BOUGHT WITH THE INCOME PROM THE SAGE ENDOWMENT FUND THE GIFT OF Henirg W. Sage 1891 /\,SLn'^m!L '^m]i^ 97a4 Cornell University Library arV17356 A treatise on dynamics. 3 1924 031 235 603 olln.anx Cornell University Library The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/cletails/cu31924031235603 A TREATISE ON DYNAMICS MACMILLAN AND CO., Limited LONDON . BOMBAY ■ CALCUTTA MELBOURNE THE MACMIIXAN COMPANY NEW YORK ■ BOHTON - CHICAGO ATLANTA • SAN FRANCISCO THE MACM[LLAN CO. OF CANADA, Ltd. TORONTO A TREATISE ON DYNAMICS WITH EXAMPLES AND EXERCISES BY ANDREW GRAY, LL.D, F.R.S. PROI'ESSOB or NATUBAI. PHILOSOPHY IN TDK UNIVERSITY OF OLASOOW AND JAMES GORDON GRAY, D.Sc. LECTOREB ON PHYSIOS IN THE UNIVERSITY OP OLASOOW MACMILLAN AND CO., LIMITED ST. MARTIN'S STREET, LONDON 19 1 1 PREFACE. This book has been written to provide a discussion of higher dynamics suitable for students of engineering, physics, or astronomy. To a large extent the examples and exercises have been drawn from practical affairs, and have been chosen more for the sake of illustration of physical principles than for their mathematical interest. With hardly an exception, the exercises given at the end of each chapter have been carefully verified, and it is hoped that but few of them are in error. A large number of examples have been worked out in the various chapters, where practical illustration seemed to be required. A considerable space is devoted to gyrostats and gyro- static action, and we have used throughout this chapter, and elsewhere, the method set forth in § 9 of calculating rates of change of directed quantities for a moving system. This method of proceeding occurred to one of us about fifteen years ago [see Gray's Physics, Vol. I.], and we have found it very useful in our teaching, as enabling solutions of difficult problems of rotational motion to be readily built up from first principles. The advantage of the method is most apparent in Chapter IX., which is an expansion of an article on Gyrostats and Gyrostatic Action in Machinery communicated to the Institution of Engineers and Ship- vi PREFACE. builders in Scotland in 1905. Some elementary accounts of gyrostatic action have appeared during the last two or three years, and it is right to say that we are not indebted to these for our method of treatment. We have derived assistance from various works, but, as was to be expected, our obligations to Sir George Greenhill's writings are especially great. Besides making additions of the most practical and valuable kind to the science of dynamics, Sir George Greenhill has long advocated the use of units of the sort employed by men to whom a com- parison with the force of gravity on a given piece of matter is the most ready means of estimating a force, and protested against the common dynamical limitation of the word weight. There can be no doubt that the ordinary use of the word in connection with the buying and selling of commodities "weighed" by a balance can never be got over, and that the connotation of the word in that connection is more frequently that of quantity of matter than that of gravity force. And it is better to take advantage of a common connotation than to do something which may tend to confuse it. Hence we have often used the so-called practical units, but without any sacrifice, for none was required, of the real advantages of the absolute system. We are under obligations also to Jacobi's Vorlesungen iiher Dynamik, Routh's two treatises, Appell's Me'canique Rationelle, Despeyrous' Mdcanique, and Herr Foppl's Tech- nische Mechanik. The aim of the last-named work is similar to our own, and the student will find in it, e.g with regard to compound vibrators of difierent kinds, some interesting developments of matters which have been treated — though generally in a somewhat different manner — in the present volume. PREFACE. vil The proofs of the first two-thirds of the book have been read with great care by our colleague, Dr. R. A. Houstoun, and Dr. Pinkerton, of Edinburgh, has kindly read the chapter on Gyrostats. We offer our thanks to these gentle- men, and also to the officials and workmen of the Glasgow University Press for their care and attention throughout the printing of the book. ANDREW GRAY. JAMES G. GRAY. CONTENTS. CHAPTER I. KINEMATICS OF A MOVING POINT. Section 1. Speed and Velocity. 2. Varjring Motion. 3. Illustrations of Varying Speed. Curve of Speed. 4. Distance traversed at Varying Speed. 5. Uniformly Varying Speed. 6. Graphical Representation of Directed Quantities. Composition and Resolution of Velocities. Relative Velocity. 7. Curve of Velocities — Hodograph. 8. Accelera- tion. 9. Angular Velocity. Directed Quantities referred to Moving Axes. Rate of Growth of Directed Quantity. 10. Examples of Acceleration. 11. Curvilinear Motion. Radial and Transverse Com- ponents. 12. Polar Coordinates in Three Dimensional Space. 13. Radial and Transverse Components of Acceleration. 14. Uniplanar Motion of a Point. Revolving Axes. Components of Velocity and Acceleration. 15. Three-Dimensional Motion. Revolving Axes. 16. Curvilinear Motion in Space of Three Dimensions. Normal ' and Tangential Accelerations. 17. Curvature of a Path in Space of Three Dimensions. 18. Examples. Motion of Point along a Moving Guide. 19. Tangential Acceleration vi^ith Space as Independent Variable. 20. Equation of Hodograph. Case of Falling Body. 21. Motion of Projectile in Uniform Field of Force. 22. Properties of Path. 23. Horizontal Range. Range with Path through Fixed Point. 24. Envelope of all Coplanar Paths with given Speed of Projection. 25. Examples of Parabolic Motion. 26. Motion under Acceleration varying inversely as Square of Distance from Fixed Point. 27. First Integral of Equations of Motion. Equation of Hodograph. 28. Equa- tion of Path. 29. Speed at Different Points of Path. 30. Resolution of Velocity into Two Parts of Constant Amount. Path Deduced from Hodograph. 31. Polar Coordinates : Differential Equation of Path of Particle under Central Acceleration. 32. Simple Harmonic Motion. 33. S.H.M. Velocity and Acceleration. Integral Equation. 34. S.H.M. Amplitude, Period and Phase. 35. A Uniform Circular Motion the Resultant of Two S.H.M.s. 36. Composition of Equal and Opposite Circular Motions gives S. H. M. 37. Composition of Two S.H.M.s in Same Line. 38. Composition of any Number of S.H.M.s in Parallel Lines. Tide-Predicter. 39. Composition of S.H.M.s in One Line but of Different Periods. 40. Composition of S.H.M.s in Perpen- dicular Lines and of Different Periods. 41. Composition of S.H,Ms. X CONTENTS. in Different Lines but of Equal Period. 42. S.H.M.s in Perpendicular Lines but not of the Same Period. 43. Resisted S.H.M. defined by Equiangular Spiral. 44. Differential Equations of Exponential Motion and S.H.M. Exponential Motion repesented Graphically. [Exercises] pp. 1-82 CHAPTER II. DYNAMICAL PRINCIPLES. Section 45. The Laws of Motion. Momentum and Rate of Change of Momentum (R.C.M.). 46. Effect of Change of Mass on R.C.M. 47. R.C.M. in Curvilinear Motion. Force. 48. Kinetic Energy. R.C.M. as Space-Rate of Variation of K.E. 49. Potential Energy. Equation of Motion for Particle under Central Force derived from Energy. 50. Discussion of First Law of Motion. 51. Second Law of Motion. Example. 52. Meaning of Equations of Motion. 53. Non-Rotational Motion of Extended Body. Systems of Varying Mass. 54. Units of Force. Dimensions. 55. Gravities of Bodies Proportional to their Inertias or Masses. 56. Third Law of Motion. Discussion. 57. Action and Reaction across a Surface of Contact or across an Interface. 58. Action and Reaction between Bodies at a Distance apart. 59. Centre of Mass (or Centroid) of a Body or System. 60. Properties of Centroid. External and Internal Forces. 61. Newton's Law of Equal and Opposite Activities. 62. Theory of Work. Units of Work. 63. Active and Inactive Forces. 64. Constant and Varying Constraints. 65. General Variational Equation of Work. Theory of Energy. 66. Forces as Derivatives of Potential Energy. 67. Work spent in overcoming Friction. Dissipative Forces. 68. Meaning of Solution of a Dynamical Problem. 69. Angular Momentum. BU)ta- tional Motion. 70. Components of Angular Momentum (A.M.).- 71. Angular Momenta about Parallel Axes. 72. Rate of Change of A.M. 73. Rate of Change of A.M. when Effective Inertia different in Different Directions. 74. Rate of Change of A.M. when Body Gains or Loses Mass. 75. Rates of Change of A.M. equal to Moments of Forces. Independence of Motions of Translation and Rotation. 76. Rigid Body, Rolling Motion of. 77. Examples on A.M. of Bodies of Varying Mass. Energy-Changes. 78. Kinetic Energies of Motions of Translation and Rotation. 79. Couples. Equivalence of Couples. 80. Effective Inertia different in Different Directions. Case of a Ship. 81. Why a Ship carries a Weather Helm. [Exercises] pp. 83-143 CHAPTER III. DYNAMICS OF A PARTICLE. Section 82. Rectilinear Motion of a- Particle in Resisting Medium. 83. Limiting Speed in Resisting Medium. 84. Resistance varying as to"" Power of Speed. Discussion. 85. Motion under Resistance varying as V. 86. Resistance varying as 1?. 87. Resistance varying as w'. 88. Examples on Resisted Motion in a Vertical Line. 89. Examples CONTENTS. xi of Rectilinear Motion under Gravity. 90. S.H.M. Motion of a Simple Pendulum. 91 . Motion of a Simple Pendulum in a Finite Arc. Elliptic Integrals. 92. Motion of a Particle in a Vertical Circle. Elliptic Functions. 93. Revolution of Particle in Vertical Circle. 94. Examples of Motion in a Vertical Circle. 95. Equilibrium of a Plummet under Gravity. Apparent and Real Gravity. 96. Plummet in Railway Carriage. Apparent Gravity. 97. Cyoloidal Motion. Cycloidal Pendulum. 98. Tautoohronous Motion. 99. Brachisto- chrones. 100. Braohistoohrone m Conservative Field of Force. Euler's Theorem. 101. Variational Method for Brachistochrone under Gravity. 102. Variational Method for Brachistochrone in Conservative Field of Coplanar Forces. 103. Brachistochrone in any Field of Force. 104. Conical Pendulum. 105. Double Pendulum. 106. Double Pendulum. 107. Double Pendulum. Discussion of Cases. 108. Physical Ana- logues of Double Pendulum. 109. Two Connected Spiral Springs in Same Vertical. 110. Three or more Connected Springs with Attached Masses. [Exercises] - pp. 144-205 CHAPTEE IV. RESISTED MOTION OF A PARTICLE IN A UNIFORM FIELD OF FORCE. Section 111. Uniform Field. Resistance fa). 112. Resistance fe. Trajec- tory. 113. Construction of Trajectory. 114. Resisted Motion. Tan- gential and Normal Resolution. 115. Resistance = fa;". 116. Particular Cases. Hodograph. Intrinsic Equation of Path. 117. Intrinsic Equation of Path for Resistance kv'. 118. Flat Trajectory when Resistance =iy^. 119. Experimental Laws of Resistance to Shot. [Exercises] pp. 206-219 CHAPTER V. FREE MOTION OP A PARTICLE UNDER A FORCE DIRECTED TO A FIXED POINT. Section 120. Path lies in a Plane. Differential Equation. 121. Effect of Force transverse to Radius vector. 122. Speed from Infinity. Exhaustion of Potential Energy. 123. Concavity or Convexity of Orbit towards Centre of Force. 124. Force varying directly as Distance. 125. Examples of Force in Different Cases. 126. Solu- • tion of Differential Equation in Various " Cases. Energy Relations. 127. Discrimination of Orbit. 128. Period of Particle in Orbit. 129. Determination of Orbit from Distance and Velocity, etc. 130. Newton's Revolving Orbit. 131. Examples. 132. Acceleration in terms of Tangential and Radial Forces. 133. Hodograph of Par- ticle describing Orbit. 134. Velocity resoluble into Two Com- ponents of Constant Amounts. 135. Deduction of Law of Force from Form of Orbit and Uniform Description of Area. 136. Kepler's Laws. Verification. 137. Newton's Dynamical Deductions from Kepler's xii CONTENTS. Laws. 138. Effect of Mass of Planet. 139. Correction of Kepler's Third Law by Theory of Gravitation. 140. Weighing the Planets. 141. Newton's Theory of Universal Gravitation. 142. Does New- tonian Gravitation extend to the Fixed Stars? 143. Experimental Illustration of Gravitational Attraction. 144. Elements of an Orbit. 145. Time in an Elliptic Orbit. 146. Time of DesoriV)ing any Arc. Lambert's and Euler's Theorems. 147. Disturbed Orbits. (1) Tan- gential Impulse. 148. Disturbed Orbit. (2) Normal Impulse. 149. Disturbed Orbit. (3) Change of Intensity of Central Force. 150. Examples of Disturbed Orbits. 151. Orbit slightly disturbed from Circular Form. 152. Theory of Apsides. 153. Centre attracting according to Inverse Cube of Distance. 154. Force varying as Inverse «"■ Power of Distance. 155. Different Centres for Same Orbit. New- ton's Theorem. 156. Hamilton's Theorem. 157. Second Statement of Hamilton's Theorem. 158. Orbit a, Conic touching Two Straight Lines drawn from C.F. 159. Particle acted on by Forces from Several Centres. Bonnet's Theorem. 160. Theorem of Curtis. 161. Examples of Multiple Centres of Force. 162. Eartlj-Moon System disturbed by Action of Sun. 163. Stability of Earth-Moon System. Hill's Theorem. [Exercises] - - pp. 220-306 CHAPTER VI. MOTION OF A RIGID BODY. Section 164. Angular Momentum (A.M.) of a Rigid Body. 165. Moments of Inertia about Parallel Axes. 166. Calculation of Moments of Inertia. Momental Ellipsoid. 167. Principal Axes of Momental Ellipsoid. 168. Meaning of a Product of Inertia. 169. Reactions of an Unsymnietrical Rotating Body on its Bearings. Free Axis of Rotation. 170. A.M. about any Axis. Equations of Rotational Motion. 171. Moments of Inertia in Different Cases. 172. M.I. of a Lamina. 173. M.I. of Triangular Plate. 174. M.I. about Axes at any Point parallel to Principal Axes at Centroid. 175. Examples of M.I. 176. Condition that an Ellipsoid may be a Momental Ellip.soid. 177. Foci of Inertia. 178. Ellipsoid of Gyration. 179. Equimomental Cone. Theorem of Binet. [Exercises] - pp. 307-337 CHAPTER Vn. APPLICATIONS OF DYNAMICAL PRINCIPLES. Section 180. Practical Applications. 181. Acceleration in the Direction of Motion. 182. Motion of a Railway Train. Time lost in Stop- pages. 183. Work done on Trains. Tractive Force. 184. Effect of Nature of Road Surfaces on Vehicular Traffic. 185. Efficiency of Brakes. 186. Time of Train from Station to Station. 187. Dynamics of Self -Propelled Vehicle on Straight Road. 188. Dynamics of a Vehicle on a Curve. 189. Bicycle on Curve ^ncl s becomes Jr( h ^^^ intei-vals, we have by the last example. But clearly r=2a, and v = at-i^=rt^, so that Now the distance is 10560 feet, and so taking foot-second units, we get «= 10560/(30 +150 + 15) = 54-15. and therefore a = "902, »•= 1-805. Thus, indicating the units in the manner explained in § 1, we ^^■^^ v = 54-15/7s, a = -902//s2, j-=l-805//«2. Ex. 4. A bullet from a service rifle has a speed at the muzzle of 2500//S. If it is shot vertically upwards, find, on the supposition of zero resistance, how far the bullet will ascend, its speed when at half that height from the. point of projection, and the interval of time after which it will just have returned to that point.' Ex. 5. It is recorded of Hiawatha that " He could shoot ten arrows upwards, Shoot them with such strength and quickness That the tenth had left the bowstring Ere the first to earth had fallen." Supposing that he shot off' an arrow every four seconds, find the initial speed of the first arrow, and the height to which it ascended. 6. Graphical Representation of Directed Quantities. Com- position and Resolution of 'Velocities. Relative Velocity. Any directed quantity can be represented by a straight line drawn in the specilied direction, and made as many units in length as there are units in the numerical measure of the quantity. Hence we may represent a velocity in this manner by a straight line so drawn from any convenient point 0. Let, then, OA (Fig. 3) represent a displacement in direction and magnitude, and OB, 00 be adjacent sides of a parallelo- gram of which OA is the diagonal passing through 0. If we consider a point displaced along the line OA, it is easy to see that the step OA is not merely equivalent in result to the two steps OB, BA, or the two OB, 00, or the 12 A TREATISE ON DYNAMICS. [CH. I. two OG, GA, taken in succession; but that when it is taken any one of these pairs may be regarded as effected simultaneously. For let the point move along the line OG, and at the same time let the paper with this line upon it be carried in the direction OB in such a manner that the Ffg. 3. motion of the point is along the line OA in space. The displacements effected are OG and GA, where GA is the displacement relative to the point moving along OG. Similarly, BA is the displacement relative to the point moving along OB. [See Relative Motion in our Elementary Dynamics^ Similarly, if OA represent a velocity, that is the dis- placement per unit of time in that direction, OB and BA, or OB and OG, or OG and GA, represent three pairs of velocities made simultaneous or coexistent in the same way, and each pair is equivalent to the single velocity OA. Let a. be the angle AOB, and let the other angle OAB, ^ say, be also given. To find OB and OA, we have sin /3 r,r, r, A ^ , sin a OB=GA = 0A . ; . ^, . OC=BA = OA sin(a+|8) sin(o(. + ;S) since OB A = tt — (a + ;8). If the second condition assigned be not the angle /3, but the length of OB, we have OG=BA = -JOA^+OB' -20 A. OB cos a. ■^x. . o OB . with sin p = 7jp sin a.. §6] COMPOSITION AND RESOLUTION OF VELOCITIES. 13 If for OA we write v, for OJB, or OA, v^ , and for OG, or BA, v^, we put these equations in the more compact form sin B sin a. ^ sin(a. + ^) ^ sin(a + ;8)' ^ ' ^'2 = ^/^'^ + ^''^ — 2i;t;jCosa, (2) sin /3 = — sin a (3) and ds/ dz 9dr fdx Most frequently the resolution is rectangular, that is OB, OG are taken at right angles. Then the equations are simply v^ — v cos a, v^ = vwa.cL (4) It is clear that in the case of rectangular resolution the problem is definite if one angle a, or one component OB, is given, but that in the more general case, either both angles a. and /3, or one angle and one component, must be given. In both cases the plane of resolution must also be specified, as the resolution may be made in any plane containing OA. Further, a given velocity in any direction OA may be resolved into components along three directions not all in one plane. It is usual to consider only three directions which are mutually at right angles, OX, Y, OZ, say. components v^, Vy, v^ of v are given by Vx = vcos OL, Vy = vcos^, '?;3 = 'WC0Sy, (5) where a., (8, y are the angles which the direction of v makes with OX, OY, 0^ respectively. These give v^ = vl+v^^ + vl, (6) since the condition cos^a + cos^/3 + cos^y =1 (7) Fig. 4. The 14 A TREATISE ON DYNAMICS. [CH. I. is fulfilled for the three angles a, /3, y, which three rectangular axes make with any direction in space. The cosines of oc, ^8, y are called the direction cosines of OA. The three components v^, Vy, v^ are, when taken together, equivalent to the single velocity of speed v. Now it is a proposition in geometry of space, easily proved, that the projection of any line OA (of length v say) upon any other line OB, inclined at an angle 6 to OA, is equal to the sum of the projections upon OB of the components {v^, Vy, v^) of OA parallel to the axes OX, OY, OZ. We have thus, if a', /3', y' be the angles which OB makes with these axes, i;cos Q=VxC080i^ +VyCos ^' +VzCOBy', (8) or, by the values of v^, Vy, v^ found above, cos d = cos a. cos oc' + cos ^ cos /3' + cos y cos y , (9) a value of cos0 which will be frequently of service in what follows. The notation I, rti, n for cos a, cos ^, cos y is commonly used for brevity. Then (9) is written cos e = ll' + mm' + nn' (9') Any number of coexisting velocities can be compounded so as to give an equivalent system of velocities. Thus the two velocities of speeds ■yj, ■^2, discussed above, are equivalent to the single velocity of speed v given by equation v = ^t;l+vl-2v^v^cose, (10) where 6 is the angle of inclination of the direction of V2 to that of v.^. The single velocity found is the resultant of the two given velocities. Its direction is in the plane of the given velocities and inclined to the direction of -Wj at an angle oc given by ,^ •"1 + ^2^08 ^11) cosa= V Now take the more general case of a number n of coexisting velocities of speeds v^, v^, v^, ...v^. Take three axes at right angles to one another, OX, OY, OZ, and let these axes make with the direction of v^ angles 04, jSj, yj, with the direction of v^ angles ocj, (82, yg, and so on. Then §§6,7] HODOGRAPH. 15 taking the sum of the components of all along OX, of all along OY, and of all along OZ, we get Vy=v^cos^^ + v^coa^^+...+v^co8^A (12) Vi = ■i;iC0S yi + ^;2C0S y^+...+ v„cos yj Thus we get three coexisting velocities of speeds v^, Vy, v^ which are equivalent to the given system. These have a resultant of speed v given by v==Jvl+vl + vl. (13) The direction of v makes with the axes angles the cosines of which are v^ Vy v,_ v' v' V These results apply also to other directed quantities which are capable of being resolved and compounded in the same manner. If the speeds of two particles A, B with reference to chosen axes be v, v', in specified directions, then the velocity of A relatively to B is obtained by compounding with the velocity of 4, a velocity equal and opposite to that of B. Thus, for example, the component velocities of the motion of A with respect to B have the speeds, v^ — v'x, Vy — v'y, v^ — v'^. Similarly the components of the velocity of B relative to A have the speeds v'^ — Vx, v'y — v^, v'^ — v^. 7. Curve of Velocities — Hodograph. Fig. 2 is a diagram of speeds, that is the ordinates of the curve v =f(t) represent successive numerical values of the velocities, which may be in different directions ; and the area, taken as specified, gives the distance traversed between any limits of time proper to the motion. But now let us suppose that a point is moving in a curve, and let tangents be drawn to the curve at successive positions P, Q, R, ... of the moving point. The directions of motion at these positions are shown by the arrow heads on the tangents. Now, from a point 0, let lines Op, Oq, Or, ... be drawn parallel to the tangents at P, Q, R, ... and in the directions of the arrows, 16 A TREATISE ON DYNAMICS. [CH. I. and let each line be made as many units in length'as there are units in the velocity which it represents. By taking the points P, Q, R,... sufficiently close together we can determine, as nearly as may be desired, a curve jjgr . . . which might be called with more propriety than the former a cwrve of velocities. It is usually called the hodograph of the motion of the point. J Fig. 5. The hodograph gives to the eye a picture of the mode of variation of the velocity both in direction and magnitude, and its chief use is in the determination of the rate of this variation. In the general case of the motion of a point along any curve in space of three dimensions it is itself a three-dimensional curve; but for many of the motions considered in elementary dynamics its form is simple. For example, in undisturbed planetary motion, the hodo- graph of the planet is a circle in a plane parallel to the orbit, with an eccentric point within the circle as the origin from which the lines Op, Oq, Or, ..., representing the velocities at different points in the orbit, are drawn. Other cases will be discussed later when the subject of acceleration has been dealt with. 8. Acceleration. At a given instant the velocity of a point is represented by Op, and at a subsequent instant it is represented by Oq (Fig. 5). By the principle of composition of velocities, explained above, the velocity represented by Oq is equivalent to the two coexisting velocities represented by Op, pq. It is reasonable to take as the change of velocity (not change of speed) that has occurred in the interval, t say, between the two instants, the velocity pq, which, coexisting with the initial velocity, gives the final. §§7,8] EXAMPLES. 17 We now define the average acceleration during the interval t as the ratio, (velocity pq)lT. It will be observed that the average acceleration for the interval t, as thus defined, has direction (that of the chord pq of the hodograph) as well as magnitude. It is also clear that as t is made smaller and smaller without limit, the direction of the chord pq approaches more and more, without limit of closeness, to that of the tangent to the hodograph at p. Now the limiting value of the ratio pq/r, as q is brought more and more nearly into coincidence with p, is defined to be the acceleration at the position P of the point in its path, that is the acceleration when the velocity is represented by Op. Hence the acceleration of the moving point at the instant when it is at P is in the direction of the tangent to the hodograph at the corresponding point p. Now suppose a second point to move in the hodograph, so that as the first point moves in the path the second is, at every instant, at the extremity of the line representing the velocity for that instant. The rate at which the second point moves along the hodograph is then, at each instant, both in magnitude and direction the acceleration of the particle ; or, as it is sometimes, though not quite properly, put, the velocity in the hodograph is the acceleration in the path. We insert here some examples of rectilinear motion and acceleration. Ex. 1. A crank OA turns with uniform angular speed m about 0, and a connecting rod AB pivoted at A communicates rectilinear motion in the fixed direction to a cross-head B : to find the speed of B at any instant. Let X denote the distance OB, a the length of the crank OA, I the length AB ol the connecting rod, 6 the angle A OB, and 4> the angle ABO. We have x=: a cos 6 + 1 cos , asm6=lsm, and therefore x= —a sin 6.6—1 sin (^ . <^, which, since a cos 6 . 