of contact. Let V, V be tho resolved velocitie.s of G just Defore impact, parallel to the tangent and normal respectively at .4 ; u, V the resolved velocities at any time t after the com- mencement of the impact, but before its termination. Then t is indefinitely small. Let VL be the angular velocity of the body, whose centre of gravity is G, just before impact, oi the angular velocity after the interval t. Let M be the mass of the body, k its radius of gyration about G. Let GN be a perpendicular from G on tho tangent at A, and let AN = x, NG=y. Let accented letters denote corresponding quantities for the other body. 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, NA respectively. Then they must be supposed to act in the opposite directions on the body whose mass is M'. Since II represents the Avhole momeutum 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. 12(j, 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. 152 M{u-U)=-F M{v-V)=R ' (1), ]\W {(o - n) = Fi/ + Rx . M'{u'-U')=F M'{v'-V')=-R [ (2). The relative velocity of sliding of the points in contact is by Art. 127 S = u — yoi — u — y'a>' (.3), and the relative velocity of compression is by the same article C=v'+a;'o)' — v — xm (4). Substituting in these equations from the dynamical equations we find S=8,-aF-hR (5), G just )ectively le com- 'hen t is le body, angular le body, ndicular : y. Let le other body M let i^ be re. We 1/ in the iposed to r. cated to um com- 11 ly push 20, when »f R may ro at the imum at nvenient riable. ...(1), Lct is by ...(3), tide ...(4). quations .(6). IMPULSIVE FORCES. 143 C=C,-bF-a'R (G), where 8,= U- yn- U' -y'Cl' (7), C, = V'+x'il' - V-xil (8), _ 1 1 j/ y" .Q. '^~iM'^M''^MIc''^ilW' ^^^' ""^M^ M'^TW^MV' ^^ ^' M¥ M'k' ^ '' These may be called the constants of the impact. The first two 8^, Cq represent the initial velocities of sliding and com- pression. These we shall consider to be positive ; so that the body If 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 aa > V. 165. "When 6=0, the discussion of these equations, as in Art. 163, does not present any difficulty, but in the general case it is more easy to follow the changes in the forces, if we adopt a graphical method. Let us draw two lengths AB, AF along the normal and tangent at A in the directions NO, -4 -AT respectively, to represent the magnitudes of R and F at any moment of the impact. Then if we consider AR and AF to be the co-ordinates of a point P, referred to 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. It will be convenient to trace the two loci determined hy 8=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 these, we must find their inter- cepts on the axes of F and R. Take ^C= ■^, A8=^, AG' = ^\ a a b AS' = -^ b then *Si^', CG' will be these straight lines. Since a and a' are necessarily positive, while b has any sign, we see that their 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' > 6*, 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 i^than the latter. It easily follows that the two straight lines cannot iU, t I V f it 144 MOTION IN TWO DIMENSIONS. intersect in the quadrant contained by HA produced and FA produced. 106. In the beginning of the impact the bodies slide over each other, hence, as explained in Art. 144, the whole limiting friction is called into play. The point P therefore moves along a \ R . > \ c \ s- ^^ \ T A. S c straight line AL, defined by the equation F=fili, where fi is the coefficient of friction. The friction will continue to be limiting until F reaches the straight line 88'. If R^ be the abscissa of this point we find 72.= — " , . This gives the whole normal ^ '^ aji + h ^ blow, from the beginning of the impact, until friction can change from sliding to rolling. If R^ is negative, the straight lines AL and 88' will not intersect on the positive side of the axis of F. In this case the friction will be limiting throughout the impact. If R^ is positive the representative point P will reach 88 '. After this onlj so much friction is called into play as will suffice to prevent sliding, provided this amount is less than the limiting friction. If the acute angle which 88' makes with the axis of ^ be less than tan"^ fi, the friction dF necessary to prevent sliding will be less than the limiting friction /ic?Z?. Hence P must travel along 88' in such a direction that the abscissa R con- tinues to increase positively. In this case the friction will not again become limiting during the impact. But if the acute angle which 88' makes with the axis of i2 be grbater than tan'^/t, the ratio of dF to dR will be numerically greater than fx, and more friction is necessary to prevent sliding than can be called into play. The friction will therefore continue to be limiting, and P, after reaching 88', must travel along a straight line, making the same angle with the axis of R that AL does. But this angle must be measured on the opposite side of the axis of R, for when the point P has crossed 88' the direction IMPULSIVE FOllCES. 145 I ■: incl FA ide over limiting along a 3 fi is the limiting bscissa of 5 normal n change lines AL ixis of F. impact. After uffice to limiting axis of R it sliding P must R con- will not 3 of -R be merically it sliding continue along a that AL side of direction of relative sliding and therefore the direction of friction is changed. In this case it is clear that the friction will continue limiting throughout the impact. When P passes tlie straight line CC, compression ceases and restitution begins. But the passage is marked by no peculiarity except this. If R^ be tlie abscissa of the point at which P cro.sse8 CC, the whole impact, for experimental reasons, is .supposed to terminate when the aKscissa of P is R,^ = R^ (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 straiglit lines AL, SS' and CC. But in all cases the progress of the impact may be traced by the method just explained, which may be briefly stated thus. 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 axis of R as AL does, but on the opposite side. The one along which it proceeds is the steeper to the axis of F. It travels along this line in such a direction as to make the abscissa R increase. The complete value of R for the whole impact is for.ud by multiplying the abscissa of the point at which P crosses CC by 1+e. The complete value of F is the corresponding ordinate O'f P. Substituting these in the dyna- mical equations (1) and (2), the motion just after impact may be easily found. If the bodies be smooth, the straight line AL coincides with the axis of R. The representative point P must travel 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. 167. It is not necessary that the friction should keep the same direction during the impact. The friction must keep one sign when P travels along AL. But when P reaches SS', its direction of motion changes, and the friction dF called into play in the time dt may have the same sign as before or the opposite. But it is clear that the friction can change sign only once during the impact. It is possible that the friction may continue limiting through- out the impact, so that the bodies slide on each other throughout. The necessary conditions are that either the straight line SS' must be less steep to the axis of F than AL, or the point P must not reach the straight line SS' until its abscissa has be- come greater than R.^. The condition for the first case is, that h must be greater than fia. The abscissae of the intersections S of AL with SS' and CC are respectively R^ = — ^ and R. D. 10 M irmsmmsim 116 ' bfi+a MOTION IN TWO DIMENSIONS. The condition for tlio second case is necessary, that i?, must be positive, and R^ either negative or positively greater than R^ (1 + e). 168. Ex. 1. Show that the reprosentative point P as it travels in the manner directed in the text must cross the line of greatest compression, and that the abscissa R of the point at which it crosses this straight line must be positive. Ex. 2. Show that the conic whose equatidii referred to the axes of R and F if) aF^ + 2bFR + a'lP=f, where e is some constant, is an ellipse, and that the straight lines of no sliding and greatest compression are parallel to the conjugates of the axes of P and R respectively. Show also that the intersection of the straight lines of no sliding and greatest compression must lie in that angle formed by the conju- gate diameters which contains or is contained by the first quadrant. Ex. 3. Two bodies, each turning aboiit a fixed point, impinge on each other, find the motion just after impact. Let 0, G\ in the figure of Art. 164, be takon as the fixed points. Taking moments about the fixed points, the results will be nearly the same as those given in the case considered in the text. Initial Motions. 169. Suppose a system of bodies to be in equilibrium and that one of the supports suddenly gives way. It is required to find the initial motion of the bodies and the initial values of the reactions which exist between the several bodies. The problem of finding the initial motion of a dynamical system is the same as that of expanding the co-ordinates of the moving particles in powers of the time t. Let (x, y, 6) be the co-ordinates of any body of the system. For the sake of brevity let us denote by accents differential coefficients with regard to the time, and let the suffix zero denote initial values. Thus x^' cPx denotes the initial value of -^ • ^7 Taylor's theorem we have a; = a + <'i2 + <"^+ the term x^ is omitted because we shall suppose the system to start from rest. First, let only the initial values of the reactions he required. The dynamical equations will contain the co-ordinates, their second differential coeflficients with regard to t, and the unknown re- actions. There will be as many geometrical equations as re- actions. From these we have to eliminate the second differential .(1) INITIAL MOTIONS. 147 iccssary, )8itively le manner that the ive. and F is le straight bes of the light lines the conju- ach other, ). Taking iiose given ium and [uired to s of the i^namical of the he the brevity d to the have ..(1) : jTstem to 'equired. second own re- as re- 'erential coeflflciv^nts r.nd find the reactions. Tlio process will bo as follows, which is really the same as the first method of solution described in Art. 125. Write down the geometrical equations, differentiate each twice and then simplify the results by substituting for the co-ordinates their initial values. Thus, if wt use Cartesian co-ordinates, let (x, y, 6) =0 be any geometrical relation, we have since a?,' = 0, " rfv "" '" ~ dx dd The process of differentiating the equations may sometimes be much simplified when the origin has been so chosen that the initial values of some at least of the co-ordinates are zero. We may then simplify the equations by neglecting the squares and products of all such co-ordinates. For if we have a term a?, its second differential coefficient is 2 [xx" -f x'^), and if the initial value of 03 is zero, this vanishes. The geometrical equations must be obtained by supposing the bodies to have their displaced position, because we require to differentiate them. But this is not the case with the dynamical equations. These we may write down on the supposition that each body is in its initial position. These equations may be obtained according to the rules given in Art. 125. The forms there given for the effective forces admit in this problem of some simplifications. Thus since r^ = 0, ^^ = 0, the accelerations along and perpendicular to the radius vector take the simple forms r^ ' and rt/>o". So again the acceleration — along the normal vanishes. If, for example, we know the initial direction of motion of the centre of gravity of any one of the bodies, we might conveniently r ^solve along the normal to the path. This will supply an equa- tion which contains only the impressed forces and such tensions or reactions as may act on that body. If there be only one re- action, this equation will suffice to determine its initial value. We may also deduce from the equations the values of x^', y"> ^o"> ^^^ ^^^"^ ^y substituting in equation (1) we have found the initial motion up to terms depending on f. 170. Secondly, let the initial motion he required. How many terms of the series (1) it may be necessary to retain will depend 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 10—2 tavm 148 MOTION IN TWO DIMENSIONS. 4 ' • I fl and by differentiating equation (1) in ^ «' =a:;'< + <"f^ + ajr,^ + X &c. =&c.; results which may also he obtained by a direct use of Taylor's theorem. If then the body start from rest, the radius of curvature is zero. But if cc^'y^' ~" ^o "Vo" = ^> "^^ ^^^^ p = 3 v,'*'o +yn ^0 Vo ~^o Vo 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 so, y, 0, &c. their initial values, and for x, y, ff nxo. Differentiate each geometrical equation four times and then reduce each to its initial form. We shall thus have sufficient equations to determine x^', x^", x^^, &c., B^, R^, iJg", &e., where M is any one of the unknown reactions. It will often be an advantage to eliminate the unknown reactions from the equations before differentiation. We shall then have only the unknown coefficients w^', x^", &c. entering into the equa- tions. If we know the direction of motion of one of the centres of gravity under consideration, we can take the axis of « a tangent V* to its path. Then we have p = ^ , where x is of the second order, y of the first order, of small quantities. We may therefore neg- lect the squares of x and the cubes of y. This will greatly sim- plify the equation?. If the body start from rest we have a?/ = 0, and if x^' = 0, we may then use the formula 171. Ex. A circular disc is hung tip by three equal strings attached to three points at equal distances in its circumference, and fastened to a peg vertically over the centre of the disc. One of these strings is suddenly cut. Determine the initial circumstances of motion. INITIAL MOTIONS. 149 Let bo the peg, AB the circle seen by an eye in its plane. Let OA bo tbe string which is cut and C be the nuddle point of the chord joining the points of the K a Taylor's vature is ied thus. I reduce il values, equation ^6 shall , &c., i?„, ions. It reactions len have he equa- } centres tangent ad order, 'ore neg- atly sim- e 4' = 0, ed to three ically over the initial m circle to which the two other strings are attadied. 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 i2=22'cosa. Let I be the length of OC, /3 be the angle GOO, a be the radius of the disc. Let («, y) be the co-ordinates of the displaced position of the centre of gravity with reference to the origin Or x being measured horizontally to the left and y vertically downwards. Let d be the angle the displaced position of the disc makes with AB. By drawing the disc in its displaced position it will be seen that the co-ordinates of the displaced position of C7 are x - 2 sin /3 cos and y-lwt\.^m\.$. Hence since the length OC remains constant and equal to I we have a;S + y» _ 21 sin /3 (o! cos tf + y sin tf) = Z" cos' /3. Suppose the initial tensions only to be required. It will be sufficient to differ- entiate this twice. Since we may neglect the squares of small quantities, we may omit a;", put cos 0=t,e,m.$ = e. 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', ff) to zero, and put for (x, ^,^) their initial values (0, I cos ft 0). We get ^o" cos /3 = sin /3 (V -^ I cos /35o") • This equation may also be obtained by an artifice which is often useful. The motion of Q is made up of the motion of C and the motion of Q relatively to C. Since C begins to describe a circle from rest, its acceleration along CO is zero. Again, the acceleration of relatively to C when resolved along CO is QC -r^ cos ft The resolved acceleration of G is the sum of these two, but it is also j/o" cos p - Xff' sin ft Hence the equation follows at once. In this case wo require the differential equations only in their initial form. These are ma!o"=i?osin/3 my(^"=mg - Rq cos /3 }■ mk%' = i?o I sin /3 cos /3) where m is the mass of the body. Substituting in the geometrical equation we find I?0 = Wflf. jij cos L i V i ■ I l + Ti,sin''j9cos''/3 150 MOTION IN TWO DIMENSIONS. The tension of any string, before the string OA was cut, may be found by the [es of Statics, and is clearly T, change of tension can be found. rules of Statics, and is clearly T, = 5 — — , where 7 is the angle AOO. Hence the ' 3 cos y 172. Ex. 1. Two strings of equal length have each an extremity tied to a weight G and their other extremities tied to two points ^, J5 in the same horizontal line. If one be cut the tension of the other is instantaneously altered in the ratio l:2cos''?. [St Pet. Coll.] Ex. 2. An elliptic lamina is supported with its plane vertical and transverse axis horizontal by two weightless pegs passing through the foci. If one pin be released show that if the eccentricity of the ellipse be a/ ^ , the pressure on the other pin will be initially unaltered. [Coll. Exam.] Ex. 3. Three equal particles A, B, C repelling each other with any forces, are tied together by three strings of unequal length, so as to form a triangle right- angled at ^. If the string joining B and C be cut, prove that the instantaneous changes of tension of the strings joining BA, CA are J TcosB and ^ jf cos C respec- tively, where B and C are the angles opposite the strings joining CA, AB respec- tively, and T is the repulsive force between B and C. Ex. 4. Two uniform equal rods, each of mass m, are placed in the form of the letter X on a smooth horizontal plane, the upper and lower extremities being con- nected by equal strings ; show that whichever string be cut, the tension of the other is the same function of the inclir.ation of the rods, and initially is | mg sin a, where a is the initial inclination of the rods. [St Pet. Coll.] Ex. 6. A horizontal rod of mass m and length 2a hangs by two parallel strings of length 2a attached to its ends : an angular velocity w being suddenly communi- cated to it about a vertical axis through its centre, show that the initial increase of tension of either string equals —^ , and that the rod will rise through a space &g [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 is instantaneously changed in the .. 3a''-46a r^ „ ,, ratio 2 (ct"- ft") ' [Coll. Exam.] Ex. 7. A sphere resting on a rough horizontal plane is divided into an infinite number of solid lines and tied together again with a string ; the axis through which the plane faces of the lines pass being vertical. Show that if the string be cut the pressure on the plane is diminished instantaneously in the ratio 45t^ : 2048. [Emm. Coll.] RELATIVE MOTION. 151 a space ; 2048. On Relative Motion or Moving Axes. 173. In many dynamical problems the relative motion of the different bodies of the system is frequently all that is required. In these 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 these in order. 174. Let it be required to find the motion of any dynamical system relative to some moving point C We may clearly reduce C to rest by applying to every element of the system an accelera- tion equal and opposite to that of C. It will also be necessary to suppose that an initial velocity equal and opposite to that of C has been applied to each element. Let /be the acceleration of C 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 f%m acting at the centre of gravity. Hence to reduce any point 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 C a force measured by Mf, where M is the mass of the body and/ the acceleration of C. ■The point G may now be taken as the origin of co-ordinates. 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 (7 as origin. The accelerations of the particle ^^^^ j^a — ''(751) ^^^ ~'JfV~^)' *^°°S ^^^ perpen- dicular to the radius vector r. Taking moments about C, we get [moment round G of the impressed forces ,8 dO\ _ plus the moment round G of the reversed dt) effective forces of G supposed to act at the centre of gravity. If the point G be fixed in the body and move with it, -j- will be the same for every element of the body, and, as in Art. 88, 2w dt V we have tm ;^ (*'" 77) = ^^^' de w '\ \ I I 11. i i V i i-il i 152 MOTION IN TWO DIMENSIONS. 175. From the general equation of moments about a moving point G we learn that we may use the equation day _ moment of forces about C dt moment of inertia about G in the following cases. First. If the point G be fixed both in the body and in space ; or, if the point u being fixed in the body move in space with uniform velocity ; for the acceleration of G is zero. Secondly. If the point G be the centre of gravity ; for in that case, though the acceleration of G is not zero, yet the moment vanishes. 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 oida-. Hence in the time dt the velocity of G has increased from zero to oada; therefore its acceleration is (o -j:. To obtain the accurate equation of moments about G we dt dc must apply the eflfective force Xm . to -7- in the reversed direction dt at the centre of gravity. dcr But in small oscillations a> and -r- are dt both small quantities whose squares and products are to be neglected, and in an initial motion &> is zero. Hence the moment cf this force must be neglected, and the equation of motion will be the same as if G had been a fixed point. It is to be observed that we may take moments about any point very near to the instantaneous centre of rotation, but it will usually be most convenient to take moments about the centre in its disturbed position. If there be any unknown reactions at the centre of rotation, their moments will then be zero. 176. If the accurate equation of moments about the instan- taneous centre be required, we may proceed thus. I-et L be the moment of the impressed forces about the instantaneous centre, * If a body be in motion in one plane it is known tbat the actual displacement of every particle in the time dt is the same as if the body had been turned through some angle udt about some fixed point O. This may be proved in the same way as the corresponding proposition in Three Dimensions is proved in the next Chapter. See Art. 183. The point C is called the instautauoous centre of rotation, and w is called the instantaneous angular velocity. See also Salmon's Higher Plane Curves^ 1852, Arts. 246 and 2C4. BELATIVE MOTION. 163 the centre of gravity, r the distance between the cent: o 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 C is L-Mw^^.r cos GC'G: at If k be the radius of gyration about the centre of gravity, the equation of motion becomes writing for cos QC'C its value -j- . dt 177. Ex. 1. Two heavy particles ■whos& masses ( •• m and ta' are connected by an inextensible string, which is laid over the vertex of a double iTicUned 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 of the string is m(i/6ina+/cosa). By considering the particle m', we find the tension to be also m' {g sin a' -/cos a'). Equating these two we have ,_ m sin a - m sin a ^ ~ ml cos a' + m cos a Hence F is found. 178. Ex 2. A cylindrical cavity whose section is any oval curve and wlwse generating lines are horizontal is made in a cubical mass tohich 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 vditcal plane through the centres of gravity of the mass and the sphere is perpendicular to the generating lints of the cylinder. A momentum B is communicated to the cube by a blow in this vertical plane. Find the motion of the sphere relatively to the cube and the least value of the blow that the sphere may not leave the surface of the cavity. Simultaneously with the blow B there will be an impulsive friction between the cube and the sphere. Let M, m be the masses of the cube and sphere, a the radius of the sphere, k its radius of gyration about a diameter. Let Fq be the initial velocity of the cube, Vq that of the centre of the sphere relatively to the cube, w„ the initial angular velocity. Then by resolving horizontally for the whole system, and taking moments for the sphere alone about the point of contact, we have wi(i'o+Fo) + il/Fo = i?| ■(1), 'i t] il l\ !i1 !■ ; i 164 MOTION IN TWO DIMENSIONS. and sinoe there is no eliding ro-a«o=0 (2). To find the Bubsequent motion, let {x, 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 fimotiou of X. Let ^ be the angle the tangent to the cavity at the point of contact of the sphere makes with the horizon, then tan^=-^. Lot V be the velocity of the cubical mass, then, by Art. 131, m{^ + V\-k-MV=B (3). If Tg be the initial vis viva and y^ the initial value of y, we have by the equation of vis viva '"!(§+ ^y+(fy+^"W+^^''=^«-^'"^^2^"2'») (^)' where u is the angular velocity of the sphere at the time t. If v be the velociiy of the centre of the sphere relatively to the cube, we have since there is no sliding v=au. Eliminating F and » from these equations, we have (|)*.j(UtenV)(u^:)-^! = C'.-2«, (5). where Cg= +'^ffyo' (M+m) ^M + (M+m)^} This equation gives the motion of the sphere relatively to the cube. To find the pressure on the cube, let us reduce the cube to rest. Let R be the pressure of the sphere on the cube, then the whole effective force on the cube is JZ sin 1^ parallel to the axis of x. By Art. 174 we must therefore apply to every particle an acceleration — ^r^-^ opposite to this effective force. The sphere will then be acted on by ^^ 72 sin ^ in a horizontal direction in addition to the reaction E, the friction and its own weight. Besolving the forces on its centre along a normal to its path we have vhere p is the radius of curvature of the path of the centie of the sphere. Elimi' nating -y- by the help of the equation of vis viva, we have <'-^.»+p- along and perpendicular to OM. • ii 1 ,.'• ^•-* ^"-^d > r-l X The resolved parts of the velocity of N are in the same way -^ and 7}oi along and perpendicular to ON. By adding these with their proper signs we have 156 MOTION IN TWO DIMENSIONS. velocity of P I d^_ paraUelto 0^)~di~"^'^' velocity oi P\_dri ,, parallel to Orj) dt In the same way by adding the accelerations of M and N we have acceleration of P ) ) be the polar co-ordinates of P, we have acceleration of PI accelerp 'ion of < parai ' i along rad. vect. acceleration of P) M d'r fd^ V perp. to rad. vect I =Jlf*(f-)} 180. Ex. 1. Let the axes 0{, Orj be oblique and make an angle a with each other, prove that if the velocity be represented by the two components u, v parallel to the axes, dt ^ ^ u=^ - «f cot a-uT] coseo a, Cut » = ^+ w>7 cot a + wf coseo a. In this case PM is parallel to Ot/. The velocities o! M and N are the same as before. Their resultant is, by the question, the same as the resultant of u and v. By resolving in any two directions and equating the components we get two equa- tions to find u and v. The best directions to resolve along are those perpendicular to 0% and O17, for then v, is a'usent from one of the equations and v from the other. Thus tt or V may bo found separately when the other is not wanted. Ex. 2. If the acceleration be represented by the components X and F, prove Jt = 3- — cim cot a - wv cosec a, at „ i/o , r= 7- + &w cot a + wu coseo o. dt These may be obtained in the same way by resolving velocities and accelerations perpendicular to 0^ and O17. I '^l RELATIVE MOTION. 157 prove 181. Ex. A particle under the action of any forces moves on a smooth curve which is constrained to turn with angular velocity u about a fixed axis. Find the motion relative to the curve. Let na suppose the motion to be in three dimensions. Take the axis of Z as the fixed axis, and lot the axes of (, t) be fixed relatively to the curve. Then the equations of motion are 7] dt dt' Z + Rn .(1). where X, Y, Z are the resolved parts of the impressed accelerating forces resolved parallel to the axes, R is the pressure on the curve, and (Z, m, n) the direction- cosines of the direction of R. Then since R acts perpendicularly to the cutvp ,d^ dn dt . Suppose the moving curve to be projected orthogonally on the plane ' ! j, % let a be the arc of the projection, and v'= — be the resolved part of the velocity ^ -allel (tt to the plane of projection. Then the equations may be written in the f ""m dt^ = Z + Rn. The two terms 2uv' ^ and - 2&«/ ^ may be regarded as the resolved parts of a force 2uv' acting in a direction whose direction-cosines are j,_dri m' = - da »'=0. These satisfy the equation I'-^+m' -p+n' t^=0« Hence the force is perpendicular to the tangent to the curve, and also perpen- dicular to the axis of rotation. Let R^ be the resultant of the reaction R and of the force 2(iw'. Then Rf also acts perpendicularly to the tangent, let {I", m", n") be the direction-cosines of its direction. The equations of motion therefore become dt^ da , dt' •(2). Hi ;!! ',h I' \ 158 MOTION IN TWO DIMENSIONS. These are the equations of motion of a particle moving on a Jixfd curve, and acted on in addition to the impressed forces by two extra forces, viz. (1) a force wV tending directly from the axis, where r is the distance of the particle from the axis, and (2) a force -j- r perpendicular to the plane containing the particle and the axis, and tending opposite to the direction of rotation of the curve. In any particular problem we may therefore treat the curve as fixed. Thus suppose the curve to be turning round the axis with uniform angular velocity. Then resolving along the tangent we have dv_ dx ydj/ dz J ^ d»~ da ds da da ' where r is the distance of the particle from the axis. Let V be the initial value of V, rg that of r. Then r2- V^=2f(Xdx+ Ydy + Zdz) + u^r*-r^'). Let t'o be the velocity the particle would have had under the action of the same forces if the curve had been fixed. Then Hence vj* -V^=2 f(X dx+Ydij + Z dz). r'-VQS=w«(,.3-ry'). The pressure on the moving curve is not equal to the pressure on the fixed curve. The pressure R on the moving curve is clearly the resultant of the pressure J?' on the fixed curve, and a pressure 2e V (7 (6 - a), a being the radius of the ball, and h 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 sud- denly fixed, show that the initial angular velocity ol ABii three times that of BC. Also show that in the subsequent motion of the rods, the greatest angle between them equals cos'^ f , and that when they are next in a straight line, the angular velocity of BO 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, jeined 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 angiilar velocity may be given to the third. EXAMPLES. 161 intal plftne, oiul weight et fttU from lor that the id vertically states freely libriiim, the avity of the the angular Explain the lix which is g line of the iclix, starting , the angular h an angle & aation of the lass M, and a 1 on a smooth [lining in cou- ,rden-roller at ,th a uniform the roller, if I moving with end A be sud- 3S that of BG. angle between e, the angular in the same ^pplied at the length ; show th of the given rm one beam, qualinl'jngth, irpendicular to ,t the greatest RetuU. If m bo the masn of either of the outer rods, /9m that of the inner rod, P the momentum of the blow, w the angular velocity communicated to the third 8 rod, then mawf- + q + -jj = P. Hence when u ia a maximum p=\j3. 15. Two rough rods A, B are placed parallel to each other and in the same horizontal plane. Another rou^h rod C is laid across thom at rit,'ht angles, its centre of gravity being half way V)etwoon them. If C be raised through any angle a and let full, detcrmlno the conditions that it may oscillate, and show that if its length be equal to twice the distance between A and D, the angle through which it will rise in the n"> oscillation is given by the equation sin tf = I - ; . sin a. 16. A rod moveable in a vertical plane about a hinge at its upper end has a given uniform rod attached to its lower end by a hinge about which it can turn freely in the same vertical plane as the u]>per rod ; at what point must the lower rod be struck horizontally in that same vertical plane that the upper rod may initially be imaffected by the blow ? 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 10 + 14 tan' ^ : -? + 49itan'' 6, where e is the elasticity of the ball and 6 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 angl(<. 19. A rectangle is formed of four uniform rode of lengths 2a and 2& 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 len^h 2a suddenly becomes fixed. Show that the angular velocity of the sides of length 26 immediately becomes ^ "tjv «• Find 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 ('.!) in opposite directions. Prove that in the first case, when the outer rods make . ue greatest angle with the direction of the middle one pro- duced on each side the common angular velocity of the three is — , and in the second case after the impact of the two outer rods the triangle formed by them will move with uniform velocity — - , 2a tbeing the length of each rod. o 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 uinI the vertex downwards. If after falling through any height, the middle point of i/.a R. D. 11 \\ 162 MOTION IN TWO DIMENSIONS. upper rod be suddenly stopped, the impulsive strains on the upper and lower hinges will be in the ratio of sjl3 to 1. If the lower liinge would just break if the system fell tnrough a height —p , prove that if the system fell through a height -.— the lower rods would just swing through two right angles, 22. A perfectly rough and rigid hoop rollir.g 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 - and F= velocity just before impact, then the condition that the hoop will surmount the spike is F*> V fl''' 1 1 - sin («+/:)(. a being the inclination of the plane to the horizon. Show that unless V^<'^t^ gr.sm(a + ^\ , the hoop will not remain in contact with the spike at all. If this inequality be satisfied the hoop will leave the spike when the diameter through the p^lut of contact makes an angle with the horizon =sm' ■MsLt+^^^K''-*-?)! 23. A flat circular disc of radius a is projected on a rough horizontal table, which is such that the friction upon an element a is c F** ma 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 Wo<^c' prove that tlu velocity m and angular velocity u at any sub. equent tune satisfy the relation ( „ , — , . I = -^r- • 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 Mj'iarj. 25. A string without weight is coUed round a rough horizontal cylinder, of which the mass is M and radius a, and which is capaule of turning roimd 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 if d be the angle tlu'ough which the cylinder has turned after a time t before the chain is fully stretched, Mae= j ( % - «^ ) 26. Two equal rods AG, BC, are freely connected at C, and hooked to / and B, two points in the same horizontal line, each rod being then inclined at an angle a to the horizon. The hook B suddenly giving way, prove that the direction of the strain '1 + Csiu«a 2-3( at C is iustantaneouply shifted through an angle tan' ■i/l±« Vl + 6 cos'' a 8 sin icos'aX a COB a J ir binges B system ^k the V3 comes in us of the )act, then In contact B diameter mtal table, be velocity of the disc. city of the sub, equent inside of a the motion. )ulse which )t going oil cylinder, of id its axis. Iss is m and Itbat if e be the chain ia EXAMPLES. 163 U 27. Two particles A , B are connected by a fine string ; A rests on a rough hori- zontal table and B hangs vertically at a distance I below the edge of the table. If A be on the point of motion and B be projected horiaontally with a velocity u, show that A will begin to move with acceleration - -r- /U+l I and that the initial radius of curvature of B's path will be (/x + 1) I, where n 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 tlio string is horizontal ; their distances from the ring being a and a', they are let go. If p, p' be the initial radii of curvature of their paths, prove that = — , , and -- + - = -+-. p P p p a a 29. A sphere whose centre of gravity is not in its centre is placed on a rough table ; the coefficient of friction being p,, determine whether it will begin to slide or to roll. 30. A circular ring is fixed in a vertical position upon a smooth horizontal plane, and a small ring is placed on the circle, and attached to the highest point by a string, which subtends an angle o at the centre ; prove that if the string be cut and the circle left free, the pressurep on the ring before and after the string is cut are in the ratio M+m sin^ a : il/coso, 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 chcumference of a circle of radius a, while the rod itself is made to revolve in the opposite direction with the same angular velocity about that extremity. The rod initially coincides with a diameter, and a smooth ring capable of sliding freely along the rod is placed at the centre of the circle. If r be the distance of the ring from C at the time t, prove »*=-v (e"'+e~"*) + t cos 2nt. 32. Two equal uniform rods of length 2« are joined together by a hinge at one extremity, their other extremities being connected by an inextensiblo 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 acceleration of either rod will be g • 8an-' ■62a*c'> d + ■ l^ - 8a-cl 33. A smooth horizontal disc revolves with angular velocity sjp. about a verti- cal axis at which is placed a material particle attracted to a certain point of the disc by a force whose acceleration is /u x distance ; prove that the path on the disc will be a cycloid. to / and B, In angle a to If the strain I cos'' ' I a COB a/ 11—2 I CHAPTER V. MOTION OF A EIGID BODY IN THREE DIMENSIONS, r. Translation and Rotation. 182. 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. 183. One point of a moving rigid body being fixed, it is re- quired to deduce the gen^-al 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 repre- sented by that of P. If the displacements of two points A, B, on the sphere in any time be given as AA'y BE, then clearly the displacement of any other point P on the sphere may be found by constructing on A'B' as base a triangle A'F 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. Let D and E be the middle points of the arcs AA\ BB', and let DC, EG he arcs of great circles drawn perpendicular to AA', f;v I IS re- of the mtre of body, sphere repre- in any of any med as lie base on the tremity atcd in W, and to AA', TBANSLATION AND ROTATION. 165 BB' respectively. Then clearly CA = GA' and CB- Cff, and therefore since the bases AB, A'B' are equal, the two triangles ACB, A'CB are equal and similar. Hence the displacement of C is zero. AIpo it is evident since the displacements of and G are zero, that the displacement of every point in the straight line OCia 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 POP' such that any one point P is brought into coincidence luith its new position P'. Then every point of the body will be brought from its first to its final position. 184. A body is referred to rectangular axes x, y, z, and the origin remaining tlie same the axes are changed to x, y', s*, accord- ing 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~\)x + a.^y + 0.^2=0, hiX+ {\-l) y + h^z=0, c^x+dy -\:{c^-\)z=0, K, y', / X Oil "2, flj y K K &3 z Cl, Ca» c. through an angle $, where 8-4sin'- = ai + Ja + C3. What is the condition that these three equations are consistent 7 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) {a^h, b^h, c^h) therefore their distance is V2(l-( But it is also 2h sin 7 siu - ; .0 2 sin" -sin' 7=1-^3. We have by a a similar reasoning 2 siu2-8in3tt=l- a^ and 2 siu2,cBin'/3 = l-&j, whence the equa- tion to find follows at once. 185. When a body is in motion we have to consider not merely ita first and last positions, but also the intermediate posi-> 16d MOTION IN THREE DIMENSIONS. tiona. Let us fhevi suppose AB, A'B' to be two positions nt any indo^ litoly sraaM interval of time di. We see that wlien a bodv I'uyvos about a fixed point 0, there is, at every instant of the motion, a straight line OG, 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 d9 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 dO 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 ribout the same axis throughout that unit with the angular velocity it had at the proposed instant. 186. Let us now remove the restriction that the body is moving with some one point fixed. We may establish the fol- lowing proposition. Every displacement of a rigid body may he represented hy a combination of the two folloiuing 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 luith the body and through the same space. (2) A motion of rotation of the wliole body about some axis through this assumed point P. 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 movin^- any two points of the body not in one straight liac with P into their final positions. This last motion has bef^^ ■ iu od 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 rotate the body fi'fst and then translate it. We may even suppose them to take place si multaneou sly. It is clear that any point P of the body may be chosen as the base point of the double operation. Hence the given displace- ment may be constructed in an infinite variety of ways. 187,. To fnd the relations between the awes and angles of rota- tion when different points P, Q are chosen as bases. Le - the displacement of the body be represented by a rotation 6 about in axis Pli and a translation PP'. Let the same dis- piaceraeni l>e also represeutud by a rotation 6' about an axis QS tind ■ trvwlatio I Q(J'. It is clear that any point has two dis- TRANSLATION AND ROTATION. 167 ition (lis- QS dis- K placements, (1) a translation oqual and parallel to PF', 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 displace- ments are the same as 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 Pl^', and the rotation 6 round PR, are the same as the corresponding displacements for the point Q. Hence this parallel must bo the axis of rotation correoponding to the base point Q. We infer that the axes of rotaiinn corresponding to all base points are parallel. 188. The axes of rotation at P and Q having been proved parallel, let a be the distance between them. 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 2a sin ^ and is the dis- placement due to rotation. The whole displacement of Q is the resultant of Qq and the displacement of P. In the same way the rotation & about QS will cause P to describe an arc, whose chord Pp is equal to 2a sin — . The whole displacement of P is the resultant of Pp and the displacement of Q. But if the displace- ment of Q is equal to that of P together with Qq, and the dis- placemexit 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 rotat'ons 9, & about PR and QS should be equal and in the same direccion. We infer that the angles of rotation corresponding to all base points are equal. 189. 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 ' are given. Since Qq, the displacement due to rotation roimd PR, is per- pendicular to PR, the projection of QQ' on the axis of rotation is the same as that of PP'. Hence the projections on the axis of rota- tion of the displacements of all points of the body are equ■ V >^V'i?^,J hil m\'- 1G8 MOTION IN THREE DIMENSIONS. mast be parallel to PB. Hence a rotation about any axis may he replaced by an equal rotation about any parallel axis together with a motion of translation. 191. When the rotation is indefinitely small, the proposition can be enunciated thus, a motion of rotation todt 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 awdt perpendicular to the plane containing the axes and in the direction in which Q8 moves. 192. It is often important to choose the base point so that the direction of translation may coincide with the axis of rotation. Let us consider how this may be done. Let the given displacement of the body be represented by a rotation 6 about PR, and a translation PP'. Draw PN perpendi- cular to PR. If possible let this same displacement be represented by a rotation about an axis Q8, and a translation QQ' along QS. By Arts. 187 and 188 QS must be parallel to PR and the rotation about it must be 6. This translation will move P a length Q Q' along PR, and the rotation about QS will move P along an arc perpendicular to PR. Hence Q Q' must equal PIf 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 p; , or, which is more conveniont, at a distance y from the plane NPP' where NP' = 2y tan ^ . The rotation 6 round QS is to bring ^to P' and is in the same direction as the rotation Q rouiid PR, Hence the distance -?/ must be measured from the » •' , ^ ■ ; lif! 1 !'■ \ nay he ',r with (osition m axis parallel islation in the so that otation. 3d by a 3rpendi~ resented ong QS. rotation gth QQ' f an arc P' must I a plane R where e y from »und QS )tation :om the TRANSLATION AND EOTATION. 169 1' middle point of NP' in the direction in which that middle point is moved by its rotation round Pit. Having found the only possible position of QS, it remains to show that the displacement of Q is really along QS. The rotation round PE will cause Q to describe an arc whose chord Qq is g 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 only moved by the translation PN, i.e. Q is moved along QS. 193. It follows from this reasoning that any displacement of a body can be represented by a rotation about some straight line and a translation parallel to that straight line. This mode of con- structing the displacement is called a screw. The straight line is sometimes called the central axis and sometimes the axis of the screw. The ratio of the translation to the angle of rotation is called the pitch of the screw. 194. 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 CB by Art. 187 are parallel. The displacement of any point Q on CB 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 CB. 195. 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 (odt about an axis PR and a translation Vdt in the direction PP. Measure a distance VsmP'PR y= from P perpendicular to the plane P'PR on that side of the plane towards which P' is moving. A parallel to PM through the extremity of y is the central axis. 196. 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, 0/3, O7 parallel and equal to A A', BB', CC If Op be the direction of the axis of rotation, the projections of Oa, 0/3, Oy on Op are all equal. Hence Op is the perpendicular drawn from on the plane a/37, ^^is O'^^o 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, find the translation along it. It is clearly equal to Op. 170 MOTION IN THREE DIMENSIONS. Ex. 8. Given the diflplncoments A A', BB' of two pointR A, B ot the body and the direction of the contnil axis, find the position of the central axia. Draw planes tlironph AA', BB' parallol to the central axis. BiHcct A A', UB'hy planes i)erpen- diciila:' to these planes respectively and parallel to the direction of the central axis. The'3e two last planes intersect in the central axis. Composition of Rotations. 197. It is often necessary to compound rot.ations 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- ti alarity, and then indicate generidly the mode of proceeding when the rotations are of finite magnitude. 198. To explain what is meant by a bodij having angular velocities about more than one axis at the same time. A body in motion is said to have an angular velocity to about a straight line, when, the body being turned round this straight line through an angle oodt, 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 (o^dt, to.jdt, o)/H. Then we shall first prove tha,t the final position is the same in whatever order those rotations are effected. Let P be any point in the body, and let its distances from OA, OB, C, respect- ively be Tj r^, r^. First let the body be turned round OA, then P receives .% displacement a>j\dt. By this motion let r^ be in- creased to 1\ + di\, then the displacement caused by the rotation about OB will be in magnitude w^ (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 becomefii ta^rjit. So also the displacement due to the remaining rotation will be wjrAt. 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 parallelepiped 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 Dody and w planea ( perpen- )ral axis. mt axes 3 which ons are ne par- iceediug angular ft) about straight brought + dt. ■ time dt, ght lines [dt, (o./it, he same be any respect- )A, then , be in- rotation )rding to he Umit acement jmaining le same manner will be diagonal ides, and nee then said to \\i three ermined mall in- die body COMPOSITION OF ROTATIONS. 171 round the three axes successively, through angles (o^dt, (o.jdt, m^dt. Then when df diminishos without limit the motion during the whole time will be accurately represented. 199. 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 di- rection of OA is called the positive direction of the axis. 200. If tv)o ancfidar velocities about two aires OA, OB he represented in magnitude and direction by the tiuo lengths O A, OB ; then the diagonal 00 of the parallelogram constructed on OA, OB as sides will be the resultant axis of rotation, and its length will represent the magnitude of the residtant angidar velocity. This Prop, is usually called " The parallelogram of angular velocities." Let P be any point in OG, and let PM, PN be drawn per- pendicular to OA, OB. oince OA represents the angular ve- locity 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 . sin COA - OB . sin GOB] = 0. Therefore the point P is at rest and 0(7 is the resultant axis of rotation. Let «D be the angular velocity about OG, then the velocity of any point A in OA is perpendicular to the plane AOB and is represented by the product of o) into the perpendicular distance of A from 00= ot . OA sin COA. But since the motion is also VI 172 MOTION IN THllEE DIMENSIONS. »1 :^1 f »f 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 sin BOA ; .-. o> = 0B sin BO A sin COA OG. Hence the angular velocity about C is represented in mag- nitude by OG. From this proposition we may deduce as a corollary "the parallelogram of angular accelerations." For if OA, OB repre- sent the additional angular velocities impressed on a body at any instant, it follows that the diagonal OG will represent the resultant additional angular velocity in direction and magnitude. 201. This proposition shows that angular velocities and an- gular accelerations may be compounded and resolved by the same rules and in the same way as if they were forces. Thus an an- gular velocity to about- any given axis may be resolved into two, (0 cos a and to sin a, about axes at right angles to each other and making angles a and „ — a with the given axis. If a body have angular velocities w^, w^, w^ about three axes Ox, Oy, Oz at right angles, they are together equivalent to a single angular velocity w, where w = Vwi* + «/ + w^, about an axis making angles with the given axes whose cosines are re- spectively ft). ft)„ w. This may be proved, as in the corre- _j ft) ft) ft) sponding proposition in Statics, by compounding the three angular velocities, taking them two at a time. It will however be needless to recapitulate the several propo- sitions proved for forces in Statics with special reference to an- gular velocities. We may use " t!.e triangle of angular velocities " or the other rules for compounding several angular velocities together, without any further demonstration. 202. A body has angular velocities a, w 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"G lying in the plane containing OA, O'B. m OB, the L of OB ,nBOA; in mag- iry "the B repre- body at esent the rnitude. 3 and an- the same us an an- into two, 3ther and ihree axes dent to a about an s are re- le corre- e angular ral propo- ce to an- elocities " velocities 'parallel resulting e limit of Lstance, it it the two velocity ning OA, COMPOSITION OF ROTATIONS. 173 Let X be the distance of this axis from OA, and suppose it to be on the same side of OA as OB. Let fl be the angular velocity about it. Consider any point P, distant y from OA and lying in the plane of thr three axes. The velocity of P due to the rotation about OA is wy, the velocity due to the rotation about OB is o)'{y — a). But tlicso two together must bo equivalent to the velocity due to the resultant angular velocity 11 about 0"G, and this is fl (y — x), .". 601/ + to' (y — a) =n (y — x). This equation is true for all values of y, .*. H = w + co', x=-^ . This is the same result we should have obtained if we had been seeking the resultant of two forces w, co' acting along OA, OB. If 0) = — (I)', the resultant angular velocity vanishes, but x is in- finite. The velocity of any point P is in this case wy-'ca!{y— a) =a&), which is independent of the position of P. The result is that two angular velocities, each equal to w 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 aoi. 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 «a about an axis OA is equivalent to an equal motion of rotation about a parallel axis OB plus a motion of translation aw perpen- dicular to the plane containing OA, OB, and in the direction in which O'B moves. 203. 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 Fj), where p is the distance between the forces. Corresponding to these wo 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. fl IMAGE EVALUATION TEST TARGET (MT-3) •'/ / .^■- 1.0 I.I ■50 IL25 i 1.4 WJA |25 ■^ Uii 12.2 m -.„ |2Q u 1.6 FholDgraphic .Sciences Corporation 23 WIST MAIN STMIT WIUTIi.N.Y. MSM (71«) t73-4S03 ^ '^1^ W'^ V ^t^ V" k 174 MOTIO^T IN THREE DIMENSIONS. 2. An angular velocity « is equivalent to an equal angular velocity about a parallel axis together with a linear velocity equal to cop, where p is the distance between the parallel axes. It follows that every proposition in Statics relating to forces has a corresponding proposition in Dynamics relating to the motion of a rigid body, and these two may be proved in the same way. To complete the analogy it may be stated (i) that an angular velocity like a force in Statics requires, for its complete determina- tion, five constants, and (ii) that a velocity like a couple in Statics requires but three. Four constants are required to determine the line of action of the force or of the axis of rotation, and one to determine the magnitude of either. There will also be a conven- tion in either case to determine the positive direction of the line. Two constants and a convention are required to determine the positive direction of the axis of the couple or of the velocity and One the magnitude of either. It is proved in Statics that a system of forces and couples is generally equivalent to a single force and a single couple, and that these may be reduced to a resultant JR, acting along a line called the central axis, and a couple about that axis. Or they may also be reduced to a resultant R of the same magnitude as before, acting along any line parallel to the central axis at any chosen distance c from it, together with a couple 0' about an axis perpendicular to the line whose length is c, and in- clined to the resultant It at an angle 0. Then we know that G' = '^ G^ + M^c*, and is a minimum when c = 0, and also that tan = -7y . The same train of reasoning by which these results ware established, will establish the following proposition. The ins'tan- taneous motion of a body having been reduced to a motion of translation and one of rotation, these are equivalent to a motion of rotation to about a line called the central axis, and a trans- lation V along that axis. Or they may also be reduced to a rotation « of the same magnitude as before about any line par- allel to the central axis, and at any chosen distance c from it, together with a translation V along a line perpendicular to the line c, and inclined to the axis of w at an angle 0. Then we know that 1^' = V !'''*+ cW, and is a minimum when c = 0, and C(0 V' may be established. X e f r a n &] to as ar also that tan = -^, In a similar manner many other propositions i ft 204. Ex. 1. The locus of points in a body moving about a fixed point which at any proposed instant have the same actual velocity is a circular cylinder. m » I :■ COMPOSITION OF ROTATIONS. 175 Ex. 2. The geometrical motion of a body is represented by angular velocities inversely proportional to ^-y, y-a, o - /3 about three lines forming three edges of a cube which do not meet nor are pai-allel. Prove that the body rotates about the line {P-y)x-aa = {y-a)7j-ap={a-p)z-ay, 2a being an edge of the cube, the centre being the origin, and the axes parallel to the edges. Ex, 3. A body has an angular velocity u about the axis as-a_y-/3_ z-y I m ~ n * where ?' + m^ + n^ = l. The motion is equivalent to rotations lu, mu, nu about the co-ordinate axes, and translations (my-np)u, (na-ly)2ri'-ff-»^-» 178 MOTION IN THREE DIMENSIONS. on the one side or the other of OT according to the direction of the rotation, equal to the supplement of % and produce RB to B so that TB=^ and join AB. By the triangle of rotations the rota/on is now represented by a rotation about OA which we may call d, followed by a rotation about OB which we may call ff. By Art. 211 the rotation 6' is equivalent to an equal rotation 0' about a parallel axis CD, 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 ir-J' 2 d' and if the distance r between AB, CD be such that 2r sin -^OZ*. The whole displacement has thus been reduced to a rotation aboiit OA followed by a rotation ff about CD. 213. Analytically, we might reason thus — A screw motion is given when we know (1) its axis, (2; the rotation about it, (3) the translation along it. The axis is known when its inclination to two of the axes and the two co-ordinates of the point in which it cuts the plane of xy are given. Thus six constants are required to determine a screw. Let a given screw be resolved into two screws. We have then twelve constants, but since they are together equivalent to the given screw there are six relations between the constants. We are therefore at liberty to choose any six relations we please between these twelve constants. We might, for example, resolve a given screw into two screws of any given pitches, the remaining four constants being chosen to make the axis of one screw coincide with any given straight line. If the given pitch of each screw be zero, the screws are reduced to simple rotations, and thus any displacement can be reduced to two conjugate rotations. It has been shown in the preceding article that the two rotations are real. 214. Ex. Show that any screw may be resolved into two real screws having the axis of one in a given direction and the axis of the other intersecting the first at a given angle. 215. Any two successive displacements of a body may he 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 V COMPOSITION OF ROTATIONS. m join AB, on about i parallel anslation B, CD be I rotation, 1 followed 1 when we Che axis is ' the point equired to constants, X relations 3lationB we ve a given ants being ae. If the ations, and t has been ews having g the first \ted by two lent a and M angle of rotation 0, and secondly along the axis CD with displacement a' and angle ff. Let OC be the shortest distance between OA and CD, and for the sake of the perspective let it be called the axis of y. 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 a' AOx=a. Draw a plane xOT making with the plane of xz an angle ^ , and let it cut Draw another plane AOR making with xz an angle ~ , and cutting the yz in OT. plane xOT in OR. Produce ^0 to a poijit P, not marked in the figure, so that PO=a, and let us choose P as a base pouit to which the whole displacement of the body may be referred. The rotation 0' is equivalent to a rotation 0* about Ox together with a 0' translation along 02'=2rsin~ by Art. 190 By Art. 205 the rotation about OA followed by 0' about Ox is equivalent to a rotation about OR where is twice the The whole displacement is now repre- angle ART, so that sm ^=8m :r . . „ 2 2 siuKx sented by (1) a translation of the base point 'P to 0, (2) the rotation fi, (3) a further 0' linear translation which is the resultant of the translations 2r sin - along OT and o' along Ox. By Art. 186 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 rule given in Art. 192. 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 and its axis is parallel tc OR by Art. 187. It follows by Art. 206 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. 189 add together the projections on OR of the three displacements OT, a, a'. The 0* projection of 0T=2r sin - cos TR = 2r cos Ty . cos TR which is twice the projection of the shortest distance r on the axis of rotation. If T be the Unear displace- ment, we have T=2r cos Ry + a cos RA + a' cos Rx. 216. If the component screws be simple rotations we have o=0, a'—O, and it may be shown without diflSculty that T sin ^j = 2r sin ~ sin -j,- sin a. It has bean shown in Art. 212 that any displacement may be represented by two conjugate rotations in an infinite number of ways, but it now follows that in all these r sin - sin -= sin a is the same. When the rotations are indefinitely small, and equal to udt, w'dt respectively, this becomes \ ruu 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 equa- tions to the axis of the resultant screw are 0' 0' 0' 0' 0' 0' -actan^' + j/Bin ^ + 2C085- = r8in-, ycos^ -28in-=rsin - cos^'cot-g , 12 2 180 MOTION IN THREE DIMENSIONS. where ^' is the angle xOR and is the resultant rotation. The first eqiiatiou expresses the fact that the central axis lies in a plane which bisects at right angles a straight line drawn from perpendicular to OR in the plane xOR to represent the linear tranBlation in that direction. The second expiesses that the central nxis lies in a plane parallel to TOR at a distance from it determined by Art. 192. These equations may also be deduced from those of Bodrigues given in Art. 223. To effect this we must write for (a, 6, c) the resolved parts of the translation along OT. Since however the positive direction of the rotation in Bodrigues' formulas 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 0, ff 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 OT, CP be drawn in the plane DCO such that the angle POC= ^ and tan PCO= sin" s" cot ^ . Then if P bo moved backwards by the rotation or forwards by the rotation d\ in either case its new position is a point on the central axis. Ex. 3. If OA , OS 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 ^t^ii^ and ^f l + cof'^'sin^g j , where , ^' ho angles the -t ultant axis of rotations makes with OA and CD. Then u ^ screwed back- Y r Is by the first screw or forwards by the sec.nd, in either case its new position is n point on the central axis. 217. Ex. 1. If the instantaneous motion of a body be represented by two con- jugate rotations udt and u'dt, the axis of the resultant screw intersects at right angles the shortest distance between the conjugate axes. Let y, y' be the angles the conjugate axes make with the axis of their resultant, a the angle they make with each other ; c, c' the shortest distances between the conjugate axes and the axis of the screw, V and C the linear and angular velocities of the screw, then prove that sin a* V sm7' cos 7' sm7 c'w' cos 7 sma c tan 7'= c' tan 7 = — . The first line follows from Art. 201. The second expresses the fact that the direction of the linear motion of the point where the axis cuts the shortest distance is along the axis of the screw. Ex. 2. If one conjugate axis of an instantaneous motion is at right angles to the central axis, the other meets it, and conversely. Ex. 3. If one conjugate axis of an instantaneous motion is parallel to the central axis, the other is at an infinite distance, and conversely. \> \: I I [[nation t angles ipresent ;ral nxis Irt. 223. >u along Eormulsa Trite fot either of taken in shortest rn in the 1 if P be ir case its inear dis- e straight ictively at ngles the wed back- position is two con- at right ngles the make with ;he axis of ove that t that the it distance ; angles to lei to the FIXED AXES. 181 Ex. 4. The locus of tangents to the trajectories of different points of the s ime straight line in the instantaneous motion of a body is a hyperbolic paraboloid. Let AB he the given straight line, CD its conjugate. The points on AB axe turning round CD and therefore the tangents all pass through two straight lines, viz. AB and its consecutive position A'B', and are also all parallel to a plane which is perpendicular to CD. Ex. 5. If radii vectores be drawn from a fixed point to represent in direction and magnitude the velocities of all points of a rigid body in motion, prove that the extremities of these radii vectores at any one instant lie in a plane. [Coll. Exam.] Motion referred to fixed axes. 218. The general equations of motion given in Art. 71 of Chapter II. involve the diflferential coefficients t- , -r , -r- . -t-5 » ^ dt at at dtr &c. It will now be necessary to express these in terms of the instantaneous angular velocities of the body. 219. Let us suppose in the first instance that one point in the body is fixed. Let us take this point as the origin of co- ordinates, and let the axes Ox, Oy, Oz be any directions fixed in space and at right angles to each other. The body at the time t is turning about some axis of instantaneous rotation. Let its angular velocity be fl, and let this be resolved into the angular velocities w^, w,, Wj about the co-ordinate axes. We have to dx dii dz find the resolved velocities -^ , -^ , -^ of a particle whose co- ordinates are x, y, z. These angular velocities are supposed positive when they tend the same way round the axes that positive couples tend in Statics. Thus the positive directions of Wj, «,, w, are respectively from y to z, from z to a;, and from a; to y. I' I I 1; 182 MOTION IN THREE DIMENSIONS. Let US determine the velocity of P parallel to the axis of z. Let PN be the ordinate z, and let PM be drawn perpendicular to Ox. The velocity of P due to the rotation about Ox is clearly (oPM. Resolving this along ^P we get w, Pi/ sin NPM = u)^y. Similarly that due to the rotation about Oy is — &),^a; ; and that due to the rotation about Oz is zero. Hence the whole velocity of P parallel to Oz is dz dt = (0^1/-(0^X, and the velocities parallel to the other axes are dx dy f^ = co,x-i — y^f^it by substituting for -^ , -^ , their values just found ; die dV xz yz «3-«l7--«2^« Hence the angular velocity of a particle about Oz is the same as that of the body when the particle lies in the plane of xy, or ,, (o^fjind the equations to the central axis. Let the same motion be also represented by the linear ve- locities u', v', w' parallel to the axes, of some other point 0' and by angular velocities &>,', w^, rUg' about axes parallel to the co- ordinate axes and meeting in 0'. Let (f, rj, f) be the co-ordinates of 0'. We have now two representations of the same motion, both these must give the same result for the linear velocities of any point. Hence u + w^z-(o^ = u' + «; (z-^)- to^' {y-v)^ V + co^x — (a^z = v' + (o^ (a;-|) — (u/ (2 — ^ r (1)> 1^+ w,y - (i)^x = w'+ to^' (y-v)- <»a' (^ - ^) . must be true for all values of x, y, z. This gives o)j'=6),, (o^ = to^, lo^' =a>^, so that whatever origin is chosen, the angular velocity is always the same in direction and magnitude. See Art. 188. Also (f, r}, f) may be so chosen that the vek uty of 0' is along the axis of rotation ; in this case we have {u', v, w) proportional to (a),, Wj,, tUg). The equation to the locus of 0' is therefore u + ft).^ g" - 6)3?; _ v + w^^—a>X _ w + to, 7; — . .(2). By multiplying the numerator and denominator of each of these fractions by g>j, &>,, 0)3 respectively, and adding them to- gether, we see that each of them is The motion of the body is thus represented by a motion of translation along the straight line whose equations are (2) and an angular velocity equal to fl about it. This straight line has been called the central axis, and the fraction just written down is equal to the ratio of the velocity of translation along the central axis to the angular velocity about it, i. e. the pitch of the screw. If the motion be such that mWj + vwjj + ^6)3 = 0, and «,, a>^, 6)3 do not all vanish, each of the equalities in (2) is zero, and hence by equation (1) it' =0, w' = 0, t«'= 0. The motion is there- fore equivalent to a rotation about the central axis, without translation. This is also evident from the analogy explained in Art. 203. 222. When the rotations are finite the corresponding formulae are somewhat more complicated. Let the given displacement of the body be a rotation through a '■^" 184 MOTION IN THREE DIMENSIONS. ii I ; 1 finite angle about an axis passing through the origin whose direction cosines are (I, m, n). It is required to find the changes produced in the coordinates [x, y, z) of any point P. Let PP* be tlie chord of the arc described by P and let Q be the middle point of PP'. Let x + Sx, y + ij/, « + & be the co-ordinates of P' and f, ?j, f those of Q. Since the abscissas of Q is the arithmetic mean of those of P and i" we have dx i'J 6z f=a!+rr; similarly 7;=y+^, f=j+ „ . Let QM be a perpendicular from Q on 2* e the axis, then PP' = 2 QM tan ,. Let (\, fi, v) be the direction cosines of PP', then since PP' is perpendicular to the axis, we have A+m/t + nv = 0, and since it is also perpendicular to OQ we have i\ + rifi + S;» = 0, hence \ ft _ V mt-nii~ nii-li~ lit-m^' The sum of the squares of the denominators is (f ' + »?' + f«) (J" + m» + n") - (/H »»»; + nf )«, which is OQ' - 03f*= ^37*. Hence each of these latios is = ^^ . Now ix is the projection of PP" on the axis of x, 8 Q .'. Zx=2Q,M . tan 5 \ = 2 tan ^ (mf - nij) ; similarly 5y=2tan ^ (n^-l^;), &=2 tan 5 {l-ri-m^), which are the required formulso. If the origin have a linear displacement whose resolved parts parallel to the axes are (a, 6, c), we must add those displacements to the values of 8ar, iy, 8« found by solving these equations. Let the co-ordinates of the middle point of the whole dis- placement of Pbe represented by f, 1;', f . Then we have, as before, k'=x + -^ &o., but since 805, Sy, Sz, are increased, by a,, b, c we must write f-H»'?'-o>f'~o ^0' f 1 ij. f« We thus obtain 8«.=a-!-2tan| jn^^f -|j -n ^ij'-0j , with similar expressions for Sy and Sz. 223. 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 (^', yj, f) the co-ordinates of the middle point of the displacement are co-ordinates of a point in the axis, and 8x, Sy, Sz are proportional to {I, m, n) the direction cosines of the axis. Hence a + 2tan|j^(r-|)-n(v-|)j i + 2tan|] n(r-|) -^(^-0 j I m c^!iun|j.(^-|).-^(f-;)j n Each of these is evidently equal to la + nib + nc, which is the linear displacement along the central axis. The results of this and the preceding Article are due to Bodrigues. FIXED AXES. 185 coslneA are I {x, y, z) o! iddle point those of Q. P' we have From Q ou ndionlar to iQ we have id formulae. to the axes iz found by ) whole dis- Sx , b ., e "2'* ~2 ns without along the sement are m, n) the placement ire due to 1 224. Ex. Let the restraiuta on a body be such that it admits of two motions A and B each of which may be represented by a screw motion, and let m, m' be the pitches of these screws. Then the body must admit of a screw motion compounded of any indefinitely small rotations udt, u'dt about the axes of these screws accom- panied of course by the translations mudt, m'u'dt. Prove that (1) the locus of the axes of all those screws is the surface z {x' + y'*) = 2ary. (2) If the body be screwed along any generator of this surface the pitch is c + a cos 20, whore c is a constant which is the same for all generators and is the angle the generator makes with the uxis of X. (li) The size and position of the surface being choben so that the two given screws A and B lie on the surface with their appropriate pitch, show that only one surface can be drawn to contain two given "rews. (4) If any three screws of the surface be taken and a body be displaced by being screwed along each of these through a small angle proportional to the si^e of the angle between the other two, the body after the last displacement will occupy the same position that it did before the first. This surface has been called the cylindroid by Frof. Ball, to whom these four theorems are due. 225. Ex. 1. If an instantaneous motion be given by the linear velocities (u, V, w) along and the angular velocities {ui, w,, u^) about the co-ordinate axes, show that the equations to the conjugate of -/. I y-g m z-h are X w, I X / m y 9 z W3 n z h =/u + ni» + nw,. = (/-a;)u + (5f-y)r + (/t-2)«. The first equation follows from the fact that the direction of motion of any point on the conjugate is perpendicular to the given axis, and the second from the fact that the direction of motion is also perpendicular to the straight line joining the point to (/, g, h). Ex. 2. If an instantaneous motion be represented by a screw along the axis of 2, the linear and angular velocities being V and 0, prove that the equations to the X - / y-g conjugate of —j^ = - — - = , - aiemx-hj + n-^=0 aaigx-fy--^(z-h)-0. I m n II if Ex. 8. The locus of the conjugates of all axes of- instantaneous rotation which are parallel to a fixed straight line is a plane parallel to the central axis and to the fixed straight line. Ex. 4. The locus of the conjugates of all axes of instantaneous rotation which pass through a given point is a plane. If two axes intersect, their conjugates also intersect. 226. If the instantaneous motion of a body b« represented by two conjugate rotatioas about two axes alright angles, a plane can be drawn through either axis perpendicular to the other. The axis in the plane has been called the characteris- tic of that plane, and the axis perpendicular to the plane is said to cut the plane in its focus. These names were given by M. Ghasles in the Comptes Rendus for 1843. Some of the following examples were also given by him, though without demonstra- tions. 186 MOTION IN THREE DIMENSIONS. Ex. 1. Show that every plane has a characteristic and a focaa. Let the central axis cut the plane in 0. Besolve the linear and angular veloci- ties in two directions Ox, Oz, the first in the plane and the second perpendicular to it. The translations along Ox, Oz may be removed if we move the axes of rotation Or, Oz parallel to themselves, by Art. 202. Thus the motion is represented by a rotation about an axis in the plane and a rotation about an axis perpendicular to it. It also follows that the chaiacteristic of a plane is parallel to the projection of the central axis. Ex. 2. If a plane be fixed in the body and move with the body, it mtersects its consecutive position in its characteristic. The velocity of any point P in the plane when resolved perpendicular to the plane is proportional to its distance from the chMacteristic, and when resolved in the plane is proportional to its distance from the focus and is perpendicular to that distance. Ex. 3. If two conjugate axes cut a plane in F and G, then FG passes through the focu;. If two conjugate axes be projected on a plane, they meet in the characteristic of that plane. Ex. 4. If two axes CM, CN meet in a point C, their conjugates lie in a plane whose focus is C and intersect in the focus c ' the plane CMN. This follows from the foct that if a straight line cut an axis the direction of motion of every point on it is perpendicular to the straight line only when it also cuts the conjugate. Ex. 5. Any two axes being given and their conjugates, the four straight lines lie on the same hyperboloid. Ex. 6. If the instantaneous motion of a body be given by the linear and angu- lar velocities (m, v, lo) (wp Wj, a?a)i prove that the characteristic of the plane is its intersection with A (u + u^z- w^y) +B (v+ u^x- uiz) + C {w + u^y - u^) = 0, and its focus may be found from w + Ujy - WjO! ■ ^- . For the characteristic is the locus of the points whose directions of motion are perpendicular to the normal to the plane, and the focus is the point whose direction of motion is perpendicular to the plane. What do these equations become when the central axis is the axis of z ? Ex. 7. The locus )f the characteristics of planes which pass through a given strai-iiht line is a hyperboloid of one sheet ; the shortest distance between the given straight line and the central axis being the direction of one principal diameter, and the other two being the internal and external bisectors of the angle between the given straight line and the central axis. Prove also that the locus of the foci of the planes is the conjugate of the given straight line. Ex. 8. Let any surface A be fixed in a body and move with it, the normal planes to the trajectories of all its points envelope a second surface B. Prove that if the surface B bo fixed in the body and move with it, the normal planes to the M + WgZ - Wjy _v+ u^ - Wj^s A ~ B ular velooi- jndicular to of rotation sented by a mdicular to rojection of it intersects it /* in the iitauce from its distance euler's equations. 187 trajectories of its points will envelope the surface A : so that the surfaces A and B have conjugate properties, each surface being the locus of the foci of the tangent planes to the other. Prove that if one surface is a quadric the other is also a quadric. Ex. 9. A body is moved from any position in space to any other, and every point of the body in the first position is joined to the same point in the second position. If all the straight lines thus found be taken which pass through a given point, they will form a cone of the second order. Also if the middle points of all these lines be taken, they will together form a body capable of an infinitesimal motion, each point of it along the line on which the same is situate. Gayley's Report to the Brit. Assoc, 1862. 3es through icteri&tic of ) in a plane iirection of 'hen it also •aight lines and angu- le notion are direction ;h a given the given leter, and the given )ci of the e normal 'rove_ that 28 to the Eulers Equations. 227. To determine the general equations of raotion of a body about a fixed point. Let the fixed point be taken as origin, and let x, y, z be the co-ordinates at time t of any particle ni referred to any rectangular axes fixed in space. Let Xm, Ym, Zm be the impressed forces acting on this element parallel to the axes of co-ordinates, and let L, M, N be the moments of all these forces about the axes. Then by D'Alembert's Principle, if the effective forces m dt\ X m -A , m, -^ a ^6 applied to every particle m in a reversed direc- tion, there will be equilibrium between these forces and the im- pressed forces. Taking moments therefore about the axes, we have H^^^-y^y^- ■ «. and two similar equations. To simplify these equations, let a>^, w^, w, be the angular velo- cities about the axes. dz ^ = <.^y-a>^x; d'x Then -£ = co^z-(o^, dy G),a; — a)j.«, Je ~^ dt d(o do), , . , . y-^^^y \^^y ~ ^*^) ~ "» ("'^ ~ ^'^h dhi dw, da>- . . . , . 183 MOTION IN THREE DIMENSIONS. ■. I Substituting in equation (1) we get Sm (a^ + /) — - tmyz 1. K — Xmxy . (ft)/ — cDj,') + Sm (cc' — j/^) m^o)^ — Xmyz . 6)^6), + Xmxz . &)j,ft), 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 ei)j, ft)^, 0)3 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, OG are passing through them at the moment under consideration. Then 0)3, = a),, w^ — a)^, co,= (o^. If the principal axes at the fixed point have been chosen as the set of axes fixed in the body, and A, B, C be the moments of inertia about them, the equation takes the form C di ft). dt -(A-B)co,a,, = N, in which all the coefficients are constants. 228. "We shall now show that -~ = -— . This may appear at first sight to follow at once from the equation 0)3 = w^ But it is not so; 0), denotes the angular velocity of the body about OG fixed in the body, while ft>, denotes the angular velocity about a line Oz fi^ed in space and determined by the condition that at the time t 0(7 coincides with it. At the time t-\-dt OC will have separated from Oz and we cannot therefore assert a priori that the angular velocity about OG will continue to be the same as that about Oz. We have to prove that this is the case as far as the first order of small quantities. Let OR, OR' be the resultant axes of rotation at the times t and t-\-dt, i.e. let a rotation ^dt about OR bring OG into coincidence with Oz at the time t, and let a further rotation ^'dt about OR' bring OG into the position OC' in space at the time t+dt. Then according, to the definition of a differ- ential coefficient da> dt 5 = X'of ft). ^'=Z'of dt n' cos R'C'-n cos RG dt CI' COS R'z — n COS Rz ~dt Since a rotation about OR' brings OG from the position Oz to OG', EG' and R'z differ by quantities of the second order, and therefore these two diflferential coefficients are ultimately equal. J 11 euler's equations. 189 ^ = N. ! manner. d products ce and are en take a body, and es. Since ! so chosen 3m at the >, = «8- If as the set I of inertia lay appear But it is 1 00 fixed a line Oz the time t separated le angular about Oz. 3t order of if rotation OR bring a further in space a differ- jon Oz to Irder, and 1 equal. 229. The following demonstration of this equality has been given by the late Professor Slesser of Queen's College, Belfast, and is instructive as founded on a different principle. Let A, B, Che 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 fl be the angular velocity about it. Let the angles LOA, LOB, LOG be called respectively a, /8, 7. Then by Art, 201 fl = &)j cos a + ftjg co^ fi + f^s cos 7 ; do. d w. i^ dt 230. The three equations of motion of the body referred to the principal axes at the fixed point are therefore ^da>. dt .d(o„ {B-C)co,co, = L, B"^'-{C-A)co,co,=^M, C^^f-iA-B)co,ay, = K These are called Euler's equations. 231. 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 moment of the effective force about the principal axes at the fixed point are expressed by the left-hand sides of the above equations. If there is no point of the body which is fixed in space, the motion of the body about its centre of gravity is the same as if that point were fixed. In this case, if A, B, G be the principal moments at the centre of gravity, the left-hand sides of Euler's equations give the moments of the effective forces about 190 MOTION IN THREE DIMENSIONS, 'I 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. 232. Ex. 1. If 2T=-Awj' + Buf^' + Cu)^' and be the moment of the impressed forces about the instantaneous axis, the resultant angular velocity, prove that dT dt = GQ. Ex. 2. A body turning about a fixed point is acted on by forces which tend to produce 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. 233. 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 P, Q, R be the resolved parts of the pressures on the body in these directions. Let /it be the mass of the body. Then we have 1"^ = ^-^^''^ d^x.. and two similar equations. Substituting for -5-j its value in terms a>„ 6)^, G), we have , 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 —j-" and -y- from Euler's equat.c as. We then have with similar expressions for Q and R. 234. 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 O** . GH along GH which is a perpendicular on the instantaneous axis 01, being the resultant angular velocity, and the other 0'*. GK perpendicular to the plane OGcK, where GK is a perpendicular on a straight B-C C-A line OJ whose direction cosines are proportional to - w«w, J "81 "S"!' A-B Wjw,, and 0'* is the sum ot the squares of these quantities. i euler's equ.mions. 191 want the I the fixed uts by the lie impressed ', prove that hich tend to is, show that iments at the ce rotation is t rest. ty referred point, and le body in 1 we have le in terms •+2wX le with the considera- I equat.c as. JV \ the terms on ted point are jendicular on |nd the other ^n a straight C-A 235. To determine the geometrical equations connecting the motion of the body in space with the angidar velocities of the body about the three moving axes, OA, OB, 00. Let the fixed point be taken as the centre of a sphere of radius unity ; let X, Y, Z and A, B, C be the points in which the sphere is cut by the fixed and moving axes respectively. Let ZC, BA produced if necessary, meet in E. Let the angle XZG = y(r, ZG = 0, EGA = , y^, and Wj, tu^, Wj. Draw CN perpendicular to OZ. Then since •x^ is the angle the plane GOZ makes with a plane XOZ fixed in space, the velo- city of G perpendicular to the plane ZOG is GN -T , which is the same as sin ^ -^, the radius OG of the sphere being unity. Also the velocity of C along ZG is de dt' Thus the motion of G is re- presented by -J- and sin 6 -^ respectively along and perpendi- cular to ZG. But the motion of G is also expressed by the angular velocities Wj and »„ respectively along BG and GA. These two representations of the same motion must therefore be equivalent. Hence resolving along and perpendicular to ZG we have dO sin ^-i'- = — dt J = ft>, sin + «i>a ^^^ Wj cos ^ + Wj sin n 192 MOTION IN THREE DIMENSIONS. i i ' Similarly by resolving along CB and CA we have (»j = -^ sin 9 — -r- sm cos 9 dt ■dO dyjr cOjj = -^ cos , "^y we should have obtained other equations. But as we cannot have more than three independent relations, we should only arrive at equations which are algebraic transformations of those already obtained. 236. Ex. lip, q, r be the direction cosines of OZ with regard to the axes OA, OB, 00, show that these equations may be put into the symmetrical form dr - dp Any one of these may foe obtained by differentiating one of the expressions p = -Bin.0coa, g = 8intf8in^, r = coBO. The others may be inferred by the rule of symmetry. 237. It is clear that instead of referring the motion of the body to the principal axes at the fixed point, as Euler has done, we may use any axes fixed in the body. But these are in general so complicated as to be nearly useless. When, however, a body is making small oscillations about a fixed point, so that some three rectangular axes fixed in the body never deviate far from three axes fixed in space, it is often convenient to refer tlie motion to ii ! EULERrt EQUATIONS. 193 Qt to each )raic trans- :on O^we -^ sin ZE, at f A relative this is the e along AB is also ex- of the same tion of any as we cannot should only ons of those to the axes OA, form ,1 = 0. the expreflsions infened by the ,nof thehody has done, we in general so er, a body is it some three tr from three tie motion to these even though they are not principal axes. In this case «,, ft)j, Wg are all small quantities, and we may neglect their products and squares. The general equation of Art. 227 reduces in this case to at at at where the coefficients have the usual meanings given to them in Chap. I. We have thus three linear equations which may be written thus : dt dt dt^ dt at -E (U, dt dt dt 238. It appears from Euler's Equations that the whole changes of Wj, u.^, wj are not due merely to the direct action of the forces, but are in part due to the centrifugal force of the particles tending to carry them away from the axis about which they are revolving. For consider the equation du, N A-B = ?. + — /?- '^'Wj- dt C N Of the increase du^ in the time dt, the part -r^ dt is duo to the direct action of wjWjj dt is due to th. centrifugal the lorces whose moment is X, and the part — ^^ force. This may bo proved as follows. If a body he rotating about ■. n axis 01 with an angulir velocity w, then the moment of the centrifugal forces of the -hole body about the axis Oz is {A -B) wiWj. Let F 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 wh-m tending from 01. II. D. 13 } ; ri \ , i .': «•( !' ■« ■ 1 1 194 MOTION IN THREE DIMENSIONS. The force u'rm is evidently equivalent to the four forces bPxm, uhjm, u*zm, and - w'um acting at P parallel to x, y, z, and u respectively. The moment of ui^xm round Oz= - u'rym \ uhjm = u^xym y, w'zm =0 ) these three therefore produce no effect. The force - u'um parallel to 01 is equivalent to the three, - ww, um, - wwj um, -UW3UVI, acting at P parallel to the axes, and their moment round Oz is evidently wum(«iy-«ax). Now the direction cosines of 01 being — , —7, -f, we get by u u u <•>, Wfl projecting the broken line *, y, z on 01, u=-^x+ — y+ —z; therefore sub u u Btituting for u, the moment of centrifugal forces about Oz is = (ujij - Wjx) {ujx + w^y + u^z) m, = (wi'iry + WjWay' + u^"^^ - w^w^a;^ - w^xy - WjjWaiKa) m. Writing S before every term, and supposing the axes of a;, y, z, to be principal axes, then the moment of the centrifugal forces about the principal axis Oz = WiW32»i(y* - a:*) = w^Wj (A — B). Let the moments of the centrifugal forces about the principal axes of the body be represented by L', M\ N', so that L' = (B-C} W4W3, M'={C-A) W3W1, N'=(A- B) Wj Wg, and let G be their resultant couple. The couple G is usually called the centrifugal couple. Since L'w-^-\-M'u^-\-N'u.^=Q, it follows that the axis of the centrifugal couple is at right angles to the instantaneous axis. Describe the momeutal 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 Au^, Bua, Cwg. Since Aw^L' + BuiM' + CW3N' =0, it follows that the axis of the centrifugal couple is at right angles to the perpendicular OH. The plane of the centrifugal couple is therefore the plane lOH. If /ik^ be the moment of inertia of the body about the instantaneous axis of rotation we have *"= -y^, and T=ijJc^u)^ is the Vis Viva of the body. We may then easily show that the magnitude G of the centrifugal couple is G=T tan , where is the angle lOH. This couple will generate an angular velocity of known magnitude about the diametral line of its plane. By compoun-liug this with the existing angular velocity, the change in the position of the instantaneous axis might be found. Expressions for Angular Momentum. 239. We may now investigate convenient formuloB for the angular momentum of a body about any axis. The importance of these has been ah-eady pointed out in Art. 77. In fact, the general equations of motion of a rigid body as given in Art. 71, I EXPRESSIONS FOR ANGULAR MOMENTUM. 195 ym, u*zm, aud urn, -wu^um, Oz is evidently -3, we get by u therefore sub- to be principal ixis Oz es of the body the centrifugal fugal couple is instanianeous tangent plane Cwg. Since al couple is at aneous axis of )dy. We may is G'^Ttan^, iude about the gular velocity, lie for the importance n fact, the in Art. 71, 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 Mk^o) where AW is the moment of inertia, and w the angular velocity about that point. Our object is to find cor- responding formulae when the body is moving in space of three dimensions. Following the same order as in Euler's Equations, we shall first find the angular momentum about any fixed straight line in space, taken as the axis of z and passing through the fixed point; secondly, the momentum about any fixed straight line in the body and also passing through the fixed point, and lastly, we shall show how the angular momenta about other axes may be found. 240. A body is turning about a fi, be the angular velocities of the body about the fixed axes. Then the moment of the momentum about the axis of z is L = 2m \x dy 8» (o, — a>,, and the moment of the momentum about the moving axis of z will be expressed by the form where C7=2w(ar" +y«). E = Xmx2\ D^Xmy'z. These will be constant throughout the motion, and their values may be found by the rules given in Chapter I. If the axes fixed in the body be principal axes, t\. i die pro- ducts of inertia will vanish. The expressions for the moments of the momentum will then take the simple forms hi — -4a), A,' = Bu)^ K = ^^z where A, B, C are the principal moments of the body. Let the direction-cosines of the axes fixed in space but moving with reference to axes fixed in the body be given by the following X y z a:, y, o,, a, K c. a' a. &o, K '8' 1' 'U* diagram ; where, for example, \ is the cosine of the angle between the axes of z and y'. It has just been proved that the resultant of the mo- menta of all the particles of the body is equiva- lent to the three "couples" h^, h^, h^ about the axes Ox', Oy\ Oz'. Hence the moment of the momentum about the axis of z which is fixed in space may be v/ritten in the form which will be frequently found useful. 242. It may be required to find the moment of the momen- tum about axes neither fixed in space nor in the body, but moving in any arbitrary manner. This will be expressed by the same form as if the axes were fixed. If at^, (Oy, tw, be the angular velocities about these axes, the moment required will be = Xm (a;* i-y^) w, — (Zmxz) a>^ — {Xmyz) a^. If the axis of z coincide with the instantaneous axis of rotation, «,, = (), (Uy=0, and ft)j is the resultant angular velocity. The ex- pressions for the moments of the momentum or areas conserved about the axes of x, y, z become respectively - i^mxz) w„ - (Xmyz) &>,, tm {x^ f ?/") w,. The axis of the couple which is the resultant of the moments of the momentum about the axis is sometimes called the resultant axis of angular momentum and sometimes the resultant axis of areas. It is to be remarked that this axis does not in general ON MOVING AXES AND RELATIVE MOTION. 197 1 about the their values . i cUe pro- moments of but moving be following X, y, z ace may be he m omen- but moving Y the same le angular of rotation. The ex- conserved e moments e resultant mt axis of in general coincide with the instantaneous axis of rotation. The two are coincident only when the axis of rotation is a principal axis. If a body be turning about a straight lino, which we may call the axis of z, as instantaneous axis, it is a common mistake to suppose that the angular momentum about a perpendicular axis is zero. We see from the last remark that this is not generally true. If it be required to find the moment of the momentum about the axis of ^ of a rigid body moving in any manner in space, we may use the principle proved in Chapter II. Art. 76. In the case of a system of rigid bodies, the moment of their momenta may be found by adding up the separate moments of the several bodies. Ex. 1. A triangnlar nroa ACB whose mass is M is turning roimd the side CA with an angular velocity u. Show that the angular momentum about the side CB is -^ Mob sin- Cw, where a and b 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 u about a vertical straight line through so that the two rods are in a vertical plane. If 0,

.] Ex. 3. A right cone, whose vertex is fixed, has an angular velocity u com- municated to it about its axis OC, while at the same time its axis is set moving in space. The semi-angle of the cone is - and its altitude is h. If be the inclii tion of the axis to a fixed straight line Oz and f the angle the plane zOC makes with a fixed plane through Oz, prove that the angular momentum about Os is f Mh^u (sin' -j^ + i cos 0), where M is the mass of the cone. Ex. 4. A rod AB is suspended by a string from a fixed point and is moving in any manner. If {I, m, n) {p, q, r) be the direction cosines of the string and rod referred to any rectangular axes Ox, Oy, Oz, show that the angular momentum about the axis of z is dp .,.«/, <^'» dl\ .,a^ I dq dp\ , .,ab f dm dp , ,dq dl\ '^^Vw-'''dt)'-^jVi-^i)-'''2[pdt-'''i+'Tr^di}' where M is the mass of the rod, and a, b the lengths of the rod and string. On Moving Axes and Relative Motion. 243. In many cases it will be found convenient to refer the motion of the body under consideration to axes moving in space in some manner about a fixed origin. If we refer the motion of these axes to other axes fixed in space we shall have the inconvenience of two sets of axes. For this reason their motion at any instant is sometimes defined by angular velocities {0^, 6^, 0^) about them- I I ! ; :< 198 MOTTON IN THREE DIMENSIONS. selves. In this case wo are to regard the axes as if they were a material system of three straight lines at right angles whose motion at any instant is given by three coexistent angular velo- cities about axes instantaneous! v coincident with them. When the axes are moving we may suppose the motion of the body to be determined by the three angular velocities a>,, ta , tw, about the axes, in the same manner as if the ax'^s were fixed for an instant in space. The position of the body at tho time t + dt may be constructed from that at the time t by turning the body through the angles (o.dt, (o^dt, (o,^dt successively round the instan- taneous poHition of the axes. liut it must be remembered that w^dt does not now give the angle the body has been turned through relatively to the plane xz, but relatively to some plane fixed in space passing through the instantaneous position of the axis of z. The angle turned through relatively to the plane o*" xz is {o>,- 0,) dt 244. To find the resolved part of the velocity of any particle parallel to the moving axes. The resolved parts of the velocity of any point whose co- ordinates are {x, y, z) are not given by ^ » T/f » ^ • These are the resolved velocities of the particle relatively to the axes. To find the motion in space we must add to these the resolved veloci- ties due to the motion of the axes themselves. If we supposed the particle to be rigidly connected with the axes, it is clear that its velocities would be expressed ^y the forms given in Art. 219 with ^,, Q^, 6^ substituted for w,, w^, «g. So that the actual resolved velocities of the particle are dX y. n *'=^-«^i + «^^3. dt dZ y, ^ 245. To find the accelerations of any particle p)arallel to the axes we may proceed thus. The velocities of the particle at the time t resolved parallel to the axes Ox, Oy, Oz are respectively («, v, w). At the time t-\- dt, the axes have been turned into the position Ox, Oy, Oz by rotations equal to d^dt, d^dt, d^dt round the axes Ox, Oy, Oz respectively, and the velocities of the particle parallel to the axes in their new position are , dn ,. , dv ,^ dw , w + -J- «<, V 4- -Tzdt, w + y dt. at dt dt ON MOVINQ AXES AND RELATIVE MOTION. 199 <^,„ (o. xz the Describe a sphere of unit radius whose centre is at the fixed origin and let all these axes cut the sphere in the points x, y, z, x\ y , z respectively. Thus we have two spherical triangles xyz and x'y'z', all whose sides are right angles. The resolved part of the velocity of the particle at the time t + dt along the axis of z is H'r w + ^dt) COS zz . at dt) COS zx' + (v + -£dt) COB zy' + (v., ^^ By the rotation round Oy, x' has receded from z by the arc O^dt, and by the rotation round Ox, y' has approached z by the arc 6^dt. Therefore zx =zx + $,^ dt, zy' = zy — 0^ dt. Also the cosine of the arc zz differs from unity by the squares of small quantities. Substituting these we find that the compo- nent velocity of the particle at the time t + dt parallel to the axis of z is ultimately w + -r-dt — u9^ dt + vd^ dt. But the acceleration is by definition, the ratio of the velocity gained in any time dt to that time. Hence if Z be the acceleration resolved parallel to the axis of z, we have Similarly if X and Y be the accelerations parallel to the axe ^ of X and y, we have r= *-,„(>.+<. 246. Ex. 1. Let the motion be referred to oblique moving axes so that the sides of the spherical triangle xyz are a, b, c and the angles A, £, C. Let the equal quantities sin a sin 5 sin C, sin 6 sine sin J, sin c sin a sin ^ be called /i. Prove that if the velocity be represented by the three components u, v, lo parallel to these axes, then the resultant acceleration parallel to the axis of z is _ dw du , dv ^ ^ Z = -TT + -i7C0s6 + 3- cosa-«tf„u+vtf,u, at at at with similar expressions for X and T. This may be done by the use of the spherical triangles xyz, afy'sf, by first proving that za! =b + O^dt 3in c Bin A, zy'=a-6idt sine Bin B, and then substituting as before. Ex. 2, Prove in the same way that it x, y, z be the co-ordinates referred to oblique axes, and u', v', w' the resultant velocities parallel to the axes, , dz dx , dy w =^+ -TrCOsJ+^cosa-x^gM + y^iA*, with similar expressions for «' and f '. (' I, '. ( < ' I r \A • :-!■ 1 ii^; 1:1 I • 200 MOTION IN THREE DIMENSIONS. Ex. 3. Prove also that the equations connecting u, v, w with the co-orJinatea are sin'c -cotfi -cot^ h Ox e. z X y with two similar expressions for u and v. Since w' is the resolved velocity parallel to 2 of (m, v, w,) we have « cos 6 + V cos a + to= w', with similar expressions for u' and 1/, By solving these we get the required values of W, I', w. Ex. 4. If the whole acceleration be represented by the three components X, Y, Z parallel to the axes, prove that the expressions for these in terms of uvw, may be obtained from those given in the last example by changing x, y, z into u, v, w and M, V, to into X, Y, Z. 247. To express *lie geometrical conditions that a straigitt line whose equations with reference to the moving axes are given is fixed in direction in space. Let the equation to the given straight line be V 9. r and let the equations be so prepared that {p, q, r) are the direction cosines of the line. Let a straight line be drawn through the origin parallel to this given straight line and let a point Pbe taken on this at any given distance L from the origin 0. Then the co-ordinates of P are pL, qL, rL respectively. Since the straight line OP is fixed in direction in space, the resolved parts of the velocity of P parallel to the axes are zero. Hence we have dip dt - Lq9^+ Lrd,^^ 0, and two similar equations. The required geometrical conditions are therefore f^-qe,+re, = o, §->-^. + ?'^3 = o, %-pe,+qe, = o. When it is necessary to refer the motion of those moving axes to otl.cr axes fixed in space, we may cither use tlie equations of this article or those of Art. 23o. Taking the notation of the ON MOVING AXES AND RELATIVE MOTION. 201 30-orJinates lirecl values omponents ns of uvw, into M, V, w \ight line is fixed diti ous g axes oils of f the are the through Dtp be Then ice the ',/ d parts . '■ e have ■■' '■:] article referred to, it is obvious (the axes being treated as a body consisting simply of three straight lines) that we shall have the results -^ sin ^ = — ^j cos + 0^ sin . do). n , a Similarly — ^j, hc^ce da^ dt d(o^ ~dt d(o,, da)„ n n If we substitute these expressions in the given general equation we shall have the corresponding equation referred to moving axes. If the moving axes be fixed in the body, and move with it, we have ^j = o>j, 6^=-(o^, d^=o)^. In this case the relations will become -"^^ -—' d^y_d^, ^^ _ ^ as in Art 290 become ^^- ^^, dt~ dt' dt - dt ' ^^ '"^ ^^^- '^-'^• The preceding proof of the relation between —j^ and -~ IS a simple corollary from the parallelogram of angular velocities. The result will therefore be true for any other magnitude which obeys the " parallelogram law." In fact the demonstration is exactly the same. Now linear velocities and linear accelerations do obey this law. Hence the expressions obtained in Arts. 244, 245, for the velocities (m, v, w) and the accelerations {X, Y, Z) may be deduced from the one proved above. If the general equation i/r = should contain the velocity or acceleration of any particle of the body, then to obtain the corre- sponding equation referred to moving axes, we must substitute for these velocities or accelerations the expressions found in Arts. 244 and 245. 251. If the general equation sliould contain - — j' or any other second differen- tial coefficients, the expressions to be substituted for them become more compl • catod. 'I I 204 MOTION IN THREE DIMENSIONS. '■. u; i ii Since -^ , -r^ , -y-' , being angular accelerations, follow the parallelogram law, ftt »C («C we have dt = (-^ - Wa^3+ ta-iOa) COS0+ f ^ - wA + '^1^3) ''"^ ''^ ( ^' " "1^8 + <^A) COS 7. We may repeat the same reasoning and we shall finally obtain So we may proceed to treat third and higher differential coefficients. 2.52. A body is tuiiiing about a fixed point in any manner, to determine the moments of the effective forces about the axes. Let {x, y, z) be the co-ordinates of any particle m of the body referred to axes fixed in space and meeting at the fixed point, and let h^, h^, h^ be the moments of the momentum about the axes. The moment of the effective forces about the axis of z is 0. and this may be written in the form dh^ It Thus the moments of the effective forces about axes Ox, Oy, Oz fixed in space are respectively ,S ->,-, -7.-, where A,, \, h^ have the values found in Art. 240. Let A,', ^/, /ig' be the moments of the momentum, found by Art. 242, about axes Ox', Oy, Oz moving in space about the fixed origin. Let d^, 6^, 0^ be the angular velocities of these i xes about their instantaneous directions. Then since moments or *.uaples follow the parallelogram law, we see by the proposition of Art. 250 that the moments of the effective forces about the moving axes are respectively ff-V^3 + W. ^-h:e,^hx, 'J^-h;d, + h^e,. If the moving axes be fixed in the body, wo have 0^ = ro^ , 0g=ft)j, ^g — (Wg, and the equations admit of some simplification. If the axes be the principal axes we have h^ — Aw^, h^'=Ba)^, |1 (: ON MOVING AXES AND RELATIVE MOTION. 205 Ag' = Ca)g, and the moments of the effective forces take the simple forms dt (A -B) (0^(0^, where A, B, C are the principal moments. See Art. 230. If it be required to find the moment about the axis of ^ of the effective forces on a rigid body moving in any manner in space, we may use the principle proved in Chap. ll. Art. 72. In the case of a system of rigid bodies, the moment of their effective forces may be found by adding up the separate moments of the several bodies. 253. To obtain the general equations of motion of a system of rigid bodies. These equations have been already obtained in Chap. Ii. Art. 83, when the system is referred to axes fixed in space. If the axes be moveable we must replace the accelerations -n > 'jit ~jf. t>y the corresponding forms in Art. 245 and the couples -y,*, -y^, -^ by the expressions in Art. 252. Thus, suppose we refer the motion to three axes moving in space about a fixed origin 0. Let X, Y, Z be the impressed forces on any rigid body of the system, including the unknown reactions of the other bodies of the system. Let L, M, N be the moments of these forces about axes drawn through the centre of gravity of the body parallel to the co-ordinate axes. Let m be the mass of the body. Then if we adopt the notation of Arts. 245 and 252, the equations of motion for the rigid body under con- sideration will be at ^ VI dv n n Y dt * ' m dw a y a ^ dt * ' m -:i r 4 ^i^.tM'uJlm I \l >.ll 206 and MOTION IN THKEE DIMENSIONS. dh; dt -Ke, + h:0, = L, dt - h:d, + k;e, = M, dh: dt - hX + K^^ = ^^. where h^, hj, h^ have the values given in Art. 240 *. Similar equations will apply for each body of the system. Besides these dynamical equations there will be the geome- trical equations expressing the connections of the system. As every such forced connection is accompanied by some reaction, the number of geometrical equations will be the same as the number of unknown reactions in the system. Thus we have sufficient equations to determine the motion. 254. If two of the principal moments at the fixed origin are equal, it is often convenient to choose as axis of z the axis OG of unequal moment, and as axes of x, y two other axes OA, OB moving in any manner round OG. Let ;^; be the angle the plane oi AOG makes with some plane fixed in the body and pass- ing through OG. Then we have 0^= Wj, 6^= w^, and d^ = 0)3+ -^ . Also by Art. 241, we have h^'=A(o^, A„' = Aco^y h^ = Cto^ equations of motion of Art. 253 now become dt The dt In this case the most convenient geometrical equations to express the relations of these moving axes to straight lines fixed in space will be those given in Art. 235. flfv Since -^ is arbitrary, it m?.y be chosen to simplify either the dynamical or the geometrical equations. • The equations of Art. 253 were first given in this form by Prof. Slesser to whom the equations of Art. 254 had been shown by the author. It appears however that similar results had been previously published in Liouville's Journal in 1858. em. 5 geome- ,em. As reaction, e as the otion. origin ' the axis axes OA, ingle the and pass- dy '3' The Itions to les fixed ther the ISlesser to h however In 1858. ON MOVING AXES AND RELATIVE MOTION. 207 First, we may put -^ — ~^v The dynamical equations ihen become ^'dt --=iV. dx Secondly, we may so choose -^ that = 0. In this case the plane COA always passes through a straight line OZ fixed in space. The geometrical equations then become, dd d^ . ^ _dx d± dr dt""^^^- (O. S" d(o^_M dt~ A' da)^_N 255. If three principal moments at the fixed origin be equal, there are three sets of axes such that when the motion is referred to them, the equations take a simple form. First. We may choose axes fixed in space. Since every axis is a principal axis in the body, the general equations of motion become da>^ _ L ~dt~J.* dt ~ A' dt ~ A The geometrical equations of Art. 235 are not required. Secondly. We may choose one axis as that of OC fixed in space and let the other two move round it in any manner, then as in Art. 254, the equations of motion become da). dy M dt d CO. dt N A Thirdly. We can take as r xes any three straight lines at right angles moving in space in any proposed manner. The equations of motion are then by Art. 253 dt day, dt -0)3(9,4-0)^^3=^, . - 7V" n ■ 'til 20S MOTION IN THREE DIMENSIONS. i' : The geometrical equations will then be the same as those givon in Art. 235 or Art. 247. 256. Ex. An ellipsoid, whose centre is fixed, contracts by cooling and being set in motion in any manner is under tbo action of no forces. Find the motion. The principal diameters are principal axes at throughout the motion. Iiet us take them as axes of reference. The expression? for the angular momenta about the axes are by Art. 241 h^'=Auy, hs=Bu^, h^'=Cu^. The equations of Art. 263 then become d dl d dt d dt {Bu)^-{C -A) W3Wi = Multiplying these equations by A Wj, Bu^, C'wg, adding and integrating we see that A^u^-\-B^(j)^ + C'^, where w is the angular velocity of the sphere, the angle its instantaneous axis makes with the axis of 0, and the plane of the couple is parallel to the axes of n and u. On Motion relative to the Earth. 2G4. The motion of a body on the surface of the earth is not exactly the same as if the earth were at rest. As an illustration I of the use of the equations of this chapter, we shall proceed to I determine the equations of motion of a particle referred to axes of I co-ordinates fixed in the earth and moving with it. Let be any point on the surface of the earth whose latitude lis \. Thus \ is the angle the normal to the sr^rface of still water lat makes with the plane of the equator. Let the axis of z be irertical at and measured positively in the direction opposite to [gravity. Let the axes of x and y be respectively a tangent to the leridian and a perpendicular to it, their positive directions being Irespcctively south and west. In the figure the axis of y is dotted m v,** jiT.rcssaKt-. -s^rxr^-^z 5ra=::.-SiL,K-B.t-i.i_,. !;„L..»j.-i.ji-i.i..'_'.'-,'...i'-..i..„jvj.:;.'j..- 214 MOTION IN THREE DIMENSIONS. ! '! ): Iff to indicate that it is perpendicular to the plane of the paper. Let ft) be the angular velocity of the earth, b the distance of the point from the axis of rotation. We may reduce the point to rest by applying to every point under consideration an acceleration equal and opposite to that of 0, and therefore equal to (o^b and tending from the axis of rotation. We must also apply a velocity equal and opposite to the initial velocity of 0. This velocity is tab. The whole figure will then be turning about an axis 01, parallel to the axis of rotation of the earth with an angular velocity m. When the particle has been projected from the earth it is acted on by the attraction of the earth and the applied accelera- tion oy'b. The attraction of the earth is not what we call gravity. Gravity is the resultant of the attraction of the earth and the centrifugal force, and the earth is of such a form chat this resultant acts perpendicular to the surface of still water. If it were not so, particles resting on the earth would tend to slide along the sur- face. It appears, therefore, that the force on the particle, after O has been reduced to rest, is equal to gravity. Let this be repre- sented by g. Besides this there may be other forces on the par- ticle, let their resolved parts parallel to the axes be X, Y, Z. Since the earth is turning round 01 with angular velocity w, the resolved part about Oz is a sin\, since the angle lOz is the complement of w; since the rotation is from west to east, the resolved angular velocity is from ?/ to x, which is th« negative direction, hence 0^ = — (i) sin X. The resolved angular velocity round Ox is &> cos \ and is from y to z, which is the positive direction, hence d^ = (o cos \. Also since 01 is perpendicular to Qj^^ 0^ = 0. Hence, by Art. 244, the actual velocities of any particle whose co-ordinates are {x, y, z), are i I the paper. Let ice of the point ilying to every Ltid opposite to Tom the axis of md opposite to le whole figure 1 to the axis of le earth it is )plied accelera- e call gravity. earth and the this resultant it were not so, along the sur- larticle, after O this be repre- ces on the par- X, Y, Z. w t liar velocity w, gle lOz is the st to east, the s the negative gular velocity s the positive rpendicular to ocities of any ON MOTION RELATIVE TO THE EARTH. M = -^- +0) sin\y 213 dt v = -^ — a) coaXz— a sinXa? at dz , - «; = -,- + &) cosX?/ dt '' To find the equations of motion it is only necessary to substitute these in the equations of Art. 245. The resulting equations may be simplified if we neglect such small quantities as the difference between the force of gravity at dif- ferent heights. If a be the equatorial radius of the earth and g' the force of gravity at a height z, we have g' =g\\ j nearly. Now ft)*a is the centrifugal force at the equator, which is known to be 1 z -— g. Hence if we neglect the small terra ^r - we must also neglect ti^z. The equations will therefore become <«> 2a) cos X ^r — 2g> sin \ ^r = Y \ , at dt de ^ + 2a,cos\^ = -^ + ^ .^ = -. where the terms (X, Y, Z) include all the accelerating forces, except gravity, which act on the particle. These equations agree with those given by Poisson, Journal Polytechnique, 1838. 265. If we do not neglect the term containing to, the equa- tions of motion are -jTa^ + 2g> sin \ -^ — ft)' sin'Xa? — m^ sin \co%\z — X, de' 2(0 cos X -57 — 2a) sin X -rj — a)'y = Y, dt dt TTg + 2w cos X s? — w' cos'Xa — o)' sin X cosXaj = — ^r + Z. 266. As an example, let us consider the case of a particle dropped from a height h. The initial conditions are therefore «, «, -^, ~, -r^ all zero, and dt at at z=h. As a first approximation, neglect all the terms containing the small factor w. Thenwehave«=0, y=0, « = h-5»/<». \\\ v\ ).-;M m i n 216 MOTION IN THREE DIMENSIONS. For a second approximation, we may substitute these values of (x, y, z) in the small terms. We have after integration fi 1 a;-0, 2/= - wcosX^ , z = h- gf^. Thus there will be a small deviation towards the east, proportional to the cube of the time of descent. There will bo no southerly deviation, and the vertical motion vviU be the same as if the earth were at rest. An elementary demonstration of this resiJt will make the whole argument clearer. Let the particle be dropped from a height li vertically over 0. Then being reduced to rest, the particle is really projected eastwards with a velocity w/i cos X. Hence, if the direction of gravity did not alter owing to the rotation of the earth about 01, the particle would describe a parabola and the easterly deviation would be (w/t cos X) t where t is the time of falling. Since h=-\()t^, this deviation is u cos \fj - . The rotation w about 01 is etiuivalent to w sin X about Oz and -'%^>=^' d'y de dz dx -25''.+^s'''=^> of (at, y, 2) in the )out Oz and u cos \ = -uh cos X wlien ON MOTION RELATIVE TO THE EARTH. 217 d'z dx dy ^ _ For example, if we wished to determine the motion of a projectile, it will be convenient to take the axis of z vertical and the plane of xz to be the plane of projection. Let the axis of x make an angle ^ with the meridian, the angle being measured from the south towards the west. Then ^i=wcosXcos/3, Sj = - w cos X sin j3, tf3=-wRinX. These equations may be solved in any particular case by the method of continued approximation. If we neglect the small terms we get a first approximation to the values of {x, y, z). To find a second approximation we may substitute these values in the terms containing w and integrate the resulting equations. As these equations are only true on the supposition that &>* may be neglected, we cannot proceed to a third approximation. 268. Ex. 1. A particle is projected with a velocity F in a direction making an nngle a with the horizontal plane, and such that the vertical plane through the direction of projection makes an angle /3 with the plane of the meridian, the angle j3 being measured from the south towards the west. If x be measured horizontally in the plane of projection, y be measured horizontally in a direction making an angle j3 + - with the meridian, and z vertically upwards from the point of projection, prove that x=7cosa<+( r sin at* -^firt'j wcosXsin/S, y= [ Vsinot'-;r(7<^ j wcosXcos/3+ Vcosot'wsinX, z— Fsinoi-ggf**- Fcosat'wcosXsin/S, where X is the latitude of the place, and w the angular velocity of the earth about its axis of figure. Show also that the increase of range on the horizontal plane through the point of projection is y^ /I \ 4w -J sm /3 cos X sin a ( ^ sin' a - cos' a 1 , and the deviation to the right of the plane of projection is r sin Q. 4w -J sin' a (cos Xcos /3 — j.— + sin X cos a). Ex. 2. A bullet is projected from a gun nearly horizontally with great velocity BO that the trajectory is nearly flat, prove that the deviation is nearly equal to iJtosinX, where R is the range, and the other letters have the same meaning as in the last question. The deviation is always to the right of the plane of firing in the Northern hemisphere, and to the left in the Southern hemisphere. It is asserted (Qom-^iti Rendus, 1866) that the deviation due to the earth's rotation as calculated by this formula is as much as half the actual deviation in Whitworth's gun. 'i |: \ r:ii . i.i 'I ! Ml 218 MOTION IN THBEE DIMENSIONS. Ex. 3. A spherical bullet is projected with so great a velocity that the resistance of the air must be taken into account. The resistance of the air being assumed to be i^— , and the trajectory to be flat, prove that, neglecting the effects oi the k rotation of the earih, 2wRinX * X ,. z=xtma-§-yl-2l-l)- ?-"-?^-"^-^H«(el-|-l). These are given by Poisson, Journal Poly technique, 1838. 269. Let us apply these equations to determine the effect of the rotation of the earth on the motion of a pendulum. In this as in some other cases, it will be fouud advantageous to refer the motion to axes not fixed in the earth but moving in some known manner. Let the axis of z be vertical as before and let the axes of X and y move slowly round the vertical with angular velocity to sin \ in the direction from the south towards the west. In this case we have 6^ — a) cos \ cos j3, 6^= — a) cos \ sin fi, and ^3 = — G> sin \ + 0) sin X, = 0, where /9 is the angle the axis of x makes with the tangent to the meridian, so that -^ = a> sin \. If, as before, we neglect quanti- ties which contain the square of a> as a factor, the terms which riff (10 contain -jJ and -r^ must be omitted. Hence the required equa- tions may be obtained from those of Art. 267, by putting 0^ = 0. If m be the mass of the particle, I the length of the string, and T the tension ; these equations are d'x „ -y ' adz T X -J- — 2a> cos X sm iS ^ = -y dv dt ml ^ — 2a)COs\cosi8 J- = ^ dr "^ dt m I d^^ , n % • yo^^ . o N ady T Z -Ts + "«o COS \ sm a -j- -1- 2o) cos \ cos iS -^ = — <7 ^ dv dt dt ^ m t the origin being taken at the point of suspension. If the oscillation be sufficiently small z will differ from I by small quantities of the order a* where n is the semi-angle of oscil- lation. The last equation then shows that T differs from mgi by quantities of the order aa at least. If then we neglect terms of the ON MOTION RELATIVE TO THE EARTH. 219 order wa' and a', we may put mg for i in the two first equations and neglect the uerms containing to -^ . The equations of motion thus become the same as for a pendulum attached to a fixed point. The solutions of the equations are clearly x = A cos (\/f'+^)' y-BA.y\t^D). The small oscillations of a pendulum on the earth referred to axes turning round the vertical with angular velocity w sin \ are therefore the same as those of ai imaginary pendulum suspended from an absolutely fixed point. Let us then suppose the pendulum to be drawn aside so as to make with the vertical a small angle a and then let go. Relacively therefore to the axes moving rou. \ the vertical with angular velocity o) sin \ we must suppose the particle to be projected with a velocity Z sin a o) sin \ perpendicular to the initial plane of dis- placement. We have then when i = 0, x = hy y = 0, -,- = 0, ^ = laco sin \. It is then easy to see that in the above values clt of a; and y, G and D are both zero and that the particle de- scribes an ellipse, the ratio of the axes being to sin ^ a/"* "^^^ effect of the rotation of the earth is to make this ellipse turn round the vertical with uniform angular velocity w sin \ in a direction from, south to west. If the angle a be not so small that its square may be neglected, it is known by Dynamics of a particle that, independently of all considerations of the rotation of the earth, there will be a progression of the apsides of the ellipse. It is therefore necessary for the success of the experi- ment that the length I of the pendulum should be very great. This motion of the apsides depending on the magnitude of a is in the opposite direction to that caused by the rotation of the earth and cannot therefore be mistaken for it. It also appears that the time of oscillation is unaffected by the rotation of the earth, provided the arc of oscillation be so small that the effects of forces whose magnitude contains the factor coa* may be neglected. 270. In Chapter iv. we have considered the motion of a system ?* bodies constrained to remain in a fixed plane. Since no plane can be found which does not move with the earth, it is important to determine what effect the rotation of the earth will have on the motion of these bodies. Let us treat this as an example of the method of Coriolis given in Art. 257. Let the plane make an angle a with the axis of the earth. Let a point in this plane be on the siuface of the earth and let it be reduced to rest. Then, as M 1 ,}'; 220 MOTION IN THREE DIMENSIONS. proved iu Art. 2C4, the moving bodies w^ile in the neighboiirhood of are acted on by their weights in a direction normal lo the surface of the eai'th. The earth ia now turning round an axis through parallel to the axis of figure with a constant angular velocity w. Let this angular velocity be resolved into two, viz., u sin a about an axis perpendicular to the plane and w cos a about an axis in the plane. Now the square of w is to be rejected, hence by the principle of the superposition of small motions, we may determine the whole effect of these two rotations by adding together the effects produced by each separately. It is a known theorem that if a particle be constrained to move in a plane which turns round any axis in that plane with a constant angular velocity u> cos o, the motion may be found by regaiding the plane as fixed and ''"pressing an accelera- tion ui^r cos^ a on the particle, where r is the distance of the particle from the axis. This may be deduced, as iu Art. 260, from the theorem of Coriolis. This impressed acceleration is to be neglected beeaiise it depends on the square of w. The angular velocity u cos a has therefore no sensible effect. If the bodies be free to move in the plane, the effect of the rotation u sin a is to turn the axes of reference round the normal to the plane drawn through the point 0. If then we calculate the motion without regard to the rotation of the earth, taldng the initial conditions relative to fixed space, the effect of the rotation of the earth may be allowed for by referring this motion to axes turning round the normal with angular velocity w sin a. For example, if the body be a heavy particle sus- pended by a long string from a point fixed relatively to the earth, it is really constrained to move in a horizontal plane, and the reasoning given above shows that the plane of oscillation will appear to a spectator on the earth to revolve with angular velocity a sin a round the vertical If the bodies be constrained to revolve with the plane, it " vill be required to find the motion relatively to that plane. We must therefore apply to each particle the force of moving space and the compound centrifugal force. If r be the distance of any particle of mass mi from 0, the former is mrw" sin^ o. This is to be neglected because it depends Oxi the square of u. The latter is therefore the only force to be considered. By Art. 262, the compound centrifugal forces on all the particles of a body are equivalent to a force at the centre of gi'avity and three couples. In our case these couples are easily seen to be zero. For if the plane be taken as the plane of x>j, we have 0^=0, ^,=0, «i=0, w, = 0. Hence L, M, N are all zero. If, there- fore, m be the mass of a body, V the relative velocity of its centre of gravity, the effect of the rotation of the earth may be found according to the rule given in Art. 2o7, by impressing on the body a force equal* to 2nirwsina, acting at the centre of gravity, iu the plane of motion and perpendicular to the direction of motion of the centre of gravity. The ratio of this force to gravity for a particle moving S2 feet per second, is at most , which is less than a five thousandth. This is so small that, except under special circumstances, its effect will be imperceptible. 271. Ex. 1, In Foucault's experiment, a long pendulum is suspended from a point over the centre of a circular table, and the arc of oscillation is seen to pass from one diameter to another. Show that the arc of the circidar rim of the table described by the plane of oscillation iu one day is equal to the difference in length betwcr two parallels of latitude one through the centre and the other through the ON MOTION RELATIVE TO THE EARTH. 221 northern or southern extremity of the rim. This theorem in due to the late Prof. Young. Ex. 2. A heavy particle is suspended from a Jixed point of support by a string of length a. It performs elliptic oscillations whose major and minor semi-axes are b and c. If 6 and c be small compared with a, prove that the apses will advance, 3 be in each complete revolution of the particle, through an angle — 2jr nearly. If b O U" and c be not small compared with a but be very nearly equal, the apse will advance through an angle ^ -1^2^, V^l-^sin«« / b . vhere sina= in each complete revolution of the particle. Ex. 3. A pendulum, at rest relatively to the earth, is started iu any direction with a small angular velocity, show that the oscillations will take place in a vertical plane turning uniformly round the vertical so that the pendulum becomes vervical once iu each half oscillation. Ex, 4. Let be the angle a pendulum of length I makes with the vertical, and ^ the angle the vertical plane containing the pendnlum makes with a vertical plane which turns round the vertical with uniform angular velocity w sin \ in a direction from south to west. Prove that when terms depending on u^ are neglected the equations of motion become (§)•"'"'« \dtj I COS0 + A, — I sm" ^ ^ 1 = dt\ dtj do 2 sin- 6 cos (0 4-/8) w cos \ -j- , where A is an arbitrary constant, and the other letters have the meanings given to them in Art. 267. See M. Quet in Liouville'a Journal, 1853. These equations will be found convenient in treating the motion of a pendulum. They may be easily obtained by transforming those given in Art. 239 to polar co- ordinates. Ex. 5. A semi-circular arch ACB is fixed with its plane vertical on a horizontal wheel at A and B, and may thus be moved with any degree of rapidity from one azimuth to another. A rider slides along the inner edge of the arch which is graduated and may be fixed at any degree marked thereon. A spiral spring by means of which a slow vibration is obtained with comparatively a short length ia attached at one end to a pin in the axis of the semicircle so that the point of attachment may- be in the axis of rotation and at the other end it is fixed to a similar pin in a paraUel position fixed to the rider. The vertical semicircle is not placed in a diameter of the horizontal wheel but parallel to it at such a distance as not to interrupt the eye of the observer from the vertical plane passing through the diameter, and in which plane the wire in all its positions remains. If the rider be placed at an angular distance $ from the highest point of the arch and the wire set in vibration in any plane, show that the plane of vibration of the wire will make a complete revolution relatively to the arch while the arch turns round sec 6 complete revolutions. This is best observed by fixing the eye on a line i 1- .)! :f : 6 ' I'^i 1 ■ it';| 1 '. A m 222 MOTION IN THREE DIMENSIONS. in the eame plane with the wire while walking round with the wheel during its rotation. This apparatus was devised by Sir C. Wlieatstone to illustrate Foucault's mechanical proof of the rotation of the earth. Proceedings of the Royal Society, May 22, 1851. 272. Hitherto we have considered chiefly the motion of a single particle. The eftect of the rotation~of the earth on the motion of a rigid body will be more easily understood when the methods to be described in the following chapters have been read. If, for example, a body be set in rotation about its centre of gravity, it will not be difficult to determine its motion as viewed by a spectator on the earth, when we know its motion in space. It seems, therefore, sufficient here to consider the peculiarities which these problems present, and to seek illustrations which do not require any extended use of the equations of motion. 273. The effect of the rotation of the earth is in general so small compared with that of gravity, that it is necessary to fix the centre of gravity in order that the effects of the former may be perceptible. Even when this is done, the friction on the points of support and the other resistances, cannot be wholly done away with. If, however, the apparatus be made with care that these resistances should be small, the effects of the rotatiuu of the earth may be made to accumulate, and after some time to become sufficiently great to be clearly perceptible. If a body be placed at rest relatively to the earth and free to turn about its centre of gravity as a fixed point, it is actually in rotation about an axis parallel to the axis of the earth. Unless this axis be a principal axis, the body would not continue to rotate about it, and thus a change would take place in its state of motion. By referring to Euler's equations, we see that the change in the position of the axis of rotation is due to the terms (^— ^)&),ft)g, (5 — (7) tOgWg, {C — A)(o^a>^. The body having been placed apparently at rest, m^, eo^, Wj are all small quan- tities of the same order as the angular velocity of the earth ; these terms are, therefore, all of the order of the squares of small quan- tities. Whether they will be great enough to produce any visible effect or not will depend on their ratio to the frictional forces which could be called into play. But since those frictional forces are just sufficient to prevent any relative motion, these terms will in general be just cancelled by the frictional couples introduced into the right-hand sides of Euler's equations. The body will, therefore, continue at rest relatively to the earth. In order that some visible effect may be produced, it is usual to impress on the body „ very great angular velocity about some axis. If this be the axis of w^, the terms in Euler's equations, which are due to the centrifugal forces, and which contain co^ as a factor, The gr will be, body 1 If .sufficie 24x()0 In thes( of the velociti€ lected. As a selected an elem< 274. while th( relatively aocis of J Let I gravity b the earth an angle on the pli called the about its with the given by Let ft), [moving a [centre of figure is i [parallel t( lapplicatio: [axes are i ISubstituti ON MOTION RELATIVE TO THE EARTH. 223 factor, become greater than when Wj had no such initial value. The greater this initial angular velocity, the greater these terms will be, and the more visible we may expect their effects on the body to be. If the angular velocity thus communicated to the body be sufficient to turn it only once in a second, it will be still 24 X GO X 60 times as great as the angular velocity of the earth. In these problems, therefore, we may regard the angular velocity of the earth as so small, compared with the existing angular velocities of tlie body, that the square of the ratio may be neg- lected. As an example of the application of these principles, we have selected one case of Foucault's pendulum, which seems to admit of an elementary solution. 274. The centre of gravity of a solid of revolution is fixed, while the axis of figure is constrained to remain in a plane fixed relatively to the eaHh. The solid being set in rotation about its axis of figure, it is required to find the motion. Let us refer the motion to moving axes. Let the centre of gravity be the origin, the plane of yz the plane fixed relatively to the earth. Let the axis of figure be the axis of z, and let it make an angle ^ with Ihe projection of the axis of rotation of the earth on the plane oi yz. Let this projection, for the sake of brevity, be called the axis of %. Let p be the angular velocity of the earth about its axis, a the angle the normal to the plane of yz makes with the axis of the earth. The motion of the moving axes is given by j 6^=p co8a + -^, ^2 ^i' s^"^ * ^^'^ X' C^=p sin a cos X- Let Wj, o)j, Wgbe the angular velocities of the body about the [moving axes; A, A, G the principal moments of inertia at the centre of gravity. Let Ji be the reaction by which the axii of figure is constrained to remain in the fixed plane, then R acts parallel to the axis of x. Let h be the distance of its point of I application from the origin. The angular momenta about the [axes are respectively h^ = A(0^, h^ = Aa>^y h^=Ca>^. [Substituting in Art. 230, the equations of motion are A^^^^^-C<.,e, + A^J, = Rh fl 9 dt ^ft>,^,+ ^a),^, = ,iii ! ■'^m 224 MOTION IN THREE DIMENSIONS. > i" Since the axis of z is fixed in the body, we see by Art. 243, that o), = ^,, (a^ = 6^. The last equation of motion, therefore, shows that w, is constant. It should however be remembered that eog is not the apparent angular velocity of the body as viewed by a spectator on the earth. If VL^ be the angular velocity relatively to the moving axes, we have by Art. 243, llj = 0)^—6^, so that fig + p sin % cos X — constant. Thus the body, if so small a difference could be perceived, would appear to rotate quicker the nearer its axis approached the pro- jection of the axis of the earth's rotation on the fixed plane. The first equation of motion after substitution for w^, w^, 6^, 0^, their values in terms of x> becomes A ~j^ — Ap' sin'^ a sin % cos ;^ + Cnj) sin a sin ;^ = 0, where n has been written for tWg. The second term may be rejected as compared with the third, since it depends on the square of the small quantity p. We have, therefore, d'x G . . By Art. 92, this is the equation of motion of a pendulum under the action of a force constant in magnitude, and whose direction is along the axis of v, i.e. the projection of the axis of rotation of the earth on the fixed plane. The body being set in rotation about its axis of figure, we see that that axis will imme- diately begin to approach one extremity or the other of the axis of X with a continually increasing angular velocity. When the axis of figure reaches the axis of ^t its angular velocity will begin to decrease, and it will come to rest when it makes an angle on the other side of the axis of v equal to its initial value. The oscilla- tion will then be repeated continually. The axis of figure will oscillate about that extremity of the axis of x> which, when ^ is measured from it, makes the coefiBcient on the right-hand side of the last equation negative. This extre- mity is such, that when the axis of figure is passing thro"g^ i^ the rotation n of the body is in the same direction as the resolved rotation p of the earth. 275. If we compare bodies of different form, we see that the 1 C time of oscilldtion depends oidy on the ratio -^ . It is otherwise independent of the structure or form of the body. The greater M this ratio the quicker will the oscillation be. For a solid of m revolution, it appears from the definitions in Art. 4, that this ON MOTION RELATIVE TO THE EARTH. 225 ratio is greatest when Swu' = 0. In this case tho ratio is equal to 2, and the body is a circular disc or ring. 27G. If we compare the different planes in which the axis may be constrained to remain, we see that the motion is the same for all planes making the same angle with the axis of the earth. It is therefore independent of the inclination of the plane to the horizon at the place of observation. The time of oscillation will be least, and the motion of the axis most perceptible when a= ^,i.e. when the plane is parallel to the axis of rotation of the earth. If the plane be perpendicular to the axis of the earth, the axis of figure will not oscillate, but if the initial value of -7^ is zero, it will at remain at rest in whatever position it may be placed. 277. Ex. 1. Show that a person furnished with the particular form of Fou- caiUt'H pendulum just described, could, without any Astronomical observations, determine the latitude of tho place, the direction of the rotation of the earth, and the length of the sidereal day. This remark is due to M. Quet, who has given a different solution of this problem in Liouville'a Journal, vol. xviii. Ex. 2. If the body be a rod, and its centre of gravity supported without friction, prove that it could rest in relative equilibrium either parallel or perpendicular to the projection of the earth's axis on the plane of constraint. If it be placed in any other position, its motion will be very slow, depending on j)', but it will oscillate about a mean position perpendicular to the projection of the earth's axis. Ex. 3. If the axis of figiwe be acted on by a frictional force producing a retarding couple, whoso moment about the axis of x bears a constant ratio ft, to the moment of the reactionol couple about tho axis of y, and if the fixed plane bo 1 parallel to the axis of the earth, find the small oscillations about the position of equilibrium. Show that the position at any time t is given by X=Zc-^'cos[(^-X«)*<+m]. jwhere 2A\=fk{Cn~2Ap) and L and M are two constants dependhig on the initial conditions. Ex. 4. The centre of gravity of a solid of revolution is fixed, while the axis of Sgure is constrained to remain in the surface of a smooth right cone fixed relatively \q the earth. Show that the axis of figure will oscillate about the projection of the is of rotation of the earth on the surface of the cone, and that the time of a com- A Bine ^.r31 be proportional to those distances, then in all the three cases the same conditions will hold at the end of a time dt, and so on con- tinually. The three particles will therefore describe similar orbits about the centre of gravity in a similar manner. First, let us suppose that the three particles are to be in one straight line. To fix our ideas, let m' lie between m and w", and between m and r/i. Then since the attraction on any particle must be proportional to the distance of that particle from 0, the three attractions m m m m m m {PPy^ {PFY {F'P'f {PF) k> {ppy {ppy PF' PF must be proportional to OP, OF, OF'. Since 'ZmOP^ 0, these two equations amount to but one on the whole. Let z = OP _ m'+'m"{l + z) OF _ -m + m"z sotbat-pp- ^^^'^^'' > PP'-m + m' + m"' Then we have 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 k = 2 and the equation becomes ms» {(1 + zY - 1} - m' (1 + zY (1 - «') - ni!' {(1 + zf - z'] = 0. This is an equation of the fifth degree, and it has therefore always one real root. The left side of the equation has opposite signs when z = and ^ = oo , 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, and m" the moon, we have very nearly z = a/ — K = TTxTL • If then originally the earth and moon had been placed in the same straight line with the sun at distances from the sun proportional to 1 and 1 + rrrr^ , 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 d la Gonnaissance des Temps, 184o, that such a motion would be un- stable. i; Pi t ', I t t i.ll I i;'' 232 MOMENTUM. ! if The paths of the particles will be similar ellipses having the centre of gravity for a common focus. Secondly. Let us suppose that the ' iw of attraction is " as the distance." In this case the attraction on each particle is the same as if all the three particles were collected at the centre of gravity. Each particle will describe an ellipse having this point for centre in the same time. The necessary conditions of projec- tion are that the velocities of projection should be proportional to the initial distances from the centre of gravity, and the directions of projection should make equal angles with those distances. Thirdly. Let us suppose the particles to be at the angular points of an equilateral triangle. The resultant force on the par- ticle m is ^, cos FPO + ^cos F'PO. P P The condition that the forces on the particles should be pro- portional to their distances from shows that the ratio of this force to the distance OF is the same for all the particles. Since m'p" cos FPO + m"p' cos P"FO ={m + m' + m") OP, it is clear f^iat the condition is initially satisfied when p = p = p". Hence, by the same reasoning as before, if the particles be pro- jected with equal velocities in directions making equal angles with OP, OP, OP' respectively, they will always remain at the angular points of an equilateral triangle. Ex. 1. Show that if the three particles attracted each other according to the law of nature, the paths of the particles, when at the comers of an equilateral triangle, are equal ellipses having for a common focus. Find the periodic time. Ex. 2. If four particles he placed at the eorners of a quadrilateral whose sides taken in order ore a,,h,c,d and diagonals p, p', then the particles could not move under their mutual attractions so as to remain always at the corners of a similar quadrilateral unless (/)y » - 6»d») (c™ + o») + («"(!» - /jV) (6" + d") + (6"d'» - a'^c") (p^ + />'") = 0, where the law 'J attraction is the inverse (w- l)ti» po'^ jt of the distance. Show also that the mass at the intersection of b, e divided by the mass at intersection of c, dia equal to the product of the area formed by a, p', d divided by the area formed by a, b, p and the difference -7ji--j^ divided by the difference P Cv p» ~ i" ■ 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 bo 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 well the subject of the chapter. MOMENTUM. 233 283. When the system under consideration consists of rigid bodies we must use the results of Art. 75 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. 76 in Chap, ii, combined with Art. 123 in Chap, iv, if the motion be in two dimensions, or Art. 240 in Chap. V, if the motion be in three dimensions. 284. Ex. A disc of any form is moving in its own plane in any manner. Suddenly any point O in the disc is fixed, find the angular velocity of the disc about O. Let us suppose that just before became fixed the centre of gravity O was moving with velocity V, and that p is the length of the perpendicular from on the direction of motion. Also let to be the angular velocity of the body about its centre of gravity. Just after has become fixed, let the body bo turning about with angular velocity w'. Let M¥ be the mou:ont of inertia of the disc about the centre of gravity, and 'et 00 = r. The change in the motion of the disc is produced by impulsive forces acting at during a short time t^ — t^. These forces have no moment about 0. Hence the moment of the momentum about is the same just after and just before the impact. Just before became fixed, the moment of the momentum about G was Mk^(o (Art. 123), and the moment of the momentum of the whole mass collected at was MVp. Hence the whole moment of the momentum about is the sum of these two (Art. 76). Just after has become fixed the body is turning about 0, hence by Art. 123 the moment of the momentum about is M{k'^ + r") w'. Equating these we have M (F + O 0)' = Mk'co + MVp ;, ,_ k^co+Vp ••«- ;fc« + ^8 . Ex. A circular area is turning about a point A on its circumference. Suddenly A is loosed and another point B also on the circ amference is fixed. Show that if AB is a quadrant, the angular velocity is reduced to one-third its value, and if il ^ is a third of the circumference, the area will be reduced to rest. 285. Ex. A disc of any form is turning about an axis Ox sitiwited in its own plane with an angidar velocity co. Suddenly this axis is let free and another axis Ox, also situated in the plane of the disc, becomes fi^ed, it is required to find the new angular velocity to' about Ox'. The change in the motion of the disc is caused by the action of the impulsive forces due to the sudden fixing of the axis Ox'. These act at points situated in Ox' and have no moment about ,i I ili 1 'iii Ii m f t • h 1 t {■'? n 1 1 :•■ IS avMiMMi 234 MOMENTUM. 1 Ox'. Hence the moment of the momentum about Ojj' is the same just before and just after Ox' is fixed. tauces Let d"ty'^d(T = a> {Xy'da — JiXyda] . Let A, A' be the moments of inertia of the disc about Ox, Ox' respectively, y the distance of the centre of gravity from Ox, M the mass of the disc. Then we have A'(o' = (o{A-Mhy). Secondly, let Ox, Ox' not be parallel. Let be the origin and the angle xOx = a, then y =ycosa. — x sin a. Let F be the product of inertia of the disc about Ox, Oy where Oy is perpen- dicular to Ox. Then by substitution in (1) we have A'a>' = G) (^ cos a — jPsin a). Ex. 1. An elliptic area of eccentricity e is turning about one latiis rectum. Suddenly this latus rectum is loosed and the other fixed. Show that the angular velocity is of its former value. Ex. 2. A right-angled triangular area ACB is turning about the side AC. Sudde ily ^C is loosed and BC fixed. If C be the right angle, the angular velocity is q-.p of its former ynlue. 286. A rigid body is moving freely in space in a known manner. Suddenly either a straight line or a point in the body becomes fixed. To determine the initial subsequent motion. MOMENTUM. 235 This proposition will include the last two articles as par- ticular cases. It is obvious that all the impulsive actions on the body pass through the fixed straight line or the fixed point. Hence the moments of the momentum of the body about the fixed axis in the first case or about any axis *arough the fixed point in the second case are unaltered by the impulsive forces. First. Let a straight line suddenly become fixed. Let it be taken as the axis of z. Let MK* be the moment of inertia of the body about the axis of z, and H the angular velocity after the straight line has become fixed. Suppose that the body when moving freely was turning with angular velocities w^., at^, w, about three straight lines Ox, Gy' Oz' throuQfh the centre of gravity parallel to the axes of co- ordinates. And let the co-ordinates of the centre of gravity be i, y,z. Then 0'o>. - (Sm^V) 0). - {tmz'y') a>^^M(x^^-y^^ = MK\ fl, where C is the moment of inertia of the body about Oz', and "^mz'x, %mz'y' are calculated with reference to the axes Ox , Oy', Oz'. Secondly. Let a point in the moving body be suddenly fixed in space. Take any three rectangular axes Ox, Oy, Oz, and three parallel axes Ox', Oy, Oz through the centre of gravity 0. Let (o^, w^, ©, be the known angular velocities of the body about the axes Gx, Oy', Oz before the point became fixed, Ilj,, Oj,, O, the unknown angular velocities about Ox, Oy, Oz after nas become fixed. Then, following the same notation as before, we hav« by Art. 240, A'ta^ - (Sm x'y) >, - (2m xz) w, ■^rtm\v~-z -J j = A^^ - {tm xy) Oy - (2m xz) H,. B'a^ - (2m y'z') (o, - (2«i y'x) o>^ + 'Zm\z~~x ^ j = B[ly - (Zmyz) SI, - (tm yx) fl^. O'w, - (2m z'x) m^ - (2m z'y') ^, to , a,). In the same way if we construct the momental ellipsoid at 0, the right-hand sides are proportional to the direction-cosines of the diametral plane of the axis (O^, fl^,, H,). Thus the instantaneous axes of rotation, before and after is fixed, are so related that their diametral planes with regard to the momental ellipsoids at G and respectively are parallel. "We may also deduce this result, without difficulty, from Art. 117. The motion of the body about the axis GI may be produced by an impulsive couple in the diametral plane of GI with regard to the momental ellipsoid at 0. Let us then suppose the body at rest and fixed, and let it be acted on by this couple. It follows from the same article, that the body will begin to turn about an axis OT which is such that its diametral plane with regard to the momental ellipsoid at is parallel to the plane of the couple. The direction of the blow at may also be easily found. The centre of gravity being at rest suddenly begins to move perpen- dicular to the plane containing it and the axis 01'. This is obviously the direction of the blow. 288. Ex. 1. A sphere in co-latitude 6 is hung up hj a point in its surface in equi- librium under the action of gravity. Suddenly the rotation of the earth is stopped, it is required to determine the motion of the sphere. [Math. Tripos, 1867, j Let be the centre of the sphere, its point of suspension, and a its radius. Let C be the centre of the earth. Let us suppose the figiire so drawn that the sphere is moving away from the observer. Let w= angular velocity of the earth, then if CQ^/m, the sphere is turning about an axis Gp parallel to CP, the axis of the earth with angular velocity u, while the centre of gravity is moving with velocity ^o- sin ^ . u. Let OC, Op, and the perpendicular to the plane of 00, Op be taken as the axes of X, y, z respectively, and let Oj., Oj,, 0, be the angular velocities about them just after the rotation of the earth is stopped. set of MOMENTUM. 237 By Art. 286, the angular momenta about Ox, just before and juat after tbo rotation was stopped, are equal to each other ; where Mk* is the moment of inertia of tlie sphere about a diameter. Again, the angular momenta about Oy are equal to each other ; .-. - Mk^w Bisi0 + Mna^ w sm0=M {k'<' + a^)Qy. Lastly, the angular momenta about Oz are equal.; .•• 0=Mk'''il,. Solving these equations, we get .0],=wsintf -,» — ~ = uBm.O — ^r— ^. * F + a« 7 But Ox= u cos 9. Adding together the squares of Qj.,ily, 0, we have O" = and D the densities of the dust and earth respectively. If the density of the dust be twice that of water and A= ^V express this in numbers. The InvariaUe Plane. 290. It is shown in Art. 72 of Chap, ii, that all the momenta of the several particles of a system in motion, are together equi- valent to a single resultant linear momentum at any assumed origin 0, represented in direction and magnitude by a line O'V, together with an ungular momentum about some line passing through 0, represented in direction and magnitude by a line OH. Let h^, h^, hg be the moments of the momenta of the particles about any rectangular axes Ouj, Oy, Oz meeting in 0, so that with similar expressions for A-,, h^, and let h' K -T? and n the an- Then the direction-cosines of OH are gular momentum itself is represented by h. If no external forces act on the system then by Art. 72 or Art. 279 h^, h^, Ag are constant throughout the motion, hence OH is fixed in direction and magnitude. It is therefore called the in- variable line at 0, and a plane perpendicular to OH is called the invariable plane at 0. If any straight line OL be drawn through making an angle 6 with the invariable line OH at 0, the angular momentum about OL is ^cos^. For the axis of the resultant momentum-couple is OH, and the resolved part about OL is therefore OH cos 0. Hence the invariable line at may also be defined as that axis through about which the moment of the momentum is greatest. At different points of the system the position of the invariable line is different. But the rules by which they are connected are the same as those which connect the axes of the resultant couple of a system of forces when the origin of reference is varied. These pi I ■ i ii^: f; i \\\ 240 MOMENTUM. have been already stated in Art. 203 of Chap. V, and it is un- necessary here to do more than generally to refer to them. 291. The position of the invariable plane at the centre of gravity of the solar system may be found in the following manner. Let the system be referred to any rectangular axes meeting in the centre of gravity. Let co be the angular velocity of any body about its axis of rotation. Let Mk^ be its moment of inertia about that axis and (a, /9, 7) the direction-angles of that axis. The axis of revolution and two perpendicular axes form a system of principal axes at the centre of gravity. The angular momentum about the axis of revolution is Mk^oa, and hence the angular mo- mentum about an axis parallel to the axis of z is Mk''oi cos 7. The moment of the momentum of the whole mass collected at the centre of gravity about the axis of « is ilf [ a? -^ — y -^ j , have hence we A3 = Sl/F(BC0S7 + Sif^a;^- dx y-dt lx\ dtj' The values of h^y h^, may be found in a similar manner. The posi- tion of the invariable plane is then known. 292. The Invariable Plane may be used in Astronomy as a standard of reference. We may observe tbe positions of the heavenly bodies with the greatest care, determining the co-ordi- nates of each with regard to any axes we pilease. 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 con- siderable. 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 would 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 THE INVARIABLE PLANE. 241 that tlio centre of gravity is either at rest or moves uniformly ia 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, there- fore, 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 otate 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 also fixed in space. Tliie, as Poisson has suggested, is supplied by the projection on the Invariable Plane of the direction of motio.i of the centre of gravity of the system. If the centre of gravity of the solar system were at rest or moved perpendicularly to the Invariable Plane, this would fail. 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. 293. If the planets and bodies forming the solar system can be regarded as spheres whose strata of equal density are concen- tric spheres, their mutual attractions act along the straight lines joining their centres. In this case the motions of their centres will be 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 laHer motions altogether. This is what Laplace has done, and in his formula the terms depending on the rotations of the bodies in the precedii)g values of A,, h.^, h^ are omitted. This plane might be called the Astronomical Invari- able Plane to distinguish it from 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 emitted in the expressions for A,, h^, h^ are not abso- lutely 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 Astro- nomical Invariable Plane must be slightly altered. A collision between two bodies of the system, if such a thing were possible, R. D. 16 ■i ■ n . i ti : PI' 242 MOMENTUM. or an explosion of a planet similar to that by which Olbers sup- posed 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 fosition of the Astronomical Invariable Plane, but the Dynamical nvariable 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 form- ing our system are not all known, so that the position of the Dynamical Invariable Plane cannot be calculated to any near degree of appro x'mation, while we do know that the terms into which these vmknown 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. Al- though 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 in course of time. In the M^canique 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 1"." 089, and longitude of the ascending node 114''.3979; at the second epoch the inclination will be the same as before, and the longitude of the node 114!''.3934!. 294. Ex. 1. Show that the invariable plane at any point of space in the straight line described by the centre of gravity of the solar system is parallel to that at the centre of gravity. Ex. 2. If the invariable planes at all points in a certain straight line are parahei, then that straight line is parallel to the straight line described by the centre of gravity. Impulsive Forces in Three Dimensions. 295. To deterw.'He the general equations of motion of a body about a fixed point undir the action of giuen impulses. Let the fixed point be taken as tlie origin, and let the axes, of co-ordinates be rectangular. Let (il„, ft^, flj, (tw^, Wy, m,) bo the angular velocities of the body just before and just after the impulse, and let the differences w^ — D,^, tw,^ — 11^, &>, — H, bo called ft)/, ft)/, ft)/. Then ft)/, coj , coj are the angular velocities generated by the impulse. By D'Alombcrt's Principle, see Art. 87, IMPULSIVE FORCES. 24*3 the difference between the moments of the momenta of the par- ticles of the system just before and just after the action of the impulses is equal to the moment of the impulses. Hence by Art. 240, Aw J — (Xmxj/) (oj — (Xmxz) oaj = L \ Bwy - (Zrmjz) coj - (Xmi/x) coj =mI (1), Cft)/ — (Zmzx) coj — (%mzy) coy = N J where L, M, N are the moments of the impulsive forces about the axes. These three equations will suffice to determine the values of (oj, (Oy, ft)/. These being added to the angular velocities before the impulse, the initial motion of the body after the impulse is found. 296. Ex. 1. Show that these equations are independent of each other. This follows fi'om Art. 20 where it is shown that the climiuant of the equations cannot vanish. Ex. 2. Deduce those equations from the general equations of motion referred to moving axes given in Art. 253. Ex. 3. 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. 297. It is to be observed that these equations leave the axes of reference undetermined. They should be so chosen that the values of A, Xmxy, &c. may be most easily found. If the posi- tions of the principal axes at the fixed point are known they will in general be found the most suitable. lu that case the equations reduce to the simple form AcoJ = L B(o.:=M (2). The values of coj, cdJ, &)/ 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, G, H, be the pressures of the fixed point on the body d; ,(3). XX + F=M.'~ by Art. 8Q = il/(«;i-a,;^)byArt. 219 tY+G = M{i i r\ !l i 1 \M ln]< 244 MOMENTUM. to its plane at the other extremity P of the curved boundary. Supposing the disc to be at rest before the application of the blow, find the initial motioii. Let the equation to the parabola be y^=iax and let the axis of zbe perpendicular to its plane. Then S>fta;2=0, i:myz = 0. Let /i be the mass of a unit of area and let ON=c. Also 2mxy—ixjjxydxdy--/il x—dx=2/xl ax^dx=-nac^. 16 " * ^=i/* / y^dx=:^iMa\"' , B = ^^H £=fi rxhjdx=- AuiV and C=A+£ by Art. 7. The moments of the blow B about the axes are L = bJ^, M=-Bc, iV=0. The equations of Art. 295 will become after substitution of these values 16 It 2 , oTji^T ^ixa c uy--iJMC^u„ = -Bc I «,=0 J i ■■ I I I '■] From these «,, uy may be found. By eliminating B we have — =7rz — — . Hence Ujl iO C 7 if NQ, be taken equal io^rpNP, the disc will begin to rotate about OQ. The re- Jo 75 B Bultant angular velocity will be -^ — -^ OQ. 299. 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 con- sideration 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 iu Art. 295. The ioUowing 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. 117. IMPULSIVE FORCES. 245 Ex. 2. Let 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 Q. Let fi be the angular velocity generated, r the radius vector of the ellipsoid which is the axis of fi. Let Mi* be the parameter of the ellipsoid. Prove that 7: = — . Q pr Ex. 3. If Qx> fij/i ^a l>e angular velocities about three conjugate diameters of the momental ellipsoid at the fixed point, such that their resultant is the angular velocity generated by an impulsive couple G, A', B', C the moments of inertia about these conjugate diameters, prove that 4'fije=Gcosa, B'Qg=0 coap, C'Q^ = G cob y, where o, ft 7 are the angles the axis of G makes with the conjugate diameters. Ex. 4. If a body free to turn about a fixed point be acted on by an impulsive jouplu 0, whose axis is the radius vector r of the ellipsoid of gyration at 0, and if p be the perpendicular from on the tangent plane at the extremity of r, then the axis of the angular velocity generated by the blow will be the perpendicular p and the magnitude fi is given by G = MprQ. 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. i ,ii . % \'4 111 s ■ 4^ ':< 300. Ex. 1. When a body turns about a fixed point the product of the moment of inertia about the instantaneous axis into the square of the angular velocity is called the Vis Viva. Let 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 tiirn about a fixed axis passing through the fixed point be 2T'. Then prove that T'=Tcos^d, where d is the angle between the eccentric lines of the two axes of rotation with regard to the momental ellipsoid at the fixed point, Ex. 2. Hence deduce Lagrange'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. Ex. 3. If a body be moving in any manner about a fixed point and an axis through the fixed point be suddenly fixed, show that if the vis viva 2r be changed into 22", we have T=Tcoa^$, where 6 has the same meaning as before. 301. To determine the motion of a free body acted on by any given impidse. Since the body is free, the motion round the centre of gravity i:i 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 Ai't. 295. To find the motion of the centre of gravity, let {U, V, W), {> , V, vj) be the resolved velocities of the centre of gravity just :t Jl i:-: 1 u ;■■! ii ! 111 ii. 'I 246 MOMENTUM. before and just after the impulse. Let X, F, Z be the com- ponents of the blow, and let M be the whole mass. Then by resolving parallel to the axes we have M{u-TJ)^X, M{v- V)=Y, M{w-W) = Z. If we follow the same notation as in Art. 295, the differences u— U, V —V, w — Wma,y be called u, v, w'. 302. 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 bome axis, A (B - C)'pYZ + B{G- A) qZX+ C(A-B) rXY=0. Is this condition sufficient as well as necessary ? See Art. 221. Ex. 2. A homogeneous cricket-ball is set rotating abotit a horizontal axis in the vertical plane of projection with an angular velocity iJ. V,1ien 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 -, where a is the radius of the ball. tan~i 6 Fcosa 303. The equations of Art. 301 completely determine the motion of a free body acted on by a given impulse, and from these by Art. 219 we may determine the initial motion of any point of the bod3^ Let (p, q, r) be the co-ordinates of the point of appli- cation of the blow, then the moments of the blow round the axes are respectively qZ — rY, rX—pZ, pY—qX. These must be written on the right-hand sides of the equations of Art. 295. Let ip'> q > '>"') he the co-ordinates of the point whose initial velocities parallel to the axes are required. Let {u^, v^, wj, {u^, v^, w^) be its velocities just before and just after *he impulse. Let the rest of the notation be the same as that used in Art. 295. Then Mjj - w, = w' -f ft)/r' - o,'q, with similar equations for v^—v^, lu^ — w,. Substituting in these equations the value of u', v, w', aj, S' cos ^ = U 4 C(oJ — M + CO), (5), Si^md = v' ~c'(oJ —V + Cft)j, (6). Let C be the relative velocity of compression, then C=w' — w (7). Substituting in these equations from the dynamical equations we have Scoa e= 8, cos e,~pP. (8), /S'sin^=/8;sin^„-l7^ (9), C=C,~rR (10). If'.S where IMPACT OF ROUGH ELASTIC BODIES. 8, cos 0,^u'+ c'n; -u+cci s,sme,= v'-c'aj-v+cnl C = W'-W 249 .(11), ^=i+ ,'«-« 1^ M (12). These are the constants of the impact. S^, C^ are the initial velocities of sliding, and 0^ the angle the direction of initial sliding makes with the axis of x. Let us take as the stando.rd case that in which the body M is sliding along and compressing the body M, so that /S'j and G^ are both positive. The other three constants p, q, r are independent of the initial motion and are essentially positive quantities. 307. Exactly as in two dimensions we shall adopt a graphical method of tracing the changes which occur in the frictions. Let us measure along the axes of x, y, z three lengths OP, OQ, OB 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 by /Si = 0, C=0. The locus given by /S> = is a straight line parallel to the axis of R, which we may call the line of no sliding. The locus given by C= Ois a plane parallel to the plane P, Q, 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. 144 the friction called into play is limit- ing. Since P, Q, R are the whole resolved momenta generated in the time t\ dP, dQ, dR will be the r-esolved 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 fidR, ■where ^i is the coefficient of friction. We have therefore .(13), e^Q " -So sin ^„ - jQ. {dPf+{dQy = fi^{dRf (14). The solution of these equations will indicate the manner in which the representative point T approaches the line of no sliding. [I t I Vm 1 : 1 '* \ j I 1 1 • ■ . 'm ; i i ,1 il 250 MOMENTUM. The equation (13) can be solved by separating the variables. We get 11 {8, cos 0, -pPy> = OL [S, sin e, - qQ) 5. where a is an arbitrary constant. At the beginning of the motion P and Q are zero, hence we have / 8, cos 8, -pP \i _ f 8, sin 0,-gQ \\ . . V 8,cose, J [ 8,Bin6, J ^ ' ^' which may also be written (Br'^Am)^ (-)• This equation gives the relation between the direction and the velocity of sliding. 308. If the direction of sliding does not change during the impact 6 must be constant and equal to 6^. We see from (16) that if p = q, then d = 0Q; and conversely if 6=6^, /S would be constant unless jp — q. Also if sin 6^ or cos 6^ be zero, 8 woidd be zero or infinite unless Q=6^. The necessary and sufficient condition that the direction of frict^'or. should not change during the impact is therefore ^ = 2^ or sir 2^o = 0. The former of these two conditions by (12) leads to «' (i- 2) +<'"(t -!'}=» (i«)- If this condition holds, we have by (13) P= Q cot 6^ and therefore by (14) ^""f"^^"} (19). It follows from these equations that when the friction is limit- ing, the representative point T moves along a straight line making an angle tan"' /a with the axis of P, in such a direction as to meet the straight line of no sliding. 309. If the condition p^q docs not hold, we may, by dif- ferentiating (8) and (9) and eliminating i*, Q, and 8, reduce the determinaLcn of It in terms of Q to an integral. By substituting for 8 from (17) in (8) and (9), we then have P, Q, li expressed gs functions of 9. Thus we have the equations to the curve along which the representative point T travels. The curve along which 2' travels may more conveniently be I l! IMPACT OF ROUGH ELASTIC BODIES. 251 defined by the property that its tangent by (14) makes a constant angle tan"^/u,with the axis of R and its projection on the plane of FQ is given by (15). And it follows that this curve must meet the straight line of no sliding, for the equation (15) is satis- fied by 2>P =■ ^0 cos ^0 , qQ= 8^ sin 0^. 310. 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 li increases. The complete value R^ of R for the wliole impact is found by multiplying the abscissa R^ of the point at which T crosses the plane of greatest compression by 1 + e so that R.^ = R^{l+e), if e be the measure of the elasticity of the two bodies. The complete values of the frictions called into play are the ordinates of the position of T corresponding to the abscissa R^R^. Substi- tuting these in the dynamical equations (1), (2), (3), (4), the motion of the two bodies just after impact may be found. 311. Let us consider an example. 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 there- fore 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 7 ■ Q = -«l^-^«. 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 or < q. Show also that the representative point reaches the line of no sliding when has either of those values. I'l "i X Pi I '•! V ' ;' i ,i 252 MOMENTUM. Ex. 2. If the bodies be such that the direction of sliding continues unchanged diu'ing the impact and the shdiug ceases before the termination of the impact, the roughness must be such that u> 7^ — ,^ — . CXl + 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 billard, 1835. Ex. 4. If two inelastic solids of resolution impinge on each other, the vertex of each being the point of contact, prove that the direction of sliding is the same throughout the impact. This and the next proposition have been given by M. Phillips in the fourteenth volume of Liouville's Journal. Ex. 5. If two bodies having their 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 masses impinge on each other at the extremi- ties of their axes of c, c', and if aa'=bb' and ca'~bc', prove that the direction of friction is constant throughout the impact. 313. Two rough hodie$ moving in any manner impinge on each other. Find tJie 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 i after the beginning, but before the termination of the impact. Let fi^, Qy, Q, be the angular velocities of the same body just before impact abo^it axes parallel tc the co-ordinate axes, meeting at the centre of gravity; w„ Uy, 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-ordi- uate axes meeting at the centre of gravity. Let JTbe 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 (a;, 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 (w^ -U^-F (wy -Qy)-E (w, - Q,)= -yR + zQ. -F(u,^~n^)+B(u,y-ny)-D(w,-n,)=-zP+xR[ (i). -JJK-O^)- I>{u,y-Qy) + C(w,-Q,)=-xQ + 7jP) M(v- r) = Q\ (2). M(w-W) = R) We have six similar equations for the other body, which differ from those 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 (1). IMPACT OF ROUGH ELASTIC BODIES. 253 Let S bo the Telocity with which one body Rlides along the other and let $ be the angle the direction of sliding makes with the axis of x. Also let C be the relative velocity of compression, then (S cos = U' - ujz'+ W.y - M + UyZ-UJ/K Ssin0 = v' - Wj'a;' + u^'z' -v +u^-Ujz\ C=V)'- Wx'y' + Vy'sC^ - \W + Uj^ - WyX] If we imbstitute from (1) (2) (3) (4) in (5) we find SQeoa0o-Scose = aP+fQ + eR-. SaBm0o-SBme=:fP + bQ + dRl... Co-C=cP + dQ + eR) (5). (6), where Sg, $ot C'o ^^^ ^^^ initial values of S, 0, G and are found from (5) by writing for the letters their initial values. The expressions for a, h, c, d, e, f are rather complicated, but it is unnecessary to calculate them. 314. 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 thb forces. The equations to the line of no sliding are found by putting ^j=0 in the first two of equations (G). We see that it is a straight line. The equation to the plane of greatest compression is found by putting (7=0 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 before, by dP dQ cos sin =HdR. •(7). When the representative point T reaches the line of no shding, the sliding of one body along the other ceases for the instant. After this, only so much friction is called into play as will suffice to prevent sliding, provided this amount is less than the limiting friction. If therefore the angle the line of no sliding makes with the axis of R be less than tan~Vi the point T wUl travel along it. But if the angle be greater than tan"^/*, more friction is necessary to prevent sliding than can be called into play. Accordingly the friction will continue to be limiting, but its direction will be changed if S changes sign. The point T will then travel along >i, curve given by equations (7) with d increased by ir. The complete value B^ of i? for the whole impact is found by multiplying the ab- scissa R of the point at which T crosses the plane of greatest compression by 1 -f c, where e is the measure of elasticity, so that Ri=Ri (1-f f). The complete values of P and Q are represented by the ordinates corresponding to the abscissa R^. Sub- stituting in the dynamical equations, the motion just after impact may be found. 315. The path of the representative point before it reaches the line of no Blidiug must be found by integrating (7). By differentiating (C) we have d (S cos e) _ adP+fdQ + edB _ an cos 6 +fix sin g -f e ^ d{SBW.e) ~ fdP + bdQ + ddit ~ fn cos e + b/iBind + d' H :?! 254 MOMENTUM. which reduces to , .„ ^^ + '~ coB20+fBin29 +- COB 0+ -sine S dd _ a-b ^^ 2^ ^.ycoa 2tf + - cos ^ - * bin From this equation wo may find S as a function of in tho form S=Af(d), tho constant A being determined from tho condition tliat 8=8^ when 0=Oa- Diileron- tiatiug tho first of equations (0) and substituting from (7) we got -Ad {ooa0f (0)] = {na 003 + fif am + c)dIl, whence wo find R=AF{0) + B, tho constant B being dotormiued from the condition that 22 vanishes when 0=0^, By substituting these values of >Si and R in tlie first two equations of (6) we find P and Q in terms of 0. The three equations giving P, Q, R as fimctions of are the equations to the path of the representative point. It should be noticed that the tangent to the path at any point makes with tho axis of i2 an angle equal to tan~' fx. 816. If the direction of friction does not change during tho impact, is con- stant and equal to 0,,, so that d cannot be chosen as the independent variable. In this case P=nRooa0Q, Q=iiRBia0Q and the representative point moves along a straight line making with the axis of R an angle tan~^ n. Substituting these values of P and Q in the first two of equations (6) we have — ^ sin 2^0 +/COS 2^0 + - cos ^0 - - sin ^o = 0, 2 " M A* as a necessary condition that the direction of friction should not change. Conversely if this condition is satisfied the equations (6) and (7) may all be satisfied by making constant. In this case it is also easy to see that the path of the representative point intersects the line of no sliding. If Sq be zero, and if more friction is neces- sary to prevent sliding then oan be called into play, the initial value of is im- known. But if 0^ be taken equal to that root of the above equation which makes S positive, and if d be supposed constant, the equations (6) and (7) are all satisfied. 817. Ex. 1. Let 0= A -F -E yR- -zQ - -P B -D zP- ■xR - -E -D C xQ- ■yP yR -zQ zP-xR xQ-yP and let A be the determinant obtained by leaving out the last row and last column. Let G', A' be the corresponding expressions for the other body. Then a, b, c, d, e, f are the coelBcients of P», Q", Ji«, 2QR, 2RP, 2PQ in the quadrio (F + i)(^'+^''+^')+^^^-*--^'- ■ 2E, A" ■ A' where 2E is a constant, which may be shown to be the sum of the vires viva) of the motions generated in the two bodies, as explained in Art. 304. This quadric may be shown to be an ellipsoid by comparing its equation with that given in Art. 28, Ex. 3. Show also that a, b, e are necessarily positive and ab>f', bc> d\ ca > e^. Show that by turning the axes of reference round the axis of R through the proper angle we can make / zero. EXAMPLES. 