6=1 cos cfi . (j>, becomes x= —x tan <^ . 6= —u>x tan <^. This simple expression for the speed of the cross-head suggests the following construction, for which the student should draw his own figure. Produce BA to meet in C a perpendicular to BO drawn from 0. Then, if OA be taken to represent the speed of A, that is G.D. B 18 A TREATISE ON DYNAMICS. [CH. I. ad lit right angles' to OA, OC will represent the speed of B, which is at right angles to OC, from B towards 0, or towards B, accord- ing to the position and direction of motion of the crank. Or if a distance OD=OG be laid off from along the crank, the speed of B is numerically equal to the speed of the point D. According as B is moving towards or in the opposite direction, OD may be laid oif from towards A or in the opposite direction. At the " dead-points," where i^ is zero, OD is of -zero length. When the crank is at right angles to the line of motion of B, cb has a maximum numerical value, for then xta.n = a. The angle we obtain x= —CO {x tan <^ + r(H- tan^ ) (f>} = - lo^x {(1 + tan2.^) ^1 - tan2<^]- , /, Zcos I, since sin Ojsin =lja. But (Z cos <^ - a cos d)ll={xl cos (j>-xa cos 0)1x1 =(J? — a^)jxl, since x = aQ,os 9 + lcoa , and Psm^=a^airv'd. Thus we have — x = (o'\x y — se&(pj- This expression for x suggests the following construction. Along BA produced, and backward along AB, lay off ^1^ and A F each equal to OA. Then BE=^l + a, BF=l-a. Along BO lay oflT BG = l and BF' = BF= I - a. Join EQ, and through F' draw F'E parallel to OE. Then we have BHIBE=BF'jBG, that is BH/(l + a) = (l-a)/l. Hence Bff=(P-a^)/l. It may be noticed that H is the point at which the connecting rod is met by a perpendicular let fall from to the rod when OA is at right angles to BO. Now draw HI, IK, KL, perpendiculars to BE, BI, BK, respectively, so that K is on BA, and /, X are on BO. Then BI= BR sec cj), BK=BIsec <^, and BL=BKaec, so that BL =B II see' = {l^- a'') sec^ 4, /I. Hence LO=x-(P-a^)se(fict>ll and -x = b?.LO. Thus, on the scale on which AO{ = a) represents the acceleration of the point A in the circular motion, LO represents the acceleration of the cross-head B. §§ 8, 9] MOVING AXES. 19 Ex. 3. A steamer sails at a speed of 30 feet per second in the direction from North to South, and a wind blows from West to East with a speed of 12 miles per hour. If the particles of smoke are supposed to come to rest relatively to the air just above the funnel mouth, find the speed of a particle of smoke relative to the steamer. The relative speed is 12 miles per hour or 17'6 feet per second from West to East, and 30 feet per second from South to North, that is a speed of 34-78 feet per second in a direction to the North of East, inclined to the Easterly direction at the angle tan-i(30/17'6). This is the direction of the stream of smoke with reference to the steamer. Ex. .4. To find the motion of the cross-head B in Ex. 1 relative to the crank-pin A. The cross-head has speed o)^tan<^ in the direction from B towards 0. The motion of the crank- pin at right angles to OA gives a component wa sin 6 in the direction from B towards and a component laa cos 6 in the direction of the perpendicular drawn through A from the line OB. Thus we have for the motion of B relative to A the components (ax tan (jy — wa sin 6 from B towards and oia cos 6 along the perpendicular from A on the line of stroke. The resultant is ia{x^t3,n^(j>-2ax;ta:Ji ^sin d + a^)' and makes the angle tan~^{ a cos d/{x tan /l}—o}'^acos6 in the direction from B towards 0, and o^sin 6 in the direction from OB towards A along the perpendicular let fall from A on OB. The resultant relative acceleration is therefore oi^ix - '^^ see's ^y + ^2 _ 2„(^ _ EjZ^ sec3(^)cos 6}*, and is inclined to the line of stroke at the angle tan-i {a sin 6l[x - (l^ - cf,2)sec5 (j>/l - a cos 0] } on the side of that line towards A. 9. Angular Velocity. Directed Quantities referred to Moving Axes. Bate of Growth of Directed Quantity. If a straight line be turning about one extremity, the angular speed of the line is measured by the speed of, the point at unit 20 A TREATISE ON DYNAMICS. [CH. I. distance from the fixed end. The specification of the plane and direction of turning is required to complete the idea of angular velocity. The following simple theorem, which is easily proved, will be of great service in what follows. If any directed quantity (of amount L say), characteristic of the motion of a body, be associated with a line or axis 01 (Fig. 6), which is changing in direction, it causes a rate of pro- duction of amount wL of the same quantity for a line or axis, Om, at right angles to 01, towards which 01 is turning with angular speed m. If M be the amount of the same quantity already associated with this latter line or axis, the total rate of growth of the quantity in that direction is M+wL. To prove this, let 01 have turned towards Om, in the short interval of time dt, through an angle d6, from the position at right angles to Om. The extremity of the vector L has moved a distance Ldd parallel to Oth. There is now a component of L along Oon of amount L sin dO, or simply Ldd, since dO is small. This is produced in time dt, and /Ld0 therefore the rate of produc- tion is Ldd/dt, or Lw, if w denote the angular speed dd/dt, or 6, as we shall usually write it. To Loo falls to be added M, the rate of growth of M, the amount of the quantity already associated with Om,. The student may easily convince himself that the rate of variation i of Z contributes nothing to the rate of growth of the quantity .along Om,. It will be observed by the student that L is the amount of the directed quantity for the position of 01 at the instant under consideration, and M that for Om in its position at the instant, since Otn may also be a line of reference for the motion of the body, and be itself in motion. In short, L and M are the amounts of the directed §9] MOVING AXES. 21 quantity for fixed axes coinciding with the instantaneous positions of 01 and Orn, and the rate M+Lw is associated with the fixed axis with which Om at the instant coincides. It is to be remembered that L + Ldt, M+Mdt are components of the directed quantity for the axes 01, Om, in the new positions which they occupy after the lapse of the short interval of time dt, from the instant considered. M+Loo is the rate of growth of the quantity for the direction which Orn, occupies at that instant. The same process may be applied to other vectors of the same kind turning during dt towards O'yn, and the total rate of growth of the quantity obtained for Om, by addition. The theorem just stated is, as we shall see in the dis- cussion of the motion of tops and gyrostats, sufficient to deal with complicated cases of motion ;* but it may be regarded as a particular case of the following theorem re- garding a system of three moving axes Ox, Oy, Oz (Fig. 7). \<«3 Fig. 7. Let these make at time t angles a, /3, y with a fixed axis Ok, and be in motion about the fixed point, 0, so that oc, p, y are changing at the time-rates 6c, (8, y. Then the component K of the quantity associated with Ok is given by .g'=Z/Cosa-t-ifcos/3 + A'"cosy, (1) * See a paper by A. Gray in the Transactions of the Institution of Engineers and Shipbuilders in Scotland for 1905. 22 A TREATISE ON DYNAMICS. [CH. I. SO that ^ = i cos a + ilf cos /3 + iV" cos y — La. sin a. — M^ sin/3 — Jfy sin y (2) If now we suppose that Oy coincides at the instant con- sidered with Ok, then /3 = 0, a = y = 7r/2, and we have k=M-La.-Ny. .(3) But, clearly, if the system of axes be turning, as shown in the diagram, with the angular speeds indicated, namely ftjj about Ox, Wg about Oy, and oj^ about Oz, Ox is turning towards Ok with angular speed w^, and so w, = — d ; simi- larly Oz is turning away from Ok with angular speed ftjj^, and so ft'i = yi. Thus we have K=M+Lu)^-Nw^. .(4) The motion of the system of axes causes growth of the component associated with the fixed axis Ok, with which Oy coincides at the instant, at rate Lw^ — Nw^, of which the part L(j}^ arises from the rate of approach of Ox to Ok, and — iVftjj from the rate of recession of Oz from Ok. [See further with regard to moving axes, § 15.] 10. Examples of Acceleration. As examples of this theorem we may take the following. The line 01, of length r say, is turning about with angular speed in a given plane, say that of the paper. The rate at which the point I is moving parallel to Om, that is the rate of growth of the distance of I from any straight line drawn parallel to 01 in the plane of the paper, and to the right of 01 in the diagram, is rQ. This result has already been found in § 9, if we take L there as representing the length r of 01. Fig, 7 a. §§ 9, 10] ACCELERATION. 23 ■ Again, let us consider a point moving in a given curve, and endeavour to determine its acceleration. If P be the position of the point at the instant, and the speed be v, the component of acceleration along the curve at P is v. But since the path is curved at 'P this is not the only- component of acceleration. Let PC (Fig. 8) of length R be the radius of curvature of the path at P, that is let C be the centre of the circle passing through P and two points Pg, Pj infinitely close to P, and situated one on the left, the other on the right of P, as in the diagram. By the theorem stated above the rate of growth of velocity in the direction from P towards G is V(j), where ^ is the angular speed with which the tangent at P is turning round towards the direc- tion PG. But clearly (f, = v/R, and therefore velocity of the point in the direction PC is growing up at rate v^/R. The two components, i) along the curve and v^/R towards the centre of curvature, thus found are the totjal components for these directions, and therefore when compounded give the re- sultant acceleration. This result holds whether the curve lies in a plane or in space of three dimensions. The result may also be derived from the hodograph. The resultant acceleration, a. say, in the path is represented by the velocity of the imaginary particle" (Fig. 5, § 7) in the hodograph. Resolving ot. into two components, one per- pendicular, the other parallel to Op, we see that the amount of the former is the speed perpendicular to Op, of the imaginary particle. This is clearly V(p, that is v^jR, since (p, the angular speed of the radius to the imaginary particle, is that with which the tangent at P i^ turning. The other component, that parallel to Op, is obviously i). Ex. 1. A particle moves in a plane curve witli varying speed v, and a second particle moves so as always to be at the centre of curvature of the path for the position of the first.,. Find the accelerations of the second particle parallel to the ta,ngen,t. ai\cl normal of the path. The speed of the second particle along' the normal is dp/dt or v dpjds. 24 A TREATISE ON DYNAMICS. [CH. 1. where p is the radius of curvature, and the speed at right angles to this is zero. But the tangent to the path turns round in time dt through the angle dO=ds/p, and.the tangent to the evolute, the radius of curvature of the path, turns in the same time through the same angle, while the point of contact moves a distance dp. The curvature of the evolute is therefore (ds/dp)/p. The acceleration of the second particle towards the centre of curvature of the evolute, that is parallel to the tangent to the path, is therefore (dpldtf(ds/dp)/p or (v^dp/ds)/p, and is clearly in the direction opposed to the motion of the first particle. Again, the acceleration along the evolute is v'dv'jdp, where v' = vdplds. But dv'/dp = {dv'/ds)ds/dp, so that we obtain v'dv'ldp = vd(v dplds)lds. Ex. 2. A particle moves in a cycloid in such a manner that its resultant acceleration is always perpendicular to the base. Prove that the acceleration is inversely proportional to the fourth power of the radius of curvature at each point. Eefer to Fig. 41. Resolving the acceleration normally and tan- gentially, and calling the component perpendicular to the base ol, we get 9 7 «.= Bind) +11^- cos , and so the equation for a. becomes 4ai>2 64a' , , ,„ a.= --^ = --^(const.)2 by the relations « sin <^ = const, and p = 4asin<^. Ex. 3. A particle moves in a catenary of which the intrinsic equation is s = c tan <^ [Gibson's Calculus, § 142] ; the direction of its acceleration at any point makes equal angles with the tangent and normal to the path at that point. If the speed at the vertex be u, find the speed and the acceleration at any other point. The normal and tangential accelerations must have equal values, that is ii = Plp. But by the equation of the path, we have p = dsld = {c'^ + s'^)/c, §§ 10, U] ACCELERATION. 25 i so that we get -= s- Integrating, we obtain logs = tan-i- + C=<^+C, where C is a constant. Now let be the angle between the radius-vector and the tangent at the position P of the particle at time t, p the perpendicular from the focus on the tangent, and p the radius of curvature. Eesolving normally and tangentially, we get — =(as-a',)sin^, v-j-=(a + ar)cos = dr/ds. Also, in the parabola, if a. be the distance of the focus from the vertex, p^=a.r. Hence the equations just written become Thus we get „l dp dv p dr dr „\ dp dv _ dv o\ dp i=v' — f-+v^r' 2,a,=v-j — v^ — f-. p dr dr dr p dr But the relation p^ = a:r gives {dpjdr)jp = ll2r, and therefore 1)2 fjy Id,,. dv y2 1 d fv^A ^''•'=''dr-2r=rd-r[v)' which prove the proposition. 11. Curvilinear Motion. Radial and Transverse Components. As another example we find the components of acceleration in two directions chosen as follows : a line OP drawn from 26 A TREATISE ON DYNAMICS. [CH. I. any origin to the position P of the moving point, and a line FT drawn at right angles to OP in the plane determined by OP and the direction of motion at P. Denoting the length of OP by r, and resolving v into two components, one along V OP and the other along the ^ transverse PT, at right angles to OP, as in the diagram (Fig. 9), we get for the former r and for the latter rw, if P, * as we may suppose it to do, '°' ■ accompanies the point in its motion and w is the angular velocity with which OP then turns. We have thus v = (r^- + w^r^f (1) If the motion is in one plane, w may be expressed as the rate of growth 6 of the angle d which OP makes at the instant under consideration with some fixed line, Ox say, in the plane of motion. We have then v = {r^ + rW)^ (2) The reciprocal 1/r, that is the shoHness of r, is fre- quently employed in such expressions as these for the speed in the path. Putting u = 1/r, we have v^ = \(u^ + u^e^) (3) If, as is the case in a class of motions which we shall have to consider later, 6 = hu^, where A. is a constant, uW = h^u^ ^^ = {pJ = h^u^Q\ (4) so that ^2 = ;,2|(^gy + ^2|_ (5) a form which will be of great service in the discussion of the class of motions referred to. 1 11, 12] ACCELERATION. 27 If p be the length of the perpendicular let fall from the origin on the tangent to the path at P, then it is clear that h = vp. This value of h substituted in (5) gives 1 _/^Y .=( p^ \ddJ +u\. •<6) cos 0d(t> a geometrical relation which is also of much service in the discussion of orbital motion. 12. Polar Coordinates in Three Dimensional Space. Lastly, in the case of three dimensional motion, the speed of the moving point can be expressed as follows. Take coordinates (Fig. 10) according to the following specification : (a) the distance, OP = r, from the origin to the moving point, (b) the angle 6 which OP makes with a fixed plane through 0, and (c) the angle ^ which the projec- tion OM of OP on this latter plane makes with a fixed line OX in the same plane. For exarnple, the position of a point on the earth's surface is fiied by the distance r of the point from the centre 0, the geocentric latitude 6, that is the inclination of OP to the plane of the equator, and the longitude (p, that is the angle which the meridian plane of the point — a plane through the poles and the point — makes with the meridian plane of some specified place, e.g. a certain point in the Greenwich Observatory. If, then, r, 6, be the coordinates, as thus defined, of the first extremity of the element of path ds described in the element of time dt, r+dr, G + dd, + df }^. '"^^"-l/rcosedip Fig. 10. 28 A TREATISE ON DYNAMICS. [CH. I. Hence v={r^ + r^e^ + r^cos^e.^^^ (1) The change from these coordinates to coordinates x, y, z with reference to rectangular axes OX, OY, OZ, is to be made by the relations, which are obvious from the diagram, •(2) X = OM cos (j) = r cos Q cos y — OM Bin (p=rcoa 6 sin

x, (1) which we shall have occasion to refer to in the solution of various problems of the motion of a point. If now we take lines equal in length to u, v, as the x, y coordinates of another point, with reference to the same moving axes, the motion of this point will give the §§ 13, 14] ACCELERATION. 31 component acceleration of the first point in the directions of the instantaneous positions of OX and OY. Calling the values of these accelerations U, V, we obtain by the same process as before V = u — uiv = x — 2wy — u)'^x — (ay\ V=v + u!U = y + 2oDX — ui^y + wx) The terms —Ay, dix vanish if the angular speed w is constant. If U, V be each zero, and to be constant, the equations become x — 2u>y — u>^x = 0, y + 2u>d; — w^y = 0, which are the equations of motion, referred to uniformly revolving axes, of a particle moving in the plane of the axes under no forces.^ The particle therefore, as we shall see later, moves in a fixed straight line ; and hence, if we turn the whole diagram of axes and moving particle round in its own plane, with angular speed —w, the axes will be brought to rest, and the particle will describe a spiral of Archi- medes. We infer that the component accelerations x, y of the particle referred to the fixed axes are given by the equations x = 2ccy + a,^x, y=-2wx + a>^y (3) Ex. 1. If the spiral of Archimedes, r=a9, where 6 = /u^ + v' These give at once the two relations im — uv = ui(u^+ v^), uii +vi= — i^gslu^ + 1^. Differentiating now the values of u, v suggested above, and substi- tuting in the equations, we get v'U — uv = to{V—ii^tY, uu+vi= —iig{V—iigt). But {V- iigif = u^ + v\ and V-ij^t=^Jv? + v\ so that the equations are satisfied. Ex. 3. The motion of a particle P in a plane is referred to axes Ox, Oy, inclined at an angle .J3, of the component associated with a fixed axis i3oinciding with the instantaneous position of Ox. Hence the total rate of growth is F-w^G + w^ff. Similarly we obtain for fixed axes coinciding with the instantaneous positions of Oy, Oz, rates of growth of the components associated with them, G-WiH+oisF, H-w^F+w-^G. If we call these three rates L, M, N, we have the equations L=F-^F+fi] In precisely the same way as before (§ 14) we get for the time-rates of variation of the components of the quantity {L, M, N) for fixed axes coinciding with Ox, Oy, Oz, V=M-3L \ (2) W=N-iM] in which the values of L, M, N are to be inserted from (1), o,D, C! 34 A TREATISE ON DYNAMICS. [CH. I. The resultant of i, M, N or U, F, W is the rate of displacement of the outer extremity of the vector representing F, G, H or L, M, N, as the case may be. If for example F, O, H=x, i), z, the speeds of a particle with reference to the moving axes, and u, v, w be the components with reference to fixed axes coinciding with the moving axes at the instant, u=i) — (i>^y + w^, 1 v=^ -ojjz +(>>^, \ (3) w=z — 0)2^ +(Oiy.J Example. To find the components of velocity and acceleration along the radius-vector, the tangent to the meridian, and the tangent to the parallel of latitude, for the instantaneous position of the point of Fig. 10. These directions are to be regarded as fixed axes, with which the moving OP and the tangents carried with it coincide at the instant considered. The speed-components are i; rO, r cos d., if 6 is taken as shown in Fig. 10, and r, rO, r sin 6. if the angle POZ, the colatitude, is taken as 6. In the latter case rO is in the opposite direction to r6 in the former. Now take the acceleration-component along the instantaneous position of OP. We have first the part r of this component. Next we observe that as P makes in dt the step rdd in the plane POZ that transverse step turns through the angle dd away from the fixed outward direction OP, and therefore, by § 9, furnishes — rd dO, increase of speed along OP, that is a rate — rd' of growth of speed along OP is caused by the turning. Again, as P moves in dt through rcoaO-dcf) along the parallel, the direction of the parallel turns towards the first perpendicular from P on OZ, and a rate r cos O.^"^ of growth of speed along that perpendicular is the result. This has components rcoa^d.^ along the fixed direction PO, and r sin 6 cos 6 ■ 4>^ in the direction of the transverse rdO. The total acceleration along the fixed direction OP is therefore r - r^ - ?■ cos^ e.^=r + {r^- ■i^)jr. The acceleration along -the meridian at P is found in the same way to be , ^{rO) + r6 + r cos d . ^^ sin 6, that 13 \~ (rm -t- r sin e cos Q . i?. r dt If the colatitude is taken as Q, this must be changed to - -^ (r^^) - J- sin Qco&Q . 4>\ r dt and the direction is opposed to the former direction. §§ 15, 16] NORMAL AND TANGENTIAL ACCELERATIONS. 35 Along the parallel of latitude we have (1) the part d{r cos 6 .4>)/dt of the total component of acceleration, (2) the part due to the moving OP along which the speed is r, (3) the part due to the moving transverse to OP in the plane POZ, along which the speed is rd. Now, with respect to (2), we observe that r along OP resolves into r sin 6 along 02 and rcos 9 along OM, of which the latter only changes direction with respect to the fixed position of the parallel at P. The result is acceleration rcos6.4> along that fixed direction. Again, for (3), rd resolves into j-^cos^ along OZ and -rdsind along OM. The latter gives acceleration - r6 sin 6 along the parallel, the former gives nothing. The total acceleration along the parallel is thei'efore d(r cos . )/dt+r coa 64> - rd sin 9, that is -J-^l{r^cos^9.4>), r cos 9 dt or, if the colatitude is taken for 9, -^^(r^mn^9.//R^ + v^) '; in the osculating plane at the point, that is the plane containing two consecutive tangents at the position of the point, or one tangent there and the radius of curvature. But the components of velocity along any system of fixed axes Ox, Oy, Uz are x, y, z (which are such that v^ = x^-\-y'^-\-z''), and the accelerations along these axes are therefore x, y, z. Hence the resultant acceleration is (x^ + y^ + z^) . Thus we have the equation or ■^+s^ = x^+y^ + z^ ■0) 36 A TREATISE ON DYNAMICS. [CH. I. in complete fluxional notation. In the notation of the diiferential calculus it is -w+w ^\W) ^\dp) ^\w) ^^^ From the discussion above it will be seen that the acceleration s along the curve coincides with the resultant acceleration only when R = oo. It is not unusual for students to assume that s is the resultant acceleration, chiefly because too frequently the only cases considered in elementary dynamics are those in which there is no acceleration except in the line of motion. The resultant acceleration is the square root of the right-hand side of (2), while dx..dy..dz.. ' Equation (1) can be transformed as follows. If, as we have already supposed, ds be an infinitesimal step along the curve, taken by the moving point in the correspondingly small interval of time dt, we have x^sdxjds. Hence ^^s^_+,_=_E^+,_ (4) and similar results are derived in the same way from y, z. These substituted in (1) give the transformation in question [(3), §17]. 17. Ciirvature of a Path in Space of Three Dimensions. We infer from (4) of last section that Rd^x/ds^ is the ^-direction-cosine of the radius of curvature. For x is the acceleration parallel to the axis of .r ; this must be equal to the sum of the two rectangular components s^/R, s, each multiplied by the cosine of the angle which its direction makes with the axis of x. Hence the direction-cosines of the radius of curvature are /^ ^ ^s Again dxjds, dy/ds, dzjds are the direction-cosines of the tangent to the curve at the element ds, and s)"-ar-ey- o !lr d^x dy dh/ dz d^z ds ds^ ds ds^ ds ds^ dxd^jdyd^jdz^d^^^ il6, 17] CURVATURE OB" PATH IN SPACE. 37 The expression on the left of the last equation should not be confused with the similar in appearance but quite difiFerent expression on the right of (3), § 16. From (1) and (4), § 16, we see that if both sides of (4) be squared, and also both sides of the two similar equations for d'^yjdt'^, dhjdfi, we get by addition ..,..«.