255 Ex. 2. Provo that the line of uo sliding is parallel to the oonjngato diameter of the plane containing the frictions I', Q. And the plane of greatest compression is the diametral plane of the reaction li. Ex. 3. The line of no sliding is the intorsoction of the polar planes of two points situated on the axes of P and Q and distant respectively fiom the origin 2E 2£ and „ • a • The plane of greatest compression is the polar plane of a 2E pomt on the axis of li distant -p^ frrm the origin. Ex. 4. The plane of PQ cuts the eUipsoid of Ex. 1 in an ellipHo, whose axes divide tlie plane iiito four quadrants; the lino of uo sliding cuts the plane of PQ io that quadiant in which the initial sliding Sq occurs. Ex. 6. A parallel to the line of no sliding through the origin cuts the plane of groatost compression, in a point whoso abscissa It has the same sign as C^, Honce 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 circiunference become fixed, find the new angular velocity. 3. A imiform rod of length 2a lying on a smooth horizontal plane passes through a ring which permits the rod to rotate freely in the horizontal piano. Tho middle point of tho rod being indefinitely near the ring any angular velocity ia impressed on it, show that when it leaves the ring the radius vector of the middle point will have swept out an area equal to 6 4. An elliptic lamina is rotating about its centre on a smooth horizontal table. If Wj, Wj, W3 be its angular velocities respectively when tho extremities of its major axis, its focus, and the extremity of the minor axis become fixed, prove 7 6 w, Wo 5 + — w. 6. A rigid body moveable about a fixed point at which the principal momenta are i, 5, C is struck by a blow of given magnitude at a given point. If the angular velocity thus impressed on the bod' be the greatest possible, prove that (a, b, c) bemg the co-ordinates of the given p int referred to the principal axes at 0, and (I, m, n) the duection cosines of the blow, then al + bm cii = 0, I \B-' c) "^ m \G^ Ay ■*" n \A-' B-) ' * These examples are taken from the Examination Papers which have been set iu the University and in tho Colleges. I ! t . W ||! l'.\ '* '-I !■...■ i 256 MOMENTUM. 6. Any triangular iamina ABC has the angular point C fixed and is capable of free motion about it. A blow is struck at B perpendicular to the plane of the triangle. '^' ow that the initial axis of rotation is that trisector of the side AB which is furthest from £, 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 with the generating lines of the cone. A heavy particle of mass P moves along the groove under the action of gravity, the system being initially at rest with the particle at a distance c from the vertex. Show that if be the angle through which the cone has turned when the particle is at any .distance r from the vertex, then mk'' + Pir^siia?a _ jie sin o . cot^ mk''' + Pc'^ sin* a = € it being the radius of gyration of the cono about its axis. 8. A body is turning about an axis through its centre of gravity, a point in the body 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 fixod. Show that the angular velocity about this edge as axis =7- ._, 10. Two masses m, m! 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 « by the two unw apped portions of the string, ^A [2^1 _ 1 /I 1 \ 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 wliile that centre describes a ciicle in space itb imiform angular velocity. 12. A uniform circular wire of radius a, moveable about a fixed point in its circumference, lies on a smooth horizontal plane. An insect of mass equal to that of the wire crawls along it, starting from the extremity of the diameter opposite to the fixed point, its velocity relative to the wire being uniform and equal to V. Prove that after a time t the wire will have turned through an angle 5 P tan"i I — 2a ^-6 \J'i tan '2a)' 13. A small insect moves along a uniform bar of mass equal to itself, and length 2a, the extremities of which are constrained to remain on the circumference ol a fixed circle, whose radius is — - Supposing the insect to start from the middle ble of )f the leAB s, and j3 with Qg the th the hrough vertexi ,t in the pal axis tangular ,ne of its that the jh passes on in one ute areas jn contact dng along [round its in space bint in its lal to that [)posite to 3ual to V. litself, and lumference Ithe midiile EXAMPLES. 257 point of the bar, and its velocity relatively to the bar to be uniform and equal to V; 1 Vt prove that the bar in time t will turn through an angle —.- 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 n^ 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 will have revolved through an angle — log ( 1 + - | , where a is the angle between the tangent and radius vector at any point of the epiral. 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 w. A small rocket whose weight is an Hth of the weight of the disc is placed at the inner extremity of the groove and discharged ; and 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 we~'. 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 he in the line represented by •62)-=0; the prin- the equations aVx + Vmy + c'm = 0, (6'' .c^)^+(c^-a^)l + (a^. cipal axes through the centre of gravity being taken as axes of co-ordinates, a, b, c 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 w about a fixed axis contains a closed tubular channel of small uniform section filled with an incomprecdible fluid in relative equilibrium ; if the rotation of the solid body were suddenly destroyed the fluid would move in the tube with a velocity - - , where A is the area of the t 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 veloci .y of the gate as it passes through the position of rest is impulsively dimin- ished in the ratio sin!" a - tan" fi , and the time between successive impacts when tho Bin«o-htan»^ oscUIations become small decreases in the same ratio, the weights of the bar and joint being neglected. R. 1). 17 -1 1 i iM ' ■rv. * '-' \ t ■ - : ( : ! i| liii " s II ■i I I ! CHAPTER VII. VIS VIVA. The Force-function and Work. 318. If r>. particle of mass m be projected along the axis of x with an initia^ velocity V and be acted on by a force i^ 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 {y- V)=fFdt. •'0 If we multiply both sides of the differential equation of the dx second order by -7 and integrate, we get* dt ^m {v' - V) = f Fdw. * It is seldom that Matliematicip'" , can be foimd engaged in a controversy such as that which raged for forty years in the last century. The object of the dispute was to determine liow 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 into its velocity. Leibnitz, however, tlionght he perceived an error in the con;mon opinion, and undertook to show that the proper measure should be, the product of the mass into the square of the velocity. Shortly all Europe was divided between the rival theories. Germany took part with Leibnitz and Bernoulli ; while Eng- land, true to the old measure, combated their arguments with great success. France was divided, an illustrious lady, the Marquise du Chatulet, being first a warm supporter and then an opponent of Leibnitzian opinions. Holland and Italy wore 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, had the same solution. However the force was measured, whether by the first or the second power of the velocity, the result was the same. The argu- ments and replies advanced on both sides are briefly given in Montucla's Ilistory, and are most interesting. For this however we have no space. The controversy was at last closed by D'Alembort, 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 only the property that certain resistances can be overcome by the moving body. It FORCE-FUNCTION AND WORK. 259 The first of these integrals shows that the change of the mo- mentum 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. 279, and afterwards to its application to determine the changes produced by very great forces acting for a very short time. The s'^oond 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. .319. For the purposes of description it will be 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 tlie kinetic energy of the particle. Many names have been given, to the right-hand side at various times. It is now commonly called the woi'k of the force F. When the force does not act in the direction of the motion of its point of application the term "work" will require a more extended definition. This we shall discuss in the next article. 320. Let a force F act at a point A of a body in the direction AB, and let us suppose the point A to move into any other po- sition A very near A. If be the angle the direction AB oi the force makes with the direction AA' of the displacement of the point of application, then the product i^ . ^^' . cos is called the work done by the force. If for ^ we write the angle the direction AB of the force makes Avith 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 hy the force is also ecjual to the product F.AM, where AM is to be estimated as positive wh^n 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. A A'. 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. is not then by any simple coiisiilorations of merely the mass and the velocity of the body that we must estimate this force, but by the natm'e 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. I^Alembert then points out that there are different kinds of obstacles and examines how their different Iduds of resistances may be used as measures. It will perhaps be sufficient to observe, that the resistance may in some cases be more conveniently measured liy a space-integral and in others by a time-integral. See Jlontucla's Hhlorij, Vol. III. and Whewell's Hhtory, Vol. ii. 17—2 i Ffjl ; 1 ■f: ■ . '. l-i . '■ 'I i i! W I ^iM ! i 260 VIS VIVA. I !.. I' I ■ ' ; Thus defined, tho work done by a force, corresponding to any indefinifply small displacement, is the same as the virtualmoff* int r.f trc force. In ,Statics, we are only concerned -with the small ljy[»otbetical displacements, we give the system in applying the priiiciple of Virtual Velocities, and this definition is therefore suihcient 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 sbould be noticed that tbe work done by given forces as the body moves from one given position to another, is independent of the time of transit. As stated in Art. 318, the work is a space- integra'l and not a time-integral. 321. If two systems of forces be equivalent, the work done hy one in any small displacement is equal to that done hy the other. This follows at once from the principle of Virtual Velocities in Statics. For if every force in one system be reversed in di- rection 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 tlie 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. 322. We may now find an analytical expi'cpsion for the work done by a system of forces. Let {x, y, z"\ the rectangular co-ordinates of a particle of the system ai «i !< ohe mass of this particle be m. Let (A' Y, Z) be the accLluiating forces acting on the particle resolved parallel to the axes of co-ordinates. Then mX, mF, mZ are the dynamical measures of the acting forces. Let us suppose the particle to move into the position x -f- dx, y -t- dy, z-\- dz; then according to the definition the work done by the forces will be 2 {mXdx + m YJy + m.Zdz) (1 ), the summation extending to all the forces of the system. If the bodies receive any finite displacements, the whole work will be '.m j{Xdx+ Ydy + Zdz) (2), tlie limits' v'.' the integral bf vng determined by the extreme positiunr of liie sys'cTfi. FCxvClil-FUNCTION AND WORK. 261 < >. 323. Vr hen the forces are such as fff^nerally occur in riature, it will be proved that the summation (1) of tb'^ last Article is a t,omplete differential, i.e. it can bo iiit«^grated independently of any relation between the co-ordinates x, ;y, z. The summation (2) can therefore be expressed as a function of the f^o-ordinates of the system. When this is the case the indejhiite integral of the summation (2) is called the force-function. This name was given to the function by Sir W. R. Hamilton and Jacobi independently of each other. If the force-function be called U, the work done by the forces when the bodicf^ move from one given position to another is the definite integral b\— U^, where tf^ and Z/^ ave 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 posi- tion. When the forces are such as to possess this property, i.e. when they possess a force-function, they have been called a con- servative system of forces. This name was given to the system by Sir W. Thomson. 324, There will he a force function, first, iiilien the external forces tend to fixed centres at finite distances and are functions of the distances from those centres ; and secondly, when the force 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 ?n^ (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. 322 is clearly Xm, z) be the cyliudrical 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 z, prove that dU'=^m(Pdp + Qpd(p + Z(]z). Ex. 2. If (r, 0, (/>) be the polar co-ordinates of the particle m ; P, Q, R the resolved parts of the forces respectively along the radius vector, perpendicular to it in the plane of and perpendiciUar to that plane, prove that d f7= 2m {Pdr + Qrd0 + Itr sin edr d\ sJ(a'^ + \)(b^ + \)(c' + \} 330. Ex. 1. An envelope of any shape and whose volume is v, contains gas at a uniform pressure p. Assuming that the pressure of the gas per unit of area is some function of the volume occupied by it, prove that the work done by the fb pressures when the volume increases from v = o to t' = 5 is I pdv. reasoning as FORCE-FUNCTION AND WORK. 265 Divide the surface into elementary areas each equal to d. When the radius r increases by dr, the distance between these sides is increased by drdO. Hence the work done by these tensions is Tr sin 6d(t> . drdd. The work done by tbe tensions on the four sides of the element is therefore 2Trdr sin 6ddd. Integrating this from 0-=O to 2w, 6=0 to w, we find that tbe work done over tbe whole sphere when the radius increases by dr is 8irTrdr. If the membrane be such that we may apply Hooke's law to the tension T, wo have T=E , where a is the natural radius of the membrane and E is the co- a efficient of elasticity. Substitutiug this vaVie of T we find that the work done by 4:E tbe tensions when the radius increases from a to 6 is - - (6 -a)* (26-ha). O Of If we assume that for a sopp-bubble T is constant, we find that the work done when the radius increases from ;* io 6 is iwT (6' -a'). If we suppose the spherical membrane to be slowly stretched by filling it with gas at a pressure^, we have by a theorem in Hydrostatics j)r=2r. In this case the r 4 work required has been shown to be pdv, and since v = q7rr' this leads to the same resulL as before. M FORCE-FUNCTION AND WORK. 207 833. Ex. 1. A roil originally straight Ih bent in one piano, if L be the stresfi couple at any point, p the radius of curvature, it is known both by experiment and thcoiT that Z = - where ^ ia a constant depending on the nature of the material P and the form of a ticction of the rod. ABsuming this prove that the work done iu t '* f 'i bending the rod is o I tt £?*■ Let PQ be any element of the rod and lot its length be dn. As PQ is being bent, let t// be the indefinitely small angle between the tangents at its extremities, then the stress couple ia E J-. Ah f increases from to — the work done is -^ / ^dr//, which is the same as dn' The work done on the whole rod is therefore Ij?''- Ex. 2. A uniform heavy rod of length I and weight w is supported at its two extremities so as to be horizontal. Show the work done by gravity in bending it IS 240E ' Conservation of Vis Viva and Energy. n34<. Def. The Vis Viva of a particle is the product of its mass into the square of its velocity. If a system he in motion under the action of finite forces, and if the geometrical relations of the parts of the system he expressed hy equations tvhich do not contain the time explicitly, the change in the vis viva of the system in passing from any one position to any other is equal to twice the corresponding work done hy the forces. In detei-mining the force-function all forces may be omitted which would not appear in the equation of Virtual Velocities. Let X, y, z be the co-ordinates of any particle m, and let X, Y, Z be the resolved parts in the directions of the axes of the impressed accelerating forces acting on the particle. The effective forces acting on the particle m at any time t are m dt;' a » m d^y di a » m d^z df If the effective forces on all the particles be reversed, they will be in equilibrium with the whole group of impressed forces by Art. 67. Hence, by the principle of virtual velocities, Xm (X- df Sx+iY -f)v + Z- dh di Zz\ = 0, where hx, By, Sz are any small arbitrary displacements of the par- ticle m consistent with the geometrical relations at the time t. 'H^ Mi I 1 i! I •7 m 1 4 I IMAGE EVALUATION TEST TARGET (MT-3) 1.0 1.1 11.25 [jKii 122 ^ "^ 1^'^ ■it u HiotograiJiic Sciences Corporaaoii 23 VVIST MAIN STMIT WItSTIR.N.Y. I4SM (71«)I72-4S03 \ S' V <^ ^. ^.\ 4^ \ hi ■I ' u ', \i 268 VIS VIVA. Now if the geometrical relations be expressed by equations which do not contain tlie time explicitly, the geometrical relations which hold at the time t will hold throughout the time Bt ; and, therefore, we can take the arbitrary displacements Bx, By, Bz to be respectively equal to the actual displacements -^ Bt, -^ Bt, -j- Bt of the particle in the time Bt Making this substitution, the equation becomes fd'x dx . d^y dy d?z dz [dt;* dt "^ df dt ^ dt^ dt Integrating, we get Sm(=^-^- + = 2?n 4l+^ ^ A. 7^\ dt^ dt) : ^» |(s)* + (IT + ©} = <^+ 2S»/(X^+ Yd,^Zd.). where C is a constant to be determined by the initial conditions of motion. Let V and v be the velocities of the particle m at the times t and t'. Also let U^, U^ be the values of the force-function for the system in the two positions which it has at the times t and <'. Then 335. The following illustration, taken from Poisson, may show more clearly wliy it is necessary that the geometrical relations should not contain the time explicitly. Let, for example, ^ [xy y, z, t) = 0. .(1) be any geometrical relation connecting the co-ordinates of the particle m. This may be regarded as the equation to a moving surface on which the particle is constrained to rest. The quanti- ties Bx, By, Bz are the projections on the axes of any arbitrary displacement of the particle m consistent with the geometrical relations which hold at the time t They must therefore satisfy the equation ts-+^s,+^-*a.= o. dx dy dz The quantities -j- Bt, -^ Bt, -vr Bt are the projections on the axes of the displacement of the particle due to its motion in the time Bt. They must therefore satisfy the equation « dx dx dt d 0. 't I.: ii ■ T'lf , m I ■: ''il I ';•■ 270 VIS VIVA. '. 111 fr 1 i 'I' (cPiJ (TxS X -^.^ — y -jTi ] — ^in {x Y— i/X), which represents any one of the three last general equations of motion in Art. 71. 339. The principle of Vis Viva was first used by Huyghens in his determination of the centre of oscillation of a body, but in a form different from that now used. See the note to pige 69. The principle was extended by John Bernoulli and applied by his son, Daniel Bernoulli, to the solution of a great variety of problems, such as the motion of fluids in vases, and the motion of rigid bodies under certain given conditions. See Montucla, Histoire de Mathematiquef Tome ill. The great advantage of this principle is that it gives at once a relation between the velocities of the bodies considered and the variables or co-ordinates which determine their positions in space, so that when, from the nature of the problem, the position of all the bodies may be made to depend on one variable, the equation of vis viva is sufficient to determine the motion. In general the principle of vis viva will give a first integral of the equations of motion of the second order. If, at the same time, some of the other principles enunciated in Art. 278 may be applied to the bodies under consideration, so that the whole number of equa- tions thus obtained is equal to the number of independent co- ordinates of the system, it becomes unnecessary to write down any equations of motion of the second order. 340. Ex. If a system in motion pass through a position of equilibrium, t. e. a position in which it would remain in equilibrium under the action of the forces if placed at rest, prove that the vis viva of the system is either a maximum or a minimum. Courtivron's Theorem, Mem, de VAcad. 17i8 and 1749. 341. Suppose a weight mg to be placed at any height h above the surface of the earth. As it falls through a height z, the force of gravity does work which is measured by mgz. The weight has acquired a velocity i>, half of its vis viva is ^mw" which is known to be equal to mgz. If the weight fall through the re- mainder of the height h, gravity may be made to do more work measured by mg{h—z). When the weight has reached the ground, it has fallen as far as the circumstances of the case permit, and no more work can be done by gravity until the weight has been lifted up again. Throughout the motion we see that when the weight has descended any space z, half its vis viva, together with the work that can be done during the rest of the descent is constant and equal to the work done by gravity during the whole descent h. If we complicate the motion by making the weight work some machine during its descent, the same theorem is still true. ! n 3 re- work the case jight that viva, the iring kvork Itrue. VIS VIVA AND ENiSRGY. 271 By the principle of vis viva, proved in Art. i334, half the vis viva of the particle, when it has descended any spiice z, is equal to the work mgz which has been done by gravity during this descent, diminished by the work done on the machine. Hence, as before, half the vis viva together with the difference between the work done by gravity and that done on the machine during the re- mainder of the descent is constant and equal to the excess of the work done by gravity over that done on the machine during the whole descent. Let us now extend this principle to the general case of a system of bodies acted on by any conservative system of forces. 342. Let us select some position of a moving system of bodies as a position of reference. This may be an actual final position passed through by the system in its motion, or any position which it may be convenient to choose, into which the system could be moved. Suppose the system to start from some position which we ma}' call A, and at the time t, to occupy some position P. Then at the time t, half the vis viva generated is equal to the work done from A to P. Hence half the vis viva at P together with the work which can be done from P to the position of refer- ence is constant for all positions of P. To express this, the word energy has been used. Half the vis viva is called the kinetic energy of the system. The work which the forces can do as the system is moved from its existing position to the position of reference is called the potential energy of the system. The sum of the kinetic and potential energies is called the energy of the system. The principle of the conservation of energy may be thus enunciated : — When a system moves under any conservative forces, the sum of the kinetic and potential energies is constant throughout the motion. 343. The distinction between work done and potential energy may be analytically stated thus. The force-function has been defined in Art. 323 to be the indefinite integral of the virtual moment of the forces. As the system moves the work done is the definite integral taken with its lower limit fixed and its upper limit determined by the instantaneous position of the system. The potential energy is the definite integral taken with its upper limit fixed and its lower limit determined by the instantaneous * Coriolis, Helmholtz and others have suggestecl that it would be more con- venient if the Via Viva were defined to be half the sum of the products of the masses into the squares of the velocities. See Phil. Trans. 1854, p. 89. But this change in the meaning of a term so widely established in Europe would bo very likely to cause some oonfusion. It seems better for the present to use another name, such as kinetic energy. Pill :1^ , 1, Ml ill I ■ I H I • '■;i 272 VIS VIVA. position of the system. The terms potential energy and actual energy are due to Prof. Ra kine. 844. Ex. 1. A particle describes an ellipse freely about a centre of force in its centre. Find the whole energy of its motion. Let m be the masp of the particle, r its distance at any time from the centre, nr the accelerating force on the particle. If coincidence of the particle with the centre of force be taken as the position of reference, the potential energy by Art. 343 18 -l> nifir) dr = 5 m/ir^. If r' be the semi-conjugate of r, the velocity of the particle is ojfii^ and the kinetic energy is therefore -mur'^. As the particle de- scribes its ellipse round the centre of force, the sum of the potential and kinetic energies is eqnal to -mn (a'+ 6'') where a and 6 are the semi-axes of the ellipse. Ex. 2. A particle describes an ellipse freely about a centre of force in the centre. Show that the mean kinetic energy during a complete revolution is equal to the mean potential energy; the means being taken with regard to time. Ex. 3. If in the last example the means be taken with regard to the angle described round the centre, the difference of the means is ^mn{a- 6)'. Ex. 4. A mass M of fluid is running round a circular channel of radius a with velocity «, another equal mass of fluid is running round a channel of radius 6 with velocity v, the radius of one channel is made to increase and the other to decrease until each has the original value of the other, show that the work required to pro- duce the change is ^ f Ij - ^^ ] (6" - a") M. [Math. Tripos, 1866.] 345. In applying the principle of vis viva to any actual cases, it will be im- portant to know beforehand what forces and internal reactions may be disregarded in forming the equation. The general rule is that all forces may be neglected which do not appear in the equation of Virtual Velocities. These forces may be enumerated as follows : A. Those reactions whose virtual velocities are zero. 1. Those whose line of action passes through an instantaneous axis ; as rolling friction, but not sliding friction nor the resistance of any medium. 2. Those whose line of action is perpendicular to the direction of motion of the point of application; as the reactions of smooth fixed surfaces, but not those of moving surfaces. B. Those reactions whose virtual velocities are not zero and which therefore would enter into the equation, but which disappear when joined to other re- actions. 1. The reactions between particles whose distance apart remains the same ; as the tensions of inextensible strings, but not those of elastic strings. 2. The reaction between two rigid bodies, parts of the same system, which roll on each other. It is necessary however to include both these bodies in the same equation of vis viva. li I VIS VIVA AND ENERGY. 273 rolling jtion of Ithose of lerefcre Ibher re- fime; as liioh roll le tame C. All tensiona which act along inextensibld striags, even though the strings are bent by passing through smooth fixed rings. For let p. string whose tension is T connect the particles m, m', and pass through a ring distant respectively r, r' from the particles. The virtual velocity is clearly Tdr+rSr', because the tension acts along the string. But since the string is iuextensible 5r + 5/=0; tlierefore the virtual velocity is zero. 346. To determine the vis viva of a rigid body in motion. If a body move in any manner its vis viva at any instant is equal to the vis viva of the whole mass collected at its centre of gravity, together with the vis viva round the centre of gravity con- sidered as a fixed point : or The vis viva of a body = vis viva due to translation + vis viva due to rotation. Let X, y, z be the co-ordinates of a particle whose mass is m and velocity v, and let x, y, 2 be the co-ordinates of the centre of gravity of the body. Let x = x + ^, y=y + r}, z = 'z+^. Then by a property of the centre of gravity 2m| = 0, "Zmrj = 0, %m^= 0. Hence S»n ^ = 0, Sw J? ="0, 2m J* = 0. Now the vis viva of a body is Smij' = 2m {§)'-(l)'-(S)}- Substituting for x, y, z, this becomes 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 2wt, 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. 347. First. Let the motion be in two dimer^sions. Let v he the velocity of the centre of gravity, r, 6 its polar co-ordinates referred to any origin in the plane of motion. Let ?•, be the distance of any particle whose mass is m from the centre of gravity, and let Vj be its velocity rela/tively to the centre of gravity. Let to be the angular velocity of the whole body about the centre of gravity, and Mk^ its moment of inertia about the same point. . J I ii H. D. 18 -TS^TTTrMTVSm 274 VIS VIVA. The vis viva of the whole mass collected at is Mv*, which may by the Diflferential Calculus be put into either of the forms «'=^l(S)"-(l)}=^{(i)'-ni)}- The vis viva about G is Swv,'. But since the body is turning about O, we have v^ = r^a. Hence Xmv' = eo" . '^mr' = w" . Mk\ The whole vis viva of the body is therefore 'S^mv' = Mv' + MkW. If the body be turning about an instantaneous axis, whose distance from the centre of gravity is r, we have v = ra>. Hence Smr» = ilf a>» (r« + A;') = Mk'W, where Mk'' is the moment of inertia about the instantaneous axis. 348. Secondly. Let the body he in motion in space of three dimensions. Let V be the velocity of G ; r, 6, its polar co-ordinates re- ferred to any origin. Let a>x, (o^, co^ be the angular velocities of the body about any three axes at right angles meeting in G, and let A, B, C be the moments of inertia o^ 'he body about the axes. Let ^, ;;, ^ be the co-ordinates of a ^le m referred to these axes. The vis viva of the whole mass.^ collected at G is Mv\ which may be put equal to according as we wish to use cartesian or polar co-ordinates. The vis viva due to the motion about G is x«v=.»{(D'.(§)V(f)]. Substituting these values, we get, since A = Sm {rf + ^), 5 = 2m(r + r), G=Xmi^-' + v% Xmv,' = AcoJ" + Bco^' + Ceo,' — 2 (%m^T)) (OgWy — 2 i^mr]^) w^w, - 2 (Sw^|) w.o)^. If the axes of co-ordinates be the principal axes at G, this re- duces to Swy^' = Jw/ + Ba>J' + Co)/. t)l 1 fixed way when point about 849, its posij 0, , }f/ in the n 21 where ac Show takes the This resu Ex.2, fluence of If the law perature that the where A, 2 Ex. 3. equation Let the that if X, y, axes fixed or with regard Thus the coefficients a semi vis viv round the cej Ex.4. same express] the origin is VIS VIVA AND ENERGY. 275 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 where A\ B\ C are the principal moments of inertia at the point 0, and w^, w^, w, are the angular velocities of the body about the principal axes at 0. 849. 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, if>, ^ which the principal axes at the centre of gravity make with some fixed axes in the manner explained iu Art. 235. Show that its vis viva is given by 2r = Jf (x'« + y'« + 2'2) + C{' + y}/ cos tf)» + {A sin" + 5 cos« 0) 0'* + BVD?0{A cos''0+5 sin2 0) ^'« + 2 (B-A) sm S sin ^ cos ^^'f', where accentj denote differential coefiScients with regard to the time. Show also that when two of the principal moments A and B are equal, this takes the simpler form 2r = 3f («'!>+ y* + z'S) + C (0' + y}/ cos BY + A {0" + sin« tf ^'«). 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 the body is always similar to itself. If the law of expansion be that the distance between any two particles at the tem- perature d is equal to their distance at temperature zero multiplied hyf{0), show that the vis viva of the body =AwJ' + Buy'> + Cu,' + ^(A + B + C)(^^^~^\ , where A, B, '^ 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=Au^'+Buy' + Cu,^ - 2Du)yU,-2EuiUg,-2FugUg. Show that the angular momenta about the axes are 5 — , dT dT dUy dT dwg' Let the body be moving freely and let 27*0 be the vis viva of translatioc. Prove that if X, y, z be 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, then the linear momenta parallel to the axes will be dT, dx" d7\ dy- dTo dz' Thus the vis viva, like the force-function, is a scalar function whose differential coefficients are the components of vectors. See Art. 240 and 326. In the case of the semi vis viva, these are the resultant linear momentum and angular momentum round the centre of gravity. Ex. 4. A body is moving about a fixed point and its vis viva is given by the same expression as in the last example. Show that if the axes are fixed in space and the origin is at the fixed point, the equations of motion may be written iu the form ■dt dwx ' !,: (I 18—2 27G VIS VIVA. ^Mi i\l " : i I i f; ',■: .1 'I with two similar equations for the axes of y and t. In these eqnations A, B, &e. will gonerally bo variable. If the axes move in the manner explained in Art. 243, tht equations of motion are d dT dT . dT '•Og diOy * "Stdiir d(i>« * du, " 01, = L, with two similar equations. See Art. 253. If the centre of gravity of a body moving freely bo referred to axes moving about a fixed origin and if 27^ be the vis viva of translation, show that the equations of motion of Art. 245 may be written ddTo dT„ dr„ aidx' ' dy'^^'^'dl"''-'^' with two similar equations. 850. Ex. 1. A circular wire can turn freely about a vertical diameter as a 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, u 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 be the angle the radius drawn from the centre of the wire to the bead makes with the axis of y. 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 Jf Pw + ma' sin* 0u=h. And by the principal of vis viva, Mh^uP + »/i I a" ( -^ j + a' sin" ^w' | = C+ 2mga cos 0. These two equations will suffice for the determination of j- and w. Solving them, we get %m~. — TT ■ -t n + w*"^ ( J. 1 =C-\- 2mga cos 0. Mk^ + ma^ sm* \ dt) '' This equation cannot be integrated, and hence cannot be found in terms of f . To determine the constants h and C we must recur to the initial conditions of motion. Supposing that initially 0=it, and ^ = and w=a, then A = A'ifc'o and 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 is given by r-=c' aTijt ' 'wliere r is the distance of G from the point of contact, and i is the radius of gyration of the body about an axis through G perpendicular to its plane, and c is 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 VIS VIVA AND ENEROY. 277 line, the distance between which in b. Show that, if the beams ho perpendicular to each other at the commencement of the motion, tlie velocity of their centre of gravity, when in the line joining the pegs, is equal to a/ t^ ' , where k is the radius of gyration of either beam about a line porpeudioular to it through its centre of gra\'ity. Ex. 4. A uniform rod is moving on n horizontal table about one extremity, and driving before it a particle of mass equal to its own, which starts from rest in- definitely near to the fixed extremity ; show that when the particle has described r distance r along the rod, its direction of motion makes with the rod on angle * [Christ's Coll.] tun~i Vr'+'A' Ex. 5. A thin uniform smooth tube is balancing horizontally about its middle point, which is fixed; a uniform rod such rs 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 siich a horizontal velocity that its middle point shall only just reach that of the tube ; supposing the velocity of projection to bo known, find the angular velocity of the tube and rod at the moment of the coincidence of their middle points. [Math. Tripos.] Bciult, If wi be the mass of the rod, m! that of the tube, and 2a, 2a' theu* re- spective lengths, V the velocity of the rod's projection, « the • oquired angular velocity, then w«=- „ , ,, . Ex. 6. The centre C of a ciicular 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 veiy small section, and a hea\y 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 0=lBCm, and if n be the angular rate at which 27 describes the cu'cular trace on the horizontal plane, prove that ( 77 ) = ~ cos a cos ^ - n* cos- o cos' 6 + const, where a is the radius of the wheel. Aimales de Gergonne, Tome xix. Ex. 7. If an elastic string, whose natural length is that of a uniform rod, be attached to a 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 B 8 angle $, which satisfies the equation cot' ;, - cot - - 2n=0, where the tension of the string, v/hen stretched to double its length, is n times the weight. [Math. Tripos.] Ex. 8. A regular homogeneous prism, whose normal section is a regular polygon of n sides, the radius of the circumscribing circle being a, rolls down a perfectly rough inclined plane whose inclination to the horizon is o. If w„ be the angular velocity just before the n* edge becomes the instantaneous axis then g sm g 8 + cos 2ir n asm- 5 + 4cos — n n . 8-hC03^''\ , gsing 11 1 a sni -5+1 cos — / « u I I ■ I \ I it Si' I 1:1 m ^■.*r\ *■ il y^ *lil 278 VIS VIVA. I I '! 881. The eqnation of Vis Viva may be applied to the case of relative motion in the following manner*. Suppose the system at any imtant to become fixed to the set of moving axes relative to johich the motion is required, and calculate what would then be the effective forces on the system. If we apply these as additional impressed forces to the system bxit reversed in direction, we may use the equation of Vis Viva to determine the relative motion €u if the axes were fixed in space. We may reduce the origin of the moving axes to rest by applying to every particle an acceleration equal and opposite to that of 0, in the manner explained in Art. 174. As these will be included as part of the additional forces mentioned in the enunoiotion it will be sufficient to prove the theorem for axes moving about a fixed point. If we follow the notation of Art. 259, the accelerations of any point P resolved paroUol to rectangular moving axes having a fixed origin are with two similar expressions for y and z. The three last terms, with the corre- sponding terms in the othsr expressions, are the resolved accelerations of a point Pg rigidly attached to the axes but occupying the instantaneous position of P. Let us call these A'o, Yq, Zq. Recurring to the proof of the principle of vis viva given in Art. 334 we see that we d"x have to substitute these expressions for -r-^ , &c. in the general equation of virtual velocities. After substitution for dx, Sy, Sz, it is clear that the terms containing — , J-, -J ail disappear. The equation after integration then becomes, as before, ^'^\(j)'+(^^y+{^y\=^^^f^(^-^o)dx+{Y-T,)dy + (Z-Z,)dz} + C. The theorem of Coriolis really follows at once from that of Clairaut given in Art. 257. The above mode of proof has the advantage of recurring to first principles. 352. Ex. 1. A sphere rolls on a perfectly rough plane which turm with a uniform angular velocity n about a horizontal axis in its own plane. Supposing the motion of the sphere to take place in a vertical plane perpendicular to the axis of rotation, find the motion of the sphere relative to the plane. Let Ox be the trace described by the sphere as it rolls on the plane, and let Oy be drawn through the axis of rotation perpendicular to Ox in the plane of motion of the sphere. Let nt be the angle Ox makes with a horizontal plane through the axis of rotation. Let ^ be the angle that radius of the sphere which was initially perpendicular to the plane makes with the axis of y. Let {x, y) be the co-ordinates of P the centre of the sphere, and Mk^ the moment of inertia of the sphere about a diameter. If the sphere were fixed relatively to the plane its effective forces would be Mn^x and Mn-y parallel to the axes, and Mk'^ ^7 =^ round the centre of gravity. Also the * This theorem is due to Coriolis, see the Journal Polyteeh. 18ul. VIS VIVA AND ENERGY. 279 impressed foroe, (gravity, is equivalent to ffeinnt and -pcoant parallel to the moving axes. Houce tbo cciaation of Via Viva for relative motion beoomoa Id {/dxy /duy ,,/dy) , dx , dv , dx rftf Here -. =a — and -t(=0. Wo have therefore at Ut dt 04:) d^x dt^ r^ni n^x+g Bin nt. This equation miglit also have been derive " from the formulae for moving axes 2 given in Art 179. If i'=g a», this equation leads to where A, £ are two constants which depend on the initial conditions of the question. 353. To determine the change in the vis viva of a moving system produced by any collisions between the bodies or by any explosions. (Carnot's Theorem.) Let v^, v^y v„ vj, vj, V,' be the resolved parts of the velocities of any particle m of the system before and after the impulse. Then the momenta m {vj — vj, m {vj — v^), m (v/ — u,), being reversed and taken throughout the whole system, are by D' A.lembert's Principle in equilibrium with the impulses. But these last are themselves in equilibrium. Hence the former set are also in equilibrium. Therefore by Virtual Velocities, tm {(vj - V,) tx + (V - vj hj + « - V.) Zz] ^ 0, where Zx, Zy, Zz arc any small arbitrary displacements of the particles impinging on each other, which are consistent with the geometrical conditions of the system during the time of action of the impulse. During the impact, it is one geometrical condition that the particles impinging on each other have no velocity of separa- tion normal to the common surface of the bodies of which they form a part. First Let the bodies be devoid of elasticity. Then the above geometrical conditon will hold just after the moment of greatest compression as well as during the impact. Hence we can put Zx = vJBt, Zy = vJZt, Zz = vJZt. The equation now be- comes Sm {{vj - V,) vj + {vj - v^) vj + « - V,) <} = ; .-. %m «' + <» + v,") = Im (v^vj + ty-; + v.v.'). i ' i'ij ' I ii I ; iJj ' I I ! n ^1 11 1! r i 280 VIS VIVA. This may be put into the form tm («;» + vj' + v,") - tm (y/ + v; + v') = - 2m {(vj - v^)' + « - v^y + {v; - tO«|. Therefore in the impact of inelastic bodies vis viva is always lost. Secondly. Let an explosion take place in any body of the system. Then the geometrical equation above spoken of will hold just before the impulse begins as well as during the ex- plosion, but it will not hold after the particles of the body have separated. Hence we must now put hx = v^St, By = v^,St, Bz = v,Bt. As before, we have and r^ = + 2m {{vj - v:f + [v^ - v,Y + {v: - vn Therefore in cases of explosion vis viva is always gained. Thirdly. Let the particles of the system be perfectly elastic. Then the whole action consists of two parts, a force of compres- sion as if the particles were inelastic, and a force of restitution of the nature of an explosion. The circumstances of these two forces are exactly equal and opposite to each other. By examining these two expressions it is easy to see that the vis viva lost in the compression is exactly balanced by the vis viva gained in the restitution. 354. It should be noticed that Oarnot's demonstration does not exclusively apply to collisions, but to all impulses which are such as do not appear in the equation of Virtual Velocities. Let a system be moving in any way, and let us suddenly intro- duce some new restraints, by which some of the particles are compelled to tak": new courses. The impulses which produce this change of motion are of the nature of reactions, and are such that in the subsequent path their virtual moments are zero. It follows from Carnot's first theorem, that vis viva will be lost, and the amount of vis viva lost is equal to the vis viva of the relative motion. Let there be two systems at rest, in a,ll respects the same except that one is subject to some restraints from which the other is free. Let both these be set in motion by equal im- pulses, and let X) K, Z be the components of these. Then, if 1, .' VIS VIVA AND ENERGY. 281 accented letters I'efer to the more free system and twice accented letters to the other, we have 2m (vJSx + &c.) = t {XBiv -{- &c.)) %m {vJ'Bx + &c.) = E {XBx -f &c.)j ' where Bx, By, Bz are any arbitrary displacements consistent with the geometrical conditions. Since both systems may be displaced in the manner in which the less free system actually begins to move, we mc!,y put Bx = vJ'Bf, &c. We therefore have Xm {vjvj' + &c.) = Sm (vj" + &c.). It again follows from Carnot's first demonstration that the vis viva of the constrained system is less than that of the free. Generally, the greater the constraints impressed on a system at rest, the less will he the vis viva generated by any given impulses. Tliis theorem is in part due to Lagrange, it has been generalized by Bertrand in his no-tes to the MeGanique Analytique. 355. Let two systems be in all respects the same and moving in the same manner. Let us suppose that suddenly some of the constraints are removed from one system and at the same instant let both be acted on by equal impiUses. Then following the same notation as before, we have 2 m {{Vx - Vt) &>; + &C.1 = S {XSx + &c.), 2wi {(t'j;" - Vx) Sx + &c.} = S (A'5j! + Ac). If we make Sx=Vg"St, &c. we obtain 2wi (f 2 V' + &c.) = 2?>i {v/^ + &c.), and we may deduce from this equation theorems similar to those of the last article. Let us now give these two systems any other displacement which is permitted by the geometrica,l relations common to both. Let this displacement be represented by Sx=Vx"'U, &c. Then as before we have 2ot (f>/' + &c.) = Sm {vj'v^" + Ac). From this and the last eq.uation we easily find 2hi {« - v^y + &c.} = 2/(t {{vj - vj'f + &c.} + 2m {(r/ - v/O** + &c.}. Let Oj, a^, &c. be the positions of the particles m^, m.^, &c. just bef^^re the action of the impulses ; a/, a./, &c. , o/', aj", &c. their positions just after, in the more free and constrained systems respectively, a^'", a^"', &o. their positions after any hypo- thetical displacement. Then Zvi (aWy = 2?» {a'a"y + 1m, (a"a"')«. Hence we infer that the motion of the more constrained system is such that 2nt [a.'a"Y is less than if the particles took any other coiu'ses, consistent with all the geometrical relations. If we suppose the systems to be acted on by a series of indefinitely small im- pulses, these impulses may be regarded as finite forces. We may therefore infer the following theorem, which is called Gauss' principle of least cntmtra int. The motion of a system of material points connected by any geometrical nln- tious is always as nearly as possible in accordance with free motion; i.e. if the 1 1 : ': M 'H ill 1 m X I i! i -aU ili i t • n ^. m 282 VIS VIVA. constraint during any time dt is measnred by the stira of the products of the mass of each particle into the square of its distance at the end of that time from the position it would have taken if it had been free, then the actual motion during the time dt is such that the constraint is less than if the particles had taken any other positions. M. Gauss remarks that the free motions of the particles when they are incom- patible with the geometrical conditions of the system are modified in exactly the same way as geometers modify results, which have been obtained by observation, by applying the method of least squares so as to render them compatible with the geometrical conditions of the question. 356. To determine the mean vis viva of a system of inaterial points in stationary motion. Clausius' Theorem*. By stationary motion is meant any motion in which the points do not continually remove further and further from their original position, and the velocities do not alter continuously in the same direction, but the points move within a limited space and the velocities only fluctuate within certain limits. Of this nature are all periodic motions, such as those of the planets about the sun and the vibrations of elastic bodies, and further, such irregular motions as are attributed to the atoms and molecules of a body in order to explain its heat. Let X, y, z be the 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 We have by simple differentiation, and therefore dt^ ~ m 2 „df dx\ „ /rfx\« „ d'x -''dt[^di)=^[dt)+^''dr^' fdxy 1 m#(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 m 2t [i/dxy,^ If^,, mrd(x^) fd{x')\~\ 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 1 and - - xX during the time t. For a periodic evidently the mean values of -^ ( 77 ) 9 • 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 liavo taken place, and the above expressions of the mean values have become sufiiciently 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 * This and the next article are an abridgement of Clausius' paper in the Phil, Mag., August, 1870. 1 . '- VIS VIVA AND ENERGY. 283 le Phil not periodic, bnt 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 with very great values of t. The same reasoning will apply to the motions parallel to the other co-ordi- nates. Hence adding together our results for each particle, we have, if v be the velocity of the particle m, 1 1 mean - Smt>*= - mean ^ S (Xx + Yy + Zz). The mean value of the expression - ^ S (Zx + Yj/ + Z£) has been called by Clausiua the virial of the system. His theorem may therefore be stated thus, t)ie mean temi vis viva of the system is equal to its virial. 357. In order to apply this theorem to heat, let us consider a body as a system of material particles in motion. The forces which act on the system will in general consist 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 virial. If (r) be the law of repulsion between two particles whose masses are m and m'. we have Xx + X'x'= -^(r) X X J f \ ^ ^ f , t \ \P^ ~ ^) x-^{r) -—-x'=(t>(r) And since for the r ■ - ■ r • • - J. two other co-ordinates corresponding eqtuations may bo formed, we have for the mtemal vii-ial - ,-j S {Xx+ Yy + Zz) = - 2r0 (r). As to the external forces, the case most frequently to be considered is where the body is acted on by a uniform pressure normal to the surface. If p bo this pres- sure, d,&e.) ...(1), with similar equations for y and z. It should be noticed that these equations are not to contain -^ , ~ , &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 could, if required, be expressed in terms of them by means of equations which do not contain any differ- ential 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 spoken of as the number of the degrees of freedom of that body. Sometimes it is referred to as being the number of independent motions which the system admits of. In the following investigations total differential coefficients with regard to t will be denoted by accents. Thus -r. and -vj will be written x' and x'. But laoranoe's equations. 289 If 2T be the vis viva of t.ie system, we have 22'=Sm(aj"+y+0 (2); we also have, since the geometrical equations do not contain & t , &c., rim: ///*• _. ffm (3), dx dx yy da; ., g dt dd with similar equations for y' and z'. In these the differential co- efficients jt » jB » &c. are all partial. Substituting theie in the expression for 2 T, we find 2T=F{f,e,,&c.d',',&c\ When the system of bodies is given, the form of F will be 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 i'^are dynamically equivalent. It should be noticed that no powers of ff, ', &c. above the second enter into this function, and when the geometrical equa- tions do not contain the time explicitly, it is a homogeneous function of ff, ^', &c. of the second order. 368, To find the virtual moments of the momenta of a system, and also of the effective forces corresponding to a displacement prO' duced by varying t>ne co-ordinate only. Let this co-ordinate be 0, and let us follow the notation al- ready 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 will be Xm (x'Sx + y'By-^ z'Bz), where Sx, By, Bz are the small changes produced in the co-ordinates of the particle m by 'I variation Bd of 6. This is the same as s™(4^^'l+4.)«''- If 22' be the vis viva given by (2) of the last article dT ^ f 'dx' . \ But differentiating (3) partially with regard to 0', we see UiOS dot* that -jTv = -j;; . Hence the virtual moment of the momenta is dd do equal to -Tff^^' R. D. 19 I- i\' '.' i I 1 ■ ■ \'\ 1 ■ \\\ :m I'/ 290 VIS VIVA. The virtual moment of the effective forces will be This may bo written in the form 4&c.)-2m(ar'^^J + &c.), d ^ f ,dx where the t- represents a total differential coefficient with regard to t We have already proved that the first of these terms is -j: jQi- I^ remains to express the second term also as a differ- ential coefficient of T. Differentiating the expression for 22* partially with regard to d dT = 2w [■^' -ja +&C.J, But differentiating the expression for x' with regard to 6 dx d^x . d' X (fa; dd ~ dddt ^de'^"^ ded '^' "*■ *°' and this is the same ^s ^ -^^ . Hence the second term may be written -52 , and the virtual moment* of the effective forces is WJ therefore (^§'-§)s^. i'l • The following explanation will make the orgnment clearer. The virtual moment of the effective forces is clearly the ratio to dt of the difference between the virtual moments of the momenta of the particles of the system at the times t + dt and t, the displacements being the same at each time. The virtual moment of the momenta at the time ( is first shown to be -73 h9. Hence I t:;; + ^ ^-r; dt ) 55 d9 \M dt do' J is the virtual moment of the momenta at the time t + dt corresponding to a dis- placement SO consistent with the positions of the particles at that time. To make the displacements the same, we must subtract from this the virtual moment of the momenta for a displacement which is the difference between the two displace- dx ments at the times t and t + dt. Since Sx=^r:S$, this difference for an abscissa is dO -|- f ^ J dt 55. We therefore subtract on the whole 2m J a/ -r [ -j|] dt + &c. 1 55, Jind dT this is shown to be ,- d( SO. do LAORANOE*S EQUATIONS. 291 virtual Between |e times lomeiit a dis- 18. To lomeut displace- scissa is SS.tod 869. To deduce the general equations of motion referred to any co-ordinatea. Let U he the force-function, then CT" is a function of 6, , &c. and t. The virtual moment of the impressed forces corresponding to a displacement produced by varying only is -rrf^O' But by D'Alembert's principle this must be the same as the vutual moment of the effective forces. Hence ddT dtdef dT dd' dU dd' o. M , , d dT dT dU 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 dff dd '^ de~ ' with similar equations for <^, -^, &c. In using these equations, it should be remembered that all the differential coefficients are partial except that with regard to t. These are called Lagrange's general equations of motion. Lagrange only con- siders the case in which the geometrical equations do not contain the time ex- plicitly, but it has been shown by Yieille, in Liouville'i Journal, 1849, that the equations are still true when this restriction is removed. In the proof given above \v6 have included Vieille's extension, and adopted in part Sir W. Hamilton's mode of proof, PhU. Trant., 1834. It di£Fers from Lagrange's in these 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 2mx'*. 370. To deduce the general equations of motion for Im- pulsive forces. Let 8Z7j be the virtual moment of the impulsive forces pro- duced by any displacement of the system. Then from the geo- metry of the system, we can express BU^ in the form BU, = F8d+QS+.„ The virtual moment of the momenta imparted to the par- ticles of the system is Xm{{x:-x:)Bx+{y:^y:)8y + {z:-z:)Bz}, where (a-,', y/, z^), (aj/, y/, z^) are the values of {x, y\ z) just before and just after the action of the impulsive forces. 19—2 ii :^'!| rf: \ li! ^•1 m 292 VIS VIVA. t Let 0^, ^;, &c. 0^, ^, &c. bo the values of ^, ', &c. nist before and just after the impulse, and let T^, T^ be the values of T when these are substituted for ff, «f>', &c. Then as in (fJT tIT \ j^\ — 1/1° ) 8^. The Lagrangian equations of impulses may therefore be written de; dd;~-^* with similar equations for ^, and ^, &e. 371. If we compare this equation with the general principle of Art. 295, viz. that the momenta of the particles just after an impulse compounded with the reversed momenta just before are equivalent to the impulse, we sec that it will be convenient to call jTv the component of the momenta with regai'd to 6, a name only slightly altered from that suggested in Thomson and Tait's Natural Philosophy. More briefly we may say that the ^-com- dT ponent of the momentum is -^, . In the same way we may d dT dT define the 6 component of the effective forces to be ^ itv n? . ^ dt da do 872. These eqnations for impulsive forces are not given by Lagrange. They seem to have been first deduced by Proi, x'tiven from the Lagrangian equation ddT dTdU dt dff~ dd~ de' We may regard an impulse as the limit of a very large force acting for a very short time. Let Iq, t^ be the times at which the force begins and ceases to act. Let us integrate this equation between the limits t = tQ to 1=1^. The integral of the first r" dT~^ti d'P term is I :j^, I which is the difference between the initial and final values of -jr, . L"" J^o d9 The integral of the second term is zero. dT For Tfl is a function of d, ', Ac. which though variable remains finite during the time t^ - tf.. If A be its greatest value during this time, then the integral is less than A (<^ - to) which ultimately vanishes. Hence the Lagrangian equation becomes I jw I ' = -i^ • S^o a paper in the Mathematical Messenger for May, 1867. 373. Other expressions for the virtual moments of the momenta and of the effective forces may be found when T is expressed in terms of the angular velocities of the bodies of the system instead of the co-ordinates. Thus taking any one body, if {x,y, z) be the co-ordinates of its centre of gravity, w,, w^, u, the angular velocities about rectangular axes meeting at the centre of gravity, M its mass, A, B, C, Ac. its moments and products of inertia, v the velocity of its centre of gravity, then by Art. 348, 2 r= Mv^ + A'ug^ + £uy^ + Cw,* - 2 Z>«^w, - 2^w,w, - 2Fu^Uy. I I M laoranoe's equations. 293 Tho Tirtaal moment of the momouta will then be by Ex. 8. of Art. 819 dT, (IT, dT, dT ,^ dT ,^ dT , , dx dy •' dz du)ji dug dw, and by Ex. 4 the virtual moment of the ufleotive forces will be if the directions of tho axes arejixed in space d dT, ^ d dT ,„ , dtdUji Jtdaf' where ix, 8y, tz are the linear displacements of tho centre of gravity and S0, 80, 8^ tho angular diuplacoments of tho body a^out the axes of Uj^, Wy, u,. If tho axes be moving wo have merely to substitute for the coefficients of Sx, &c. the corresponding expressions given in Axo example just rotcred to. 874. Before proceeding to discuss some properties of Lagrange's equations, let us illustrate their use by the following problems. A body, two of whose principal moments at the centre of gravity are equal, turns about a fixed point situated in the axis of unequal moment under the action of gravity. To determine the conditions that there may be a simple equivalent pendulum, Jkf. If a body be suspended from a fixed point under the action of gravity, and if the angular motion of tho straight line joining to the centre of gravity be the some aa that of a string of length / to the extremity of which a heavy particle is attached, then I is called the length of the simple equivalent pendulum. This is (in extension of the definition in Art. 92. liot OC be the axis of unequal moment, A, A, the principal moments at the fixed point, and let the rest of the notation be the same as in Art. 819, Ex. 1. Then 2T=A(e'^-^ sin* ^f «) + C (0' + ^' cos d}*, V= Mgh cos 6 + 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 z. Lagrange's equations will be found to become ^ {Ae')-A sin $ cos e^'* + C^' (0' + f cos 0) sin fl = - Mgh sin 9, |^{(7(0' + f cose)} = 0, d ^j {C (0' + f cos e) cos tf + ^ sin" tf f } = 0. Integrating the second of Lagrange's eqi;ation3 we have 0'+ ^'cos B=n, where n is some constant expressing the angular velocity about the axis of unequal moment. Integrating the third we have dip Crt cos tf + 4 sin» e -j^ = o, where a is another constant expressing the moment of the momentum about the vertical through 0. ' ' ' ■\l i „ I'i w\ t . 1 !' t m m II 29-^/ VIS VIVA. ; I r.' I'l B There is an error, Bometimes made in nsing Lagrange's equations, which we should here guard against. If u^ be the angular velocity about OC, we know by Euler's equations, Art. 230, that u^ is constant. If n be this constant, the Vis Viva of the body might have been correctly written in the form 2T= A (^» + sin" eip'^ + Cn\ But if this value of T be substituted in Lagrange's equations, we should obtain results altogether erroneous. The reason is, that, in Lagrange's equations, all the differential coeffcients except those with regard to t are partial. Though Wg is constant, and therefore its total differential coefQcient with regard to t is zero, yet its partial differential coefficients with regard to 6, , &c. are not zero. In writing down the value of T, preparatory to nsing it in Lagrange's equation, no properties of the motion are to be assumed which involve differential coefficients of the co- ordinates as indicated in Art. 367. But we must introduce into the expression any geometrical relations which exist between the co-ordinates and which therefore ro- duoe the number of independent variables. Instead of the first equation^ we may use the equation of vis viva, which gives To determine the arbitrary constants a and /3 we must have recourse to the , ~ be the initial values of d, f, y ; initial values of 6 and i^. Let Og, ^o> 777 » 'jT ^ *^^ initial values of ^, ^i ^ . -^ , then the above equations become dt On A Bin' ^ -7; + "r cos 9 = sin'-* 60 ^" + ^ cos $0 dt Cn 1 em' •(f)^(f)*=*-.(t)^ (f )'^^^'<— «l .(1). I id These equations, when solved, give and ^ in terms of t, 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^MP, aiiih=l, where I is the length of the pendulum. This gives sin«j = 0'sin^-^Bin0cos0 \ Wj=tf'cos^ + ^sinff sin^ \. u^=~'d(p' „ dT _ dw. ^ , dT . dw, _ da>a . „ Now^,=C«3^^»=C7«3, and -=i«,_V5«.-=i«,«,-5a;,«„ a8 may be seen by differentiating the expressions for u^, wj. Also by Art. 326, if if be the moment of the forces about the axis of (7, ■t-=N. d Substituting we have j^(Cw3)-(^-5)WiWg = i\r, which is Euler's equation. Ex. 2. A body turns about a fixed point and its vis viva is given by 2 r= J wi« + Bw^ + Cms* - 2DuiO)^ - 2^WjWi - 2 f Wjo;,. Show that if the axes are fixed in the body, Euler's equations of motion may be generalized into d dT dT dT dt rfwi dwj ' du. U^r=L, with two similar equations. This result is given by Lagrange. 376. Ex. Dcdwe the equation of Vis Viva from Lagrangc''s equations. If the geometrical equations do not contain the time explicitly, 7 is a homo- dT dT goneous function of 6', 0', &c. of the second degree. Hence 2T=-t^$' + -t-, , yp, &c. If we now substitute on the right-hand side from Lagrange's equations, we have „d7' dT^. dT .„ dU ., , ^di = do''-de''^de'^'''' ^n 1 ,! t : 1 1 i ilf 296 VIS VIVA. dr dT dT But since T ifl a fonction of 0, 6', 4>, ', &e., t, = j^ ^ + ,t5' ^' + *"•» '3^' subtracting this from the last expression we have dT dU^dU^,, di=de^^d^'^^- Integrating, we have the equation of Vis Viva T- U=h, where A is an arbitrary constant, sometimes called the constant of Vis Viva. 377. Ex. As an illustration of the application of Lagrange's equations to impulsive forces, let us consider the example already discussed in Art. 154. Let X be the altitude of the centre of gravity of the rhombus at any time, then « and a may be taken as the independent variables. We have Let P be the impulsive action between the rhombus and the plane, then the virtual moment of the impulsive forces is 8 ?7= /'3 (as - 2a cos a) = P5j! + 2a sin a PJa. The Lagrongian equations are therefore 4(Xi'-0=P 4 (fc« + a") (o/ - O = 2a P sin o !• Now the initial and final values of x' are x„'= - V, x^'— - 2a sin au ; those of a' are Oo'=0, ai'=w. Hence eliminating P we have la = 3 V sin a 2 al + Ssin^a' the same result as before. 378. Sir W. R. Hamilton has put the general equations of Lagrange into another form, which is found to be more con- venient for the investigation of the general properties of a dyna- mical system. This transformation may be made to depend on the following lemma. Let Tj he a function of 6, , &c., 0', , &c. and u, v, &c. from these equations of the first order. Let T, = -r, + M^'+i'f +&C., ' and let T^ he expressed in terms of 0, ' + &c. = 22\, and therefore T^= T^, but diffeicntly expressed, T^ being a function of 6', ', &c. and 0, (f>, &c., T^ a function of u, v, &c. and 0, ', &c. and introducing M, V, &c. will have to be frequently performed, it will be con- venient to have a name for the result. We shall call T^ the reciprocal function of T^, because 2\ may be derived from T^ by a nearly similar process. If T, be the vis viva of a dynamical system, this process is equivalent to changing from the component velocities to the com- ponent momenta and conversely. 879. Ex. If (0', (f>', \f/), (m, V, w) be regarded as the Cartesian co-ordinates of two points and T^ be a homogeneous quadratic function of ($', 4>', ^), then 2^=^ is the equation to a quadric. Prove that its polar reciprocal, with regard to a sphoro whose radius is ijh, may be found by eliminating {d', ip', ^') by means of the equa- tions-n^; = dT. dT, — ?=« i .„,—«, -j^=v, j-7/='*'' Hence show geometrically that, if r,r=A be the reciprocal quadric, — -=^', — '=', -r-' =•(/''. dvL dv dw ^ S80. To express the Lagrangian equations in the Ilamiltonian form. If a system be acted on by any impulses, the Lagrangian equations of motion may be written in the typical form f ,^, )=Py where the bracket implies that 0' — 0^, ,' — ^^, «&;c. are to bo written for ff, <^', &c. after differentiation, using the rame notation sis, before. Let H be the reciprocal function of T. Then these equations take the typical form 0^ — 0J= ( y- ) > where the bracket on the right-hand side implies that (P, Q, &c.) are to be written for (w, V, &c.) after differentiation. ^ 5 rt; i' ii \ In (' I i )■■ \ i (l i V ! t. ; !il i^*l y'i ill 'I 293 VIS VIVA. 381. If a system be acted on by any finite forces, tlie La- grangian equations of rilotion may be written in the typical form d^ dL dL _^ ~dt dd'~W~ ' where L = T+ U, so that L is the difference between the kinetic and potential energies. Since U does not contain {d\ ', &c.) the equations of transformation may be written in the form _dL_dT ^~ d&~ d&* _dL_dT ^~d'~d'* Also Lagrange's equations may be written in the form «' = dL dd' / dL . Let II be the reciprocal function of L, then these equations 6' change into dH du' dv , dH , dH which are called the Biamiltonian equations. When the geometrical equations do not contain the time ex- plicitly, r is a homogeneous quadratic function of {ff, ', &c.), and therefore uB' + vf + &c. = 2r. Hence n=- L^uO' -^-vj) +&.c. = T-U. Thus H is the sum of the kinetic and potential energies, ar^d is therefore the whole energy of the system. 882. Ex. To deduce the equation of Vis Viva from the Hamiltonian equa- tions. Since ZT is a function of (0, 0, &c.), (m, v, &c,) we have, if accents denote total differential coefficients with regard to the time, „, dH dll„, All , ^ dn dt do du dt so that the total diiierential coefficient of B 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 n=h, where A is a con- stant. 383. Ex. 1. To deduce Euler's equations of motion from the Hamiltonian equations. LAGRANGE'S EQUATIONS. 299 Taking the samo notation as in the corresponding proposition for Lagrange's equations, Art. 376, we have u=^^,=AuiBm,p + Buncos =(-ilwiC03 + £wjsin^) sin^ + Cw^cos 0. To express T in terms of (m, v, w) we must find (wj, Wg, Wj). Wo have 1 • . I /. V COS Au, =« sin + (V cos O-w) —. — ^ , Rin0 Also £ci>.=:U cos A - (v cos ^ - W) -;— ^ . " ^ Sin tf An As we only require one of Euler's equations, let us use -,- = -v', The former of these gives ^Wi-r-i + 5wa -7^ - s-r = - G -^l % dll dv = 0'. ^Wj ^Wi rfU , dWa which is the same as Au, ^' - Bu/-^ _ '-^ = - c ~ , * ^ * a d(f> dt and this leads at once to the third Euler's equation in Art. 230. Tiie latter of the two Hamiltonian equations leads to one of the geometrical equations of Art. 235. Thus the six Hamiltonian equations are equivalent to all the three dynamical and the three geometrical Eulerian equations. 334. Ex. 1. The position in space of a body, of mass M, is given by (a;, y, z) the rectangular co-ordinates of its centre of gravity, and (6, ,y 1 1 ! , i • 1 . hi 1 i i] N#.l I I! i I ) i| '--■■11 iii. itt > ■ a i :J I'i' t I'li 300 VIS VIVA. tual moment obtained by varying ^ only and so on, it ia clear that Lagrange's equations may be written in the typical form t- ^-, ~ 'dO~^' 386. It will often be convenient to separate the forces which act on the system into two sets. Firstly those which are conservative. The partb of P, Q, &o. due to these forces may be found by differentiating the force-function with regard to 0, (p, &c. Secondly those which are non'Conservative, such as friction, some kinds of resistances, &c. The parts of P, Q, &o. due to these must be found by the usual methods given in Statics for writing down virtual moments. Though these non-conservative forces do not admit of a force-function, yet sometimes their virtual moments may be represented by a differential coefficient of another kind. ThuF suppose some of the forces acting on any 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 moment of these forces is S {/j^x'Sx + fi^'Sy + fi^z'Sz), where ni, /j^, fi^ 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 fi^, fi.^, /Xg are each equal to - k. Put ^^'° %=^ (^'»'^+&«-) =2:(At,x'g+&c.) . by Art. 3C8. Hence dP dff 5^-i.^304.&c.=sj^x'(g5^-Hg80+...) + &c.j =S(/ttja;'8a; + &c.). H! In this case, therefore, if U be the force-function of the conservative forces, F the function just defined, 055, 50, ») + Aug* + Buy' + Cw,» - 2DuyU, - 2^w,w, - 2Fu;cUy), where c is the area; A, B, &c. the moments and products of inertia of the surface of the body and (u, v, w) the resolved velocities of the centre of gravity of a. 388. To explain how Lagrange's equations can he used in some cases when the geometrical equations contain differential coefficients with regard to the time. It has been pointed out in Art. 3G7, that the independent variables 6, '+...^0 (1), the other two being derived from this by writing 2 and 3 for the suffix. These three equations express the fact that the resolved " 'll \ I ■}'i V Mm. I li '' ( , ) !: >i m ■1 , ? i *'li !■ .'T'l ^ ili II !■ 302 VIS VIVA. velocities in three directions of the point of contact are zero, equation of virtual velocities may be written The \dt dff " dd) dU hd + &c. = "" S^ + &c (2), where V is the force-function of the impressed forces. Since the virtual moments of the reactions at the point of contact have been omitted, this equation is not true for all variations of 6, (j), &c., but only for such as make the body roll on the rough surface. But the geometrical equations i,, L^, L^ express the fact that the body rolls in some manner, hence B6, B(f), &c. are connected by three equations of the form A^Bd + B^S(f>+...=0 (3). If we use the method of indeterminate multipliers*, the equa- tions of virtual velocities will.be transformed in the usual manner into d dT dT dU.^dL,. dL„ . dL^ .^. dt d(y dd dd' !•••••••• with similar equations for the other co-ordinates cf), yjr, &c. These joined to the three equations L , L^, L^ are sufficient to determine the co-ordinates of the body and \, fi, v. This process will be very much simplified, if we prepare the geometrical equations 2/,, L^, L^ by elimination, so that one dif- ferential coefficient, as 6', is absent from all but the first equation, another, as ^', absent from all but the second, and so on. When this has been done, the equation for 6 becomes d^dT^ dt dd' dT^_dU dd~ dd' dL, (5). Thus \ is found at once. The values of //. and v may be found from the corresponding equations for ^, '^. We may then sub- stitute their values in the remaining equations. 389. The method of indeterminate multipliers is really an introduction of the unknown reactions into Lagrange's equations. * If we multiply the geometrical equations (3) by X, ii, v respectively and sub- tract them from (2) we get ^ldtdd'-de-T0-^d¥-''dff-''de'y^=^- Now there will be as many indeterminate multiples X, /*, v as there are geome- trical equations (3) connecting the quantities S6, 50, &c., i.e. there are as many multipliers as there are dependent variations. By properly choosing X, n, v the coeflScients of these variations may be made to vanish, and then the coefficients of the independent variations must vanish of themselves. Hence the coefficient of each variation in this summation will be separately zero. LAGRANGE'S EQUATIONS. 303 an IS. Ibhe of of Thus let B, i» -Sa» -^3 be the resolved parts of the reaction at the point of contact in the directions of the three straight lines used in forming the equations L , L^, L^. Then L^, L^, L^ are propor- tional to the resolved relative velocities of the points of contact. Let these velocities be /c,2/,, k^L^, k^L^. Then if 6 only be varied the virtual velocity of R^ is k^A^O which may be written dL dff K, B0. Similarly the virtual velocities of i2, and B^ are dL, %' ^^ ^^^ "^ dff B6. Hence, by Art. 385, Lagrange's equa- tions are d^dT dT_dU J. dt dff dd~ dd^ "' ' -^ + kB dff ^ « " dL^ P dL^ dff ^ '"» « dff Comparing this with the equations obtained by the method of indeterminate multipliers we see that X, fjL, 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. The method of indeterminate multipliers may sometimes be used with advantage when the geometrical equations do not contain ff, ', &c., but are too complicated to be conveniently solved. Thus if f{t,e,,...) = be a geometrical equation, connecting 6, , &c., we have, as in Art. 335, |8« + Js^-|....=0. This may be treated in the same manner as the equations Z^, Xg, Lg in the preceding theory. We thus obtain the equation d^dT_dT dt dff dd '' ^+X^ + with similar equations for <^, '^, &c. 390. Ex. Form by Lagrange's method the equations of motion of c lomoje- 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 2 normal to the plane. Let (x, y, a) be the co-ordinates of the centre of gravity of the sphere, d, (f>, ^ the angular co-ordi- nates of three diameters at right angles fixed in the sphere in the manner explained in Art. 235. Then, if the mass bo taken as imity, the Via Viva is by Art. 319 2r =«" + 7/'« + *« {(0' + ^' cos ^)2 -h «'«-(- sin" 5f 2}. ! ! V ; n i (I r. 1 I J '\ m li.^i! Ill !l 304 VIS VIVA. The resolved velooities parallel to the axes of x and y of the point of the sphere in contact with the plane are to be zero. These conditions will be found to lead to the equations L^= xf - a6^ ooa^- a^' sin sin ^ = 0, Zj=y + a0'Bin^-a^'Bin0co8 0=O. Also if ^ be the resolved part of gravity along the plane and C any constant U=gx+C. The general equation of motion is dtd. Taking these in turn we have j!'=g + \, y"=n, ** {$"+- $' + V + w<«), and the equations of condition are x-a(i>y=0, y' + aux=0. Displace the sphere by rolling it along a small arc parall&l to the axis of x through an angle dd. Then we have dtdx' dtduy dx ' .'. ax" + h*-^=ga. Similarly rolling the sphere parallel to the axis of y and twisting it round tha axis of w„ we have -ay" + l*'^'=0, andA;»^'-« dt dt These, by elimination of Ug, wg, u,, lead to the same result as before. ..I*,-.' ■ ■ . '.-.fc^ ^.-rrr- . .-»-•*■ ■ a--' + Jfc« Ind tiie LEAST ACTION AND VARYINa ACTION. 305 Principles of Least Action and Varying Action. 391. Let (q^, q^, q^, &c.) be the co-ordinates of a system of bodies, and let q stand for any one of these. Let 2 T be the vis viva of the whole system and U the force-function, and let L = T+ U. As before let accents denote differential coefficients with regard to the time. Let us imagine the system to be moving in some manner, which we will call the actual motion. Then q^, q^, &c. are all functions of t, and it is generally our object to find the form of these functions. Let us suppose the system to move in some slightly different manner, i.e. let q^, q,^, &c. be functions of t slightly different from their actual forms. Lot us call the motion thus represented a neighbouring motion. We may pass, in our minds, from the actual motion to any neighbouring motion by the process called variation in the calculus of that name. By the fundamental theorem in that calculus ^.[s|(8,-,'s*)];;. where the letter S implies summation for all the co-ordinates q^, q^, &c. and, as implied by the square brackets, the terms outside the integral sign are to be taken between limits. The co-ordinates being independent of each other, each sepa- rate term under the integral sign vanishes by Lagrange's equa- tions, and we have therefore -[- mt + t where H is the reciprocal function of L, by Art. 378. The integral I L dt has been called by Sir W. R. Hamilton the principal function, and is usually represented by the letter S. If the geometrical equations do not contain the time explicitly, we have H=T — U. In this case the equation of vis viva will hold, and if h be the constant of vis viva we have hi'^L dt = -h {8t, - SO + h ^rkT- R. D. 20 I v\ A m 306 VIS VIVA. 392. Otlier functions may be used instead of 8. Let us put The function Fis called the characteristic fiincti(m. If the geometrical equations do not contain the time explicitly, we have // = h, where h is a constant which may be used to repre- sent the whole energy of the system. In this case V=8+h{t,-t,) =.f\T+U)dt+r(T-U) = 2 r TJt. dt The function V therefore expresses the whole accumulation of the vis viva, i.e. the action of the system in passing from its position at the time t^ to its position at the time t^. 393. In the proof of these theorems we have supposed that all the forces are conservative. If in addition to the impressed forces there are any reactions, such as rolling friction, which cannot be taken account of by reducing the number of independent co-ordinates, wo must use Lagrange's equation in the form d dL dL dtdq'~dq~ ' where, as explained in Art. 385, PSq is the virtual moment of these reactions corre- sponding to a displacement dq. In this case the quantity under the integral sign will not vanish unless the variations are such that SP(Sq-q'5t)=0. Now q being the value of any co-ordinate in the actual motion at the time t, q + Sq is its value in a neighbouring motion at the time t + St. But q'St is the change of q in the time St, hence q + Sq- q'St is the value of the co-ordinate in the neighbouring motion at the time t. The neighbouring motions must therefore be such that the virtual moments of the reactions corresponding to a displacement of the system from any position in the actual motion into its position in a neighbour- ing motion at the same time is zero. With this restriction on the variations, the two equations which express the variations of S and V will still be true. 394. The two fundamental equations, giving the values of B8 and 6V, will be found to lead to many important theorems which we shall now proceed to considei*. Let us call the positions of the system at the times t^ and t^ the initial and terminal positions, and let us suppose these fixed, so LEAST ACTION AND VARYING ACTION. 807 that the actual motion and rll its neighbouring motions are to have the same initial and terminal positions. In this case Bq vanishes at each limit, and the two fundamental equations take the form* ■'!': 8 Ldt^-h{Zt^-hQ, 28 {t-t,)hh. V I • We may easily establish these theorems without the use of Lagrange's theorems. Let (x, y, z) be the rectaufrular co-ordinates of any particle and let m be the mass of this particle. Lot -Y, }', Z bo the components of the impressed accele- rating forces on it. Then and by the fundamental theorem in the Calculus of Variations «j;>=i"o^x>(£-*")"'-^"""*[4'(''-'''"]::- If vre substitute for L and remember that T is a homogeneous function of «', y', z', this becomes i f''Ldt = [(U-T)8t + Tmx'5r:{' + f*'Zm{X-x") {Sx-x'St)dt. ''to to •'U If we consider the positions of the system at the times (q and t^ to bo given, 8x is zero in the part taken between limits. If the time of transit be given it is unnecessary to vary the time. Putting St=0, the part under the integral sign vanishes by the principle of virtual velocities. The part outside the integral sign is also zero and therefore 8 / ^Ldt=0. J to If the time be varied, Sx - x'St is the projection on the axis of x of the displace- ment of the particle m from its position in the actual motion at time t to its position in a neighbouring motion at the same time. Hence the part under the integral sign vanishes as before by the principle of virtual velocities. Lot us suppose that the geometrical conditions do not contain the time explicitly, then T - U— h and L=2T-h. The equation then becomes 28 f^'Tdt - [S(ht)]*' = [ - hStf . •'to to to ptl If h be giveii ;ri have 8 / Tdt^O. •'to From the general value of the variation in Cartesian co-ordinates we can also deduce the values of 5-(f-*">«i may be completely found. Hence this expression must be a perfect differential with regard to t, quite independently of the operation 5. But this cannot be unless it vanishes, because it contains only the variations Sx, dq, &c. and not the differential coefficients of these variations. We have therefore the general equa- tion of transformation d dL dtd^'^^')^^' \dx dtdx' ') \dq where the ^' implies summation for all the variables x, y, &c. or q^, q^, &o. If a;, y, and Q„. We have by differentiation dff provided i + 1 is not n. In that case (IH _ dL^ dq(^> dQ^~dq^"UQ„ _g(fc+l)=: _ d dt' >(«:) ■(2), but the first and third of these terms destroy each other, so that the theorem (2) is also true when /t + 1 = jj. Also dH ^ dL_ dL_ d^^ n -O ^ d«f>~ dq'-''^'^ df''^dq'i^^~^'' ^'dg<*>* Here the second and fourth terms destroy each other. The first and third, by d - (1), become Q\+^ or -7 Qj^Jrv Thus all the equations may be written in the typical dt Eamiltonian form dH ^-=-dt^ (It) *-di^''+^ dH dgi*' which are true for all values of k from k=Q to k=n-l. Thus there are 2n equa- tions corresponding to each g. "We may show in the same way as in Art. 382, that the total differential coeffi- cient of H with regard to t is equal to its partial differential coefficient. Bo that when L, and therefore H, are not explicit functions of t, we have as one integral if =/t, where A is a constant. Writing this at length it becomes L = 2(Q,q'+Q^q"+...) + h, which is the integral continually used in the Calculus of Variations. We see the'; this integral corresponds to the equation of 'Vis Viva in Dynamics, LEAST ACTION AND VARYING ACTION. 311 for all variations. The Bq^s being all arbitrary and independent, each coefficient under the integral sign must vanish separately, and this leads to the typical Lagrange's equation. Ex, 1. There is another method of deducing Lagrange's equations from the principle of Least Action which is worthy of notice. We are to make f ' Tdt a minimum, subject to the condition T- U—h, By Lagrange's rule in the Calculus of Variations we are to make sf{T+\{T-U--h)]dt=0, without regard to the given condition, and afterwards make \ such a function of I that the given condition is satisfied. This will be found an excellent exercise in the Calculus of Variations. The solution maybe indicated as follows. Putting W—T+\{T-U) vre have with the same notation as before and this must be equal to h8 jxdt. The integrals are to be taken between the limits, which are omitted for the sake of brevity. First, let us consider the part outside the integral sign. Tlie initial and final positions being given 52=0, and wo have WSt - S ^p q'St ■= hif\dt = h\St. This equation is satisfied by 8t=0, but since the time of transit is not to be the same in the actual and varied motions, this factor must be rejected. Also 2* is a dT homogeneous function of the ^'s, hence li ~—,q'=2T. Substituting for W its value and using this equation we find (1 + X) 2'+Xl7+AX=0. But X is such that T-U=h, hence{l + 2\)2'=0 and .-. Xrn-^. Next, let us consider the part under the integral sign. By the rule in the Calculus of Variations this gives at once the typical equation dW _ ddW^ dq dt dq' Substituting for W we have the typical Lagrange's equation. Ex. 2. If we add to the conditions given in the 'principle of Least Action, the condition that the time of transit is to be always the same, show that the minimum does not in general lead to Lagrange's equations. Following the notation of the 1 A last Article, show that the minimum for a given time is determined by X= - ^ -f »,• I I where, if the geometrical equations do not contain the time ex- plicitly, we may put h for H, h being a constant. In this case i li. ! i 'I 1 II- ! i ' I hf 1 :S 1 ■ ,■ . !*' ■ ■ ::i 1 hn J ■';:l : ' ii ! \\\ II- 314 VIS VIVA. the integrals may be used to connect the constant of vis viva with the constants (a, a, &c.). Comparing Art. 394 with these results we see that ;S^ is such a function, that all the equations of motion and their integrals are included in the statement that 8S is a known function of the variation of the limits. If we keep the limits fixed, we get Lagrange's equations; if we vary the limits we get the integrals. 400. In just the same way, if we regard q^', q', &c. as functions of t, the initial co-ordinates r-ud their initial velocities, we may eliminate t also by means of the equation which reduces to H=T— ?7 when the geometrical equations do not contain the time explicitly. Let us suppof'e V to be expressed in this manner as a function of the initial co-ordinates, the co-ordinates at the time t, and of H. Then, by the equation for 8 F, dV dT dV dT^ dV da da" dli = t. dq dq Supposing V to be known, the first of these equations gives in a typical form all the first integrals of the equations of motion. The second supplies as many equations as there are co-ordinates {q^, q^, &c.). When the geometrical equations do not contain the time explicitly these do not contain t, but they all contain h. One of them, therefore, reduces to the relation between this constant and the constants (a, a', &c.). The equation —rj- = t will give another second integral of the equations of motion containing the tim"^. 401. Ex. ^iQ=f (Sqp' + E) dt, vrliere p=j-,, jirove th&t SQ = imt + 'SqSj)]* . Thence show that if Q be expressed as a function of the initial and terminal components of momentuji, viz. (6^, b^, &c.) an;' (py, p^, &c.) and of the time, then -- = «, ^=-a. ^=H. ThisresuItisduetoSirW.R. Hamilton. dp do at 402. Ex. 1. A homogeneous sphere of unit mass rolls dovm a perfectly rough fixed inclined plane. If the position of the sphere is defined by the distance q of the point of contact from a fixed point on the inclined plane, show that „ 7 (q-a)^ 1 , V , 5 , , where g is the resolved part of gravity down the plane and tQ=0. GENERAL EQUATIONS OF MOTION. 315 Thence obtain by substitution the Hamiltonian first and second integrals of the equation of motion. B 7 We easily find, as in Art. 133, that q=a + a't + jjgt''. Also ^=iq9''> ^=92- pi To find S, we substitute in &= / (T+ U) dt. After integration we must eliminate *'o a' by means of the equation for q, Ex. 2, Taking the same circumstances of motion as in the last example, show 2 /ii " P that r= — W -=-^iOi + '0' - (f/« + '0^}' Thence also deduce the Hamiltonian first and second integrals. Ex. 3. Show how to deduce the equation (f vis viva, from the Hamiltonian integrals. We have V a function of 2' ^^- ^^^^ eliminate q^\ q^, &c. Let the reciprocal function of H thus found be H:=F{q„p^,q^,p,,&c.). I i I i < I r i ■It '.' i I j ■■Ik ' Pi t ^1 1 H M ' ■ ■ -: .i , li;i 816 VIS VIVA. But Pi = ;t— > Pa=-r— ' *^^' ^^^ ^~~"^' Hence 8 must satisfy the equation dS ^ r^f dS dS , \ ^ In just the same way, p^= -j- , p^ = - , &c. and the equa- tion of vis viva gives H = h. Hence Fmust satisfy the equation T.( dV dV , \ , If we consider the initial value of T, we shall have another equation of a similar form with a^, a^, &c. written for q^, q^, &c., and ^, for t. It is necessary that the functions should satisfy both these equations. Ex. Taking the same circumstance of motion as in Ex. 1 of Art. 402, show that the difierential equation to find ^ ^'^ yI\ t) ~91 — ^^' Integrate this equa< tiou and thence find the motion. 404. When there are several independent variables, the equation to find V is of the form .2^"(^J-'^"^¥."''^''=^-''' <^^' where (5^, B^^, &c.) are functions of q^, q^, &c. only. The left-hand side of this equation, by Ex. 2 of Art. 384, may be written in the form of a determinant. We dV dV have only to replace w, v, &c. by their values — , -j— , &c. We thus have, in general, a partial differential equation to find F, and Sir W. Hamilton gave no rule to determine which integral is to be taken. This rule has been suppUed by Jacobi in the following proposition. Suppose a solution to have been found containing n-1 constants* besides h, and the constant lohich may he introduced by simple addition to the function V. These need not be tlie initial values o/q,, q2- but may be any constants whatever. Let them be denoted by Oj, aj...an-i» *° *'"** V = f (qi, qa-qn. Oi, aj...On_i) + on (2). Then the integrals of the dynamical equations will be rar^- '''■u-^r^^-^ (3)' ffi=*+' <4). • An integral of a partial differential equation has been called by Lagrange "complete," when it contains as many arbitrary constants as there are independent variables. It is implied that the constants enter in such a manner into the inte- gral that they cannot by any algebraic process be reduced to a smaller number. For instance, if two of the cougtauts enter in the form aj + a,, they amount on the whole to only one. where the equi Let where tl iber. the GENERAL E'^UATIONS OF MOTION. 317 where Pi, /9j.../9n_i ""'^ « ^^^ " w^'" arbitrary constants. And the first integrah of the equations may be written in the form dqi dqi' ' dq^ ~ dqj' ' Let the expression for the semi-vis-viva be 5). ^=s^ii'?i"' + ^ii.'7iV+&c. .(6), where the coefficients A^^, A^^, &o. are functions of q^, jj, &c. only. Let Qi, Qg...Q^he such functions of 17,, ^g.-.^n and the constants, that they may satisfy identically the n equations df ,(7). ^=AiiQi+A,^Q^+... &C. = &C. Then from the mode in which the differential equation to find V has been formed, in Art. 403, we know Qj, Q.^ will also satisfy identically the equation ^ + '^=2^iiQi' + ^i2„ > dq. q{q,-... * We may also show that the Jacobian integrals satisfy the Hamiltonian form of the equations of motion. The peculiar relation of the differential equation to the Hamiltonian function H adapts it to this process. If we substit.ute the value of F given by (2) in the differential equation (1), the result is an idantical equation. dV Differentiating this identity with regard to each of the n constants and replacing ,— dq I, * I' , ^x, t dH dH dH dH A X ^ J by p, we get n equations of the form -j—^-—- + -,— -^ — ~ + ...=0 to find dp, dq, da op, dqg da J- , y- , &0. These are the same as the equations (9) in the text, hence ——=q'. Again, differentiating H partially with regard to g,, we have dp dH dHd^ dH d^f ^^ dq, dp,dq;^ dp^dq^dq^ But all the terms of this equation except the first are together equal to the total dp. differential coefficient dt Hence t- = ~ -jz' The investigations of Hamilton and dq, dt Jacobi apply to a system of free particles mutually attracting each other referred to Cartesian co-ordinates. In the text the reasoning has been applied to a system of bodies referred to any co-ordinates. GENERAL EQUATIONS OF MOTION. 319 Next let U8 consider the expression for T\ we see that the partial differential coefficient f,. -9 .1:7 9i + ,,„ QiQvth Ci=fi{Pi, 2vPi, flu."- m (1). &o. =&c. ) where p, q, ... are some variables which determine the position an \ motion of the system, and which are such that the equations of motion may bu written in the forms , dH , dll ,„. ^ = -d^' «=rf^ ^2>' in the manner explained in Art. 381. Let the equations of motion of a second dynamical problem be dH dK , dH dK ... ^^-Tq-di^ ^ = d^ + d^ (^^' I m 320 VIS VIVA. where K is nome function of p,q,...t. If we consider fj, r„ ... the constants of the solution of tlie first problem to be functions of p, q, and (, we may suppose the solution of the second problem to be represented by integrals of the same form (1) as those of the first problem. It is therefore our object to discover what func- tions fp c.^, ... are oi p, q, and t. The function K is called "the disturbing func- tion," and is usually small as compared with //. Since the equations (1) are the integrals of the diflorontial equations (2), we shall obtain identical expressions by substituting from (1) in (2). Hence dif- ferentiating (1), and substituting for y and q' their values given by (2), we got Q^_dc,dJI_^dr,dn^^^dc, dp dq dq dp dt (4). 0=«&c. uj, ... are considered as variables, the equations (1) are the integrals of the differential equations (3). Hence repeating the same process, we have But when e,, c. rfCj _ dc^ dn dc^ dH dt dp dq dq dp dc, dK de, dK — ' f. — 1 J. dp dq dq dq dc^ "di de^_ dt = &0. where the differential coefficients on the left-hand side are total, and those on the right-hand side partial. Hence, using the identities (4), we get •(5), dcy _ rfcj dK dc^ dK dt ~ dp dq dq dp dc with similar expressions for -j^ , &o. If K be given as a function of p, q^ &c. and f, we have -f^ , &c. expressed ad functions of p, q, &c. and t. Joining these equations to those marked (1) we find Cj, Cj ... as functions of t, continue thus, dK If K be given as a function of c^, c^, ... and t we may dK dc, dK dc, dp rfcj dp rfcj dp dKdKdc^ dK de^ dq ~ dc^ dq dc, dq Substituting in the expression for -^, we get dc» de^ d£g~j dK dqj dc. dt Ldq dp ~ dp dq J dcg \_dq dp dp dq. where the 2 means summation for all values of ^, q, viz. p^, q^, p,, q^, &c. Since by hypothesis Cj, Cj,... are supposed expressed as functions of pj, q^, &c. and «, these coefficients may be found by simple differentiation. It will, of course, be more convenient to express them in terms of Cj, Cj, &c. and t by substituting for jjj, ji, &c. their values given by the integrals (1). 407. On effecting this substitution it will be found that t disappears from the expressions. This may be proved as follows. Let A be any coefficient, so that /1 = S of2)i, . "l. Cj, ...(5). ^1, &c. course, titutiDg rom the BO that GENERAL EQUATIONS OF MOTION. 321 ^ = S I . '- .---/' ^- I * wo have to prove that A heing regarded as a (tmctioQ d A of j)j, 2i, &o. and t, the total differential coefficient -* — ia zero. Now d.A dA dA , dA ,, dt dp^ dq^ dt The letters p^, 9j, &o. enter into the expression for A only throngh e^ and e,. Let us consider only the part of -\ due to the variation of c^, then the part duo to the variation of c, may be found by interchanging «j and c„ and changing tho sign of the whole. The complete value of d.A d.A dt ia the sum of those two parta. The part of - ' - due to tho variation of c^ ia \_dp {dq dt dpdq dq dq^ dp "") dq (dp dt dp^ dq dpdq dp *"iJ* dc If we substitute for -.^ its value given by the identity (4), we get I, dp \dp dq* dq dpdq)~ dq \ dp dp dq dq dp* ) J ' If we now interchange c^ and c, we get the same result. Hence when the two d. A parts of -~~ are added together, the sigps being opposite, the bu:j1 is zero. 408. Let the expression S ["^ ^ - ^i J?"] , where tho S means summa- tion for all the values of p, q, be represented shortly by (Cj, Cj). Then in any dynamical problem if iT be the disturbing function, the variations of bhe parameters "i* Cj, dcj_ , dK dK are given by -r} = [c^, Cj) ^— + (01,03)^— + ..., where all the coefficients are dt dc dc. functions of the parameters only and not of (. This equation may be greatly simplified by a proper choice of the constants e^, c,, ... In the M^canique Analytique of Lagrange, it is shown that if the con- stants chosen be tho initial values of pj, j)^, ... and q^, Jt,..., viz. a, /3, 7,... and \, H, y, ... respectively, then the equations become da_ _ dK d^_ _ dK dt~ d\' dt~ dfi' d\ dK d/i dK &0. dt da' dt d^' &c. It is assumed in the demonstration that iT is a function of 7^, 17,,... only. This simplification has been extended by Sir W. Hamilton and Jacobi to other cases, but for this we must refer the reader to books which treat on theoretical dynamics. 409. It foUows from the investigation in Art. 407, that if two integrals of a dynamical problem be found, viz, Cj=o, Cj=/3, where c^ and c, stand for some functions of p^, Qi, jpj, q^, ... and t, and a and /3 are oonstaats, then (Cj, Cj) is also constant. So that (c^, Cj) = 7, where 7 is a constant, is either a third integral of the equations of motion or an identity. If it is an integral it may be either a new integral or one derivable from the two c^ and Cj already found. R. D. • 21 ilH m '4 m 322 VIS VIVA. I 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 commimicated to the screw. Sesult. Let n be the ratio of the mass of the screw to that of the particle, a=the angle the tangent to the scrpv' makes with the horizon, h the height descended by the particle. Then the angular velocity generated is v: 2(lh cos'' a a^(/H-l)(n4- sin's aj' 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, carryint^ 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. 4. A straight tube of given length is capable of turning freely about one ex- tremity in a horizontal plane, two equal particles are placed at different points within it at rest, an angular velocity is 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 ai i connected by an elastic string whose natural length is f of the circumference. The string is stretched until the particles are in contact and 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 particle.^ 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 for unit length is fi through one continuous half and jw' 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 «hat the time of a small oscillation of the chain under the action of gravity is , />/ + ( ^W — i ira+b) (fi + fJ.') 2 {n-fi)g ' 7. Two particles of masses »i, in' are connected by an elastic string of length «. The former is placed in a smooth straight groove and the latter is projected in a * These examples are taken from the Examination Papers which have been set in the University and in the Collegesi. (' ^of length a. loted in n been set EXAMPLES. 323 direction perpendicular to the groove with a velocity V. Prove that the particle m will oscillate tlxrough a space — , , and if m he large compared with m' the time of oscillation is nearly 2wa m + m 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 tliis plane at a subsequent time t before the sphere leaves the plane, then 2^35 (||'_84«-60c)(eV?"'-e- v ^nt , 5 ft . ; c being the distance from the axis to the plane measured upwards. 9. The extremities of a imiform heavy beam of length 2a slide on a smooth wire in the form of the curve whose equation is r=a (1-cos^) 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 1 ( iJ^ I iJ^i bring it into a horizontal position tan^=- ]e''2a - p-^ •!«.' through which the rod has turned after a time t. 2a - e~^ ao ' I , where B is the angle 10. A rigid body whose radius of gyration about G the centre of gravity is Ic, is attached to a fixed point C by a string fastened to a point A on its surface. CA{=b) 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 be the angle between A G and A C at time t, show that it m y^k'^-iabain^^ b^ 2 sin" 2sl^ be very small, the period is 2nbk and if the amplitude of 0, i.e. V^a{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 'm + ix rH' + 2Vat, where b is the uiitial value of s, m and n 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 w, 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) 5(w--wr ■« u^ + u' * 21—2 between them is given by the equation cos 2a ill 1 i!! ;lf i 1 ' \ \ ; 324 VIS VIVA. 18. 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 a, a' from the ring and are pro- jected with velocities v, v' at right angles to the string. Prove that if »iv'a'=mW' their second apsidal distances from the ring will be a', a respectively. 14. If a imiform 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 Telocity varies inversely as the difference between the sum of tho 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. A small bead can slide freely along an equiangular spiral of equal mass and angle a which can turn freely about its pole as a fixed point. A centre of repulsive force F is situated in the pole and acts on the particle. If the system start from rest when the particle is at a distance a, show that the angular velocity of the spiral when the particle is at a distance i from the pole is where mJc^ is the moment of inertia of the spiral about its pole. m*2{l + 2cot''a) 16. The extremities of a uniform beam of length 2a, sUde 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 - j t;o the horizon. The system is set rotating about the vertical line through the point of intersection of the rods with an angular velocity u, prove that if be the inclination of the beam to the vertical at the time t and a the initial value of d m- (3cos'a + Bin''a) » 3cos''tf + Bin''^ > 6(7 w'= (3oos'a + sin'o)w' + — (sino-sintf). 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- tween them. Prove that the vertical velocity of its centre in any position will be sin a cos jlOgr (a +c)(l- sin «/>))* I 7-5coB'»0cos''ar" , where

h a a The extent of the oscillations on each side of the central position may be found by substituting the values of t given by this equa- tion in the expression for x — j. Since these must occur at a constant interval equal to IT y-? we see that the extent of the oscillation continually decreases, and that the successive arcs on each side of th:^ position of equilibrium form a geometrical pro- ait gression whose common ratio is e ^i'>-a'. a If 5 — -J is negative, the sine must be replaced by its expo- nential value, and the integral becomes X .. ^,/-|.Vf.).^^/-«-V.%)._ where A' and B' are two undetermined constants. The motion is now no longer oscillatory. If a and h are both positive, or if the initial conditions are such that the coefficient of the exponential which has a positive index is 2ero, x will ultimately become equal to r and the system will ultimately continually approach the position determined by this value of x. a' If J — — = 0, the integral takes a different form and we have aj = |+(^"< + 5")e-?, where A" and B" are two undetermined constants. If a is positive, the system will ultimately continually approach the position determined by a; = r . ONE DEGREE OF FREEDOM. 327 When the value of x as given by these equations becomes large, the terms depending on a;* which have been neglected in forming the equation may also become great. It is possible that these terms may alter the whole character of the motion. In such cases the equilibrium, or the undisturbed motion of the system as the case may be, is called unstable, and these equations can represent only the nature of the motion with which the system begins to move from its undisturbed state. d^x dx Ex. Show that the complete solution of -j-j + a-Tr + bx =/(«) is x=e hat I .Bin 67 , ,,, a . ,,, ) 1 /*« -%{t-t') . ,,,, .,..,.,. ,, ^ ' ~ ,' -^,— + .^0 (cos b't + 2^, sm i") + j, j « sm 6' (« - 1') / (f) dt', a? d" where i'* = 6 - — and oja, x'a ai*o the values of x and - ,- "hs^" c -: 0. [Math. Tripos, 1876.] 412. It will be often found advantageous to trace the motion of the system by a figure. Let equal increments of the abscissa 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 F will exhibit to the eye the whole motion of the In the case in which a and h — — are both positive the system. curve takes the form 4 i; t' \'\ » :: i ; at The dotted lines correspond to the ordinate ±Ae ^ . The repre- sentative point P oscillates between these, and its path alternately touches each of them. In just tiie same way we may trace the representative curve for other values of a and b. I'M 328 BMALL OSCILLATIONS. The most important case in dynamics is when a = 0. The motion is then given by x-l=±Asm(^/bt + B). The representative curve is then the curve of sines. In this case the oscillation is usually callea harmonic. 'i 1 418. Ex. 1. A system oscillates about a r^ean position, and its deviation is measorcd by x. If x^ and Xq be the initial valacs of c and ^- , show tbe system will never deviate from its m«an position by so much as f ? 'r\ 9 P if 4& is 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, h, c. Ajiy three determinations of the co-ordinate x at three different times will gene- rally 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 con- veniently observed, because, owing to i;he slowness of the motion, it is diiHcult to dx determine the precise moment at which — vanishes. If three successive values of x thus found be x-^, «j, x^, the ratio ©I th« tw« suc- cessive arcs x^-Xi and x^ - x, is a known function uf a and h and one equation can thus be formed to find the constants. If the position of equilibrium is unknown, c c we may form a second equation from the fact that the three arcs a^-r, a^-ri x^-rr also form a geometrical progression. In this way we find t whieh ffi the value of X corresponding to the position of equilibrium. The position of equilibrium being known, the interval bet ffeen two successive passages of the system through it is also a known function of a and &> and thui) 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 arep, q, r; prove that the co-ordinate of the position of equilibrium and the time of vibration if there were no resistance are respectively f^S,r''V*^^{'<^^V ■ [Math. Tripos, 1870.] ONE DEGREE OF FREEDOM. 829 414. When the coefiSoients are functions of the time, the equation can be integrated only by some artifice suited to the particular case under consideration. Let the equation be d*x dx then a few nseful methods of solution will be indicated in the following examples. Ex. 1. n^ I dp If g-j-g-^isa positive constant, viz. n', prove that the successive oscillations of the system will bo performed in the same time, though the extent of the oscillations may follow any law. This may be proved by clearing the equation of the second term in the usual way, i.e. put x=^e-^-^'^' o d 1 Ex. 2, If r=0 and -.z. - y. — r=a, where a is a constant> prove that V2 ''^ Vg x^e'^^y^^A sin j ^1 - jf*s/qdt + b\ . Thence show that if / i^Jqdt does not become infinite, the time of oscillation is independent of the arc of oscillation but the successive oscillations are not per- formed in the same time. This may be proved by writing «=^(\), and then so choosing the form of \p that the coefficient of x in the differential equation becomes unity or some constant. Ex. 3. A system oscillates about a position of equilibrium and its motion is determined by the equation ■-^ + qx-0, whore g is a known function of t, which during the time under consideration always lies between /S' and /y, the latter being the greater. If the system be started with an initial co-ordinate tK, and an initial velocity Xq in a direction away from the position of equilibrium, show that the system will begin to return before x becomes so great as */ x^' + "^j-. If ± »i, TWl ^. be two ^ccesfiive maximum values of st^ prove that m' cannot be bo great as ^- m, and that the time trom one maximum to the next lies between ^ and -^ • P P 415. When the arc of oscillation is not small, the equation cannot always be reduced to the linear form, and no general rule can be given for its solutiooy In many caaes it is important to ascertain if the mo ion of the system is tautochro- nous. Various methods of determining this will be shown iu the following examples. Ex. 1. Show that if the equation of motion be dt« = ( a homogeneous function of -j- and x of the first degree) , then, in whatever position the system is placed at rest, the time of arriving at the position detei mined by iii;=0 is the same. • i I; 1: > \'l i. ■ 1 ; , i |.:l I ' ? I :|' 1 1 1 ■ 1 1 ' I i^^; i I «■' I ill ^ - A.'] !i ■.)■ 1;H 11 I! I < I 1 Y .i! 330 SMALL OSCILLATIONS. Let the homogeneous function be written ^f {--/}] • ^^^ ^ ^'^^ f ^^ t^^ co- ordinates of two systems starting from rest in two different positions, and let a! = a, f = (ca initially. It is easy to see that the differential equation of one system is changed into that of the other by writing | = Kir. If therefore the motion of ono system is given by x — (t, A', B'). To determine the arbitrary constants, A, B and A', B', we have exactly the same conditions, viz. when t = 0, tj> = a and ~- ■■ =0. Since only one motion can follow from the same initial conditions wo have A'=A, and B'—B. Hence throughout the motion }^ — kx and therefore as and { vanish together. It follows that the motions of the two systems are perfectly similar. This result may also be obtained by integrating the differential equation. If we put ";^=p, we find x=A(t>{t + B). When ( = 0, -;,=0, and therefore 0'(«) = O. 30 (tt etc Thus B is known and x vanishes when (p(t + B) = whatever be the value of A . Ex. 2. If the equation of motion of the system bo = - ( -r^ I >V r + ( ^ homogeneous function of — and/(.c) of the first degree j, where /(«) is any function of x, show that in whatever position the system is placed the time of arriving at the position determined by a; = is the same. This is Lagrange's general expression for a force which makes a tautochronoua motion. The formula was given by him in the Berlin Memoirs for 17C5 and 1770. Another very complicated demonstration was given in the same volume by D'Alem- bert, which required variations as well as differentiations. Lagrange seems to have believed that his expression for a tautochronous force was both necessary and sufficient. But it has been pointed out by M. Fontaine and M. Bertrand that though sufficient it is not necessary. At the same time the latter reduced the demonstration to a few simple principles. A more general expression than Lagrange's has been lately given by Brioschi. In practice it will be more convenient to apply Bertrand's method than d'X I (2'jc\ Lagrange's rule. Suppose the equation of motion to be - = i^* f ar, y J . Put x—ipiy) and if possible so choose the form of <(>, that "^ becomes a homogeneous function of y and ~ of the first degree. If this can be done, the motion is, by Ex. 1, tautochronous. Ex. 3. 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 propor- tional to the velocity. This theorem is due to Lagrange. Ex. 4. A particle, acted on by a repulsive force varying as the distance and tending from a fixed point, is constrained to move along a rough curve in a medium rorsisting as the velocity, find the curve that the motion may be tautochronous by Lagrange's rule. Let V bo the velocity, s the arc to be described, r the radius vector of the particle, it the perpendicular on the tangent, p the radius of curvature. Let ar bo FIRST METHOD OF FORMING THE EQUATIONS OF MOTION. 331 the repulsive force, b the coeflSoient of friction. Then omitting the resistance by Ex. 3, the equations of motion are j^-ap + It J Eliminating the pressure It, we have -^^ = h~ + ahp-ajr^-p\ By Lagrange's rule, the motion is tautochronous if, when f{s) = ab2>- a Jr'^-p'*, we b f (s) find - = - -J— I . This will be found to give p = {l + b')p, which is on epicycloid. ( I ' .{ 1 First Method of forming the Equations of Motion. 416. 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. First, let the motion be in two dimensions. It has been shown in Art. 175, 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, these perpendiculars must meet in the instantaneous centre of rotation. The equation will, in general, reduce to the form Mh'^ (/*^ _ /moment of impressed forces about\ dt^* \ 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 the left-hand side of the equation contains the small factor -j.^ we may here suppose the instantaneous centre to have i I ) ! ^i 332 SMALL OSCILLATIONS. its mean or undi&turbed position. On the right-hand side there is 710 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 eqnilibriam in the same position under two different syBtems of forces, and if Lj, 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 111 417. Ex. A homogeneous hemisphere performs small oscillations on a perfectly rough horizontal plane : find tJie motion. Let C be the centre, O the centre of gravity of the hemisphere, N the point of contact with the rough plane. Let the radius = a, CG=c, 0=^ NCO. Here the point N is the centre of instantaneous rotation, because the plane being perfectly rough, sulBcient friction will be called into play to keep N at rest. Hence taking moments about N {k^ + GN*)'l^^ = - go. Bine. Binoe we can put GN=a-e in the small terms, this reduces to {i? + (a-ty]^,+gt.0=O. illation is = 2ir a. / ^'- , ^ eg Therefore the time of a small oscillation 2 g It is clear that fc' + c' = (rad.)' of gyration about C= -= a' and c = ^a. o o If the plane had been smooth, M would have been the instantaneous axis, GM being the perpendicular on CN. For the motion of iV is in a horizontal direction, because the sphere remains in contact with the plane, and the motion of is vortical by Art. 79. Hence tho two perpendiculars GM, NM meet in the instanta- neous axis. By reaseniug similar to the above the time will be found to bo ^ eg 418. A cylindrical surface of any form rests in stable equi- librium on another perfectly rough cylindrical surface, the axes 1, riRST METHOD OP FORMING THE EQUATIONS OF MOTION. 383 of the cylindera being parallel. A small disturbavce 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, CA! be normals at those points -4, A' which before disturbance were in contact, and let a be the angle A makes with the vertical. Let OPG be the common normal at the time t. Let Q be the centre of gravity of the moving body, then before disturbance A'O was vertical Let AQ=.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 A OP, A' CP be denoted by 0, ' respectively. Let d be the angle turned round by the body in moving from the position of equilibrium into the position B'A'P. Then since before disturbance, A'G and AG were in the same straight line, we have = ^ CDE= + ', where GA' meets OAE in 3. Also since one body rolls on the other, the arc J.P=arc^'P, .•. p^=p'^', -,e. P+P Again, in order to take moments about P, we require the horizontal distance of Q from P; this may be found by projecting the broken line PA' +A'G on the horizontal. The projection of PA' = PA' cos {oL + 6) = p(j} cos a when we neglect the squares of small quantities. The projection of A'G is rd. Thus the hori- zontal distance required is [ ——> cos a - r j ^. 1 H (I I V r :i :4i 1 ■m'i [■■n-i if II ii \' >i v'l 'i i 334 SMALL OSCILLATIONS. If k be the radius of gyration about the centre of gravity, the equation of motion is (i'+0^')^f = -^s(V^,cosa-r). If L be the length of the simple equivalent pendulum, we have L p + p -, cos a—r. 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 -4^, ^ P P. produced if necessary, cut this circle in N. Then GN= s cos a — r, the positive direction being from N towards A. The length L of the simple equivalent pendulum is given by the formula k' + GA' = GK It is clear from this formula, if G* lie without the circle and above the tangent at -4, X is negative and the equilibrium is * Let Jl be the radius of curvature of the path traced out by G as the one cylinder rolls on the other, then we know that R= - . f >, . so that all points with- NG 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 con- vexity 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 eircle. FIRST METHOD OF FORMING THE EQUATIONS OF MOTION. 335 unstable, if within L is positive and equilibrium is stable, circle is called the circle of stabiliti/. This 419. It may be noticed that the prccodinp; problem is perfectly general and may be used in all cases in wliich the locus of the instantaneous axis is known. Thus p is the radius of curvature of the locus in the body, p that of the locus in space, and a the incli- nation of its path to the horizon. If dx be the horizontal displacement of the instantaneous centre produced by a rotation cW of the body, then the equation to find the length of the simple equivalent pendulum of a body oscillating under gravity may be written L dd This follows at once from the reasoning in Art. 418. It may also be easily seen that the diameter of the circle of stability is equal to the ratio of the velocity in space of the instantaneous axis to the angular velocity of the body. Ex. 1. A homogenoous sphere makes small oscillations inside a fixed sph tg bo that its centre moves in a vertical piano. If the roughness be sufficient to prevent all sliding, prove that the length of the equivalent pendulum is seven- fifths of the difference of the radii. If the spheres were smooth the length of the equivalent pendulum would be equal to the difference of the radii. Ex, 2. A homogeneous hemisphere being placed on a rough fixed plane, which is inclined to the horizon at an angle sin~i — ;- , makes small oscillations in a vertical plane equivalent pendulum is ( 2^/2 Shew that, if a is the radiui of the hemisphere, the length of the 46 J^' 5 ~ 4 i). 420. If the body be acted on by any force which passes through the centre of gravity, the results must be slightly modi- fied. Just as before the force in equilibrium must act along the straight line joining the centre of gravity O to the instantaneous centre A. When the body is displaced the force will cut its former line of action in some point F, which we shall assume to be known. Let -4i*^=/, taking / positive when O and F are on opposite sides of the locus of the instantaneous centre. Then it may be shown by similar reasoning, that the length L of the simple equivalent pendulum under this force, supposed constant and equal to gravity, is given by 1^ + r^ pp fr ^ P + P /+»• where a is the angle the direction of the force makes with the normal to the path of the instantaneous centre. ■• 1 i 1 ; if,. 1. 336 SMAIL OSCILLATIONS. If we measure along the line AG & length AO' so that 111 -T-Ty, = j-75 + -jc'> tlisn the expression for L takes the form i" + r» = G'iV. ^1 ■I f The equilibrium is therefore stable or unstable according as G' lies within or without the circle of stability. 421. Two points k, "& of a body are eonstrained to describe given curves, and the body ia in equilibrium under the action of gravity. A small disturbance being given, find the time of an c '. Since the body may be brought from the position AB into the position A'B' by turning it about through an angle 6, we have -^ '- = ~-?r~- = 0. Also GG' is ultimately perpen- (JA V-tS dicular to 00, and we have GG' = 0G,6i. Also let x, y be the projections of 00' on the horizontal and vertical through 0, Then by projections X cos j + y sin j = distance of 0' from OD = OD . ; 0D.suii.. Then, by M'Laurin's theorem, y=yo+yo4>+yo"-2 + •••' where 3/0', y,," are the values of — , -r~ when = a. But in the position of equili- brium y is a maximum or minimum; .'.yg'^O. Hence the equation of Vis Viva becomes ■4 >\ ere ;ii >t 0, that vity, then ; .-. the idependent the equa- qaautities, lall quanti- iiderations. BtionB, and sample, by iinates x, y makes with f gravity it brium, and I of equili- [)f Vis Viva FIRST METHOD OF FORMING THE EQUATIONS OF MOTION. 339 dx where x^' is the value of -^ when 0=a; differentiating we get d9 (*o''+*»)^t=-fl^VV- ir L be the length of the simple equivalent pendulum, we have where for 9 we are to write its value a after the differentiations have been effected. It is not difficult to see that the geometrical meaning of this result is the same m that given in the last article. This analytical result was given by Mr Holditch, in the eighth volume of tho Cambridge Tramactiom, It is a convenient formula to use when the motion of the oscillating body is known with reference to its centre of gravity, 424. When a body moves in space with one independent motion there is not in general an instantaneous axis. It has, however, been proved in Art. 186 that the motion may always be reduced to a rotation about some central axis and a translation along that axis. Let I be the moment of inertia of the body about the instan- taneous central axis, fl the anfrular velocity about it, Fthe velocity of translation along it, M the mass of the body, then by the prin- ciple of vis viva ^ ICl" + ^ MV'^ = U+ C, where U is the force- function, and C some constant. Differentiating we get dt '^2 dt'^ il dt ~lldt' Lot L be the moment of the impressed forces about the in- stantaneous central axis, then L= ^r-r- by Art. 326. 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 instantaneous axis p = 0, and we see that we may take moments about the instantaneous axis exactly as if it vere fixed in space and in the body. 22—2 i' i; l^'Y ;f^ I ■ m 1 !■ m\9 ■'■■j I n i I . i m lii' 340 SMAWi OSCILLATIONS. Mi ;! 425. Ex. A heavy hody oscillates in three dimensions, with one degree of freedom, on a fixed rouyh surface of any form in such a manner that there is no rotation about the common normal. Find tlie motion. (1) Let be the point of contact, Og the common normal, Oy a tangent to the arc of rolling determined by the geometrical conditions of the question, 01 the instantaneous axis. Then 01, Oy are conjugate diameters in the relative indi- catrix. The relative indicatrix is a conic having its centre at and lying in the com- mon tangent plane at 0, such that the difference of the curvatures of the normal sections through any radius vector OR varies as - .-^ . (2) Let p, p' be the radii of curvature of the normal sections through Oy, taken positively when the curvatures are in opposite directions, and let - = - + - . Then s mny be called the radius of relative curvature. Measure a length « . sin^ yOI along the common normal Oz, and describe a cylinder on it as diameter, the axis being parallel to 01. If the centre of gravity of the body be inside, the equilibrium is stable; if outside and above the plane of xy, imstablo. This cylindei may therefore be called the cylinder of stability. (3) Let G be the centre of gravity, and let 00 produced out the cylinder of stability in V; then if K be the radius of gyration about 01, the length L of the simple equivalent pendulum is given by -^ = GV. sin* GOI. This equation may also be written in tho form -jr =s coa Goz. Bin' yOI- 00 .sin^ GOI. M This result may be obtained by taking moments about the instantaneoiis axia, TiCt 0' be the point of contact, G' the position of the centre of gravity at the time t and let 07' be the instantaneous axis. In the small terms we may con- Hider these as coincident with 0, G and 01 respectively. If be the angle turned round the instantau')ous axis, it may be shown that the arc 00' rolled over is Os^inyOI. Let this be called '.2L^sin^co8^ .(i). d^ "" "^ 3 We have also the geometrical equations x = lcos0j y = lsin0 Eliminating R, K^ from the equations (1), we get rfV d^x . ,^d^0 df y'W^^^^l^~9^'~ ^"^y ~ **' 5 sin ^ COS 6 di dt .(2). .(3). • If a body in one plane be turning about an axis in its own plane with an angular velocity w, 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 G as origin and the axis of y parallel to the fixed axis. Let c be the distance of G from the axis of rotation. Then all the centrifugal forces are equivalent to a single resultant force at G =/(ir' (c + x) dm - ul^ . Mc, since 5 = 0, and to a single resultant couple =/«' (e + x) ijdw., = uPJxydm, since y = 0. ? SECOND METHOD OF FORMINO THE EQUATIONS OP MOTION. 343 To find the position of red. We observe that if the rod were placed at rest in that position it would always remain there, and that therefore •;r^ = 0, -j^ = 0, "rj=0. This dt df gx — (axy de gives 0)* s sin ^ cos ^ = 0. .(4). Joining this with equations (2), we get ^ = ^ , or sin 6 =-t^j, and thus the positions of equilibrium are found. Let any one of these positions be represented by ^ = a, a: = a, y = 6. To find the motion of oscillation. Let x = a + x\ y — h-\-i/, 6 = 0. + ff, where x, y', & are all small quantities, then we must substitute these values in equation (3). On the left-hand side since -^ , — ^, -^ , are all small, we have simply to write a, h, a, for X, y, 6. Ou the right-hand side the substitution should be made by Taylor's Theorem, thus da db dx We know that the lirst term f{a, h, a) will be zero, because this was the very equation (4) from which a, h, a were found. We therefore get ^'^~^Tfi^'^~M='^-' ^^) ^ " ^^li - w Q cos 2a . d . de de de 3 But by putting ^ = a + ^ in equations (2), we get by Taylor's Theorem as' = -- i sin a . ^, y' = ^ cos a . ff. Hence the equation to determine the motion is (P + T^)~ + {ghina + | oiT cos 2a) ^ = 0. 4 Now, if gl sin a + K w'^ cos 2r = w be positive when either of the o two values of a is substituted, that position of equilibrium is stable, and the time of a small oscillation is 27r a/ — — . If n be negative the equilibrium is unstable, and there can be no oscillation. li(o*>~ 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 " ! . II i ■ ; I ■ :m m - »t, u m--m»-^ ir 344 SMALL OSCILLATIONS. 1/1 If ©' < -^ 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 eqailibrium. To determine the small oscillations we must retain terms of an order higher than the first. By a known transformation we have de y d('~dt V dtj' jia Hence the left-hand si'^a of em itioi (3) becomes (i' + A;') t,j . The right-hand side becom, *^^ • tylor's Theorem 50" (^' cos a — sm 1.2 + &C. TT When M = 0, we have a = ^ and <»' = -^ . Making the neces- sary substitutions the equation of motion becomes Since the lowest power of & on the right-hand side is odd and its coefficient negative, the equili' ,/ium is stable for a displace- ment on either side of the position of equilibrium. Let a be the initial value of ff , then the time T of reaching the position of etiuilibrium is j0L*-d' put ^ = a^, then V gl 'J,jl-(b*'a' Jl -<}>* Hence the time of reaching the position of equilibrium varies inversely as the arc. When the initial displacement is indefi- nitely small, the time becomos infinite. Thia definite integral may be otherwise expressed in terms of the Gamma .04 function. It raay be easily shown that Jo 'Ji-'P* 4n/27i 429. 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 mo- ments about Nf we have the equation e neccs- OSCILLATIONS WITH TWO OR MORE DEGREES OF FREEDOM. 345 (f + k') -~ = gl COS ^ _ J^ «« (i + ry 4P sin 6 cos 6 dr = gl cos ^ — -^ to' sin ^ cos ^ =f(e). . Then the position of equilibrium can be found from the equa- tion /(a) = Oand the time of oscillation from the equation ^,^,^^o;^m,_ df doL 430. Ex. 1. If the mass of the rod AB is M show that the magnitude of the couple which constrains the system to turn round Oy with uniform angular velocity is Would the magnitude of this couple be altered if Ox or Oy had any mass ? Ex. 2. The upper extremity of a uniform beam of length 21 is constrained to elide on a smooth horizontal rod without inertia, and the lower along a smooth vertical rod through the upper extremity of which the horizontal rod passes : the system rotates freely about tha vertical rod, prove that if a be the inclination of the beam to the vertical when in a position of relative equilibrium, the angular velocity of the system will be ( /r— "^ — ) . m^^ if t^e beam be slightly displaced from this position show that it will make a small oscillation in the time 47r -y (sec o+ 3 cos a) I 4 [Coll. Exam.]. In the example in the text the system is constrained to torn round the vertical with uniform angular velocity, but in this example the system rotates freely. The angular velocity about the vertical is therefore not constant, and its email variations must be found by the principle of angular momentum. ^-ti-f I' I : I i' ') jl , ,1 I Oscillations with two or more Degrees of Freedom. 431. When the position of a system of bodies depends on several independent co-ordinates, the equations to determine the motion become rather complicated. In order to separate the difficulties of analysis from those of dynamics, we shall consider the case in which the system depends on two independent co- ordinates, though the remarks about to be made will be for the most part quite general, and will apply, no matter how many co-ordinates the system may have. In the sequel we shall con- sider Lagrange's general method of forming the equations when the system has ?i co-ordinates. Iriiii-I i.i , f 1 I 11 I, I 346 SaiALL OSCILLATIONS. 432. The equations of motion of a dynamical system per- forming small oscillations with two independent motions are of the form „cPx df dt de dt A'-.^ + B'^+C'x + F'^UG'^^+H'y = 0. dt de dt To solve these, we eliminate either a? or y ; \i D stand for we have dt' AD" +BD + C, FI)'+ GD + H A'I/ + B'D+C\ rD^+O'D+H' x = 0, with a similar equation for y. If AB stand for the determinant A B . . a' jy this biquadratic becomes, when x is omitted. AFD*+(A G+BF)D'+(AII+Ba+CF)D'+{BH+CG) D+GH=0. If the roots of this biquadratic be m^, m^, m^, m^, we have by the theory of Linear Differential Equations X = Mj^^^* + M^el^ii + M^€l^>^ + M^e"^**, where J/,, M^, Jfg, M^ are arbitrary constants. Similarly we have The ilf's are not independent of the Jf s, for by substituting in either differential equation and taking any M and M as typical of all, {Am^ + Bm+C)M^--{F'm? + Gm-\-E)M'. There are therefore just four arbitrary constants, and these are to be determined by the initial values oi x, y, -jj , -^ . 433. If the position of the system depends on three indepen- dent co-ordinates x, y, z, we shall have three equations of motion similar to the two at the beginning of this article. These may be solved in the same way. In this case we obtain a subsidiary equa- tion of the sixth degree to determine the exponentials which occur in the variables. The relations between the coefficients of corresponding exponentials can be, found by substitution in any two of the equations of motion. In certain cases it may be more convenient to choose x or y to be itself a differential coefficient of a co-ordinate. In this case the biquadratic or sextic equation will reduce to a cubic or quintic. A dt' OSCILLATIONS WITH TWO OR MORE DEGREES OP FREEDOM. 347 ' 434. It appears from this summary that the character of the motion depends on the forms of the roots* of this biquadratic. * If the general character of the motion is required it will be necessary to analyse the biquadratic. Rules by which this is made to depend on a cubic equation are given in most of the books on the theory of equations, but aa the final results are not stated, it will be useful to give here a short analysis for reference. Let the biquadratic be ox* + 46iic» + 6 ~P^l' »/-^' We then have o« ~ 2 I ^=(r^) -v (!>'»+ 2'") J If JT is positive or zero, it is easy to see that K must be negative. If therefore // and K are both positive, the four roots are real, if either is negative or zero, tho four roots are imaginary. If the discriminant A is zero, but / and / not zero, it is known that the biquadratic has two roots equal. If two of the roots are real and equal and the other f ^ \v I I It V ■ ,' i^ 1: >. v..,-. Ill 343 SMALL OSCILLATIONS. If any one of the roots is real and positive, x and y will ultimately become large, unless the initial conditions are such that the term depending on this root disappears from the values of x and ;/. If the roots are all real and negative, the motion will gradually disappear and the system will come to rest at the end of an infinite time. If two of the roots are imaginary, we have a pair of imaginary exponentials with imaginary coefficients, which can be rationa- lized into a sine and a cosine. This rationalization will be however unnecessary if, as usually happens, only the character of the oscil- lations is required. Suppose the roots to be o ± jj VC- 1). we have X = c"' (iVj cos 2yt + N"^ sin pt) + &c., where iVj, N^ are arbitrary constants. There will be a similar expression for y with N' written for N". Thus the period of the . 27r oscillation is — . The oscillation will ultimately become very large or vanish away, according as a is positive or negative. If a = 0, the oscillations will continue throughout of the same mag- nitude. > I: 1-1 i If it be required to find not merely the character of the motion, but also the motion resulting from given initial conditions, it will be necessary to determine the relations between the arl>itrary constants which enter into the expression for x and y. This may be effected very easily in the following manner. Let D^ +fO + be the factor which eqiiated to zero gives the imaginary roots, then /and g are known in terms of a and p. Iict us now substi- tute —fD — g for D* in the two first equations of Art. 432. They reduce to equations of the form dt dt (B;| + a.> + ((j,'^+ff;)2, = o t » Hi - 1 two imaginary, we see by putting g' zero that if IT is positive or zero, A' must bo negative. Hence if // and K are both positive all the roots are real, if // or K is negative or zero, two roots are real and two imaginary. If G is zero, there are then two pairs of equal roots. In this case K is zero, and these roots are all real if // is positive, all imaginary if H is negative. Lastly if A is zero and also both I and J zero. The biquadratic has three roots equal, and therefore all the roots are real. If H=0 also, the four roots are all equal and real. OSCILLATIONS WITH TWO OR MORE DEGREES OF FREEDOM. 349 where B^, C,, &c. are some constant coefficients. Eliminating -~ from these equations, we have an equation of the form where K and L are constants, so that when the two terms of x, which depend on this factor are known, tlie corresponding terms of 1/ can be found immediately. If there be another pair of imaginary roots, wo obtain by a similar process a similar e(juation with different constants for A' and L, to find the corresponding terms in i/. If two of the roots are equal, say m^ = m^, then, by the theory of Linear Equations, we know that where N^ and N,, &c. are arbitrary constants. If three roots are equal, there will be a term with f and so on. The expres- sion for ^ will of course contain similar terms. Let it be The terms containing i as a factor will at first increase with t, and if ??^, is positive or zero will become very great, but if wi, is negative, they will ultimately vanish. The motion will, in the latter case, be stable if the initial increase of the terms is not such that the values of x and y become large, i e, if the system is not at first so much disturbed that the motion cannot be considered as a small oscillation. In some cases the relations between the constants in the ex- pressions for X and y are such that the coefficients of both the terms containing the factor t vanish*. When this occurs the four * To prove this let us find the relations between the constants. Substituting the values of x and y in the two first equations of Art 432, we find {A nti" + Bm^ + N^ = - (Fm^^ + Gm^ + H) N.^, (4 mi" + Z?j»i + C)Ni + (2iwii + B) N^ = - {Fiiii^ + Cfm^+E) iV/ - {2Fmi + G) N^, with two Rimilar equations which may be obtained from these by accenting the letters A, B, C, F, 0, H. If the. ylHii" + 2?% + C - 1 JPmjS + Om^ + // = ) A'mi' + B'm^ + C" = \ ' F'm^^ + U'n\ + i?'= i ' while the two expressions (2^1mi + B) (2 F'm^ + 6') and (24 'nt^ + B) {^Fm^ + G) are unequal, we have N„, N^' both zero, and A',, N^' both arbitrary. If the two expressions just written down were equal also, it may be shown that the biquadratic tij liiid D would have three equal root^. t J ! II i 1'' P' 4 . Ill; i; I i ii V 350 SMALL OSCILLATIONS. arbitrary constants will be JV,, N' il/g and 31 ^. In such cases the motion is stable for all initial conditions. 435. lue most important case is that in which there are no real exponentials in the values of x and y. If AG+ BF and BH-\- CG both vanish, there v/ill be no odd powers in the sub- sidiary biquadratic. The biquadratic may now be regarded as quadratic in L^. If its roots are real and negative, let them be —p^ and — j". The expression for x will then take the form a; = JV, sin (jpt + v^ + N^ sin {qt + v^, where iV^, N^, v,, v^ are arbitrary constants. The corresponding terms in y may be found by the rule just given. Eliminating — between the two given equations of motion, let the result be A'^t^S^+G'x+F^ + H'y^^O. de dt d? df Then writing —p^ for -7-g , we have df C'-Ay X — B dx y~ ' H'-Fp*'" JI'-Fy dt C'-Ay .r - f ., ^ •»> XT / = - w^y ' ^'" ^^ "*" "^^ - W^fy ^' "^^^ ^^* "^ ''»^ C'-AV xr • / *^ N ^^ AT / . N 436. In many cases it will be found impracticable to solve the biquadratic on which the character of the motion depends. If however we only wish to ascertain whether the position of equilibrium, or the steady motion about which the system is iu osr lation, is stable or unstable, we may proceed without solving tht, Diquadratic. With the reservations in the case of equal roots mentioned in Art. 434, the necessary and sufficient conditions for stability are, that the real roots and the real parts of the imaginary roots should be all negative. It is proposed here to investigate a method of easy application to decide whether the roots are of this character. Let the biquadratic be written in the form Let us form that symmetrical function of the roots which is the OSCILLATIONS WITH TWO OR MORE DEGREES OF FREEDOM. 351 product of the sums of the roots taken two and two. If this be called V, we find* a X=hcd-ad^-eV = i 2a h c b d c d 2e It will be convenient to consider first the case in which X is finite. Suppose we know the roots to be imaginary, say o ± pV— 1, and ^±q V^. Then f = 4a/3 {(a + /3)» + (p + qY] {(a + ^)» + (;> - 2/}. Thus, oijS always takes the sign of — , and a + /9 always takes the Of sign of — . Thus the signs of both a and fi can be determined ; and if a, h, X have the same sign, the real parts of the roots are all negative. Suppose, next, that two of the roots are real and two imagi- nary. Writing c[ V— 1 for q, so that the roots are o.±p \f—\ and P + ^'i we find X ~ = 4a^ {[(a + /3)« +/ - g'7 + ^pY)- a * This value of X may be found in several ways more or less elementary. If we substitute J)ssE±Z in the given biquadratic and equate to zero the even and odd powers of Z, we have aZ*+{loaE*+SbE + c)Z*+AE*+bF' + cE^ + dE+e = 0) {iaE + b)Z3 + (iaE^+QbE^-^2cE + d)Z = o\' Rejecting the ix>ot Z = and eliminating Z we have 64o»JS«+ +bcd-acP-eb^=0, where only the first and last terms of the equation are retained, the others not being required for our present purpose. Since x = ^± Z it is clear that each value of E is the arithmetic mean of two values of x. We have an equation of the sixth degree to find E because there are six ways of combining the four roots of the biquadratic two and two. The product of the roots of the equation in E may be deduced in the usual manner from the first and last terms, and thence the value of X is seen to be that given in the text. If we eliminated E we should obtain an equation in Z whose roots are the nrithmetic means of the differences of the roots of the given equation talicu two ftud two. 1 I -1 ■f n 'M I U =M U fi, >i \H i 352 SMALL OSCILLATIONS. );i i Just as before, 0/3 takes the sign of — , and a + ^ takes the sign it b e of . Also, 0^ — q" takes the sign of the last term - of the bi- quadratic. This determines whether /8 is numerically greater or less than q\ If, then, a, h, e, and X have the same sign, the real roots and the real parts of the imaginary roots are all negative. Lastly, suppose the roots to be all real. Then, if all the coefficients are positive, we know, by Descartes' rule, that the roots must be all negative, and the coefficients cannot be all posi- tive unless all the roots are negative. In this case, since X is the product of the suras of the roots taken two and two, it is clear that will be positiv >. Whatever the nature of the roots may be, yet if the real roots and the real parts of the imaginary roots are negative, the biquad- ratic must be the product of quadratic factors all whose terms are positive. It is therefore necessary for stability that every coeffi- cient of the biquadratic should have the same sign. It is also clear that no coefficient of the equation can be zero unless either some real root is zero or two of the imaginary roots are equal and opposite. Summing up the several results which have just been proved, we conclude that if X is finite, the necessary and sicfficient con- ditions that the real roots and the real paHs of the imaginary roots shoidd he negatine or zero are that every coefficient of the biquad- ratic and. also X shoidd have the same sign. Tlie case in which X = does not present any difficulty. It follows from the definition of X, that if X vanishes two of the roots must be equal with opposite signs and conversely if two roots are equal with opposite signs X must vanish. Writing — D for Z) in the biquadratic and subtracting the result thus obtained from the original equation we find bl)^ + dD = 0. The equal and opposite roots are therefore given by Z) = + a/ — j- . If h and d have opposite signs these roots are real, one being positive and one negative. If b and d have the same sign, they are a pair of imaginary roots with the real parts zero. js The sum of the other two roots is equal to — and their product be ad' We therefore conclude that if X = 0, the real roots and the real parts of the negative roots luill be negative or zero if every coefficient of the biquadratic is finite and has the same sign. e sign the bi- ater or le real ive. ill tlio at the 11 posi- '. is the ar that il roots biquad- rms are ^ cooffi- ; is also s either ual and proved, nt con- •?/ roots ^iquad- Ity. It of the if two Writing lit 0. thus Th(! If d b' positive re a pair product and the if every OSCILLATIONS » WITH TWO OR MORE DEGREES OF FREEDOM. 353 If either a or e vanishes, the biquadratic reduces to a cubic. Putting e zero, we have -v, = hc — ad. ad If the coefficients have all the same sign it is easy to see that it is necessary for stability that be — ad should be positive or zero. If a and e be not zero and one of the two b, d vanish, the other must vanish also, for otherwise X could not have the same sign as a. In this case X vanishes, and the biquadratic reduces to the quadratic aD* + cD' + e = 0. As this equation admits of an easy solution, no difficulty can arise in practice from this case. It is necessary for stability that the roots of the quadratic should be real and negative. The con- ditions for this are, ^firstly the coefficients a, c, e must all have the same sign, secondly that c' > 4ae. 437. If the equation on which the motion of the system depends is of the fifth degree, we may proceed in the same way. Let the equation be {D) = aD^ + bD* + cD3 + dD^+sD+f^0, and let us sitppose the coefiBcient a to he positive. Form, as before, the product of the sums of the roots taken two and two. If this , X „ , X= be -ad be-af be -; , we find , , , \, a*' he-af de-cf ' Lot us consider first the case in which X is finite. Suppose that there are four imaginary roots a±p^-l, /3=fc7^-l, and one real root y. Then y has the sign opposite to /, and o/3 takes the sign of X, while 2(a + ^)+7=--. If then / be positive, y is negative. If b be positive and (p ( — ) negative, the root y is numerically less than - , so that a + /3 is negative. If therefore a, b, f, X, and -i>i — j be all positive, a, |3, y will bo all negative. Suppose that there are two imaginary roots aJt^piJ-1, and three real roots /3, 7, 5. Then, if all the coefiicients are positive, /3, y, 5 are nrf^ative, and X takes tlio sign opposite to a; so that, if X be also positive, a, /3, y, S vill bo all negative. Suppose all the roots to be real; then, if all the coefiicients be positive, the roots will be all negative, and not otherwise ; and it is also clear that X, being the product of ten negative quantities, will be positive. In both those cases, if the real roots and the real parts of the imaginary roots be negative, it is clear that <;!> ( — ) must have the sign opposite to a. Conversely, if all the real roots and the real parts of the imaginary roots bo negative, the factors of tho equation, and therefore the equation itself, must havo all tho coefiicieutu of the same sign. R. D. 88 m ■;ti p^ 1 : '■ ' ii 1 ■ t| ] Hi M i u !■ < r *>-,4 iij* SMALL OSCILLATIONS. We therefore conclude that n is necessary and sufficient for stability of equili- brium that every fneffioient of tTie equation, ■ li[BM I'i |l ' i III) I! ^^1 '•111 I t I; I i; i 356 SMALL OSCILLATIONS. taneonn value of the first trigonometrical term, Tims the oscillationa tcill appear to be. hirmomc to the eye, while the apparent mean position will travel Jirst to one side and then to the otJier of tlie real mean. 443. Ex. Investigate the following geometrical construction to represent the motiou i= X = N^ Bin pt + N^ din qt. Let q be gieater than p in the standard case and let x have a sign such that N^ is positive. Describe a circle with centre and radius equal to ^^^iV,. Let another circle with centre C and radius equal to - N^ toucli the first circle externally at a point A. Measure CP equal to AT^ in the direction 00, so that if N^ is negative CP must be measured in the opposite direction. If the second circle be now made to roll on the first, the point P traces out an epitrochoid. If C and P' be cor- responding positions of the centre of the moving circle and the generating point, then the distance of P' from the fixed straight line OA is the value of x, while the angle CO A is equal to pt. Apply this to trace the motion when p and q are nearly equal. The third or Lagrange's metliod of forming the equations of motion. 444. Let a system of bodies be in equilibrium under any con- servative forces. When disturbed into any otlier position let Z7be the force function, 2jrthe vis viva. Let the position of the system be defined by n co-ordinatos 6, (j), &c., which are such that they vanish in the position of equilibrium. Then if the system oscillate about the position of equilibrium, 0, ' + A,,"-\-&c (1). Here the coefficients A^^, &c. are all functions of 6, <}>, &c., and 've may suppose them to Lo expanded in a series of some powers of these co-ordinates, "f the oscillations of the system are so small that we may reject a?! pow is of the small quantities 0, , &c. except the lowest which orcMr, v e may reject all except the con- stant terms of these stvies. Wo shall therefore regard the coeffi- cients A^^, &c. as constants. In the same way we ukiv expand U in &, series of powers of 0, = ^'^'- . The n equations (3) therefore become &c. = &c. (4). These are Lagrange's equations to determine the small oscillations of any system about a position of equilibrium, under any conserva- tive forces, provided the geometrical equations do not contain the time explicitly, and are not functions of the differential coeffi- cients of the co-ordinates. These equations may be obtained in a variety of ways. In many cases it is more convenient to use the process of taking moments and resolving. The advantage of using Lagrange's method is that the whole motion is made to depend on one function only, viz. T^-U. 445. We shall now proceed to the solution of the equations. We notice that these equations do not contain any differential coefficients of the first order. This will be the case when a dyna- mical system oscillates about a position of equilibrium under con- servative forces. This peculiarity greatly simplifies the solution. Instead of using exponentials, as in Art. 432, which (when we want anything moi'e than the periods) have afterwards to be ration- alized, we may now conveniently introduce the trigonometrical expressions at once. Let us then put 6 = L^ sin {ii^ -h a,) -t- L^ sin (j)./ + 7^) + &c. = il/j sin (;?,< -I- a,) -I- M^ sin \i\t + a,) -f &c. ^ (5), &c. = &c. r ,1 '-«MnM|iW* K If fi i n ^V ' Si I' ! ;4l 358 SMALL OSCILLATIONS, which may be written in the typical form 6^ = Z sin (2)t + a), = M sin {jit -f a), &c. If we substitute in equations (4) we have (J„ p' + aj L + (^l,,p'^ + a J Af+ &c. = 01 (0). &c. &c. = EHminating L, M, &c., we have a detcrminantal e(juation A^^p''■Va^„A^,Jy' + a^^,kc. = (7), &c. &c. &c. which, it will be observed, is symmetrical about the leading diagonal. This equation Js of the ?i*'' degree to find JJ^ It will be presently shown that its roots are real. Taking any root positive or negative, the equations (6) determine the ratios of J/, N, &c. to L ; and we notice that these ratios will also be all real. If all the roots are positive, the equations (5) will give the whole motion, with 2m arbitrary constants, viz. Zj, L^...L,^\ a,, a5,....a„. These have to be determined by the initial values of 6, (f), &c,, 6', ^', &c. If any root be negative, the corresponding sine will resume its exponential form, the coefficient being rationalized by giving the coefficient L an imaginary form. That the determinant should contain no odd powers of p is just what we should have expected a piiori. In our preliminary assumption (5) each sine is really the sum of two exponentials with indices of oppc-ite signs. The equation therefore of Art. 432 to determine p shoulu here give pairs of equal roots of opposite signs. The equation (7) may be written down without difficulty as soon as the values of T and U have been expanded in powers of 6' , &c., 6, (fee, respectively. In finding the times of oscillation of a system about a position of equilibrium, it is not necessary to go through all the intermediate steps; we may, if we please, write down at once the detcrminantal equation. The rule will be as follows. Omitting the accents in the expression for' T, and the canstant term in U, equate to zero the discriminant of p'^T + U. 2'he roots of the equation thus formed are the values of p. If we require the motion as well as the periods, we shall require e(|ua- tions (6). But these may be also very simply found in the follow- ing manner. Omitting accents as before and taking any of the values of ^ j^ist found, form the equations* * These cqimtioii.s arc given by Lagrange. ■i ■t \ r« laorange's method. 359 (8). The 0, ^, bo the angles the string and the rod make with the vertical. Proceed- ing as in Art. 136, we find that when powers of and higher than the second are neglected, r=^ m {l^0'^ + 2al0''+ (r f flS) , &c.), prove that rjyj2+ Ui=o, T,p.,^+ u^=o, &c. This follows from equations (8) by Euler's theorem on homogeneous functions. 446. In order to determine the values of p^, it will often be necessary to expand the determinant. This may be done by the use of Taylor's theorem. Let A be the discriminant of T and let IT represent the operation n=a 11 d ilA + «, 11 d dA, + a, m d dA + &C., 23 then the determinant when expanded becomes A^)'-^" + n (A) ^2.»-2 + n* (A)2)2'«-i + . . . = 0. If A' be the discriminant of U and 11' the operation 11 when the great and small letters are interchanged, we may write the equation in the form A' + n' (A')23H n'2 (A')i)* + . . . = 0. When ther<3 are only three independent co-ordinates, we may adopt the notation used in tte chapter on Invariants in Dr Salmon's Conies. IS- ' riil ■.III 111 i' i I ■'V :^ i! i • 360 SMALL OSCILLATIONS. Ex. 1. If a system be in a position of equilibrium, sbow thnt the equi- librium will be stable if - n(A), IFCA), -Il'HA), &o. be all powitive. Firstly, we may show that A is necessarily positive, and secondly that these are then the conditions that the roots of the equation (7) are all real. Ex. 2. If S^ bo the sum of the products of each itth minor of the discriminant A' into the conjugate minor of A, prove that .S'^ is the coefficient of /»*. Ex, 3. The same dynamical system can oscillate about the same position of eciuilibrium under two different sets of forces. If p^, pa... and o-j , (T3 . . . be the jieriods of the oscillations when the two sets act separately, i?i, Ri... the periods when they act together, prove that S „ + S — , = S -^^ . p' a^ ii" This follows from the fact that ri(A) contains A^-^ Ac. only in their first powers. Ex. 4. Two difTeront Kystems of bodies wlien acted on by the same set of forces oscillate in periods p,, pa... and ctj, arj ... If JJj, Eg... be the periods when they are both set in oscillation by the same set of forces, prove that Zp' + So-'^SiJ'. Ex. 5. Prove that the equation giving the periods of the oscillations may bo expressed as a determinant of 2/i rows and columns by using Sir W. Hamilton's equations given in Art. 381. 447. If we refer the motion of the system to any other co-ordinates {, ij, f, i&o. which vanish in the position of equilibrium, it is clear that when d, (f>, ^, &c. aro expressed in terms of ^, <&c. and the squares of small quantities neglected, wo shall have equations of the form .(0). = /"if + ^•^'7 + »'«■'<'• [ &c. =&c. J Now 0, , &c. being expanded in a series of sines as in equations (5) it is clear that f, t), &c. will bo expanded in a series of the same sines but with different co- efficients. Hence the values of p^ found from the determinantal equation will bo the same whatever co-ordinates the system is referred to. The ratio of tho coefficients of the several powers uf p are therefore invariable. If fjL be the determinant of transformation, wo know that A becomes /x^A. Henco all the other coefficients will be altered in the same ratio. The quantities A, 11(A), n°(A), &o. are therefore called the invariants of the dynamical system. 448. To show that the values ofp^ are all real*. Since T is essentially a positive quantity for all values of 6', <^', &c. the coefficients of 6"\ 0'^ &c., viz. A^^, A^, &c., must be all positive. Let us collect ^•--^: ether the terms containing Q'^, 6', and complete the square by adding and subtracting the proper qua- dratic function of 0', i/r', &c. We have This theorem seems to have been first discovered by Sii' \V. Thomson. 1 . I wher^ LA.aRANGB:'S METHOD. S61 u 'It and since A^^ is positive, this transformation is real. In the same way B„ muwst be positive, and we may repeat the process. We thus have where 23 and it is clear that this process may be repeated continually. We may take f, rj, &c. as co-ordinates of the system because they arc; independent of each other and vanish in the position of e(juilibrium. We thus have 2r=r + V"+... 1 2(f^-f^„)=/„r+2/,f^+...|' where /„, /,,, &c. are all real constants. The detorminantal equation now takes the form &c. &c. &c. = 0. When there are only three co-ordinates, this is the discrimi- nating cubic used in Solid Geometry, and we know that its roots are all real. When there are more than three co-ordinates, it is proved in Dr Salmon's Higher Algebra, Lesson VI., that the roots are all real. 449. 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, (f), &c. be changed into ^, rj, &c. by the equations (9) of Art. 447, we observe that 6', ', &c. will Ije changed into f', r)', &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 force function in the form ff !'!f li .' I I ■■( ■ (ii J ru ; i' u :Hm ^. IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I us ■u u 11-25 W 1.4 ISA I 1.6 ^ FhotDgraphic Sciences Corporalion 23 WIST MAIN STRHT WIISTn,N.Y. 14StO (71«)«72-4503 e^ -I ^k I s K i 3C2 SMALL OSCILLA'nONS. These new co-ordinates ^, ij, &c. are called the principal co- ordinates of the dynamical system. A great variety of other names have been given to these co-ordinates ; such as Harmonic, simple and normal co-ordinates. 450. When a dynamical system is referred to principal co- ordinates, Lagrange's equations take the form rf^f ef77 so that the whole motion is given by ^=Esm{pJ; + aJ, rj = F sin (pjt + a^), &c., where E, F, &c., Oj, a^, &c. are arbitrary constants to be deter- mined by the initial conditions and p^ = — 6„, p^ = — b.^, &c. When the initial conditions are such that all the principal co-ordinates are zero except one, the system is said to be per- forming a principal or harmonic oscillation. 451. The physical peculiarities of a principal oscillation are : 1. The motion recurs at a constant interval, i.e. after this interval the system occupies the same position as before, and is moving in exactly the same way. 2. The system passes through the position of equilibrium, twice in each complete oscillation. For taking f as the variable co-ordinate, we see that ^ vanishes twice while p^t increases by 27r. 3. The velocity of every particle of the system becones zero at the same instant, and this occurs twice in every complete dB . oscillation. For -^ vanishes twice while /),< increases by 27r. These may be called the extreme positions of the oscillation. 4. The system being referred to any co-ordinates, 0, <^, yjr, &c., which are all variable, the ratios of these co-ordinates to each other are constant thi'oughout the motion*. For referring to the equations (9) of Art. 447, we see that when r), f are all zero, and only ^ is variable, _

  • , &c., or else there must be an indeterminateness in the coefficients L, M, &c. given by equations (8). Referring the system to principal co-ordinates we see that the first alter- native is in general excluded. If two values of jt>' were equal, say J„ = 6- J, the trigonometrical expressions for ^ and tf have equal periods, but terms which contain < as a factor do not make their appearance. The physical peculiarity of this case is that the system has more than one set of principal, or harmonic oscillations. For it is clear that, without introducing any terms containing the products of the co-ordinates into the expressions for T or If, we may change ^, t) into any other co-ordinates ^,,1;,, which make ^ + »?* = f ,* + Vx^ *he other co-ordinates 5", &c. re- maining unchanged. For example we may put f = ^. cos ol — tj^ sin a and 17 = ^1 sin a -I- 1;^ cos a, where a has any value we please. These new quantities ^j, rj^, ^, &c. will evidently be prmcipal co-ordinates, according to the definition of Art. 449. One important exception must however be noticed, viz. when one or more of the values of p are zero. If, for example, &„ = 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 /JTT fiTT of equilibrium are -^ = 0, — = 0, &c., where U has the value given at the end of Art. 449. These in general require that f, VI, &c. should all vanish, but if 6„ = they are satisfied whatever ^ may be, provided 97, f, &c. are zero. These values of f must however be very small, because the cubes of ^, rj, &c. have been rejected. It follows therefore that there are other positions of equilibrium in the immediate neighbourhood 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 the higher order to obtain an approximation to the motion. 453. The motion being referred to any co-ordinates 6, .., it may be required to find the principal oscillation. This may be done by finding \, /i, &c. in equations (9) Art. 447, by the analy- tical process of clearing the two quadratic expressions of the terms containing the products, in the manner explained in Art. 449. We may also proceed thus. Let the system be performing the principal oscillation whose period is — . Then in the equations (5), Zjj, ilig, &c., Z3, il/3, &c. are all zero, hence 0, , sp-, &c. arc in I: ^'i f t It^ ■ '■ 1 * i'i' Ik h M i , I 364 SMALL OSCILLATIONS. the ratio X,, M^, &c. But these ratios are given by (6) or (8), in the form .(8). j0{p^'T+U)=O, ^^{p,'T+U) = 0,&c where the accents in T have been omitted. These equations give the relations between 0, , \f/ as the Cartesian co-ordinates of a point P, it is clear that the position of P at any instant will give the position of the system. Omitting the accents in T and the constant term in Z7, the equations T=a, 17= -j8, where a and /3 are any constants, represent two quadric surfaces which have their centres at the origin. These have a common set of conjugate diameters wluch may be found by the following process. Let 0, , \jf be the co-ordinates of any point on one of the three conjugates. Then, since the diametral planes of t^<'s point in the two qnadrics are parallel, we have dT _ dU dd' dT MjT = ^dJ7 dT^dU 'd~d, ^ contain only a single trigonometrical term (Art. 450) when the system is performing a principal oscillation, we see that the representative point P moves with en acceleration tending to the origin and varying as the distanc3 there- from. 455. As an example of this geometrical analogy let us consider the following problem. A rigid body, free to move about a fixed point 0, is under the action of any forces and makes small oscillations about a position of equilibrium; find the principal oscillations. Let OA, OB, OC be the positions of the principal axes in the position of equilibrium, OA', OB', OC their positions at the time t. The position of the body maybe defined by the angles between (1) the planes AOC, AOC, (2) the pla'^ds BOG, BOC, (3) the planes GOA, COA'. Let these be called 0, , ^ aro angular displacements of the body about OA, OB, OG. Taking theise as the axes of co-ordinates in the geometrical analogy ; a small displacement of P from the origin to a point 0, f, f represents a rotation of the body about the • I le oommon LAGRANGE'S METHOD. 365 straight line desoribed by P and whose magnitude is measured by the distance traversed by P. U A, B, C he the principal moments of inertia at O, the vis viva of the body is clearly 2T=Aff* + B !1H I- ' ' .'t ' f; 3G6 SMALL OSCILLATIONS. horizontal section of tho oylindor, it is clear that two vertical planes each contain- ing one of the principal or harmonic axes are at right angles to each other. 457. Ex. Show that thd mean kinetic energy of a dynamical system oscillating about a position of equilibrium is equal to the mean potential energy, tho mean being taken for any long period, and the position of eqoilibrium being the position of reference. Befer the motion to principal co-ordinates and let 22'=f'» + »,"' + <&c„ 2{U- f7o)= -Pi^'^-PtW-Ao- Then we find ^=E8in{p{t + aj), ti = F sin {p.2t + a). Substituting these in T and Z/q - 17 we have the instantaneous kinetic and potential energies. Tho means of these are obviously the same, and equal to j (E^Pi^ + F^p^^ + Sec.). If the system remain in the position of eqmlibrium the Hamiltonian character- istic funoLon 5= UqU If the system be disturbed and after any time t again pass through the position of equilibrium, the value of S for these two neighboimng modes of passing from one position to another in the same time must be equal. Hence / (T+V)dt= U^t, i.e. the mean values of the kinetic and potential energies •'0 arc equal. 458. Ex. Find the energy of a dynamical system oscillating about a position of equilibrium referred to any co-ordinates. By referring the system to its principal co-ordinates, we can easily show that the energy is the sum of the energies of its principal oscillations. Let the system be referred to any co-ordinates 9, ip, &o. and let it perform the principal oscillation whose type is, by equation (5), j^ = ^ = &0. = sin {p^t + oi). Substituting in the expression for T, we have T=TiPi^coB!^{pit + aj). Bepeating this for all the principal oscillations, we have kinetic energy = T^pj^ cos^ (Fi* + Oi) + T^p^^ cos' {p^t + Og) + &c. where Ti, Tj, Ac. are the values of 2* when L^, M^, Ac, Xj, M^, &c. are substituted for 0', ', &o. Similarly we find when the position of equilibrium is taken as the position of reference potential energy = - t/j sin' (^, ( + a^) - f/g sin' (j) j< + Og) -H &o. Adding these two, we have by Art. 445, Ex. 3, whole energy = T^p^' + T^p^^+... 459. Ex. 1. A new constraint is introduced into a dynamical system, so that the general co-ordinates 0, ^, &o. are constrained to vary in the ratio I, m, &o. If we put d — lain{p't + a), , &c. which however do not necessarily vanish in the positioa of rest. As in Art. 444, let where J.,^, &c. are functions of $, <^, &c. Since the system starts from rest, 6', ', &c. will all be very small quantities in the be- ginning of the motion. If we reject all powers of ff, 0', &c. except the lowest which occur, we may regard -4„, &c. as con- stants whose values are found by substituting for 0, 0, &c. their initial values. Further, since the initial position of the system is not a position of equilibrium, the first differential coefficients of U with regard to d, , &c. will not be zero. Let the initial values of these differential coefficients be respectively a^, a^, &c. The equations of motion are now AJ"+AJ>" + ...=a^ &c. = &c. From these equations we may determine the initial values of $", \ &c. If X, y, z be the co-ordinates of any particle m of the system referred to any rectangular axes fixed in space, we have, by the geometry of the system, these co-ordinates ex- pressed as known functions of 0, , &c.), we have initially X db' with similar expressions for y and z. The quantities oj", y'\ z" are evidently proportional to the direction cosines of the initial direction of motion of m. In this way the initial direction of motion of every part of the system may be found. Ex. A systeni has three co-ordinates 0, f:^? %9i i i;i t! (I « 368 SMALL OSCILLATIONS. '*'fil. When the geometrical equations contain differential coefficients with regard to the time, or when we do not wish to express T and U in terms of independent co-ordinates, the La- graugian equations must be modified in the manner explained in Art. 38S. The equations (3) of Arc. 444 must be replaced by the equations (4) of Art. 388. Since we reject all powers of the small quantities 6, <^, &c. except the lowest which occur, we may still use the expression for T given in (1) Art. 444, and treat the coefficients as constants. But, in making the position of the system depend on the quantities 0, <^, &c. (Art. 3G7), we may not have used all the available geometrical conditions, and therefore the first powers of 0, ^, &c. in the expansion of U may not be absent. Let U=Uo + a^0 + aj^ + &c. + J a„^+ a^^0 + &c. Also let the geometrical equations which are to be introduced by the method of indeterminate multipliers be H& + K'+... = (A (10), &c. = where E, H, &c. are in general functions of 0, , &c., each of which may be expanded in the form i:=E^ + i;^0 + E^^ The equations of motion of Art. 388 iO A^^0" + &c. = aj + a J- + &c. + \E+ fiH+ &c. A,^0"-^&c. = a, + aJ + &c. + XF + fjtK-^-&eX (11). &c. = &c. Since the system has been disturbed from a position of equi- librium, these equations are satisfied by ^ = 0, ^ = 0, &c. We thus obtain the equilibrium values of X, fi, &c. Let these be \, H^, &c., then 0=a, + \E,+fjL^IT^ + &c.l = &c. J ^12> Let \ = \„ + \j, fi=fi^+fi^, &c. so that X,, /*,, &c. are small quantities of the same order as 0, (f>, &c. The equations of oscil- Ifition then become A^/'+&c. = aJ+&c.+X, {E^0 + E^ + &c.) + \E, + &c. &c. = &c. }...(13). Joining these to equations (10) we have a sufficient number of linear equations to find 0, , &c., X^, /Aj, &c. in terms of t. The solutions of these equations may evidently be conducted as in Art. 445. ENERGY TEST OF STABILITY. 3G9 [ferential t wish to the La- lained in iaced by rs of the we may treat the le system not have efore the )e absent. itroduced ....(10), , each of .. (11). a of eqiii- &c. We these be ,...(12). are small of oscil- ...(13). it number of t The ted as in i The equations will be greatly simplified if the equilibrium values of \, /*, &c. are all zero. This will generally be the case if 6, <^, &c. can be so chosen that the first powers in the expansion of U are absent. In this case E^, E^, &c. disappear from the equations, so that it is unnecessary to calculate the geometrical equations (10) beyond terms of the first order. The coefficients will then be constant, and the equations can be integrated. As explained in Art. 388, we may now reduce the number of variables B, <}>, &c. to the proper number of independent co-ordinates. We may therefore proceed as in Art. 444, without introducing \, /*, &c. into the equations. If, however, we prefer to retain the quantities X , /a,, &c., we Boe by equations (10) and (13) that we may obtain the periods exactly as in Art. 445, by equating the discriminant oi p'T+ U' to zero, where . u'=^u+ \ {E^e + F,+...)+^^ {H^e + iir„<^+ ...) +«&c. The determinant thus obtained has as many rows as there are quantities 0, (f>, &c., \, fi^, &c. The Energy test of Stability. 462. The principle of the Conservation of Energy may be conveniently used in some cases to determine whether a system of bodies at rest is in stable or unstable equilibrium. Let the system be in equilibrium in any position and let V^ be the potential energy of the forces in this position. Let the system b*^ displaced into any initial position very near the position of equilibrium and be started with any very small initial kinetic energy T^, and let V^ be the potential energy of the forces in this position. At any subsequent time let T and V be the kinetic and potential energies. Then by the principle of energy T+ v= 7;+ V, ; (1). Let V be an absolute minimum in the position of equilibrium, so that Fis greater than V^ for all neighbouring positions. The initial disturbed position being included amongst these, it follows that Fj — F, is a small positive quantity. Now the kinetic energy T is necessarily a positive quantity, and since F is > F^, the equation (1) shows that T is < T^ + F, — F„. Thus throughout the subsequent motion the vis viva is restricted 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 that F should become greater than T^ + Fj. These two results may be stated thus. n. D. 24 w 11 I'i '< \\ ■ si -.1 ! I m 'hi ^' i' :"l Irvm [■:r' I !;; i'l I I M i 370 SMALL OSCILLATIONS. If a system he in eqnilihnum in a position in which the potential enerrfy of the forces is a minimum or the work a maximum for all displacements, then the system if slightly displaced will never acquire any large amount of vis viva, and will never deviate far from the position of equilibrium. The equilibrium is then said to be stable. 4C3. If the potential energy be an absolute maximum in the position of equilibrium, V is less than V^ for all neighbouring positions. By the same reasoning we see that T is always greater than 7^^+ V^— V^, and the system cannot approach so near the position of equilibrium that V should become greater than 7\ + V^. 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 could exist, let the system be placed at rest in an extreme position of that oscillation, then the sys- tem will describe that principal oscillation and will therefore pass through the position of equilibrium. But if T^ be zero, V can never exceed V^, and can therefore never become equal to V^. 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 proposition. 4G1<. 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, (j), &c. Then we have 2T=d^ + "+ 2iU-U,)=^h,e' + b,^'+ where b^, h^, &c. are all constants, and U^ is the value of U in the position of equilibrium. Taking as a type any one of Lagrange's equations • .ddT_dT_dU dtdd' dd~dO' we have e"-b,0 = O, • This demonstration is twice given by Lagrange in his Mecanique AnahjUquf. In the form in which it appears in the first part of that work, 7 is expanded in powers of the co-ordinates, which arc supposed very small ; bnt 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 proposition is simplified from that of Lagrange by the use of principal co-ordinates. EXERQY TEST OF STABILITY, 371 with similar equations for , \ , (%c. If J, is positive, this equation will give d in terms of real exponentials, and the equilibrium will bo unstable for all disturbances which affect $, except such as make the coefficient of the term containing the positive exponent zero. If i, is negative, d will be expressed by a trigonometrical term, and the equilibrium will be stable for all disturbances which affect only. In this demonstration the values of b^, \,&c. are supposed not to be zero. If in the position of equilibrium U is a. maximum for all possible displacements of the system, we must have 6,, J,, &c. all negative. Whatever disturbance is given to the system, it will oscillate about the position of equilibrium, and that position is then stable. If Z7 is a maximum for some displacements and a minimum for others, some of the coefficients b^, \, &c. will be negative and some positive. In this case if the system be dis- turbed in some directions, it will oscillate about the position of equilibrium; if disturbed in other directions, it may deviate more and more from the position of equilibrium. The equilibrium is therefore stable for all disturbances in certain directions, and un- stable for disturbances in other directions. If f^ is a minimum in the position of equilibrium for all displacpimento, the coefficients ij, 6 , &c. are all positive, the equilibrium >vill then be unstable for displacements in all directions. Briefly, we may sum up the results thus, The system will oscillate about the position of equilibnum for all disturbances if the potential energy is cu minimum for all dis- placements. It will oscillate for some disturbances and not for others if the potential energy is neither a maximum nor a minimum. It will not oscillate for any disturbance if the potential energy ia a maximum for all displacements. It appears from this theorem that the stability or instability of a position of equilibrium does not depend 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 determine 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 equilibrium is unstable. [Earnshaw^s Theorem.] Let be the position of equilibrium. Ox, Oy, Oz any three rectangular axes, then if V be the potential of the bodies, 6j = — , 62 = -v-^ , h- s^' ^^^ ^^'^^'^ the sum of these is zero, &j, b^, 63 cannot all have the same sign. Ex. 2. Hence show that if any number of particles, mutually repelling each 24—2 1 i M 'i \ i t ) 1 1 1 ( 1 ■ 1 i i 11 li I 'i^ vifc '4 372 SMALL OSCILLATIONS. )! r. I h other, be contained in a vcBSfl, and be in eqnilibrinm, the equilibrium will be unstable udIchr thoy all lie on the containing surface. [Sir W. Thomson, Camb. Math. Journal, 1845.] 