=^{(S)V©)V(S)>(gj. (3, Hence, substituting in (1), we obtain ±__fd^\\{dW,fdhy . - li^'Kdsy ^\dsy "'"UsV' ^ ' » purely geometrical equation for the curvature of the path at the element ds. This result may also be obtained as follows without the introduction of the idea of motion. If the direction- cosines of the tangent, at a point P in the curve, be dxjds, ..., those of the tangent at a point distant ds from the former are dx/ds + d^x:/ds^ .ds, .... Hence if we lay off from an origin two lines OA, OB, ■ each of unit length, in the directions of the two tangents, the coordinates of A and B will be simply the direction-cosines in each case. This gives for the distance AB the expression {m-m-mr^- which, since OA = OB = 'l, is also the measure of the small angle AOB. But since OA, OB are parallel to the tangents at the extremities of ds, this angle has also the measure dsjR. Thus we obtain ^={(£)^©)"-(S)f={(S)"-(S)'-(S)}*-*) the same equation as before, with I, m, n put for dxjds, dyjds, dzjds. If I, m, n thus denote the direction-cosines of OA, and l+dl, m+dm, n+dn those of OB, we see from what precedes that the angle AOB has the measure {{diy + {dmf+(dn)^)^ or {{dlldsy+(,dmldsy+(,dnldsy}^ds, and the cosines of AB have the values Rdl/ds, Rdm/ds, Rdnjds. Further, as the point moves along the path its direction of motion 38 A TREATISE ON DYNAMICS. [CH. I. turns in the plane of the consecutive tangents at the point with angular speed = {P + m^ + n^)i (6) The same discussion shows, that (i^ + m^ + n'yi is the angular speed of change of any direction to which direction-cosines I, m, n, which vary with the time, apply, whether successive directions of motion of a moving point or successive diiections of an " axis " with which some directed quantity is associated. If we call this angular speed to, and take two lines OA, OB as above, then the direction-cosines oi AB are 'lju>, m/ o , D' i fy^ ■\m "A N \ Fig. 13. This is the equation of a parabola of latus rectum, 4o(. in length, LFM in Fig. 13. The new origin is the vertex of the curve, and a _ vertical through is called the axis, from the fact that for every value of y there are two values of x, viz. ± 'ijanj, which are numerically equal and op- posite in sign, so that the curve lies symmetrically on the two sides of the axis. The co- ordinates of F are i/ = a, a; = 0, and so LF=FM= 2a.. The distance of F from the point, of projection is {a^-f-(6 — a)^}^ that is V^/2g, and the coordinates of F from that point as origin are V^ sin 26/2g, V^ cos 20/2^. 22. Properties of Path.. If a horizontal line DD' be drawn in the plane of the curve at a height a above the origin, it can be shown that the distance of any point of the curve from the line is equal to the distance of the point from F, which is therefore called the focus of the curve. For the former distance is a.+y, and the latter is {(■y-(x.)2 + a;^}^=a.+y, since x^ = 4:a.i/. The line drawn as specified is called the directrix of the curve. The distance II of the point of projection from the directrix, being equal to the distance of that point from the focus, is V^/2g, that is we have V'=2gff. The speed of projection is therefore equal to that which a particle would acquiie in falling to Pg from rest at the directrix, and therefore II is called the " head " for the speed V. The distance H of P,, from the directrix, it will be seen, is independent of the angle of elevation d„. H may of course be the distance of any point P on the curve at which the speed is d, and we have theni;2 = 2^Z/; ^ is then the "head" for the speed V. If P be any point on the curve, the tangent of the inclination of PF to the vertical is xl(%/-a.) = AaJicl{x'^ — 4,aJ), by the value of y. The tangent of the inclination of the tangent to the curve at P to the vertical is dx/dy = 2a.lx. Now 2{'2,a./x)/{l-{'2a.lxy} = Aa.x/(x^- 4a.^), and therefore the tangent at P bisects the angle between PF and the vertical, a well-known property of the parabola. Again, take any two points on the curve, say P,,, which may be taken as the point of projection, and P, which may be regarded as the point which the moving particle has reached after the lapse of an interval of time t. Draw a tangent to the curve at P„ and let it meet the vertical through P in <2 ; then Q is the point which the 44 A TREATISE ON DyNAMIOS. [CH. I. particle would have reached in time t if there had been no acceleration, that is P„§2= VH\ But it is clear that QP=igfi, so that P.§2 = 2(P/^).§P. Hence P„Q'^4P,F.QP, (1) since, as has been proved above, Pf,F= V^/2ff. At any point P on the curve (Fig. 13) draw a tangent PT, a normal PJV, and a line /'M' at right angles to the axis, and let these lines intersect the axis in T, N, W. The distance M'N is called the subnormal, and has a constant length. For its length is x dxldy, and this by the equation of the curve is 2a.. This is a characteristic geometrical property of the parabola. Again, by the diagram, if ck be a step along the curve from P, and - dy, since y is measured downward, be its projection on the axis of the curve, we have sin ^= -c?y/fl?s. But also sin 0=a;/PiV. Hence PN= -x^= -x%= --io.-. (2) dy y X But X is constant, and therefore PiV may be taken as representing the speed s of the particle in the path, with direction turned through a right angle. The subnormal represents on the same scale, and with the same change of direction, the constant horizontal speed. It may be noticed that two paths coplanar with that here dis- cussed, having the same point /q, and speed V, of projection, but inclinations ^0+"-; ^o~«-> where a. is very small, will intersect the path for inclination dg, at the point where the direction of projection is perpendicular to that of motion. This follows from the fact, which the student may easily verify, that if two particles be projected from the same point at the same instant in any two directions, the line joining the particles remains perpendicular to the line bisecting the angle between the two directions. Thus the two particles in the case supposed must cross together the trajectory for do, as stated above. 23. Horizontal Range. Range with Path through Fixed Point. Denoting the range on the horizontal plane through the point' of projection by R, and the latus rectum, or parameter, of the curve, that is 'iV cos^ dolff, hyp, and putting y=0 in (2) of §21, we have x=0 and x=R, where iJ= — sin2(9„=ptan^o (1) Thus we may write the equation (2), § 21, of the curve in the forms ,y=a;tan^o-^ = (*-|)tan^„, tan^o=|+;^ (2) The last form is important. Here y is any ordinate of the curve and yjx, yl{R-x) are the tangents of the angles <^, ' which the §§22,23] PROJECTILE IN UNIFORM FIELD. 45 ordinate subtends at P„, the beginning of the range R, and P'o, the end of the range. Thus we can write the last equation as tan ^Q = tan^ + tan^', (3) which gives the " elevation " required to enable the projectile just to clear a wall of height y at distance x from the firing point, and reach an object at a distance R-x beyond. The head H of the speed V and the range R may be employed to give the equation of the path in a form which is sometimes useful. We have seen that if the point of coordinates x, y — the top of the wall in the last article — lie on the path, tan 0o=2/{l/,i;+l/(fl-a')}. Substituting in (2), § 21, and reducing, we obtain (/j_^)2_4|^(^_^) + gyj2=o, (4) the form referred to. Here it is to be understood that if x, y are fixed, R varies with H according to this relation ; but if R and H are assigned, that is if the speed V of projection and the angle of elevation Q^ are given, then x, y are the coordinates of any point which lies on the path. It is easy to show that for a given value of H and a given point X, y on the path (the top of the wall, say, just referred to) there are two values of R — x at which the shot will reach the horizontal through the point of projection. That this must be true is evident from the fact that for the given initial speed there are in general two elevations which will enable the shot to reach a given point, if the point can be reached at all with the given speed of projection. Writing Z) for ^-.r in (4), we get, after reduction, {x'^+y'')D^-'2,xy{2H-y)D+xY=0, (5) from which we obtain D=J^,{^H-y±sl{%H-yy-{x^+f)\, (6) There is thus a value below which H cannot be taken if the point X, y is to be reached at all by the shot, that is J3'=i{N/^+p+y} (7)^ For any value of H above this there are two values of Z>, given by (6), at which the shot will strike the horizontal plane, and there are two corresponding values of R. A point inside the smaller of these distances is in no danger of being struck by the shot. The foregoing problem may be discussed geometrically as follows. It has been noticed that the head II( = V^/2g), that is the distance of Pq from the directrix, is independent of 6g, so that all paths from P^, with speed Fof projection, are parabolas which have the same directrix. Their foci 46 A TREATISE ON DYNAMICS. [CH. Fig. 14. are all at the same distance V^j^g from Pg, and therefore lie on a circle described from P^ as centre with V^j^g as radius. This construction, made in Fig. 14, enables the path to be found which passes through a given point P, the problem just considered analytically. With the dis- tance of P from the directrix as radius and P as centre, describe a circle. If P can be reached at all by the projectile, with the given speed of projection, this circle will intersect the former circle in two points, or at least touch it. Either of the points of intersection F-^, F^ is the focus of a path by which the projectile will pass through P, and thus in general there are, as we have seen above, two possible paths for a given V. As P is carried further off towards the right in the diagram, the two points F-^, F^ come closer and closer together until at last they coincide; if P be carried further off there is no path on which it lies. 24. Envelope of all Coplanar Paths with given Speed of Projection. Consider the position of P (Fig. 15) for which the circles just touch in a point R. Draw the line PM perpendicular to the directrix and produce it to N, so that PN= OP, where is the point of projection. Thus P lies on the parabola of focus and directrix NL. This para- bola is the envelope of all the paths which correspond to different inclinations, all in one plane, of the direction of projection with velocity V from 0. To prove this take as origin. The equation of a path is 2/ = -tan 00-1-^,^^.2 or, as we can write this equation. .(1) 2F2 ia,n^Of, tan gx ?o=-(] + 2ty V- gx^ §§23,24] PROJECTILE IN UNIFORM FIELD. 47 where we take cc, y as the coordinates oi; the point P^. The roots of this equation in tan Q^ are equal when ^F_2 g_ 2 y 2g 2V^^' ■(2) Thus the point x, y lies on the parabola of which (2) is the equation. N L The value of dy/dx is zero when x = 0, and therefore lies on the axis. When x = 0, y= V^/2g, hence the point of coordinates 0, V^/2g is the vertex, and lies on the directrix of the family of parabolas which are the paths for as point of projection and V as speed of projection. If this point be made the origin, and y be measured down- ward, the equation of the path becomes ^/3y, .(3) where ^ = V^/2g. Thus the distance of the focus from the vertex is V^/2g, that is is the focus. The directrix of this parabola is at a distance V^/2g above the former directrix, which agrees with the construction stated above. That this parabola is the envelope of all the paths may be seen at once by diiferentiating the right-hand side of (1) with respect to 6^, equating the result to zero, and then eliminating 0,, between the equation so obtained and (1). [See Gibson's Calculus, § 145.] The result is (2), and the proposition is proved. 48 A TREATISE ON DYNAMICS. [CH. I. 25. Examples on Parabolic Motion. Ex. 1. The speeds at the extremities of a focal chord of the path of a projectile are v, v', and u is the horizontal speed : prove that By the hodograph, if 6 be the inclination of the tangent at one extremity of the focal chord to the horizontal, v = u/cos 6, v' = wlain 6. Thus 111 1 -9 + -;o = -5 (sin2 ^ + cos^ ^) = -J . Ex. 2. At three points P, Q, li on the path of a projectile the inclinations of the tangents to the horizontal aie oc + jS, ol, a. — j3, and the speeds are v, xf, v". If the time from P to § be i and from § to .S be 11, prove that i/'t=vtf, and that ljv + l/v" = 2coa fS/v'. If 11 be the horizontal speed, we have, by the hodograph (Fig. 12), D = M/cos(a + /8), v' = u/coscL, v" = ulcos(a.- ^), while the vertical com- ponents are Mtan(a.+/3), tttancx., m tan (a. — ;8). Hence g'< = M{tan(a.-|-y8)-tana.}, gri!'=?t{tan(X — tan(a. — ^)}, and we obtain „"2<2= iL { tan (o(. + /3) - tan a.}2{ 1 + tan2(a. - /3)} and rY2 = ^{tano(.-tan(a.-;8)}2{l+tan2(a.-|-/3)}. But {tan (a. + /3)- tan a.}2=(l + tan2a)2tan2/3/(l- tana, tan )8)2, and similarly { tan oc-tan (a.- ;8)}2=(l-|-tan2a.)nan2^/(l +tan a. tan /3)2, while 1 + tan2(o(. =F ;8) = (1 + tan^o. + tan^/S + tan^a. tan2/3)/(l ± tan o. tan fif. Hence, substituting, we get identically v"H^=v^t'^. Moreover, we have i+l=J{cos(a + ^) + cos(-6'')cosa. i^+r^ sin(e+^)cosa.-2cosecos^sino(.' Ex. 8. Prove that the oblique range up a plane inclined at the angle a. to the horizontal is 2 F2sin(0-') ~ff Fcos^+ F'cos^" which of course may be either positive or negative according to the values of F, F', 6, ff. Ex. 10. If t, f be the times of flight for the two directions 6, ff of projection by which a particle shot off from P with initial speed F can reach a given point P, and t, t' be the times in which the particle in the two cases reaches the highest point of its path ; show that {tT + fT')/(ta.ne+ta,ne') depends only on the distance P^P and on the inclination of the line Pf,P to the horizontal. [Math. Trip. 1876. The statement has been altered.] By the previous example, ^_2Fs in(l9-«.) ^^2F sin(g'-(x) g cos OL ' g cos a. V V Also T=— sinft t' = — sin^', 9 9 so that we obtain 2F2 «t + «'t'=-5 {sin(^-a.)sin0+sin(^-tx.)sin0'}. ^^^COSOL ^ ' But if R be the horizontal range for the elevation 6, we have, § 23, fl=?'cosa.tan ^/(tan^-tano(.)=rcos2(x.sin ^/sin(^-a) or sin(^ — a.)=»'cos^a.sin ^/^ = (7rcos2asin ^/F^sin2ft XT ■ /n -, ■ n 'f cos^oLsin^^ Hence sm(^ - a.)sm 8= ^ ^-^ — 52 A TREATISE ON DYNAMICS. [CH. Similarly, we obtain sin(^ — a.)sin ^ = ^cos^a. sin'' [/2-"° ""sin 26^ Hence we have tT + i!t' = - cos a. (tan 6 + tan &). 9 Thus \/2/»JtT+t'T'/(ta,n d + ta,n ff) is the time in which a body falls freely from rest under gravity g through a distance rcosa.. Ex. 11. To find the maximum range of an unresisted projectile on a slope inclined at an angle a. to the horizontal. Let X, y be the coordinates of a point P on the path for elevation Q, and R be the horizontal range, then it is easy to prove (see § 23) that ■y tan0=-^ + X R — x If then P be the point on which the shot meets the slope after projection, and r be the range on the slope, we have , „ , , rsino. tan o = tan a. + -n , U — r cos a. and therefore r= R ^'^"^-tan'^. cos a. tan d Now R={V^AnW)lg, and therefore r = (sin '2,6-% cos^ tan a.), g cos a. " from which we find for a maximum value of r, dr V^ 55=^^^(^<=°^^^+2 sin 2etanoc)=a Thus tan2^= -l/tana.= -cotoc or In this result regard must be had to the sign of a, which is to be taken positive if the shot is fired up the slope, and negative if the shot is fired down. If a.=0, we get 2^=|7r or 6 = iir, which is obvious from the value of R. Ex. 12. A gun is placed on a plane hillside : prove that the area commanded on the slope by the gun is bounded by an ellipse of which the position of the gun is a focus, the major-axis is along the line of greatest slope, the eccentricity is the sine of the angle of greatest slope, and the semi-latus rectum is of length equal to twicfe the greatest distance to which the gun can send a shot vertically upwards. §25] EXAMPLES OF PARABOLIC MOTION. 53 Let /3 be the angle of greatest slope, then the angle of slope of a line on the hillside, inclined to the line of greatest slope at an angle (j>, is sin-^sin /3 cos <^). Now, by last example, 1T9. - (sin 26-2 cos^ tan a). g cos a. But, since for the maximum range 26=oi+^, we have sin 2^= cos OL, cos' 6 =^{1- sin cl). Hence, after reduction, the last equation becomes yi yi 1 r= s-(l-sina-) = — -r- — ; — g -;, ffcos'a. g 1 -+ sin /3 cos (f) which is the polar equation of an ellipse of eccentricity sin/? and semi-latus rectum V^Jg, as stated above. The major axis is plainly along the line of greatest slope (<^ = 0), and the range in the hori- zontal direction (<^ = j7r) is V^/g, the maximum horizontal range B, as it evidently ought to be. The range along the line of greatest slope is thus V^/(Jl +sm I3)g, upwards, and V^j(l - sin fi)g, downwards. The total length of the major axis is thus 2 F^/(l - ain^^)g. Ex. 13. The curve r=f{ff) is in a vertical plane, and particles slide from the curve to the origin along radii-vectores, and then pursue free paths under gravity with the velocities so acquired as velociti.es of projection : to find the locus of the foci of the paths. We suppose the angle measured from the horizontal through the origin. The speed of projection is then given for a particle by the equation V^=2gr sin 9=2g sin 6 f{ff). The coordinates of the focus of the path are, taken positive, x= V^&in^Oj^g, y= V^cos2dl2g. The radius-vector to the focus has length V^/2g, and the angle which it makes with the axis of x is 26. Calling this (f>, we have for the equation of the locus, p= V^/2j = sin 6/(6), that is ^=^^tf{l)- Ex. 14. Find the locus in Ex. 13 if the curve is a circle and the radii-vectores be chords drawn to the lowest point. The equation of the circle is r = 2a sin 6 if a be the radius. Hence the locus of the foci of the path is that is a cardioid. p = 2a sin^ ^=a(l-cos (f)), Ex. 15. A tennis ball is projected from a point A with speed V at elevation 6, and rebounds from a vertical wall B at horizontal distance a, then from a iloor at distance h below B. If the normal component of speed of rebound from the wall be e times that of 54 A TREATISE ON DYNAMICS. [OH. I. approach, find the time of leacliing the floor and the time of return to the vertical through A. [The ball is supposed to have no rotation.] Before the ball impinges on B its horizontal speed is Fcos d. Hence the time from projection to the first impact is a/ Fcos 0. After the rebound the horizontal speed is - e Fcos 6, and as this is not affected by the impact with the floor the time of returning to the vertical through A is ajeV cos d, and hence the whole time from projection is a(l + e)/e Fcos 9. The time, «2 say, from the instant of projection to that of reaching the floor is (since the impact on- B does not affect the vertical speed) given by Fsin 9 .t2 — \gt\= —h, that is by h=^^^-^^+-'JVHm'9 + 2gh, for the negative root given by the solution of the quadratic refers to the case of the ball arriving with upward vertical speed Fsin 6 at A from the floor, and gives the previous instant at which the ball was at the floor. Ex. 16. It is required to find the condition that the ball in the last example may return to A, on the supposition that the vertical component of the speed of rebound is e' times that of the speed of approach. The vertical speed of the ball after leaving the floor is e'( Fsin 9 - gQ =eWVHm^e+ 2.gh, and the horizontal speed is e Fcos ft The time, t^ say, required to Tiie; from the floor to the height h is therefore given by e^t3\/V^sm^9 + 2gh -igtl = h, that is <3 = -{ e'^J W^^9 + 2gh± \/e'2( V^sin" 9+2gh) - 2gh}. This interval of time added to t.^ must just make up the whole time of flight. Hence the required condition is ^ eVmsl ^ ^sin^+(l+e')^^^'^sin2e + 2^/i±^/e'2(F2sin2ft+27A)-2^^. 26. Motion under Acceleration varying inversely as Square of Distance from Fixed Point. If the equations of accelera- tion, or, as we say, of motion, are *=-% y=-^> 0) where /i is a constant, we have the case of a point moving under an acceleration fijr^ directed towards the origin of coordinates, and varying inversely as the second power § 25, 26, 27] ACCELERATION TOWARDS FIXED POINT. 55 of the distance, r, of the moving particle from the origin. For if we denote the angle between the line OP and the axis of X by Q, we have cos0 = a'/r, sm6 = y/r, and therefore the components are as stated in (1). It is supposed that i = 0, so that the motion is in the plane of x, y. If we multiply the first equation by y, the second by x, and subtract the first product from the second, we obtain xy-yx = 0, which gives by integration xy — yd) = h, (2) where A is a constant. This last equation expresses the so-called " law of conservation of areas," that is the fact that the radius- vector (of length r=Jx^ + y^), drawn from the origin to the moving point, sweeps over equal areas in equal times in the plane of motion. For if 6 be the angle which the radius-vector makes with a fixed straight line in the plane of the path, the equation may be written r^ = h, (3) which renders it obvious that h is twice the rate of descrip- tion of area. It is interesting to notice that the angular speed 6 with which the radius-vector is turning varies inversely as r\ is, in fact, h/r^. We shall see later that the equation expresses the dynamical fact that the angular momentum, about the origin, of a particle moving in the path remains constant throughout the motion. 27. First Integral of Equations of Motion. Equation of Hodograph. Now by means of (2), the equations of motion (1) of last section can be transformed to h dt f). *=+fl© « or, as we may write them, if the axis of x be taken along the fixed line from which 6 is measured, 56 A TREATISE ON DYNAMICS. [CH. I. For equation (3) gives ljr'^ = 6jh. This value of l/r^ substi- tuted in equations (2) transforms them to h r h or --fl<»'° ")--!*© <^) Similarly, we obtain *=fi<-«=fie) w From the relation 1/r^ = 6/h, we see also that the resultant acceleration ju/r'^, which by the equations of motion is along Fig. 16. the radius-vector towards the origin, is fiO/h, and is there- fore proportional to the angular speed of the radiusTvector. Integrating (3) and (4) and putting f, ;; for x, y, we obtain f=-fl + «' H>' (5) §§ 27, 28] ACCELERATION TOWARDS FIXED POINT. 57 and therefore ii-ciy+(r,-bf = ^. (6) But ^, tj are the coordinates of a point on the hodograph, and the equation (6) just found is that of a circle of radius fi/h, and coordinates of centre a, b. The hodograph is therefore a circle (see Fig. 16, where an elliptic orbit and the corresponding hodograph are shown separately). That the velocities of an undisturbed planet at the different points in its orbit are represented in magnitude and direction by lines drawn from a chosen fixed point to a circle, is a very remarkable and interesting result, and an elementary geometrical proof of' it will be given later, in Chapter V. 28. Equation of Path. From equations (3) and (4) of last section, we can easily find the path and the relation to it of the hodograph. For (5), derived from them, can be written in the form *=-|sin0 + a', 2/ = |cos0 + 6' (1) (where a', b' are put for a, b to avoid confusion in what follows) : multiplying the first hy y=r sin 6, the second by a; = rcos0, and subtracting the first product from the second, we obtain 'h = ^r — (a' sin 6 — b' sin 6)r, that is T^ — = r(a'siii0 — ?>'cos0), (2) the polar equation of a curve of the second degree, or a conic section, as it is commonly called. If we write b' = A cos a, —a' = A sin a, we get ^=TrV^ • (3) £,+-^cos(e-(x) For 0-a = O, r = h^l{fj. + Ah), and f or 9 — OL = TT, r = h^jifi — A h). 58 A TREATISE ON DYNAMICS. [CH. I. Calling the first of these a(l-e), and the second a(l+e), where a and e are constants, we find fi/h^=l/a{l-e^), A/h = ela(l-e^). Hence the equation of the path can be written in the form '^~l + ecoa{e-a.)' *■ ' which is the equation of a conic section of parameter 2a{l—e^), of length of major axis 2a, and of eccentricity e. For e -< 1 the curve is an ellipse. For e >- 1 the curve is a hyperbola, and we then change the sign of a and write the equation as a(e^ — l) '^^l+ecosid-a.y ^'*^ For the limiting intermediate case e=l, the curve is a parabola. 29. Speed at Different Points of Path. The hodographic origin has so far been taken coincident with the origin for the path, but this is not neceaaarj, and it ia more convenient to give a separate diagram of the hodograph as in Fig. 16, where also an elliptic orbit is shown for compalrison of directions. Now by (1) of last section, we obtain „ xdx + ydy= —^(a'cosO + b'mn d)dd, (1) and this vanishes when the speed \lx^+y' in the path is a maximum or a minimum. Hence, when this is the case, sin 5/cos d= —a'jb', and two values of differing by tr, for one of which sin 6 is negative and cos 6 positive, and for the other sin 9 is positive and cos 6 negative, satisfy this condition. A second differentiation shows that in the former case the speed is a maximum, in the latter a minimum. But (2) of last section gives — dr = r {a' cos 6 + b' am d)d6, (2) and we have the same condition, sin Ojcos 0= —a'/b', as before, but in this case for a maximum or minimum of r. But the sign on the right of (2) is different in this case ; and we see that the speed is a maximum when the length of the radius-vector is a minimum, and vice versa. Now let us measure from the minimum radius-vector (OAf, in Fig. 15). If we do this we have initially x—r=0, and so we must puta' = 0. Thus we have ^=-^sine, y = l^cosd + b', (3) which, if ^, rj be put for ±, ■if and 6 be eliminated, give again the hodograph (6), § 27, but with a==0, b = b'. |§ 28, 29, 30] ACCELERATION TOWAEDS FIXED POINT. 59 30. Resolution of Velocity into Two Parts of Constant Amount. Patli Deduced from Hodograpli. Equations (3), §29, show that the velocity of which the components are x, ^ — b' is perpendicular to the radius- vector, the direction of which is defined by the angle 0. Thus when ^=0, the speed for this is fijh + h'. The velocity thus consists of the constant part b' in the direction of the y-axis, and a part of constant amount /x/A, always at right angles to the radius-vector. This is shown in Fig. 16, v?here oCf^ represents V, and CqAs, cJ>, etc., each the velocity of amount iijh, according to the position of the point in the path. Multiplying in (3), § 29, &\)y x and y by y, and adding, we get axi+yy=rf=b'r sin 6= -h'-rx. (1) Therefore r=c-h'-x '■=*'^(aP''-^) •••••(^) Thus the distance r of any point on the path from the origin is equal to the distance cij,/hb' — xoi the point from a fixed straight line parallel to the y-axis, multiplied by hb'Ifi,. This is the focus and directrix condition fulfilled by the conic sections, and hence again we see that the path is one of these curves. The same thing is obvious from (1), for r, the rate of growth of r, bears a constant ratio to x, the rate of increase of the distance of the point on the curve from a fixed line perpendicular to the axis of x. This can be seen also from the hodograph. Draw a line from o perpendicular to any of the lines c^fl, cjb, etc., say c^p, produced back- ward from Co, and let the lines meet in e. Then oe is that component of the velocity op which is at right angles to Cf^p, that is parallel to the tangent at p, and therefore parallel to the resultant acceleration at P in the path, that is parallel to r. It therefore represents f. Again, the perpendicular pd let fall from p on the axis of y in the hodograph represtents *. Now, since the triangles oecQ,pc^ are similar, we have oelpd=ocJCfjP, a constant ratio. The nature of the path may also be deduced from the circular hodograph thus: Let op produced backward from o meet the circle again in q. Then qo.op is constant, since o is fixed and p, q lie on a circle. Now if p be the length of the perpendicular let fall from the origin of the path (the point to which the acceleration is directed) on the tangent drawn to the path at the point P, where the speed is v, we have vp=h. But qo.op=v .qo, and thus v.qo also is constant. Hence qo is proportional in length to p. The locus therefore of the feet of the perpendiculars let fall from the origin on tangents drawn to the path at different points is a circle. This is a geometrical property of the conic sections, and of no other class of curves. Hence again we see that the path is a conic section. 60 A TREATISE ON DYNAMICS. [CH. I. 31. Polar Coordinates : Differential Equation of Path of Particle under Central Acceleration. The equations of motion in polar coordinates may be written down at once from the values of the accelerations along and at right angles to the radius-vector found in § 13 above. They are since the second of these equations is equivalent to the equation ^rio + u)r=0, which must hold in the present case since there is no acceleration transverse to the radius-vector. The first of these may, by the second, be written f-'l=-!L (2) It is convenient to eliminate the time from this equation. This can be done by remembering that since r and 6 vary together, r is a function of 6, and that d=hjr\ We have ■ _dr i_h dr ~i^d(F r'Kdel' Thus we obtain, instead of (2), ,Jd\ 2fdrY ) 2 ,->x ^{dffi-Ade) -T-^"' (^> It is convenient to write \\u instead of r. When this substitution is made the equation becomes, as the reader may verify, w^''=w (^) This only holds in the case of acceleration =/;(./r' ; but in the general case in which the acceleration is along the radius-vector and has value R, the equation is W^'^'hH^' ^^> Here R is taken as positive when towards the origin : the outward acceleration is — R. This equation will be established in a totally different manner in Chapter V., where many examples will be found. Ex. A particle moves in a plane so that the components of its acceleration along and at right angles to the radius-vector drawn to i 31, 32] HARMONIC MOTION. 61 the particle from a fixed point in the plane are respectively f(r) and fi,r\ where r is the length of the radius-vector. Prove that if the particle move once round a closed curve, the square of its speed is increased by 4/x, times the area of the curve. We are here given r -rff^=f{r), d(r^6)/dt=fj,r^ (see §13). The square of the speed at any point is r^ + r^^. Now we have, since .. dr /: h dr ... fy, h df l? ., , 'r = -.T7\Q=—^n\ and r-rl92 = — ty7,-TT =/(»•)• 'd6 dO Thus we obtain, since r Again, by the problem, d{r'^ff)ldt = ix,r\ that is h dh ■2 dQ ■fi {hlr^)drlde, and drldd=f(,ry/h+h/r, = fj/r' or dk_ r» dd~^'h' r^ dQ' But d{r^eP')lde=ddhlde + hd{hlr^)ldd. This gives, with the result already obtained for d{r^)jd6, d ,.„ . „Ms ^i-/ sdr , hdh . h dh dO +- = 2/(r)g + 2;.r2. Integrating this expression round the closed curve, we get zero for the first term of the integral, and for the second 2iJijr^d6=4ij,A, where A is the area swept over by the radius-vector. Thus v^ increases by 4/x4. 32. Simple Harmonic Motion. We now pass to the considera- tion of simple harmonic motion of a particle, that is to the Irinematical study of vibra- tions, a species of motion of which we have examples in all parts of physics. To define the motion let a particle describe the circle AGBD (radius r) of Fig. 17, with uniform speed v. We call this circle the auxiliary circle. Then, by § 10, the particle has no acceleration in 62 A TREATISE ON DYNAMICS. [CH. I. the direction of motion at any point, but has everywhere acceleration v^jr toward the centre. The time T, in which the particle describes the circle once, is iirrlv. Hence also r'^'W''- ^^^ But 2Tr/T is the uniform angular speed, n say, with which the radius drawn from the centre of the circle to the particle ■ turns round as the particle moves, and therefore we have also 2 - = 'n,V. (2) Now let Pj be the position of the particle at the zero of reckoning of time, and P its position after the lapse of an interval t. Let fall a perpendicular from each position of the particle to the diameter AB, and let pg, p be the feet of these perpendiculars for the positions Pq, P- As the particle moves round the circle the perpendicular p moves to and fro along the diameter AB. The motion of p is called simple harmonic. 33. S. H. M. Velocity and Acceleration. Integral EcLuation. The velocity and acceleration of p are the components, along the line of motion of p, of the velocity and ac- celeration of the particle in the circular motion. Now taking the position of P in the diagram, and denoting the displacement of p from the centre by x, we have, by the diagram, for the'displacement, velocity, and acceleration of p, x = rcoaPOA, x=—vsmPOA, x= —~ cos POA. ...(1) The values of sb, x can of course be got from that of x by differentiation. If further we denote the angle PfiA by e, we get POA='nt — e, and therefore £c = — t) sin int — e), V a X— cos(nt — e)= —n\cos(nt — e) (2) Thus we have t + v"x = (.3) The last equation shows that the acceleration of p is §§ 32, 33, 34] HARMONIC MOTION. 63 directed toward the centre of the range of motion, and is proportional to the distance Op from that point. It is to be noted that v? = 4nryT^ Now, for X itself we have x — rcos(nt — e) (4) or, as we may write it if we put A =r cos e, S = r sin e, x=A cosnt+Bsinnt, (5) and this is the complete integral equation corresponding to the differential equation (3). The two constants r and e in (2) or A and B in (5) are called a/rhitrary constants, for the reason that their values are immaterial so far as the satisfaction of the differential equation is concerned. They must be determined to suit the circumstances of any given case of motion. For ex- ample, the displacement aij and speed v^ of p, when ^ = 0, may be given, and from these we can determine A and B. When ^ = 0, (5) gives x = A; therefore A =Xq. Again, x= —nA sin nt + nBoosnt, (6) and therefore Vf^ = nB or B = vjn. Hence (5) becomes x = Xn cos nt-i — ^ sin nt (V) Thus the value of x at time t is made up of two parts, one depending on the initial displacement, the other on the initial speed of the point. This analysis of the motion at time t is of importance in the theory of waves. 34. S.H.M. Amplitude, Period and Phase. The two con- stants r and e of (4) of § 33 are called respectively the amplitude and the epoch of the simple harmonic motion. The epoch is sometimes referred to as the time in the circular motion from P^ to J. ; it is then e/n. This is also the time in the s.h.m. from p„ to A. The period T of revolution of the particle in the auxiliary circle is also called the period of the motion. The phase of a simple harmonic motion at any instant is the fraction of the period T which has elapsed since the last passage of the moving point through the middle of its 64 A TREATISE ON DYNAMICS. [CH. I. range of motion in the direction regarded as positive. For example, in the motion along AB the phase at time t is t/T -(e-Tr/2)/2Tr or «/T+i-e/27r. For this is the ratio of the angle DOP to 27r, and therefore also the ratio to T of the time taken by the radius of the auxiliary circle to turn through that angle. The difference of phase of two motions of epochs e^, e^ is (eg — ei)/2Tr. It is convenient to remember that —ccjx = 4nr^/T^, which brings out the fact that the ratio of the positive value, —x, of the acceleration, to the displacement x, has always, in a simple harmonic motion, the value i-Tr^/T^. This enables the period to be readily calculated in experimental work. The reader should notice that when the displacement has its greatest value +r or — r the speed of p is zero, and the acceleration is v^/r=n^r= inr^r/T^, towards 0, and is thus a maximum. When the point p is at the centre the speed of p is v and the acceleration is zero. 35. A Uniform Circular Motion the Besultant of Two S.H.M.s. Now let fall a perpendicular from P on the diameter GD, and let q be its foot. Then the point moving in the circle may be regarded as having at P the two displacements Op, Oq at the same instant. As it moves round the circle from P towards G, its displacement Op diminishes and the other Oq increases, and clearly the motion of q is also simple harmonic with the same period and the same maximum and minimum magnitudes of velocities and accelerations as for p. We take as the epoch for the motion of q the angle P^OG, which we denote by /. Obviously we have here e =/— ttJ^. Denoting Oq by y, we get y = r coa lCOP = r cosint — e — ^j =r sm(nt — e) (1) Hence y = nrcoaint — e) = v cos {nt — e), •(2) |2 V y= sm {nt — e) = — wV sin {nt — e) or y + n^y = 0, where, as before, n^ = i-Tr^/T^. §§34,35] HARMONIC MOTION. 65 These results show that a uniform circular motion is the resultant of two simple harmonic motions of the same period, in lines at right angles to one another, and of epochs differing by ■7r/2. The difference of phase is 1/4. This proposition, that two equal simple harmonic motions, of the same period, and differing in phase by 1/4, give by composition uniform circular motion, has many applications in the theories of sound and light. If the diagram (Fig. 16) be projected by lines perpen- dicular to its plane on a second plane inclined to the former at any angle between zero and x/2, the projection of the circle will be an ellipse and the projections of the lines of the two simple harmonic motions will be two conjugate diameters of the ellipse. The projections of the motions are clearly also simple harmonic motions. Thus we get the important theorem that two simple harmonic motions, in two lines inclined to one another at any angle and differ- ing in phase by 1/4, give by composition an elliptic motion. Ex. A particle moves in a plane curve, with speeds wy (a2 - 62)/(o2 + W), x{a^ - V')l{d'' -1- h\ along axes Ox^ Oy, at right angles to one another and turning with uniform angular speed (o : to find the path relative to the axes. Here we have £52 _ 52 a^ _ J2 and therefore x—^v/u—^, — 5^, w=— 2(oar-5 — ^,. so that -^ = 5 . ax y a' Thus, integrating, we get 52 where c is a constant. Hence the equation of the relative path is h^x^ + dhp = a V, that is an ellipse. In reality we have here relative to the revolving axes two simple- harmonic motions parallel to these axes. For differentiating x, y again, we find ^252 (^252 The periods of these simpJe-harmonic motions have the same value ■!c{a^ + ¥)l, amplitudes and phases. y \n Draw the auxiliary circles / /' "^s^ for the two motions from the / / ^ — "-^^^V same centre 0, and let the / / /^ /^^\\ points Pj, Pg be in the posi- \ [ 1 ^r^ 1 — 44 tions shown in the diagram ', \ \ J '/ I (Fig. 19) at time t. Describe \ \ ^- / / on OP-^^, OP2 as adjacent sides \ ^^^^ ^_^/ / a parallelogram, and draw the ^^, diagonal OQ. Then jOj, p^ ^^^ ^^"~~ ,-''' the positions of the harmoni- Fig '19 cally moving points at the same instant. As Pj, Pg describe their circles, with uniform speed, the point Q also describes a circle with uniform 68 A TREATISE ON DYNAMICS. [CH. I. speed, that is the auxiliary circle for q, which clearly has displacement equal to the sum of the displacements of P^ and Pg. The motion of q is thus simple harmonic, of amplitude equal to OQ, in the hne AB coinciding with A^B^ and A^B^ in direction and position. The period is clearly that of the component motions. The result of the composition of the two motions is clear from the construction in the diagram, but the analytical solution may be stated. We write x = a-^cos{nt — e^ + a^cos{'nt — e^, (1) where on the right we have the two simple harmonic dis- placements. We can write this x = A cos,{nt — e), (2) if A = {a\+al+ia^a^co^{e^-e^))^''\ ^^^^^Ojjinej+^o^sin^ j ^^^ a, cos gj + ctg tios e^ ' ) It will be seen from the diagram that the value of 4 as given by (2) is OQ, and that nt — eis the angle QOq. 38. Composition of any Number of S.H.M.s in Parallel Lines. Tide-Predicter. From this it follows that if we have any number of simple harmonic motions in parallel lines, of any amplitudes and phases, but of the same period, and a point 'p be made to move in a straight line in such a way that its displacement from a fixed point is the sum of the displacements in the different motions from the middle points of the different ranges, the motion of p is itself simple harmonic. The values of A and e given in (3), § 37, may be easily generalised for this case. A simple mechanism, the elements of which are shown in the diagram (Fig. 20) is used to impart to the writing style of Lord Kelvin's Tide-Predicting Machine, a vertical displacement equal at each instant to the sum of the displacements of a number of points describing simple harmonic motions in parallel lines. Each of the slotted T-pieces shown in the diagram is moved up and down by a pin at the outer end of an arm which revolves about a §§37,38,39] HARMONIC MOTION. 69 pivot at the other end. The pin works in the slot, and lateral motion of any part of the T-piece is prevented by guides properly placed. Each T-piece carries a pulley at the upper end, and over these pulleys passes a thin chain, which is fixed at one end and carries the marking pen at the other. It is clear that the rise or fall of the pen in any time is twice the sum of / T, ToT, the vertical displacements of all the T-pieces in that time. This mode of adding to- gether displacements is applicable to any system of simple harmonic motions whether of the same period or not. The various revolv- ing arms may be geared together so as to have any required relation of periods. As a matter of fact it is applied to the Tide-Predicter to add together the displace- ments in a large number of tidal motions which are of widely different periods not all connected by any simple relationship. The use of the pulleys and chain, or cord, for this summation was suggested to Lord Kelvin by Mr. Beauchamp Tower; but the arrangement seems to have been previously used for purposes of integration. Fis. 20. 39. Composition of S.H.M.S in One Line but of Different Periods. If the two motions in the same line which are to be com- pounded are not of the same period, we can write their resultant -ei) + a2eos{{n + v)t-e2) (1) 3;=a,c.os{ in the same form as before ; but now the amplitude A and the epoch e vary with the time. The periods are here ^Tr/n and ^/(n + v). The frequencies are nj^ir and (m-f-v)/27r, so that the difference of frequency is v/^tt. By (2) and (3) of § 37, we have x=A cos{nt — e), •(2) 70 A TREATISE ON DYNAMICS. [CH. I. where now A = {al+a^+2aia2Cos{vt — e2 + e^)}^, | _a!isinei — a2sin(v< — 62) ( «! cos 61 + aj cos ( i/i - «2)' ' Thus the amplitude oscillates in the period Stt/i/ from the value a^ + a^ (at an instant when vt - 6^+ ei = 2mir, where m is any integer) to the value ai-a^ (at the instant when vi-e2 + ei = {2m + l)Tr). At each of these instants tane=tanei. FiQ. 21. An excellent example of this is afforded by the solar and lunar tides at any place. The amplitude of the lunar equilibrium tide is about 2"1 times that of the solar. Thus (on the " equilibrium theory ") spring tides are about 3'1, neap tides about 1"1, times the solar tide. Again, when two musical notes which differ slightly in frequency are sounded together the ear perceives an alternate swelling out and dying away of the sound : the notes are said to beat. If the frequency of either note is known, the frequency of the other can be inferred by counting the beats in a given time. The slower the beats the more nearly the notes are in unison. Pig. 21 shows s.H.M.s (ordinates = displacements, abscissae = times, periods in the ratio 1 : 2, 61 — 62 = 0) compounded. 40. Composition of S.H.M.s in Perpendicular Lines and of Different Periods. The composition of motions of different periods in lines at right angles to one another can be worked out easily in the case in which the relation of the periods is simple. For example, let the displacements at time t be ^ = ocos(2?ii-e), ^=bcosnt, (1) so that the ratio of periods is 1 : 2, and an arbitrary difference of phase i 39, 40] HARMONIC MOTION. 71 e/ 27r exis ts between the motions. Substituting yjh for cosrei and \/l -y'^jV'' for sin nt in the expression for x expanded, we get a; = raf 2|j - 1 j cos e + 2|y 'Y 1 •(2) If now cose=0, so that e= ±ir/2, we get or for either sign ^fWi-ft (3) ^^ = 4^^(1-1) (4) Clearly this curve is represented at the origin by the two straight lines y=±x. b/2a, that is it has there the form of a St. Andrew Cross of vertical angle ta,a^^{iab/(b^-4M^)}. Any line parallel to the axis of 2/ on either side of the origin at a less distanc'e than a cuts it in four points, and the pairs of lines y= ±6 and x= ±a are tangents. O Fig. 22 (1). Fig. 22 (2). Each of the latter lines touches the curve at two points for which the values of y are equal but of opposite sign. The curve is thus a " figure of eight," as shown in the diagram [Fig. 22 (1)]. Again, if sine = 0, so that e= ±ir, we obtain (^p/-"). = ±2^ .(5) which represents a parabola with its axis in the direction of x and its vertex to the left or the right of the origin, according as the plus or the minus sign is taken [Fig. 22 (2)]. If the periods are not exactly in the ratio 1 : 2, that is, if we have x = acoB{2(n + v)t-e}, y = bcosnt, (6) where v is small compared with n, we may take as the epoch in the expression for x, e~ 2ft, and we see that the difference gradually alters 72 A TREATISE ON DYNAMICS. [CH. 1. with the time and the resultant curve passes through all the varieties of form shown in Fig. 23, and back again, in the reverse order, with reversal also of the direction of motion in the intermediate curves. Fig. 23. Other cases, say the case of periods in the ratio 2 : 3 or 1 : 3, may be worked out and considered by the student. The resultant curves are shown in works on Acoustics. 41. Composition of S.H.M.s in Different Lines but of Equal Period. We can find a motion that combines displacements for any number of simple harmonic motions of any amplitudes and epochs in different lines if they are all of the same period. Let li, wij, n^, I3, m^, n^, ... be the direction cosines of the different lines of motion, and the displacements in these lines be ri = ajCos(?i< — e,), ?'2=a2Cos(?!< — gj), ... ; (1) then resolving along rectangular axes of x, y, z, we get x=A cos nt + A' sin nt,\ y=B cosnt+B' smnt, \ (2) z=C cos nt+ C sin nt, ] where .4=2(a;cose), A' ='Z{al sin e), B = '2(amcoae), ..., (3) and the summations are taken for all the motions. Then taking the first terms on the right of these equations we obtain a simple harmonic motion, j^ , ^='/A^+W+C^. COS nt. (4) The second terms on the right give in the same way a simple harmonic motion, , r,=^A'^+B'^ + C'KBiiint. (5) The displacement ^ has the direction-cosines (A, B, C)/-Ja^+W+C', and the displacement t) has the cosines (A', B, C')l'jA'^^ff^+C^. These two harmonic motions are equivalent to the original system in the sense that they give the same sum of component displacements in any direction as the system gives. They differ in epoch by 7r/2, and their lines are inclined at the angle (V) It will be observed that, if in this equation we put ±acos(j) for ?/, it reduces to (^Tasin< ^)^=0 ; (8) and that, if we put ±a^sin<^cos<^/\/6^-l-«^cos'''<^for x, it reduces to ?/= iN/PTo^oos^. (9) Thus the ellipse touches each of the lines .v^asinrf), x= -asintj), and also each of the lines y='Jb'^ + a^cos^, y= -'Jb'' + a'coB'' = o.l3, so that a.±/3=\/a^TF±2a6sm^, (10) from which ot. and f3 can be found. Again, if 6 be the angle which either of the rectangular axes used in (6) makes with a principal axis of the curve, say that between the two axes of .a;, a^sinZdi ,„- a^cos2+b' Thus the axes may be regarded as determined. 42. S.H.M.S in Perpendicular Lines, but not of the Same Period. Now consider shortly the case in which two given sirnple harmonic motions parallel to x and !/ are not of the same period. We may take as their equations x=acoant, y = bcos{(>i + v)t-e} (1) The part bcoa(vt-e)cosnt of ?/, compounded with x, gives the simple harmonic motion u = {a^ + b^cos^{vt-e}\^cosnt (2) 74 A TREATISE ON DYNAMICS. [CH. I. This motion combined with the second part of y, -h%\ti.{vt — e)A'ant, gives, for the instant t, an elliptic motion, of which the directions of the components and their amplitudes, namely, {a^ + h'^coa'^{vt-e)Y and 6sin(v< — e), are the directions and lengths of a pair of conjugate axes ; and the angle between the axes is 7r/2 + sin-H6cos(v«-e)}/{a2 + 62cog2(yi_g)p. Since the difference of phase vt — e varies with the time, the ellipse continually changes in form and position. The axes of the ellipse, and their position at time t, can be found easily. Solving equations (1) for cosnt, sinnt, squaring and adding, ''^eget {x:h^^^^). 44. Differential Equations of Exponential Motion and S.H.M. Exponential Motion represented Graphically. In the foregoing §§ 32-42, the subject of simple harmonic motion has been discussed, and all contained in these sections may be regarded as illustrative of the properties of the complete integral of the differential equation of the form -c^+n^x^C) (!) The other differential equation which we obtain when the acceleration x is in the direction of x increasing, !i-n^x=0, (2) has as its complete integral x=Ae''* + Be-'^, (3) where A and B are constants, the value of which, so far as the differential equation is concerned, may be chosen at pleasure. Take for example the first term and write 30= A^. If, as we suppose, n be positive, the motion may be supposed produced in the following manner. Consider an equiangular spiral of which the equation is, for n positive, r = Ae^ (4) Here nt is the angle which the radius- vector r makes with a fixed line in the plane of the curve, and n is the cotangent of the angle which the curve makes at each point with the radius-vector. Now suppose the axis of x to coincide with this radius-vector, and to remain fixed in space while the spiral revolves with constant angular speed n about the pole. If the spiral turns so that the point of intersection of the curve with the axis of x moves outward, we have x varj'ing exactly as expressed in (4). The (outward) speed of the point of intersection is x=nA «»' = nx, and the acceleration, also outward — the distance between two suc- cessive convolutions of the spiral increases with distance from the pole— is ^ = „2^gn«^^2^_ In order to represent the motion expressed by x = Be-'"\ (5) 78 A TREATISE ON DYNAMICS. [CH. I. where n is positive, we must suppose a spiral drawn with x=B for «=0 and ^=0 for <=oo, in fact a spiral the radii-vectores of which are the reciprocals of those of the curve ,r=e"*/5. Then, if we suppose the axis of x to coincide with the initial direction of the radius- vector, and imagine the spiral to revolve with angular speed n in its own plane about the pole, in the direction to cause the point of intersection to move in towards the pole, the speed will be and the acceleration, which must be positive, will have the value x=n^Be~'^ ='r?x. The sum of these two motions is that represented by the equation (3). Exam/pies of Exponential and Vibratory Motions. 