405. We may in certain cases apply the energy criterion to determine when a given motion is ntable. Let a dynamical system be in motion in any manner under a conservative system of forces, and let E be its energy. Then J? is a known function of the co-ordinates 6, 4>, Sea. and their first differential co- efficients ef, 0', iS;c. ; this is constant and equal to h for the given motion. Sup. pose that either some or all of the other first integrals of the equations of motion are also known, let these be /"i {e, ff, Sea.) = Cj , F, (», ff*, &c.) = C,, *c. =&c. For the purposes of this proposition, lot us regard 6 and 0', and 4>', &o. as inde< pendent variables, except so far as they are connected by the equations just written down. Then if E be an absolute maximum, or an absolute minimum, for all variations of 0, ff, &c. (those corresponding to the given motion making E con- stant), the motion is stable for all dibturbances which do not alter the constants Ci,C„ Ac. This result follows from the same reasoning as in Art. 462, which we may briefly recapitulate thus. Let as many of the letters as is possible be found from the first integrals in terms of the rest, and substituted in the expression for E. Let ^, \(/', &o. be these remaining letters, then we have •E = / (^, f , &c., Ci , C, , &c.) = h. Let the system be started in some manner slightly different from that given, then the constant k is altered into h + S/t. First let f be a minimum along the given motion, then any change whatever of the letters \j/, f, ko. increases E, and it follows that the disturbed motion cannot deviate so far from the given motion that the change in E becomes greater than Sh. Similarly, if £ be an absolute maximum, the same result will follow. The same argument will apply to any first integral of the equations of motion, besides the energy integral. If any one of the functions F,, F^, Ac, which con- tains all the letters, be an absolute maximum or miuimiim, tlion the motion is stable for all displacements which do not alter the constants of the other integrals used. When the system is disturbed from a position of equilibrium which is defined, as in Alt. 444, by the vanishing of the co-ordinates 0, , &o., we have E=^Aiie'^ + Ai^B'4>' + &c.~ U, where A^^, Ai^, &o. are all constants, and U is independent of ff, ', &o. Here the terms which constitute the kinetic energy, being necessarily positive and vanishing with ff, ' &c., are evidently a minimum for all variations of ff, 0', &c. We see, without the use of any other integrals, that if - Z7 be a minimiim for all variations of 6, , &c., £ will be an absolute minimum, and that therefore the eqiiilibrium is stable. 466. It often happens that the expression for the energy is not a function of some of the co-ordinates, though it is a function of the differential coefficients of all the co-ordinates with regard to the time. When this is the case, the system admits of what we shall call a steady motion. Let x, y, &o. be the co-ordinates which ore absent from the expression for the energy E, and let ^, ■>!, &o. be the ilNEROY TEST OF STABILITY. 373 renifiininr; co-ordinates, then E in & function of f, 17. Ac, f, V. Ac., x', y', Ac. If we form the 0(iuation8 of motion by Lagrange's rule (Art. 309), these equations will contain (, ri, i',if\ f", V'. x',y', x'\if, Ac, Ac. Since those equations do not contain t o\i»licitly, they may be satisfied by putting x'=^a, rj = h, Ac, f = a, i;-/3, Ac, where a, b, Ac, a, /3, Ac. are constants to be determined by substituting in the equations. If 6 stand for any one of the co-ordinates, it is evident that ,^ and ..,, will both bo constants after the substitution is made. The constanta av ad must therefore satisfy the tj-pical equation -— j^ — ' = (Art. 3C9). Siuce «, y, Ac. da are absent from the expressions for T and U, this is an identity if we write any of these co-ordinates for 0. Hence we ^ave as many equations, viz. d(T+U)_ d{T+U) 0). fts there are co-ordinates (, 17, Ac. present in the expressio is for T and U. The quantities a, b, Ac. are therefore undetermined except by the initial conditions, while a, ff, Ac. may be found in terms of a, b, Ac by these equations. These equations may be conveniently remembered by the following rule. In the Lagran- gian function, which is the difference between the kinetic and potential energies, trrite for the differential coefficients, their assumed constant values in the steady motion, viz. x'— a, &o., ^=0, &o. Differentiating the result partially with regard to each of the remaining co-ordinates, we obtain the equations of steady motion. 467. To determine if this motion is stable, we must by Art. 465 use the integrals Let -r-7=«, x-/ = ''i &c., where «, v, Ac. ore constants, dx dy T = ^ {XX) x'^ + (a;^) x'f' -f Ac. .(2), where the coefficients of the accented letters, viz. the quantities in brackets, are all known functions of {, 17, Ac, but not of x, y, Ac. The integrals may then be written in the form (sKc) a;' -I- (a;y) y -f . . . = u - (a;f ) f ' - (a!ij) V - &c. I («y)a!'+(yy)y'+- = ''-(ys^)£'-t'/'»)V-&4 (3). Ac. =Ac. 7' -Ac J For the sake of brevity, let us call the right hand sides of these equations u-X, v-Y, Ac Since T is a quadratic function of the accented letters, we may write it in the form T=lmr+{iv)^r,' + &o. + lx'{u+X) + ly'{v+Y) + &o. If we substitute in the terms after the first Ac. the values of x', y' given by (3) we obtain the determinant 2A 0, u-\-X, v+Y, Ac u-X, (xx) {xy), Ac. v-Y, {xy), {yy), &o. Ac where A is the discriminant of T, when {', ij', Ac. have been put zero. If we change the signs of A', Y, Ac, this determinant is unaltered, hence when expanded such terms as uX, vX, Ac. cannot occur. If therefore, we put •i! i t 1 ! \ 1 if !! f- i , 'i I m *., :«i m m ■ fl;: 1' ■| ■I 1|i. ■ .*^r ' . I i I i 374 SMALL OSCILLATIONS. F= 1^ 2A M W u (xx) (xy) .(4), aud expand the first determinant, we have (5), vhere the terms after F express some homogeneous quadratic function of (', if, &o. When f , 7)', &e. are prt zero, the process of finding F is exactly that described in Art. 378, as the Hamiltonian method of forming the reciprocal function. Following the same proof* as in that Article, we may show that if ^ be any letter JTT JET contained in T, we have ^ = - j^ • Hence the equations of steady motion (1) may also be written in the fjrm d{F-V) ._rHF-U) ''- du ' d{F-U) dr, =0 y = d(F-V) do (0). where F - U is the energy expressed rs a function of u, v, &c. instead of x', y', Ac, the other accented letters, viz. ^', rj, &c. heing put equal to zero either before or after differentiation. Further T is essentially positive for all values of a/, y', &c. and therefore for such as make m, v, &o. all zero. Hence the quadratic expression Bu^'* + &c. is a minimum when ^', n)', &c. are zero. If then the function F -TJ is a minimum for all variations of f, ij, &c., the steady motion given by (6) is stable for all disturbances which do not alter the momenta u, v, &c. 468. If the energy be a function of one only of the co-ordinates, though it is a function of the differential coefficients of all of them, we may show conversely that the steady motion will not be stable unless F -^ U wo minimum. Let { be this single co-ordinate, then following the same notation as before, we have by Vis Viva Is^.^'^ + F-U^h. Differentiating with regard to t, and treating J5n as constant because we shall neglect the square of f*, we obtain * Taking the notation of Art. 378, the proof is as follows. The total differential of T^ when all the letters vary is ^-^^de-'^l^- do t/| dT,= -'^de- -— » di+(-~^ + tA dff+ 6'du + &c. ; as before, the quantity in brackets vanishes, and hence when T, is expressed as a (IT function of 9, , &o., w, v, &e. and {, wo have --.'= "4 dT^ di ' w, (5), )f {', V. &0. it desoribed 1 function, e any letter ion (1) may (fi), fx',y',&o., '.r before or herefore for '» + &c. is a mum for all iiattirbances ough it is a versely that i before, we se we sball differential rcsBcd as a . OSCILLATIONS ABOUT STEADY MOTION. To find the oscillation, let f =a + p, then by (6) we have 3V5 d^p r dHF-U) -t ^^^dr^ + l—de~V' where a is to ,be written for { after differentiation in the quantity in square brackets. The motion is clearly stable or unstable according as the coefiSciont of p is positive or negative, i.e. according as i^- U^ is a minimum or maximum. 469. Ex. 1. Let us consider the simple case of a particle describing a circular orbit about a centre of attraction whose acceleration at a distance r is /ur". If 6 be the angle the radius vector r makes with the axis of x, we have her« a steady motion in which /=0 and ^ is constant. Also 1 ur" n + i* We notice that is absent from this expression, hence by the rule we eliminate 0' also by the integral rW=h, where h is the constant called u in Art. 467. We have then 1 ,„ 1 /i" /tr"+i £=tir" + -■-,, + ' 2r^ ' n + 1 Putting the remaining accented letters equal to zero according to the rule, we have in steady motion dr J..1 <- ' and since this steady motion is stable or unstable according as n + 3 is positive or negative for all disturbances which do not alter the angular momentum of the particle. Ex. 2. Taking the example considered in Art. 374, show that a state of steady motion is given by $ constant and that it is stable if C^ii^ + iMgJiA cos d is positive. Hence ii d < ^ the motion is stable for all values of n. Ex. 3. A solid of revolution moves in steady motion on a smooth horizontal plane, so that the inclination of its axis to the vertical is constant. Prove that the angular velocity fi of the axis about the vertical is given by Cn Mg dz '* Adoso'^'^ ABVuOao^edd =0, where z is the altitude of the centre of gravity above the horizoital plane, n the angular velocity of the body about the axis, C, A and A the principal moments of inertia at the centre of gravity and M the mass. Find the least value ol n which makes /x real and determine if the steady motion is stable. Oscillations about Steady Motion.' 470. The oscillations of a system about a state of steady motion may be found by methods analogous to those used in the oscillations about a position of equilibrium. Let the general equa- tions of motion of the bodies be formed by any of the methods already described. If any reactions enter into these equations it 4 ■ m ''if Wi Ml I ! \ 1 ' : 1 376 SMALL OSCILLATIONS. will be generally found advantageous to eliminate them as ex- plained in Art. 428. Let the co-ordinates used in these equations to fix the positions of the bodies be called 0, <^, &c. Suppose the motion, about which the oscillation is required, to be determined by =f\t), = F(t), &c Then exactly as in Art. 428, we substi- tute =f[t) +x, ^ = F{t) + y, &c., in the equations of motion. The squares of x, y, &c. tjeing neglected, we have certain linear equations to find x, y, &c. These equations can, however, seldom be solved unless we can make t disappear explicitly from them. When this can be done the linear equations can be solved by the usual known methods, and the required oscillations are then found. In what follows we shall first illustrate the method just de- scribed by forming the equations in a few interesting cases from the beginning. We shall then generalize the process and obtain a determinantal equation analogous to that given by Lagrange for oscillations about a position of equilibrium. This equation will be adapted to all cases which lead to differential equations with constant coefficients. 471. Ex. 1. To find the motion of the balls in WatVs Oovernor of the steam engine. The mode in which this works to moderate the fluctuations of the engine is well kno\vn. A somewhat similar apparatus has been used to regulate the motion of clocks, and in other cases where uniformity of motion is required. If there be any increase in the driving power of the engine, or any diminution of the load, so that the engine begins to move too fast, the balls, by their increased centrifugal force, open outwards, and by means of a lever either cut off the driving power or increase the load by a quantity proportional to the angle opened out. If on the other hand the engine goes too slow, the balls fall inward, and more driving power is called into action. In the case of the steam engine the lever is attached to the throttle- valve, and thus regulates the supply of steam. It is clear that a complete adapta- tion of the driving power to the load cannot take place instantaneously, but the machine will make a series of small oscillations about a mean state of steady motion. The problem to be considered may therefore be stated thus : — Two equal rods OA, OA', each of length I, are connected with a vertical spindle by means of a hinge at which permits free motion in the vertical plane AOA'. At A and A' are attached two balls, each of mass m. To represent the inertia of the other parts of the engine we shall suppose a horizontal fly-wheel attached to the spindle, whose moment of inertia about the spindle is /. When the machine is in uniform motion, the rods are incIiueJ at some angle a to the vertical, and turn round it with uniform angular velocity n. If, owing to any disturbance of the motion, the rods have opened out to an angle Q with the vertical, a force is called into play whose moment about the spindle is - /3 (0 - a). It is required to find the oscillations about the state of steady niotiou. Let be the angle the plane AOA' makes with some vertical piano fixed in space. The equation of angular momentum about tlio spindle is l^^,2,n„^0)'^. ■P(0-a). (1). THE GOVERNOR. 377 where mk^ is the moment of inertia of a rod and ball about a perpendicular to the rod through 0, the balls being regarded as indefinitely small heavy particles. The semi Vis Viva of the system is and the moment of the impressed forces on either rod and ball about a horizontal through perpendicular to the plane A OA' is g -j^ = -mghBin 0, where h is the dis- tance of the centre of gravity of a rod and ball from O. Hence by Lagrange's d dT dT dU dt ' where a has been written for sin cos 6 i2 m-i'^' (% This equation might also have been obtained by taking the acceleration of either ball, treated as a particle, in a direction perpen- dicular to the rod in the plane in which d is measured. To find the steady motion we put 0=a, 37=n> ^^^ second equation then gives n» cos tt = - . To find the oscillations, we put d = a + x,-^=n-\-y. The two equa- tions then become (1+ 2mlc^ sin« a) :^ + 2mJfln sin 2a -j- = - fix * ' /** dt dt • n sin 2oy = ( n* cos 2a - - cos o j « To solve these equations, we must write them in the form (sin2a2> + 2-'|^) nx + (^-^^ sin»a)z>,=0) ^ (Z)' + n* sin* a)x-n sin 2ow = 0.' (Z)' + n* sin* a)x-n sin 2oy = ( where the symbol D stands for the operation ^ . Eliminating y by cross multipli- cation we have \_\2mh ^^.^ + sin''a)D'> + n*sin''a^l + 3cos«a + 2^)D + 2l«F^"""^"]*=^- The real root of this cubic equation is necessarily negative because the last term is positive. The other two roots are imaginary because the term D* has dis- appeared between two terms of like signs. Also the sum of the three roots being zero, the real parts of the two imaginary roots must be positive. Let these roots therefore be - 2p and j)±:qj -1. Then X = He-^»* + AV sin (g« + L), where H, K, L are three undetermined constants depending on the nature of thg initial disturbance. Thus it appears that the oscillation is unstable. The balls will alternately approach and recede from the vertical spindle with increasing violence. i' :fi X js V l\ i: / if PlA\ "Ay I: I '% m ^ ;'i I •! 1 378 SMALL OSCILLATIONS. 472. A common defect of governors is that they act too quickly, and thua produce considerable oscillation of speed in the engine. If the engine is working too violently, the governor cuts off the steam, but owing to the inertia of the parts of the machinery, the engine does not immediately take iip the proper speed. The consequence is that the balls continue to separate after they have reduced the supply of steam to the proper amount, and thus too much steam is cut off. SimQar remarks apply when the balls are approaching each other, and a con- siderable oscillation is thereby produced. This fault may be very much modified by applying some resistance to the motion of the governor. In the same way when the motion of clock-work is regulated by centrifugal balls, it is found as a matter of observation that there is a strong tendency to irregularity. If the balls once receive in the slightest degree an elliptic motion, the resistance p {0-a) by which the motion of the balls is regulated may tend to render the elliiiso more and more elliptical. To correct this some other resistance must be called into play. This resistance should be of such a character that it does not affect the circular motion and is only produced by the ellipticity of the movement. One method of effecting this has been suggested by Sir G. Airy. The elliptic motion of the balls may be made to cause a slider on the vertical spindle to rise and fall. If this be connected with a horizontal circular plate in a vertical cylinder of slightly greater radius, and filled with water, the sUder may be made to move the plate up and down by its osl Illations. Thus the slider may be subjected to a very great resistance, tending to diminish its oscillations, while its place of rest, as depending on statical, or slowly altering forces, is totally un- affected. Memoirs of the Astronomical Society of London, Vol. xx., 1851. Tho general effect of the water will bo to produce a resistance varying as the velocity, and may therefore be represented by a term -y-fr on the right hand of equation (2). form rit The solution beuig continued as before, the cubic will now take the /3 If the roots of this cubic are real, they are all negative, and the value of x takes the form x = Ae-''^ + Be-''i+Ce-''*, where -X, -/i, -v are the roots, and A, B, C are three undetermined constants. If one root only is real, that root is negative, ana if the other two be jp ± g v' - 1 tho value of X takes the form X = lie. - ♦■« + Ke^t sin (2< + L) , where 11, K, L as before are undetermined constants. In order that the motion may be stable it is necessary that p should be negative. The analytical condition* of this is • If the roots of the cubic aji? + bx'^ + cx + d=Ohex=a:i^PyJ{-l) and y, we have -- = 2a + 7,-=27a + a2 + |3=, _ - == (aS + /S^) 7, whence wo easily deduce ^-^~ = - 2a{(a-f7)* + /32}; hcuco be - ad and a have always opposite signs. See Art. 436. ' * • THE GOVERNOR. 3^ (9 7(l + 3coB«a + 2-i;j^,)>2,^^, 2coto, If 7 be sufficiently great this condition may be satisfied. The uniformity of motion of the rods round the vertical will then be disturbed by an oscillution whose magnitude is continually decreasing and whose period is — . By properly choosing the magnitude of I when constructing the instrument, the period may sometimes be so arranged as to produce tlie least possible ill effect. If the period bo made very long the instrument will worlf smoothly. If it can be made very short tliere will be less deviation from circular motion. In tliis investigation no notice has been taken of the frictions at the hinge and at the mechanical appliances of the Governor, which may not be inconsiderable. These in many cases tend to reduce the oscillation and keep it within bounds. 473. In the case of Watt's Governor if any permanent change be made in the relation between the driving power and the load, the state of uniform motion which the engine will finally assume is different from that which it had before the change. Thus, when the engine is driving a given number of looms, let the rods OA, OA' of the Governor be inclined to each other at an angle 2a and be revolving about the vertical with an angular velocity n. If some largo number of the looms is sud- denly disconnected from the engine, the balls will separate from each other, and the rods will become inclined at some other angle 2a'. In this case, if n' be the angulir velocity about the vertical, n'*cos a' = Mucosa. The rate of the engine is therefore altered, it works quicker with a leua load than with a greater. This is a great defect of Watt's Governor. For tliis reason it has been suggested that the term Governor is inappropriate, the instrument being in fact only Vi, moderator of the fluctuations of the engine. This defect may be considerably decreased by the use of Huyghens' parabolic pendulum. In this instrument the centres of gravity ^ , ^4' of the balls are made to move along the arc of a parabola whose axis is the axis of revolution. Let AN ho an ordinate of the parabola, A the normal, then NG is constant and equal to L, where 2Z is the latus rectum. Regarding the balls as particles, and neglecting the inertia of the rods which connect them with the throttle valve, we see by tlio triangle of forces that the balls will rest in any positions on the parabola, if n^L=g, where n is the angular velocity of the balls about the vertical through 0. It is also clear that when the angular velocity is not that given by this formula, the balls ^unless placed at the vertex) must slide along the arc. Let us now consider how this modification of the governor affects the working of the engine. When the load is diminished the engine begins to quicken ; the balls separate and the steam is cut off. It is clear that equilibrium will not be established until the quantity of steam admitted is just such as to cause the engine to move at exactly the same rate as before. Ex. Show that when the inertia of the rod and ba^^s are taken account of, the centre of gravity of either ball and rod must be constrained to describe a parabola whose latus rectum is independent of the radius of the ball, if the Governor is to cause the engine always to move at a given rate. 474. The reader who may be interested in the subject of Governors may refer to an article by Sir G. Airy, Vol. XI. of the Memoirs of the Astronomical Society, 18-10, where four different constructions are considered Ho may also consult an ' \' I 111 ; f P.n Il ll \ * 380 SMALL OSCILLATIONS. article by Mr Siemens in the Phil. Tram, for 1866, and a brief sketch of several kinds of governors by Prof. Maxioell in the Phil. Mag. for 1868. An account of some experiments by Mr Ellery, on Huyghens* paraboUo pendulnm, may be found in the A8tro7wmical Notices for December, 1875. 475. Ex. 2. It has been shown in Art. 282 that if three particles be placed at the comers of an equiangular triangle and properly projected, they will move under their mutual attractions so as always to remain at the angular points of an equi- lateral triangle. These we may call Laplace's three particles. It is our present object to determine if this motion is stable or unstable*. Let the mass M of the particle to be reduced to rest be taken as unity, and let m, m' be the masses of the other two. Let r, r', R be the distances between the particles Mm, Mrri, mm'; and let 0', ^, \j/ be the angles opposite to these distances. If 0, d' be the angles r, r' make with a straight line fixed in space, and if the law of attraction be the inverse xth power of the distance, the equations of motion are ' rdt\ dt) \b m' cos d) „ R" m' sin \f/ m' sin (f> ~~r^ ^^ =0 'I with two similar equations for the motion of nt'. Let us now put r=a+x, r'=a+x+ X, and let the angle between these radii vectores be ^ + T, also let 0==nt+y, where x, y, X and Y, are all small quantities whose squares are to be neglected. It should be noticed that a variation of x, y alone, X and Y being zero, will represent a variation of steady motion in which the particles always keep at the corners of an equilateral triangle, while a variation of X, Y will represent a change from the equilateral form. The former of these we know by Art. 282 is a possible motion, hence the equations can be satisfied by some values of x, y joined to X=0, Y—0. By this choice of variables we may hope to discover some roots of the fundamental determinant previous to expansion, and thus save a great amount of numerical labour, tions will now become If D stand for ^ , the four equa- l6Z)'- (K+l)(l + TO + m')U-2a6»i)i/-|»i,'(K + l)2C- jmV + l)ai'=0, V3 3 1 2hnBx-k- a62)>--^-m'((c + l)X+jm'(K + l)ar=0, 6Z)»-(K + l)(l+OTH-j>i')|«-2«6>i2)i^+}62)--(/c+l)(l+| + m')jX-J2a6wZ)+^m(/c + l)ajr=0, 2hnDx+ abD''y+ \2bnD-^~(K+l)mlx+ |a62)2- Jm(K + l)aj F.-^O. * In a brief note in JuUien's Problems, Vol. ii. p. 29, it is mentioned that this question has been discussed by M. Gascheau in a These de M^canique, the particles being supposed to attract each other according to the law of nature. The result arrived at is that the motion is stable when the square of the sum of the masses is greater than 27 times the sum of the products of the masses taken two and two. No reference is given to where M. Gascheau's work can be found, and the author is therefore unable to give a description of the process employed. these radii Laplace's three pauticles. 381 476. To solve tbese we put a; = .4 e^', y = Be**, X=Gc'^', r=i7e^'. Substituting and eliminating the ratios ot A, B, and H we obtain a dcterminantal equation whose constituents are the coefficients of x, y, X and 1" with X written for D. This equation will give six values of \. We see at once that one factor is \, This m:ght have been expected, because we know that a variation of y with x, X and Y all zero, is a possible motion. Again, some variation of x and y with X and Y both zero is also a possible motion, hence some factor of the determinant can be found by ex- amining the first two columns. By subtracting from the first 2n times the second column we find that this factor is 6\* - (k - 3)(1 + m + m')=0. To find the other factors we divide the determinant by the factors alrea-ly found. Then subtracting the first row from the third and the second from the fourth wo have three zeros in the first column and two in the second. The expansion is then easy. We see that there is another factor X, also 6«XH 6X''(3 - /c)(l + m + m') + 1(1 + K)2(m + m' + mm') = 0. The two zero roots give x=Ai + A^t with similar expressions y, X and Y. But K + 1 A by substitution in the equations of motion we see that x=A^, y—Bj^ — x- * nt, X=0 and F=0. These roots therefore indicate merely a permanent change in the size of the triangle. On examining the other values of X*, we find (1) The motion cannot be stable unless k is less than 3. (2) The motion is stable whatever the masses may be, if the law of force be expressed by any positive power of the dis- tance or any negative power less than unity. (3) The motion is stable to a first approximation if (Af+wi+m'js :-GHy. Mm + Mm' + mm' where M, m, m,' are the masses. To express the co-c*dinates in terms of the time, we must return to the diffe rential equations of the s> cond order. The results are rather long, and it may be Ki>f .' lient to state that when, as in the solar system, two of the masses are much smallev than the third, the inequalities in their angular distances, as seen from the large body, have much greater coefficients than their linear distances from the same body. 477. To form the general equations of oscillation of a dynami- cal system about a state of steady motion. Let the system be referred to any co-ordinates 6, <^, ■^, &c. Let the state of motion about which the system is oscillating be determined by 6 =f (t), = F (t), & c, then as explained in Art. 470 we shall put d=f{t)+x, ^ = F't)+y, &c. Let the Lagrangian function L (see Art. 381) be exjjanded in powers of x^ y^ &c., as follows : i = i„ + A^x + A^y + &c. 4- B^x + B^' + &c. ^ \ {A,,x- + ^A,,xy + &c.) + \ {BJ' + ^IBJy' + &c.) + C.^xx + C^^xy + C^^yx + &c. ■ i I > I ! 1 ; i I 1 si M j lit' . J f^ i. III I- 'm 382 SMALL OSCILLATIONS. We shall now define a steady motion to be one in which all the coefficients in this expansion are independent of the time. The physical characteristic of such a motion is that when referred to l^roper co-ordinates the same oscillations follow from the same dis- turbance of the same co-ordinate at whatever instant it may be applied to the motion. If the coefficients are not constant for the co-ordinates chosen it may be possible to make them constant by a change of co-ordinates. There are obviously many systems of co-ordinates which may be chosen, and a set may generally be found by a simple examination of the steady motion. If there are any quantities which are constant during the steady motion, such as those called ^, 17, &c. in Art. 466, these may serve for some of the co-ordinates, others may be found by considering what quanti- ties appear only as differential coefficients or velocities, for example those called x, y, &c. in the same Article, If none of these are obvious, we may sometimes obtain them by combining the existing co-ordinates. Practically these will be the most convenient methods of discovering the proper co-ordinates. 478. To obtain the equations of motion we must now substi- tute the value of L in the Lagrangian equations ddL_dL dt dx dx = 0, &c. = 0, and reject the squares of small quantities. The steady motior. being given by x, y, &c. all zero, each of these must bo satisfied when we omit the terms containing a;, y, &c. We thus obtain the equations of steady motion, viz. A^ = 0, ^2 = 0, &c. = 0, which by Taylor's theorem are the same as the equations (1) of steady motion give i in Art. 466. Omitting these terms and retaining the first powers of all the small quantities we obtain the equations of small oscillations, of which the following is a specimen : + |b„|' + (C., - C„) ^ - ^..} 2 + &c. = 0. To solve these we write x = L^*, y = Me^^, &c. Substituting and eliminating the ratios of L, M, &c. we obtain the following deter- minantal equation ABOUT STEADY MOTION. 383 ^u'^' - ^u &c. ■??.^' - K &c. As^^ - ^33 &c. &c. &c. &c. &c. = 0. If in this equation we write — X ^or \ the rows of the new doterminant are the same as the columns of the old, so that the determinant is unaltered. When expanded the equation contains only even powers of \. 479. Regarding this as an equation to find X", we notice that if the roots are all real and negative, each of the co-ordinates cc, y, &c. can be expressed in a series of trigonometrical terms having different periods; the motion will therefore be stable. If any one of the roots is imaginary or if any one is real and positive, there will be both positive and negative real exponentials entering into the expressions for x, y, &c. and therefore the motion will be un- stable. The condition of dynamical stability is therefore that the roots of this equation must all be of the form \ = + fjbs/ — 1, where /A is some real quantity. 480. It follows also that when a system, under the action of forces which have a potential, oscillates about a stable state of steady motion, the oscillations of the co-ordinates are represented by trigonometrical terms of the form A sin (\t + a.) which are not accompanied by any real exponential factors such as those which occurred in the problem of the Governor. We see further that there will in general be as many finite values of X" and therefore as many trigonometrical terms of differ- ent periods as there are co-ordinates. It often happens, as ex- plained in Art. 477, that some of the co-ordinates are absent from the expression for L, appearing only as differential coefficients. Suppose for example 6 to be absent; then A^^, A^^, &c. are all zero, and we may divide X both out of the first line and the first column of the fundamental determinant. We therefore have two zero values of X, while at the same time the number of finite values of X** is diminished by unity. Hence the number of trigo- nometrical terms of different periods cannot exceed the number of 1^: . '\ ! a I i '' ' I' Mf V ■I.. I, i -hfl 384 SMALL OSCILLATIONS. co-ordinates which explicitly enter irito the Lagrangian function. For example in Art. 374, the function T— f^has only the co-ordi- nate 6 explicitly expressed, the others 0' and >^' appearing only as differential coefficients. It follows that if a top is disturbed from a state of steady motion, there will be but one period in the oscillation. 481. The relations between the coefficients / ice. in the exponential values of x, y, &3. may bo obtained wi>.iiout difficulty if we remember that the several lines of the fundamental determi- nant are really the equations of motion. Taking any one line ; multiply the first constituent by L, the second by M, &c. and equate the sum to zero. We thus obtain as many equations as there are co-ordinates. On the whole we shall have, exactly as in Art. 445, twice as many arbitrary constants as there are co-ordi- nates, all the other constants being determined by the equations just found. The arbitrary constants are determined by the initial values of the co-ordinates and their differential coefficients. But, unlike Art. 445, the quantity \ occurs in the firsf power in each of these equation.s, so that the ratios of L, M, Sic. thus found may be imaginary. The expressions for the co-ordinates when rationalized may therefore take the form ■x=A^ sin {\t + a,) + A^ sin (\< -f- ot^) + . .. y=B^ sin {\t -F )9J -h i?., sin (\< + ^J + . . . z = &c. where a^ is not necessarily equal to ^^, nor ofj, to y3^, &c., though they are connected together. 482. When the initial conditions are such that every co- ordinate is expressed by a trigonometrical term of one and the same period, the system is said to be performing a principal or harmonic oscillation. Thus each trigonometrical term corresponds to a principal oscillation, and any oscillation of the system is therefore said to be compounded of its principal oscillations. The physical characteristic of a principal oscillation is that the motion of every part of the system is repeated at a constant interval. 48.3. The stability of the motion depends on the nature of the roots of the fundamental determinant. If we expand the determi- nant we may use the methods given in the theory of equations to discover if the roots are all of the proper form. This however is often tedious and we may sometimes settle the point by a simple examination of the determinant as it stands. ;i! ABOUT STEADY MOTION. 385 In practice it frequently happens that the determinant is reducecl to two rows. If the invariants be written A = A^,A„ ■^ii> s=^.A,-K'> = AA + AA-2AA the conditions of stability are (1) A is positive. (2) (C^ai ~ CJ' - is positive and greater than 2 VZ/y. These conditions may also be expressed thus. Let a and /S be the roots of the quadratic formed by omitting the terms containing (7,g and C„. Then by Art. 448, a and ^ are real. If a and /3 are both negative the motion is stable. If both are positive, the C " C motion is stable or unstable according as -",— " is numerically greater or less than sja + i^^, the roots being taken positively. If a and /3 have opposite signs, the motion is unstable. Whatever maybe the number of co-ordinates, it may be shown that the motion cannot be stable unless the discriminant of A^^x^ + ^A^^xy + &c. is positive or negative according as the number of rows is even or odd. The following theorem is also useful. Beginning with the fundamental determinant we may form a series of determinants, each being obtained from the preceding by erasing the first lino and the first column. As we may supplement the fundam ital determinant with a row and a column of zeros added on at the bottom and right-hand side with unity at the right-hand bottom corner, we may suppose the series of determinants to terminate with unity. Let us substitute in the series any negative value of X"" and count the number of Variations of sign in the series. Then as \' changes from — oo to 0, there cannot be fewer negative roots between any two given values of \' than there are losses in the number of variations of sign corresponding to the two values of \'. If there be more negative roots than losses the excess nmst be an even number. 484. Ex. A homogeneous sphere of unit mass and radius a is suspended from a fixed point by a string of length h, and is set in rotation about the vertical diame- ter. When the sphere is slightly disturbed, let hx, hy and b be the co-ordinates of the point on the surface to which the string is attached; hx+af, by -i-arj, and b + a the co-ordinates of the centre, the fixed point being the origin and the axis of z being vertical and downwards. Also let x='P + ^ where ^ and ^ have the same meaning as in Art. 235, so that before disturbance x'=n. Prove that the La- grangian function is R. D. 25 11 i 1 , 1 , 1 1 j 1 'i i ' 1 1 ! i ii i it[| H ) ;i I M 386 SMALL OSCILLATIONS. If the motion of tho centre of gravity be roprosontod by a BoriM of tonua of tho form 31 COB (jit T N), prove that tlio voIuub of m are given by (.•-») (^'---rO'i-- Bliow that wliatovor sign n may have thia equation has two positivo and two negative roots, which ore separated by tho routs of either of the factors ou the loft- hand side. f! 1 Application of the Calculus of Finite Differences. 485. We shall give some examples to illustrate the use of the Calculus of Finite Differences in cases in which there are an in- definite number of bodies similarly placed. 48G. Ex. A string of length (n + 1) 1, and insensible mass, stretched between two fixed points with a force T, is lauded at intei'vals 1 with n equal masses m not under the influjnce of gravity T ... aiid is slightly disturbed ; if f~ = c', prove that the periodic times of the simple transversal vibrations which, in general coexist are given by the formula — cosec-x-. — — -rv on putting in succession issl, 2, 3...n. Let At B be the fixed points; y,, ^^,'..t/^ the ordinates at time t of the n particles. The motion of the particles parallel to AB is of the second order, and hence the tensions of all the strings must be equal, and in the small terms we may put this tension equal to T. Consider the motion of the particle whose ordinate is y^ The equation of motion is ^ J/tc _ .Vn-1 *" Vk rp Vk "" Vk-i rp . ^dt'" I ^ r ^' .•.g* = c'(2/,,.-22r, + y,J (1). Now the motion of each particle is vibratory, we may therefore expand y^ in a series of the form y, = -^1 sin (pt + a) (2), where 2 implies summation for all values of ^x lUB of tho '0 and two 1 the loft- Be of the ■e an iu- hle mass, oiided at f gravity )dic times \€odst are siLccession inates at arallel to he strings is tension ordinate .... (1). therefore ,...(2), CALCULUS OF FINITE DIFFEBENCES. 387 As there may be a term of the argument pt in every y, let L^, L^, ... ho their respective coefficients. Then substituting, wo have A+i ~ 2Zft + A-i == - ^ A ■(»). To solve this linear equation of (liffcrencoa wo follow the usual rule. Putting L^ — Aa^, where A and a are two constants, we get after substitution and reduction a — 2 + - = — ( ^- ) , or ^a-^ =^ V^J and y/a + -)~ = ± 2 a/i - f^'V; \Ja c ^ \/a y \2cJ Let these roots be called a^ and a^, then is a solution, and since it contains two arbitrary constants, it is the general solution ; .-. y, = 2[^a,* + Z?a,*]sin(p« + a) (4). The equations (1) and (3) will represent the motion of every particle from ^ = 1 to ^• = n, provided wo suppose y^ and t/,,^, both zero, though there are no particles corresponding to values of k equal to and « + 1. Since y = when A; = for all values of t, every term of the series must vanish; .•. -4+i? = 0. Alsoy=0 when A; = « + 1 for all values of « ; .'. Ja,"*' + i^V' = 0. These equations give a^*^ = a^*^. But if |- > 1, the ratio of a^ to a,^ is 2c real and different from unity. Hence wo must have |- < 1. Let 2c then ^ = sin ^ ; and therefore a = cos 2^ + sin 2^ V— 1. 2c Hence, by what we proved before, (cos 20 + sin 29 V- I)'"* = (cos 26 - sin 26 V^)"'' ; W ITT .•.sin2(n+l)^ = 0, or |^ = sin ^^-^^ , and the period of any term = If m and I be indefinitely small and n indefinitely large, tho loaded string may bo regarded as a uniform string of Ici.gtli {n-\-l) 1= L and mass nm = M stretched between two fixed points 25—2 111 m •I! 1 ?li ,1 i. iit I I II m m ivi ! ll \ 388 SMALL OSCILLATIONS. with a tension T. In this case the expression just found reduces toi)=7rty^. 487. If we substitute these values of in the expressions for a^ and a^, we easily find y4=SC = -— - Ei. n + 1 2 2 (n+1) 488. Lagrange in his Mdcanique Anahjtique has applied his general equations of motion to the solutioii of the preceding problem. He has also determined the THE CAVENDISH EXPERIMENT. 389 oscillations of an inextensible string charged with any number of weights, and suspended by both ends or by one only. Though several solutions of these pro- blems had been given before his time, he considers that they were all more or less incomplete. 489. Ex. 1. A light elastic string of length nl and coeflScient of elasticity E ia loaded with n particles each of mass vi, ranged at intervals I along it beginning at one extremity. If it be suspended by the other extremity, prove that the periods of its vertical oscillations will be given by the formula ir a/ i=0, 1, 2 ... n- 1 successively. Hence show that the periods of vertical oscillation of a heavy elastic string will be given by the formula „. — r- k/-~ 2i + X ▼ iJ length of the string, M its mass, and i is zero or any positive integer. Tripos, 1871.] Im 2t + 1 IT , ■ , where L is the [Math. Ex. 2. An infinite number of equal particles, each of mass m, are placed in a row at distances each equal to I and mutually repel each other so that the force between any two is nfifip), where D is tlie distance between those two, A disturb- ance is given to the system such that each particle makes oscillations in the direc- tion of the row whose extent is very small compared with I. Show that the disturbance of the li^ particle, counting from any one particle, is given by the series Stt cos Y" (fti ftJ), where S implies summation for all values of \, and 'wa.1(i\^ K = ls!Tn jlV'W (~)%2V'(2/0 ('-^ ) +&C. J +&C. \ and 9= ;r . h Thence show that all very long waves travel with the same velocity. If /(2)=/x2~", show that V is infinite unless n is greater than 3. [Phil. Mag.] The Cavendish Experiment. 490. As an example of the mode in which the theory of small oscillations may be used as a means of discovery we have selected the Cavendish Experiment. The object of this experiment is to compare the mass of the earth with that of some given body. The plan of effecting this by means of a torsion-rod was first suggested by the Rev. John Michell. As he died before he had time to enter on the experiments, his plan was taken up by Mr Cavendish, who published the result of his labours, in the Phil, Trans, for 1798. His experiments being few in number, it was thought proper to have a new determination. Accordingly in 1837, a grant of £500 was obtained from the Government to defray the expenses of the experiments. The theory and the analytical formuliB were supplied by Sir G. Airy, while the arrangement of the plan of operation and the task of making the experiments were undertaken by Mr Baily. Mr Baily made upwards of two thousand experiments with balls of different weights and sizes, and suspended in a variety of ways, a full account of which is I ! : I ! m iil-i; .■pill 390 SMALL OSCILLATIONS. given in the Memoirs of the Astronomical Society, Vol. xiv. The experiments were, in general, conducted in the following manner. 491. Two small equal balls were attached to the extremities of a fine rod called the torsion-rod, and the rod itself was sus- pended by a string fixed to its middle point C. Two large spherical masses A, B were fastened on the ends of a plank Avhich could turn freely about its middle point 0. The point was vertically under C and so placed that the four centres of gravity of the four balls were in one horizontal plane. i ^ I ! A 1 1 r ^, i First, suppose the plank to be placed at right angles to the torsion-rod, then the rod will take up some position of equilibrium" called the neutral position, in which the string has no torsion. Let this be represented in the figure by Col. Now let the masses A and B be moved round into some position J5,-4,, making a not very large angle with the neutral position of the torsion-rod. The attractions of the masses A and B on the balls will draw the torsion-rod out of its neutral position into a ncAV position of equi- librium, in which the attraction is balanced by the torsion of the string. Let this be represented in the figure by CE^. The angle of deviation EJ^x and the time of oscillation of the rod about this position of equilibrium might be observed. Secondly, replace Hhe plank AB at right angles to the neu- tral position of the rod, and move it in the opposite direction until the masses A and B come into some position AJi^ near the rod but on the side opposite to B^A^. Then the torsion-rod will perform oscillations about another position of equilibrium CE,^ under the influence of the attraction of the masses and the torsion of the string. As before, the time of oscillation and the deviation EjOa might be observed. In order to eliminate the errors of observation, this process was repeated over and over again, and the moT,u results taken. •I ill THE CAVENDISH EXPERIMENT. S9l The positions B^A^ and A^B^, into which the masses were alter- nately put, were as nearly as possible the same throughout all the experiments. The neutral position Ca of the rod very nearly bisected the angle between -BjJ.j and A^B^, but as this neutral position, possibly owing to changes in the torsion of the string, was found to undergo slight changes of position, it is not to be considered in any one experiment coincident with the bisector of the angle -4j0^2- Let Cx be any line fixed in space from which the angles may be measured. Let 6 be the angle xCci, which the neutral position of the rod makes with Gx ; A and B the angles which the al- ternate positions, B A and A^B^, of the straight line joining the A + B centres of the masses, make with Cx ; and let a = — ^ — • -A^so let oo be the angle which the torsion-rod makes with Cx at the time t. Supposing the masses to be in the position A^B^, the moment about GO oi their attractions on the two balls and on the rod will be a function only of the angle between the rod and the line A^B^', let this moment be represented by ^ (A— x). The whole appa- ratus was enclosed in a wooden casing to protect it from any currents of air. The attraction of this casing cannot be neglected. As it may be different in different positions of the rod, let the moment of its attraction about GO be "^{x). Also the torsion of the string will be very nearly proportional to the angle through which it lias been twisted. Let its moment about CC^ be E{oa—h). If then / be the moment of inertia of the balls and rod about the axis CO, the equation of motion will be df {A'-x) + ylr(x)'-Ji:{x-h). Now a-'X is a small quantity, let it be represented by f. Substituting for ca and expanding by Taylor's theorem in powers of ^, we get -/^|='(^-a)-.|r'(a) + ^}|. Let < f>'(A^a) -^lr'{a) + E It — ' -r " 1 fi' i 11 hi II -11 \ [ and .- . a. < l>(A-(i) + ir(a)-E{a-h) Then x = e + Lsin {nt + L'), where L and L' arc two arbitrary constants. Wo sec therefore that in the position of equilibrium the angle the torsiou-n d % !i m 1 '..-V 1 '^1 I 392 SMALL OSCILLATIONS. makes with the axis of x is e, and the time of oscillation about the position of equilibrium is n Let us now suppose the masses to be moved into their alternate position A^B ; the moment of their attraction on the balls and rod will now he —^{x — B). The equation of motion is therefore if=-^(.. B) + ylr(w)^i:(x-b). Let a = £B — ^, then substituting for B its value 2a — A, we find by the same reasoning as before x = e' + Nsia(nt + N'), where n has the aame value as before and /-„ , -(A-a)+ ylr{a)^Eia-b) In In these expressions, the attraction yjr (a) of the casing, the coefficient of torsion E and the angle b are all unknown. But they all disappear together, if we take the difference between e and e. We then find (A — a) _e — e m- •(A), 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 two positions of equilibrium and also the time of oscillation about either can be observed. 492. The function ^(A—a) is the moment of the attraction of the masses and the plank on the balls and rod, when the rod has been placed in a position Cf, bisecting the angle A,CB^ be- tween the alternate positions of the masses. Let M be the mass of either of the masses A and B, m that of one of the small balls, m that of the rod. Let the attraction of M on m be represented by [I ra- , where D is the distance between their centres. If {p, q) be the cf'-ordinates of the centre of A^ referred to Cfam the axis of X, the moment about C of the attraction of both the masses on both the balls is = 2iJiMm\- ^-^r oq \[{p-cr + q'\^ {{p + cf-i-qfr where c is the distance of the centre of either ball a, b from the centre C of motion. Let this be represented by fiMmP. The moments of the attraction of the masses on the rod may by into- n about Itemate alls and lerefore -A, we iing, the m. But between ....(A), Irsion-rod bus the ;veen the on about ttraction the rod ,^J5, be- the mass all balls, )resented )tres. If tyas the e masses from the IP. The by intc- THE CAVENDISH EXPERIMENT. 393 gration be found =tiMm'Q, where ^ is a known function of the linear dimensions of the apparatus. The attraction of the plank might also be taken account of. Thus we find <^{A-a)= fiM{mP + mQ). If r be the radius of either ball, we have /=2»i H'] + w ,{o-rY 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 _ e-^ (W ^^'mF^mq~ 2 \t)' Let E be the mass of the earth, JB its radius and g the force of gravity, then g = ii-ai' Substituting for /^, we find E e-e' /27rY _1^ 2 '[Tj-gP'' m m P' + Q: m P + Q m The ratio — , was taken equal to the ratio of the weights of the ball and rod weighed in vacuo, but it would clearly have been more accurate to have taken it equal to the ratio weighed in air. For since the masses attract the air as well as the balls, the pres- sure 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. 493. 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 'orsion-rod, and (2) the angle e — e' between its two positions of equilibrium when under the influence of the masses in their alternate positions. To observe these, a small mirror was attached to the rod at C with its plane nearly perpendicular to the rod. A scale was engraved on a vertical 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 fur- * In Baily's experiment, a more accurate value of g was used. If e be the ellip- ticity of the earth, m the ratio of centrifugal force at the equator to equatoreal gravity, and \ the latitude of the place, we have ^ = M^il-2e + (^nt -el cos' \ i, : 'i\ ii H -:!' 394 SMALL OSCILLATIONS. nished with three vortical 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 ^5 • tt^ • o '='73" 4 6. But the lo lUo A 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 sqiiare of its circular measure could be neglected ; but as it extended over several divisions it is clear that it could be obsei-ved with accuracy. A minute description of the mode in which the observations were made would rot 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. 494. The density of water in which the weight of a cubic inch is 252725 grains (7000 grains being equal to one pound avoirdupois) was taken as the unit of density. The final result of all the experiments was that the mean density of the earth is 5-6747. 495. Two other methods of finding the mean density have been employed. In 1772 Dr Maskelyne, then Astronomer Royal, 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 tlie 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 I'hil. Trans. 1778 and 1811. From some observations near Arthur's Saat, the mean density of the earth is given by Lieut.-Col. Juuies, of the Ordnance Survey, as 5'316. See Phil. Trans, 185G. 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 the mean density of the earth was found to be GoGG. Sec Phil. Trans. 1856. i •sion-rod ielescope ;aiit the istituted lirteenth negative le image interval But the raally by division the frac- B torsion- could be ; is clear ascription 1 rot find refer the the effect This was, of taking allowance rsion-rod. a cubic ne pound nal result the earth sity have ler Royal, ired with iced in a tin chosen ind to be il Trans. gsat, the es, of the nparc the le surface, this way See FliiL OSCILLATIONS OF THE SECOND ORDER. 395 OscillaiiQ"'^ of the Second Order. 496. The equations of small oscillations are formed on the following principle. Some small quantities are selected as the co-ordinates of the system, and all powers of these above the first are neglected. The assumption is tacitly made that the order of magnitude of the terms is not materially altered by the process of solving the equations ; so that a small term, which should by the rule be neglected in forming the differential equations, cannot become of importance in the final integrals. This assumption, however, is not strictly correct. In the Lunar and Planetary theories, where something more is wanted than the mere periods of oscillations, there are many instances of small terms in the differential equations, which become of great magnitude in the result. Wo require some rule to dis- tinguish the small terms v/hich become of importance from those which remain insignificant. For the sake of simplicity we shall consider the case in which the system depends on two independent co-ordinates, though the remarks are for the most part quite general. 497. Referring to Art. 432, let PsinXt be some small periodic term which occurs on the right-hand side of the first of the two differential equations of motion. To simplify the solution, let us write for the trigonometri al term its exponential value, and fix our attention on the part — p=. P^^"^ or, as we shall 2 1^ — 1 write it, Qe**'. Let/{Z)) stand for the determinant which is the operator on as in the third equation of Art. 432. Also let F(D} be the minor of the leading con- stituent ; the value of x is then known to be The term Qe*^' in the differential equation is the analytical representation of some small periodical force which acts on the system. The first term of the expression for x is the direct effect of the force, and is sometimes called the forced vibration in the co-ordinate x. The quantities m,, mj, &c. being generally imaginary, the remaining terms are also trigonometrical and are sometimes called the free or natural vibrations in the co-ordinate. In the analytical theory of linear differential equations, the forced vibration is called the particular in'egral and the free vibration the compleiiumtary function. 498. If we examine the coefficient of the forced vibration in x we shall see that it is large only if /(/x) is very small or zero. Since the roots of the equation / (/t) = are m^, m^, &c. the rule may be simply stated thus : any sviall periodical term lohose coefficient in the dijf'erential equation is less than the standard of quantities to be neglected may rise into imporvance if its period is nearly equal to one of the free vibrations of the system. Suppose the dynamical system to have two of its free periods equal and let it be acted on by a small force whose period is nearly equal to this free period. The divisor/ (/t) of the forced vibration will be a small quantity of the second order and the magnitude of the terra may be much greater than if the free periods were unequal. When such a case occurs in the Lunar theory, the term is said to rise tivo orders. I I' I "I i I ^jllll t. i !.j u ■iff Mi 396 SMALL OSCILLATIONS. Is • 499. This principle admits of an elementary explanation in some cases. Let a system oscillating with one degree of freedom be acted on by a small periodical force at some point A, The force will act sometimes to accelerate the motion of A and sometimes to retard it, and thus the maguitiide of the vibration will not become very great. But if the period of the force be equal to that of the point A, the force may continually act to increase the motion of A in whatever direction A is moving. Thus the extent of the vibration will be continually increasing. For example, every one knows how a heavy swing can be set in violent oscillation by a series of small pushes and pulls applied at the proper times. If the period of the force be only nearly equal to that of the point A, a time will come when the force acts continually to decrease the motion of A. Thus the oscillation will not increase indefinitely/ but will alternately slowly increase and as slowly decrease. 600. A remarkable nse of this principle was made by Gapt. Eater in his experiments to determine the length of the seconds' pendulum. It was important to determine if the support of his pendulum was perfectly firm. He had recourse to a delicate and simple instrument invented by Mr Hardy a clockmaker, the sensibility of which is such that had the slightest motion taken place in the support it must have been instantly detected. The instrument consists of a steel wire, the lower part of which is inserted in the piece of brass which forms its support, and is flattened so as to form a delicate spring. On the wire a small weight slides by means of which it may be made to vibrate in the same time as the pendulum to which it is to be applied as a test. When thus adjusted it is placed on the material to which the pendulum is attached, and should this not be perfectly firm, the motion will be communicated to the wire, which in a little time will accompany the pendulum on its vibrations. This ingenious contrivance appeared fully adequate to the purpose for which it was employed, and afforded a satisfactory proof of the stability of the point of suspension. See Phil. Trans. 1818. 601. It generally happens that the small terms rejected in the equations of motion are functions of the co-ordinates and their differential coefficients. To take account of these terms we proceed by successive approximation. Suppose the co-ordinates x, y to determine the oscillation about some state of steady motion, and to be zero for that motion. As a first approxunation we obtain (Art. 432) a; = JJ/^e*"'* + JJ/,c'"'* + with a corresponding expression for y, where m^, mt, &c. give the free periods, and jl/j, 3/a, &c. are all small quantities of the first order. If we now substitute these values of x and y in any small term of a high order which occurs in the differential equation, it becomes a series of exponentials of the form where p, q, &c. are positive integers whose sum is equal to the order of the term. By the principle explained in Art. 498, the corresponding forced vibration cannot be important unless pwij + ^nij + . . . is very nearly equal to one of the quantities «ii, Wig, &c. In the same way, in any approximation, if the periods of the terms are not such that an equality of this nature can be very nearly true, the next approximation to the motion will not produce any important terms. Even if such a relation does approximately hold, yet, if the order of the term to be examined is great, the term will probably remain insignificant. OSCILLATIONS OF TEE SECOND ORDER. S97 502. As an oxample let Tis consider the case of a planet describing a circle about the sun, considered as fixed in the centre. If slightly disturbed the changes in the radius vector and longitude will be very small and will correspond to what we have called x and y. From the theory of elliptic motion, we know that these will be approximately «=a-a'. I I I i I I " ■ 'ft 400 8MALL OSCILLATIONS. i,'i il:; f:'i \i in the form 9^f{t), tlion if wo negloot all the Hmall torme, ^ vanisboB when mt= ± „ . Put then mt= -^ + x where jc id a small quantity, we have Now f.fji f Off Qtn \ /'(e) = ae-«'(mcosmt-/{sinn»t)-^^-c~2«'( - 2(c + -^ cos 2m« + - Bin2mn + „„ -«"'*'{ -m sin mt-8/c cos mf). A fluiBciently near approximation to the valne of f"(t) may be fonnd by differ- K 4 iiCtfC 7t CL entiatincr the first term of the value of/' (t). We thus find x= ^- _„ - ; the second of those terms being smaller than the other two might be neglected. We also find as the arc of descent Kir Kir Kir Smr Similarly to find the arc of ascen^^^ we put mt=^ + y. This gives w = -—— , and the arc of ascent is = ae Kir " Sni 2 (fir * « >« I ~ am , w'a" ~ am 3*" Ism \ ' In those expressions for the arcs of descent and ascent the terms containing x and y are very small, and assuming k not to be extremely small, these terms will be neglected *. Kow a is different for every swing of the pendulum, we must therefore eliminate Kir a. Let M„ and w„4.j be two successive arcs of descent and ascent, and let \=e ^m , so that X is a little less than unity. Then we have 1 2 „ 1 ^ 2 ,,, eliminating a we have very nearly where c = 5- r--^ = -. — nearly. 2/x 1 + X" 4/xm • If these terms are not neglected tho equation connecting the successive arcs of descent and ascent becomes M« Mn4.i 3 ^ ' 32Kni \ 'n "n+l 2Kir Now 1-X*= — nearly, so that this additional term is very small compared with m that retained. !S8ivc axes of SMALL OSCILLATIONS. 401 The auoeeosiTe »roB are, thereforoi anob tb»t - 4- ia the general term of n geometrical eeriea whoae ratio ia e^ . Tbe ratio of any arc u^ to tbe following arc «. '»+i is which ooutinually decreases with the arc. In any scrioR of oscillations tbe ratio is at first greater and af terwarda lesa than its mean value. This result seems to agree with experiment. Tojlnd the time of oneillation. Let r„ f, bo the timfs nt which the pendulum is at the extreme left and right of its arc of oscillation. Then ir K HI 'a TT 2 K m n-a* 32w« ' The time of oscillation from one extreme position tu the other is f, - t^ which is equal to — . This result is independent of the arc, so that tbe time of oscillation m remains constant throughout the motion. The time is however not exactly thn same as in vacuo, but is a littlo longer ; the difference depending on the square of the small quantity k. Ex. 2. If in Art. 418 a first approximation to the motion in $=A sin {nt + B), show that a second will bo <>= .4 sin («e + ^) + J (ft + c) i4« + K36 + c) ^' cob 2 {at + B) where b = rs sm a 1 COR a da sin 2a sin 7 na) k^ + r^ ' ^~2 scoBa-r and ff is the length of the arc of either cylinder. A general method of solving problems of this kind, both for two and three dimensions, is given in tbe Proceeding* oj the London Mathematical Society, Vol. v. page 101, 1874. Ex. 8. A rigid body is Rnspended by two eqnal and parallel threads attached to it at two points symmetrically situated with respect to a principal axis through the centre of gravity which is vertical, and being turned round that axis through a small angle is left to perform small finite oscillations. Investigate the reduction to infinitely small oscillations. [Smith's Prize.] EXAMPLES*. 1. A uniform rod of length 2e rests in stable equilibrium with its lower end at tbe vertex of a cycloid whose plane is vertical and vertex downwards, and passes through a small smooth fixed ring situated in the a;as at a distance h from the vertex. Show that if the equilibrium be slightly d'sturbed, the rod will perform * These examples are taken from the Examination Papers which have been set in the University nud in the Colleges. R. D. 26 *!: r 4 \:m 1 ■MRem sl'i hi fiii !■! 402 SMALL OSCILLATIONS. email oscillations with its lower end on the arc of the cycloid in the time 47r cf} ac) , where 2a is the lengtl: of the axis of the cj'cloid. 2. A small smooth ring elides on a circular wire of radius a which js con- strained to revolve about a vertical axis in its own plane, at a distance c from the centre of the wire, with a uniform angular volocity a/ _l_Vr_ • show that the ring ^ Cs/2 + a will he 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 ; and that if the ring be slightly displaced, it wiU perform a small oscillation in the' time < 9 c\/8 + a' V8 + < 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 throiigh a small angle about the vertical line through the middle point, show that the time of a small oscillation is 27r , 4. Two equal heavy rods connected by a hinge which allows them to move in a vertical plane rotate about a vertical axis through the .hinge, and a string whose length is twice that of either rod is fastened to their extremities and bears a weight at its middle point. If M, M' be the masses of a rod and tlio particle, and 2a the length of a rod, prove that the angular volocity about tho vertical axis when the rods and string form a square is \/ — --— M + 2M' and 2a J2 ' M if the weight bo slightly depressed in a vertical direction the time of a small '" V 15^ 'M + 2M'' oscillation is 2?! 5. A ring of weight W which slides on a rod inclined to tho vertical at an angle a is attached by means of an elastic string to a point in the plane of the rod so situated that its least distance from the rod is equal to the natural length of tho string. Prove that if 6 be the inclination of the string to the rod when in W equilibrium, cot - cobO = coso, where to is the modulus of elasticity of tho 10 string. And if the ring be slightly displaced the time of a small oscillation will be 2ir A / — :— — , where I is the natural length of the string. V wff 1 - mu?d 6. A circular tube of radius a contains an elastic strmg fastened at its highest point equal in length to v: of its circumference, and having attached to its other o extremity a heavy particle which hanging vertically would double its length. Tho system revolves about the vortical diameter with an angular velocity a /•-• Find the position of relative equilibrium and prove that if the particle be slightly dis- turbed tho time of a small oscillation is - ^ * / - • ^/7r + 4 V J/ he time ill is con- from the ,t tlie ring le circular hat if the igfl each of ^h a small 5 time of a m to move id a string amities and •od and tho f about tho ^W M of a small and at an angle tho rod so Ingth of tho rod when iu picity of tho ition will be It its higliest to its other length. Tho Find a sligbtly dia- EXAMPLES. 403 7. A heavy uniform rod AB has its lower extremity A fixed to a vertical axis and an elastic string connects B to another point C in the axis such that AO = — - = 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 4^ 21ach of these strings a particle of mass M is attached ; show that the time of a small oscillation of tho system is 2ir \ V- . \ n 0/ 10. In a circular tube of uniform bore containing air, slide two discs exactly fitting the tube. The two discs are placed initially so that tho lino 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 contro is a small quantity 7i'. If the inertia of the air be neglected, prove that tho point on the axis of the tube equidistant from tho centre of tho discs moves uniformly -T-p- , where M is the mass of each disc, a the radius of the axis of tube, P tho pressure of air on the disc in its natural state. and that the time of an oscillation of each disc is2.y^ 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 particlo of mass m. If, during a small oscillation of tho system, the inclination of the string to the vertical is always twice that of tho beam, then M(3l-a) — Cmi(l + a). 12. A conical surface of semivertical angle a is fixed with its axis inclined at an angle to the vertical, and a smooth cono of semiverticnl- angle /3 is placed within it so that the vertices coincide. Show that the time of a small oscillation =:27r / f "^"("~P ) where a is the distance of the centre of gravity of tlio cono from the vertex. 13. A number of bodies, tho particles of which attract each other with forces varying as the distance, are capable of motion on certain curves and surfaces. Trove that if A, B,C bo the moments of inertia of tho system about xhreo axes mutually at right angles through its cent'-o of gravity, tho positions of stable equiUbrium will bo found by making A+B + C si minimum. 20—2 [' i lli*'' ) m ■ m fmmmmm CHAPTER IX. MOTION OF A BODY UNDER THE ACTION OF NO FORCES. ■J ; Solution of Elder's Equations. 608. To determine the motion of a body about a fixed point, in the case in which there are no impressed forces. The equations of motion are by Art. 230, ^^|i-{5-0)a>.a,3 = multiplying these respectively by «»,, «„ w^; adding and inte- grating, we get ila>,« + 5<+Ca)3«=!r. (1), where T is an arbitrary constant. Again, multiplying the equations respectively by -4a),, Ba>^, Cwg, we get, similarly, A\' + ffco,' + Cto* = (P (2), where is an arbitrary constant. To find a third integral, let o>i' + Wj* + ©a" = a>' 9 9 ■(3); G). da) ~dt day, d(o. ' + ««-:7r + '»»8-:7*' = <" dt dt aoy di' (0. &)„ O). then multiplying the original equations respectively Tt)y -f , -^i yf , and adding, we get dot fB- C . C^A A-B\ da I B-C ^ C^A A C -ja),a),a)3 W {B-C)(C-A){A'-B) ABC WjWjWg. SOLUTION OF EULEllS EQUATIONS. But solving the equations (1), (2), (3), we get 405 < = < = CA .(-\ + a,») 8 {B-A){B-C) AB / ^ . .N iC-B){C-A)'^^^''^'^^J T (B + C) — G* where \ = — ^^ — ^^^ , with similar expressions for X, and \. (5), BG Substituting in equation (4), we have da to ^ = V(\-«^)(\-o)^)(\-a,«) dt (6). The integration of equation (6) * can be reduced without diffi- culty to depend on an elliptic integral. The integration can be effected in finite terms in two cases ; when A=B, and when G* = TB, where B is neither the greatest nor the least of the three quantities A, B, C. Both these cases will be discussed further on. Ex. If right lines are xneasored along the three principal axes of the hody from the fixed point, and inversely proportional to the radii of gyration round those axes, the sum of the squares cf the velocities of their extremities is constant throughout the motion. 509. It will generally be supposed that A, B, C are in order of magnitude, so that A is greater than B, and B than C. The axis of B will be called the axis of mean moment. If we eliminate w^ from the equations (1) and (2), we have AT-G^=B{A-B)u^^+C[A-C)u^, which is essentially positive. lu the same way we can show that CT- CP is nega« tive. Thus the quantity -^ may have any value lying between the greatest and least moments of inertia. The three quantities Xi , Xg, \ in Art. 508 are all po^dtive quantities; for since B-^C-Avi positive, and -ys or < B. It follows from equations (6) that throughout the motion w' must lie between X, and the greater of the quantities \ and X3. * Euler's solution of these equations is given in the ninth volume of the Quarterly Journal, p. 361, by Prof. Cayley. Kirohhoff's and Jacobi's integrations by elliptic functions are given in an improved form by Prof. Oreenhill in the fourteenth volume, pages 182 and 265. 1876. il V Irta ,1 ' '"'f 11 Mil i' ' ^ r -I ';'l ] 1 ; • Ml 406 MOTION UNDER NO FORCES. 510. The solution in terms of elliptic integrals has been effected in the follow- ing manner by Kircbboff. If we put '' sin'* A{i>)-^Jl-ii'smP) =Jo Vl-A^i then k is called the modulus of F, and must be less than unity if F is to be real for all values of 0. The upper limit is called the amplitude of the elliptic integral J*' and is usually written am F. In the same way sin , cos , . , > dA (0) h? sin cos rf0 , ., . These equations may be made identical with Euler's equations if wo put i''=X(«-T)and o)i—aAam\(t-T) W3 = 6sinam\ (<-t) I (2), Wj^ccos amX (t-r) J A-B c\ A-C b\ B-C ,.a\ .(3). C ah' B cd" A -;&» he- We have introduced here six new constants, viz. a, 6, c,\ h and t. With these we may satisfy the three last equations and also any initial values of Wj, w.,, w^. The I olution if real will also be complete. Whcn<=r wehavefrom (2) Wj=a, Wj, = 0, and W3=c. Hence by Art. 508 Aa^ + Cc'^=T, A-^a'^ + C-'c^ = G"'\ G^-CT c'= AT-G* A [A-C)' ~C(A-q' _ ' Dividing the second of equations (3) by the first, we have ■ ' b^_A-CO ,3_ AT-G' c" A-BB' •■• '' ~£{A-B)' Multiplying the first and secondof equations (3), we obtain The ratios of the right-hand sides of (3) are as c^ : b^ : khi^, and those have just been found. Hence ii' the signs of a, b, c, \ be chosen to satisfy any one of the tliree equalities, the signs of all will bo satisfied. Dividing the last of equations (3) by either of the other two, we find ' ■ A-BG-'-CT' •'■ ^-(rr77)((,«-cT)' poinsot's and mac cullagh's constructions. 407 If 0^ > BT and A, B, C are in descendiUg order of magnitude, the values of a', 6", c* and X" rre all positive. Also I? is positive and less than unity. The solution is therefore real and complete. If G' < BT we must suppose J , B, (7 to be in ascending order of magnitude to obtain a real solution. If we may anticipate a phrase used by Poiusot, and which will be explained a little further on, we may say that the expression for Wj in this solution is to be taken for the angular velocity about that principal axis which is enclosed by the polhode. UCP=BT we have F= 1 and Jo cos 1 . 1 + sin (f> 2 ^^'l-siu^' sin amF= pf-i e*'+e- Snbstiiutin{j in equations (2) the elli^itic functions become exponential. If B — we have F=0 and in this o tse F=(t>, so that amF= F. If we again substitute in equations (2) the elliptic functions become trigonometrical. The geometrical meaning of this solution will be given a little further on. Poinsot's and MacCallaglis comtructiom for the motion. 511. The fundamental equations of motion of a body about a fixed point are V + ^V/+CV=<^*. V(o^ + Bw, (1), + c<=r. (2). These have been already obtained by integrating Euler's equations, but they also follow very easily from the principles of Angular Momentum, and Vis Viva. Let the body be set in motion by an impulsive couple whose moment is O. Then we know by Art. 279, that throughout the whole of the subsequent motion, the moment of the momentum about every straight lino which is fixed in space, and passes through the fixed point 0, is constant, and is equal to tho mo- ment of the couple G about that line. Now by Art. 241, the moments of the momentum about the principal axes at any instant are A(t\, Ba,^, Ga^. Let a, /8, 7 be the direction angles of the normal to the plane of the couple G referred to these principal axes as co-ordinate axes. Then we have -4ft)j= 6^ cos a ' i?w^ = (? cos /9 • (3), C(W3= G cosy adding the squares of these we get equation (1). s ; I i ' 1 ,; 1 .■; I: . 1 ' j! ! ^; ■ I ■ i !ki S , J'! W ] H MM (•r 408 MOTION UNDER NO FORCES. Throughout the subseque^it motion the whole momentum of tlie body is equivalent to the couple O. It is therefore clear that if at any instant the body were acted on by an impulsive couple equal and opposite to the couple O, the body would be reduced to rest. 512. It follows from Art. 290, that the plane of this couple is the Invariable plane and the normal to it the Invariable line. This line is absolutely fixed in space, and the equations (3) give the direction cosines of this line* referred to axes moving in the body. It appears from these equations, that if the body be set in rotation about an axis whose direction cosines are (^ m, n) when referred to the principal axes at the fixed point, then the direction cosines of the invariable line are proportional to Al, Bm, Cn. If the axes of reference are not the principal axes of the body at the fixed point, the direction cosines of the invariable line will, by Art. 240, be proportional to Al — Fm — En, Bm— Dn — Fl, and On — El — Dm, where the letters have the meaning given to them in Art. 15. 513. Since the body moves under the action of no impressed forces, we know that the Vis Viva will be constant throughout the motion. Hence by Art. 348, we have where T\ is a constant to be determined from the initial values of Wj, 0),, Wj. The equations (1), (2), (3) will suffice to determine the path in space described by every particle of the body, but not the posi- tion at any given time. * That the straight line whose equations referred to the moving principal axes na 9J 2 are -t— = -~- = jz- is absolutely fixed in space may be also proved thus, if we Au-^^ JSu^ C(>>3 assume the truth of equation (1) in the text. Let x, y, z be the co-ordinates of any point P in the straight line at a given distance r from the origin, then each of the equalities in the equation tc the straight line is equal to ^ and is therefore con- stant. The actual velocity of P in space resolved parallel to the instantaneous position of the axis of x is _dx ~ (ft' But this is zero, by Euler's equation. Similarly the velocities parallel to the other axes are zero. f It should be observed that in this Chapter T represents the whole vis viva of I'ji body. In treating of Lagrange's equations in Chapter vii. it was convenient to let T represent halj the vis viva of the system. ■yuz + zaa=-AA -^'-{B- C) WjWj,| to the other POINSOTS CONSTRUCTION. 409 514. To explain Poinsot's representation of the motion hi/ means of the momental ellipsoid. Let the momental ellipsoid at the fixed point be constructed, and let its equation be Let r be the radius vector of this ellipsoid coinciding with the instantaneous axis, and p the perpendicular from the centre on the tangent plane at the extremity of r. Also let a be the an- gular velocity about the instantaneous axis. The equations to the instantaneous axis are — = ^l = — and ®i <»a <»8 if (a?, y, z) be the co-ordinates of the extremity of the length r, each of these fractions is equal to — . Substituting in the equation to the ellipsoid, we have I T r Again the expression for the perpendicular on the tangent plane at (a;, y, z) is known to be —^ = ^^j^ , substi- tuting as before we get JfV 1 _^V + ff<-}-CV t-J^ ¥t. P' JfV TiJi a> a MY ' T ' p ^MT , Q . €. The equation to the tangent plane at the point (a;, y, z) is Ax^ + Byr)+ Cz^=M€\ substituting for (a?, y, z) we see that the equations to the perpen- dicular from the origin are A(o^ Boii^ Co),' but these are the equations to the invariable line. Hence this perpendicular is fixed in space. From these equations we infer (1) The angular velocity about the radius vector round which the body •"" turning varies as that radius vector. H til i.'i^ I If Ji I ) I' •I ^'1 fi^i) kil: m ii 410 MOTION UNDER NO FORCES. *' .1 U4h ; (2) TJie resolved part of the angular velocity about the per- pendicular on the tanr/ent plane at the extremity of the instan- taneous axis is constant. Tliis theorem is due to Lagrange. For the cosine of the angle between the perpendicular and n tlie radius vector = - . Hence the resolved angular velocity is n T . . = &)- = >., which is constant. r G (3) The perpendicular on the tangent jtlane at the extremity of the instantaneous axis is fixed in direction, viz. normal to the invariable plane, and constant in length. The motion of the momental ellipsoid is therefore such that, its centre being fixed, it always touches a fixed plane, and the point of contact, being in the instantaneous axis, has no velocity. Hence the 'motion may he represented by supposing the momental ellipsoid to roll on the fixed plane with its centre fixed. 515. Ex. 1. If the body while iu motion be acted on by any impulsive couple whose plane is perpendicular to the invariable line, show that the momental ellipsoid will continue to roll on the same plane as before, but the rate of motion will be altered. Ex. 2. If a plane be drawn through the fixed point parallel to the invariable plane, prove that the area of the section of the momental ellipsoid cut off by this plane is constant throughout the motion. Ex. 3. The sum of the sqiiares of the distances of the extremities of the princi- pal diameters of the momental ellipsoid from the invariable line is constant through- out the motion. This result is due to Poinsot. Ex. 4. A body moves about a fixed point under the action of no forces. Show that if the surface Ax^ + Bif + Cz' ?= M{x^ +y^ + z^Y ^^ traced in the body, the principal axes at being the axes of co-ordinates, this surface throughout the motion will roll on a fixed sphere. 51G. To assist our conception of the motion of the body, let us suppose it so placed, that the plane of the couple G, which would set it in motion, is horizontal. Let a tangent plane to the momental ellipsoid be drawn parallel to the plane of the couple G, and let this plane be fixdd in space. Let the ellipsoid roll on this fixed plane, its ceotre remaining fixed, with an angular velocity which varies as the radius vector to the point of contact, and let it carry the given body with it. We shall then have constructed the motion which the body would have assumed if it had been left to itself after the initial action of the impulsive couple G*.* * Prof. Sylvester has pointed out a dynamical relation between the free rotating body and the ellipsoidal top,. as he calls Poinsot'a central ellipsoid. If a material the per- e instan- ;e. jular and Blocity is extremity >ial to the mch tliat, , and the ) velocity. momental ilsive couple Qtal ellipsoid tion will be he invariable ^ ofE by this »f the princi- ant through- rces. Show the principal motion will body, let G, which me to the couple G, 11 on this velocity t, and let nstructed lad been eG*: ree rotating a material poinsot's construction. 411 The point of contact of the ellipsoid with the plane on which it rolls traces out two curves, one on the surface of the ellipsoid, and one on the plane. The first of these is fixed in the body and is called the polhode, the second is fixed in space and is called the herpolhode. The ecpiations to any polhode referred to. the prin- cipal axes of the body may be found fronti the consideration that the length of the perpendicular on the tangent plane to the ellip- soid at any point of the polhode is constant. Hence its equations are Eliminating ij, we have A {A - B) x' + C{C-n) z'=[^^^-B\ I Me* Hence if B be the axis of greatest or least moment of inertia, the signs of the coefficients of x^ and z' will be the same, and the projection of the polhode will be an ellipse. But if B be the axis of mean moment of inertia, the projection is an hyperbola. A polhode is therefore a closed curve drawn round the axis of greatest or least moment, and the concavity is turned towards the axis of greatest or least moment according as -„,- is greater or less than the mean moment of inertia. The boundary line which separates the two sets of polhodes is that polhode whose projection on the plane pei*pendicular to the axis of mean moment is an hyperbola whose concavity is turned neither to the axis of greatest, nor to the axis of least moment. In this casv. G'=BT, and the projec- tion consists of two straight lines whose equation is A{A-B)x'-G{B- G)z' = 0. This polhode consists of two ellipses passing through the axis of mean moment,' and corresponds to the case in which the per- pendicular on the tangent plane is equal to the mean axis of the ellipsoid. This polhode is called the separating polhode. Since the projection of the polhode on one of the principal planes is always an ellipse, the polhode must be a re-entering curve. ellipsoidal top be constructed of uniform density, similar to Poinsot's central cllip- sold, and if with its centre fixed it be set rolling on a perfectly rough horizontal plane, it will represent the motion of the free rotating body not in space only, but also in time : the body and the top may be conceived as continually moving round the same axis, and at the same rate, at each moment of time. The reader is referred to the memoir in the rhilosophical Transactions for 18G6. il! i!M H'!| ^ i' i ■J ' i' ' \ ■'.- " ; . I r ■ ' ; ■■>■ ' yii - . I ^lil'l #11 !^"::l:l '1 1) w 412 MOTION UNDER NO FOKCES. 517. To find the motion of the extremity of the instantaneous axis along the polhode which it describes we have merely to sub- stitute from the equations w, _ a)^ _ Wg _ w _ / y 1^ in any of the equations of Art. 508. For example we thus obtain dx_ ITB-Cyz X ,• — M A BG ~,&c., &c., {A-a){A-B) (-V + r"), &c., &c. Ex. 1. A point P moves along a polhode traced on an ellipsoid, show that the length of the normal between P and any one of the principal planes at the centre is constant. Show also that the normal traces out on a principal plane a conic bimilar to the fooal conic in that plane. Also the measure of curTature of an ellipsoid along any polhode is constant. Ex. 2. Show that the line OJ used in Art. 234 to find the pressnre on the fixed point is at right angles to the invariable line, and parallel to the tangent plane to the momental ellipsoid at the point where the invariable line cuts it. 8howalsothatO'^=-c^ + a,'^^^'-^'^^-^''^-<^'^«7»)^^^^^»^%herep,.^„l>3 are the sum of the products A, B, C taken respectively one, two and three together. 518. Since the herpolhode is traced out by the points ef contact of an ellipsoid rolling about its centre on a fixed plane, it is clear that the herpolhode must always lie between two circles which it alternately touches. The common centre of these circles will be the foot of the perpendicular from the fixed centre on the fixed plane. To find the radii let OL be this perpendicular, and / be the point of contact. Let LI= p. Then we have MAC cullagh's construction. 413 The radii will therefore be found by substituting for w' its greatest and least values. But by Art. 509, these limits are \ and the greater of the two quantities X,, Xj. The herpolhode is not in general a re-entering curve ; but if the angular distance of the two points in which it successively touches the same circle be commensurable with 27r, it will be re- entering, i.e. the same path will be traced out repeatedly on the fixed plane by the point of contact. 619. To explain Mac Cullagh's representation of the motion hy means of the ellipsoid of gyration. This ellipsoid is the reciprocal of the momental ellipsoid, and the motion of the one ellipsoid may be deduced from that of the other by reciprocating the properties proved in the preceding Articles. We find, (1) The equation to the ellipsoid referred to its principal axes is A^ B^ C~ M' (2) This ellipsoid moves so that its superficies always passes through a point fixed in space. This point lies in the invariable line at a distance -r- from the fixed point. By Art. 509 we know that this distance is less than the greatest, and greater than the least semi-diameter of the ellipsoid. (3) The perpendicular on the tangent plane at the fixed point is the instantaneous axis of rotation, and the angular velocity of the body varies inversely as the length of this perpendicular. 1 /T lip be the length of this perpendicular, then ^ — 'K/ll' (4) The angular velocity about the invariable line is constant and = ^ . The corresponding curve to a polhode is the path described on the moving surface of the ellipsoid by the poin*j fixed in space. This curve is clearly a sphero-conic. The equations to the sphero- conic described under any given initial conditions are easily seen to be ^ x^ t/" z^ 1 MT ^+f + ^=U^n A'^ B'^C M' These sphero-conics may be shown to be closed curves round the axes of greatest and least moment. But in one case, viz. 5 ; ; if (f: ; ■' 4U MOTION XTNDER NO FOllCEH. » i: I . i 'i I m i ' ' when 7„ = Ji, whore B is neither the greatest nor least mo- ment of inertia, the sphero-conic becomes the two central circular sections of the ellipsoid of gyration. The motion of the body may thus be constructed by means of cither of those ellipsoids. The momcntal ellipsoid resembles the general shape of the body more nearly than the ellipsoid of gy- ration. It is 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. C20. MacCullagb has uncd tlio ellipsoid of g^'ration to obtain n gcomotrioal intcrprotatiou of the solution of Euler'H equations in terms of elliptic integrals. The ellipsoid of gyration moves so as always to touch a point L fixed in space. Let us now project the point L on a plane passing through the axis of mean moment and making an angle a with the axis of greatest moment. This projection may be effected by drawing a straight line parallel to either the axis of greatest moment or least moment. We thus obtain two projections which we will call P and Q. Those points will bo in a plane PQL which is always perpendicular to the axis of moan moment. As the body moves about the point L describes on the surface of the ellipsoid of gyration a sphero-conic KK\ and the points P, Q describe two curves pp', qq' on the plane of projection OBD. If the sphero-conic as in the figure enclose the extremity A of the axis of greatest moment, the curve inside the ellipsoid is formed by the projection parallel to the axis of greatest moment, but if the sphero-conic enclose the axis of least moment, the inner curve is formed by the projection parallel to that axis. The point P which describes the inner curve will obviftiisly travel round its projection, while the point Q which describes the outer curve will oscillate between two limits obtained by drawing tangents to the inner projection at the points where it cuts the axis of mean moment. . . ... inner curve is of mean maccullagh's construction. 415 Since the direction cosines of OL are proportional to Au^, liu^, Cu^ it is easy to see that, it x, y, zaxo the co-ordinatca of L, Au^ liu.i Cwj G ^MT Let OP=p, 0(1= p', and let the antjlcs those radii vcctores make with the pluno contiiining the axes of greatest and leaat moment ho