1. Prove that the hodograph of the motion of the point P (Fig. 24) is a Jogarithmic spiral of the same angle as the path, and that, if <^ denote the constant angle between the radius-vector and the tangent to the curve, the acceleration along OA is -)-r0^cos(2<^- ^)/sin2<^. Derive the differential equation (5), § 43. [Here r diminishes as Q increases.] 2. Prove that if x = e~^^, the differential equation reduces to £ -)- (m^ — ^^) ^ = 0, and show that .r = e"** (4 cos ■Jn^-W' . i + 5 sin 'Jn^ - F . t\ according as n^ is greater or less than k'^. 3. Prove that if the equation of a logarithmic spiral be written r = Ba^, and the radius turn in the direction of r increasing, cot <^ = logja, [Ex. 1] and that the curvature at any point P, to which the radius is r, is sin (^jr. Prove also that the components of acceleration towards the centre of curvature and along the tangent are respectively ^V/sin 1^ and j-^^cos <^jsw?<^. 4. Prom Ex. 3 find the accelerations along and at right angles to OA (Fig. 24), and also those along and at right angles to OP. Prove that these are equivalent to a component rd^/sin^ along PC and a component 2r^2cos^/sin^^ in the direction of motion. 5. A radius revolves about one extremity 0, with constant angular speed n, and alters in length while revolving, so that the other extremity traces out the spiral of Archimedes, r = ad, where d=nt. Prove that the displacements from 0, taken along and at right angles to the initial position of the turning radius, satisfy the differential equations x + 2n'j- n^x = 0, jj - 2nx -nh/ = 0. § 44] EXERCISES. 79 EXERCISES I. 1. A boat on a river is distant 300 feet from the shore and 400 feet from a water-fall directly down-stream. If the speed of the stream be 4 miles an hour, find the least velocity with which the boat must be propelled in order to avoid the fall. Show also how to find the direction in which the boat will have the least distance to travel to reach the bank, supposing its speed sufficiently greater than this minimum. 2. A railway passenger seated in one corner of a carriage looks out of the windows on the far side and observes that a star near the horizon is traversing these windows in the direction of the train's motion, and that it is obscured by the partition between the corner window at his end of the carriage and the middle window while the train is moving through the seventh part of a mile. Prove that the train is on a curve, the concavity of which is directed towards the star, and which, if it be circular, has a radius of nearly 3 miles, the breadth of the carriage being 7 feet and the breadth of the partition 4 ins. 3. Two points describe concentric circles uniformly, the time of describing the outer being m times that taken to describe the inner. If V is the speed in the former circle and u that in the latter, show that if the angular velocity of the one point relatively to the other is zero, the actual velocity of the one relatively to the other is V: "^-liu^-v^). 'to-M 4. In one of Bashforth's experiments three screens, equally spaced apart at a distance of 150 feet, were penetrated by a projectile 0'5569, 0'634 and 0'7069 seconds respectively after projection. Assum- ing that the motion of the shot may be represented by the equation t =0-6314 + as + bs^, where s is the distance of the shot from the middle screen at time i, prove that the resistance to the motion at the middle screen is about eleveri times the weight of the shot. 5. In certain experiments on the resistance of the air to the motion of cannon balls it was found that the number s of feet travelled by the shot in t seconds was given by the equation t = as + bs^, where a and b are constants. Find the relation between the velocity and the tangential retardation. 6. A crank OA rotates uniformly about an axis through ; the end A is pivoted to a rigid rod which slides in an oscillating cylinder. The axis about which the cylinder turns passes through its centre 0' and is parallel to the axis of the crank. Show that the angular speed of the rod and cylinder is given by 01= -(jor(r — dcos 6)/(r^ + cP- 2rd cos 0), 80 A TREATISE ON DYNAMICS. [CH. where w is the angular speed of the crank, r the length of the crank, d the distance between the two fixed axes, and v the angle between the crank and the line OU. 7. A crank OP of length a rotates with uniform angular velocity 0) about an axle through a fixed point ; a connecting rod of length b joins P to the end Z) of a crosshead which is constrained to move in a straight line through at right angles to the axle. If 6 is so large compared with a that all powers of a^/b above the first may be neglected, show that the acceleration of B when the angle BOP is equal to ^ is — aio^ cos 6 - a'd)^ cos 9,d/b. 8. A particle describes the circle r=2aeoad, the component of acceleration towards the origin being always zero. Show that the transversal component varies as cosec°ft 9. 4 is a fixed point on a plane curve, B is the position at time i of a point which is moving along the curve, and on the tangent at B a point C is taken. If the arc AB=s, BG=r, and 6 be the angle through which the tangent revolves as the point passes from A to B, show that the accelerations of C in the direction BO and in the direction perpendicular to BC (in the sense in which d increases) are respectively , , s+r-rff', L^XrWHW. T at 10. Prove that in the case of a particle moving in a groove which is made to rotate in its own plane about a fixed point in the plane, the motion of the particle relative to the groove can be obtained by superposing on the external forces on the particle the following system : otcdV along the radius- vector outwards, — mrio perpendicular to the radius-vector, and 2ot»'o) perpendicular to the groove, where v' is the velocity of the particle relative to the groove. Indicate by a figure the directions of the forces. [If r be the radius-vector from the origin to the particle, 6 the angular speed of r with reference to a given point of the groove, <^ the angle between the forward tangent and r, and R the inward normal reaction of the tube, the equations of motion are m{f-r{d+o>f} = -Rsm4,, - ^{r^0+u>)} = Rcos-S>. T at Along the tangent and normal these give v-j-=tii^rcosd>, — = -(oVsind)-) 2aii.l as ^' p ^ m ■' A groove in the form of a parabola (latus rectum 4a) is initially at rest with a particle at the vertex. It is suddenly made to rotate about the focus with constant angular velocity ^(Sj^2 - 4). [There is no force on the particle except that applied by the groove.] I.]. EXERCISES. 81 11. A smooth conical cup, whose semi-vertical angle is o., revolves with angular velocity at about a vertical axis parallel- to the axis of the cone and at a distance e from it ; show that if a particle be moving on the surface of the cup, the component of its acceleration along the generating line is r-»-sin^a.((^ + co)^H-cft)^cos(^sina., where r is the distance of the particle from the vertex and the angle between a plane through the two axes and a plane through the particle and the axis of the cone. 12. If the position of a point moving in a plane be determined by the coordinates r and <^, where r is measured from a fixed circle (radius a) along a tangent which has revolved through an angle <^ from a fixed tangent, show that if a. and /8 are the accelerations along and perpendicular to r respectively, a.=r-r

JTJg. 22. A particle of mass M rests on a smooth horizontal plane and is attached to one end of a light elastic string, the other end of which is fastened to the plane. The unstretched length of the string being I, show that if the particle be moved along the plane until its distance from the point of attachment is l'(l'>l), and is then let go, it will pass the point of attachment after a time given by '=Vf(l+r^z)' if A, be the force required to produce unit extension of the string, and force vary as extension. 23. Describe the rectilinear motion of a particle whose distance from a fixed point in its line of motion is given by a: = a + b cos ait. A and B are two points at a distance d apart. A particle moves in the line AB, its speed at time t being given by (c sin co<)/r/, c and + Moi in the direction of a, together with a component Mv sin in the direction v and a component Ma. sin ^ at right angles to a. in the plane of v and a.. The resultant rate of change of momentum R is {M^v^ + M^a.^ + 2MMva. cos ^)*, and makes an angle the cosine of which is (Mv + Ma. cos (l>)/R with the direction of v. These results are easily verified by means of the com- ponents parallel to the axes of x, y, z, wliich are Mx+Mx, My + My, M^ + Mz. Thus the resultant rate of change of momentum is {{Mx + Mxf + (My + Myf + (Mi + Mzf)^, which expanded gives at once (MH"- + M'^o^ + IMMva. cos ^)*. The loss or gain of momentum, through loss or gain of mass, is an important consideration in various cases of motion; and care must be exercised in taking it into account. For example, the rapid burning away of the powder charge of a rocket propels the rocket forward and upward. Again, a tank on wheels may lose mass in a jet of water from a hole in one end of the tank, and a reaction will be exerted on the tank by the jet, either aiding or hindering the motion of the former. But the tank may lose matter by a jet through a hole in the bottom, in which case only a vertical reaction exists. [See § 52 below.] 47. R.C.M. in Curvilinear Motion. Force. From the results stated in §§ 8-11 above for accelerations, we see that a moving particle of constant mass has, at each instant, rate of change of momentum mv^lR towards the centre of curvature of its path, and ms in the direction of motion. We notice that if the curvature of the path at any point 86 A TREATISE ON DYNAMICS. [CH. II. be very great, that is if the radius R be very small, the rate of change of momentum mv^jR is very great. In fact a particle cannot be made to turn a perfectly sharp corner in its motion. Again, the rate of change of momentum may be resolved along the radius- vector drawn from a chosen origin, and at right angles to the radius-vector in the plane of motion. In the general case the components are m (r — ft)V), wi( 2ft)r + wr) ; in the case of motion in a plane curve they are m,ir — d^r), m(2rd + Or). For brevity we shall now call the rate of change of momentum in any direction the force in that direction, in the case for the most part in which the mass is not subject to change. Each more general case will be considered as it arises. 48. Kinetic Energy. B.C.M. as Space-Bate of Variation of K.E. For a particle moving with a velocity v, the product ^Tnv'^ is called the kinetic energy of the particle. For an aggregate of particles (of total mass M), all of which have t\\e. same velocity v, the product ^Mv^, where M is the total mass, is called the kinetic energy. In the case of a system of particles, of masses m^, on^, ..., the speeds of which are Vj, Vg'--- ™ different directions, the. kinetic energy is defined to he the sum i(my^ + m^vl+...), usually written ^'S,{mv^), of the products obtained by mul- tiplying half the mass of each particle by the square of its speed- This case will be considered later. It will be seen that the rate of change of momentum of a particle in the direction of motion may be written in the form mv dv/ds, for this is simply ms. But it is also the rate of variation of the kinetic energy ^mv^ in the direction of motion. Thus the time-rate of change of momentum, or force, in the direction of motion is equal to the space-rate of change of the kinetic energy in the same direction. §§ 47, 48, 49] ENERGY. 87 It is convenient when a space integration is required to write vdv/ds for dv/dt or s, and use v when a time integration is convenient. For example, take the case of a shot which is resisted according to the cube of its speed, so that v = v dv/ds = — kv^. We can at once integrate over a time or a space as may be required. 49. Potential Energy. Equation of Motion for Particle under Central Force derived from Energy. For a material system in motion, each part of which is acted on only by other parts of the system, and is not affected in its motion by frictional resistances, we have JE(mt;2)+F= const., .*. (1) where F is a single-valued function of the masses and the coordinates of the parts of the system. Thus if we put T for the kinetic energy ^1,{7nv^), we have T+F=const (2) F is what is called the potential energy of the system, and it is such that the total rate of change of momentum of the system in any direction, that of x say, is —dV/dx; that is the total force On the system, in the direction of x, is the space-rate of diminution of the potential energy in that direction. But by (2), if the vs are supposed to be ex- pressed as functions of the coordinates, ■ ?2:__9F .dx~ -dx' ••■■• ^'^' and so the force in the direction of x is the space-rate of increase of the kinetic energy in that direction. [See also § 66.] As an example, take the equation for v^ in the case of a particle under a central acceleration (§11 above). Fis here a function of the distance of the particle from the centre 0, towards which the acceleration is directed. Assuming that in this case the equation (2) holds, and taking the mass of the particle as unity, we have t '^'i^'HO'^'^] <« -^ s— 'f-'-^^-") (^) 88 A TREATISE ON DYNAMICS. [CH. II. since d(du/de)/du = r^fv + ^Trpr^. The first of these terms is the force required in consequence of the addition, at rate 47rprV, of matter which must be made to take up the speed v, while the second term is the force required to give acceleration v to the drop as it exists at the instant. These two forces must equal ^irfyfig, since no momentum is brought with the water deposited. Hence . , Zrv r " is the equation of motion, which can be written also in the form d'o ,^'» 9 ~7~"ro- — — , or r c since c=r. Multiplying this equation by g^'"?'', and integrating, we get since r-a=ct. Division by r^ gives Ex. 2. A tank is mounted on a truck and water issues horizontally from an orifice in one end. If the truck be moving with speed v in one direction and the water leave the truck, from an orifice in the hinder end, with speed v' in the opposite direction, and the effective inertia of the truck be M, find the equation of motion. The jet exerts a reaction on the truck. Since the momentum of the truck and its contents is Mv, the r.c.m. is Mv + Mv, where, if m be the mass of water which issues per second, —M=m. Momentum is given to the jet by the truck at rate m(v + v'), and the reaction due to the jet has this value. The r.c.m. Mv + Mi is due to the net forward horizontal tractive force F applied from without, the reaction of the jet, and the rate of flow of momentum, conjointly. Hence we get Mv+Mv = I''+m.{v + v')-mv, or, since ]if= — m, Mv = F+m.(v + v'): Ex. 3. A light open carriage runs on horizontal rails. A heavy uniform vertical rain falls, and water is received by the truck on a horizontal area A : find the effect of the deposition of water on the motion. Let the mass of water which comes down per unit area per second be m, then the rate of gain of mass by the carriage is mA, whatever the speed, may be. If the total mass at time theJf and the speed ■;;, the R.C.M. is Mv + Mi = Amv + Mv. The rain exerts no horizontal 94: A TREATISE ON DYNAMICS. [CH. 11. -action, but a forward force Amv is required in order that each small addition of mass may take up the speed v. Thus, if F be the balance of tractive force over resistances, we have Mv + Amv=I\ If water at the same time flows out through an orifice in the bottom at rate /x, the r.c.m. is Mv + Amv — fiv, and this is due to the force F and the flow of momentum conjointly, that is, Mv + Amv — fibV = F— fiv or Mv+Amv=F, so that the equation of motion is not affected. This is of course on the supposition that all the water which enters takes up the motion. If F=0, we get v/v= — AmjM, and Mis a, function of t. If we take this case, we have if= Mfj + {Am — fj.) t, if /a be constant, so that v_ Am Am Am — ft, V Mfj+{Am — ii.)t Am — fj,M^ + {Am — fj.)t Integrating, we find But when t=0, v=Vg, and therefore log ISL = ^'^ log ^o+(^™-m)< , V Am — IX, ' Mfj. If li. be zero, that is, if the case be the first stated above, log ^ = log- , ■"o l._-^0 + -^™' V that is, M^Vf^=(Mf^-\- Am()v, as of course could have been stated at once, since the total momentum at time t must be equal to the initial momentum. Ex. 4. A thin uniform flexible chain of small links is hung vertically from its two ends. One of the ends is then let go : to find the tensile force at the bi^ht where the chain passes over from the free side to the stationary side. , In the first place, there is no tensile force in the chain on the side that is let go, for every portion is at each instant falling freely under gravity, and has therefore the same downward speed and acceleration. In time t the free end has descended a distance ^gfi, and acquired a speed v{=gt), which is also the speed of each part of the chain between the free end and the bight. If 9,1 be the whole length of the chain, the falling side has length l-^s, while the part on the other side, which is stationary, has length l + \s, and is therefore increasing in length at rate ^v. § 53] EXAMPLES. 95 Thus mass is passing across from the falling to the stationary side at rate Ju-w, where tr is the mass of the chain per unit length, and each element as it passes across has its downward speed destroyed. To effect this upward r.c.m,, an upward pull must be exerted by the lower end of the fixed part of the chain of amount io-w^, since this is the momentum produced per second. Thus (see §57) on the fixed side the tensile force at the bight is The downward pull P of the chain on the support of the fixed end is \(TgH'^-'rg(T{l-\-\s)=gfTl-^\forces applied at its ends. 'If the force on D, in either case, is greater at one end than at the other, motion of the matter D will take place .unless the difference is balanced by external forces, those due to gravity for example. Equality of action and reaction does not provide that the force on D at one cross-section shall be the same as that on D at the other; it makes certain, however, that the two aspects of the stress at each cross- section shall be equal and opposite. The forces at different cross-sections are all equal, if the matter between is unacted on by external forces and does not suffer change of motion. The agreement of the results which flow from the third law of motion with those of experience over a wide range of physical phenomena, is the best proof of the validity of the law. Fig. 25 (1). Fis. 25 (2). 100 A TREATISE ON DYNAMICS. [CH. II. 58. Action and Reaction between Bodies at a Distance apart. In the case of bodies which cannot be regarded as being in contact, such for example as the sun and the earth, to say that the pull exerted by the sun on the earth is equal to the pull exerted by the earth on the sun is the only way of expressing the law; and the validity of the law is here again to be regarded as established by the agreement of theoretical results with observation and experience. That the force on the earth and the equal and opposite force on the sun are here referred to as pulls is not material; the earth may, in consequence of the presence of the sun in the gravitational field (whatever may be the cause of gravitation), be pushed toward the sun, and in the same way the sun pushed toward the earth. The material fact, which is beyond cavil, is that each body experiences a force toward the other, and that these forces are equal and opposite, and that fact re- mains whatever mode of speech is adopted regarding it. 59. Centre of Mass (or Centroid) of a Body or System. It will be convenient to define here the centre of inertia or centre of mass (or shortly, the centroid) of a system of particles, and deduce some of its properties. Let the posi- tions of the particles be referred to rectangular coordinates, and denote the coordinates of the first, of mass m,^, by iCj, 2/j, z^, of the second, of mass m^, by x^, y^, z^, and so on. ' Then the centroid of the system is the point whose co- ordinates are given by the equations _ 7n^Xj^+7n^x^+... 2(ma;) , , 00 — — — = , (J-) ■m.j-|-m2+--- ^'^ ^ mi + TOg-l-... 2(m) ' ^' m^ + m2+ ... ^ 2m ' ^ ^ that is the a;-coordinate is equal to the sum of the products TTijCCi, mgCCg, ... , obtained by multiplying each mass by its distance from the plane of yz, divided by the sum of the masses, and similarly for the y- and ^-coordinates. Thus §§ 58, 59, 60] DYNAMICAL PROPERTIES OF CENTROID. 101 each coordinate is the mean, when account is taken of the masses, of the corresponding coordinates of the particles. The student may easily satisfy himself, by changing the origin and turning the axes of coordinates round through any angle, that the centroid as thus determined is a definite point in space, the position of which depends only on the positions of the particles and not at all on the choice of axes. The equations therefore enable us to define the centroid as that point the distance of which from any plane whatever fixed in space is the average distance of the particles from that plane. We do not here devote space to the calculation of the positions of centroids for different bodies : such calculations form properly a chapter of the Integral Calculus. The student is referred to Gibson's Calculus, § 137, and to Ex. 7, Exercises XXX., of the same work. 60. Properties of Centroid. External and Internal Forces. Differentiating the equations (1), (2), (.3) of last section, by which x, y, z are defined, we get, using the abridged notation there indicated, ■^_ 2(m^) ^ _ ^{my) . 2(TOi) . *~ 2m ' 2/- 2m ' 2m ' ^ ' and putting M for 2m., Mx = I,{mx), My = 'E(my), Mz = 'E{mz). (2) Now, on the right in each case we have the total mo- mentum of the system of particles in the direction of the axis referred to, and on the left the momentum in that direction which a particle of mass equal to the total mass of the system would have if it moved with the centroid. Hence, if the momentum of the system in any direction is zero, the centroid has no motion in that direction. Again differentiating, we obtain Mx = l^(mx), My = 'E,(my), Ml=2(m0), (3) which asserts that the rate of change of momentum of the particle just referred to as moving with the centre of mass, is for every direction equal to the total rate of change 102 A TREATISE ON DYNAMICS. [CH. II. of momentum of the system. Now, going back to the equations of motion of a particle (§ 53), and writing for X, Y, Z in the equations of any particle Xe-\-Xi, Yi-\- Yi, Ze+Zi, where X, denotes the force on the particle, in the direction of x, produced by matter external to the system, and Xi the corresponding force on the same particle pro- duced by the other particles of the system, we have for the equations of motion, 'mx = Xe + Xi, my=Y,+ Yi, mz = Z,+Zi (4) Writing the equations for all the particles in this way and equating the sum of the left-hand sides of the flj-equations to the sum of the right-hand sides, and doing the same for the other axes, we get 2(m«)=2Z„ 2(mi/) = 2F„ 2(mz) = 2.^o (5) for the sums SXi, SF,, '2,Zi must each vanish, since the contribution to each of the forces Xi, Yi, Zi, on the particle considered, made by any other particle, is accompanied by an equal and opposite force on the latter, which comes into the account when the equations of motion are added. Hence we have the very important result that if no forces from without act on the particles of the system, that is if 2:Ze = 0, SF, = 0, SZ, = 0, we have M = 0, 'y = Q, z = 0; (6) and therefore we get by integration x = at + e, y = bt+f, z = ct-\-g, (7) where a, b, c, e, f, g are constants ; that is the centroid moves with constant component velocities a, b, c in a straight line. We see moreover that if external forces do act on the system of particles, the internal forces cannot affect the motion of the centroid. Thus if, for example, a shell bursts in , the air, the motion of the centroid is sensibly the same just after the explosion as before, except so far as the gas into which the powder is changed has been affected by the resistance of the air. The motion of the centroid of the solid casing which contained the powder sustains §§60,61] LAW OF EQUAL AND OPPOSITE ACTIVITIES. 103 little or no change in its motion, since the increased action of the air on the matter now in fragments is practically negligible. We shall find many other examples in what follows. 61. Newton's Law of Equal and Opposite Activities. In a scholiuTn appended to the third law of motion Newton gives another view of action and reaction. 'J'o understand this it is necessary to go back to the forces exerted in opposite directions across a cross- section of a tie or strut (§ 57). Let F denote the force exerted by the matter G, which is on one side of the section AB, on the matter D on the other side ; then —F is the force exerted by D on G across the same section Let now the cross-section be in motion with speed v in the direction of a line drawn from G to D. Then we may call the product Fv the action oi G on D. The reaction of D on G is now —Fv, and is equal and opposite to Fv. The product Fv is what we shall call in future a rate of working, or an activity ; it is the rate at which work is being done by G. On the other hand, while G advances at the section AB, D there recedes, and work Fv is done on D, that is D does work —Fv on G. In the same way, when a piece of matter is acted on by force the matter reacts on the agent. The reaction may be due only to the inertia of the body; and the reaction on the agent, when the acceleration produced is what it is agreed shall be the unit of acceleration, and there are no resistances such as friction to be overcome, is the proper measure of the inertia of the body. It may therefore be, as it is sometimes, called the inertia-tesistance of the body. And everywhere, when matter has force applied to it, there is an equal and opposite force applied to the agent; and therefore, if we regard the acting forces on any system as one group, and the reacting forces as another group, these two groups of forces if applied together to the same body or system would give zero rate of change of momentum in any direction, that is the two groups would, as it is usually put, form a system of forces in equilibrium. 104 A TREATISE ON DYNAMICS. [CH. XI. This statement may be taken as an expression of the principle known as that of D'Alembert. In the case of the attraction of the earth by the sun (or the vis a tergo, exerted on the earth in consequence of the existence of the sun in the gravitational field, or whatever the cause of the action may be) there is work done on the earth when the earth moves in the direction of the attraction ; the attraction then does positive work ; the earth, by the resistance which its inertia offers, and which is overcome, does negative work. The two rates of work- ing, that by the force and that by the resistance, are equal and opposite. And so for any complex of forces applied to a material system. 62. Theory of Work. Units of Work. The work dpne by a force in any displacement of a body acted on, or as we may put it, in any displacement of the poifit or place of application of a force to a body (generally some particle or part of the system), is measured by the product of the force into the component of the displacement in the direction of the force. Thus, if the displacement is from A to B, and the force, supposed of constant amount during the dis- placement, act in the direction AG, the work done by the force in the displacement is F. AB cos l BAG =Fs cos 0, if s = AB and 6=lBAG. In any finite displacement, under a variable force, the work done is I Fcos 6 . ds, where d is the angle between the directions of ds and F when the step of displacement ds is being taken, and the integral is taken along the whole displacement. It is not necessary that these directions should remain the same throughout the displacement. Thus the work done in a displacement along any curve, along which the force acts at each step, is XFds. If I, m, n be the direction-cosines, and X, Y, Z the components of F, and V, m! , n' the direction-cosines, and dx, dy, dz the components of ds, then (see § 6) F cos 6. ds = F(U' + mm' + nvf) ds = Xdx+ Ydy + Z dz. §§ 61, 62, 63] WORK AND ACTIVITY. 105 Hence jFcosd .ds=\{Xdx+Ydy + Zdz), (1) where the integrals are taken for the whole finite dis- placement. When the chosen unit of force acts over a displacement of unit distance in its own direction unit of work is done. Thus a force of 1 dyne in a displacement of 1 centimetre does the c.G.s. unit of work, the erg. A force equal to that of gravity on a pound of matter does work of amount 1 foot-pound in a displacement of 1 foot; and so on for other units. [For further particulars as to units of work see Elementary Dynamics.] Again, if s be the rate of displacement at any instant, the product Fs cos 6 is the time-rate of working, or, as it is often called, the activity. For this we may write also Xx+Yy + Zz. The whole work done in any interval of time t is, if A be the activity, \Adt=\(Xd;+Yy+Zz)dt, (2) where the integral is taken over the interval of time t. The unit of activity is that rate of working in which unit of work is done per unit of time, e.g. one erg per second is the c.G.s. unit of activity. Another is one foot- pound per second (f.p.s.), still another is 550 foot-pounds per second, or, which is the same, 33,000 foot-pounds per minute. This last unit is called a horse-power, and is based on estimates made by James Watt for use in deciding the power of steam-engines required for different practical purposes. The dimensional formula for work and energy is that of force X displacement, or ML^T-\ The dimensional formula for activity is that of work/time, or ML^T'^ The mode of using such formulae for change of units has been explained in § 54. 63. Active and Inactive Forces. Now consider any system of forces acting on a material system; the forces are partly internal forces between the dififerent parts of the system, and partly forces exerted on its parts by matter 106 A TREATISE ON DYNAMICS. [CH. II. outside the system. For the system as a whole the former forces constitute what has sometimes been called, not quite properly, an equilibrating system : they produce no change or the total momentum in any direction, but they produce relative displacements of the parts. Hence, in estimating the effects of the forces in changing the momen- tum of the system, we may disregard the whole group of internal forces. Not so, however, when we consider the work done by the different individual forces. The works done by the equal and opposite forces between a pair of particles do not necessarily give a zero sum. For example, consider two particles united by a stretched band of india- rubber. Neglecting any force necessary to set the matter of the band in motion, or to change its motion, we see that there are equal and opposite forces applied by the band to the particles, on which act also in general other forces. Let each particle be displaced towards the other, one particle, A, a distance a, and the other, B, a distance h. If F be the force on A, —F is the force on B. The dis- placement a is in the direction of F, the other, h, in the opposite direction, and therefore ought to be reckoned a negative displacement. Whatever the other forces do, F does work Fo,, —F does work (—F)x{ — b) = Fb, and the whole work done by these two forces is F(a + b); and so, even if the displacements were equal, which they are not necessarily, the work of these forces would not be zero, but 2Fa. Or, to take an example from the dynamics of extended bodies, two carriages of a train are in contact by their buffers. If one carriage is urged against the other, as for example in stopping the train, the buffer springs are compressed in opposite directions, but the work done in compressing one spring has the same sign as the work done in compressing the other, and the two quantities of work must be added together. If, however, instead of an elastic band between two particles, we had a connection of invariable length, then whatever small displacement parallel to the length of the link one end sustained, would have to be accompanied by an equal displacement of the other end in the same direction. Hence, if equal and opposite forces were applied §§63,64] WORK AND ACTIVITY. 107 by the link along its length to the particles, work would be done on the particle at one end by the bar, and by the particle at the other end on the bar, and these works, having opposite signs and the same numerical value, would cancel one another. We have therefore to distinguish in considering work done, not between internal and external forces, but between forces which do work and those which do none — between active forces and inactive forces. Denoting the components, parallel to the axes, of the force F acting on a particle of the system by X, Y, Z, and supposing the particle to sustain any small displacement of components Sx, Sy, Sz parallel to the axes, then the work done by F in the displacement is X8x+YSy + ZSz. If, similarly, all the particles are displaced, the work SW done is the sum of all such expressions as that just found, that is SW=I,iXSx+YSy+ZSz). In the sum on the right no component of the inactive forces appears, since each of these must appear twice in equal and opposite contributions to S W. 64. Constant and Varying Constraints. The displacement (8x, 8y, Sz) of the specimen must be such a displacement as the conditions of the system, as they exist at time t, permit. With this restriction it may be any displacement that can be imagined. It is therefore called an arbitrary displace- ment. In their motions the particles may fulfil conditions of constraint, which may or may not be expressed by equations. For example, the particles may constitute what is called a rigid body, that is they may fulfil the condition of invariability of their relative positions and distances, however the body which they compose may be displaced or turned. This condition is not directly expressed by equations, but only, as we shall see later, gives a certain form to the equations of motion. It is to be noticed that the system may be under varying conditions of constraint, so that at the time t+dt, the conditions may have changed from those which held at time t. Thus the arbitra/ry displacement Sx, Sy, Sz, though possible under the conditions which hold at the instant t. 108 A TREATISE ON DYNAMICS. [CH. 11. may be a displacement which the system cannot actually sustain in its motion. Whatever the conditions of constraint may be, their fulfilment involves the application to each particle of forces of constraint, over and above the forces which are applied by external bodies, or by particles of the system so distant from any particle considered as not to have any influence on its constraint. We shall return to this point in the chapter on General Dynamics. 65. General Variational Ectuation of Work. Theory of Energy. The equations of motion of a particle give l,{m{xSx+y8y+zSz)} = ^{XSx+Y8y+ZSz) (1) This is not a mere identity, for it is to be observed that the components of acceleration of every particle appear on the left, while all the corresponding forces for each particle do not appear on the right. The inactive forces have disappeared, those applied from the outside, ea/ih by itsdf, on account of its zero amount of work, and those mutual actions within the system which do no work, in pairs. But it is not to be forgotten that when we have to find the motion of a particular particle, all the force on that particle, whether of external origin or arising from the constraints to which the system is subjected, must be taken account of. So far we have considered only an arbitrary displacement (Sx, Sy, Sz); now let the displacement considered be the actual displacement sustained in the interval dt by the specimen particle in the motion. Thus, instead of Sx, Sy, Sz, we have components of displacement, which we shall usually denote by dx, dy, dz (reserving the symbol S for arbitrary changes) an'' which have the values xdt, ydt, zdt respectively. Then (1) becomes, if we denote active forces by Xa, etc., 'L{m{xx+yy+zz)] = l:{Xa!b+7ay+Z^z) "'■ |^2;{m(*H2/Hi^) = S(X„*+F„y+^„i) (2) The expression |2{m(a;^ + 2/^ + i'^)}, §§64,65] THEORY OF ENERGY. 109 of which we have the time-rate of variation on the left- hand side of the last equation, is called the kinetic energy of the system. We shall usually denote it by T. In consequence of possessing kinetic energy, the system can do work on other bodies, losing, in whole or in part, its motion in doing so, and the work so done will, as can be seen by (2), be equal to the work done on the system in building up the kinetic energy ; so that the kinetic energy is a real and useful equivalent of the work done in creating it. , In a considerable number of cases, indeed in almost all those with which we have to deal in nature, the expression on the right of (2) is derivable from a function V of the coordinates of the particles in the following manner. Let V be such a function, if one exists, that -dV=^{Xa,dx+Yady + Zadx) (3) Here — c?F is understood to be a. perfect differential of a single-valued function of the coordinates of the particles, or of a sufficient number of them for the specification of the work done in the displacements considered. The meaning of a perfect differential is explained in Gibson's Calculus, §§ 94, 165 ; but it is important to remark that, for all displacements for which V thus exists, the work done by the forces Xa, Y^, Z^, in the transference of the system from one given configuration to another, is independent of the paths followed by the particles in the passage, that is the excels of the initial value of V above the final value depends only on the initial and final coordinates. From (3) we obtain -'^=i:(X,x+Yay+ZaZ), (4) so that (2) becomes, when T is written for the kinetic energy, |(^+^) = 4 (5) or T+V=h] where A, is a constant. V is usually called the potential energy of the system and T+ V its total energy. Here the system is supposed no A TREATISE ON DYNAMICS. [CH. II. to be taken large enough to include all the bodies effectively acting, so that the forces concerned are only internal forces ; the kinetic energy is also in strictness that of all the bodies of the system. In some cases, for example that of a stone falling to the earth or a planet moving under the sun's attraction, the changes of motion of "the larger body — the earth in the former case, the sun in the latter — are so small that the corresponding variations of the kinetic energy is left out of account, and we refer to the kinetic and potential eiiergies as of the stone or the planet ; but this reference to only one of the bodies is not quite just, and the results, though accurate in a high degree, are not absolutely correct. If the system is not uninfluenced by other systems, and also if all the forces are not related to the potential energy, we may be able to refer part of the sum on the right of (4) to the potential energy of the system under consideration, while leaving the remainder under the sign of summation as above. Thus we may write ^{X^dx+ Y^dy+Zadz)= -dV+l.{X'dx+ Y'dy + Z'dz) + X(Xedx+Yedy+Zedz), ....(6) where X', ¥', Z' are the components of an active force which exists within the system, but has no relation to the potential energy, and Xg, Ye, Z^ are components of force on a specimen particle exerted by matter outside the system. We obtain ^^{T+V) = liX'x+Y'y + Z'z) + i:(X,x+Yy + Zj),(7) where T and V on the left refer to the limited system under consideration. Thus we see that the rate of increase of the energy of the system is equal to the rate at which work is done on the system by the forces which arise from matter outside the system, by external forces as we call them, and by the forces, if such there be, which exist within the system and are unrelated to any energy-function. As a rule, no forces of the latter kind, except frictional forces (§67), at present excluded, have to be taken account of. A system on which external forces do not act we shall call a self- contained system. ,.(2) §§65,66] THEORY OF ENERGY. Ill It is here assumed that no frictional forces exist: they will be found dealt with in § 67 below. They always resist the relative motions of the parts of the system, and so diminish the energy. 66. Forces as Derivatives of Potential Energy. The expres- sion for F as a rule will not contain the coordinates of all the particles of the system, but as usually known will suffice only to enable the forces on certain parts, into which the system is divided, to be found. For such parts of the system the forces which are derivable from the function V will be found by the relations Z.-l^, F=-f. ^=-1^; (1) dx dx dz ^ ^ and in the most general case we shall have dV This is to be regarded as a specimen set of forces acting at a point x, y, z. A similar set is to be regarded as existing for each part of the body, and the coordinates in each case are those of the point at which the forces are regarded as applied — the point of application of the force. The differential coefficients —dV/dx, ... are partial, that is the differentiations are carried out in each case with reference to the variable (x, say) indicated, supposed appear- ing in the expression for V either explicitly or through given functions of the coordinates, while the other variables (y and z) are kept unvaried. If he has any difficulty, the student should here read §§ 89-91 of Gibson's Calculus. It will be noticed that if we write T in the form ^'E('mv^) we have for any coordinates x, y, z, when there are no external forces and none underivable from V. ^r^ash-^' ^rv^"v ^r^3^h-3j- ^^^ Here the v of each part of the equation is regarded as a 112 A TREATISE ON DYNAMICS. [CH. II. function of the coordinates of some or all of the parts, in which, if the variables were made explicit, we should have T equal to the function — F of the coordinates together with the constant h. It is to be remembered that the forces thus obtained are those only of the field of force in which the part considered is placed, and have nothing to do with the reactions of fixed guides or with other inactive forces. The following are examples of partial diiferentiation : Ex. 1. F=/i/\/^+pT^- This is the case of a repulsive force directed from the origin towards the p oint x, y, z, and varying inversely as the square of the distance r=\/j?+y+^. We have 'dx~ 'fi~ r^' 'dy~ "fi" r^'' 'dz r^ /•"' as might hav3 been written down at once from Euler's theorem [Gibson's Calculus, § 158, 2]. so that ^^+3/ +3^=°- Ex. 2. ?'=^(r2+r^^^). This is the kinetic energy of a planet of mass m, when at a point in its orbit for which the radius-vector is r and the vectorial angle d. The speed along the radius-vector is r, and the speed at right angles to the radius- vector is rO (see § 11 above). We have or ^ or 2)6 The coordinate 6 is in this case absent from the expression for the kinetic energy, but if it had been present the fact that r for every point of the path is a function of 6 would not have affected the differentiation with respect to r. The student should notice that here that is the rate of change of momentum in the direction outwards along the radius- vector. The expression on the left belongs to a theory which we shall explain and illustrate later. Again, to illus- §§66,67] DISSIPATIVE FORCES. 113 trate the distinction between partial and total differentiation, take d(m/r^d)ldt. We have -^ (rrvfiQ) = 2mrrd + mrW. This is the rate of change of angular momentum of the planet about the origin, which must vanish if no force transverse to the radius- vector act upon the body. 67. Work spent in overcoming Friction. Dissipative Forces. So far we have supposed that frictional resistances to the motion of the system do not exist; and the theory of energy explained above is not applicable without correction to systems in which friction is present — dissipative systems as they are often called. For a long time it was supposed that work done against friction — unlike that done against inertia-resistance — was without equivalent ; but the experi- ments of Joule have shown that when work is so done an amount of heat proportional to the work expended is generated ; and the dynamical theory of heat, which was worked out mainly during the latter half of the nineteenth century, proves that under certain ideal conditions the heat so generated can be made to do an amount of work equal to that expended. Thus the heat generated is the energy- equivalent of the work done in overcoming friction. The laws of friction are stated in § 201. The equations written above can be modified so as to include frictional or dissipative forces. Let, as before, Xa, Ya, Za be the component forces actually applied to the particle chosen for consideration, and Xy, Yf, Zp be the frictional or dissipative parts of these, and so for other particles. Then, for the system, we have S{m(,« 8x+y Sy+zSz)} = 2(Z<, Sx+ Ya Sy+Za Sz) -2(XJx+Y^Sy+Z^Sz), ..."(1) where only the active non-frictional forces are included in the first expression on the right. If, now, we can write, as before, 2{Xadx+ Yady + Zadz)= -dV+2{Xedx+ Y^dy+Zedz), where dx, dy, dz are the components of the actual displace- ment of the system in the element of time dt, and —c^F is 114 A TREATISE ON DYNAMICS. [CH. II. a perfect differential of a function of the coordinates, we have ^(r+ F) = 2(Z,x' + Y,y + Z,z) - ^{Xjx + Y^y + Zjz). . . .(2) On the right we have first the rate at which the energy of the system is being increased by the action of external systems, and in the second line the rate at which the sum of the kinetic and potential energies of the system is being diminished by the dissipative forces. If forces of the sort referred to in § 65, and denoted there by accented letters, exist, a term must be included, as there explained, to represent their activity. The differentiation on the left with respect to t is total, that is it includes the rate of change of the quantity differentiated, arising through the rates of change of the coordinates, as well as the rate of change (if such there be) due to the explicit appearance of t in the expression of the quantity. 68. Meaning of Solution of a Dynamical Problem. It is to be remembered that the solution of a dynamical problem consists in expressing the coordinates which determine the configuration of the system at any time as explicit functions of the time and of the initial coordinates and the initial velocities. The simple result expressed in (7), § 60, is an example in point. The function F has been assumed to be an explicit function of the coordinates only; but it may also be an explicit function of the time t, as well as of the coordinates for that time. This more general case will be dealt with later. (See Chapter XI., where the integration of the equations of motion of a material system is more fully considered.) 69. Angular Momentum. Rotational Motion. It is con- venient to consider here another application of the laws of motion, namely to the motions of the particles of a system about a straight line, or axis as we shall call it, given in position. In the first place, let a single particle P of mass m be moving at the instant considered along the line PQ, in the plane of the paper with speed v, and let be the point in i'67, 68, ( ANGULAR MOMENTUM. 115 Fig. 26. which an axis at right angles to that plane intersects it. If p be the length of the perpendicular let fall from on PQ, the product mvp is called the Tnoment of nnoTnentum, or the angular wxymentwrn of the particle about the axis. Taking first the speed v and the perpendicular p as both positive, we attach the positive or negative sign to the product according as the radius-vector OP appears to an observer, regarding the motion as here shown to be turning as in the diagram (Fig. 26) with the motion of P in the direction in which the hands of a watch appear to turn, or in the contrary direction. The product mtyp is twice the rate of description of area by the radius-vector just refer- red to multiplied by m. For let the particle go from P to Q in time dt, then the radius- vector sweeps over the area of the small triangle POQ, which is clearly l^pv dt by the diagram. Hence vp is twice the rate of description of area. Now at P resolve the velocity into two components in the plane of the paper — we suppose for the present that there is no component perpendicular to the paper. Let the aj-component be x, the ^/-component y. The student can easily convince himself from the diagram that by the con- struction there given, area POiJ- area PO*Sf=area POQ, that is that 7n{yx — xy) = mvp. Now myx is the angular momentum about the axis through at right angles to the plane xOy and due to the component velocity y, while mxy is that due to the component x, and the signs are chosen according to the convention stated above. If the axis be not, as it is taken here, at right angles to the direction of motion, we resolve the momentum into two components in a plane containing the line of motion 116 A TREATISE ON DYNAMICS. [CH. II. at the instant and parallel to the given axis, taking one component parallel to the axis, the other perpendicular to it. The angular momentum of the particle about the axis is now defined as the product of the latter component of momentum into the distance of the axis from the plane just defined. 70. Components of Angular Momentum (A.M.). Let a, b, e be the direction-cosines of the axis, which we suppose as above to pass through the origin, and x, y, z be the components of v parallel to the axes Ox, Oy, Oz. Then the direction-cosines of a normal to the plane parallel to the axis and containing the line of motion at the instant ^'^^ (cy — bz, az — ccb, bx — ay)lvsva.6, where Q is the angle between the directions of the axis and the line of motion. If x, y, z denote the coordinates of P (or indeed of any point in the plane just referred to), the distance of the origin from the plane is {{cy — bz)x + {az — cx)y + {b!i; — ay)z) jv sin 6. But the component of momentum at right angles to the axis is m/VBYsiQ, and hence the angular momentum, as defined above, m {{cy — bz)x+{az — cx)y + (bx — ay)z}, which may be written as a {7n(zy — yz)} + b {'m{xz — zx)} -f c {m,(yx — xy) }. Clearly this may be regarded as the result of resolving along the given axis (direction-cosines a, b, c) three com- ponents, m(zy — yz), ... , of angular momentum associated with the axes Ox, Oy, Oz respectively. In point of fact they are, as the student will see from §69, the angular momenta of -the particle about these axes. We shall denote them by F, Q, H. If we measure, from along Ox, Oy, Oz, distances representing F, 0, H, and project these upon the given axis through 0, we obtain a distance along it which represents the angular momentum about it. The distance for each component is drawn in the positive or negative §§69,70,71] ANGULAR MOMENTUM. 