    , BOQ- -0': we then have - p sin ) pco3' =y=Bu.,{M'I')-n ^ '' It is proved in treatises on solid geometry that, if tlio plane on which the projection is made is one of the circular sections of the ellipsoid, the projections will he circles. This result may ho verified by finding p or p' from these equations, licmcmboring that p and p' are constants, let us substitute in Eulcr's equation from (2) and the first of equations (3). We have P -ji= jp iJmT pp' sin a cos a cos 0'. Since p' cos 0' is the ordinate of Q, we see that the velocity of V varies as the ordinate of Q, and in the same way the velocity of Q varies as the ordinate ofV, To find the constants p, p' we notice tl I p is the value of y obtained from the equations to the sphero-conic when s=0. Wo thus have s = '^AT-Ct'>')B ,j ^ {,G" - CT)B ^ MT{A-Ji)' ^ MT{Ii-C)* the latter being obtained from the former by interchanging the letters A and C. Hence ( velocity \ ^A -B i^, — j^ /ordinate \ 521. Since p' sin 0' = p sin (p, wo have by substitution where X' has the same value as in Art. 510. Let us suppose ^ expressed in terms of t by the elliptic integral • X((_r)= \ , so that 0=amX(<-T). Substituting this value of

    ^ i 7 11 '. II i! whose centre is at the fixed point, and which is either fixed in the body or fixed in space at our pleasure. This will be found con- venient when we wish to use a diagram. 525. Let the principal axes at the fixed point be taken as the axes of co-ordinates. The axes of reference are therefore fixed in the body but moving in space. By Art. 512, the direction-cosines of the invariable line are L«. G B(o. a G (o. a to. ft). cosmes of the instantaneous axis are — , -^ (0 (O equations (1) and (2) of Art. 511, we easily find and the direction- From the Wo 0) A(o^' + Bay,' + Geo,' = (^ V + ^"< + ^O ^-2 . • If we take the co-ordinates x, y, z to be proportional to the direction-cotines of either of these straight lines and eliminate w,, Wg, &)g by the help of this equation, we obtain the equation to the corresponding cone described by that straight line. In this way we find that the cones described in the body by the invariable line and the instantaneous axis are respectively AT-G' , BT-G' « + r GT-G' ,2 — = 0, A *" ' B ^ ' G A {A T- G') x'' + B{BT-G')y'+C {CT- G') z' = 0. These cones become two planes when the initial conditions are such that G' = BT. Ex. 1. Show that the circular sections of the invariable cone are parallel to those of the ellipsoid of gyration and perpendicular to the asymptotes of the focal conic of the momental ellipsoid. 526. There is a third straight line whose motion it is sometimes convenient to consider, though it is not nearly so important as either the invariable line or the instantaneous axis. If x, y, z be the co-ordinates of the extremity of a radius vector of an ellipsoid referred to its principal diameters as axes and if a, 5, c be the semi- X tJ z axes, the straight line whose direction-cosines are - , r > - is called the eccentric line a c of that radius vector. Taking this deAnition, it is easy to see that the direction- cosines of the eccentric line of the instantaneous axis with regard to the momental ellipsoid are "j. / = , <>>ax/f> '^»\/f' ^^^^^ ^^^ ^^^° *^^^ directioi-cosir of the eccentric line of the invariable line with regard to the ellipsoid of gy/ation. This straight lino may therefore be called simply the eccentric line and the c( no described by it in the body may be called the eccentric cone. Ex. 1. The equation to the ecoentiic cone referred to the principal axes at the fixed point is (A T - cr^) x^+{nr-G^) i/ + (ct - c') c« = o. ■cosines 1 in the nd con- in as the fixed in i-cosines irection- rom the lal to the linate Wj, on to the this way invariable = 0. litions are parallel to of the focal onvenient to le line or the radius vector be the semi- cccentric line le direction- le momental ictior -cosines of gy^'ation. and the C( no 1 axes at the THE INVARIABLE AND INSTANTANEOUS CONES. 419 This cone has the same circular sections as the momental ellipsoid and cuts that ellipsoid in a sphero-conic. Ex. 2. The polar piano of the instantaneous axis with regard to the eccentric cone touches the invariable cone along the corresponding position of the invariable line. Thus the invariable and instantaneous cones are reciprocals of each other with regard to the eceontric cone, 6.27. Let a sphere of radius unity be described with its centre at the fixed point about which the body is free to turn. Let this sphere be fixed in the body, and therefore move with it in space. Let the invariable line, the instantaneous axis, and the eccentric line cut this sphere in the points L, I, and J5/ respectively. Also let the principal axes cut the sphere in A, B, C. It is clear that the intersections of the invariable, instantaneous, and eccen- tric cones with this sphere will be three sphero-conics which are represented in the figure by the lines KK\ JJ', DD\ respectively. The eye is supposed to be situated on the axis OA, viewing the sphere from a considerable distance. All great circles on the sphere are represented by straight lines. Since the cones are co- axial with the momental ellipsoid, these sphero-conics are sym- metrical about the principal planes of the body. The intersections of these principal planes with the sphere will be three arcs of great circles, and the portions of these arcs cut off by any sphero- conic are called axes of that sphero-conic. If we put a = in the equations to any one of the three cones, the value of - is the tangent of that semi-axis of the sphero-conic which lies in the plane of xy. Similarly, putting y = 0, we find the axis in the plane of xz. If (a, h), (a, J'), (a, /3) be the semi-axes of the invariable, instantaneous, and eccentric sphero-conics respectively, we thus find 27—2 I I.. It • iHlli >Sif lil';'! f v or < BT. sin^ b Ex. 1. If we put l-e3=-r-s— we may define e to be the eccentricity of the Bin" a •' sphero-conio whose semi>axes are a and b. If e and e' be the eccentricities of the AB-C BA-C and invariable and eccentric sphero-conics respectively, prove that e^ = B — C ^—'J~n ^^ *^*t ^oth theue eccentricities are independent of the initial conditions. Ex. 2. If the radius of the sphere had been taken equal to ( wy,) instead of nnity, show that it would have intersected the ellipsoid of gyration along the invari- —7f^- \ , it would have intersected the momental ellipsoid along the eccentric ellipse. Ex. 3. A body is set rotating with an initial angular velocity n about an axis which very nearly coincides with a principal axis 00 at a fixed point O. The motion of the instantaneous axis in the body may be found by the following formulae. Let a sphere be described whose centre is 0, and let / be the extremity of the radius vector which is the instantaneous axis at the time t. If {x, y) be the co-ordinates of the projection of I on the plane AOB referred to the principal axes OA, Ob, then * = V-B (B - C) i sin {pnt + M), y=jA(A~ C) L cos {pnt + M), IB — Cf\ (A — C) where p'*=- -^ , and L, if are two arbitrary constants depending on the initial values of x, y. Ex. 4. If in the last question L be the point in which the sphere cuts the invariable line, if (p, B) be the spherical polar co-ordinates of C with regard to L as origin, and a the radius of the sphere, then P^ r.n^^^ L^ \2AB^C (A + n) + (A -D)Cco62 (pnt + M)U = ^^t + CO J p aPdt neAOB, le imagi- t cut the aeir con- : 00, i.e. , moment icity of the eitiea of the AB-C and BA-C I conditions. instead of g the invari- jrsected the bout an axis nt 0. The le following le extremity (x, y) be the rincipal axes ding on the lere cuts the th regard to THE INVARIABLE AND INSTANTANEOUS CONES. 421 528. To find the motion of the invariable line and the instan- taneous axis in the body. Since the invariable line OL is f.xed in space and the body is turning about 01 as instantaneous axis, it is evident that the direction of motion of OL in the body is perpendicular to the plane 10 L. Hence on a sphere whose centre is at the arc IL is normal to the sphero-conic described by the invariable line. This simple relation will serve to connect the motions of the invariable line and the instantaneous axis along their respective sphero- conics. 529. Lot V be the velocity of the invariable line along its sphero-conic, then since tbe body is turning about 01 with an- gular velocity &», and OL is unity, we have t; = w sin LOT. But T . . T by Art. 514 ^ = to cos i OL Elirctittatmg o) we have v = ^ tan LOT. 530. Produce the arc IL lo cut the axis AK in N, so that LN \s &. normal to the sphero-conic described by the invariable line. Taking the principal axes at the fixed point as axes of reference, the direction -cosines of OL and 01 are respectively proportional to ^w,, Bm^, Cq>^, and Wj, a^, Wg. The equation to the plane LOT is {B - C) a^w^x + {C-A) (0^(0 J/ + {A-B) co^a^z = 0. This plane intersects the plane of xy in the straight line ON^ hence putting 2 = 0, we find the direction-cosines of ON to be proportional to {A — G)o)^, {B— C) a>^, and 0. Hence ,o,LON^i^^.zSM±m^S>L. Gsl{A-Cfo>,' + {B-Crftoy The numerator of this expression is easily seen to be 0^ — CT. Expanding the quantity under the root we have A\' + B'co^'-2G{Aa>^'+B(o^')+C'{,% which is clearly the same as G« _ C V - 2 C (T - (7a,/) + C (a,» - to,'). Substituting we find coaLON= G'-CT G^/G''-2GT+G Q> Um L ON = C\/OW~ T' G'-CT' ' t! Ill * ni il ii l|!f 1 If I. i J t ■ ' it' f i- . i ii -.1 jrf Ji l! 422 MOTION UNDER NO FORCES. But jy = 0) COS L 01, '. tan LOI^'^l^'—^. Hence the T .. iBxiLOI G^-CT , . ,, . . . .1 I . ratio ~ — y^^ = — j:^ — , and is therefore constant throughout the motion. Treating the other principal planes in the same way, we see that this proposition supplies us with a geometrical meaning for G^ G^ G* the three expressions -r-ji— 1, 'vfff~'^> ^^^ 'PT~ ^' Combining this result with that given in the last Article, we see that the velocity of L I ^ G'-CT along its conic] ~ CQ ' where n is the angle LON. If we adopt the conventions of spherical trigonometry, n is also the length of the arc normal to the sphero-conic intercepted between the curve and the principal plane AB oi the body. Ex. 1. If the focal lines of tlie invariable cone cwt the sphere in S and S', these points are called the foci of the sphero-conic. Prove that the velocity of L resolved perpendicular to the arc 8L is constant throughout the motion and equal l\(G^- BT)(AT-(P) \k If LM be an arc of a great circle perpendicular to the axis containing the foci, and p be the arc SL. prove also that *°0r AB dp_ G \ (A-C){B-0 )i dt " c\ AB \ sm LM. 'if if ; J i Ex. 2. Prove that the velocity of L resolved perpendicular t& the central radius AT-GP vector AL is — -,-pi — cot AL. Ex. 3. If r, /, r" bo the lengths of the arcs joining the extremity A of a princi- pal axis to the extremities £, I, E of the invariable line, instantaneous axis, and eccentric line respectively ; 0, &, 0" the angles these arcs make with any priucipftl plane A OB, prove that CO?: " _ cos r' _ cos r" tan (y tan ff' B sjBC' where f^^^aroi/. Tliis theorem will enable us to discover in what manner the motions of the three points L, I, E are related to each other. Ex. 4. Show that the velocity of the instantaneous axis along it» sphero-conic is TT, — -T-jT' ^^^ '*' '^'^^ ^' ^^^^'^ '*' ^^ ^^^^ length of the normal to the instantaneous sphero-couio intercepted between the curve and the arc AB, and f-arc LI. liculor to the sentral radins maiiuer tlio THE CONE OF THE HERPOLHODE. 423 Comparing thia result with the corresponding formula for the motion of L given in Art. 630, we see that for every theorem relating to the motion of L in its sphere- conic there is a corresponding theorem for the motion of /. For example, if S' be a focus of the instantaneous sphero-conic, we see that the velocity of / resolved per- pendicular to the focal radius vector S'l bears ;' constant ratio to cos LI, This constant ratio is^j. - 534. Let be the fixed point, 01 the instantaneous axis. Let the angular velocity oi about 01 be resolved into two, viz. T a uniform angular velocity -p about the invariable line OL, and an angular velocity to sin lOL about a line OH lying in a plane fixed in space perpendicular to the invariable line, and passing through the fixed point 0. Let this fixed plane be called the invariable plane at 0. As the body moves, OH will describe a cone in the body which will always touch this fixed plane. The velocity of any point of the body lying for a moment in OH is unaffected by the rotation about OH, and the point has therefore only the motion due to the uniform angular velocity about OL. We have thus a new representation of the motion of the body. Let the cone described by OH in the body be constructed, and let it roll on the invariable plane at with the proper angular velocity, while at the same time this plane turns round the in- T variable line with a uniform angular velocity yy . The cone de- scribed by OH in the body has been called by Poinsot the Boiling and Sliding Cone. 535. To find a construction for the sliding cone. Its generator OH is at right angles to OL, and lies in the plane lOL. Now OL is fixed in space ; let OL' be the line in the body which, after an interval of time dt, will come into the position OL. Since the body is turning about 01, the plane LOL' is perpendicular to the plane LOT, and hence OH is perpendicular to both OL and OL'. That is, OH is perpendicular to tlie tangent plane to the cone 426 MOTION UNDER NO FORCES, ,: 4 ■!t: described by OL in the body. Tlio cone described by OH in the body is therefore tlie reciprocal cone of that described by OL. The equation to the cone described by OL has been shown to be AT- O^ , . BT- G" , CT- CP , ^ — :r— ^ + — 5— 2/' + — 77— «'= 0. Hence the equation to the cone described by Oil is A . B . G AT-G ,a!' + BT-G' y' + CT-(P «' = 0. The focal lines of the cone described by OH are perpendicular to the circular sections of the reciprocal cone, that is the cone described by OL. And these circular sections are the same as the circular sections of the ellipsoid of gyration. Hence the focal lines lie in the plane containing the axes of greatest and least moment, and are independent of the initial conditions. This cone becomes a straight line in the case in which the cone described by OL becomes a plane, vi/. when the initial con- ditions are such that G^ = BT. 53G. To find the motion o/OH in space and in the body. Since OL, OH and 01 are always in the same plane the motion of OH in space round the fixed straight line OL is the dt m same as that of 01, and is given by the expression for Art. 532. To find the motion of OH in the body it will be convenient to refer to the figure of Art. 532. Produce the arcs PL, PL to H and H' so that LH and L'H' are each quadrants. Then // and H' are the points in which the axis OH intersects the unit sphere at the times t and t + dt. We have therefore /velocity\ _ /velocityN V of // ; ~ V of Z } sm ( P + 2; T ^ = -p tan if cot p. smp Substituting for tan p as before we may express the result in terms of §" or n at our pleasure. Since the cone described by OH in the body rolls on a plane which also turns round a normal to itself at 0, it is clear that the angular velocity of OH in the body is loss than the angular velocity of OH in space by the angular velocity of the plane, i. e. T G' /velocity\ _ rZ0 V of ^ J~dt r/ in the by OL. n to bo mdicular the cone same as the focal md least hich the itial con- dy. lane the •Z is the dt ir m nvenient PL, PL' Then ects the result in a plane that the angular ne, i. e. MOTION OF THE PRINCIPAL AXES. Motion of the Piinclpal Axes. 427 537. To find the angular motions in space of the pnncipal axes. Since the invariable line OL is fixed in space it will be con- venient to refer the motion to this straight line as axis of z. Let OA^ OB, OG be the principal axes at the fixed point 0, and let, as before, a, /3, 7 be their inclinations to the axis OL or OZ. Let X, fi, V be the angles the planes LOA, LOB, LOC make with some fixed plane LOX passing through OL. Our object is to find J- and -r- with similar expressions for the other axes. This problem is really the same as that discussed in Art. 235, but it will be found advantageous to make a slight variation on the demonstration. Describe a sphere whose centre is at the fixed point, and whose radius is unity. Let the invariable line, the instantaneous axis and the principal axes cut this sphere in the points L, I, A, B, C respectively. The velocity of A resolved perpendicular to LA will then be sin a -^ . But since the body is turning round 01 as instantaneous axis, the point A is moving perpendicularly to the arc lA^ and its velocity is w sin lA. Resolving this per- pendicular to the arc LA, we have sin a -rr = ft) sin AI cos LAI at = 0) cos LI— cos LA cos I A sin Lxi ' by a fundamental formula in spherical trigonometry. But w cos LI is the resolved part of the angular velocity about OL, which is T equal to -^ > ^^^ ^ cos lA is the resolved part of the angular ! til m f i?< I. -^ !>■ 428 MOTION UNDEB NO FORCES. Mi velocity about OA, which is a>^, Wo have therefore . ^ tl\ T 8in a Ti = -7=; — w, cos a, at O ^ a result wliich follows immediately from Art. 249. G cos a = Au)^, we have dX^ T G'cos'g This result may also be written in the form AT-a'' Since sin' a .(1). d\_T dt a"^ AG cot' a .(2). da. 538. To find -^ we 'may proceed in the following manner. We have cos a = lO), G' cos /3 = -^, cos 7 = -Tj\ Substituting in Euler's equation ^ Tt dfx we have sm a dt = ( -^ - -^JG cos ^ cosy (3). But by Art. 508 cos a, cos/9, cos 7 are connected by the equations G" cos'a cos^ cos''7 AT ■*■ ~B~ "^ n7~ cos'a + cos'/S + cos''7 = 1 .(4). If we solve these equations so as to express cos /8, cos 7 in terms of cos a, we easily find Bin , fday G^ (CP-CT A-C „ \/ G^-BT A-B G« A COS' .).. (5). 539. Since the left-hand side of equation (6) is necessarily real, we see that the values of cos* a are restricted to lie between certain limits. If the axis whose motion we are considering is the axis of greatest or least moment let B be the axis 0^-CT A of mean moment, G^~BT A In this case cos^ a must lie between the limits and G« A-G if both be positive. By Art. 509 the former of these two is positive G* A-B and less than unity ; this is easily shown by dividing the numerator and the de- nominator by A C(P. If the latter is positive the spiral described by the principal axes on the surface of a sphere whose centre is at the fixed point lies between two concentric circles which it alternately touches. If the latter limit is negative cos a lias no inferior limit. In this case the spiral always lies between two small circles on the sphere, one of which is exactly opposite the other. MOTION OP THE PRINCIPAL AXES. 429 COS 7 If tltA axis couHidoroil is tho axiH of moan moment, coh> a muHt lio outiide tlio Bamu two liuiitH an befui-o. Both these are positive, but one In greater and the otlier leHH tbau unity. Tho spiral thoroforo lies between two amall circIoB oiJpoHito each other. In order that ,. may vanish we must havo G'co8'a = iir, but this by substitu- tion makes t- imaginary. Thus t- always keeps one sign. It is easy to see that G" if tho initial conditions arc stoh that -=^ is less than the mcmont of inertia about tho aids which describes tlio hi iral wo are considering, tho angular velocity will bo greatest when the axis is nearest tho invariable hue and least when tho axis is furthest. The reverse is the case if -yp is greater than tho moment of inertia. 640. Ex. 1. Let OM be any straight line fixed in tho body and passing through and let it cut the eUipsoid of gyration at in Ihe point M. Let OM' bo the perpendicular from on the tangent plane at Hf. If OM~r, OM'-p, and if i, i' be the angles OM, OM' make with the invariable lino OL, prove that Bin* 1 4^ ■■ at TO.., ; cos I cos 1 , Q pr where j is tho angle the plane LOM makes with some plane fixe i in space passing through OL. This follows from Art. 249 or from Art. 537. Ex. 2. If KLK' be the spiral traced out by the invariable line in the manner described in Art. 527| show that % r^ ^A /vectorial area\ where X is the angle described by the plane containing the invariable line and the principal axis OA, Ex. 3. If xj/ be the angle described in space by the plane containing the invari- able line and any straight line OM, fixed in the body, passing through and cutting the sphere in M, prove that . T -A /vectorial area \ where MN is any spherical arc fixed in the body and cutting in N the sphero-conic described by the invariable line. Ex. 4. If we draw three straight lines OA, OB, OC along the principal axes at the fixed point of equal lengths, tho sum of the areas conserved by these lines on the invariable plane is proportional to tho time. [Poinsot.] Ex. 5. If the lengths OA, OB, OG bo proportional to tho radii of gyration about the axes respectively, the sum of the areas conserved by these lines on the invariable plane will also be proportional to the time. [Poinsot.] ill I ;i i'll ti'sli " .-i 430 MOTION UNDER NO FORCES. il I r I.' Motion of the hody when two principal axes are equal. 541. Let the body be rotating with ,an angular velocity ta about an instantaneous axis 01. Let OL be the perpendicular on the invariable line. The momental ellipsoid is in this case a spheroid, the axis of which is the axis of unequal moment in the body. Let the equal moments of inertia be A and B. From the symmetry of the figure it is evident that as the spheroid rolls on the invariable planes, the angles L OG, L 01 are constant, and the three axes 01, OL, OC are always in one plane. Let the angles LOC = %IOC=i. Following the same notation as in Art. 508, we have ft). = ft) cos I, ft), + &>a = ft) sm I, T=(AsmU+CcosU)o>\ AVe therefore have Cftjg _ C cos t cos 7 = jA^shiH+V'coFi' This result may also be obtained as follows. In any conic if i and 7 be the angles a central radius vector and the perpendicular on the tangent at iio extremity make with "^he minor axis, and if a, b be the semi-axes, then tan 7 = — a ta,n i. Applying this to the momental spheroid, we have tan 7=7^ tan i. The angle i being known from the initial conditions, the angle 7 can be found from either of these expressions. The peculiarities of the motion will then be as follows. The invariable line describes a right cone in the body whose axis is the axis of unequal moment, and whoso semi-angle is 7. The instantaneous axis describes a right cone in the body whose axis is the :\xis of unequal moment, and whose semi-angle is I. The instantaneous axis describes a right cone in space, whose axis is the invariable line, and whose ^emi-angle is i ~ 7. The axis of uneqiial moment describes a right cone in space whose axis is the invariable line, and whose semi-angle is 7. The angular velocity of the body about the instantaneous axis varies as the radius vector of the spheroid, and is therefore constant. MOTION WHEN A = B. 431 542. The rate of motion of the invariable line and the instantaneous axis in the body may be found most readily by referring to the original equations of motion in Art. 508. We have in this case >-=ol A-r^ — {A — C) (0^(0 COS 1 = ■ A-^-\-{A-C)(o^cocosi=0 Solving these by differentiating the first and eliminating w^,' we find a)^ = i cos I — -^ — (ot COS z ft)j=-Fsinr ^ (otco&z + n, A A-C where i^ and /are arbitrary constants. Let the projection of either the instantaneous axis or the invariable line on the plane per- pendicular to the axis of unequal moment make an angle ;^ with any fixed straight line which may be taken as axis OA. Then tan ;^ = — ^ Hence we find __ 2 dt A- G A (o cos I. 543. To find the common rate of motion in space of the instantaneous axis and the axis of unequal moment. Let G be the extremity of the axis of figure of the momental ellipsoid, and let H be the rate at which the plane LOG is turning round OL. Let CM, CN be perpendiculars on GL and CI. Then since the body is turning round GI, the velocity of G is GN.(o. But this is also CM M. Since GM=OGsmy, CN= (9(7 sin i, we have at once fl sin 7 = G) sin i, whence fl can be found. 544, Ex, 1. If a right circular cone whose altitude a is double the radius of its base turn about its centre of gravity as a fixed point, and be originally set in motion about an axis inclined at an angle a to the axis of figure, the vertex of the cone will describe a circle whose radius is -r asin o, 4 [Coll. Exam.] Ex. 2, A circular plate revolves about its centre of gravity as a fixed point. If an angvdar velocity w were originally impressed on it about an axis making an angle a with its plane, a normal to the plane of the disc will make a revolution in space in 27r time — ; — [Coll. Exam.] w>/l + 3sin*o :!■ Il!| lir ! n 432 MOTION UNDER NO FORCES. Ex. 3. A body wliich can turn freely about a fixed point at wbich two of the principal moments are equal and less than the third, is set in rotation about any axis. Owing to the resistance of the air and other causes, it is continually acted on by a retarding couple whose axis is the instantaneous axis of rotation and whose magnitude is proportional to the angular velocity. Show that the axis of rotation will continually tend to become coincident with the axis of unequal moment. In the case of the earth therefore, a near coincidence of the axis of rotation and axis of figure is not a proof that such coincidence has always held. Astronomical Notices, March 8, 1867. Motion when G' = BT. ' if I H !r 1 : I 545. The peculiarities of this case have been already alhuleil to in Art. 508. When the initial conditions are «uch that this relation holds between the Vis Viva and the Momentum of the body the whole discussion of the motion becomes more simple*. The fundamental equations of motion are Solving these, we have B-C G^-B'ay.^ = bt] (o' = A-C AB , A-B G' B'- '' ft)„ BC But d(o„ C-A dt B <»i<»3; .(1). (2). k\ do), dt --w^ B) { B- G) G' - R AG ' B* <»„ When the initial values of w^ and w, have like signs, {G- A) &>,&•, d CO. is negative and therefore -rj' must be negative, hence in this expression the upper or lower sign is to be used according as the initial values of cd,, Wg have like or unlike signs. B' " G'- B% } dt = V {A-B){B~G ) AG » •' III * This case appears to have been considered by nearly every writer on tins subject. As examples of different methods of treatment the reader may consult Lvrjmdrc, Traite den Fonctious EllqHiques, 1825, Vol. I. page 382, and Poimot Theorie Nonvelle dc la Rotation des coqjs, 1852, patrc 104. tvo of the about any tally acted md whoso f rotation ment. In a and axis ronomical alliKle,'. Taking the principal axes at the fixed point as axes of refer- ence, the equations of the invariable line are -j — = -^r— = 77— . ^ A(o^ xjo), C«»3 Eliminating a>^ and a^ the locus of the invariable line is one of the two planes /A-B , /B-G OS 71 Si The equations of the instantaneous axes are — = -^ = — . ^ ft), G)j 6)3 Eliminating ft), and g), the locus of the instantaneous axis is one of the two planes ^A{A-B)x=± ^G{B-C) z. In these equations since — follows the sign of -^ the upper or lower sign is to be taken according as the initial values of ft),, 6)3 have like or unlike signs. These planes pass through the mean axis, and are independent of the initial conditions except sofarthat (?' = i?r. R. D. 28 ill r. , '! 1 n f T' 1 t 1^ 3i> t 434 MOTION UNDER NO FORCES. The rolling and sliding cone is the reciprocal of that described by the invariable plane, and is therefore the straight line perpen- dicular to that plane which is traced out by the invariable line. Ex. 1. Sbow that the planes described by the invariable line coincide with the central circular sections of the ellipsoid of gyration and are perpendicular to the asymptotes of that focal conic of the momeutal ellipsoid which lies in the plane of the greatest and least moments. Ex. 2. The planes described by the instantaneous axis are perpendicvdar to the umbilical diameters of the ellipsoid of gyration and are the diametral planes of the asymptotes of the focal conic in the momental ellipsoid. 547. The relations to each other of the several planes fixed in the body may be exhibited by the following figure. Let A, By C be the points in which the principal axes of the body cut a sphere whose centre is 0, and radius unity. Let BLK', BIJ' be the planes traced out by the invariable line and the instantaneous axis respectively. Then by the last Article tanCA =V6"2^i5'*^^^'^=V2-^5- Hence v sin lOL about a perpendicular to OL, viz. OH, it follows that the plane fixed in the body turns round the line fixed in space with a T C uniform angular velocity ^ or ^ . At the same time the plane moves so that the line fixed in space appears to describe the plane with a variable velocity w sin lOL, If /8 be the angle BL, T this has been proved in the last Article to be ^ n sin fi. 549. The cone described by OH in the body is the reciprocal cone of that described by OL, and from it we may deduce re- ciprocal theorems. The motion is therefore such that a straight line fixed in the body, viz. OH, describes a plane fixed in space, viz. the plane perpendicular to OL. The straight line moves 28—2 ' ! I'M M' 436 MOTION UNDER NO FORCES. ! \: is ■ i ' if! T G along this plane with a uniform angular velocity equal to ^ or ^ , ■'vhile the angular velocity of the body about this straight line is +-^Jisin^. 550. The motion of the principal axes may be deduced from the general results given in Art. 537. But we may also proceed thus. Since the body is turning about 01, the point B on the sphere is moving perpendicularly to the arc IB. Hence the tangent to the path of B makes with LB an angle which is the complement of the constant angle IBL. The path traced out by the axis of mean moment on a sphere whose centre is at is a rhumb line which cuts all the great circles through L at an angle whose cotangent is ± n. 65i. To find the motion of the instantaneous axis in spac3. This problem is the same as that considered in Art. 532. We may however deduce the result at once from Art. 548. The angle ILB is always a right angle, it therefore follows that the angular velocity of / round L is the same as that of the arc BL round L. T But the angular velocity of the latter is constant and equal to ^. If then be the angle the plane hOI containing the instanta- neous axis arid the invariable line makes with some fixed plane passing through the invariable line, we have 7^ = 75 • 652. To find the equation of the cone described by the instantaneous axis in space, we require a relation between if and <^, where f is the arc IL on the sphere. From the right-angled triangle ILB we have n sin y9 = tan 5", and by Art. 547, cot| = V£'e ^ • Eliminating ^, we shall have an expression for §" in terms of U We find o -?^ = cotf+tanf = V:^e tan ^22 By the last Article (f> = ^t + F, where F is some constant. Let us substitute for t in terms of j_and let us choose the plane from v.hich is measured so that s/Ee^^^— 1. The equation to the cone traced out in space by the instan- taneous axis is 2/icotf=e«* + e-»*. T G ight line iced irom proceed B on the ence the ich is the aced out is at is L at an i32. We rhe angle e angular round L. T ual to ^. instanta- ced plane by the 1 ^ and <}>, ht-angled jrms of t. constant, ihe plane le instan- CORRELATED AND CONTRARELATED BODIES. 437 When ^ = 0, we have tan(;'=n. Therefore the plane fixed in space from which (f> is measured is the plane containing the axes of greatest and least moment at the instant when that plane contains the invai'iable line. On tracing this cone, we see that it cuts a sphere whose centre is at the fixed point in a spiral curve. The branches determined by positive and negative values of are perfectly equal. As increases positively the radial arc ^ continually decreases, the spiral therefore makes an infinite number of tuins round the point L, the last turn being infinitely small. 2mb Ex. In the herpolhode '=/-«» +c-'»*, if the looua of the extremity of the polar subtangent of this curve be foand and another carve be similarly generated from this locus, the curve thus obtained will be similar to the herpolhode. [Math. Tripos, 1863.] On Correlated and Contrarelated Bodies, 553. To compare the motions of different bodies acted on hy initial couples whose planes are parallel. Let a, /S\ 7 be the angles the principal axes OA, OB, 0(7 of a body at the fixed point make with the invariable line OL. Then by Art. 511, Euler's equations may be put into the form dcosa , ^/l 1\ „ f. .-V — ^^-+G'f-g--^jco3/3cos7 = (1), with two similar equations. Let \, fi, v be the angles the planes LOA, LOB, LOG make with any plane fixed in space, and passing through OL. Then . » 1anes are al axes of imon axis ...#.(C). (7), ' O" ) and (4) dies being 1 to each, y turning > axis of tho described by NO ellipsoids 10 thing, the erbolic focal )f the couplo of Art. r>53 lion of Prof. CORRELATED AND CONTRARELATED BODIES. 43.9 the body whoso principal axes are A', B, C about tho cora- mon axis of the impulsive couples through an angle [jy — Trijt in the direction in which positive impulsive couples act*. 554. When the couples G and 0' are equal the condition (6) becomes A A' B lj:~ C Cf ~ G-' ' the bodies are then said to be correlated. If m omental ellipsoids of the two bodies be taken so that the moment of inertia in each bears the same ratio to the square of the reciprocal of the radius vector these ellipsoids are clearly confocal. When the couples G and G' are equal and opposite, the equation (6) becomes 1 1 _ 1^ 1 _ 1 1 _ r+ r A^ A:~ B^ B~'G^C~ G* * and the bodies are said to be contrarelated. 555. To compare the angular velocities of the two hodit xt any instant. Let ft) be the angular velocity of one body at any insta ^ M:en following the usual notation we have If the same letters accented denote similar quantities for the other body '« r"2 /cos (o =G (-^ a cos* /3 r + cos' 7'\ 2?" ' 6"* Bat remembering the condition (G) these give ..-.■.=(f4)[..«(^f,).co.,(«4).cosv|H-g;)]. * Since the cones described by the invariable line in the two bodies are identical, their reciprocal cones, 1. e. Poinsot's rolling and sliding cones, are also identical in the two bodies. Thus in the two bodies, the rolling motions of these cones are equal, but the sliding motions may be different. The si. ding motions represent T T' angular velocities about the invariable line respectively equal to ^ and ^, . Hence we have dt ~ (It ~ dt " (it dt~' dt G~ G" This remark on the former note is due to Prof. Cayley. '_ ' t m M ?5i 440 MOTION UNDER NO FORCES. ! 11 By referring to (7) the quantity in square brackets is easily T T seen to be ^ + T77 1 Ex. If two bodies be so related that their ellipsoids of gyration are confooal, and bo initially so placed that the angles (a, /3, 7) (o', ^, 7') their principal axes mako with the invariable lino of each are connected by the equations cos a cos a' cos /3 Cos/S* cos y cos 7' J A' ' Jb Jb' ' Jc ^/C" ' and if these bodies bo set in motion by two impulsive couples 0, 0' respectively proportional to iJaBG and Ja'B'C', then the above relations will always hold be- tween the angles (a, /3, 7) (a', /3', 7'). If p and p' be the reciprocals of -3; and -r- , then Op-Q'p' will bo constant throughout the motion, where \ X', &o., are the angles the planes LOA, L'O'A' make at the time t with their positions at the time (=0. 556. When a body turns about a fixed point its motion in space is represented by making its momental ellipsoid roll on a fixed plane. This gives no representation of the time occupied by the body in passing from any position to any other. The preceding Articles will enable us to supply this defect. To give distinctness to our ideas let us suppose the momental ellipsoid to be rolling on a horizontal plane underneath the fixed point 0, and that the instantaneous axis 01 is describing a polhode about the axis of A. Let us now remove that half of the ellipsoid which is bounded by the plane of BG, and which does not touch the fixed plane. Let us replace this half by the half of another smaller ellipsoid which is confocal with the first. Let a p^ane be drawn parallel to the invariable plane to touch this ellipsoid in /' and suppose this plane also to be fixed in space. These two semi-ellipsoids may be considered as the momental ellipsoids of two correlated bodies; If they were not attached to each other * This result may also bo obtained in tho following manner. By Art. 534 the T angular velocity w of one body is equivalent to an angular velocity ^ about the invariable line and an angular velocity 12 about a straight liuo Oil which is a gene- rator of the rolling and sliding cone. Hence w^ = ^o + 0". A similar equation with accented letters will hold for the other body. Since in the two bodies the angles between the principal axes and tho invariable line are equal each to each through- out the motion, the rolling motions of the two cones must be equal, hence Q=R'. It follows immediately that w'-«'»= -p, - ;^t„. Or' Cr ^ is easily mfooal, and axes mako respectively ,y3 hold be- i\ , d\' &c., are the tious at the notion in roll on a occupied ler. The nomental the fixed a polhode ellipsoid not touch f another a plane ellipsoid !'hese two ipsoids of ich other Irt. 534 the r about the 1 is a gene- uation with 3 the angles h through- ence Si=0'. CORRELATED AND CONTRARELATED BODIES. 441 and were free to move without interference, each would roll tho one on the fixed piano which touches at /, and the other on that which touches at /'. By what has been shown the upper ellipsoid (being the smallest) may be brought into parallelism with tho lower by a rotation ^M j ~ "^') about the invariable line. If then the upper plane on which the upper ellipsoid rolls be made to turn round the invariable line as a fixed axis with an angular velocity ^( t~'t)' *^® ^^^ ellipsoids will always be in a state of parallelism, and may be supposed to be rigidly attached to each other. Suppose then the upper tangent plane to be perfectly rough and capable of turning in a horizontal plane about a vertical axis which passes through the fixed point. As the nucleus is mado to roll with the under part of its surface on the fixed plane below, the friction between the upper surface and the plane will cause the latter* to rotate about its axis. Then the time elapsed will be in a constant ratio to this motion of rotation, which may be measured off on an absolutely fixed dial face immediately over the rotating plane. The preceding theory, so far as it relates to correlated and contrarelated bodies, is taken from a memoir by Prof. Sylvester in the Philosophical Transactions for 1866. He proceeds to in- vestigate in what cases the upper ellipsoid may be reduced to a disc. It appears that there are always two such discs and no more, except in the case of two of the principal moments being equal, when the "olution becomes unique. Of these two discs one is correlatetx and the other contrarelated to the given body, and they will be respectively perpendicular to the axes of greatest and least moments of inertia. Poinsot has shown that the motion of the body may be con- structed by a cone fixed in the body rolling on a plane which turns uniformly round the invariable line. If, as in the preceding theory, we suppose the plane rough, and to be turned by the cone as it rolls on the plane, the angle turned through by the plane will measure the time elapseu. * As the ellipsoid rolls on the lower plane, a certain geometrical condition must be satisfied that the nucleus may not quit the upper plane or tend to force it upwards. This condition is that the plane containing 01, 01', must contain the invariable line, for then and then only the rotation about 01 can be resolved into a component about Or and a component about the invariable line. That this condition must be satisfied is clear from the reasoning in tho text. But it is also clear from the known properties of coufocal ellipsoids. ' 1 I I: I 442 MOTION UNDER NO POUCES. • EXAMPLES*. 1. A right cone the base of whioli is an ellipso is supported at O the centre of gravity, and has a motion oommnnicatcd to it about an axis through per])cndicu- lar to the line joining G, and the extremity li of the axis minor of the base, and in the piano through B and the axis of the cone. Determine the position of the in- variable plane. liegult. The normal to the invariable plane lien in the plane passing through the axis of the cone and the axis of instantaneous rotation, and mokes uu angle 2. A spheroid has a particle of mass m fastened at each extremity of the axis of revolution, and the centre of gravity is fixed. If the body be set rotating about any axis, show that the spheroid will roll on a fixed plane during the motion provided — = r^fl--jj, where 31 is the mass of the spheroid, a and c are the axes of the generating ellipse, e being the axis of figure. 8. A lamina of any form rotating with an angular velocity a about an axis through its centre of gravity perpendiciUar to its plane has an angular velocity a \/ B^^p impressed upon it about its principal axis of least moment, A, B, C being arranged in descending order of magnitiide : show that at any time t the angular velocities about the principal axes are respectively „o< and that it will ultimately revolve about the axis of mean moment. 4. A rigid body not acted on by any force is in motion about its centre of gravity: prove that if the instantaneous axis be at any moment situated in the plane of contact of either of the right circular cylinders described about the central ellipsoid, it will be so throughout the motion. If a, b, c be the semi-axes of the central ellipsoid, arranged in descending order of magnitude, Cj, e^, e^ the eccentricities of its principal sections, Oj, 0^, R, the initial component angular velocities of the body about its principal axes, prove that the condition that the instantaneous axis should be situated in the plane above - ., - . 0, ahit, described is -^ = -= — = . 5. A rigid lamina not acted on by any forces has one point fixed about which it can turn freely. It is started about a line in the plane of the lamina the moment of inertia about which is Q. Show that the ratio of the greatest to the least angular velocity is hJa + B : Jb + y, where A,Boxq the principal moments of inertia about axes in the plane of the lamina. * These examples are taken from the Examination Papers which have been set in the University and in the Colleges. e contro of peqiemlicn- ittBu, and iu of the iu' [ng through IS un angle f tho axis of g about any on provided axes of tho )nt an axis lor velocity !ut, A,B, C r time ( the EXAMPLES. 443 ;s centre of ated iu the the central nding order fig, fig the , prove that )lane above 6. If tho earth were a rigid body acted on by no force rotating about a diameter which is not a principal axis, show that tho latitudes of places would vary and that (ho values would recur whenever J A - li J A - V Ju^dt is a multiple to 2wJli(J. If a man were to lie down when his latitude is a minimum and to rise when it be- comes a maximum, show that he would iucrease the vis viva, and so cause the polo of the earth to travel from the axis of greatest moment of inertia towards that of least moment of inertia. 7. If do bo the angle between two consecutive positions of the instantaneous sxf!t, prove that 8. If n be the angular velocity of the plane through the invariable lino and the instantaneous axis about the invariable line and X tho compouout angular velocity of the body about the invariable line, prove that as)'^<»-'("-!)("-i)(-')=»- 0. If a body move in any manner, and all the forces pass through tho contro of gravity, prove that T-^2|(loga,,)^^aogc.4jlog«,)=0. where w,, «g, Wj are the angular velocities about the principal axes at tho centre of gravity, and w is the resultant angular velocity. ! M ; i ibout which the moment 3ast angular aertia about ivo been set 1' m Il < il CHAPTER X. MOTION OF A BODY UNDER ANY FORCES. 557. In this Chapter it is proposed to discuss some cases of the motion of a rigid body in three dimensions as exo.mples of the processes explained in Chapter V. The reader will find it an instructive exercise to attempt their solution by other method," , for example, the equations of Lagrange might be applied with advantage in some cases. i f Motion of a Top. 658. A body two of whose principal moments at the centre of gravity are equal moves about some fixed point in the axis if unequal moment under the action of gravity. Determine the motion. See Art. S?-*. To give distinctness to our ideas we may consider the body to be a top spinning on a perfectly rough horizontal plane. Let the axis OZ be vertical. Let the axis of unequal moment at the centre of gravity be the axis OG and let this be called the axis of the body. Let h be the distance of the centre of gravity of the body from the fixed point and let the mass of the body be taken as uuity. Let OA be that principal axis at which lies in the plane ZOO, OB the principal axis perpen- dicular to this plane. If we take moments about the axis OC we have by Euler's equations (Art. 230), C^-{A-B)co,^*» \ / ,' r-s/ \ \ / ^' /\/\ • ^ ' / jction OQ ^ It is clear that the moment of the momentum about OZ will be constant throughout the motion. Since the direction- cosines of OZ referred to OA, OB, OG are — sin^, and cos^, this principle gives -Aw^amd + Gncos0 = E (2), where E is some constant depending on the initial conditions, and whose value may be found from this equation by substituting the initial value of a, and 0. The equation of Vis Viva gives A {(o * + (o^') + Cn^=F-2gh cos (3), where F is some constant, whose value may be found by substi- tuting in this equation the initial values of w^, w,, and ^ t, * To avoid confusion in the figure, the body which is represented by a top is drawn smaller than it should be. t If we eliminate Wj, Wj from equations (1), (2), (3) we have two equations from which and ^ ^^7 be found by quadratures. These were first obtained by Lagrange in his Mccanique Analijtique, and were afterwards given by Poisson in his Trait4 de Blecanlquc, The former passes them over with but slight notice, and proceeds to discuss the email oscillations of a body of. any form suHpeudod under the action of gravity from a fixed point. The latter limits the equations to ;;lf! t< !h . I'' 446 MOTION UNDER ANY FORCES. 650, Let ns measure along the vertical OZ, in the direction opposite to parity as the positive direction, two lengths 0^/^= T^, 0F=-^ ".-"-'. These lengths Cn 2gh we shall write briefly OU=a, and OV=b. Draw through U and V two horizontal planes, and let the vertical through P intersect these planes in M and N. Then the equations (2) and (3) give by (1), ■(4). horizontal velocity) Cn , „„., ofP \=-f^tmPUM (velocity of i')2= 2*; PJV (5). Thus the resultant velocity of P is that due to the depth of P below the horizontal plane through V, and the velocity of P resolved perpendicular to the plane ZOP is proportional to the tangent of the angle PU makes with a horizontal plane. It ap;"iears from this last result that when P is below the horizontal plane through U, the plane POT turns round the vertical in the same direction as the body turns round its axis, i.e. according to the rule in Art. 199, OF and OP are the positive directions of the axes of rotation. When P passes above the horizontal piano tiirough U, the plane POV turns round the vertical in the opposite direction. If P be below both the horizontal planes through O and U these results are still true, but if a top is viewed from above, the axis will appear to turn round the vortical in the direction opposite to the rotation of the top. lu all the cases in which P is below the plane UAf the lowest point of the rim of the top moves round the vertical in the same direction as the axis of the top. If we substitute for u^, Wj, E and F in (2) and (3) their values, we easily obtain P hi sin" e '/ + Cn cos e = Cn^ at I (»)• These equations give in a convenient analytical form the whole motion. We sec from the last equation, >vliat is indeed obvious otherwise, that b - 1 cos 6 is always positive. The horizontal plane through V is therefore above the initial position of P and remains above P throughout the whole motion. Ex. 1. If w be the resultant angular velocity of the body and v the velocity of P show that a»*=n' + (y) . Ex. 2. Show that the cosine of the inclination of the instantaneous axis to the ^. , . £+ (A -C)ncQS0 vertical is ^-^ , Au 560. As the axis of the body goes round the vertical its inclination to the vertical is continually changing. These changes the case in which the body has an initial angular velocity only about its axis, and applies them to determine directly the small oscillations of a top (1) when its axis is nearly vertical, and (2) when its axis makes a nearly constant angle with the vertical. His results arc necessarily more liinitsd than those given in this treatise. to gravity ese lengths horizontal IN. Then (4). (5). 3 horizontal plane ZOP plane. ontal plane ition as the and OP are e horizontal te direction, ilts are still a round the 11 the cases e top movea isily obtain (6). notion. We - 1 COR is G the initial velocity of P IS axis to the ertical its se changes ita axis, and when its axis gle with the ivcn in this MOTION OF A TOP. 447 dt may be found by eliminating -J^ between the equation (6). We thus obtain (i^^ 9 /; T m C^i' fa -I cos e\ .(7). I am 6 It appears from this equation that 6 can never vanish unless a = l, for in any other case the right-hand side of this equation _ would become infinite. This may be proved otherwise. Since J is equal to the ratio of the angular momentum about the vertical to that about the axis of the body, it is clear the axis could not become vertical unless the ratio is unity. Suppose the body to be set in motion in any way with its axis at an inclination i to the vertical. The axis will begin to approach or to fall away from the vertical according as the initial value of -77 or a^ is negative or positive. The axis will then oscillate between two limiting angles given by the equation = 2ghr (h ~ I cos 0) (1 - cos"*^) - CV (a - I cos fff (8). This is a cubic equation to determine cos 6. It will be neces- sary to examine its roots. When cos ^ = — 1 the right-hand side is negative; when cos ^ = cose", since the initial value of [-Ji) is essentially positive, the right-hand side is either zero or positive ; hence the equation has one real root between cos ^ = — 1 and cos ^=cos i. Again, the right-hand side is negative when cos^= + l and positive when cos d= oc . Hence there is another real root between cos 6 = cos i, and cos ^ = 1 , and a third root greater than unity. This last root is inadmissible. 5C1. These limits may be conveniently expressed geometrically. The equation (7) may evidently be written in the form v2 . _.. C-'h"-' /P3I\-> ('3" -^■"--^iZf Describe a parabola with its vertex at I', its axis vertically downwards and its Intus rectum equal to —r-., . Ijet the vertical PMN cut this parabola in H, wc then have ^ff ('")'■ 20.MN 1 1 PM "*" PR .(10). The point P oscillates between the two positions in which the harmonic mean of PM and PJi is equal to - 2 . MN, In the figure T is drawn above U, and in tliis case one of the limits of P is above CM, and the other below the pnrabola. If wc take U as origin and UO the axis of x, we have PM — r, I'M-y. Let 2^)1 be the f •1 1 1; ■ * ■ 1 iMf 1 r '!^ 448 MOTION UNDER ANY PORCF^. latas rectum o^ the parabola, and lJV=e, then the axis oi the bcV; bptw3»?n the two ;;;«nition8 in which P liea on the cubic curve o-:.CiJi-'M'''f y«(a!+c)=2pfo« (11) When c is positive, i. e. when V is above 17, the form of the carve is Lcdica+'j.' in the figure by the dotted line. The tangents at U cut each other at a finite angle and the tangent of the angle either makes with the vertical is f — j . When e is negative the curve has two branches, one on each side of the vertical, with a conjugate point at the origin. It is clear from what precedes that the upper branch will lie above, and the lower branch below, the initial position of P, and that P must always lie between the two branches. 662. In the case of a top, the initial motion is generally given by a rotation n about the axis. We have initially oa =0, u>^ — 0, and therefore by (2) and (3) E= Cn cos i, and F— Cii = 2gh cos i. ■■ 2pl, as before, the roots This gives a = 5 = Z cos i. Putting ^gft' of equation (8) are cos 6 = cos i, and cos ^ = ja — Vi — 2^ cos i + ^/^ The value co3d=p + '^1 — 2pcosi+p^ is always greater than unity, for it is clearly decreased by putting unity for coai, and its value is then not less than unity. The axis of the body will therefore oscillate between the values of just found. Since a=b, the horizontal planes through 17 and V coincide, and c— 0. T'jo cubic curve which determines the limits of OBcillation becomes the parabola ril and the straight lino UM. The axis of the body will thon oscillato b(;tween the two positions in which P lies on the horizontal through C and on the parabolrt. Generally the angular velocity n about the axis of figure is very great. In this case p is very great, and if we reject the squares of - we see that cos 6 will vary between the limits cos i f and cos i — ,, cjIii' i. 2p If the initial value of i is zero, we see that the two limits of cos { are the same. The axis of the body will therefore remain vertical. 663. Ex. 1. When the limiting angles between which varies are equal to each other, bo that 6 is constant throughout the motion and equal to a, show that tan' (p - tan rf> tan a H — ^- tan' o =0, where

    ' Afi'— + Bin a (gh- A fj? cos o) '^ + n^A sin» a0'=O\ To solve these, put ^= Psin (pt+f), and ^'= G cos (pt+f). Substituting, we have - An am a. pG ={gh-Afi* cos a) F \ {A ftp* - iJ?A sin" o) F= - {gh - Ay? cos o) sin a . , i.p. fgh sin a 4-.1 /gh sin _ t T^ " u— { ''—, — cot IC+ - \ An A -C \ sin 70 A J sin IZ ' ve, and there- But when the motion is steady OZ, 01 and OC are all in one plane. Now the angular velocity of C round I is w, and therefore its angular velocity round Z is But wcos IC=n, hence, tan7C= — ". Substituting this value of fi=U smJC ^mZO' n gh tan IC in the value of n, we get ~-Cn-An cos o, the same expression as before. 567. Ex. A top two of whose principal moments at are ennal is set in rota- tion about its axis of figure viz. OC with an angular velocity «, the point being fixed. If OC be horizontal, and if the proper initial angular velocity be communi- cated to the top about the vertical through O, prove that the top will not fall down, but that the axis of figure will revolve round the vertical, in steady motion, with an angular velocity fi=~ , where h is the distance of the centre of gravity of the top from 0, and C is the moment of inertia about the axis of figure. Show also that if the top be initially placed with OC nearly horizontal and if a very great angular velocity be communicated to it about OC without any initial angvdar velocity about OA or OB, then OC will revolve round the vertical remaining very nearly in a hori- zontal plane with an angular velocity /* given by the same formula as before, and the time of the vertical oscillations of OC about its mean position will be — -^ . Cn 568. A body tvhose principal momenta of inertia are not neces- sarily equal has a point fixed in space and, moves about O under the action of gravity. It is required to form the general equations of motion. Let OA, OB, OG he the principal axes at the fixed point 0, and let these be taken as axes of reference. Let h, k, I be the co-ordinates of the centre of gravity G, and let the mass of tiie body be taken as unity. Let F be drawn vertically upivard^ n'j ^-.-^^ '-' f lii:'l;J :yi 452 MOTION UNDER ANY FORCES. and let p, q, r be the direction-cosines of OF referred to OA, OB, OC. Then we have by Euler's equations .(1). ,(2). A^^^{D-C),=:~g{hq-kp) Also p, q, r may be regarded as the co-ordinates of a point in OV, distant unity from 0. This point is fixed in space, and therefore its velocities as given by Art. 248 are zero. We have dp da dr ^^ = a>,p-^^q It is obvious that two integrals of these equations are supplied by the principles of Angular Momentum and Vis Viva. These give A(o^p + Bco^q + Cw^r = E, 2g{ph+qk + rT), where E and F are two arbitrary constants. The first of these might also have been obtained by multiplying the equations (1) by p, q, ^ respectively, and (2) by Ato^, Ba>^, Gto^, and adding all six results. The second might have been obtained by multiplying the equations (1) by «j, Wj,, c»g respectively, adding and simpli- fying the right-hand side by (2j. 669. A body whose principal moments of inertia at the centre of gravity G are not necessarily equal, has a point in one of the principal axes at Q fixed in space and moves about under the action of gravity. Supposing the body to he performing small oscillations about the position in which OG is vertical, find the motion. Referring to the general equations of Art. 568, we see that in this case/i=0, jfc=0. Since OC remains always nearly vertical, w^ and u^ are small quantities', wo may therefore reject the product w^w,, in the last of equations (1). This gives Wj constant. Let this constant value be called lu For the same reason r = 1 nearly and p, q are both small quantities. Substituting we get the following linear equations, Ato^' + B(o^^+C(o^^ = F- A~'-(B-C)nio, = lgq J^-t!- {C-A)nu^= -Igp at .(3), dp_ di~' dq di = -pn + ui .(4). MOTION OF A TOP. 453 To solve these, assume wi = ii'8in(\<+/)| Wj,=Gcos(\t+/) \' Substituting, we get A\F-{B-a)nG=glQ) n\0~(A-C)nF=glp\ .(5). p = PBin(\t+f)) q = Q COS (\t+f))' \P = Qn-G \Q=Pn-F •{«)• Eliminating the ratios F : G : P : Q vfo have X'n'>{A + B-C)^={gl+A\^+ {B-C)n^^ {gl + B\^ + (A~C)n']. If the values of X thus found should be real, the body will make small oscillations about the position in which OG is vertical. If C be the greatest moment, and n' Bufiiciently great to make bothjjfZ- (C - A) n'^ and gl - [C - B) n^ negative, then all the values of X are real and the body will continue to spin with Off vertical. If G be beneath 0, I is negative and it will be sufficient that OC should be the axis of greatest moment. In order that the values of X' may be real, we must have {gl(A + B)+n^AC+BC-2AB-C^)}^>M(B-C)n''+gl]{{A-C)n^+gl\ AB, and in order that the two values o* a^ may have the same sign we must have the last term of the quadratic positive ; .'. {(B-C)n'^+gl}{(A-C)n!'+gl]= a positive quantity, and in order that the values of X" may be both positive, we must have the coefficient of X^ in the quadratic negative ; .:gl(A-\-B) - ■ ■ . By referring to equations (5) and (f)) it will be seen that when ^1 = .fi we have F = G and P=Q. If X^, \ be the two values of X found above, we have jp = Pi sin (\t +/i) + Pa sin (X,* +/») ) 2 = Pi cos (\t +/i) + Pj cos (Xa< +/») J " Let 9 be the angle OC makes with the vertical, then r^=coa* = 1-0^, ..nd lienco 0» = ^s + 23^Pj2+P.^2+ 2PiP, cos J(\i - X,) (+/i -/,}. ■I ! I il •'1 i J. 4.H MOTION UNDER ANY FORCES. I' I ^;i •\-t Also if, an in Art. 285, we let be the angle the plane containing OA, OC makcH with the plane containing OC and the vertical OV, we have j>= -biutfcos^, and q = sinO sin ^, and henoe 'jCOS^(Vj-/,) ,8in(V+/,)' Also since $ ia very small we have, following the notation of the Bume Article, where o is some constant, depending on the position of the arbitrary plane from which f is measured* . • In order to understand the relation which exists between these results and those of Art. 56-5, it will be necessary to determine the oscillations by some process which holds both when a is large and very small. This may be done as follows. We have by Vis Viva the equation (see Art. 558) fdey f E - Cn cone y _ F'- 2qh cos $ \dt) ■'"V ^Biu<# ) ~ A ^^'' where F' has been put for F- C«*. If we put 3= cos d, this takes the form ^'(^O +(^'-tW = ^(i^'-V«'2){l-s'') (2). Let us assume as the solution of this equation 2 = cos o + P cos (\t -I-/) (3), where P is so small that on substituting in the above equation we may neglect P'. Substituting and equating to zero the coefficients of th§ several powers of cos (Xt +/) we get A-P-'S? + (E - Cn cos a)^=A(F'- 2fih cos a) (1 - cos' o) ^ -(E -Cn cos a) Cn= -ghA-AF' cos a + '6(/hAcos,'^ a \. (4). - .4-\a + Cm" =-AF + &ghA cos a ) Now let us change the constant E inio another fi by putting — - — —„- '''=u+yP', A sin- a where y is to be so chosen as to remove the term A'^P^X^ in our fiist equation. Since d^ _E- Cn coiiJ9^ 'di A~^in^e ^'''' we see that, when is not small, /« differs from the constant part of /^ only by quantities depending on the squares of the small oscillation, and which are neglected in the text. Substituting for E and eliminating F' between the first and second equations we get 6rt/*=^ cos an^ + gh. Eliminating F' between the first and tliird of equations (4) and substituting for n we get . . H*A^- 2ffhA cos om" + g^h' X» == j^^ - . This process gives the period of the small oscillation in cos 0. When is finite this is the same as the oscillation in 0, since cos tf =cos o - sin a0'. When is very 0^ small, cos tf=l-- and the time of oscillation in cos ^ is the same as that in 0\ With this imderstanding it will be seen that there is a perfect agreement between the results of Arts. 565 and 569, when o is put equal to zero. MOTION OF A TOP. 455 570. A bodij whoa principal mamentt at the centre of gravity are not neceisarily equal is free to turn about a fixed point O, and it in equilibrium under the action of gravity. A small disturbance being given, find the oscillations. Beforriug to the general equations in Art. 668 we see that in this case w,, w,, Wg, are sinall, honoe in equations (1) wo may omit the terms containing the products WjWg, 01,0*3, "i'^y -^^^^ since in equilibrium OG is vertical, p, q, r are always nearly in the ratio h:k:l; hence if 00 = a, we may write-, -, - for », q, r on the a a a right-hand sides of equations (2). The six equations are now all linear. To soIto these we put Ui =H Bin (\t + /i) and|) = - + Pcos(X« + /i) (8), o>3, (i>3, q and r being represented by similar expressions with K and L written for Jf; Q, k and It, I written for P and h. Substituting these in the equations we get six linear equations. Eliminating P, Q, li we have .(4). (-AX' + k*-\-A£r-hkK-lhL=0 ■-hkH+(-B\i + P + hA F~lkL=0 - IhH - IkL + (~ C\» + fc» + fc«) £ =0 Eliminating the ratios of 11, K, L we have an equation to find X*. One root is X'=0, tho others are given by the quadratic v.(5!±i?.!!i-,*Hi.^2,.,^4^w^„ ,,, To ascertain if the roots are real we must apply the usual criterion for a quad- ratic. This requires that {A [B- C) A« + B [C-A) k^-C(A-B) ?«}«+44B(J3- C) [A-0 h^k^ (6) should be positive. Since A, B, C can be chosen to be in descending order, we see that the condition is satisfied. See also Art. 448. If G is above 0, a is positive and the values of X' are both negative. The equi- librium is therefore unstable. If G is below 0, a is negative and the values of X' are both positive. If the roots are equal, the two positive terms in (6) must be separately zero, this gives k=0 and A(B-C)h'^=C (A- B)l^, i.e. the centre of gravity lies in the asymptote to the focal hyperbola of the momental ellipsoid. In this case we find X'= --J. The case in which k=0, 1=0, B--C has been con- sidered in Art. 664. If the values of X' are written 0, "K^, \^ we have «i = Ho + -fffl't + -^1 sin (Xit + /*i) + H^ sin (\t + /i,), with similar expressions for Wj, o>a. Equations (2) then give p, q, r. But substitut- ing in (1) we find that all the non-periodic terms which contain ( are zero. Bemembering that 2>' -h 3' -f r' = 1 we have finally Wj = n ^ + Hj sin {\t -H /*i) + Jfj sin (Xj< + jUj) , ll'I I IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I 2f 124 "^ lit 140 IL25 mi 1.4 ■ 2.0 1.6 PhotDgraphic Sciences Corporalion aa WIST MAM STRHT WIUTIR,N.V. t4StO (7U)*n-4503 ^^ ^\. WrS <6f 450 MOTION UNDER ANY FORCES. w, and W3 being ropresented by similar expressions with k, K and {, L written for h, H. The values of K^, L^ and K^, L^ are determined by equations (4) in terms of H^ and Ht respectively. We also have P=a'^-^ 'cos(V + Mi)+-^^^ — ?cos(Xjt + /«j,), with similar expressions for q and r. There remain five constants viz. fi, Hi, //g, Ml, /*» to be determined by the initial values of w^ w^, W3, r and q. When the roots are equal the equations depending on p, r, Wj separate from those depending on q, u^, Wj, forming two sets; we find Wi=0- + //sin (Xt + /Wi) >, a\ ^-A'Aco8(Xt + /.,) A solution of this problem conducted in a totally different manner has been given by Lagrange in his SIScanique Anahjtique. His results do not altogether agree with those given here. If we substitute the values of «i, u.^, ua, p, q, r in the equatiou of angular momentum of Art. 568 and neglect the squares of small quantities, we evidently obtain (Ah^ + Bk^+Cl^) Q=Ea>, AHh + £Ak + CLl=0. The first of these equations shows that Q vanishes when the initial conditions are such that the angular momentum about the vertical is zero. In this case the problem reduces to that considered in Art. 455. 571. A body whose principal moments of inertia are not necessarily equal has a point fixed in space and moves about O tinder the action of gravity. It is required to find what cases of steady motion are possible in which one principal axis OC at describes a right cone round the vertical while the angular velocity of the body about OC is constant; and to find the small oscillations. Referring to the general equations of Art. 568, we see that r and w, are given to be constants. In this case the first two equations of (1) and (2) form a set of linear equations to find the four quantities p, q, u^, Wj. The solution of these equations is therefore of the form Wi=J'o + Fisin{X«+/)) Wa = G'o + G,cos(Xt+/)i ' p = Po+PiSin(Xt+/)| But these must also satisfy the last of equations (1). there will be a term on the left side of the form -^(A-Ji)F^GiBm2(\t+f). Substituting wo see that But there will be no such term on the right side. Hence we must have either A = B, Fi=0 or Gi=0. The motion in the case in which 4 = Z? has already been considered m Art. 564. Again, substituting in the last of equations (2) and equat- ing to zero the coefficient of sin 2 (X( +/) wc find , written for in terms of ;e from those ner has been ot altogether m of angtdar we evidently ial conditions this case the 'y equal hat a It is required axis OC at O \he body about f3 are given to set of linear lese equations ig wo see that It have either already been 2) and equat- MOTION OF A SPHERE. 457 Substituting in the first two of equations (1) and equating to zero the coefficients of cos {\t +/) and sin (\t +/), we find A\Fi-{B-C)nGi=glQi - B\Gi -(C-A) nFi = - glP^ ; from these equations we have F^, G^, P^, Q^ all equal to zero and therefore «i, Wj,, p, q are all constant as well as the given constants w, and r. In this case the equations (2) give p q r ' so that the axis of revolution must be vertical. Let w be the angular velocity about the vertical. Then u^^pu, w.2=qu, (03= rw. Substituting m equations (1) we get .(3). h_A(^_k Bu* I Cu^ P 9 ~Q~ 9 ~r g Unless, therefore, two of the principal moments are equal, it is necessary for steady motion that the axis of rotation should be vertical and the centre of gravity (hkl) must lie in the vertical straight line whose equations are (8). This straight line may be constructed geometrically in the following manner. Measure along the vertical a length F= ^ and' draw a plane through V perpeu- w dicular to F to touch an ellipsoid confocal with the ellipsoid of gyration. The centre of gravity must lie on the normal at the point of contact. To find the small oscillations about the steady motion, i.e. to determine whether this motion be stable or not, we must put ^ - cos a + Po sin \t + Pi cos \t, with similar expressions for q, r, Wj , «a, W3. Substituting we shall get twelve linear equations to determine eleven ratios. Eliminating these we have an equation to find X. It is sufficient for stabihty that all the roots of tliis equation should be real. Motion of a Sphere. 572. To detervtine the motion of a spliere on any perfectly rough surface under the action of any forces whose resultant passes through the centre of the sphere. Let Q bo the centre of gravity of the body and let the moving axes GC,GA,GH be respectively a normal to the surface and some two lines at right angles to be afterwards chosen at our convenience. Let the motions of these axes be de- termined by the angular velocities 0„ ^„ ^3 about their instantaneous positions in the manner explained in Art. 243. Let «, v, w be the velocities of G resolved parallel to the axes so that «>=0, and Wj, w.j, Wj the angular velocities of the body about these axes. Let F, F' be the resolved parts of the friction of the perfectly rough surface on the sphere parallel to the axes, GA, GB, and let B be the normal reaction. Let X, Y, Z be the resolved parts of the impressed forces on the centre of gravity. Let * be the radius of gyration of the sphere about a diameter, a its radius, and let its mass be unity. The equations of motion of the sphere arc by Alts. 264 and 245, ',v '■ wl 458 MOTION UNDER ANY FORCES. .(1), du It -Bm =X+i? = Y+F' -0^u + 6jV =Z+]t J and since the point of contact of the sphere and surface is at rest, we have «-awo=0) ■(2). «-aWj,=0) (3). EUminating F, F, u^, «, from these equations, we get -etv= .X+ jfc" it" O^auj du di (4). 573. The meaning of these equations may be found as follows. They are the two equations of motion of the centre of gravity of the sphere, which we should have obtained if the given surface had been smooth and the centre of gravity had been acted on by accelerating forces -^ — r. O^au^ and -r— r. 0>flu^ along the axes GA, GB, and by the same impressed forces as before reduced in the ratio d^ + k^' The motion therefore of the centre of gravity in these two cases with the same initial conditions will be the same. More convenient expressions for these two additional forces may be found thus. The centre of gravity moves along a surface formed by producing all the normals to the given surface a constant length equal to the radius of the sphere. Let us take the axes GA, OB to be tangents to the lines of curvature of this surface and let p^, p^ be the radii of curvature of the normal sections through these tangents respectively. Then Pa e,= u Pi .(5). If G be the position of the centre of gravity at the time *, the quantity 0.^dt is the angle between the projections of two successive positions of GA on the tangent plane at G. Let Xp Xa he the angles the radii of the curvature of the lines of curvature at G make with the normal. The centre of the sphere ma; be brought from to any neighbouring position G' by moving it first from G to H along one line of curvature and then from H to G' along the other. As the sphere moves from G to //, the angle turned round by GA is the product of the arc GH into the resolved curvature of GH in the tangent plane. By Meuuier's theorem, the curvature is , multiplying this by sin x\ to resolve it into the tangent plane Pi ''OS Xi we find that the part of 6a due to the motion along GH is tan Xi> Treating the Pi MOTION OF A SPHERE. 459 (1). kve .(2). (3). ■(4). hey are the ii we should gravity had g the axes th the same ir these two ig a surface eugth equal tangents to ture of the (5). ntity 0.^dt is the tangent the lines of be brought 'I along one here moves re OH into leorem, the ugeut plane 'reating the arc HG' in the same way, we have u V 0, = ~ tan Yi+ - tan Xi (6). We have also an expression for Wj given by equations (1). Substituting for Ui, U] from the geometrical equations (3) we get dt =uv ( ) \Pa Pi/ .(7). The solution of the equations may be conducted as foUo'^s. Let (a;, y, z) be tlie co-ordinates of the centre of the sphere. Then u, v may be found from the equation to the surface in terms of dx dy dz dt' dt' dt -J by resolving parallel to the axes of reference. If we eliminate «, v, 0^, 0^, 0^ by means of (4), (6), and (6), we shall get three equations containing x, y, z, a>g, and their differential coefficients with respect to t. These together with the equation to ihe surface will be sufficient to determine the motion at any time. One integral can always be found by the principle of Vis Viva. Since the sphere is turning about the point of contact as an instantaneously fixed point we have where

    be the angle the normal plane to the cylinder through a generator and through the centre of the sphere makes with some fixed plane passing through a generator, then v=/)j ^ . the first and third equations then become If -T- be not zero, at du P aw,= «" Pa rt= + k^v d^ X If X be the same for all positions of the sphere on the same generator these equations can be solved without diflBculty. For v and p, being known in terms of , we have in this case two linear equations to find w and auy If X be zero, and i« = =^, wefind o au, ^ = ABUx(/s/^i> + B\, U = AA^^(SOB(A^-i> + £\ where A and B are two arbitrary constants to t mii.ed by the initial values of u and u, a- If X be not the same for all pcsitions of the sphere on the same generator, let ( be the space traversed by the sphere measured along a generator. Then ~dt~ di/>pf' Substituting this value of u, we have two equations to find ( and au^ in terms of , ^ , and (2) the sphere starts from rest at a point where />, is zero. In such a case, the normal plane to the section through the centre of the sphere has a constant angular velocity in space and the resolved motion of the sphere perpendicular to the generators is independent of that along the generators. Ex. A sphere rolls on a perfectly rough right circular cylinder who; e radius is e under the action of no forces, show that the path traced out by the point of con- tact becomes the curve a;=ii sin . / = - when the cylinder is developed on a plane. This result shows that the sphere cannot be made to travel continnaUy in one direction along the length of the cylinder except when the point of contact de- scribes a generator. 580. If the surface on which the tphere rolls be a cone, the lines of curvature are the generators and their orthogonal trajectories. Let the axis GA be directed parallel to the generator, then p^^ is infinite and p, - a is the radius of curvature of a normal section perpendicular to the generators. Also ^i= — , ^,=0. Let the Pa position of the sphere be defined by the distance r of its centre from the vertex of the cone on which the centre always lies and by an angle (p such that d is the angle between two consecutive positions of the distance r, d

    "di ' dr v=r dt' The equations (4) and (7) become therefore HP " *■ V d«y ~ a» + fc" a^ + k> p^ '^'^^ dt rdt\ dt) a^+fc^ d (fl W3) _ r d(f> dr dt pj dt dt If the impressed forces have no component perpendicular to the normal plane through a generator, r=0, and we have r" ^ = h, where h is some constant depend- ing on the initial val\ es of r and v. If also the component X of the forces along a generator be a function of r only, another integral can be found by the principle of Vis Viva, viz. 1' f. i\ {■ V \'l »;,:• .1! J y iJiil » i' 464 MOTION UNDETl ANY FORCES. where 7t' is another constant depending on the initial values of u, v and r. If, further, the cone be a right cone, /),=r tan a wheid a is the semi-angle, and we have h cot a , „ where h" is a third constant depending on the initial values of w, and r. The equa- tions of the motion of the centre of the sphere resemble those of a particle in central forces. Hence r and ^ will be found as functions of the time if we regard them as the co-ordinates of a free particle moving in a plane under the action of a central force represented by where ta^ has the value just found. Ex. A sphere rolls on a perfectly rough cone such that the equation to the cone on which the centre G always lies is —=F(d>). If the centre is acted on bv a force Pa tending to the vertex, find the law of force that any given path may be described. If the equation to the path be -=/(^), prove that the force X is where w, is given by — — = J! 3— . a0 a a

    diit'erentiating this, we have by (iv), d^v p cos rfc d^' + '^T- d^ + ^''=<^ (vi), where de\ r )^l^' + a^\ r'J' Now p and r may bo found from the equation to the meridian curve as functions of 0. Hence P is a known function of 0. Solving this linear equation we have v found as a function of 0. Then by (iv) we have dw^ 10 _ V f p sin \ and thence having found Wj we have u by equation (iii). Knowing u and v ; and Vfr may be found by equations (v). 582. A heavy xphere rotating about a vertical axia is placed in equilihrium on the highest point of a surface of any form and being slightly disturbed mahes small oscillations, find the motion. Let be the highest point of tlie surface on which the centre of gravity G always lies. Let the tangents to the lines of curvature at be taken as the axes of X and y, snd let (x, y, z) be the co-ordinates of Q, We shall assume that is not a singular point on the surface. In order to simplify the general equations of motion (4) we shall take as the axes GA and GB the tangents to the lines of curvature at G. But since G always remains very near 0, the tangents to the lines of curvature at G will bo nearly parallel to those at 0. So that to the first order of small quantities we have e,= ~ Idy . 1 dx ''* = " rff dx dt' dif at' padt 'Pi and ^3 will be a small quantity of at least the first order. Also since the sphoro is supposed not to deviate far from the highest point of the surface, we have Wj constant, let this constant be called n. I^e equation to the surface on which G moves, in the neighbourhood of 1 fx^ y'^\ the highest point, is «=-g( — + — )• The equation to the normal at x, y, '^SPi Ps/ i— = i-^ - — - . Hence the resolved parts parallel to the axes of the normal _ i« _ 3/ - 1 Pi Pi pressui-e R on the sphere are Jt - , It and R. The equations of motion (4) Px Pi Z 13 R. 1). 80 ^n Hiii m j \ 1 4GG tbcrofuro become MOTION UNDER ANY FORCES. rf*x dfl dfl _ 36 _ I* d^ an ' a^ + k* p, o« + it«rft7», Pi i» rfoc an p, a" + ^'' (it Px (It). But 2 is a small quantity of the second order, hence the last equation gives R=g, To solve these equations, vre put a;=fcos(\e+/), y = OBin(Xf+/). These give a\n The equation to find X is therefore g'X'H* PiPj This is a quadratic equation to determine X'. In order that the motion may be oscillatory it is necessary and suillcient that the roots should be both positive. If pj, p, be both negative, so that the sphere is placed like a ball inside a cup, the roots of the quadratic are positive for all values of n. If pj, />, have opposite signn the roots cannot bo both positive. If pj, p^ be both positive the two conditions of stability will be found to reduce to , a' + F , /-, /-,, "*> -fA-OWPi+is/ps)*' If pi be infinite, it is necessary that p^ should be negative, and in that case the two values of X' are — rm ^^^ ^®'^' w^ich are both independent of «. Ct T" ft Pa If Pi=Pa, we have F=G. In this case ii 0he the inclination of the normal to the x'^ + tfl vertical, we have 6"= —~ and, as in Art. 569, we find P e'=F,' + F,^ + 2F,F, cos {{\ - Xj) t+/, -/,], where X^, X, are the roots of the quadratic i* an. a* g X«± ''-"X + aHi" p o^ + Pp = 0. This problem may also be solved by Lagrange's method in the ma? ler explained in Art. 388. Let the axes of reference Ox, Oy, Oz be the same as before. Let GG be that diameter which is vertical when the sphere is in equilibrium on the summit. Let GA, GB be two other diameters forming with GC a system of rectangular axes fixed on the sphere. Let the position of these with reference to the axes fixed in space be defined by the angular co-ordinates 0, , f in the manner explained in Art. 235. The Via Viva of the sphere may then be found as in Art. 349, Ex. 1. If MOTION OF A SPIIEl'.E. 467 (iv). tion gives notion may )th positive, e a cup, tbe )poBite BigiiH pnditions of in that case ndent of n. ormal to the ler explained Dre. Let GC the summit, angular axes axes fixed in explained in 9, Ex. 1. If yre put sin cos ^ = ^ sin sin ^ = ir, + \p = x, and reject all small quantities above the second order, we find that the Lagraugiau function ia L= l{x'' + y'') + lk^\x'*-x'{iv'-^r,) + i'* + r,'*]-^lg(^%^^y It is easy to see by reference to the fi<^ure in Art. 23S that { and i) ore the cosines of the angles the diameter OC meikea with the axes Ox, Oy. If Ug, uy, u, are the angular velocities of tlio sphere about parallels to the axes fixed in space, the geometrical equations are x' ■ 1 ( Wj, - w, - ) = These are found by making tlio resolved velocities of the point of contact in the directions of the axes of x and y equal to zero ; see Art. 219. The angular velocities Ujc, Uy, u, may be expressed in terms of d, 0, ^ by formulns analogous to those in Art. 235. Bee also the note. Thus Wj.= -tf'sin }{/ + mi cos ^j Uy= tf'cos ^ + 0'sin^ sin^> . w,= ^'costfj-f ' Substituting and expressing the result in terms of the now co-ordinates {, r), x. the geometrical equations become ^.=-^xVK'-x';,. A- | + x'|-y-x';; = ol The equations of motion are given by dtlLdL djjj dL^ dtdi ~dq'' d<^ '^ ^ dq" where q stands for any one of tlie five co-ordinates x, y, {, ?j, x- Tlie steady motion is given by x, y, f, i; all zero and x'""* Taking q =« and q-y and giving the several co-ordinates their values in the steady motion, we find that \ and /u are botli zero in the steady motion. To find the oscillations, we write for q in turn x, y, Xi ^ and ij, and retain the first powers of the small quantities, Romembpriiig that \ and n are small quanti- ties (Art. 4C1), we find X X Pi a A-V'=oJ These and the two geometrical equations L^ and L^ are all linear, and may be solved in the manner explained in Art. 432. If we put x'=m and eliminate first \ and M and then | and t) we get two equations to find x and y, which are the same as those marked (iv) in the first solution. 80—2 i\\ I i!! 1; i'm 'I I- 468 MOTION UNDER ANY FORCES. I'll ! f'^ : I ■: Ex. A perfectly rough sphere is placed on a perfectly rongh fixed sphere near the highest point. The upper sphere has an angular velocity n about the diameter through the point of contact; prove that its equilibrium will be stable if n2> _?i^iLJ , where 6 is the radius of the fixed sphere, and a the radius of the moving sphere. 683. A perfectly rough surface of revolution is placed with its axis vertical. Determine the circumstances of motion tliat a heavy sphere may roll on it to that its centre descri-hes a horizontal circle. And this state of steady motion being disturbed, find the small oscillations. In this case we must recur to the equations of Art. 581, and let us adopt the notation of that article, except that to shorten the expressions we shall put for >? its value z, a\ o dy(f To find the steady motion. We must put a, v, Wj, 0, ^ all constant. Let dt #, a, /t and n bo the constant values of 0, -J and W3. Then we have tt=0, v=hfx, where h is the constant value of r. The equation (i) becomes The other dynamical equations are satisfied without giving any relation between the constants. If the motion be steady, we have therefore 5 g ^7b . n=- — + ji-/iCota; 2 aft. 2 o thus for the same value of rt we have two values of n, which correspond to different initial values of v. We have the geometrical relation au^ = - v, so that u^ and n have opposite signs. Hence the axis of rotation which necessarily passes through the point of contact of the sphere and the rough surface makes an angle with the vertical less than that made by the normal at the point of contact. By inspecting the expression for n, it will be seen that it is a minimum when 5 g Ihn a - s ~ cot a, 2 ail 2 a and '■ lerefore hg Ad n'=35 -^' cot a, ii?=^ i tano cr lb To find the small oscillation. Tut d-a^O', J = fi-^- ^ , where a and ju are supposed to contain all the con- stant parts of and -j- , so that 6' ani -J- only contain trigonometrical terms. Let c - a be the radius of curvature of the surface of revolution at the point of contact of the sphere in stoady motion, so that p differs from c only by small quantities, and may be put equal to c in the small terms. Also we have »• - 6 + c cos a . 0'. t :' i fixed sphere n about the will be stable radius of the axis verticaL I it so that its >ing disturbed, ns adopt the lall put for Ar» )nstant. Let 3 M=0, V=bf*, lation between md to different have opposite ;h the point of le vertical less limum when lin all the con- rical terms. Let point of contact aall quantities, c cos a . 0'. MOTION OF A SPHEIIE. Now by equations (iv) and (v) of Art. 581 we have 4G0 rfwj _dd d\f^ p Bin 0-r _dff c sin a - & lU ~ dt dt a ~ dt ^ a c sin a - 6 W3 = /t -- — O' + n, de . ¥''-dr^' where n is the whole of the constant part of u^. Again, from equation (ii), we have adt\ dlj a dt dt o'+ , H dff 5dY ccosoAid^' 2 d9 ^ '•-r''''''l^--a-dt^ ^dr + 7"d*=^' integrating we have (2 _ 2nc cos a\ ^ _h d\f/ 7" a J ~a~dt' the constant being put zero because ^ and ^' only contain trigonometrical terms. Thirdly, from equation (i), we have Id f de\ r fd^y „ 2 . ^d^L 5 a . - :i; ( P j7 ) - - 1 j7 ) cos 0+s w«sin 0-J- = ^ - sin tf ; adtydtj a\dtj 7 * dt 7 a ' cd?e' b + ecoaae' . ^/. „ d\l/\ ••• aW^ a («o««-Bma<0(/*'' + 2M.J-) +|(sma+C03a<»')(M+^)(» + /i^^^^(?') = |f(sma + C03atf'). This expression must be expanded and expressed in the form Jn this case, smce 0' contains only trigonometrical expressions, we must have ^=0. Putting ^=0 in the above expression, we find the same value for n as in steady motion. After expanding the preceding equation we find A=ii^(- cob' o+I sin'a^ + /t« -I— ( 1 V 7 /cBma\ 2oos'o+ssin'o) c Bin a V 7 / 25o'sino 10 o . 10 o +^—r. -=- |smocosa+-=-^coso. In order that the motion may be steady, it *3 sufficient and necessary that this 2t value of A should be positive. And the time of oscillation is then -;= . s/A It is to be observed that this investigation does not apply if a and therefore b be small, for some terms which have been rejected have b in their denominators, and may become important. 684. The general equations of the motion of a sphere on an imperfectly rough surface may be obtained on principles similar to those adopted in Art. 306. The difference in the theory will be made clear by the following example, in which a method of proceeding is explained which is generally applicable, whenever the integrations can be effected. ■ n '^1 liii n'- r - m 1 II 470 MOTION UNDER ANY FORCES. it r f I 1 1 Hi \ 685. .4 homogeneous sphere moves on an imperfectly rough inclined plane with any initial conditions, jind the direction oj the motion and the velocity of its centre at any time. Let O be the centre of gravity of tlie sphere. Let the axes of reference GA, GD, GC have their dii-cctions fixed in space, the first being directed down the iucUned plane and the last normal to the plane. Let u, v, w be the velocities of resolved parallel to these axes, and w^, wa, u^ the angidar velocities of the body about these axes. Let F, F be the resolved parts of the frictions of the plane on the iphere parallel to the axes QA, GB, but taken negatively in those directions. Let k be tlio radius of gyration of the sphere about a diameter, a its radius, and let the mass be unity. Let a be the inclination of the plane to the horizon. The equations of motion will then be h^'^-^=-F'a] at k-%*=Fa at .(1). du di dv di = -F+gs'ma] = -F .(2). Eliminating F and F from these equations and intcgrafcuig wo have U+-5 awa = l7o+5' sin at .(3). where Uq and Vq are two constants determined by the initial values of «, v, w^, Wj. The meaning of these equations may be found as follows. Let P be the point of contact of the sphere and plane, let Q be a point within the sphere on the normal at P so that PQ= , so that Q is the centre of oscillation of the sphere when suspended from P. It is clear that the left-hand sides of the equations (3) express the components of the velocity of ^ parallel to the axes. The equations assert that the frictional impulses at P cannot affect the motion of Q, and this readily follows from Art. 119, because Q is in the axis of spontaneous rotation for a blow at P. 586. The friction at the point of contact P always acts opposite to the direction of sliding and tends to reduce this point to rest. When sliding ceases the friction (see Art. 148) also ceases to be Umiting friction and becomes only of sufficient mag- nitude to keep the point of contact at rest. If sliding ever does cease, we then have u - awj — 0, v + awi = (4). The equations (3) and (4) suffice to determine these final values of u, v, w, and W.J. Thus tho direction of the motion and the velocity of the centre of gravity after sliding has ceased have been found in terms of the time. It appears that both these elements are independent of the friction. If the equations (4) hold initially the sphere will begin to move \fithout sliding if the friction found from the equations (1), (2) and (4) is less than tho limiting friction. As in Art. 147, this requires that the coefficient of friction /«> -^ — r;, tan a. Supposing thia inequality to hold, the friction called into play will be always loss than the limiting friction and therefore equations (3) and (4) give the whole motion. MOTION OF A SPHERE. 471 plane with f its centre e GA, GB, lie inclined resolved ibout these the Lphere et k be tlio he mass be ; of motion .(2). .(3). t, V, Wi, Wj. le the point X the normal iphere vrhen (3) express assert that idily follows )w at P. ilie direction the friction fficient mag- re then have (4). u, V, Ui and gravity after vt both these hont sliding the limiting always loss hole motion. 587. If the equations (4) do not hold initially or if the ineqtiality jnst men- tioned is not satisfied, let S be the velocity of sliding and let be the angle the direction of sliding makes with OA. To fix the signs we shall take S to be positive while $ may have any value from - b- to jr. Then S cos d=u-au„ 3 Bin d=v + auy .(5). The friction is eqnal to fig cos a and acts in the direction opposite to sliding, hence F=ngeoBaco39, F" = fig eoa a Bin 9. The equations (1), (2) and (5) therefore give d (S cos , — =-( l + pj/tr^cosacostf+flrsmo d(5sine) /, a«\ . . .(6). Expanding we find dt v^ky fig cos a+9 sin a cos ,de .(7). S -TT = - fl sin a sin at If be not constant, we may eliminate t and integrate with regard to 9, this gives Ssin9=2A (tan^V (8), where n = f 1 + p j ju cot a, and A is the constant of integration. If Sq and ^o ^^ the initial values of S and determined by equations (5), we have 2A=SaBin0, (cot| .(9). Substituting the value of S given by (8) in the second of equations (7) and inte- grating we find «-l n+l n-1 n+1 A t. .(10), the constant of integration being determined from the condition that tf = ^oWhen t=0. The equations (8), (9) and (10) give S and in terms of t. The equations (3) and (5) then give u,v,u^ and Wj in terms of t. d9 The second of equations (7) shows that -:- has an opposite sign to 0, hence be- ginning at any initial value except ijr continually approaches zero. It follows that, unless a is zero, will be constant only when 0q=O as ± tt. If n > 1, i.e. fi > — — Tj tan a, we see from (8) that sliding will cease when Cb -J' fC vanishes. Tliis, by (10) will occur when g sin a ^n - 1 Ji + 1 ' {: : «1 it it ill m I t . TiiO subsequent, motion has already been found. :>). ! I 472 MOTION UNDER ANY FORCES. If n < 1 we see by (8) that S increases as $ decreases, so that sli'ling will never cease. It also follows from (10) that vanishes only at the end of an infinite time. li Sq=0, sliding will never begin if n > 1, but will immediately begin and never cease if n < 1. 588. The theory of the motion of a sphere on an imperfectly rough horizontal plane is so much simpler than when the plane is inclined or when the sphere rolls on any other surface, that it seems unnecessary to consider this case in detail. At the same time the game of biUiards supplies many problems wliich it would be unsatisfactory to pass over in silence. The following examples have been arranged eo as both to indicate the mode of proof to be adopted and to supply some results which may be submitted to experiment. The result given in Ex. 1, was first obtained by J. A. Euler the son of the cele- brated Euler, and published in theJlf^ni. de. I'Acad. de Berlin, 1758. Most, possibly all, of the other results may be found in the Jeu de Billard par 0. Coriolis, pub- lished at Paris in 1835. Ex. 1. A billiard-ball is set in motion on an imperfectly rough horizontal plane, show that the direction and magnitude of the friction are constant through- out the motion. The path of the centre of gravity is therefore an arc of a parabola while sliding continues, and finally a straight line. The parabola is described with the given initial motion of the centre of gravity under an acceleration equal to fig tending in a diiectiou opposite to the initial direction of sliding. Ex 2. If Sq be the initial velocity of sliding prove that the parabolic path lasts 1 5 If the initial velocity of sliding be one foot per second, the parabolic path lasts therefore less than a twentieth part of a second. Ex. 8. If P be the point of contact in any position and Q the centre of oscilla- tion with regard to P, prove that the velocity of Q is always the same in direction and magnitude. Thence show that the final rectilinear path of the centre of gravity is parallel to the initial direction of the motion of Q and the final velocity of the centre of gravity is = of the initial velocity of Q. If PP be the initial direction of motion and V the initial velocity of the c autre of gravity and 1 1L3 time given by Ex. 2, prove that the final rectilinear path of the centre of gravity intersects PP" in a point P' so that PP=l Vt. Ex. 4. A billiard-ball, at rest on an imperfectly rough horizontal table, is straok by a cue in a horizontal direction at any point whose altitude above the table is h, and the cue is withdrawn as soon as it has delivered its blow. Supposing the cue to be sufficiently rough to prevent sliding, show that the centre of the boll will move in the direction of the blow and that its velocity will become nniform and equal to = - 5 after a time — ^ where B is the ratio of the blow to the mass '■la 7a fig 2 S for a time = — . From some experiments of Coriolis it appears that fi=^ nearly. of the sphere and a is the radius. In order that there should be no sliding the distance of the cue from the centre of the ball must be loss than a sin e where tan c is the coefficient of friction between the cue and ball. ^ ; will never finite time. 1 and never 1 horizontal pbere rolls detail. At t would be n arranged )ine results of the cele- i?t, possibly >riolis, pub- horizontal at through- ' a parabola cribed with equal to ng path lasts =■= nearly. o path lasts '6 of oscilla- in direction re of gravity }city of the direction of me given by sects PP" in ble,isstniok e table is h, ing the cue he boll will oniform and to the mass n the centre tiou between MOTION OF A SOLID BODY ON A PLANE. 473 Ex. 5. A billiard-ball, initially at rest and touching the table at a point P, is etioick by a cue making an angle /3 with the horizon. Show that the final recti- linear motion of the centre of gravity is parallel to the straight line PS joining P to the point S where the direction of the blow meets the table, and the final velocity 5 PS of the centre of gravity is - — ^ sin /3 in the direction of the projection of the blow on the horizon. It will be noticed that these results are independent of the friction. Ex. 6. Measure Sr=^ccot/3 along the projection of the blow on the horizon- tal table, then TS measures the horizontal component of the blow referred to a unit of mass, on the same scale that PS measures the final velocity of the centre of gravity. Prove that during the impact and the whole of the subsequent motion the friction acts along PT and that the whole friction called into play will be measured by PT on the scale just mentioned. 5 PT Thence show that unless /* < = — the paraboho arc of the path will be suppressed. Show also that PT is the direction in which the lowest point of the ball would begin to move if the horizontal plane were smooth and the ball were acted on by the same blow as before. Motion of a Solid Body on a plane. 589. A solid of revolution rolls on a perfectly rough horizontal plane under the action of gravity. To find the steady motion and the small oscillations. Let be the centre of gravity of the body, OC the axis of figure, P the point of contact. Let OA be that principal axis which lies in the plane POC and GB the axis at right angles to GA,GC. Let GM bs a perpendicular from G on the hori- Hi ;1 r. { i i I; ■I 474 MOTION UNDER ANY FORCES. zontal plane, and PN a perpendicular from P on GC. Let be the angle QCtaskdi with the vertical, and ^ the angle MP makes with any fixed line in the horizontal plane. Let R be the normal reaction at P; F, F' the resolved parts of the frictions respectively in and perpendicular to the plane PQC. Let the mass of the body be unity. Let us take moments about the moving axes QA, OB, QC according to Art. 253. As in the second case of Art. 254, we put B^^w^, &t=w, and &a= -^cos 9. Bemem- bering that h^=Awi, h^=Au^, h^=Cu^ we have A^-AwJ^^GOBe+Cu,i»t=-F.GN (1). A^- CwgWi + ^Wj^cos tf= -F. OM-R. MP (2). Q/% at C^^F.PN (3). The geometrical equations are dt="« (^)- ""<'dF=-"i <«)• Let u and v be the velocities of the centre of gravity respectively along and per< pendicular to MP, both being parallel to the horizontal plane. The accelerations of the centre of gravity along these moving axes will be Tt-''dt=^ <^)' Tt^'^Tt-^ ^^^' And if 2 be the altitude of above the horizontal plane, we have 5^=-^+^ («)• Also since the point P is at rest, we have u-GMu,=0 (9), v + PNus-GN'-^i=0 (10), 2=-(?iVcos<;+PiVsintf (11). These are the general equations of motion of a solid of revolution moving on a perfectly rough horizontal plane. If the plane is not perfectly rough the first eight equations will still hold, but the remaining three must be modified in the manner explained in the next proposition. When the motion is steady, we have the surface of revolution rolling on the plane so that its axis makes a constant angle with the vertical. In this btate of motion, let tf=o, ^=/*, W8=n, GM=p, MP=q, GN=^, NP=ri, and let /5 be the radius of curvature of the rolling body at P. Then the relations between these quantities may be found by substitution in the above equations. Suppose it were required to find the conditions that the surface may roll with a given angular velocity n with its axis of figure making a given angle with the verti- cal. Here n and a arc given, and p, q, f, ij, p may be found from the equations to MOTION OF A SOLID BODY ON A PLANE. 475 (1). (2). (3). (6). 1 pel itioi ..(6), (7). ..(8). ..(9), (10), (11). the snrface. We have to find /x, Wj , w^ , u, v and the radius of the circle deacribcd by in space. Then eliminating F, F', li, we get /tt' sin a {A cob a-p^)- u/jl (C sin a + pri)-gq= 0, Wi=-^sino, (i>2=0, M=0, « = - rnj - f/t sin o. Let r be the radius of the circle described by as the surface rolls on the plane. Since Q describes its circle with angular velocity /jl, we have r/i=v, and hence r= - — -f smo. /« Eliminating n we have H^ {^ i; sin a cos a + C| sin' a + r (C sin a +pri)] = ffiV- For every value of n and a there are two values of /*, which however correspond to different initial conditions. In order that a steady motion may be possible, it is necessary that the roots of this quadratic should be real. This gives (C7 sin a +priy «' + ^02 sin o {A. cos a-p^)=a, positive quantity. If the angular velocity n be very great, one of these values of fi is very great and the other small. If the angular velocity be communicated to the body by unwinding a string, as in a top, the initial value of Wx will be small. In this case the body will assume the smaller value of (j^, and we have approximately u= Ql . •^ n(CBma+pr}) To find the small oscillation, we put $ = a-\-d', 37 = /*+ -^t wj-n + Wg'. Then we have by geometry, t=0M=p + q9', PM-=q + (^-p)e', GN=^+p9'('ma, PN='n+p8^eoBa, and substituting in (6), ['J), '\0), (6), (7) respectively, we find Ui=i - fi sin a - n cos a^ - sin a dff d£ dt ' d^|/ V = - /i sin o{ - nij - (/* cos of + up sin' a + np cos o) ^ - sin a^ —~ i;wa'. dt d^B' „ . d-i/ d^' + /i (^ cos of + up sin' o + np cos o) 6' + i;ai«8', dd' . d'-J/ do},' F'x: - (/i cos a^-p/x + ixp sin' a+npMBa)-r - sin of -j.^- - 17 -j- . Substituting these in equation (3) and integrating, we have d\f/' (C + 1;') w'j = (PM - Mf cos a -up sin' a - up cos o) r]9' - ij sin of -;- . • (A), the constant being omitted because «, a and /* are supposed to contain all the ,d^ i ' n pit' ■ 1 ■';! m constaut parts of w^, 9, and df i 476 MOTION UNDER ANY FORCES. Again substituting in (1) and integrating, we have {Cn - 2Afi cos o + { (pn - /« cos of - /* sin'o/) - np cos a)}0'-{A+ ^) sin a J =ifiu^'(ii). Also substituting in (2), wo have {A+p''+q*)-j-^+ff{Afi'(sia^a-coB,^a) + CnfiCOBa+{p-p)g^ + /t'sina fj + nuriq + /jflcoa o^ + nftfrp cos o + m' sin' "■fP ) + --^{ -?J^sino coso + C»sina + 2^^sina+npi>} '~ " '* tic + u^'{Cfi sin a+fipri) + i - il sin a COB o/t* + Cn/t sin a 4 gq + sin aft'^p^ + nupr} } d\f/ The last term 6f this equation must vanish since 0', -~ , ug'only contain periodic terms. It is the equation thus formed which determines the steady motion and gives us the value of /x. To solve these equations we may put e'=LBia{\t+f), ^ =MBm{\t+f), ua'=NBin{\t+f). If we substitute these in (A), (B), (C) we shall get three equations to eliminate the ratios L:M:N. Before substitution it will be found convenient to simplify the equations first by multiplying (A) by f and (B) by 17 and subtracting the latter result from the former, and secondly by multiplying (A) by — and adding the re- suit to (C). We then obtain the following determinant, -(A+p^ + q')\H{p-p)9 + ft? {p^- A COB 2a -qr) + n/iC cos a Afi sin a COB a n Cn{ri Bi a-p) Cn - 2A/1 cos a ^ sina Cf {p-^eoBa-p sin^a) /j, - pn cos a {sin a -(C+v') =0. 590. Ex, 1. To find the least angular velocity which will make a hoop roll in a straight line. In this case r is infinite and therefore /x must be zero. It follows from the equation of steady motion that 9=0, or the hoop must be upright. We have p=a, 2=0, {=0, 71= a, n=0, aaH C=2A. The determinant becomes ^,_ 2tfi{2A + a'')-ag ^ aTTT* ' so that the least angular velocity which will make X a real quantity is given by 2((7+a«)' =fW(B)- .=0...(C). bin periodic notion and ) eliminate to simplify g the latter ling the re< =0. I a hoop roll wa from the . We have iven by MOTION OF A SOLin BODY ON A PLANE. 477 Let the hoop ba an arc, we have C=a\ and if 7 be the least velocity of the Let the hoop be e. disc, then centre of gravity, this equation gives V> g >/"//• C= g , and we have V> /as V ¥• Ex. 2. A circular disc is placed with its rim resting on a perfectly rough horizontal table and is tpun with an angular velocity Q about the diameter through the point of contact. Prove that in steady motion the centre is at rest at an altitude - above the horizontal plane, where I is the radius of gyration about a diameter ; and, if a be the inclination of the plane to the horizon, the point of If the disc bo slightly 2ir contact has made a complete circuit in the tiiie ^ sin a. disturbed from this state of steady motion, show that tlie time of a small oscillation i«a| • {ga 3*''co8"''a + tt*sin« Ex. 3. An infinitely thin circular dit . moves on a perfectly rough horizontal plane in such a manner as to preserve a constant inclination a to the horizon. Find the condition that the motion may be steady and the time of a small oscillation. Let the radius of the disc be a, and the radius of gyration about a diameter k. Let Wj be the angular velocity about the axis, /i the angular velocity of the centre of gravity about the centre of the circle described by it, r the radius of this circle, then in steady motion (2i» + o') «a = iV cos o - — cot a, (2i' + d')r = - Ic'a cos a + ^ cot o. /* fit If 2* be the time of a small oscillation ^^ y(*« + a») = m' { fc''( 1 + 2 cos^a) + a" sin'a } - n/* cosa(6A'' + o") + 2n«{2P + o') - ^a sino. Ex. 4. A heavy body is attached to the plane face of a hemisphere so as to form a solid of revolution, the radius of the hemisphere being a and the distance of the centre of gravity of the whole body from the centre of the hemisphere being h. The body is placed with its spherical surface resting on a horizontal plane, and is set in motion in any manner. Show that one integral of the equations of motion is il sin*^ --^+C7w3 [ cos<' + -] =constant whether the plane be smooth, imperfectly rough, or perfectly rough. It is clear that the first two terms on the left-hand side of this equation is the angular momentum about the vertical through 0. Let this bo called /. Since wo may take moments about any axis through G as if (? were fixed in space, we have dl dt =F'.P3f. But PM= -PN.-, hence eliminating F' by equation (3) and in- tegrating, we get the required result. Ex. 6. A surface oi revolution rolls on another perfectly rough surface of revolution with its axis vertical. The centre of gravity of the rolliug surface lies in its axis. Find the cases of steady motion in which it is possible for the axes of both the surfaces to lie in a vertical plane throughout the motion. Ltt tf be the inclination of the axes of the two surfaces, P the point of contact, GM a perpendicular on the tangent plane at P, PN a perpen- dicular on the axis GC of the rolling body; F the friction, E the reaction at P; [X ; h ■ , i ..i I ^ i u t. ; ^ ' ;:l ..; x^ ■!l ii 478 MOTION UNDER ANY FORCES. n the anfi^lar velocity of the rolling body about itfl axis GC, ix the angiilar rate at which Q deBcribea its oiroulor path in space, r the radius of this circle. Then in steady motion Mn sin e(Cn- An COB e)=- F .QM-R. MP, R= ~ Mr/*' sin a + Mg cos a, F= -Mrft!' cos a -Mgeia a, n.PN+fABinO. QN= ~rn, where M is the mass of the body. 691. A surface of any form rollt on a fixed horizontal p^ane under the action of gravity. To form the equations of motion*. * The motion of a heavy body of any form on a horizontal plane seems to have been studied first by Foisson. The body is supposed to be either bounded by a continuous surface which touches the plane in a single point or to be terminated by an apex as in a top, whUe the plane is regarded as perfectly smooth. Poisson nses Eoler's equations to find the rotations about the principal axes, and refers these axes to others fixed in space by means of the formula; of Art. 235. He finds one integral by the principal of vis viva and another by that of angular momentum about the vertical straight line through the centre of gravity. These equations are then applied to find how the motion of a vertical top is disturbed by a slow move- ment of the smooth plane on which it rests. See the Trait6 de MScanique. In three papers in the fifth and eighth volumes of Crelle'a Journal (1830 and 1832) M. Coumot repeated Poisson's equations, and expressed the corresponding geometrical conditions when the body rests on more than one point or rolls on a*> edge such as the base of a cylinder. He aJso considers the two cases in which the plane is (1) perfectly rough, and (2) imperfectly rough. He proceeds on the same general plan as Poisson, having two sets of rectangiilar axes, one fixed in the body and the other in space connected together by the formulas usually given for transformation of co-ordinates. As may be supposed, the equations obtained are extremely complicated. M. Goiunot also forms the corresponding equations for impulsive forces. Those however which include the effects of friction do not agree with the equations given in this treatise. • In the thirteenth and seventeenth volumes of Liouviiys Journal (1848 and 1852) there will be found two papers by M. Piiiseux. In the first he repeats Poisson's equations and applies them to the case of a solid of revolution on a smooth plane. He shows that whatever angle the axis initially makes with the vertical, this angle will remain very nearly constant if a sufficiently great angular velocity be communicated to the body about the axis. An inferior limit to this angular velocity is foimd only in the case in which the axis is vertical. In the second memoir he applies Poisson's equations to determine the conditions of stability of a solid of any form placed on a smooth plane with a principal axis at its centre of gravity vertical and rotating about that axis. He also determines the small oscillations of a body resting on a smooth plane about a position of equilibrium. In the fourth volume of the Quarterly Journal of Mathematics, 1861, Mr G. M. Slesser forms the equations of motion of a body on a perfectly rough horizontal plane and applies them to the problem considered at the end of Art. 597. He uses moving axes, and his analysis is almost exactly the same as that which the author had adopted. MOTION OF A SOLID BODY ON A PLANE. 479 dar rate at , Then in he action of sms to have anded by a terminated h. Poisson , and refers ;. He finds laomentum ^nations are , slow move- al (1830 and ^responding rolls on a*^ in which the on the same I in the body ly given for obtained are quations for etion do not il (1848 and it he repeats olation on a akes with the great angular r limit to this tical. In the conditions of icipal axis at so determines a position of 361, Mr G. M. gh horizontal 597. Ho uses Lch the author Let GA, OB, OC, the principal axes at the centre of gravity, bo the axes of reference and lot the mass be unity. Lot ^ ((, 17, ^) = be the equation to the bounding surface, (f, 1;, f) the co-ordinates of the point P of contact. Let (p, 7, r) be the direction-cosines of the outward direction of the normal to the surface at the point f , tj, f, tlien a^ dift d, =0) (4). where U, V, IF are the resolved parts of the velocity of the point of contact P in the positive directions of the axes. :»■• 1^ I f) I III I il \V 4S0 MOTION UNDER ANY FORCES. 592. SfcomVy, let the plane be perfectly smooth. The oqnations (1), (2), (.1>, apply equnliy to thin cnne, but cqnationH (1) are not true. Since the rcaultaut of X, Y, Z ia & reaction 72 normal to the fixed plane, wo have X=-pR, Y=t-qR, Z=-rR (5). The negative sign is prefixed to R bccanso (p, q, r) are the directlon-coHines of the outward direction of the normal, and it ia clear that when these are taken posi' tively, the components of R are all negative. If at any moment R vanishes and changes sign the body will leave the plane. Siuce the velocity of parallel to the fixed plane is constant in direction and magnitude, it will usually be more convenient to replace the equations (2) by the following single equation. Let OM be the perpendicular on the fixed plane and lot MQ=z, then %--o^^ («)• It is necessary that the velocity of the point of contact resolved normal to the plane should be zero, this condition may bo written in either of the equivalent forms UpJt-rq^-Wr^Q dz di + (v-ia - fw*) P + (f w, - f Wa) 1 + (I", - i^Wj) r = r=o| (7). 693. Thirdly, let the body slide on an imperfectly rough plane. The equa- tions (1), (2), (3) and (7) hold as before. If /t be the coefficient of friction tho resultant of the forcos X, Y, Z must make an angle tan~' n with the normal at tho point of contact, hence {Xp+Yq + Zr)^_ 1_ ~1 + M» .(8). X-'+Y-'+Z' Also since the resultawt of {X, Y, Z), the normal at P and tho direction of slid- ing must lie in one plane, we have the determiuantal equation X(qW-rV)+Y(rU-pW) + Z{pV-qU)=0 (9). Siuce the friction must act opposite to the direction of sliding, we must have XU+ YV+ZW negative. When this vanishes and changes sign, the point of con- tact ceases to slide. If the body start from rest we must use the method explained in Art. 146 to determine whether the point of contact will begin to slide or not. Assume X, Y, Z to be tho forces necessary to prevent sliding. Then since m, v, w, Wj , Wj , Wg are all initially zero, we have by differentiating (4) and eliminating the differential coeffi- cients of u, V, w, Wp Wjj, Wj three linear equations to find X, Y, Z, in terms of the known initial values of {p, q, r) and {^, v, f). The point of contact will slide or not according as these values make the left-hand side of equation (8) less or greater than tho right-hand side. The equations to find X, Y, Z may be obtained by treating the forces as if they were indefinitely small impulses. In the time dt, we may regard the body as acted on by an impulse r/dt at and a blow whose components are Xdt, Ydt, Zdt at P. By Art. 296 we may consider these in succession. The effect of tho fir^t is to com- municate to P a velocity gdt in a direction normal to the fixed plane and outwards. If P does not slide, the effect of tho blow at P must be to destroy this velocity. Hence X, Y, Z may be found from the equations of Art. 304 if we write Ui=pff, MOTION OF A SOLID BODY ON A PLANE. 481 :i), (2), (3^. eaultaut of (5). ii-coflines of taken posi. auiobes and irection and 18 (2) by the lano and lot (6). lormnl to the 10 equivalent (7). The cqua- f friction the normal at tbo (8). rection of slid- (0). we must have e point of con- i in Art. 146 to tssnme X, 1', 2 ,, Wa, wgare all fferential coeffi- n terms of the will slide or not ) less or greater forces as if they le body as acted , YiU, Zdt at P. 3 first is to cora- and outwards. )y this velocity, ye write Mi=iV» Vi'^qy, u>i = rtf and u„ r,, tr, all equal to zero on the left-hand sides and (to suit the notation of this article) change p, q, r on the right-hand sides into {, n, f. Oeomotrically the point of contact will not slide if the diametral line of the fixed plane with regard to the ellipsoid called E in Art. S04 makes a less angle with the normal than tan~i ft. In any of these cases when p, q, r have been found, the inclinations of the prin- cipal axes to the vertical are known. Their motion round the vortical may thou be deduced by the rule given in Art. 249, When u, r, w and the motions of the axes have been found, the velocity of the centre of gravity resolved along any straight line fixed in space may be found by resolution. 594. Some integrals of these equations are supplied by the principles of angular momentum and Vis Viva. If the plane is perfectly smooth we have A Wjp -t- Bu^q + Cuar = a, Au>,*+ Bw,« -t- Cwa' + (f|)* = /9 - 2^7^. where a and /9 are two constants. If the plane is perfectly rough we have Aui' + Bu.j^ + Cua^ + u'' + v^ + w*=p-2gz. 595. Ex. 1. A body rests with a plane face on an imperfectly rough horizon* tal plane whose coefficient of friction is /x. The centre of gravity of the body is vertically over the centre of gravity of the face, and the form of the face is such that the radius of gyration of the face about any straight lino in its plane thropsh its centre of gravity is y. The body is now projected along the plane so that the initial velocity of its centre of gravity is Vq and the initial rotation about a vertical axis through its centre of gravity is Wo. If Uo be very small, prove that the centre of gravity moves in a straight line and its velocity at the end of any time t is v^ - fxgt. Show also that the anguloi: velocity at the same time is w,, [ 1 - — 1 , where Jc the radius of gyration of the body about a vertical through the centre of gravity. [Poisson, Tmiti de Mecanique,] Ex. 2. A body of any form rests with a plane face in contact with a smooth fixed plane so that the perpendicular from the centre of gravity on the plane falls within the face. If the body is then struck by a blow which passes through or begins to move from rest under the action of any finite forces whose resultant passes through 0, prove that it will not turn over, but will begin to slide along the plane, even if the line of action of the force cuts the plane outside the base. [CoumoW] 696. Wliatever the shape of a body may be we may suppose it to be set in rotation about the normal at the point of contact with an angular velocity n. If this angular velocity be not zero, the normal must be a principal axis at the point of contact, and yet it must pass through the centre of gravity. This cannot be unless the normal be a principal axis at the centre of gravity. If however n=0, this condition is not necessary. There are therefore two cases to be considered. Case 1. A body of any form is placed in equilibrium resting tcith the point C on a rough horizontal plane, with a principal axis at the centre of gravity vertical, and R. D. 31 13 • ' K: „■ 482 MOTION UNDER ANY FORCES. it then set in rotation with an angular velocity rx about GO. A small disturbance being given to the body, it is reqtiired to find tlie motion. Case 2. A body of any form is placed in equilibrium on a rough horizontal plane with the centre of gravity over the point of contact, A small disturbance being given to the body, to find the motion. i ! § i^ I' i 697. Case 1. Supposing the body not to depart far from its initial position, we have p, q, «, v, w, Wj, w, all small quantities and r=l nearly. Hence by (2), when we neglect the squares of small quantities, we see that X, Y are also small, and Z = -g nearly. It follows by (1) that Wg is constant and . . =n. Also | and i) are small and f = h nearly, where h is the altitude of the centre of gravity above the horizontal plane before the motion was disturbed. The equation to the surface may, by Taylor's theorem, be written in the form ^-h V^ + 2|« ^n where (a, h, c) are some constants depending on the curvatures of the principal sections of the body at the point C, The squares of all small quantities being neglected, the preceding equations become A~^-{B-C)nu^=-gv-hY ,dw< B^-{C-A)nw^ = hX+gi da di' dt nv=gp + X, dv di + nu=gq + Y, dp Tli = nq~Wfi, da « - Jiij + // W.J = 0, r - /( Wj + «? = 0, P abbe . = 1. Eliminating X, Y, u, v, Wj, Wj from these equations, we get d^q dp (A + h^)-^^ + {A+B + 2¥ -C)n f-{{B-C) n^ + hg + hhi^ \q=-{9 + hn'')v + hn ' df^ dt -(fi + 7»«)|^ + (J+J5 + 27t2-(7) n ^ + {{A-CW + hg + hhi^}p = {g + hn^)^ + hn^. It will be found convenient to express ^, rj in terms of p, q. The right-hand sides of each of these equations will then take the form Lp + Mq+L'^ + M''^, at dt To solve these equationH, we must then assume^;, q to be of the form p = P„ cos \t + Pj ti'in \t ) q = Po cos \t + Q, sin X( ) ' MOTION OF A SOLID BODY ON A PLANE. 483 , disturiance izontal plane e being given itial position, Hence by. (2), re also small, Also J and ij riiy above the ;o the surface the principal ling equations The right-hand e form If the tangents to the lines of ourvature of the moving body at C be parallel to the principal axes at the centre of gravity, these equations admit of corsidorable simplification. In that case the equation to the surface may be written in the form t--KM)- where a and c are the radii of curvature of the lines of curvature. The right-hand sides of the equations then become respectively - (g + An") cq + hna ~ and (g + hn^ ap + hnc -^ . To satisfy the equations, it will be sufficient to put p=Fco3(\t+f), q = Bin (\t+f). P This simpUfication is possible, because we can see beforehand that ^ = ~^ . P Substituting and eliminating the ratio - , we get the following quadratic to de- Cr termine \'. {{A-\-h'')\'' + {B-C-{-h{h-c)\n''+g{h-cMB-\-hP)\^ + {A~C+h{h-a)W-{-g{h-a)] =\H^{A-\rB + 2h'^-C-ha}{A-\-B-v21i^-C-hc). If \, \ be the roots of this equation, the motion is represented by the equations p=Pi cos {\t +/i) + Fj cos (\t -f-Za) g = (?i sin (\t +/i) + G, sm {\t +f^) f (i where tt « ^ ^™ known functions of X^, \ respectively, and Fy, F^, f^, /j are constants to be determined by the initial values of », g, ^ , -p . at at In order that the motion may be stable, it is necessary that the roots of this quadratic should be real and positive. These conditions may be easily expressed. Ex. 1 . A solid of revolution is placed with its axis vertical on a perfectly rouf,'h horizontal plane and is set in rotation about its axis with an angular velocity n. If c be the radius of curvature at the vertex, h the altitude of the centre of gravity, k the radius of gyration about the axis, k' that about an axis through tho vertex perpendicular to the axis of figure, show that the position of the body will bo stable if n > 2 — ir-h — - ' k^ + hc Ex. 2. An ellipsoid is placed with one of its vertices in contact with a smooth horizontal plane. What angular velocity of rotation must it have about the vertical axis in order that the equilibrium may be stable ? Eesult. Let a, h, c be the semi-axes, c the vertical axis, then the angular (Hg Jc*-a* + Jc*-b* velocity must be greater than /5g V c a^ + b'^i [Puiseux.] Ex. 3. A solid of any form is placed in equilibrium with tho poinl C on a smooth horizontal piano, a principal axis GC at the centre of gravity being vertical, and an angular velocity n is then communicated to it about GC. A small disturb- 31—2 ■4 ■ ■ 111 lii \ fm ! &- wn tf W M iw'r-' -.m" R r?f 484 MOTION UNDER ANY FORCES. ance being given, show that the hanuonic periods may be deduced from the qaad< ratio {A\^ + E)(B\'+F) = {A +B-C)n^\^+g^p'-p)^Bin^d COB.* S, where E = {B-C)n*+g{{h-p)sm^8 + {h-p')coa'*S\, F={A-C]n'^+g{{h-p)eoa'S+{h-p')Bin?d]. Also h is the altitude of the centre of gravity, p, p' are the principal radii of earvature at the vertex, and S is the angle the principal axis GA makes with the plane of the section whose radius of curvature is p. [Puiseux.] 698. Case 2. Supposing the disturbance to be small, we have 0^, u^, w^, u, V, w all small quantities. Hence when we neglect the squares of small quantities the equations (1) and (2) become respectively, ^|.=,z-!-F, B'^'=tx-cz. o^'=sr-,x (■). dtt ,, dv „ dw _ .... ^^=9P + X, ai=!^i+Y, -arOr+Z (u). Let fo» %> to ^^ *^^ co-ordinates of the point of contact in the position of equili- brium, and let f=fo + f'i V=Vo + v'i f=fo + f' Then in the small terms of equation (4) we may write f^, %, fo for f, 17, f. Hence differentiating these and eliminating X, Y, Z, u, v, w by help of equations (i) and (ii), we get (^ + V + fo'')^^-fo'?o'^^-Wo-~^=-ff('?r-f2) (xii), and two similar equations. Let Pq, go, Vq be the values of p, q, r ia the position of equilibrium. Then ^ = — = i^=p where p is the radius vector from to the point of contact. Now Po mating the ratios ^ , we have '.he quadratic to determinf X'. =0. (vi). Thus by virtue of the relation existing between p', q', j-*, each of these may be represented by an expression of the form Pi cos (Xjf +/i) + P, cos (Xsj« +/j). Substituting these values in equations (v) we see that w^, w^, u^ can each bo represented by an expression Oj + JSj cos (\t +/,) + E^ cos (\t+f.,), where E^, E^ are known functions of P^ P^ ... and \, \, but Q^, n^, O3 are »niall arbitrary quantities. By substituting in equations (3) and equating the coefficients of cos (\t+fi) and cos {\t+f.^), we may find the values of E^ and E^ without diffi- culty. And we also see that we must have Po 5* J-o ' so that, of the three 0„ Q^, Oj, only one is really arbitrary. We have therefore but five arbitrary constants, viz. Pj, P^, f^, f.^, and Oj. These are determined by the initial values ot Wj, Wj, W3, p' and q'. To find the motion of the principal axes round the vertical, let be the angle the plane containing GC and the vertical makes with the plane of AC. Then by drawing a figure for the standard case in which p, q, r are all positive, it will be seen that if /* be the rate at which OG goes round the vertical, /ji>Jl-j . ^ i'o<^i+9oWi : Wj COS ^+ Wj Sin ^ = /T^^T • Substituting for u^, ci>2> ^^^ takes the form At = Wj + A^i cos (Xi« +/i) + N^ cos (Xj,< +/j), where n,, N-^, N^ are all known constants. In order that the equilibrium may be stable it is necessary that the roots of the quadratic (vi) shoiUd both be real and positive. These conditions may easily be expressed. These conditions being siipposed satisfied, the expressions for p', q', r' will only contain periodical terms, and thus the inclinations of the principal axes to the vertical will not be sensibly altered. But the expressions for Wj, Wj, Wj may each contain a non-periodical term, and if so the rate at which the principal axes will go round the vertical will also contain non- periodical terms. The body therefore may gradually turn with a slow motion round the normal at the point of contact. The expressions for «, v, lo will contain only periodic terms, so that the body will have no motion of translation in space. Motion of a Rod, 599. Wlicn the body whose motion is to be determined is a rod, it is often more convenient to recur to the original equations of motion supplioil by D'AIombert's Principle. The equations of Lagrange may also bo used with advantage. These methods will be illustrated by the following problem, Hit if!? '4 : 'I ■ 1 i i f ■ M^yiM.-rW. 486 MOTION UNDER ANT FORCES. A uniform heavy rod, suspended from a fixed point by a string, ma^ws small oscillations about the vertical. Determine the motion. Let be taken as origin, and let the axis of z be measured vertically downwards ; let 2a bo the length of the rod, 6 the length of the string. Let (I, m, n) (p, q, r) be the dii-ection-coBiues of the string and rod. Then I, m, p, q are small quantities whose squares are to be neglected; and we may put n and r each equal to unity. Let w be the distance of any element du of the rod from that extremity A of the rod to which the string is attached. Let {x, y, z) be the co-ordinates of the element du, then we have »=hl + vp, y=ibm-\-uq, 8=6 + M (1). Let M be the mass of the rod, MT the tension of tho string, the equations of motion oi the centre of gravity will be dH dJ'p^ d^m d^q 0=g-T By D'Alembert's Principle the equation of moments round x will be Zdu (y^^,-z'^f^=2du{yZ-2r)=:ldu(j!,o). By equations (1) this reduces to £"du^- (b + u) (b^-^^+u^^^,y^=2ay(bm + aq). Integrating, we get which by equations (2) reduces to .(2). , d^m . - „ , 4 d'q -a — -- a dt^ Therefore the four equations of motion are . dH d^p cPl i d^p dt« ■*■ 3 " dt ' ■9P- ■(3), and two similar equations for m, q. These equations do not contain m or q, and on the other hand the equations to find in and q do not contain I or p. This shows that the oscillations in the plane xz are not affected by those in the perpendicular plane yz. 6ee Art. 450. To solve these equations , put l=Fain{M + a), p = G sin (\t + a) , we get b\^F + a\^G = .jP, b\^F+-aVG=gGi ^. ia + 3b ^. 3(;« „ ... \4 — —o\^+ 4=0, ab ab ' and the values of X may be found from this equation. i';i'(( small ivmwards ; n) ip, q, r) quantities to unity. A of the le element (!)• [uations of .(2). EXAMPLES. 487 In order to make a comparison of different methods, let na deduce the motion from Lagrange's equations. In this case we must determine the semi vis viva T true to the squares of the small quantities p, q, I, m, we cannot therefore put r=l, 11=1. Sincej!)« + g''' + r* = l, i' + m* + »"=!, we have r = l p^ + ^ n = l- I»+m" 2 ' 2 we must therefore replace the third of equations (1) by z = bn + ur=b + u-b —^ u*--~. If accents denote differential coefficients with regard to t, as in Lagrange's equations we have Sffix'3=2m(6«n + 26jyM+j>'«uS) = iH (m'' + 2hiya+'^ pA . The value of Sm/^ may be found in a similar manner. The value of Sm/' is of the fourth order and may be neglected. Hence we have 2r=6» {l'f + m'^) + 2ab{l'p'+m'fi')+ ~ (p'^+q'% 6 — ^ +a ^ „ j + constant. _,, ^. d dT dT dU , , ,,, , ,, The equation -n t;; - -r. = tt becomes oi" +.--»"= -gl; at dl dl dl to' similarly we get bl" + ^p"=-gp. These are the same equations which we deduced from D'Alembert's Principle, and the solution may be continued as before. EXAMPLES'. i It II ill! (3), ji m or g, and This shows perpendicular 1. A uniform rod, moveable about one extremity, moves in such a manner as to make always nearly the same angle a with the vertical ; show that the time of a small oscillation is 2ir /2a V 3« • i: cos a 3ff 1 + 3 cos'' a , a being the length of the rod. 2. If a rough plane inclined at an angle a to the horizon be made to revolve with unifoim angular velocity n about a normal Oz and a sphere be placed at rest upon it, show that the path in b]iace of the centre will be a prolate, a common, or a curtate cycloid, according as the polat at which the sphere is initially placed is with- out, upon, or within the circle whoso equation is a?+y^= — ^-— — x, the axis Oy being horizontal. When the sphere is placed at rest on the moving plane, it should be noticed that a velocity is suddenly given to it by the impulsive frictions. ■1 1 f * These Examples are taken from the Examination Papers which have been net in the University and in the Colleges. *ii-i: mummmmmm^ ■»«? .--^JU»L... II I L.I H—i 488 MOTION UNDER ANY FORCES. 8. A circular disc capable of motion about a vertical axis through its centre perpendicular to its plane is set in motion with angular velocity O. A rough uniform sphere is gently placed on any point of the disc, not the centre, prove that the sphere will describe a circle on the disc, and that the disc will revolve with angular velocity =-.iri- a — ;, Of where MP is the moment of inertia of the disc about its centre, m is the mass of the sphere and r the radius of the circle traced out. 4. A sphere is pressed between two perfectly rough parallel boards which are made to revolve with the uniform angular velocities Q and Q' about fixed axep per- pendicular to their planes. Prove that the centre of the sphere describes a circle a'uout an axis which is in the same plane as the axes of revolution of the boards and whose distances from these axes are inversely proportional to the angular velocities about them. Show that when the boards revolve about the same axis, their points of contact will trace on the sphere small circles^ the tangents of whoso angular radii will be c CI' " a 'W+U centre. ; , a being the radius of the sphere and c that of the circle described by its 5. A perfectly rough circular cylinder is fixed with its axis horizontal. A sphere being placed on it in a position of unstable equilibrium is so projected that the centre begins to move with a velocity F parallel to the axis of the cylinder. It is then slightly disturbed in a direction perpendicular to the axis. If d be the angle the radius through the point of cor+act makes with the vertical, prove '2 that the velocity of the centre parallel to the axis at any time ( is Fcos */ - 10 e and that the sphere will leave the cylinder when cos fi- ll' 6. A uniform sphere is placed in contact with the exterior surface of a perfectly rough cone. Its centre is acted on by a force the direction of which always meets the axis of the cone at right angles and the intensity of which varies inversely as the cube of the distance from that axis. Prove that if the sphere be properly started the path described by its ceniie will meet every generating line of the cone on which it lies in the same angle. See the SohUions of Cambridge hrohlcmi for 1860, page 92. 7. Every particle of a sphere of radius a, which is placed on a perfectly rough sphere of radius c, is attracted to a centre of force on the surface of the fixed sphere wi th a force varying inversely as the square of the distance ; if it be placed at the extremity of the diameter through the centre of force and be set ri tating about that diameter and then slightly displaced, determine its motion ; and show that when it leaves the fixed sphere the distance of its centre from the centre of force is a root of the equation 20x'< - 13 (2c + a) jc* + 7rt (2c + o)" = 0. 8. A perfectly rough plane revolves uniformly about a vertical axis in its own plane with an angular velocity n, a sphere being placed in contact with the plane rolls on it under the action of gravity, find the motion. Take the axis of revolution as axis of z, and let the axis of x be fixed in the plane. Let a be the radiuH, m the mass of the sphere ; F, F the frictions resolved EXAMPLES. 489 parallel to the axes of x and z and R the normal reaction. The equations of d'x F motion are therefore by *. 1. 179 tt:. - «'*= df* m The equations of rotation by Art. 255 are , „ dx R ,rf«z F' -an' + 2n j, = - and -r-= -g-^~. dt m iil^ " m (?», Fa dwu du, 'di ' Fa Since the point of contact has the same motion as the plane the geometrical equations by Art. 244 are — - on + awg=0, j -aug=0. Solving these equations we find that the sphere will not fall down. If the sphere sta^^ from relative rest at a point in the axis of x, we have z = - -j tan^ i { 1 - cos (nt cos i) ) where sin i = a/ -. The sphere will therefore never descend more than —^ below its original position. 9. A perfectly rough vertical plane revolves with a uniform angular velocity n about an axis perpendicular to itself, and also with a uniform angular velocity Q about a vertical axis in its own plane which meets the former axis. A heavy uni- form sphere of radius c is placed in contact with the plane ; prove that the position of its centre at any time t, will be determined by the equations i§-6n^^-2/£=o, _ d^ df' ' "■■ dt ' ''" \dt^ + QH )=o, z denoting the distance of the centre from the horizontal plane through the hori- zontal axis of revolution, and ^ that from the plane through the two axes. Prove also that 7u=7ca + 2ixb, 7v + 2)ua = 0, if a and 6 be the initial values of $ and z, u and v those of 3? and -j- . at dt 10. A hoop AGBF revolves about AB its diameter as a fixed vertical axis. GF is a horizontal diameter of the same circle which is without mass and which ia rigidly connected to the circle; DQis a smaller concentric hoop wlxich can turn freely about OF as diameter. If 0, fi', w, w', be the greatest and least angular velocities about AB, GF respectively, prove that . fi'=u'' - w'*. 11. OA, OB, 00 are the principal axes of a rigid body which is in motion about a fixed point 0. The axis 00 has a constant inclination a to a line OZ fixed in space, and revolves with uniform angular velocity fl round it, and the axis OA always lies in the plane ZOO. Prove that the constraining couple has its axis coincident with OB, and that its moment is -(A-C) 0^ sin a cos a. "I : 1 !l. I >;F' i'V ■ ' •••" CHAPTER XI. PRECESSION AND NUTATION, &C. &C. On the Potential, COO. To find the potential of a hody of any form at any external distant point. Let the centre of gravity O of the body be taken as the origin of co-ordinates and let the axis of x pass through 8 the external point. Let the distance GS= p. Let {x, y, z) be the co-ordinates of any element dm of the body situated at any point P and let QP = r, then P/S» = p" + r' - 2pa?. The potential of the body is ^ PS' 1 2/30;- r" 3 /2px—r ?si dm ( P 5 /2/: arranging these terras in descending powers of p, we get dtn U + P F=S a; 3a;' - r" 5x' - Sxr^ 35.e* - 3 0a; V -H 3/ ^^"■^"V^'^ 2p' ■•■ 8p* + ...} Let ilf be the mass of the body, then %dm = M. Also since the origin is at the centre of gravity, we have Xxdm = 0. Let A, B, G be the principal moments of inertia at the centre of gravity, / the moment of inertia about the axis of x, wliich in our case is the line joining the centre of gravity of the body to the attracted point. Then Xdmr'^^iA+B+C), -^dmx'^^dm {r' -f-z'')=\{A ^ B + C) - I. •m at any the origin le external o-ordiuates ; P and let body is so since the ,t the centre X, which in the body to ON THE POTENTIAL. 491 Let I be any linear dimension of the body, then if p be so great compared with Z that we may neglect the fraction f- J of the potential, we have ^r_M ^ A + B + C-SI If we wish to make a nearer approximation to the value of V, we must take account of the next terms, viz. 5X mx^ — 32ma;r' Let (I, t), ^ be the co-ordinates of m referred to any fixed rectangular axes having the origin at 0, and let (a, /8, 7) be the angles 08 makes with these axes. Then a; = f cos a + 17 cos /3 + (fcos 7 ; .'. "Zmx^ = cos' a Swi|' + 3 cos'a cos /3 "Zm^if + If the body be symmetrical about any set of rectangular axes meeting at G, we have Sm|' = 0, Xm^rj = 0, &c. = 0, so that this next term in the expression for the potential vanishes altogether. Thus the error of the preceding expression for V is comparable to only the fraction (-) of the potential. This is the case with the earth, the form and structure of which are very nearly sym- metrical about the principal axes at its centre of gravity. This theorem is due to Poisson, but it was put into the con- venient form just given by Prof. MacCuUagh. See Boyal Irish Transactions for 1855, page 387. 001. In the invest'gation of this value for the potential, S has been supposed to be at a very great distance. But the ex- pression is also very nearly correct wherever the point 8 be situated, provided the body be an ellipsoid whose strata of equal density are concentric ellipsoids of small ellipticity. To prove this, we may use a theorem in attractions due to Maclaurin, viz. The potentials of confocal ellipsoids at any ex- ternal point are proportional to their masses. Let us first con- sider the case of a solid homogeneous ellipsoid. Describe an internal confocal ellipsoid of very small dimeniions and let a', h', d be its semi-axes. Then because the elliptlity is very small, we can take a', h', c so small that 8 may be regarded as a distant point with regard to the internal ellipsoid. Hence the potential due to the internal ellipsoid is ^„ M' . A'^B'+C'-M' u'\\ '^■\ U 'i; 492 PRECESSION AND NUTATION. where accented letters have the same meaning relatively to the internal ellipsoid that imaccented letters have with regard to the given ellipsoid. The error made in this expression is of the /a'\* order ( - J V. Hence, by Maclaurin's theorem, the potential V of the given ellipsoid is y_M M A' ^-B'+C'-^ir and the error is of the order a* - If a, h, c be the semi-axes of the given ellipsoid, we have Similarly, B=^,B+^^ M\\ 0= ^, C + \ M\\ Also if (ot, /3, 7) be the direction-angles of the line GS with reference to the principal axes at G, we have /= J cos' a 4- 5 cos" ^+G cos' 7 = 177 / + ^ M\^. Hence, substituting, we have V= Jf . J+5+C-3J P 2p» If a, 6, c be arranged in descending order of magnitude, we can by diminishing the size of the internal ellipsoid make c as small as we please. In this case we have ultimately a = Va"* — c\ Let 6 be the ellipticity of the section containing a and c the greatest and least semi-axis. Then a' = ' V2e, and the error of the above expression for V is of the order 4 (- j e'F. The theorem being true for any solid homogeneous ellipsoid is also true for any homogeneous shell bounded by concentric ellipsoids of small ellipticity. For the potential of such a shell may be found by subtracting the potentials of the bounding ellipsoids, A^-B+ C hy Art. 5 being independent of the direc- tions of the axes. Lastly, suppose the body to be an ellipsoid whose strata of equal density are concentric ellipsoids of small ellipticity, the external boundary being homogeneous. Then the proposition being true for each stratum, is also true for the whole body. ne GS with I make c as 3f the direc- ON THE POTENTIAL. 403 This theorem was first given by Prof. MacCullagh as a pro])lcm, and was pubHshed in the Dublin University Calendar for 1834, page 268. Some years after, about 184G, he gave his proof of the theorem in his lectures, which is substantially the same as that given in this Article. See the Transactions of the Royal Irish Academy, Vol. xxii,. Parts i. and ii., Science. 602. Tlie following geometrical interpretation of the formula of Art. 600 is also due to Prof, MacCullagh. His demonstration and another by the Rev. R. Townsend may be found in the Irish Tramactiom for 1855. A system of material points attracts a point S whose distance from the centre of gravity G of the attracting mass is very great compared with the mutual distances of the particles. If a tangent plane be drawn to the ellipsoid of gyration perpendicular to GS, touching the ellipsoid in T and cutting GS in U, then the resultant attraction on S lies in the plane SOT. The coinponent of the attraction on S in the Erection TU= — ^ GU.UT. The component of the attraction on c . r ^. .. Tjr, ^I 3A + B + C-3I S in the direction UG= ^ + s ,, . These theorems are also true if we replace the ellipsoid of gyration by any confocal ellipsoid. Let a, h, c be the semi-axes of tliis confocal, and lot p be the perpendicular GU on the tangent plane. Since by Art. 26, A = Ma^ + X, £=Mb'' + \, c ^, ^ • <. L u IT ^J- M(a^ + b^ + c^-3p'') &e. where \ is some constant, we have V— — -\ ^ --^i ^—^ . P 2/)'* To prove that the resultant force on S lies in the plane SGT, let us displace Sio S' where SS' is perpendicular to this plane and is equal to pd'^. By Art. 326 \dV the force on S in the direction SS' is - — - . pdyp But after this displacement the tan- gent plane perpendicular to GS intersects along TU the former tangent plane, hence dp df To find the force P acting at S in the direction TU, let us displace S to S" where dV :0, and .\-r-r=0. d\y SS" is parallel to TU and is equal to pd^. Since OU is perpendicular to UT we have, exactly as in the Differential Calculus, TiZ^ 7^. Hence d^f/ pdtp p* ^ Lastly, to find the force S in the direction SO we have by Art. 326 _ dV M BA+B + C-3! ■»= --3- = rsT ; dp 2 Ex. Show that the product GU. TUia the same for all confocals. I'M n i i^. 494 PRECESSION AND NUTATION, ■ ! 608. Ex. If QP bo a straight line through tho oentro of prrnvity Btich that tho moment of inertia about it is equal to tho mean of the three principal momontfl of inertia at O, t}ien tho resolved attrnctiou of tho body on any point S in tho direction of gravity is a natical Journal, h, e such that where X is at does not make be constructed )unding surface, external point to the confocal emi-axes of the ited point. See VIII. page 322. is at the centre distant point as -C , placed per- M ntre of gravity. ■in shell bounded equal moments ncipal moments 1 was distributed ON THE POTENTIAL. 495 over tho focal conic of tho ellipsoid described in (4) so that tho density at any point , whore AB is tho diameter throngh P. P is proportional to - -z= ^AP . PJi Ex. 7. The attraction of any body of mass M on a distant particle may bo found iu the following manner. Lot an indefinitely tiiin shell of mass ZM bo constructed bounded by similar ellipsoids and having the ellipsoid of gyration at tho centre of gravity for one bounding surface. Also lot a particle of mass 4.1/ bo collected at the centre of gravity. Then tho attraction of M on any distant particle is tho same in direction and magnitude as if 4i>/ attracted it and 3If repelled it. 605. Ex. If the law of attraction had been - 4> (dist.) instead of tho inverse square, tho potential of a body on any external point H would have been represented by iiH^i (i'm* + w*, where I, m, n are the direction-cosines of tho radius through P referred to any rectangular axes. Show that the potential of the stratum at any external point is equal to Eja + b + c) Ef ■ 6{al^ + bm^+ ck") -a ~b- c - pf +5 p^ -' where / is the radius of the sphere and E its volume. 007. To find the Force-function dm to the attraction of any body on any oilier distant body. Let G, O' be the centres of gravity of the two bodies, and let OQ' = B, Let A, B, C; A', B, C be the principal moments of inertia of the two bodies at G and G' respectively ; I, T the moments of inertia about GG\ and let JLT, M' be the masses of the two bodies. Let m be any clement of the body If situated at the point 8, and let GS = p. Then the potential of the body 31 at m' is m < — H-5 \> where /, if, the moment of mertia of IP V ) the body M about GS. We have new to sum this expression for all values of m. This gives P -P \ ! !' I I \\ •1' i I i i If tri: :!i;! ;!-: 'msmssa '< 4^^ 496 PRECESSION AND NUTATION. The first term by the same reasoning as before gives + M — R 2R' In the second term, let x, y , z be the co-ordinates of m' Then referred to Q' as origin j[) = ^fl+-5-|- squares of oj', y , z'j , 7j = /(I + ace + ^y + yz' + squares), where a, fi, 7 are some constants. Substituting these, and re- membering that 'Zm'x = 0, Xmy = 0, Xm'z' = 0, we get Jif . A + B+G-SI 21^ ( /terms depending on the\) ( V squares of x', y, z J) Hence the required force-function is F= MM B M A' + E+C 2W 37' . ^,,A + B+C-M V M — 2R^ •11'^"" (II \ 712) ^> where I, V are any linear dimensions of the two bodies respectively. 608. To find the moment of the attraction of the sun and moon about one of the principal axes of the earth at its centre of gravity. Let the principal axes of the earth at its centre of gravity be taken as the axes of reference, and let a, /S, 7 be the direction- angles of the centre of gravity G' of the sun. Then if Fbe the potential of the sun or moon on the earth, we have F=^+j,f4>^'+^' R 2R' 37' , „,^ + 5+C-37 h if — 2R' where unaccented letters refer to the earth, and accented letters to the sun or moon. Let 6 be the angle the plane through the sun and the axis of y makes with the plane of wy, then ~rs is the required moment in the direction in which we must turn the body to increase 0. From the above expression, since 6 enters only through 7, we have dV_SM[dI de ~ 2R' dd' ON THE POTENTIAL. 497 lies of w' le, and re- 11- 1-M 7", where I, I' the sun and its centre of Df gravity be he direction- lif F be the 7-37 3ented letters through the en -ja IS the turn the body 9 enters only Now I = A cos'a + B cos'yS + C cos^, and by Spherical Trigo- nometry, we have cos 7 = sin yS sin $ cos a = sin yS cos 6 h dl .'.'^ = -2{A-C) sin*)8 sin ^ cos ^ ; .'.the moment required) « -^7' ^ .. about the axis of y | = - 3 -^t ( C - ^) cos a cos 7. In this expression the mass of the attracting body is measured in astronomical units. We may eliminate this unit in the fol- lowing manner. Let n be the mean angular velocity of the sun about the earth, R^ its mean distance, so that if M be the mass of the earth, we have M' + 3I ^0' =?= n- Now M is very small com- iM pared with M', so small that jp is of the order of terms already 31' neglected. Hence we may in the same terms put ^^ = n'^, and therefore the moment of the sun's at- traction about the axis of H = -Sn''{G- A) cos a cos 7 (^J. Let n" be the mean angular velocity of the moon about the earth, so that, if M" be the mass of the moon, B' the mean dis- M"+M tance, we have — ^7-3 — = n"^ Let v be the ratio of the mass of the earth to that of the moon, then we have 57-3 — - = w"", and • TV . ° therefore if it be the distance of the moon the moment of the moon' attraction about the axis on's ] Sir ,^ ., (R'y isofyj l + v^ ' ' \E J In the same way the moments about the other axes may be found. Putting k for the coefficient, we have moment about axis of a? = — S/c (J? — C) cos ^ cos 7, moment about axis oi z =: — Sk {A ^ B) cos a cos /3. 609. Ex. 1. A body free to move about its centre of gravity is acted on by any number of attracting particles arranged in any way at a constant distance p from the centre of gravity. If ^,, By, Cp D^, E^, F^ be the moments and products of inertia of the body referred to any rectangular axes meeting in tlie centre of gravity, R, T). 32 '11 ill il : vf ■^ Mmm 498 PRECESSION AND NUTATION. r= and if accented letters represent corresponding quantities for the particles referred to the same axes, prove that the mutual potential of the body and the particles is MM> 3(AiAi'+BiBi'+CiCj;+2FyFj^'+2DiD^'+2EiEi')-{Ai + B^ + C^){A^' + Bi' + Ci') P "^ V where HT is the mass of all the particles. If the axes of reference be principal axes for either body, this result admits of considerable simplification. Show that the numerator of the second term may be expressed in terms of the invariants of the momental eUipsoids of the body and of the system of particles. Ex. 2. The force function between a body of any form and a uniform circular ring whose centre is at the centre of gravity of the body and whose mass is Jl ' is ^ MM^ ^,A + B + C-SJ where / is the moment of inertia of the body about an axis thi-ough its centre of gravity perpendicular to the plane of the ring, and A, B, Q are the principal moments of inertia at the centre of gravity. This follows from Ex. 1. Ex. 3. Thence show iJiat Saturn's ring supposed uniform will have the same moments to turn Saturn about its centre of gravity as if half the whole mass were collected into a particle and placed in the axis of the ring at the same distance from Saturn, provided the particle repelled instead of attracted Saturn. Ex. 4. If the earth be formed of concentric spheroidal strata of small but different ellipticitieB and of different densities, show that rd{a^e). C-A fi da> where e is the ellipticity and p the density of a stratum, the major-axis of which is a ; the square of « being neglected. It follows that if e be constant C-A * - is independent of the law of density. If we assume the law of density and the law of ellipticity given in the Figure of C-A the Earth, this formula gives -^=-00313593. See Pratt's Figure of the Earth. Ex. 5. A body free to turn about a fixed straight line passing through the centre of gravity is in equilibrium under the attraction of a distant fixed particle. Show that the time of a small oscillation is 2^^^^^^^^—^-—^^, where the fixed straight line is the axis of y, the plane of xy in equihbrium passes through the attracting particle, and f, -q are the co-ordinates of the particle. Also^, B, C,D, E,F arc the moments and products of inertia of the body about the axes. If the straight line did not pass through the centre of gravity show that the time would be proportional to p. es referred irticles is e principal rms of the pai'ticles. )rm circular EisB is ilf' is its centre of ihe principal ave the same le mass were ame distance •n. of small hixt major-axis of 6 be constant, n the Fignre of •e 0/ the Earth. ig through the fixed particle. - I , where the ises through the )A,B,C,D,E,F show that the MOTION OF THE EARTH ABOUT ITS CENTRE OF GRAVITY. 499 Motion of the Earth about its Centre of Gravity. 610. To find the motion of the pole of the earth about its centre of gravity when disturbed by the attraction of the sun and moon, the figure of the earth being taken to be one of revolution. Let us consider the effect of these two bodies separately. Tlien, provided we neglect terms depending on the square of the disturbing force, we can by addition determine their joint effect. The sun attracts the parts of the earth nearer to it with a force slightly greater than that with which it attracts the parts more remote, and thus produces a small couple which tends to turn the earth about an axis lying in the plane of the equator and perpendicular to the line joining the centre of the earth to the centre of the sun. It is the effect of this couple which we have now to determine. It clearly produces small angular velocities about axes perpendicular to the axis of figure. We shall also suppose that the initial axis of rotation so nearly coin- cides with the axis of figure, that we may regard the angular velocities about axes lying in the plane of the equator to be small compared with the angular velocity about the axis of figure. Let us take as axes of reference in the earth, GG the axis of figure, GA and GB moving in the earth with an angular velocity 6^ round GG. Then following the notation of Art. 252, we have h^=A(a^, h' = A(o., h'-G i» ^a= «2- The equations of motion are therefore d(o. A~'-G^ were zero, and the -arth merely turned round its axis GC, it is clear that GC and therefore also the plane ZGC would be fixed in space. Hence 6^ is a small quantity of the same order at least as (o^ or «Dj, For a first approximation we neglect the squares of the small quantities to be found. We therefore reject the small terms a).^d^, w^d^ in the equations (1). The equations now become (2). Following the usual notation let 6 be t^^i angle ZG and ^ the angle the plane ZC makes with the fixed plane ZX. We have then the two geomv-trical equations ft). sin^ dt d0 '"»=dr ,(3). These follow at once from a mere inspection of the figure, or we may deduce them from Art. 235, by putting ^ = 0. We have now to find the magnitudes of L and M. Let S be the disturbing body and let it move in the direction X to Y. According to the usual rule in Astronomy, we shall suppose the longitude I of /S to be measured in the direction of motion Ui I. e of the earth varies very slowly, the term on the right-hand side is very nearly constant. If this be regarded as a suiHcieut approximation we have ' wi= -2^-^y-sm27, and Wj=0. But in fact these are nearly true when we take account of the periodical term provided only S moves slowly. For suppose ^ = 7lfo + 2Psin(/)« + 0, where p is small; we have in that case M'J:: Mt _. CnP . , . , ^v M neglecting the small term p^ in the denominator we have as before Wj = - -y . The motion of the axis O in space is therefore simply that duo to an angular velocity Wj about the axis A'. Since the plane 4'C moves so as always to contain the disturbing body „, where n is the mean angular velocity of the disturb- n ing be 'v about the earth. Rejecting these terms also, we have by (3), f4) and (5), d0 ^kC-A . . . „. -77 — — o p.— sni 6 sm zl dt zn G dyfr SkC-A .... „,. ■in 612. To find the motion of the pole of the earth in space referred to the pole of the orbit of the disturbing body as origin, wc have merely to iutograte the equations (7). For a point i IS N&ind ....(4), (5). low com- its axis, f this be J at once (6). I we take shown by e satisfied ity the order and such m is of the le disturb- ve have by .(7). ,h in space body as 7). For a MOTION OF THE EARTH ABOUT ITS CENTRE OF GRAVITY. 503 first approximation, in which we reject the squares of the small quantities to be found, we may regard 6 on the right-hand side as constant and equal to its mean value. If we write for I its approximate value l = nt + e, wc find by integration .(8). = const. + ~. — > — Pi — sin 6 cos 21 ^nn U ifr = const. — ^ — -, — 7=— cos (l — ^ sin 21) ^ 2nn C ^ 2 '. 613. We may also solve equations (2) in the following manner. Since wo reject the squares of the small quantities to be found, we may in calculating the values of L and ilf to a first approximation suppose d to be constant and I to be measured from a fixed point in space. We then have by the theory of elliptio motion l=n't + €' + Pi sin {pjt + gi) + i*2 sin (i^a* + (Za) + &c., where the coefiBcients of the trigonMnetrieal terms are all known small quantities, and all the coeflScients of t are very small compared with n. In the case of the sun the coefficient of t in the greatest of the trigonometrical terms is ^^ n and in the case of the moon ^ n. We may also include in thds formula the secular inequalities in the value of I. For, we shall presently find that d has no secular inequalities, and that the first point of Aries from which I is laeasured has a very slow motion which is very nearly uniform on the plane of the orbit of the disturlnug body. This slow motion m^ obviously be included in the n'. If we eliminate u^ between equations (2) we hr.ve d»wi CV 1 dL Cn^ The first term on the right-hand side we have already agreed to neglect. Sub- stituting in the expression for M given in (5) the value of I, suppose we have if=SFcos(\«+/>, where the constant part of M is given by X - and all the other values of X are very small. Then solving, we find FCn ,», , ,v w,= Since P and X" are both very small we may reject the small term X' in the denominator, we then have 1 M .,,= --Si?cos(Xt+/)=-^^. This result is strictly true for the constant term and very nearly true for the periodical terms. In the same way we may prove that w^= -,- . U 5 ; ■* 'J. ■WW P — ^— , so that by Art. 608 o = s — t^ or o = -^ — 7=j- v — ac- •' 2 G n 2 C n 1 + v cording as the sun or moon is the disturbing body, the orbit of the disturbing body being in both cases regarded as circular. Let us consider first the term —S cos 6 1 in the value of ■\|r. Let a point C^ describe a small circle round Z the pole of the orbit of the disturbing planet, the distance CZ being constant and equal to the mean value of 6. Let the velocity be uniform and equal to Sn' cos sin 0, and let the direction of motion be opposite to that of the disturbing bouy. Then 0^ represents the motion of the pole of the earth so far as this term is concerned. This uniform motion is called Precession. Next let us consider the two terms Bd=^ Ssm e cos 21, 5^ = I 5' cos 6 sin 21 I !l the help of ch \ is not tant. [lecting the theory of ding on the sun. From be effected ' are periodic L the others. 1 , it will be ng of the \k G-A m c ac- l + v rbit of the the value Z the pole GZ being le velocity irection of Then C; ar as this ission. MOTION OP THE EARTH ABOUT ITS CENTRE OF GRAVITY. 505 If we put a; = sin ^ Si/r, y = W, we have a ^ /I \l *■> ^Sfcos^sin^Y fi/Ssin^) which is the equation to an ellipse. Let us then describe round G^ as centre an ellipse whose semi-axes are „ S cos 6 sin 6 and ^ S&mO respectively perpen- dicular to and along ZG\ and let a point C, describe this ellipse in a period equal to half the periodic time of the dis- turbing body. Also let the velocity of G^ be the same as if it were a material point attracted by a centre of force in the centre varying as the distance. Then 0^ represents the motion of the pole of the earth as affected both by Precession and the principal parts of Nutation. If we had chosen to include in our approximate values of B and ^/r any small term of higher order, we might have re- presented its effect by the motion of a point G^ describing an- other small ellipse having C^ for centre. And in a similar manner by drawing successive ellipses we could represent geometrically all the terms of Q and ■^. 616. In this solution we have not yet considered the Com- plementary Functions. To find these we must solve ^^t+^^ 0, ^^-^-Ono) =0. at ^ ^* ^ dt We easily find (o^ = Hsm(-j-t+ k\ a)^ = — Hcos(^t + Kj. The quantities H and K depend on the initial values of ot^ w,. As these initial values are unknown H and K must be de- termined by observation. If H had any sensible value it would be discovered by the variations produced by it in the position in space of the pole of the earth. The period of these would be — >, , as -4 and G are nearly equal in the case of the earth, this period is nearly equal to a day. No such inequalities have been found. If however any such inequality existed we might consider these two terms together as a separate inequality to be afterwards added to that produced by the other terms of a^ w^ whose period is half a year. The effect of the complementary function on the motion of the pole of the earth has been already considered. The motion is the same as if the earth were at aay instant set in n ». i ill if ,i >00 PRECESSION AND NUTATION. '•/ I rotation about an axis whose direction-cosines are proportional to Ha'm(--7-t + K], — Hcoa [— j t + Kj and n and then left to itself. The instantaneous axis will describe a right cone of small angle round the axis of figure and also a right cone of small angle in space. Hence from this cause there can be no permanent change in the position in space of the axis of the earth. See Art. 522. 617. The preceding investigations are of course approxima- tions. In the first instance we neglected in the differential equa- tions the squares of the ratios of (o^ and (o^ to n, and afterwards some periodical terms which are an — th of those retained. We see by equations (3) and (8) that the second set of terms rejected is much gi-eater than the first, and yet when the sun is the dis- turbing body these terms are only about -^— ^ th part of those retained, and when the moon is the disturbing body these are only ^ th part of terms which themselves are imperceptibte. We have also regarded the earth as a solid of revolution so that A — B may be taken zero, a supposition which cannot be strictly correct. 3 C—A n 618. In the ca.-^o of the sun we have S=^ — /y , so that 2 C n the precession in one year is ^ SO- An' cos 9 27r. It is shown in 2 C n treatises on the Figure of the Earth that there is reason to put "^ = -0031. Also we have - = -~ , and ^ = 23°. 8'. This C n 36o gives a precession of about 15"'42 per annum. Similarly the coefficients of Solar Nutation in -^ and 6 are respectively found to be 1"*23 and 0"*53. If we supposed the moon's orbit to be fixed, we could find in a similar manner the motion of the pole produced by the moon referred to the pole of the moon's orbit. In this case 8=^ — p.- — t; . - The value of varies be- 2 G n \+v 11 1 tween the limits 23" ±5". Putting - = ^^ , j; = 80, ^ = 23°, we find a precession in one year a little more than double that pro- duced by the sun. But the coefficients of what would be the nutations are about one-sixth of those produced by the sun. 619. We have hitherto considered the orbit of the disturbing body to be fixed in space. If it be not fixed, we must take the MOTION OF THE EARTH ABOUT ITS CENTRE OF GRAVITY. 507 )portional then left t cone of b cone of in be no is of the >proxima- tial equa- bfterwards aed. We s rejected s the dis- ; of those )dy these ceptiblie. olution so cannot be — , so that n . shown in son to put 8'. This ilarly the irely found rbit to be f the pole on's orbit. varies be- = 23°, we that pro- ild be the 3un. disturbing 5t take tlie plane CA perpendicular to its instantaneous position at the moment under consideration. The quantity 0,, will not be the same as before*, but if the motion of the orbit in space be very slow, Oj will still be very small. We may therefore neglect the small terms O^m^ and 0^(o^ as before. The dynamical equations will not therefore be materially altered. With regard to the geometrical equations (8) it is clear that w,, m, will continue to express the resolved parts of the velocity of C in space along and perpendicular to the instantaneous position of ZC. To this degree of approximation therefore, all the change that will be necessary is to refer the velocities as given by equations (7) to axes fixed in space and then by integration we shall find the motion of C. This is the course we shall pursue in the case of the moon. The attractions of the planets on the earth and sun slightly alter the plane of the earth's motion round the sun, so that the position of the ecliptic in space varies slowly. It can oscillate nearly five degrees on each side of its mean position. If the earth were spherical there would be no precession caused by the at- tractions of the sun and moon. The direction of the plane of the equator would then be fixed in space, and the changes of its obliquity to the ecliptic would be wholly caused by the motion of the latter, and would be very coi.>siderable. But, as Laplace re- marks, the attractions of the sur and moon on the terrestrial spheroid cause the plane of the equator to vary along with the ecliptic so that the possible change of the obliquity is reduced to about one and a third degrees which is about one-quarter of what it would have been without those actions. At present the obliquity is decreasing at the rate of about 48" per century. After an immense number of years, it will begin to increase and will oscillate about its mean value. These in- equalities we do not propose to discuss in this treatise. We must refer the reader to the second volume of the Mecanique Celeste, livre cinquifeme. He may also consult the Connaissance des Temps for 1827, page 234. 620. Ex. 1. If the earth were a homogeneous shell bounded by similar elhpsoids, the interior being empty, the precession would be the same as if the earth were solid throughout. * The value of ^3 may be found in the foUov ng manner. The orbit at any inst;\nt is turning about the radius vector of the planet as an instantaneous axis. Let u be this angular velocity which we shall suppose known. Let Z, Z'; B, E bo two successive positions of the pole of the orbit and the extremity of the axis of B respectively. Then ZB=a right angle =Z'i5'. Hence the projections of ZZ', liB', on ZJ are equal. This gives, since ZB is at right angles to both CZ and SB, BSB' am BS=ZVZ' sin ZC. Now the angle ZCZ'- - 5^;, and the angle BSB'=u, hence 8^3 . sin 9= -n sin I. The value of 5^3 must bo added to the former value of 0^, ,4f«MM 508 PRECESSION AND NUTATION. Ex. 2. If tho earth wero a homogeneous shell bounded externally by a spheroid and internally by a concentric sphere, the interior being filled with a perfect fluid of the Hame density as tho earth, show that tho preceshiiou would be greater than if the earth were solid throughout. Let (a, a, e) be the semi-axes of the spheroid, r the radius of tho sphere. Then C - A since the precession varies as — - by Art. 615, the precession is increased in tho ratio a*c : a*c — r". Ex. 3. If the sun wero removed to twice its present distance show that the solar precession per unit of time would be reduced to one-eighth of its present value; and the precession per year to about one- third of its present value. Ex. 4. A body turning about a fixed point is acted on by forces which tend to produce rotation about an axis at right angles to the instantaneous axis, show that the angular velocity cannot be uniform unless the momentiJ ellipsoid at the fixed point is a spheroid. The axis about which tho forces tend to produce rotation is that axis about which it would begin to turn if the body were placed at rest. Ex, 5. A body free to turn about its centre of gravity is in stable equilibrium under the attraction of a distant fixed particle. Show that the axis of least moment is turned toward tho particle. Show also that the times of tho principal oscillations are respectively 2t lonfn^i^ju 8"^ ^tt ja-i/'/irr-iU • If the body be the earth and M' be the sun, show that the smaller of these two periods is about ten years. 621. To giv6 a general explanation of the manner in luhich the attraction of the Sun causes Precession and Nutation. If a body be set in rotation about a fixed point under the action of no forces, we know that the momenta of all the particles are together equivalent to a couple which we shall represent by G about an axis called the invariable line. Let T be the Vis Viva of the body. If a plane be drawn perpendicular to the axis of G at a distance — jj- e* from the fixed point, then the whole motion is represented by making the momental ellipsoid whose parameter is e roll on this plane. In the case of the earth, the axis 01 of instantaneous rotation so nearly coincides with OG the axis of figure that the fixed plane on which the ellipsoid rolls is very nearly a tangent plane at the extremity of the axis of figure. This is so very naarly the case that we shall neglect the squares of all small terms depending on the resolved part of the angular velocity about any axis of the earth perpendicular to the axis of figure. Let us now consider how this motion is disturbed by the action of the sun. The sun attracts the parts of the earth nearer to it with a slightly greater force than it attracts those more remote. by a sphoroiil I perfect fluid ;reater than if iphero. Then lOreased in tbo show tliftt the of itn preueut value. 1 which tend to ixis, show that )id at the fixed that axis about hie equilihrinm 3 axis of least times of tho {'(B-A)) ' er of these two ,er in which n. under the the particles (resent by (^ he Vis Viva le axis of G i^hole motion e parameter \e axis 01 of the axis of rolls is very is of figure, the squares the angular the axis of )y the action nearer to it nore remote. MOTION OF THE EARTH ABOUT ITS CENTRE OF GRAVITY. 509 Hence when the sun is either north or south of the equator its attraction will produce a couple tending to turn tho earth about that axis in the plane of the equator which is perpendicular to the line joining the centre of the earth to the centre of the sun. Let the magnitude of this couple be represented by a, and let us suppose that it acts impulsively at intervals of time dt. At any one instant this couple will generate a new momentum adt about the axis of the couple a. This has to be compounded with the existing momentum G, to form a resultant couple G'. If the axis of a were exactly perpendicular to that of we should have G' = \/'W^'\oLdtf = G ultimately. Let be the angle that the axis of G makes with OG, then ^ is a quantity of that order of small quantities whose square is to be neglected. Taking the case when OG, DC and the axis of a are in one plane, for this is the case in which G' will most difter from G, wo have G" = (G cos eY + {G sin e + adt)' = G^+2Gxainddt (1). Then a and being of the same order of small quantities, the term a sin is to be neglected He'ace we have G' = G. But the axis of G is altered in space by an angle — ^ in a plane passing through OG and the axis of a. Next let us consider how the Vis Viva T is altered. If T' be the new Vis Viva we have T' — T = twice the work done by the couple a = 2a (ft) cos /3) c?« (2), where to cob ^ is the resolved part of the angular velocity about the axis of a. For the same reason as before the product of this angular velocity and a is to be neglected. Hence we have T' = T. It follows from these results that the distance — ^— e^ of the fixed G plane from the fixed point is unaltered by the action of a. Thus the fixed plane on which the ellipsoid rolls keeps at the same distance from the fixed point, so that the three lines OG, 01, OG being initially very near each other will always remain very close to each other. But the normal OG to this plane has a motion in space, hence the others must accompany it. This motion is what we call Precessiou and Nutation. Lastly these small terms which have been neglected will not continually accumulate so as to produce any sensible effect. As the earth turns round in one day, the axis OG will describe ' i 4 it ■• I'i^ '■:, I! wammn 510 PRECESSION AND NUTATION. a cone of small angle 6 round OG. The axis about which the sun generates the angular velocity a is always at right angles to the plane containing the sun and OC. Hence, regarding the sun as fixed for a day, the angle 6 in equation (1) changes its sign every half day. Thus 0' is alternately greater and less than 0. Simi- larly since the instantaneous axis describes a cone about OG it may be shown that T' is alternately greater and less than T, 622. Let us trace the motion of the axis OG through a whole year. Describe a sphere whose centre is at and let us refer the motion to the surface of this sphere. Let K be the pole of the ecliptic and let the sun 8 describe the circle DEFH of which K is the pole. Let DF be a great circle perpendicular to KG, then since OG and the axis of figure of the earth are so close that we may treat them as coincident, D and i^'will be the intersections of the equator and ecliptic. When the sun is north or south of the equator, its attraction generates the couple a, which will be positive or negative according as the sun is on one side or the other. This couple vanishes when the sun ir passing through the equator at D or F. If the sun be anywhere in DEF, i.e. north of the equator, G is moved in a direction perpendicular to the arc 08 towards D. If the sun be anywhere in FHD, a has the opposite sign and hence G is again moved perpendicular to the instantaneous position of G8 but still towards D, Considering the whole effect produced in one year while the sun describes the circle DEFH, we see that G will be moved a veiy small space towards D, i.e. in the direction opposite to the sun's motion. Resolving this along the tangent to the circle centre K and radius KG, we see that the motion of G is made up of (t) a uniform MOTION OF THE EARTH ABOUT ITS CENTRE OF GRAVITY. 511 ch the sun gles to the bhe sun as sign every 0. Simi- out Oa it lan T. Lgh a whole as refer the e pole of the of which K to KG, then Iclose that we Itersections of south of the hich will be |e side or the through the 2F, i.e. north ticular to the ED, OL has the [licular to the Considering describes the small space [sun's motion. \K and radius (1) a uniform motion of G along this circle backwards, which is called Preces- sion and (2) an inequality in this uniform motion which is one part of Solar Nutation. Again as the sun moves from D to E, G is moved inwards so that the distance KG is diminished, but as the sun moves from E to F, KG is as much increased. So that on the whole the distance KG is unaltered, but it has an in- equality which is the other part of Solar Nutation. It is evident that each of these inequalities goes through its period in half a year. 623. To explain the cause of Lunar Nutation. The attraction of the sun on the protuberant parts at the earth's equator causes the pole C of the earth to describe a small circle with uniform velocity round K the pole of the ecliptic with two inequalities, one in latitude and one in longitude, whose period is half a year. These two inequalities are called Solar Nutations. In the same way the attraction of the moon causes the pole of the earth to describe a small circle round M the pole of the lunar orbit with two inequalities. These inequalities are very small and of short period, viz. a fortnight, and are therefore generally neglected. All that is taken account of is the uniform motion of C round M. Now K is the origin of reference, hence if M were fixed the motion of G round M would be represented by a slow uniform motion of G round K together with two inequalities whose magnitude would be equal to the arc MK, or 5 degrees, and whose period would be very long, viz. equal to that of G round K produced by the uniform motion. But we know by Lunar Theory that M describes a circle round K as centre with a velocity much more rapid than that of G. Hence the motion of G will be repre- sented by a slow uniform motion round K, together with two inequalities which will be the smaller the greater the velocity of M round K, and whose period will be nearly equal to that of M round K. This period we know to be about 19 years. These two inequalities are called the Lunar Nutations. It will be perceived that their origin is different from that of Solar Nutation. 