117 direction from the origin, according as to an observer, looking towards the origin from a point on the positively drawn axis, the turning of the radius OP, drawn from the origin to the projection P^ of P on the plane at right angles to the axis considered (in the diagram the axis Ox), is against or with the turning of the hands of a watch held in the plane with its face towards the observer. Thus we obtain a vector through representing the angular momentum about the given axis by its direction and its length. The resultant angular momentum of the particle is {F^+G^+H^y, and the direction-cosines of the axis are {F, 0, H)l{F^-\rG'' + mf^ The axis of resultant angular momentum, K say, for the chosen origin, passes of course through the origin, and the angle it makes with the given axis is i}> = cos-^[{aF+hO+cH)l{F'-+Q^+m)^) = cos-''K'IK, (1) if K' denote aF+bG + cH, the angular momentum about the given axis. Thus K' = Kcos^ (2) 71. Angular Momenta about Parallel Axes. Now consider how K and K' are affected by a change of origin to a fixed point 0' of coordinates h, k, I. The old x, y, z are to be replaced by their values in terms of the new, namely x-\-}i, y + k, z + l, while x, y, z remain unchanged. We have now for the given axis through the old origin, K' = a {m,{zy — yz)} + b {7n(xz — zx)) + c {m,{yx — xy)} + a{7n{zk — yl)}+b{nfn{id — z}i))+c{'m{yh — xk)]. ...(1) The expression in the first line is the angular momentum about a parallel axis through the new origin, the expression in the second line is the angular momentum about the old axis of a particle of mass m situated at the new origin and having components 7nx, Tny, irnz of momenta. A similar conclusion holds for the resultant angular momentum K. This theorem has important applications in the case of a system of particles, as we shall see later. 118 A TREATISE ON DYNAMICS. [CH. II. The student can easily prove that if 0' lie on the given axis the expression in the second line identically vanishes. For any system of particles in motion in any manner the angular momentum is obtained by summing for all the particles of the system expressions of the form just obtained for a single particle. We have, simply, F='E{m(zy — yz)}, G = 1,{m{±z — zx)}, H=-E{m(yx-xy)} (2) with [see (2), §70] K={F^ + G^+H^)^, K' = aF+bG+cH. (3) Equation (1) shows that if, instead of the axes drawn from the fixed origin 0, we take parallel axes drawn from another fixed origin 0', the coordinates of which are h, k, I, and X, y, z now denote the coordinates of a representative particle with reference to the new axes, K' = a\l,{m(zy — yz)\'\-\-l>\^{'m{xz — zx)\'\ + c[2{m(2/a;-i;2/)]} + a{/<;S(mi)-ZS(m2/)}+&{E(m*)-/i2(mi)} + c{/(,S(m2/)-/(;E(ma;)}. ...(4) The first part on the right is the angular momentum of the system about a parallel axis through the new origin 0', the second part represents the angular momentum which the system would have about the given axis through 0, if all the particles could be, and were, transferred without alteration of their component velocities to the new origin 0' ; or, which is equivalent, it is the angular momentum, about the given axis, of a single particle situated at C, and moving so that its component momenta are equal to E(ma;), S(mi/), 1,{mz). If ^, y, z be the component velocities of the centroid and M denote the total mass of the system, we have (§ 60) ilf5 = 2(m«), My = '^{my), Mz = '2{7nz). Hence, whatever point 0' may be, if we suppose placed there a single particle of mass equal to the total mass of the system, and §§71,72] ANGULAR MOMENTUM. 119 having the component velocities of the centroid, the angular momentum of this particle about the given axis added to that of the system about the parallel axis through 0', makes up the angular momentum of the system about the given axis. The point 0' here considered is, like 0, at rest; if it is in motion, then h, k, I are variable as well as x, y, z, and the component speeds for a particle are no longer x, y, z, but x + h, y + h, z+l. Equations (1) and (4) must then have terms added depending on h, k, I. These are a{il.{my) - kl,{mz) + M(lk -k)} + .... If 0' coincide then with the centroid, S(m»), l,(viy), S(mi) are now the momenta relative to axes through the centroid, and vanish by § 60 ; so that all the terms in the second line of (4) disappear, and the angular momentum is represented by the first line and the additional terms just indicated. Hence, since h, k, I are then x, y, z, and 1,{mx) = 1,(my) = 1,(mz) = 0, we get for an origin at and moving with the centroid, K' = a'E{m{zy -yz)} + ... +aM(zy -'yz)+ (5) and K-^ = ['2{mizy-yz)}+M{zy-yz)Y+ (6) 72. Rate of Change of A.M. Since, when there is no alteration of the mass of the system, d j^{'m{zy-yz)] = ^[m{zy-yz)}, (1) we have for the rates of change of angular momentum relatively to the axes with fixed origin 0, ^='2{')n{zy-yz)}, -^ = 'E{m{xz-zx)}, -^ = 'E{'m(yx-xy)} (2) If we transfer to another fixed origin 0', as before, we get for the old axes, if x, y, z be now the coordinates of a 120 A TREATISE ON DYNAMICS. [CH. II. representative particle, relative to the new axes, ^ = 2{m {zy - yz)} + lcL{mz) - i2(m^), ^=^{m{xz+zx)]+l1,{mx)-K2{mz), *• (3) ^ = 2{m(^a; -xy))+ hl{my) - /cS(m«) ; and finally, when the origin 0' is the centroid and moves with it, the values of the components of angular momentum about axes at 0, which has now the coordinates {h, k, t) = (x, y, z) relatively to the centroid, are, J EI ^ „_ -^ = S{m(% -yz)}+ M(zy - yz), da dt = E{m(a32! —zx)} +M{xz — zx), .(4) JIT -^ = I,{m(yx-xy)} +M(yx-xy), since the terms in h, k, l, arising from the motion of the centroid, are identically zero by the property of that point. 73. Rate of Change of A.M. when Effective Inertia different in Different Directions. The cancelling in dFjdt of the term {'Lnnzy) by the term 'L{myz) in the differentiation of ^{'mz)y — yz)} is worthy of a little attention. 'E{'mzy) is the angular momentum of the system about the axis Ox, arising from the motions of the particles parallel to the axis Oz, and l!,{7nzy) is the rate of growth of this angular momentum arising from the rates of change of the ^/-coordinates. Similarly, —1,(inyz) is the angular mo- mentum about Ox, arising from the motions of the particles parallel to Oy, and —1,(7nyz) is the rate of growth of this arising from the rates of change of the ^-coordinates. In ordinary circumstances these two rates of growth cancel one another, but there are cases of motion in which it is convenient to ascribe different inertias in different §§72,73,74] ANGULAR MOMENTUM. 121 directions to the body, or bodies, composing the system. The motion of a medium in which a body is immersed resulting from the displacement of the body may thus be taken account of. For example, we may, in explaining certain effects of the motion of the water, conveniently consider a ship as having a larger inertia for displacements at right angles to its length than it has for displacements along the fore and aft direction. An example of this kind is considered in detail in § 91 below. Thus, if the system consist of a single body moving without rotation in a medium, with component speeds x, y, z, it may be con- venient to regard it as having momenta M-^x, M^y, M^z, parallel to 0^, Oy, Oz- In this case we should have for the rate of growth of angular momentum about Ox, with similar expressions for the other two axes. 74. Bate of Change of A.M. when Body Gains or Loses Mass. It may be that the system ig gaining or losing mass. Thus the earth is constantly receiving meteoric matter from space, and (if we distinguish here between the earth and the atmosphere) gaining matter also by condensation of water- vapour from the atmosphere, and losing matter by evapora- tion from the surface of the sea, lakes, and rivers as well as from the surface of the land. Thus, if dnn/dt be the rate of growth of mass at any point, and if the mass gained there takes up speeds x, y, z, the total rates of gain of angular momentum from this cause are 2{^(zy-yz)], ^[^-(xz-zx)], •^[^(yx-xy)), where of course dm/dt may be either positive or negative, or positive at some places, negative at others. Taking these into account, we have ^=-2{mizy-yz)} + ^[^(zy-yz)] (1) with similar expressions for dG/dt, dHjdt. A case in point is that of a chain wound on a horizontal cylinder, or windlass, which is turning so as to unwind the 122 A TREATISE ON DYNAMICS. [CH. II. chain. Let the free end be attached to a fixed point close to the windlass so that the chain which is not on the barrel hangs down in two vertical parts connected by a short bight at the lowest point. Every element of the vertical part attached to the barrel is moving downward with speed v, and if we suppose all the moving chain on the barrel and attached to it to be at distance r from the axis, itf to be its mass, and d the angular speed of the barrel at any instant, the angular momentum of the chain about the axis is at that instant Mr^O- But chain is continually being unwound and successive elements pass from the side attached to the barrel to that attached to the fixed point, and as each element passes across the bight from the moving side to the other, it is brought to rest. The rate of transfer of mass is half the rate at which it is unwound from the barrel, namely \mrQ, if m, be the mass of the chain per unit length. Hence the rate of loss of angular momentum from the moving chain, in consequence of the transfer, is ^Tnr^Q^, being the rate of loss of mass ^imrQ multiplied by r^Q, .the moment about the axis of the speed rQ of the chain. This problem will be found fully solved in § 77 below. If the matter added to the system brings with it angular momentum, or if matter on being added or removed acts on the system (as, for example, does a jet thrown from a reaction turbine or from a hose or fire-engine), the rate of addition of angular momentum brought with the matter, and the moments of the reactions, must where necessary be entered on the other side of the account, that on which the actions producing rate of angular momentum appear. These we now go on to consider. 75. Rates of Change of A.M. equal to Moments of Forces. Independence of Motions of Translation and Rotation. Going back to the equations of motion of a specimen particle referred to any rectangular axes, say those through 0, m,x = X,-\-Xu ... , (1) where the sufiixes distinguish, as before, the internal forces of the system from the external forces on the particles, we §§74,75] ANGULAR MOMENTUM. 123 multiply the 2:-equation by y and the ^/-equation by z, and subtract the second equation from the first. We get m{zy-yz) = Z,y-Y,z+Ziy-YiZ (2) Doing this for all the particles and adding, we obtain ^{m{zy-yz)) = ^{Z,y- Y,z)+'E(Ziy-Y,z). ...(3) The products on the right are mcmients of forces about the axis of X. We suppose now that the equal and opposite internal forces between the two particles of every distinct pair in the system act along the line joining the particles. Then the pair of forces obtained by the projection of such a line, with the forces acting along it, on any plane, must obviously be a pair of equal and opposite forces. Thus, projecting all the pairs of forces on the coordinate plane yOz, we see that 1,{Ziy— Yiz) = 0. For the Yi and Zi obtained at the particle at one extremity of such a line have, taken together, the same moment about the axis of x as the force F, of which they are components, along the line, and in the same sense the forces —Yi, —Zi at the other end are equivalent to the force —Fm. the same line which acts on the other particle. Hence, extending this process to all three coordinate planes, we obtain 2{m(% - yz)) = X(Zy - Yz), | I,{m{xz-zx)} = 'E(Xz-Zx), [ (4) ^{m{yx-xy)} = -2(Yx-Xy)J where the suffixes are dropped on the right on the under- standing that only external forces are there included. With these are the equations of motion of the centroid Mx = I,X, M'y = I,Y, Mz = '2Z. (5) Now (§ 60) it has been seen that I,{mx) = Mx = I,X, I,(my) = My=;'2Y, S(mg) = i¥l = 2^. Hence, M(iy-^z) = y'EZ-z-LY, (6) with two other exactly similar equations for the axes of y and z. These show that the moments about any axis of Mx, M§, Mi, or the sums 1{mx), Himy), I,{m.z) transferred without change to the centroid, of what are often called 124 A TREATISE ON DYNAMICS. [CH. II. the effective forces, taken parallel to the axes, is equal to the sum of the moments about the same axis of the externally applied forces, supposed all similarly transferred to the same point. In (4) X, y, z denote the coordinates of a specimen particle relative to axes drawn from any fixed origin 0. If parallel axes be set up from the centroid as origin, and x', y', z' denote the coordinates of a particle relative to these axes, we have x — x-\-^, y = y + y', z = z+z', and the left-hand sides of (4) reduce to 2 {m{z'y' - y'z')) +M(iy- ^z), . . . ; for all terms of the form S(mz'y), l^imzy'), . . . are zero, since S(mz') = 0, Jl(m,y') = 0, ... . Thus, by (6) we get, dropping accents on the understanding that the axes are at the centroid, equations of precisely the same form as (4), where, however, x, y, z now stand for x', y', z\ On the left in these are the sums of moments, relative to axes through the centroid and carried with it, of the rates of change of momentum of the particles of the system relative to these axes, and on the right are the sums of moments about the same axes, of the external forces, taken exactly as they are applied. These equations are of great importance, as they enable the rotations of bodies, or systems of bodies, about axes through the centroid of the body or system, to he dealt with as if the centroid were at rest. This property is peculiar to the centroid because of the vanishing of S(ma;), 'L{mx), ^Limx), ... when x, y, z refer to the centroid as origin. Again, equations (5) are the equations of motion of a particle, the mass of which is equal to the total mass of the system, to which are applied all the external forces without change of magnitude and direction. The motion of the centroid is thus reduced to that of a single particle, and may be discussed without reference to the relative motions. The two properties, stated in the last two paragraphs, are sometimes referred to as the principle of the inde- pendence of the motions of translation and rotation. To take account of the change of mass of the system §§ 75, 76] ANGULAR MOMENTUM. 125 we must use (4) for the given axes (at the chosen origin let us suppose), with the modifications referred to above : that is, we should write l{m(&y-yz)} + ^\^(zy-yz)\ = -EiZy-Yz) + I>F, ...(7) with two similar equations. Here DF, DQ, BH denote the rate of change of angular momentum produced in each case by the matter added or removed. 76. Rigid Body, Rolling Motion of. A rigid body is defined at the beginning of Chapter VI., to which the student may refer. It is an aggregate of particles so connected that the line joining any two remains unaltered in length, and the angle between every pair of such lines remains unchanged as the body moves. But this definition is violated continually by bodies which we class as rigid : they expand and shrink with heat and cold ; in some cases they gain and lose mass, and so we obtain the idea of a body which moves at a given instant as a rigid body, but which is changing in some respect or other as time passes. For many problems regarding such bodies it is convenient to express equations of the form (7), § 75, in another way, that in fact shown in (4) below. Thus, let the body be turning about the axis of x, for example, and Q be the angle which a line fixed in the body, through the centroid, for example, and at the instant parallel to the plane of yz, makes with the axis of y. All the perpendiculars from the points of the body to the axis of x turn at the same angular speed d at the same instant in the same direction about that axis. If 6^ be the angle which the perpendicu- lar, of length r^, from a particle m, the coordinates of which are x, y, z, makes with the axis of y, we have i/ = rjcos0j, z=r■^^smQ■^, (1) and therefore zy — yz = T\6^ = r[Q. Therefore, for the whole body, 2{m(%-y2)} = 0S(mr2) (2) The quantity S(inr^) is called the Tnovient of inertia of 126 A TEEATISE ON DYNAMICS. [CH. II. the body about the axis of x. The properties of moments of inertia and their calculation are discussed in Chapter VI.; the values of E(mr^) for the bodies referred to in the examples will be stated. They will be expressed in the form MW, where M is the whole mass, and W is such a quantity that E(77ir^) = ilf/c^. The kinetic energy T of the motion of rotation about Ox is |S{m(^2 + i;2)}, and by (1) can therefore be expressed by the equation T =\Q''^{m,r') (S) Now, in calculating the rate of change of a.m. about a given axis, we have, as a general rule, cases to consider in which 2('»ir^) is constant, so that the a.m. only varies through 0, and not at all through variations of mass or its distribution. But in the general case we have, for the rate of change of a.m., ^{0E(mr2)} = Q-^i^mr') + — {2;(mr^)}. This we have to equate to the sum of the moments of the external forces about the axis, together with any rate of increase (or diminution) of A.M. directly due to action between the body and external matter, such as the inter- change of mass bringing or carrying with it a.m. We get, for the moment of the forces, Jl(Zy—Yz) = 1,(Pp), where P is the resultant of T, Z, and p is the perpendicular distance from its line of action to the axis of x. Hence we have d . j^{eT{mr^)} = l{Pp)+DF. (4) Similar equations hold for the other two axes. As examples of the value of DF in ditferent cases, we may take the following : (1) a cylinder rolling on a horizontal or inclined plane, and expanding or contracting without varia- tion of mass, so that DF=0; (2) a cylinder of ribbon (Exs. 3 and 4, § 77) rolling along a horizontal or down an inclined plane. The " rolling motion " referred to in some of the following examples is that combination of turning and translatory motion which enables a wheel to move along a rail with- out any slipping at the contact. When the axle of a wheel §§76, 77J EXAMPLES OP ANGULAR MOMENTUM. 127 is at rest, the top of the wheel' moves forward, and the bottom backward with the same speed v. If now the axle be carried forward with speed v as the wheel turns, the forward speed of the part which is the top at the instant in space will be 2v, while the speed of the part which is at the bottom at the instant will be zero. This is the motion of the wheel in space when it rolls without slipping along the rail. If the radius of the wheel be r, the centre moves forward a distance rd, when the wheel turns through an angle 6, and the forward speed v is r9. 77. Examples on A.M. of Bodies of Varying Mass. Energy- Ex. 1. If the radius of the earth is diminishing as time advances, find the effect of the contraction on the length of the day. The radius at time t will be ro(l -at), if we assume that the contraction is proportional to the time. Hence, since the a.m. then must be equal to the a.m. at time t = 0, we have JI/5?'„(1 - atyu> = M^riju}iy.* Hence, if a be small, we have nearly. The daj' is therefore shortened in the ratio of 1 to 1 + 2at. Ex. 2. A layer of cosmic dust of thickness k, small compared with a, the radius is deposited on the earth's surface ; show that if the dust brings with it no a.m. about the earth's axis, the change in the length of the day is nearly 5hp/aD of a day, where p and D are the densities of the dust and the earth respectively. Here we must have, since there is no moment of forces changing the earth's a.m., exerted by the dust, and no a.m. brought with it, (§ira^I>la^ + 4TraVipla^)b) = ^a^Dia''u)o* since the moments of inertia of a uniform sphere of mass M, and a uniform spherical shell of mass m are M'^a^ and ■m,§a' (§ 175, Ex. 6). This is equivalent to d{MJfiu>)ldt=DF=0, and gives (1 + 5^^ £). = .„. Thus the angular speed of the earth is diminished, and the day is lengthened in the ratio l+5hpjaD to 1. *It is here assumed, what is far from being exact, that the moment of inertia can be calculated by assuming the density uni form throughout and equal to the mean density. 128 A TREATISE ON DYNAMICS. [CH. II. Ex. 3. A roll of cloth of small thickness h, lying at rest on a horizontal table with the edge of the cloth along the line of contact, is propelled with initial angular speed fl, so that the cloth unrolls. If friction brings the cloth to rest as it comes into contact with the table, show, on the supposition that no work is done by the rolling cloth against friction, or against cohesion of the folds or stiffness of the cloth, that the radius of the roll will diminish from a to r in the time where ^{, and since, as will be shown below, the centroid of the roll is at a distance Sh/2Tr in front of the line of contact with the table, gravity forces have a moment ■irr'g3h/2ir about that line. We find next the acceleration of the centroid 6-' of the roll regarded as an unchanging body. Take a section S perpendicular to the length of the roll through G and consider any point P of S. Take a point ff of the section at the same distance r from the table as G, but on the normal to the table through the line of contact. Let HP=p, and make an angle 6 with the upward vertical. P has coordinates ^, -n with reference to a fixed origin on the table in the plane of S at distance x behind the line of contact, given by ^=x+psai9, ri=r+p cos 6. Hence ^=^ + pa) cos 6 - poj^sin O^ij—- p(i> sin 6- pio^cos^, since 9= (a. But }i = rih, and if P be coincident with G, p = 3A/27r, 6=Tr/2. But, as will be seen presently, h= -^Trf/w, and therefore for G i = ru> + 3rw, 7J= -3kii/2Tr. The r.c.a.m. is therefore since rj has moment - 9A%/4jr^, which may be neglected. The equation of motion (7) of § 75 is therefore ■ which can be written f Trr^o) ^ (r^w) = ^ghr'^ta = - Zirgr^'r, since - ^irflh = u>. Integrating and assigning the constant of integra- tion to suit the initial circumstances, we get We can verify this equation by the method of energy, which, on the supposition made above, is here applicable. The potential energy of the roll relatively to the table is irga? initially and Trgi^ at time t. §77] EXAMPLES OP ANGULAR MOMENTUM. 129 The kinetic energy is fwa^Q,^ at starting and firr*b)^ at time t. Hence, equating the gain of kinetic energy to the loss of potential, we get the equation to be verified. Now, if s be the length unrolled at time t, we have «= -2Trfjh = io. Thus the equation found above becomes and therefore if 4(c' - a^)ff = SQ,^a*. Hence, since r is negative, TT dr^/dt= - h'JZg{c^—fi), which gives by integration the result to be established, To prove the statement made above as to the position of the centroid of the roll, we observe first that the cross-section of the spires of cloth is an equiangular spiral of angle a. very nearly 7r/2. Hence, as the distance of the normal at any point to which the radius-vector from the pole is of length r is r cos a., and the radius increases by \h in each half turn, we have by the equation of the spiral, (r=ae^°°^'^), |A=Trrcota, that is hj2ir=r cos cL, nearly. Again, if the roll were kept at rest and the cloth unwound from it, each half turn, at radius r, would shift the centroid a distance 'ikjir, since the distance of the centroid of a semicircle of radius r from its centre is irjir. Hence the centroid would oscillate from a distance hJTT from the pole on one side to an equal distance on the other side. Therefore the centroid of the bale of cloth when the radius is ?• is at a distance hjir in front of the pole of the spiral, and, as shown above, the line of contact with the table is at a distance A/2ir, on the other side. The gravity wr^g of the cloth has therefore a moment irr^g {h'jir + hlZir) = Zirr^ghj'iTr, as stated above. It is noteworthy that the solution by the method of energy would enable the position of the centroid of the spiral to be inferred from a comparison of the equation of motion with that obtained by differentiating the equation of energy. If the roll is allowed to run completely out, a question arises as to what becomes of the energy. The potential energy has been exhausted, and there is no kinetic energy. The last part of the roll running very fast will bring the free end round on the table like a whip, and there will be commotion of the cloth which will dissipate the energy. G.D. I 130 A TREATISE ON DYNAMICS. [CH. II. Ex. 4. A ribbon of very small uniform thickness h is coiled up tightly in a cylindrical form, and placed with its curved surface in contact with a plane inclined to the horizon at an angle a.. The axis of the cylinder is parallel to the intersection of the plane with the horizon, and the outer end of the ribbon is along the line of contact. Find the time in which the cylinder will unroll from radius a to radius r, comparable with a* The ribbon is prevented by friction from sliding on the plane. The equation of motion is, by the last example, firr^o) -rJr^di) = irr^g ( r sin a. + — cos a. j, since it will be seen, partly from the last example, that the moment of forces, about the line of contact of the roll with the plane, has the value irr^gir sill a.