624. To calculate the Lunar Precession and Nutation. Let K be the pole of the ecliptic, 3f that of the lunar orbit, G the pole of the earth. Let KX be any fixed arc, KG= 0, XKG=yfr, then we have to find 6 and yjr in terms of t. By Art. 615 the velocity of G in space is at any instant in a direction perpendicular to MG, and equal to SuT G- A _1_ 2n C' l+u cos MC am MG. 11, a 1 .; V:>,!! r\ If! ;:i 1 ■ f [ ■i B H-i\ ill |! M2 PRECESSION AND NtJTATION. For the sake of brevity let the coefficient of cos MG sin MG be represented by P. Then resolving this velocity along and perpendicular to KG, we have ^ = - P sin If C cos ilf C sin ^Cif 1 sin ^ § = - P sin MG cos MG cos KGM at By Lunar theory we know that M regredes round K uniformly, the distance KM remaining unaltered. Let then KM=i, and the angle XKM= — mt + a. Now by spherical trigonometry, cos MG = cos t cos 6 + sin « sin cos MKG, • Ti*-/-/ TjryyTir cost — cos iJfC cos ^ sm MG cos KGM= r—^ sm a = cos t'sin — sin i cos cos MKG, s\ii MG. sin KGM == sin t sin MKG. Substituting these we have ^ = _ p jsin t cos i cos sin J/ZC + | sin't sin ^ sin 2MKg\ , sin ^ -^ = — P -Isin ^ cos f cos"* t — ^ sin'i j — sin tcos I cos 2^ cos MKG— ^ sin'isin ^cos ^cos 2MKG[ . For a first approximation we may neglect the variations of d0 and -^ when multiplied by the small quantity P. Hence -jr contains only periodic terms, and the inclination has no per- manent alteration. But -^ contains a term independent of MKG ; considering only this term, we have ^ = constant — Pcos ^ [cos'* — ^ sin' ijt. This equation expresses the precessional motion cf the pole due to the attraction of the moon. We may write thib liquation in the form ■'/r = ^^ —jft. To find the nutations, we must substitute for MKG its r .pproxi- mate value MKG= {-m+p) t+a-'>lr^. V sin MG dong and uniformly, M=i, and MKC, IMKC\- mkg\. •iations of 6 dd Hence ^ las no per- pendent of f the pole lib equation its f.pproxi- MOTION OF THE EARTH ABOUT ITS CENTUE OF GRAVITY. 513 We then have after integration /, . Psini cost cos ^ irr'/^ Psin'e'sin^ c^ttr^n ^ = const.-— C03 MKC — , ,- co3 2iI/AC7. m — p 4 {m — p) The second of these two periodic terms being about one- fiftieth part of the first, which is itself very small, is usually neglected. Also p is very small compared with m, hence we have - - Psin icosi cos ^ ,,rT^ — da cos MKC. This terra expresses the Lunar Nutation in the obliquity. In the same way by integrating the expression for ^, and neglecting the very small terms, we have I I D a f 2 • 1 • a A ^ n sm 2i C( >Ir = llr — P cos ^ I cos" I — rx Slli't ]t — F —r — . - ^ " \ 2 / ziii s sin 2i cos 2$ siu^ sm MKC. The angle MKC is the longitude of the moon's descending node, and the line of nodes is known to complete a revolution in about 18 years and 7 months. If we represent this period by 27r T we have MKG= — „ / + constant. The pole M of the lunar orbit moves round the point of re* ference K with an angular velocity which is rapid compared with;?, but yet is sufficiently small to make the Lunar Nutations greater than the Solar. We may also notice that if M had moved round K with an angular velocity more nearly equal to p the Nutations would have been still larger. This may explain vhy a slow motion of the ecliptic in space may produce some corresj ending nutations of very long pciiod and of considerable magnitude. R. D. nf] •h ■ 1 vl n mw m! ii j 514 PBECESSION AND NUTATION. Motion of the Moon about its centre of gravity. 625. In discussing the precession and nutation of the equinoxes, the earth has been regarded as a rigid body two of whose principal moments at the centre of gravity are equal to each other. One cons -r^aence of this supposition was that the rotation about the axis of unequal moment s not directly altered by the attraction of tho disturbing bodies. As an examplo of the ^ilect of these forces on the rotation when all the three principal moments are unequal, we shall now consider the case of the moon as disturbed by the attraction of the earth. As our object is to examine the mode in which the forces alter the several motions of tho moon about its centre of gravity rather than to obtain arithmetical results of the greatest possible accuracy, we shall separate the problem into two. In the first place we shall suppose the moon to describe an orbit which is very nearly circular in a plane which is one of the principal planes at its centre of gravity. In the second case we shall remove tho latter restriction and examine the effects of the obliqiuty of the moon's orbit to the moon's equator. 626. The moon describes an orbit ahont *lu! centre of the earth which is very nearly circular. Supposing the plane of the o bit to be one of the principal planes of the moon at its centre of gravity, find the motion of the moon about its centre of gravity. Tict uA. GB, GC be the principal axes at G the centre of gravity of the moon, and let GC be the axis pei'pendicular to the plane in which G moves. Let A, B, C be the moments of inertia about GA, GB, GC respectively, and let M be the mass of the moon, and let accented letters denote corresponding quantities for the earth. Let be the centre of the earth, and let Ox bo the initial line. Let OG=r, GOx = 6. Let us suppose the moon turns round its axis GC in the same direction that the centre of gravity describes its orbit about 0, and let the angle OGA = is the angle which GA, a line fixed hi the body, makes with Ox, a line fixed in space, the equation of the motion of the moon roand GC is d^9 dV sy'B-A dt.^ ■*■ dt 2 sin 2 - /J + npM sin {pt + a) + &c. .(3), 3 B — A where for the sake of brevity we have put n' ^ — -^ ■ 2' Now we know by observation that the moon always turns the same face towards the earth, so that amongst the various motions which may result from different initial conditions, the one which we wish to examine is characterized by ^ being nearly constant. Let us then introduce into this equation the assiimp^-ion that is nearly constant; we may then deduce from the integral how far this assumption is compatible with any given initial conditions which we may suppose to have been imposed on the moon. Tutting 0=0o + ^'> where ^^is supposed to contain all the constant part of ^, we easily find (4). 2o and unity for cos 2^,, in these equa- tions. Solving the second equation, we find, 0=7/ sin {qt + A') - ^^ + 3/ ^"^^ sin (pt + a) + &o. .(5), where II and K are two arbitrary constants whose values depend on the initial con- ditions. The angular velocity of the moon about its axis is therefore given by the formula d9 d(f> dt "*■ (it M7" = n-\-pt + IIqBm(qt + R) + M ^"'^^jnin{pt + a) + &o (6). S3— 2 II l::;1 I *:| ^' ''1 . w ml int , II 516 PRECESSION AND NUTATION. If 5' were negative or zero, the character of the sohition of (3) would be altered. In th 3 former case the expression for tf» would contain real exponentials. If the initial conditions were so nicely adjusted that the coefficient of the term containing the positive exponent were zero, the value of would still be always small. But this motion would be unstable, the smallest disturbance would alter the values of the arbitrary constants and then ^ would become large. If we also examine the solution when q'=0, we easily see that ip could not remain small. We therefore infer that of the axes 6A, oJ of the moon, the axis of least moment is turned towards the earth and that these two principal momeutb are not equal. In order that the expression ^5) for ip may represent the actual motion it if. necessary and sufficient that H when found from the initial conditions should je small. Wo see, by differentiation, that £fq is of the same order of small quantities as ^. Hence B will be small if at any instant the angular velocity, viz. TT + -17 , of the moon about GO were so nearly equal to the angular velocity, (it itt do viz. — , of its centre of gravity round the earth, that the rt.tio of the difference to q is very small. If therefore we suppose the moon at any instant to be moving with its axis of least moment pointed towards the earth and its angular velocity about its axis of rotation to be nearly equal to that of the moon round the earth, then the axis of least moment will continue always to point very nearly to the earth. The mean angular velocity of the moon about its axis will immediately become equal to that of the moon about the earth and will partake of all its secular changes. This is Laplace's theorem. It shows that the present state of motion of the moon is stable, rather than explains how the angular velocity about the axis came to be so nearly equal to the angular velocity about the earth. 627. By comparing the value of the angular velocity of the moon about its axis obtained by theory with the results of observation, wo may hope to obtain some indications of the value of q^ and thence of V-A C . If the term Ilq sin {qt + K) B- A could be detected by observation, we should deduce the value of — ^— .*fom its period. Among the other terms of the expression for the angular velocity of the moon about its axis, those will be beat suited to discover the value of q which have the largest coefficients, that is ihose in which either the numerator M is the greatest or the denominator 2' -p"^ the least possible. By examining the numerical value of B- A their coefficients Laplace has shown that if — ^ were as great as "03 the elliptic inequality could be recognized by observation, and if it were between •0011 and -003 the annual equation could be observed. 628. We may also calculate by the help of Art. 326 the radial and transverse forces which act on the centre of gravity of the moon due to the mutual attractions of the earth and moon. Since the principal moments of the moon are nearly equal and its linear size small compared with its distance from the earth, these forces are very nearly the same as if the moon were collected MOTION OF THE MOON ABOUT ITS CENTRE OF GRAVITY. 517 be altered, lis. If the containing mall. But e values of xamino the e therefore t is turned action it if, ins should ir of small lar velocity, lar velocity, iifferenoe to ;h its axis of it its axis of the axis of The mean c[ual to that res. This is the moon is ime to be so on about its pe to obtain ^2 sin (3* + A') -A from its of tho moon lich have the the greatest ■rical value of )3 the elliptic 3014 and -003 nd transverse the mutual of the moon listance from were v^oUccted into its centre of gravity. The effect of the small forces neglected by this assumption will be insignificant compared with the other forces which act ou the centre of gravity of the moon. The motion of the centre of gravity of the moon is therefore very nearly the same as if the Vviiole mass were collected into its centre of gravity. Since however there are no other forces which have a moment round GC besides those found above, the effect of these may be perceptible. The effects of tidal friction on the rotation of the moon may be omitted, at least at the present time. Ex. The centre of gravity (? of a rigid body describes an orbit which is nearly circular about a very distant fixed centre of force attracting according to the Newtonian law and situated in one of the principal planes through 0. If r=c(l + p), 6=nt + v^ ba the polar co-ordinates of G referred to 0, show that the equations of motion are 3 '2'' 3«V - Sh" '^ = - j «'7' - I n'7 cos 2^ ' dt 2 T + "-iiTf = 5 "> sill 2^ d«0 dV where 7 = 7 = 2C- A-B + ji dt^ = - ^ sin 20 •6Mc^ We may notice that the values of 7 and 7' are much smaller than 3' and might therefore be rejected in a first approximation. If the body always turns the same face to the centre of force so that is nearly constant and in small, show that there will bo two small inequalities in tho value of of the form 1 sin {pt + a), whore p ia given by (i>2 - n") (p"^ - 2") - 3/i'7 (p" + 3?i=) = 0, one of these periods being nearly the same as that of the body round the centre of force and the other being very long. If the body turns very nearly uniformly round its axis GC, so that = n't + e' nearly, show that there will be two small inequahties in the value of =w aud another in which p = 2n'. 629. E". 1. Show that the moon always turns the same face very nearly to that focus of her orbit in which the earth is not situated. [Smith's Prize.] Ex. 'z. If the centre of gravity G of the moon were constrained to describe a circle with a uniform angular velocity n about a fixed centre of force attracting according to the Newtonian law ; show that the axis QA of the moon will oscillate on each side of GO or will make complete revolutions relatively to GO according as the angular velocity of the moon about its axis at the moment when GA and GO coincide in direction is less or /jreater than n + q. Find also the exteii' of the oscillations. Ex. 3. A particle m moves without pressure along a smooth circular wire of mass M with uniform velocity under the action of a central force bituated in the centre of the wire attracting according to the law of nature. Show that this system of motion is stable if ,, > r^^-- . The disturbance is supposed to be given M 2o to tho particle or tho wire, the contre of force remaining fixed in space. i I! ilif 5 .<; ■ f •ill T lli i ! 518 PRECESSION AND NUTATION. Ex. 4. A uniform ring of mass M and of very small Bection is loaded with a heavy particle of mass m at a point on its circumference, and the whole is in uniform motion about a centre of force attracting according to the law of nature. m Show that the motion cannot be stable unless ,, lies between M + 111 •8279. •815865 and This example shows (1) that if a ring, such as Saturn's ring, be in motion about a centre of force, its position cannot be stable, if the ring be uniform ; and (2) that if, to render the motion stable, the ring be weighted, a most delicate adjustment of weights is necessary. A very small change in the distribution of the weights would change a stable combination to one that is unstable. This example is taken from Prof. Maxwell's Eatay on Saturn's Sings. Ex. 6. The centre of gi-avity of a body of mass 3f , sjmmetrical about the plane of xy, is ', and is a point such that the resultant attraction of the body on is along the line GO. Then if the body be placed with coinciding with a fixed centre of force S, and be set in rotation about an axis through perpendicular to the plane of xy with an angular velocity w, G will, if undisturbed, revolve uniformly in a circle, always tiurning the same face towards O, provided Mau^ is equal to the resultant attraction along GO, where a is the distance GO. It is required to determine the conditions that this motion should be stable. The motion being disturbed, will no longer coincide with the centre of force S. Let two straight lines at right angles revolving uniformly round fi^ as origin with an angular velocity u be chosen as co-ordinate axes, and let x be initially parallel to OG. Let {x, y) be the co-ordinates of 0, ^ the angle OG makes with the axis of x, then x, y, are all small. Let V be the potential of the body at 0, cPV dT ^ ™„. ^' dy^=^' of force. and let dj^^" ,«-,-. Let S be the amount of matter in the centre dxdy ay* Then the equations of motion of G, Art. 179, will reduce to and the equation of angular momentum about S will lead to 2uax+aj^y+ (a^+k^) j^=0, where k is the radius of gyration of the body about 0. Combining these equations as a determinant and reducing we &id that the differential equation in |, rj, or (ft is of the form The condition of stability is that the roots of this equation should be real and negative. Hence A, B, O must be of the same sign and B'^>'iAC. This pro- position is due to Sir W. Thomson and is given in Prof, Maxwell's Essay on Saturn's Rings. 630. The motion of a rigid body about a distant centre of force has been investigated on the supposition that the motion takes place entirely in one plane. We see by equation (2) of Art. 62C that the case in which the centre II ded with a rhole is in of nature. 115865 and in motion form; and )st delicate ribution of ible. This ;he plane of idy on ia ith a fixed indicular to 3 uniformly squal to the required to ire of force 5 as oriRin be initially nakes with body at 0, 1 the centre 36 equations in $, i;, or be real and This pro- / on Saturn's ae has been xely in one 1 the centre MOTION OP THE MOON ABOUT ITS OfNTRE OF GBAVITY. 619 of gravity describes a circular orbit, and the rigid body always turns the axis of least moment towards the centre of force, is one of steady motion. The preceding investigation also shows that this motion is stable for all disturbances which do not alter the plane of motion. It remains now to determine the effect of these disturbances in the more general case when the motion takes place in three dimensions. Tho whole attraction of the centre of force on the body is equivalent to a single force acting at the centre of gravity, and a couple. If the size of the body be small compared with its distance from the centre of force we may neglect the effect of the motion of the body about its centre of gravity in modifying the resultant force. The motion of the centre of gravity will then be the same as if the whole were collected into a single particle. The problem is therefore reduced to the following. A rigid body turns about its centre of gravity 0, and is acted on by a centre of force E which moves in a given manner. In the case in which the rigid body is the moon, this centre of force, i.e. the earth, moves in a nearly circular orbit in a plane which itself also has a slow motion in space. This motion is such that a normal GM to the instantaneous orbit describes a cone of small angle about a normal OK to the ecliptic. The two normals maintain a nearly constant Jn- clination of about 5". 8'; and the motion of the normal to the instantaneous orbit is nearly uniform. 631. It will clearly be convenient to refer the motion to axes OX, OY, GZ fixed in space such that OZ is normal to tho ecliptic. Let GA, GB, GC be the principal axes of the moon at the centre of gravity G. Let (p, q, r) be the direction- cosines of OZ referred to the co-ordinate axes GA, GB, GC. Then we have, since GZ is fixed in space, do ft .(I). ■^ - Ujr+ wgp = df ^-WjlJ + Wi^^O Now our object is to find the small oscillations about the state of steady motion in which OZ, GC, GM all coincide. We shall therefore havep, q, Wj, Wj all small, and r very nearly equal to unity. The equations (I) will therefore become dp dq dt -Wi + ?ip = where n is the moan value of u^. Let X, ft, V be the direction-cosines of the centre of force E as seen from Q. Then we have by Euler's equations and Art. 608, dt -(B-C) wa«a= -3n''>(/?- C7)/«i» C-^^-{A-B) wi«j= - 8rt'»(4 - B}\n (11). 11 i tit i 520 PIIECESSION AND NUTATIOf, Id the case of stead/ motion, the rigid body ulways turns the axis {GA) ot lenst momeut towards the centre of force, and w^=n'. We liave then both fi and i> small quantities, uo that in the first equation we may neglect their product /uf, and in the second equation we may put v\=v. Also, we may pat W3=Tt=n' in the small terms. If I be the latitude of the eoi-th as seen from the moon, we have Bin l=coa ZE=p\+ qn + rv~p + v nea,T\y. Hence the two first of Eoler's equations f^ake the form dt (C-A)nwi= -3n^{C-A)(-p+Bml) .(III). If the earth, as seen from the moon, be supposed to move in a circular orbit in a plane making a constant inclination tan~^ k with the ecliptic, and the longitude of whose node is -gt + /3, we shall have Bin I =kBia (n't + gt - p). In this expression g measures the rate at wliich the node regredes, and Is abont the two hundred and fiftieth part of n. We shall therefore regard - as a small n quantity. To solve these equations, it will be found convcnieDt to substitute for Wj, w, their values in terms of p, q. We then have d*q .^P A-j^l+iA + B-OnJ^-n^B-Oq^O ^%-i^+^-<^)n^f+^n^{C-A)p = Sn^{C-A)sml 1) of lenst nd V small fiv, and in the small .(III). liar orbit in B longitude md is about as a small ite for Ui, Wj MOTION OF THE MOON ABOUT ITS CENTRE OF GRAVITY. 521 To find p, q, let us put p = P sin {{n'+g)t-p], q-=Qcoa {{n' +g)t-p}, where P, Q are some constants to be determined by substHution in the equation. We have Q{A{n+g)'\- (B - On'] =P(A + B-C)n{,Hg) ) P{£(n+g)'-i(C-A)n'\-Q{A + B-C)n{n+g)=-3n*k{C-A)S' We may solve these equations to Gud P and Q accurately. In the case of the moon the ratios — —— , — - — , — -— and - are all small. If then we neglect the C A H 71 ° products of these small quantities, the first equation gives us p=l- ^. The second equation will then give n P= Snh{C-A) Sn{V-A)-2Dg' As g is very small compared with n, we may regard P and Q as equal. 632. The complementary functions may be found in the usual manner by assuming p=FBm(8t + II), q = Q COB (st+ IT), on substituting we have the quadratic AB8*-{{A+P-C)'-B{B-C)-U{A~C)]nU'' + i{A-q{B-C)n*=0, to find «>, and G _ (A + B-C)n8 to find the ratio of the coefficients of corresponding terms in p and q. If the roots of this equation were negative p and q would be represented by exponential values of t, and thus they would in time cease to be small. It is therefore necessary for stability that the coefficient of a' should be negative and the product (A - C)(£ - C) positive. Both these conditions are probably satisfied in the case of the moon. For since B-C and A-C are both small, the term (A + B- C)' is much greater than the two other terms in the coefficient of s^. Also, since the moon is flattened at its poles, we shall probably have A and B both less than C. 633. Let 31 be the pole of the moon's orbit, which is the same as that of the earth's orbit as seen from the centre of the moon. Then M is the pole of the dotted line in the figure of Art. 631. Therefore the angle EZM measured by turning ZE in the positive direction round Z until it comes into coincidence with Z}f,iB= -^-{{n + g)t~p]. Again, if the angle J?ZC be measured in the same direction, we have cos EC -COB CZ cos ZE _ v-r{ p\ + qn+r¥) sin CZaiixZE ~ J^T^emZE Hence we easily find BxaEZC= f iT~^' But BinCZM^sin EZM ooB EZV- cos EZM Bin EZC _cos{{ii + g)t ■ p\p-Bin{ (n+g)t- p\q cos EZC= -P v'p'+a' , nearly. ^^M m ^: fi ,;:.; 522 PRECESSION AND NUTATION. .1 Ml; If now wo substitute for p and q their valnen, it is clear that the terras in p and q, whose argument is n + g, disapijear. So that if F and were zero, the Hiue of the angle VZM would be absolutely zero. In this case the tlirce polos C, Z, M must lie in an arc of a great circle, or, which is the Hame thing, the moon't equator, the moon's orbit, and the ecliptic inunt cut each other in the same line of node$. It however F and G bo not zero, but only very small, wo have SF'sin (»'<+//') Bin CZM= ^//'=' + 20'" Bin («'« + //')' where F, G' contain either i!" or as a factor, and are therefore small. If then F and O be both small compared with P, llio angle VZM will remain either alwuyu small or always nearly equal to tt. The intersection of the moon's equator with the ecliptic will then oRcillato about the intersection of the moon's orbit with the ecliptic as its mean position. Since these oscillations are inHcnsible, it follows that in the case of nature, the com- plementary functions must be extremely small compared with the terms depending directly on the disturbing force. 634. If we disregard the complcmontary functions we have p = P sirup, q=P coa 4>, where = (n' + <;) t - /3. Now Hin' CZ ssp" + q''; therefore CZ= -P very nearly. The value of CZ, the inclination of the lunar equator to the ecliptic, is known to be about 1"'.28'. Hence, since - =-004, we may deduce from the ex- n C-A pression for P at the end of Art. 631 an approximation to the value of - „ C — A In tlxia manner Laplace finds —jr- = -000599. it' '■.' IS in p and bo Hiue of m a, Z, M i's equator, noilc$. If then F her alwttya illato about ion. Since 3, tbe com- s depending jj=P8in^, :=-P very e ecliptic, is rom tbo ex- , C-A ae of - D . CHAPTER XII. MOTION OF A STUING OR CHAIN. The Equations of Motion. 635. Prop. To determine the generil ejuations of motion of a stnng under the action of any forces. First. Let the string he inextensihle. Let Ox, Oji, Oz be any axes fixed ia space. Let Xmds, Ymds, Zmds bo the impressed forces that act on any element ds of the string whose mass is mds. Let u, v, to be the resolved parts of the velocities of tliis element parallel to the axes. Then, by D'Alembert's principle, the element ds of the string is iu equilibrium under the action of the forces »"* (^ - ^) ' "^^^ (^-f ) ' ^^* (^- ft) • and the tensions at its two ends. ,da} Let T be the tension at the point (a?, y, z), then T-..- , T dy ,dz ds' ^ ds' T^ are its resolved parts parallel to the axes. The resolved parts of the tensions at the other end of the element will be j,dx^d ds and two similar quantities with y and z written for x. Hence the equations of motion are du di dv m m m dt ds dw dt t) + "^^ .(1). y m ill I t ■■^ I ; 5 ) i f ( iii I W i 'i 524 MOTION OF A STKING. In these equations the variables s and t are independent. For any the same element of the string, s is always constant, and its path is traced out by variation of t. On the other hand, the curve in which the string hangs at any proposed time is given by variations of s, t being constant. In this investigation s is measured from any arbitrary point, fixed in the string, to the element under consideration. To find the geometrical equations. We have (^)'-(i)'-(iy-- Differentiating this with respect to t, we get dx du dy dv dz dw „ ,c>\ 1 ><. — I = (2). ds ds ds ds ds ds The equations (1) and (2) are sufficient to determine w, y, z, and T, in terms of s and t Ex. If V be the Vis Viva of any arc AB ol the chain ; T^, T^ the tensions at the extremities of this arc ; »/, tf,' the velocities of the extremities resolved along the tangents at those extremities, prove that Y-^= T^Ui - ^iWi' + f(Xu +Yv + Zio) mds, the integration extending oyer the whole arc. 636. The equations of motion may be put under another form. Let <^, -v/r, ^ be the angles made by the tangent at x, y, z, with the axes of co-ordinates. Then the equations (1) become ^f = |,(2^cos<^) + m.Y (3), with similar equations for v and w. dx To find the geometrical equations, differentiate cos = -^ with respect to t ; , ddi du ... '''-'"'''^dt^ds- (^)- Similarly, by differentiating cos "^ = ^ - and cos ;j^ = ^- , we get two other similar equations for -^jt and ;^. Taking these six equations in conjunction with the following cos" + cos''i^+ cos';j^ = l (5), we have seven equations to determine u, v, w, ^, >/r, -^ and T. Wll THE EQUATIONS OF MOTION. 525 nt. For t, and its land, the given by ;ion s is dt du^ ds (7). . dd> dv ''''^Tt=ds The arbitrary constants and functions which enter into the solutions of these equations must be determined from the peculiar circumstances of each problem. 637. Secondly. Let the string be elastic. Let ff be the unstretched length of the arc s, and let tndff be the mass of an element da of unstretched length or ds of stretched length. Then by the same reasoning as before, the equations of motion become m du d /^dz\ ,, i[t=d.Vdir'^'^ (')• and two similar equations for v and lo. To find the geometrical equations we must differentiate the independent variables being now and T in terms of s and t. •(3). THE EQUATIONS OF MOTION. 627 thus. We If the string he extonsihle, the dynamical equations become me plane, e tangent > element element. ikes with of motion (1). 3WS. We 6, - . Since tangent consitiera- ,...(2). ....(3). ', V, ^ and dt dt mdff at at mp da To find the geometrical equations, we may differentiate M=u'co8 ^-ti'sin with regard to a. This gives by Art. 637 . ^d4> -^^(7'cos^)=(^^^----Jcos^-(^- + --JBm0. rt. C38, this reduces _ du' v' / T\ - dcr pV X/' By the same reasoning as in Art. 638, this reduces to IdT X dt dj> dt ('-D-S'-K'-D 639. The equations (2) and (3) may also be obtained in the following manner. The motion of the point P of the string being represented by velocities w' and v' along the tangent PA and normal PO at P, the motion of a consecutive point Q will be represented by velocities u' + du and v' + dv' along the tangent QB, and normal QO at Q, Let the arc PQ = ds, and let ^i^be a perpendicular on PA. Since the string is inextensible, the resultant velocity of Q resolved along the tangent at P must be ultimately the same as the resolved part of the velocity of P in the same direction. Hence (u + du') cos d^ — (v + dv) sin dj> = u, or, proceeding to the limit, nil du — vd(^ = ; .*. -v- ^- = 0. <70 Again, .^ is the angular velocity of PQ round P. Hence the difference of the velocities of P and Q resolved in any direc- tion which is ultimately perpendicular to PQ must be equal to .'. (u + du) sin d(f> + (v + dv) cos d(f> — v'= ds -^ , or in the limit dj) _ dv u' dt ds p ' 640. Ex. 1. An elastic ring without weight, wlioso length when unstretched is given, is stretched round a circular cylinder. The cylinder is suddenly annihilated, m i I ! h 628 MOTION OF A STRING. show that ' the time which the ring will take to collapse to its natiiral length ia / Mav the natural radius. - , where M is the mass of the string, X its modulus of elasticity, and a is Ex. 2. A homogeneous light inextensible string is attached at its extremities to two fixed points, and turns about the straight line joining those points with uni- form angular velocity. Find the form of the string, supposing its figure per- manent. Hcsult. Let the straight line joining the fixed points bo the axis of x, then the form of the string is a plane curve whoso equation is 1 + ( -^ j = ( ' J , where a and h are two constants. On Steady Motion. 04)1. Def. When the motion of a string is such that the curve which it forms in space is always equal, similar, and siirii- larly situated to that which it formed in its initial position, that motion may be called steady. 642. Pkop. To investigate the steady motion of an inexten- sible string. It is obvious that every element of the string is animated with two velocities, one due to the motion of the curve in space, and the other to the motion of the string along the curve which it forms in space. Let a and h be the resolved parts along the axes of the velocity of the curve at the time t, and let c be the velocity of the string along its curve. Then, following the usual notation, we have w = a -f c cos 0| v= 6 +csin0j dtL (1). d d=■—-, cos 0= t- + g-c«)8in0J ■a). g'B= -g'8 + ficy where A and B are two arbitrary constants. At the point where the cable meets the ground, we must have either T=0 or ^=0. For if be not zero, the tangents at the extremities of an infinitely smaU portion of the string make a finite angle with each other. Then, if T be not zero, resolving the tensions at the two ends in any itircction, we have an infinitely small mass acted on by a finite force. Hence the element will in that case niter its posi- tion with an infinite velocity. First, let us suppose that ^=0. Also at the same point, y=0 and »' = 0. Hence B= -ct. IXC Putting S-=«i we get by division dy dxf' ey A-cuf+ea' This is the differential equation to the curve in which the cable hangs. To solve this equation*, let us find a' in terms of the other quantities, A^,-e^%-^eu dx dx .(2). 8 = Differentiating, we have s/^^m- ^^^.(A-cx'^ehj) 0-'^)' f. * The problem of the mechanical conditions of the deposit of a submarine cable lias been ronsidered by the Astronomer Royal in the Phil. Mag, July 1858. His solution is different from that given above, but his method of integrating the differ- ential equation (2) has been follo\Yed. tl c HI Ki- ln mm ■^i^ ON STEADY MOTION. 531 Put p for y '.rbere convenient, and put v for A -ex'+e^i/; the equation then becomes . . dp 1 dv -c dx' vdm^ {l-ep),Jl+p* in which the variables are separated, and the integrations can be effected. The equation can be integrated a second time, but the result is very long. The arbitrary constant A may have any value, depending on the length of the cable hanging from the ship at the time «=0. The curve in its lower part resembles a circular arc or the lower part of a com- mon catenary. But in its upper part the curve does not tend to become vertical, but tends to approach an asymptote making an angle cot~ie with the horizon. The as3'mptote does not pass through the point where the cable touches the ground but A below it, its smallest distance being — ; ^=^ ; the asymptote also passes below the ship. If the conditions of the question be such that the tension at the lowest point of the cable is equal to nothing, the tangent to the curve at that point will not neces- sarily be horizontal. Let \ be the angle this tangent makes with the horizon, Beferring to equations (1) we have simultaneously Hence =0, 2/=0, «'=0, r=0, and ^-X. A= --. cos X, i/= - , sin X - ct. 9 9 Tlie diJTerential equation to the curve will now become 11 dy dx'' — ; sin X + s' - ey (I ■■■^^"" * ;COSX + e«'-f»' 9 .(3), which can be integrated in the same manner as before. One case deserves notice; viz. when e=cotX. The equation is then evidently satisfied by y=-x'. The two constants in the integral of (3) are to be determined by the condition that when a;'=0, y = 0, then -y^,=tanX. Both these conditions are satisfied by the relation y=-a;'. Hence this is the required integral. The form of the cable is therefore a straight line, inclined to the horizon at an angle X=cot~^£; and the tension may be found from the formula 7= , — ^"Jl— . 1 + cos X Ex. 2. Let a cable be delivered with velocity c' from a ship moving with uni- form velocity c in a straight line on the surface of a sea of uniform depth. It the resistance of the water to the cable be proportional to the square of the velocity, the coelTicient B, of resistance for longitudinal motion being different from the coefficient A , for lateral motion, prove that the cable may take the form of a straight line making an angle X with the horizon, such that coi^\= sJ it acts in the direction in which s is mea- sured, and Q L . uritiv" when it acts in the direction in which p is measured ak .< ^ lie rormal, viz. inwards. Let m be the mass of a unit of length. Let u, V be the velocities of the element along the tangent and normal. Then the equations of motion are by Art. 638 du d(b -n 1 dT ,_. -jT-W j7 =P + --7- (1), dt dt m da ^ " |+„#=«+i?: (2), dt dt m p ^ ' where T is the tension, p the radius of curvature, and (fy the angle the tangent makes with any fixed straight line. The geometrical equations are l-r" f^)' ^p'4! <*'• Differentiating (1) and multiplying (2) by - , we get r d'u dJ'i^ dvd^dP 1 ffr-i dsdt dsdt dsdt ds m ds* \ ,^> Idv^^nd^^Q^ll^ I ^''^' p dt p dt p m p* i But by differentiating (3) we have, since - — -j, d^u ^ d^<}> 1 ^» ^ A /px dsdt dsdt p dt A^'^" Hence, subtracting the second of equations (5) from the first, we have by (4) and (6) m \ds p*/ ds p \dt J ' ON INITIAL MOTIONS. 633 rom the plane, n curve. ectively ■ce P is is mea- n which he mass tangent ^8 ...(1), ...(2), he angle Dmetrical ....(4). (•^). ....(0). the first, In the beginning of the motion just after the string has been cut we may reject the squares of small quantities, hence (-^) may be rejected. Hence we have d'T T dP Q — 3= — m-T-+m — ds' ds (7). p as p This is the general equation to determine the tension of a string just after it has been cut. The two arbitrary constants introduced in the solution of this equation are to be determined by the circumstances of the case. If both ends of the string are free, we must have T= at both ends. Since the string begins to move from a state of rest we have flit fll) initially u = 0, v = 0. At the end of a time dt, -,, dt and -^ i/t will be the velocities of any element of the string. Hence if yfr be the angle the initial direction of motion of anj el .or* of the string makes with the tangent to the element, we hn e : equa- tions (1) and (2) 1 T tan'^ = m p m ds (8;. It must be remembered that the constants of integration are necessarily constant only throughout the length of the string at the time ^ = 0. They may be functions of t and may be either continuous or discontinuous. For example, if a point of the string be absolutely fixed in space, the transverse action of the fixed point on the string may cause the constants to become discon- tinuous at that point. In this case equation (8) is not necessarily true in the immediate neighbourhood of the fixed point. 646. If the string be heterogeneous we may easily show in the same way, that the initial tension is given by ±(\dT\_\T ^_dP Q ds\mdsj m p^ ds p ' 647. A string is in equilibrium, under the action of forces in one plane. Supposing the string to he cut at any given point, find the instantaneous change of tensic n. Let Tq be the tension at any point (ic, y) just before the string was cut. Then the forces P, Q satisfy the equations of equilibrium = P+ m ds 0=^ + ir„ I ? m VI 534 MOTION OP A STIIING. Hence ds p m ds* m p* If T'bo the instantaneous change of tension, wo have T'=T-T^, The equation of the last article therefore becomes da' p' ~ '111 648. Ex. 1. A strin(j is in equilihrlum in the form of a circle about a centre of repulsive force in the centre. If the string be now cut at any point A, prove tliat the tension at any point P is instantaneously changed in the ratio of o'^ + e"'' where is the angle subtended at the centre by the arc AP. Let Fho the central force, then P=0, and mQ= - F. Let a bo the radius of tbe circle. Then the equation of Art. G45 to determine T becomea ds^ n'^ a * Let 8 be measured from the point A towards P, then s-=ad\ also F is independ- ent of s. Hence we have T=FaJrA€^ + Be-^. To determine the arbitrary constants A and B we have the condition r=0 when ^=Oand^=27r; T=^Fa.{l- e^ + e" But just before the string was cut T-Fa. tiou follows. Hence the result given in the onunoia- Ex. 2. A string is wound round the under part of a vertical circle and is just supported in equilibrium at the ends of a horizontal diameter by two fore a. The circle being suddenly removed, prove that the tension at the lowest point is instantly decreased in the ratio 4 ;e^+e ^. Ex. 3. The extreme links of a uuiform chain can slide freely on two given curves in a vertical plane, and the whole is in equilibrium under the action of gravity. Supposing the chain to break at any point, prove that the initial tension at any point is r=y (A(jy+B), where y ia the altitude of the point above the direc- trix of the catenary, fj> the angle the tangent makes with the horizon, and A, B two arbitrary constants. Explain how the constants are to be determined, Ex. 4. A string rests on a smooth table in the form of an arc of an equiangular spiral and begins to move from rest under the action of a central force F which tends from the pole and varies as the 7i"' power of the distance, show that the initial tension is given by It coB^ a + sin'' a T=-rF — « (H + l)cos*a-sin'a 'rArP+ Hi'', ON INITIAL MOTIONS. 535 wbore a is the angle of the spiral, p and q are the roots of the qnadratio a; (a; - 1) = tan* a. Show that the solution changes its form when a is such that the first term h infinite, and find the new form. 649. A string rests on a smooth horizontal table and is acted on at one extremity by an impulsive tension, find the impulsive tension at any point and the initial motion. Let T be the impulsive tension at any point P,T -\-dT i\\Q tension at a consecutive point Q, then the element P^ is acted on by the tensions T and T+dT at the extremities. Let <^ be the angle the tangent at P to the string makes with any fixed line ; II, V the initial velocities of the element resolved respectively along the tangent and normal at P to the string. Then, resolving along the tangent and normal, we have muds == (T-hdT) cos d-T mvds = {T + dT) sin d(l> therefore proceeding to the limit 1 dT m as 1 T v= . m p But by Art. 639, we have ~^ = - - Hence the equation to find T becomes ds' ~p'~^' This, as might have been expected from mechanical consi- derations, is the same as the equation in Art. 647. If the chain be heterogeneous we easily find in the same way d ndT^ ^_ Is \m ds j VI p''* ' ds \m ds The two results in this article appear to have been first given in College Examination Papers. 650. Ex. If Ti, T„ bo the imp'olsive tensions at the extremities of any arc of the chain, wi, Mj the initial velocities at the extremities resolved along the tan^ gents at the extremities, prove that the initial kinetic energy of the whole arc is This readily follows by integrating m {u' + v^)ds along the whole length of the arc. But it also follows at once from Art. 331, for the work done at either extre- mity is the product of the tension into halt the initial tangential velocity. 1" •' i 536 MOTION OF A STRING. Small Oscillations of a loose chain. 651. A heavy heterogeneous chain is suspended hy one ex- tremity and hangs in a straight line under the action of qravity. A small disturbance being given to the chain in a vertical plane, it is required to find the equations of motion*. Let be the point of support, let the axis Ox be measured vertically downwards and Oy horizontally in the plane of disturb- ance. Let mda be the mass of any elementary arc whose length PQ is ds, and let T be the tension at P. Let I be the length of the string, and let us suppose that a weight Mg is attached to the lower extremity. The equations of motion as in Art. G35 will be df ~mds\ ds)'^^ df m ds \ ds) .(1). Since the motion is very small, the point P will oscillate in a very small arc, the tangent at the middle point being horizontal. d'jtj Hence we may put -ji = ^' For a similar reason we may put dx = ds. We therefore have by integrating the first of equa- tions (1) T= constant —g jmdx. But T= Mg when x = l, hence we find T=Mg + gj mdx. (2). f.«V • In the Seventh Yulume of the Journal Poly technique, Poissc i discusses the oscillations of a heavy homogeneous chain suspended by one extremity. Putting U,-x)^i:„gh equal to « or «' according as the upper or lower sign is taken, and dy y He obtains •/' = y{l- x)i , he reduces the equation to the form , , , _ - -7 -; ^. i> J^ ' > "* dsda' 4 (8 + 8')3 the integral by means of two definite integrals and two infinite series. After a rather long discussion of the forms of the arbitrary functions which occur in the integral, he finds that a solitary wave will travel up the chain with a uniform acceleration and down with a uniform retardation each equal to half that of gravity. ' one ex- f pravity. ',al planOf measured f disturb- )se length length of led to the .(1). illate in a liorizonttil. may put of equa- (2). iscusses the ty. Patting taken, and He ottains es. After a )ccur in the a uniform lalf that of I SMALL OSCILLATIONS OF A LOOSE CHAIN. 537 Wlien the chain is homogeneous, thia equation takes the simple form T=Mf; + mff{l-x) (3). It may bo noticed that this expression is independent of the time ; the tension at any point of the chain is ccjual to the total weight of matter below that point. The secoT'.J equation may be written in either of the forms df m dx \ dxj = lr^ + i^^2/| in dx^ m dx dx J where T is a function of x given by the equations (2) or (3). (4), 652. Let us suppose that the displacements of the particles forming any finite portion of the chain during a finite time, are represented by ;/ = <^ {x, t), where is a continuous function of x and t Let P be a geometrical point within this portion of the di/ chain which moves so that the particle- velocity at P, i. e. -4^ is .always equal to some constant quantity A, Let v be the velocity with which P moves, then following in our mind the motion of P, we have d^y . d^y „ dt' dxdt .(5). Let Q be a point also within the portion, such that the tangent to the chain at Q makes with the vertical an angle whose tangent, i. e. -r- , is yp> where B is some constant quantity. Let v' be the velocity with which Q moves, then (^S)^-" dxdt dx \ ,(G). Eliminating the second dififerential coefficients of y from equa- tions (4), (5) and (6), we easily deduce that if P and Q coincide at any instant, vv' = - (7). m This reasonipg requires that all the second differential coeffi- cients should be finite, and that y should be a continuous function of x and t. It would not apply to any point P, if the discontinuous extremities of two waves were passing over P in opposite direc- tions. But the consideration of these exceptions is uunecc .ary for our present purpose. r)38 MOTION OP A STFINQ. i t f , I m Let AB be a disturbed portion of the chain travelling in the direction ^5 on a chain otherwise in equilibrium. At the con- fines of the disturbance the two portions of the string must not make a finite angle with each other. If they did, an element of the string would be acted on by a finite moving force, which is the resultant of the two finite tensions at its extremities. In such a case the disturbance would instantly extend itself further along the chain and take up some new form. Supposing we exclude any such case as this, we must have, as long as the motion is finite, both h{ — 0, and ^- = 0, at both the upper and lower ex- dt dx tremity of the disturbance. If then P be a point at which -^^ = 0, and Q a point at which -j- = Q, P and Q may be considered as taken just within the boundary of the v/ave ; P and Q Avill there- fore each travel with the velocity of that boundary. Hence putting V = v , we find for the velocity of either point ,,2^ T m ,(8). It appears therefore that if a solitary wave travel up the chain, the velocity increases as the wave approaches the upper extremity. The upper end of the wave will travel a little quicker than the lower end, because the tension at the upper end exceeds that at the lower; thus the length of the wave will gradually increase. When the wave travels down the chain, the velocity for the same reason decreases. C53. Ex. 1. If the chain be homogeneous, show that the boundaries of a solitary wave will travel up the chain with an acceleration equal to half that of gravity, and down the chain with a retardation of the same numerical amount. Ex.2. Let the law of density be m=A(l + l' -x)~i where I is the length of the chain and A, I' two constants. Also let a weight equal to 2Ag\/l' be fastened to the lower extremity, prove that This integration may be effected by writing 0=:(l + l')i -(l + l'-x)K The equation of moticu then takes the form ^-| = ^ '- .^ , which can be solved in the usual manner. Ex. 3. The chain is said to sound an harmonic note when its motion can bo represented by an expression of the form y = , Let a + be the angle the tangent at P makes with the horizon at the time t. Then coa(a + ^-)=-~^^» ds' sin (a + 0) = dy + dy ds ' sin a = ' da. = - p(f> sin o, dn A COS a = — . ds d-n ^ ^= - jp^ Bin ada-{- A, r) =J pip coa ada + B where A and B are two undetermined functions of t. The equations (2) now become 1-3 — fl = T" ( - iir<^ tan a + - cos o dt» cos' a da, \ -'^ w J d'n 1 (5); (C), (7). df co8*o da d / U . {9 + ~ sin ') ■(«). w ii 'I 1!' il i'f r i If ft I'll I 8 542 MOTION OF A STRTNG. For C 3 sako of brevity let accents denote difierentiationb th i;\;>i ' to t. Expanling Uo differentiations on the right-hand side, these eclua'.iol^! n.ay bo written in the form ■f'sina + 17' coso-^( 0sina+ j- cosa\-U } .,, ,, . , dU cos^a [' f cos a + 1; am + (7^ cos a = j_ ~;r~J _dU co8°( ~ da, w Differentiating the first with regard to o and adding the result to the second, we obtain COS( P" _ d^ d ni con a \ 3 a da'' "" da \ w J ' Differentiating the second and subtracting the first irom the result, we obtain 2/7 d d" /f/cosa> w da da" \ w J ' These equations evidently give Uco3a=u-g ( 2 /Jida-l-Co + i^ j. •(9J, dV -dt^=^ cos p -■■(g+4* + 2^) (10), where C and D are two undetermined functions of t. These are the general equations to determine the small oscillations of a slack chain. The undisturbed form of the curve being givon, p is known as a funcsion of o. We may then use the equation (10) to find (p as " function of a and t. T'i'j tension is then found from the equation (9), and the disjilacements f, 7; of any point of the chain by equations (7). 657. The determination of the whole motion depeads therefore on the solution of a single equation. Supposing the integration to have beeu otTi;rl,f>d, the ex- pression for (p will contain two new arbitrary functions of a and t. ll'hcse wo may represent by \I/(P) andx((?) where i/'and x are arbitrary iunctions of two determinate combinations P and Q of the variables, Tbo arbitrary fimctions A and U are not independent of C and B, and the ro' ., . 'letween them uiay be found by substi- tuting in equations (8). We have thus four arbitrary functions whoso values have to be determined from the conditions of the question. Let a^, Oj, be the values of a which correspond to the two extremities of the string. Then the values of and J^ are given by the etc question when ( = for all values of a from a-a,^ to o-a,; also the initial values of yl and /? are given. Thus the values of \j/(P) and x(Q) arc determined for all values of P and Q between the two limits which correspond to a= a,„ t = and a = Oj, t = 0. The forms of tp and x for values of P and Q exterior to these limits, and the values of A and li when t is not zero, ai'e to be found from the conditions at the extremities of the chain. If the extremities be fixed, we have both ^ and r/ equal to zero for all values of t when a^a^ and a^a^. It may thus happen that tlio ciVitrary functions A, H,\(/ and x are discontinuous. In many cases the circumstances of the problem will enable us to determine at once the form of C. Thus, suppose the string when in equilibrium to be symm'hical about t'. vertical line, say the axis of y, and lot the points of support be wher terms of br for sii ? Hi ay be le second, e obtain (9). (10), Ike general if'tion of a. Tli'j tenBion point of the the solution U'd, tho ox- lese wo may determinate d B are wot il liy substi- t-miucd from orrespond to ^vcn by the nitial values lined for all and a = ai, nits, and the itions at the lid ri equal to pen that tho dotermiuo at brium to be of support bo I SMALL OSCILLATIONS OF A LOOSE CHAIN. 543 xi::ud m tiie same horizontal line. Then if the initial motion bo also symmetrical about the axis of y, tho whole Bul^sequent motion will Le lym metrical. Thus ^ must be a function of o, contaming when expanded only odd powers of o. Sub- Btitnting such a series in equation (10) wo see that C must be zero. 658. There are several cases in which the equation to find the small motions of a chain may be more or less completely integrated. One of the most interesting of these is that in which the chain hangs in equiUbrium in the form of a cycloid. In this case we have, if b be the radius of the generating circle, p=ib cos a. Tho density of the chain at any point is given by vi= jr 3-, so that all the lower part of the chain is of nearly uniform density, but the density increaaes rapidly hi, ler up the chain and is infinite at the cusp. The equation to find the oscillations now takes the simple form d^ _ {I { d" ^-&i^-^*-H '")■ in which all the coefficients are constants. There are two cases of motion to be discussed, (1) when the chain swings up and down, and (2) when it swings from side to side. The results are indicated in tho two following examples. Ex. 1. A heavy chain suspended from two points in the same horizontal line hangs under gravity in the form of a cycloid. Find the symmetrical oscillations of the chain, when the lowest point moves only vp and down. In this case we have C7=0. To find the nature and time of a small oscillation, we put 0= S/2 sin Kt + SiJ' cos Kt, where ':'■ implies summation for all values of k, and B, R' arc lanotions of a only. Substituting, we have with a similar equation to find If. Therefore i?=Lsin2 ^f l + -^ja, where L is an arbitrary constant, the other constant being determined by tlie consideration that tho motion is symmetri'ial about tho axis oi y. F ihe sake of brevity, put X = 2. /(l+— j. Substituting in (7), wo find that the . jrms derived from II become | = SL 2b \'-4. { \ cos \a sin 2a - 2 sin \a cos 2a } sin Kt, ,, = 2 [- 26 2b L^„ — j{\cosXocos2a + 2smXasm2a} -L -r-cosX A^ - 4 A a + J/jsi sin Kt, where If is a constant depending on tho position of the points of support. Tho terms derived from li' n.ust bo added to these, but havo been omitted for tho sako of brevity. They may bo derived from those just written down by writing cos Kt for sin Kt and changing ihe constants L, II into two otlier constants L', 11'. :j (I ^"#^ % !--.f Mi 544 MOTION OF A STRING. Let the length of the chain be 21, then at either end 8inao= tt. At both extremities we must have f =0, i;=0. All these four conditions can he satisfied if tanXag tan2a(, This equation therefore determines the possible times of symmetrical vibration of a heterogeneous chain hanging in the form of a cycloid. 659. If a be not very large, the oscillations are nearly the same as those of a uniform cliain*. In this case since Oq is small but Xao is not necessarily small, the equation to determine X is approximately tan Xag=Xao, Sir The least value of Xa which can be taken is a little less than y Hence X now is great, and therefore k = a/( ji) ^ nearly. The expressions for f and 17 tnke the simple forms f = Si T-j |XoCOsXa-sinXa} sin ( a /h At + e) i7=SZi — {cos Xoj - cos Xa} sin ( » /^^ Xf + e ) Th(! terms depending on cos Kt have been included in these expressions for f and r) by introducing e into the trigonometrical factor. The roots of the equation tan Xoo=Xao may be found by continued approxi- mation. The first is zero, but since X occurs in the denominator of some of the small terms, this value is inadmissible. The others may be expressed by the formula Xao=(2i + l) 5 -^, where 6 is not very large vibration nearly equal to are all short. 2i + l' ^igb' This makes the time of Thus the times of vibration of the chain This result will explain why the marching of troops in time along a suspension bridge may cause oscillations which are so great as to bo dangerous to the bridge. It is clearly possible that the " marching time" may be equal to, or very nearly equal to some '^no of the times of vibrations of the bridge. If this should oocur it follows from Arts. 433 and 503 that the stability of the bridge may be severely strained. " * The rc-der ..ho may wish to see another method of discussing the small oscillations of -i uif (■ 'nsion chain may consult a memoir by Mr Rolirs in the ninth volume of the Caml>]ldge Tramactions. Mr Eohrs considers the chain to be homo- geneous, ByrainrtriciiJ iibout the vertical, and nearly horizontal from the beginning of tlie process. In the necond edition of this treatise the small oscillations wore iilfo treated lO thj same hypotliesis, but in a different manner. That method, Luwover, is not nexrly so siiaple as tlie one here given in which the approximate oscillations for a catenary are deduced from the accurate ones for a cycloid. '"""net SMALL OSCILLATIONS OF A LOOSE CHAIN. 54^ It sliould be noticed that the terms in the exprjssion for f have the square of \ in the denominator, whilo those in the expiission for ij have the first power of X. Since \ is great we might as a first approximation reject the values of f altogether, and regard each element of the chain as simply moving up and down. of the chain 660. Ex. 2. A heavy chain suspended from two points hangs under gravity in the form of a cycloid. If it swings from side to side in its own plane so thut the middle point has only a lateral motion without any perceptible vertical motion, find the times of oscillation. As in the last example, we put 0= 2J? sin Kt + S/J' cos id, where R and R' are functions of a only. Substituting in equation {11) we see that 2C=lhfim Kt + l^k Bin Kt where h and k are arbitrary coubtauts. The equation to find R becomes d^R da* + 4 (6(('\ h 1 + - I as before, we find iJ- - — + Z sin (\o+ Af). Thence taking the term of (f> which contains sin Kt, f h'-hb C0B2a . 26 ,, ,, .„ • « ^T^-^= j^ + Zj-^--^{\cos(\a + ilf)sm2a-2 8:n(Xa + .U)cos2a[, where h' is an arbitrary constant introduced on integration. Substituting in equation (8), we find h'= - ^ ( 6 + i ) • Also, we havo iu the same way Bin 26 26 -I. 5— j-{\cos{\tt + Jf)cos2a + 2sin(\a + ilf)sin2a} -L - aos{\a + M) + H. A — 4 A If we suppose the two supports to be on the same horizontal line, we must havo 1=0 aud 7]=^} when a=±0o. These conditions may bo satisfied if we take M=^, H=0, for then ^ becomes an even and 17 an odd function of a. In this case »j=0 at the lowest point of the chain. We have then two equations to find — , h equating these values, we have 2 tan 2ao - \ tan Xao tan Xoo X' - 4 cos 2a X 2a^ + bin '2ao X tan Xao '■^.n 2aQ + 2 4, 2cos''ao+j^3_^ 661. If oj be small, this equation is very nearly satisfied by Xag-iir where i is any integer. In this case the complete expressions for ^ and 7) take the simple forms ^='SL r^(cos XOfl-cos Xa-Xa sin Xa)siu( . / jy X/ +« 1 j »;=SZ— sinXa sinf . / jr Xt + e j R. I). 35 54G MOTION OF A STRING. 662. Ex. 1. If we clmngo ti.o vfiriablos from a, t to p, q whoro P = t+ r. / — " — da, q= -t+ A / — ^— da, J\/ gcoBa ' '■ JV gcoaa show that tho general equation (10) of small oscillations takes tlio form , . ff cos a , . ., where /*■• = and =fi' is a function of p + q, tho form of tho function depending on tho law of density of tho chain. This transformation may be usi^ful, hecauso it follows from Art. 055 that p is constant for the boundaries of a solitary wave travelling in one direction, and q for a wave travelling in the other direction. Ex. 2. A heavy string lianga in equilibrium under gravity in such a form that its intrinsic > quation is = -sin''(2a + c) where h and c are any constants. P 9 h^ sin'' (2a 4- c) Show that its law of density is given by m=w - '- — -^ . If such a chain be n COS' ci set in motion in any symmetrical manner, prove that its motion is given by Ex. 3. If in addition to gravity, each element of the chain bo acted on by a small normal force whose magnitude is Fg, prove that the equation of motion of the chain is ! f ~-da. J cos a y cos a dt^ da? cos a da ii! If tJio chain is nearly horizontal, so that a is very small, and if F—f^va. (at — ca), prove *.hat the denominator of the corresponding term in tho expression for is g{c''~\)-pa\ Ex. 4. A heavy chain of length 11 is suspended from two points A, B\\\ tho same horizontal line whose distance apart is not very different from 2/. Each particle of tlie chain is slightly disturbed from its position of rest in a direction perpendicular to the vertical plane through AB. Find the small oscillations of tho chain. Ex. 5. A heavy string is suspendod from two fixed points A and B and rests in equilibrium in the form of a catenary wlioso parameter is c. Lot tlic string be initially displaced, the points of support A , B being also moved, so tliat ^ = o-(l + cos 2a) + 0-' sin 2a, where a and a' are two small quantities and tho other It'ttors liavo tho sanio meaning as in Art. OSfi. If the string he placed at rest in this new position, prove? that it will always remain at rest. .. ^ U - Jt- i t » ■ „«.. ' , uimt^ ' SMALL OSCILLATIONS OF A TIGHT STRIN(J. 547 Small Oscillatiom of a tight ttring. 663. An elastic string whose xoeight mag he neglected and whose unstretched length is 1 has its extremities fixed at two points whose distance apart is V, The string beirj disturbed so that each particle is moved along the length of the string, find the equations of motion. Let A bo ono of the fixed pointa, and let AB be tlio string when unstretched and placed in a straight line. Let the extremity B be pulled until it reaches the other fixed point B'. Let PQ bo any c^imont of tho unstretched string, P'Q' the same element at the time t. Let AP=oi, and let the abscissa AP' be x'. Let T and T+ dT be the tensions a* P' and Q'. Lot 31 bo the mass of the whole string, m the mass of a unit of length of unstretched string. Then, as in Art. 637, the equation of motion is dV dT ,,, "'d^^Tx <^)- If E be the modulus of elasticity, we have by Hooke's law d^-^^E ^^^' Eliminating T, we have fPx^ _ E d'x' ,3. d«3-„i dxi '• If we put E=ma*, the integral of this equation is x'=f[at-x) + F(at-{x), where /and F are two arbitrarj' functionH. Tho discussion of this e>^uation may bo found in any treatise on Sound. Tho result is, that a function of the form (p {at - x) represents a wave which travels with a velocity equal to a. In tho case therefore of the string, the motion will be repre- sented by a series of waves travelling both ways along the string with the samo velocity. This velocity is sncli tliat the time of traversing a length I of unstretched string or a length V of stretched string is I a/ — . It should be noticed that tliia time is independent of the nature of the disturbance, and is the same whether the string be originally stretched or not. It should also be noticed, that assuming as usual tlio truth of Hooke's law, the equation (3) and these results are not merely approximations, but are strictly accurate. It is often move convtsnient to select some particular state of the string as a standard of reference and to express the actual position of any particle at the time t by its displacement from its position in this standard. Thus if the unstretched state AB ot the string bo chosen as tho standard of reference, we put x'=x+^, so that ^ is the displacement of the particle whose abscissa in the unstretched state is X. Tho equation of motion now tnkes the form f/_2^ E d'i }tt'^ ^ m dx^ ' apd tlie integral may be obtained as before. 3.-)— 2 i,-1 548 MOTION OF A STRING. 6Ci. An elastic string being stretched as in the last proposition is slightly dis- turbed in any manner, find the equations of motion. Following the same notation as before, lot {x', j/*, z') bo the co-orJinatos of P'. Then, as in Art. 637, the equations of motion are »f4(^£) ■ <». »^^^(^l) <■"■ -f=.4(^S) '". where ds' is the length of the element FQ'. If E bo tlio modulus of elasticity wo have by Hooke's law ds' . T ... dx=l + ^ <*>• Since the disturbance is very small — and -;- are very small and ," > is very da di da nearly equal to unity. Heuco the first equation takes the form d V _ dT "^ dt^~dx' and Hooke's equation takes the form dx~ ^ E' which are the same equations as in the last proposition, so that when the disturb- ance is small the longitudinal motion is independent of the motion transverse to the string. In the second equation we may regard T as constant, its small variations being multiplied by the small quantity -^ . Hence we may put T= Tq wh(!re To=E -— . cts c ds' I' This gives by equation (4) -^ =- . The equation of motion therefore becomes ax t dhf ^TpldY dt'^ ~ vi V d«« ■ The third equation may be treated in the same way. The velocity of a transverse vibration measured in units of length of unstretched lYi string is therefore */ ~y • ^^° ^^^^ °^ traversing a length I of unstretched string or I' of stretched string is */ -^ . This velocity is independent of the nature of the disturban"'^ ■ ut depends on the tightness or tension of the string. If the string be very slightly elastic we may, in this last formula, put l'=l. In this case we obtain the results given in all treatises on Sound. 605. There are two modes of applying the equations of motion to actual cases. We shall first illustrate these by solving a simple example by both methods, and we shall then make some remarks on the results. ••»■■-•»»—•- SMALL OSCILLATIONS OP A TIGHT STRINQ. 549 hthj dii- !8 of P'. (1). (2). (3), sticity vro (4). f' , is vory it le disturb- ansvcrse to tions being I' -I 0— ^ I ecomes unstretched tolled string le nature of nt l'=l. In actual cases. lodS; and wo An elastic string whose unstretched length it I rests on a perfectly smooth table and has its extremities fixed nt two yints A, B' tehose dintance apart it 1', where V is greater than 1. The extremity B' is suddenly released, find the motion. Following the samo notation as in Art. GG3, tho motion is given by the equation ^=--f(at-x)+F{at + x), where { is the displacement of the particle whose abscissa in tho unstretched string is X. The conditions to determine / and F are as follows. 1. When x=0, {=0 for all values of t. 2. When x= I, T=0 and /. ^ =0 for all values of t. dx 3. When«=0, f=rxfrom«=Otox=Z, wheroI'={r+l) J. 4. When t=0, -f=0 from a:=-0 to x=?. at From the first condition it follows that the functions F and / are the samo with opposite signs. From tho second condition we have /' (at + l)= -/' (at - 1), so that the values of the function /' recur w'th opposite signs when tho variable is in- creased by 21. If then we know the values of/' (z) for all values of z from 2=2^ to z=Zq + 21 where Zq has any value, then the form of the function is altogether known. Now tho third condition gives / ( - x) - / (x) = ra; and tho fourth gives f'{-x) =/' (as) from a:=0 to x=l. Hence f'(x)=-~ from x~-l to x~l. It follows that f (a) = - - from 2 = - 1 to ?, /' (z) = ^ from z = l to SI and so on changing sign every time the variable passes the values I, 31, 51, &o. Lot us consider the motion of any point P of the string whose unstretched abscissa is x. Its velocity is given by the formula -=/'(a«-a;)-/'(at + x). Since x '/J 550 MOTION OF A STRING. Since the motion is osoillatovy, we may suppose that all the values of n are real, and it is clear that without loss ot generality we may restrict n to be positive. We do not propose to discuss the circumstances under which these suppositions may be correctly made. For these we must refer the rej,der to Fourier's theorem. We may here regard the assumptions as justified by the result, because we con thus satisfy all the data of the question. The four conditions of the problem enable us to determine the constants. From the first condition we have /3=o+K7r, -B=(-l)*'*'^^ where k is any integer. It easily follows, by expanding, that f may be written in the form f = S (C sin nat + D cos nat) sin not, where and D are to be regarded as functions of n. From the second condition we have cos nl=0, hence nl={2i + 1) 5- where i is any positive integer. The possible harmonic periods (see Arts. 412 and 450) of the string, with proper initial dis- turbances, one end being fixed and the other loose, are therefore included in the form 7-r-. — 7-—. (2i + l)a The initial disturbance is given by the third and fourth conditions. We have 2DBmnx=rx, 2GnBuxnx=0. To find the value of 2) in any term we multiply the first equation by the coefficient of D in that term and integrate throughout the length of the string, i. e. from x-Otox=l. This gives In. , sinnl 0=''/ xam7ixdx=r 71" The other terms all vanish since /em nx Bmn'xdx=0, when n and n'are numerically unequal. Treating the second equation in the some way, we find C=0. Hence the motion is given by , _, 2r sin nl ^ , • i= 2, -, — 5— cosnatsmnas. Writing for i its values 1, 2, S, &c. successively, this equation becomes .irhcn written at length ^ 8rl ( vat . itx 1 Swat . 3irx 1 5irat . Birat , ^= ^ r^ 2r "" 2f - 3^"^^ -2T '"" •2r + 5«*'°' Tr '"" -2T -*"• This is a couvergout series for £ and it may be a sufficient approximation to the motion to take only the first few terms. For example, suppose we reject all beyond the first two terms, and in order to compare the result with that obtained in the first solution let us put at=^l. If wc trace the curve whose ordinate is - j| and abscissa x, we find that it resembles ^=0 for small values of x, then rises with a point of contrary ficxure and becomes nearly horizontal as x approaches I. This agrees very well with the former result. 667. If' wo examine these solutions, we shall see that we have two kinds of conditions to determine the arbitrary functions ; (1) There are the conditions at the two extremities of the string. The peculiarity of these is, that they hold for all values of t. (2) There are the initial conditions of motion. The peculiarity of mm , 669. Three elastic strings AB, BC, CD of different materials are attached to each other at B and C and stretched in a straight line hetiveen two Jixed points A, D. n 552 MOTION OF A STRING. If the particles of the string receive any longitudinal displacements and start from rest, find the subsequent motion. Let A be the origin, AD the direction in which x is measured. Let the nn- stretnhef'i lengths of AB, £C, CD be 2j, 2„ ^. Let B'l, E^, £, be their respective coeffici'ints of elasticity, m^, m^ m, the masses of a unit of length of each string. For the sake of brevity let £i=miai', E^=m^a^*, E^=.m^a^. Let the rest of the notation be the same as before. When the string is stretched in equilibrium between the two fixed points A and D, let To be the tension of the string. In this position the displacements of the elements of each string from their positions when nnstretched may be written At the time t after the eqnilibrinm has been disturbed, let these displacements be respectively f j + f i', fa + f j', f g + f g'. We then have fi' = SXj sin (ni* + JfJ cos n^a^t, fa' = 2La sin {n^(x-lj)+ M^] cos n^a^t, fa = Sij sin {rij {x-l^- 1^) + M^] cos n^a^t, where 2 implies summation for all the harmonics. In order to compare the coeffi- 2t cients of the same harmonic we must suppose niai=n^a^=n^a^= — , where p is the period of the harmonic. To find the constants we have the conditions when x=0, X=li, X = li + l„ X = ly + l^ + l^, f,'=o, li'={,'. I,'=f3'. f3' = 0, / -.t'-.1^'. These give IiaBin3fa=LiSin(niii + Jlfi) / EjnjL J cos itf a = ^iTJi £i cos (nrj^ + M^)\ L3 sin ilfg = ia sin ("a^j + -^i) \ E.^n^L^ cos ifg = £ jMa L, cos {n^^ + Ma) > ' Those give the following equations to find the ill's ; tan M^ _ tan(n i? i + itfi) i&nM^ _ ioxi {n ^\ + M^ . _ tan {nJ,^-\-M .^ iJjtta E{ni Solving these we find £3^3 £3^8 ■BsMg tanrij?! tan »,?, tanngJg j tannj?! tan Hj/j tannj/j iii«i SMALL OSCILLATIONS O' A TIGHT STRING. 553 art from t the nn- espeotive ih string. 3st of the its A and its of the ,tten laoements the coeffi- jre p is the 'j. + 'j + '3 > 0, ■BsWa Substituting for n^, n,, n, in terms of p we have an equation to find the har- monics. The Talnes of p being known, it is clear that the preceding equations determine all the constants except L^. We have therefore one constant undetermined for each harmonic. To find these we must have recourse to the initial conditions. The equations may be written in the forms {/=SP,, cos na«, f,'=SQ„ cos na*, {8'=2;i2„ cos na«, cPP where P„, Q„ and R^ satisfy the equation -^ = - w'P. We have therefore, after integration by parts. Similar theorems apply to Q„ and i2„. We also have the conditions when 06=0, x=li, P=0, P=Q, dx dx' dQ dR X — ^1 T frg X frjji R=0, whatever the suffixes may be, provided they are the same in each equation. If then we put 4>(m, n) =. f^' E.P^P^dx +f^'^^' E,Q^Q^dx+ f^''^^''^^ E^R^R^dx, we have mV (»». ») =nV (m, n), and therefore each is necessarily zero when m and n are difierent. A precisely similar theorem would apply if one or both ends of the string were loose, or if the string were vibrating transversely instead of longitudinally. Suppose now that we have initially ^i'=/i (x), l»'=/a {«). ^3=/^ («). We easily find / jBi/i {«) sm {njpe +Mj)dx+ f ^ E^f^ («) sin {n^ (« - U) + if,} dx •'0 «'?. + (, It " "-^a/s (*) Bin {ng {x-l^- 1,) + M^\ dx = EJj, f ' Bin2 (ni« + M.) dx + E^L^ f ' ' sin'' {n^ (x - 1^) + M^} da + E^L, fj^'^jj'^^' sin" {», (x-l,- 1,) + M.,] dx, these integrations may bo easily effected and give an additional equation to find the L, which corresponds to any value of p. If the strings did not start from rest, we should merely have to add to the expressions for |/, {,', ^3' similar functions of x but with sinnat written for cos naf. 670. Ex. 1. If the three strings vibrato transversely, and a^, CTj, a^ be the velocities of a wave along them measured in units of length of unstretched string, prove that the periods of the notes are given by the equation tan tijl^ tan nj/g tan nj/j _ ^ tan nJi tan n^l^ tan n^l^ . «l »• "s "i "« "3 554 MOTION OF A STRINQ. T 2ir where riia^ = n^a^ = n-jO-^ = — . If tlio initial disturbance ia given show bow to find the subsequent motion. Ex. 2. Two heavy strings A B, BC of different materials are attached together at B and suspended under gravity from a fixed point A. Prove that the periods of the vertical oscillatio-is ore given by the equation tan2'^^tan?-'l'' = |lfL«, the notation being the same as before. If the two strings be initially unstretched, find their lengths at any time. 671. An elastic string is stretched between two fixed points A and B' and is set in vibration, it is required to find the energy. Let the notatio** be the same as that used in Arts. GC3 and 664. First let the vibrations be longitudinal. The equation of motion is dt^ c/x3* Hence we have t -I 1= -j-x + :2[Asin{n(at~x) + a)+BBm{n(at + x) + p}]. Since ^ must vanish when x^O and be equal to I'-l when «=! we find, as in Art. 666, I' -I i=— J- 05+ 20 sin Mac sin (nat + y), where nl=iir and S implies summation for all positive integer values of f. The letters C and y are constants which may be different in every term and which de- pend on the initial disturbance. The kinetic energy of the whole string is rii , /rfA" r^i =zj - max I.I = / 2 mdx {ZCna sin nx cos (nat + y)\*. rl Now / sin nx sin n'xdx=0 when n and n' are numerically unequal since nl and «7 are both integer multiples of ir. Hence, when the square of the series is ex- panded, the integral of the product of any two terms is zero. rl 1 Also / Bio!' nxdx=„ I, hence the kinetic energy becomes = 2 mia' 2C^n' cos^ {na« + 7). To find the potential energy; we notice that the work done in stretching an element from its unstretched length dx to its length dx + d^ ia, by Art. 327, equal tu 1 /(/'\' - £ ( y J dx. Hence the whole work done in stretching the string is =f^^lEdx(^^' = f^lEdx\^^ + ^Cncosnxshxinat + y)\\ . ri 1 Now / cos nx cos n'xdx-0 or ^ I according as n and n' are numerically unequal Jo i SMALL OSCILLATIONS OF A TIGHT STRINQ, 555 or equal to cacb other ; also T coanxdx=0. Hence as before, the integral becomes 2^^ ^ + ^ EiSCV sin" (nat + 7). The first term is the work done in stretching the string from the unstretched length I to the stretclied length V. If we refer the potential energy to the position of the string when stretched in eciuilibrium between the extreme point? A and B' as the standard position, we retain the latter term only. The energy is the sum of the kinetic and potential energies. Bino E^ma', this becomes energy = J jnZa'SCV. This result might have been deduced more simply from Art. 458, where it is shown that the energy of a compound vibration is the sum of the energies of the simple vibrations into which it may be resolved. See also Art. 451. The kinetic energy of any single harmonic is easily Been by integration to be Hence the whole energy is ■mla^I,Cr'n^. We may also notice that, as in Art. 457, the mean kinetic energy is equal to the mean potential energy, the means being taken for any very long period. 672. Next, let the vibrations be transversal. Following the notation of Art. 664, the motion is given, as before, by 3/' =2(7 sin nx sin (nat + 7), where nl=iir and 2 implies summation for all positive integer values of t. The kinetic energy by fhe same reasoning as in Art. C71 is equal to jmla^SCPn^coa'inat + y). To find the potential energy, we notice that the work done in stretching an clement from its unstretched length dx to its stretched length ds' is by Art. 327 1 /|.(-)V^.(g)'i. Substituting for y' and integi'ating wo find that the work is equal to I E ^^—Jt + ^»i/1 ,1' Wo como thus to tho concluaion, tliat, taking for ovciy molecule tlio difference between tho impressed momentum-acceleration an !• oi| will become - to. *"2 0)J Thus, we have de w. dt^ -J sin i/r + "".y sin d cos \^, dO , dtt> . . . , cos i/' + -77 sin sin ip, dt dt d . dtp di dt Sometimes it will be more convenient to measure tho angular co- ordinates $, mentum- 1 momen- he actunl tlie usual liar velocl- of e, ^, yi>. 0)^ be tlio locity. If ;8 existing iiit axis of ^OX,OY, nee, in the mgular co- le, v/e wish T as co-ordi- iis 3ase, we 7, Z with IS indicated im with the from those 0) . If we ated in the for w, , W-, NOTKS. "..')() 0)1 the Impact of Bodies. Arts. 156 and 30.5. The pi-oblem of the impact of two smooth inelastic bodies is considered by PoLssoii in his Traitc de Mecanltjur, Seconde Edition, 1833. The motion of each body just before impact being supposed given, he forms six equations of motion for each body to determine the motion just after impact. These contain thii'teen un- known quantities, viz. the resolved velocities of the centre of gravity of each body along three rectangular axes, the three i-esolved angular velocities of each body about the same axes, and lastly the mutual reaction of the two bodies. Thus the equations are insufficient to determine the motion. A thirteenth equation is then obtained from the principle that the impact terminates at the moment of greatest compres- sion, i. e. at the moment when the normal velocities of the points of con- tact of the two bodies which impinge, are equal. When the bodies are elastic, Poisson divides the impact into two periods. The fii'st begins at the first contact of the bodies and termi- nates at the moment of gi-eatest compression. The second begins at the moment of greatest compression and terminates when the bodies separate. The motion at the end of the first period is found exactly as if the bodies were inelastic. The motion at the end of the secud period is found from the principle that the whole momentum communicated by one body to the other during the second period, bears a constant ratio to that com- municated during the fii-st period of the impact. This ratio depends on the elasticity of the two bodies and can be found only by experiments made on some bodies of the same material in some simple cases of impact. "When the bodies are rough and slide on each other during the impact, Poisson remarks that thei'e will also be a fiictional impulse. This is to be found from the j^rinciple that the magnitude of the friction at each instant must bear a constant ratio to the normal pressure and the direc- tion must be opposite to that of the rehitive motion of the points in contact. He applies this to the case of a sphere, either inelastic or perfectly elastic, impinging on a rough plane, the sphere tui-ning before the impact about a horizontal axis perpendicular to the direction of motion of the centre of gravity. He points out that there are several cases to be considered; (1) when the sliding is the same in direction during the whole of the impact and does not vanish, (2) when the sliding vanishes during the impact and remains zero, (3) when the sliding vanishes and changes sign. This third case, howovc r, contains an un- known quantity and his formulae therefore fail to determine the motion. Poisson points out that the problem would be vei'y complicated if the sphere had an initial rotation about an axis not perpendicular to the vertical plane in which the centre of gravity moves. This case he does not attempt to .solve, but jiasses on to discuss at greater length the im- pact of smooth bodies. M. Coriolis in his Jeu de Billard (1835) considers the impact of two rmt{}h spheres sliding on each other during the whole of the impact. He obtains the restilt given in Ai't. 312, Ev. 3. i)60 NOTES. 1^ mk H ' i 1 ;fl| i i|||j W 'il| i > iHii ^■l^^^l^^l M. Ed. Phillips in tlio fourteenth volume of Liouville' a Journal, IS-ID, considers the problem of the impact of two rouyh inelaatic hodios of any form when the direction of the friction is not necessarily the same throughout the impact, provided the sliding does not vanish during the impact. He divides the period of impact into elementary portions and applies Poisson's rule for the magnitude and direction of the friction to each elementary period. He points out how the solution of the equa- tions may be effected, and in particular he discusses the case in which the two bodies have their principal axes at the point of contact parallel each to each and also each body has its centre of gravity on the common normal at the j)oint of contact. Ho deduces from this the two results, given in Art. 312, Ex. 4 and 5. M. Phillips does not examine in detail the impact of elastic bodies, though he remarks that the period of impact must be divided into two portions which must be considered separately. These however, he con- siders, do not present any further peculiarities. The case in which the sliding vanishes and the friction becomes discontinuous, does not appear to have beeu examined by him. Sir W. R. HamiltorCa Equations. Art 378. The demonstration as given by Sir W. R. Hamilton requires that T should be a homogeneous quadratic function of the accented letters and this is generally the case in dynamics. The exten- sion to the case in which the geometrical eqiiations do not contain the time explicitly is due to Prof. Donkin. Prof. Donkin has made a further extension of this theorem which is sometimes useful. If T be' a function of any other letter, say f, as well as $, y / correct. It seems clear that since the Via Viva cannot be negative, there must be some mode of motion from one given position to another, for which the action is the least possible. When, therefore, the equations supplied by the Calculus of Variations lead to but one possible motion, that motion must make / Tdt a minimum. But v/hen there are several NOTES, 561 ])OH8iblo modes of motion, though none can bo a maximum for tho reaHon given in the text, some of these may be neither maxima nor minima. To dotermino whether tho integral is a maximum or a minimum or neither, we must examine tho terms of the second order in the variation of the integral to ascertain if their sum keeps one sign or not for all variations of the independent variables. This is a very troublesome prooes!*, and we do not propose to discuss it. It will be sufficient to call the reader's attention to some remarks of Jacobi, given in the seven- teenth volume of Crelle'a Journal, 1837, and translated in Mr Tod- hunter's History of the Calculua of Vernations, page 250. Suppose a dynamical system to start fi'om any given position which we shall call A, and to arrive at lome position B. If the time be given, the motion is found by making 8 / Ldt = ; if the energy be given, by making 8 / Tdt = 0. The constants which occur in integrating the differential equations supplied by the Calculus of Variations are to be determined by means of the given limiting values ; but as this involves the solution of equations tliere will in general be several systems of values for the arbitrary constants, so that several possible modes of motion from ^ to ^ may be found which satisfy the same differential equation and the same limiting conditions. Now let one of these modes of motion be chosen, and let the position B approach j4, so as to be always on this chosen mode of motion. Suppose that when B reaches the position G another possible mode of motion from A to B in indefi- nitely near to the chosen motion. Then C determines the boundary up to which or beyond which the integration must not extend if the inte- gral is to be a minimum. The reason seems to be as follows. If U be equal to the integral under consideration, we have along each of the motions from A to B 81/ =0. Hence when B coincides with C, we have both 817=0 and 8{U+8U) = 0. It easily follows that the terms of the second order can be made to vanish by a proper variation. When the limits of integra- tion are more extended than AC, it is not difficult to show that the terms of the second order can be made not merely to vanish, but to change sign. Jacobi illustrates his rule by considering the principle of least action in the elliptic motion of a planet. Let S be the sun, and let the particle start from A towards aphelion to arrive at a point B. The path is known to be an ellipse with aS" for focus. Since we use the principle of least action, the energy of the motion is given : hence the major axis of the ellipse is known, let this be 2a. The other focus H of the ellipse is the intersection of two circles described with centres A and B and radii 2a — SA,2a — SB respectively. The two intersections give two solutions which only coincide when the circles touch, that is when the line AB passes through the focus H. Thus if we draw a chord AC through H to cut the ellipse described by the particle in C, then the terminal posi- tion B must fall between A and C if the integral which occurs in the principle of least action is really to be a minimum for this ellipse. If ^ R. D. 36 562 NOTES. coincide with (7, then the second variation cannot become negative, but it can become zero, so that the variation of the integral is then of the third order, and may therefore be either positive or negative. If B be beyond C the second vaiiation itself can become negative. If the particle start from A towards perihelion, then the extreme point G is determined by drawing a chord AC through the focus S to cat the ellipse in C. For if A and C are the limits we can obtain an infinite number of solutions by the revolution of the ellipse round AC. If then in the last case the second limit B fall beyond G there will be a curve of double curvature between the two given points for which i Tdt is leas than it is for the ellipse. On Sphero- Conies. The following properties of a sphero-conic will be foimd useful in connexion with the theorems of Art. 527. They appear to be new. The curve is represented by the line DED'E'. As in the text, the eye is supposed to be situated in the radius through /., viewing the sphere from a considerable distance. The three principal planes of the cone intersect the sphere in the three quadrants AB, BC, CA, and any one of the three points A, B, C might be called the centre. The arcs AB and AE are represented by a and b. 1. Equation to the conic. Draw the arc PN perpendicular to AD and let PN=y, AN-x, Let NP produced cut the small circle de- scribed on Diy as diameter in F, let NP' be called the eccentric ordinate and be represented by y'. We then have tany ^ . tanJi = constants tan?/' cos a = cos y , tanoj tan a> • ' cos X f NOTES. 565 ative, but en of the If 5 be ) extreme jcus S to obtain an otind AG. I will be a for which useful in lew. The the eye is the sphere : the cone Bmy one of 3 AD and ular ix) AD [ circle de- e eccentric i 2. The projection of the normal PG on the focal radius vector SP, i. e. PL, is constant and equal to half the latus rectum. If 21 be the latus rectum, then tan I = tan* 6 tana ' Also tan GL sinPiV = constant. 3. IS QAF be an arc cutting PG at right angles, QA may be called the semi-conjugate of ^ P. Then tsinPG.tsaiPF=t&n'b. 4. The length PK cut off the focal radius vector by the conjugpte diameter is constant and eqiial to a. This follows from (2) and (3). 8111 & 5. If 1 -e' = -T-j— , e may be called the eccentricity of the sphero- conic. Then tan -4^ = e* tan ^iV. 6. Also S being a focus tsi3i(SP-a)=:etanAN. 7. Polar equation to the conic tan 2 = 1- e A cos PSA. tsinSP ^ cos* 6 8. If p be the i-adius of curvature at P, then tan'w 9. Regarding AP, AQ aa conjugate semi-diametera, sin' AP + ain'AQ = sin* a + sin*6 ainAQ . sin PF= sin a . sin 6 }• 10. If ^ be the perpendicular from the centre A on the tangent atP, tan' a tan' 6 . . . g . x a ^ « = tan* a + tan* o - tan' AP. 11. Also 12. Cor. tan'^ tan» PG - tan' Z = -An sin' P^- cos sin' a - sin' AP = sin* J^- sin' ft tan' b / 1-, ,sin'Pir. tan' PG = cos' b sin* a (coa'AP- cos' a cos* b). 564 NOTES. If sin AM =Bm. AM = -. — , the planes of the arcs BM and SM' are sin a parallel to the circular sections of the cone . Some of the properties of these arcs resemble those of asymptotes when £ is regarded as the centre of the conic. The properties which connect the sphero-conic with the arcs BM and BM' will be found in Dr Salmon's Solid Geometry. Many other properties of sphero-conics will also be found in Mr Frost's Solid Geometry. MiscellaneoiLs Notes. Art. 3. The term moment of inertia with regard to a plane seems to have been first used by M. Binet in the Journal Folytechniqiie, 1813. Arts. 19 and 182. So much has been written on the ellipsoids of inertia and on the kinematics of a solid body that it is diflScult to determine what is due to each of the various authors. The reader will find much information on this point in Prof Cayley's report to the British Association on the Special Problems of Dynamics, 1862. CAHBBIDOE: PniNTED BY C. J. CLAY, M.A. AT THE UNIVERSITY I'REBS id BM' are ies of these e centre of bh the arcs Mr Frost's e seems to 5, 1813. lipsoids of difficult to 'eader will Drt to the o ElESB,