+Zh cos a.j'i.Tr). The equation can be written 37r(r2a) + Scdtt) = 9,irg Ir sin a. + ^ cos a. j, that is, multiplying by r^ta, Zirr^to -J-, {r^bs) = 2irgr^=ga-a»: But (i> = 2x/r, and therefore Ci = 2xjr. Substituting in the equation of motion, then multiplying by 2x and integrating, we get 2{Mlfi+(T{2l-w)r^}'^^=go-(a^ -a^^-^l' (n?dt, which, since x^/r^=(a^/4, is the equation of energy. On the left is the expression of the kinetic energy ; on the right g(r{x^ - ^) is the loss of potential energy ; and the term remaining, 2 I a-afidt, which is subtracted, is the energy dissipated at the bight. For the tensile force at the bight, on the side of the stationary part of the chain, brings an element of length cbdt to rest in time dt, thus annulling momentum a-Jb dt . 2x. The tensile force is therefore So-rf-^, and it works at rate 2o-.^. The kinetic energy is thus equal to the excess 132 A TREATISE ON DYNAMICS. [CH. II. of the potential energy lost over the energy dissipated, which verifies the equation of motion. The equation ia of the form {A-Bx)x-Cx=0, or, if multiplied by x and rearranged, .■■__C. CA X B ^ B A-Bx This gives, by integration, ^2= __(.^_.^„) + „ log ___.__, and therefore t=\ — , is the time of motion. Ex. 6. To examine the energy changes in Ex. 4, § 53. It is there shown that on the stationary side the tensile force at the bight is \tn)^=\ '"o) w^ along the fixed axes, we have to increase the values of u, v, w given above by «„, v^, w,, respectively. The student will easily make out what the kinetic energy becomes, and verify that if be the centroid Of the body, 134 A TREATISE ON DYNAMICS. [CH. II. we have simply to add to the right-hand side of (4) the quantity ^Miul+ifg+w^), the kinetic energy of a particle of mass equal to that of the body and moving with the centroid. This is the kinetic energy of the motion of translation. Equation (4) gives the kinetic energy of rotation. 79. Couples. Ectuivalence of Couples. The subject of Couples will be fully considered in the chapter on Statics. But it is necessary to introduce the notion here. If two forces be equal in amount but opposite in direction, the system is called a couple. It possesses the property of producing about any specified axis a moment which depen.ds on the direction of the axis, but not on its position in space. A couple has no effect on the acceleration of the centroid of a body on which it acts ; it has no single force resultant, and can only be equilibrated by the action of an equal and opposite couple, as we shall now prove. Consider axes of X, y, z through the centroid. Let x, y, z be the coordinates of a point A on the line of action of the force P of the couple which is in the direction given by the cosines a., (3, y, and x', y', z' be a point B on the line of action of the other force. The moment of the couple about the axis of x is F{yy-^z)-F{yy'-^z'), that is p{y^y-y')-^(^-z')). Hence, only the difference of coordinates y — y', z — '^ are involved, not their absolute values. Similar expressions hold for the other two axes, so that we have the three component moments P{y{y-y')-^(,z-z'\ a.(z-z')-yix-x'), /3{x-x'ya(y-y')}. These are equivalent to the single moment P[{y(y-y')-^(^-^)}' + {a(^ - 0') - y(a; -a;')}H {/3(a: - ^') - 0^(2/ - i/')P]^ about an axis the direction-cosines of which are proportional to the component moments just written, that is about an axis at right angles to the plane containing the forces. The multiplier of P in this expression for the resultant i 78, 79] COUPLES. 135 moment is simply the distance between tlie lines in which the forces act. For it can be written ii.oc-xJ + {y-yJ+{z-zJ -{a.{x-x')+p{y-y') + y{z-z')Y-\^., where the first term in the brackets is the square of the distance AB (Fig. 27), and the second is the square of Bh. The component moments are the moments of the couples obtained by projecting the two forces in succession on the coordinate planes of yz, zx, xy. We may therefore regard a couple as a vector defined by its moment Pp, the product of either force into the distance ^ between the lines of action, (2) its " axis," that is any line at right angles to the plane of the forces, and drawn towards that side of the plane on which an eye must be situated to see the direction of turning positive, that is counter clockwise. It thus indi- cates the aspect or orientation of the plane in which the forces are situated. When the axis taken in this direction is made as many units in length as there are units of moment Pp, it represents the moment as well as the orientation and direction of turning, and thus represents the couple in all respects. The moment of a couple about any axis, as already noticed, is independent of the actual plane in which the forces act; it is also independent of the magnitude of the forces, and of their direction, provided the orientation of the plane in which they act is given and the moment. Thus a couple can be changed from any plane to a parallel plane without change of its moment about any axis. The rule for finding the resultant of two parallel forces gives for a couple a zero resultant, but at the same time prescribes for it a line of action at an infinite distance from either of the forces. The interpretation of this is the fact, already noticed, that a couple cannot be balanced except P* KiG. 27. 136 A TREATISE ON DYNAMICS. [CH. 11. by the action of a couple of equal and opposite moment in the same or in a parallel plane. In fact couples represented by their axes can be compounded like forces, as the analysis just given indicates. This rule can be experimentally illustrated by a floating stand, on which act couples in different planes, applied by strings passing over pulleys and supporting weights. 80. Effective Inertia different in Different Directions. Case of a SMp. Consider a rigid body of mass M moving without rotation parallel to a fixed plane. Take axes Ox, Oy from any origin in that plane, and let x, ^ oe tlie speeds of the body parallel to these axes. The momenta of the body in these directions are Mx, My, and the body has angular momentum M{yx — xy) about an axis of z through the origin, since we may regard the body as replaced by a particle of mass M situated at the centroid (coordinates x, y) and moving with the velocity (*, if). The time rate of change of this angular momentum is M(^x-xy), which for the present we shall suppose to be zero, through the vanishing of x, y. Now let there be matter set in motion by the body, so that the total momentum in the direction of Ox is il-^, and that in the direction of Oy is M^y. Then if we associate these components of momentum with the body, we regard it as having inertia M-^, in the direction oi Ox, and a different inertia M2 in the direction of Oy. The angular momentum about the origin is now M^y^ — M-jXr], where ^, tj are the coordinates of a point, moving with the body, the position of which it is not necessary for our present purpose to specify. The rate of change of this angular momentum (since x=y=Q) is M^y^ — M^x'r} = {M2 — ifffiy, since ^==x, r)=y, and therefore does not depend on f, -q. Or, to put the matter in another way, consider a point A of space with which a point B of the body, or moving with the body, coincides at time t. By the displacement &dt of the body, in an interval of time dt, B is carried this distance parallel to Ox from A, and angular momentum Myx dt is produced. Similarly angular momentum - Mi,'^ dt about A is produced by the displacement y dt of the body. Thus zero angular momentum is produced on the whole. But if the momentum associated with the body be M-fl parallel to Ox, and M^ parallel to Oy, the former gain of angular momentum is M^xdt and the latter M-^iy dt, and there is a gain of angular momentum in dt of amount (M^ — M^xydt, that is angular momentum about A is being gained at rate {Mi — M{)x^. This is independent of the position of A, that is it is the same for all points. This rate of gain of angular momentum about every point is wholly due to the matter set in motion by the body, and is effected by the action of a couple exerted by the body on that matter (the action of a ship, for example, on the water), which therefore exerts an equal and opposite couple on the body. ^79, 80, 81] EFFECTIVE INERTIAS OF SHIP. 137 This is the couple that tends to turn a ship at right angles to its course, and that must be counteracted by the rudder, and that actually sets a ship or plank athwart a stream in -which it is allowed to drift. A ship set on a course and left with its helm lashed would be unstable ; the helmsman has continually to prevent the ship from falling off its course, and good steering consists in correcting each infinitesimal deviation as it arises. For considering an elongated body immersed in a medium indefinitely extended in each of the directions of motion (so that we are not concerned with reactions from the boundaries), let the speed x be that of the body in the direction of its length, and y be that in a direction at right angles to the length. Let M^-Hi be positive. If either sc or i/ be zero the couple {M^-M-^Aj is zero. Let, for example, y be zero. Then if the length be allowed to swerve through the angle ^ from the direction GB (Fig. 28) in which the body is moving, " there will now exist a speed X in the direction of the length, and a speed y in the perpendicular direction, as shown by the arrows, and a couple {M^ — M^xy in the direction of the curved arrow will be exerted on the matter outside the body but in motion with it. An equal and opposite couple acts on the body and tends to turn it so to increase the angle c^, that is so as to set its length perpendicular to the course. "When the length is athwart the course the couple is again zero, but that called into play by a deviation of the body from that position is now such as to send the body back to it. The body's position relatively to the direction of motion is therefore one of instability in the first case and of stability in the second. A flat dish or plate, if let fall in water, or a card let fall in still air, with its plane horizontal, moves down, in stable equilibrium ; if it is let fall with its plane vertical, the equilibrium of position in falling is unstable. In this case we must associate M^ with the axial direction, and M^ with a perpendicular direction, and we see that M^ — Uy is negative, and there is stability in consequence in the first case. The origin of the couple may be seen in a general way as follows. Consider a ship advancing with speed i, in the direction of its length, and making leeway ij, say to starboard. The bow is continually advancing with speed x into undisturbed water, which on the starboard side, at the ship, is given speed y to starboard. There is thus a reaction thrust on the bow of the vessel in the direction to port. 81. Why a Ship carries a Weather Helm. We have here the explanation of the fact that a ship generally carries a " weather helm," e 138 A TREATISE ON DYNAMICS. [CH. that is that the rudder must be held turned to leeward to keep the vessel on her course when a wind blows across it. For, as stated above, she makes leeway, that is has a speed ^ to leeward, along with the speed io in the direction of her length. Hence, by what has been stated above, the couple {M^-M^xy, on the water, is in the direction of the arrow A, in Fig. 29, and therefore the reaction-couple, which is of equal moment, tends to turn the ship's head in the direction of the arrow A', that is to windward, and this tendency (to " gripe " as it is called) must be counteracted I I I I by a couple applied to the I I 1 i ship by means of the rudder. The tendency of a ship to ~^ *., "fall off" her course (and ^ ^ I A thereby convert her forward / motion into a component ± along her length, and another y at right angles to her length), which, as explained above, always exists, is there- ^ . fore augmented by the action A of wind, and the difficulty of Fig. 29. steering is increased. This effect of the wind is consider- able when the ship is driven by sails, and a steamer using sails as an auxiliary sometimes gripes so badly, especially with canvas on the after masts, as to make it almost impossible to steer. Thus sails on steamers used to be almost entirely confined to the foremast, and are now in large vessels completely discarded. The action here illustrated is of considerable importance as regards the equilibrium of submarine vessels, of seroplanes, and of projectiles thrown from rifled guns. We shall return to it in connection with some of these practical problems, when we shall consider also the effect of rotation of the body. EXERCISES II. 1. Two particles of mass «i, m' are attached to the ends of a uniform inextensible cord of length 2^ and mass o- per unit length. The cord is passed over a horizontal peg so that parts of the string of lengths x^, ^l — Xg, carrying m, m! respectively, hang vertically side by side, when the arrangement is left to itself. If there is no friction of the cord on the peg and no air- resistance, find the motion. At time t let the lengths of the two parts of the cord be x and 21 — X. Hence show that the equations of motion of the two parts are — {{m-\- (Tx) x} = {m-\- (Tx)g - J*! -I- (tx\ at ^[{m' + (r{2l-x)}x\=T^-{m' + (T{il-x)}g-a-x\ Cut II.] EXERCISES. 139 where Tj, T^ are the tensile forces (very nearly equal) in the cord at the peg on the two sides. Add, multiply by is and integrate, obtaining ^(ot + to' + %(TV)i?=g{x - x^{m -m! ■\- X ■\-kx=0, m. + m, representing simple harmonic motion of period 27r Wmm'/k (m+m'), combined with uniform motion of the centroid at speed v. 3. A horizontal turn-table in the form of a uniform disk is mounted on a vertical axis at its centre. The weight of the table is 280 lbs. Two men, each of weight 140 lbs., stand at the opposite ends of a diameter. Initially the system is at rest. If, now, the men move round the edge of the table in the same direction at the same speed, show that when they have gone once round the table, they have turned in space through an angle Itt. 140 A TREATISE ON DYNAMICS. [CH. 4. A turn-table of mass M rotates smoothly on a vertical axis at its centre, and a man of mass m walks on it at a uniform rate u along a radius, starting from the centre. Show that if (Oq is the initial angular velocity of the turn-table, and h is its radius of gyration about the axis, the angular displacement after time t is a)optan~'( If i» <[ X throughout the motion (which is here supposed for the present to be only upward), the integrals may be obtained by expansion of !/(!+«,") in powers of «". If during part of the motion v'^L, the integrand must be changed for that part by the substitution w=l ju. §84] RESISTED RECTILINEAR MOTION. 151 For the time T and distance H to the turning point, we L ("■ du „ !?■ f* u du , „, H=-7:\ T-r77« W S'Jol- It is interesting to take the case of infinite upward speed of projection. Then „,_Xf°° du j^_L^C'° udu ~;^JoT+^"' ~7JoT+^"' Now, it can be proved (see Gibson's Calculus, § 175) that raP-'^dx IT ■£ n ^ ^1 ~i-, — = -■ ' " 0 1 the condition as to p is fulfilled, and we have r du ^ITxP-^. iol+W^'niol+x ■ Again, if we take p = 2/n, the condition as to the value of p is satisfied if % >- 2, and we obtain r udu _ 1 rx P-^dx _ Jo l+^^" nJo 1+x i JoI+'M' ■JiJo i-+x nsirxpTT Thus we obtain for infinite speed of upward projection on the conditions stated as to the value of n, T=^^^, ^=^— ZL— (7) ^ wsin— ^ TOSin — n n If n be very great, we have {Tr/n)/am('7r/n) = 1, Tr/n sin(27r/r!,) = |, and then gT=L, 2gH=L^, that is T is, the time in which a body let fall in an unresisting medium from rest would gain the limiting speed L, and H is the distance which the body would fall in the same time. The student may imagine that the jimite values of T and H obtained here for the annulment of an infinite speed of upward projection are paradoxical ; but it is to be remembered that when the speed is very great the resistance 152 A TREATISE ON DYNAMICS. [CH. III. is correspondingly great, and so the particle is brought to rest in a finite time and space. It is to be observed that if n be even, the sign of the expression kif^ will not change with that of v, and hence that we cannot find an integral in any such case that will apply without alteration to both upward and downward motion. For, in the ascending motion, both g and lev'" tend to retard the particle. On the other hand, when the particle is descending, g acts to increase the downward speed, while /cv" acts to diminish it. The downward acceleration is then g — Jcrf: Thus if kv" changes sign with v, the equation of motion, as written in (2) or (3), will apply to both upward and downward motion, so that the limits a and h oi u may be anywhere on the whole course of the motion. When, however, n is even, we must integrate for the ascending motion up to the turning point, and then integrate separately for the downward motion, after reversing the sign of kv'^ So far we have mainly considered the motion as upward. If it is downward, and n is even, we write, taking for convenience now v as positive downward, J, L du , L^ udu ,,,, dt = - T- -, dz= — Tj -, (S) so that integrating from V/L = a to v/L = b, we get ^ LP du LH» udu * = -glT^^^- ' = jIt^^ ^^^ If V= initially, and finally v = L, these give „ L{^ du „ L^{^ udu where T and H denote the time and distance travelled downward from rest until the limiting speed L is acquired. 85. Motion under Resistance varying as v. We now consider, very shortly, the special cases oi n = T, n = 2 and n = 3. When m = I we have, by (5) of §84, the equation for the time t from the instant of projection upward with speed V=aL to that at which the speed has become hL. The displacement ^^2 §§84,85,86] RESISTED RECTILINEAR MOTION. 153 z of the particle from the point of projection in the interval t is given also by (5) of § 84, that is cf fudu , , 1+a 7 .9, /„\ In the present case, since n is odd, the differential equations apply to both the upward and the downward motion, so that we may suppose the motion to have been changed in the interval from the upward to the downward direction, or to have been wholly down- waid, that is we may suppose bL negative, or both bL and aL negative. If a is positive and 6 negative, that is if the initial speed is upward and the final speed downward, z is not the whole distance travelled ; to find that we have to calculate the upward distance and the downward distance separately and add their numerical values together. The time of ascent T to the turning point is got by making 6 = 0, in (!)■ T T T JL.V T=|log(l + a)=|log^ (3) The distance from the point of projection to the highest point is 5'=|{a-log(l + »)}=|'(J-Iog^-^) (4) If the initial upward speed be L, the equations become 7'=^log,2, ^=^'(l-log,2) (5) Equation (2) shows that if 2=0, so that aL is the upward speed, F say, and bL the downward speed F' of return, at a given point, then the time occupied in the motion is given by . -^ ; <«' that is it is equal to the time in which gravity would produce in a body falling from rest in a non-resisting medium the speed F+ F'. Of course if k were zero, the value of t would be 2 F/^ for the same speed of upward projection. The speeds F and F' are by (1), (2) and (6) in the relation y. y, L+V -ir=^°^L^- 86. Resistance varying as v^. Now let n=>2. Here we have to deal with the upward and downward motions separately. For the upward motion, we have L du J L^ iidu /, \ so that i=-(tan-'a-tan-i6), z=^log ,^ , (2) for initial speed aL and final speed bL. 154 A TREATISE ON DYNAMICS. [CH. III. The time of ascent and distance to the highest point are thus ?'=-tan'p ir=-log—^ (3) For the downward motion we take v as positive downward and obtain ,^ L du L I dv, , du \ /^\ '^'=^ i^^=¥g{TT-u+r=^h ^^^ '^^=7rr^^=-2^ j«{i°g(i-'''>} ('> Hence, integrating from the limit a for the initial point to limit 6 for the final point, we get , L, (l+6)(l-a) Z2, l-a2 ,„. '=2^'°g (l-6)(l + a)' ^=2^1°gl36^- (^> If the initial speed be zero and the final be hL, these equations give , i, 1 + 6 I?, 1 ..^^ '=2^'°grr6. ^=2^i°gr:p (0 Hence, if the terminal speed L be the final speed, that is if 6=1, the values of t and z are both infinite. 87. Resistance varying as v^ Finally, we take the case of resistance ki?. Here the downward motion need not be separated from the upward. Taking v and z positive upward, with I?=glk, so that L is the limiting speed, dt=-k^ <^^=-~~, (1) Hence, if a be the initial and b the final value of u, f_L r du, _U'fudu ,. '-gll+y?' '-gj,\W ^^' Now, by splitting 1/(1 +vF) into partial fractions, it may be verified that the integrands break up into differentials as follows : du _\ du 1 2^t-l ^ ^ 1 sjz \ + v? Z\+u 6«2-M + l "'"v/34/ --^^ ' ^' udu 1 dw 1 2m- 1 , 1 iJz ,.s + g „.2 .., ■■ '^" + -^77 TT2^ W /3 if IV 3r-2; + 1 2 1 + ^3 3 l+M^eM2-M + l"'"'^^3 4/ IV ir-2J 3V" -+^ ^86,87,88] RESISTED RECTILINEAR MOTION. 155 Hence, — ^ ri ln/2 (cos 6* -cos ft) Writing now sin|^=sin|ftsin<^, (5) we have flJ^ = 2sin|ftcos<^c?<^/\/l-Fsin^(^, (^ = sin^^o), and when d=0, <^=0, ; when 0=9^, ■ 4>, and therefore by (6) we get ^=n) is called an elliptic integral of the first kind, of modulus k and amplitude = 4'i to <^ = c^2> ^^7) is thus given by n{t,-t,)=F(k, 2)-F{k, <^,) (11) It is clear from (8) that if k (that is sin ^0^) is very small, we have T=^iJljg and T^'ivJljg, the result already obtained in § 90. 92. Motion of a Particle in a Vertical Circle. Elliptic Functions. If the pendulum start from rest from a position making an angle ^q with the downward vertical, the force toward the centre applied by the cord is at any time thereafter, when the deflection from the downward vertical is 0, mlffi+mx) cos 6, or by (2) of §91, m5'(3cos^ — 2cos6„). At the end of a swing 6= 6a, and the pull is then mg cos dg, which is negative if ft >7r/2. But if instead of a bob suspended by a string we have a particle moving in a guiding tube, bent into a circle, in a vertical plane and of radius I, the amplitude G.D. L 162 A TREATISE ON DYNAMICS. [CH. III. may have any value from up to ir. If there be no friction between the tube and the particle, the equations of § 91 apply to the motion, and the force P applied by the guide to the particle is given by P=»n^(3cose-2cos^o) (1) If the particle start with speed v^ from the position at distance W^ along the circle from the lowest point, then at deflection d the speed v is given by v' - v^ = igl{c:os 6 - cos 6^, and P is then given by tlie equation 2 P=mg{Zcoae-'icoseo) + m -^ (2) Hence, if the particle goes completely round the circle, we have when ^=7r, P=mg{ — Z -'2, COS d^+mvyi, and therefore if the value of P is not to change sign, we must have Dj>5'Z(3 + 2cos^(,), and so if 6o=ir, 1?Jl>g. If this condition be fulfilled, the particle may be suspended by a string. The reaction on the support is equal and opposite to the force P on the particle. The amplitude ^ is the angle DCQ in Fig. 36, where P repre- sents the position of the particle at time t. The circle APB is drawn with radius I, P, is the initial position of the particle, CP^ is horizontal, C is the highest point of the smaller circle, which has diameter BC. DP is drawn through P hori- zontally and intersects the smaller circle in Q. Then lDCQ is <^, as we shall prove. For join AP, OP. Then LBOP=e, LBAP=\e, LOAPo^-ieo. J ^, . ^„ , Bythediagram,Cfi=2;sin2i^„, and therefore CD=%lain^^0^cos^LBCq. But also GD.=i-Ac+oD=i-ncos^^en+icose=^i{cos^e-cos^e^). Equating these two values of CI) and reducing, we obtain XT i^r^r. J s^nidosincBCQ = smie. B.enee LBCQ = <}y. Also k=smieo=CPJAPa. If we write ^- + ^^ = 1, V is called the co-modulus, and is therefore represented by AGjAPg. The construction in Fig. 36 replaces the turning of OP, with angular speed 6, by the turning of CQ with angular speed rf. [see equations (5) and (6), § 91] and the motion of P by that of Q. Q §92] MOTION OF PARTICLE IN VERTICAL CIRCLE. 163 starts from C when P starts from Pqi ^^d coincides with P at B. If the particle just goes completely round in the guiding tube, that is if /"(, is infinitely near to A, the smaller and larger circles coincide, and =^d, 6=Tr, so that k=l. Thus t is a. function of . Conversely, <^ is a function of nt and k, called, as has been stated, the amplitude of nt. If then we write v, for nt, we have <^=amM to modulus k, and sin <^ = sinamtt, or, as it is usual now to write, sin (j> = snu. We also write cos (j> = CTiu. We have d<\>ldii = da,VQ.ujdu=\J\-lc^si.n^(^, and write this dilM. '^'^ dswic , dcnu , • — ; — =cnMdnM, — i — =-snMdnM, du au , \ (3) dAnu Baixi 4> cos 4> ddi ,« — 5 — = — T — 1 - — ^= — K'anucnu. du 'yj 1 - k^ sin^ =BCsnucnu. But BC=2lsin^i6o = 2lk% so that also DQ = 2lk^snucTiu. Thus DQIDP = BCIBPo = kcnuldnu. When the amplitude of oscillation is very small we may take k as zero, and we have then u = F{k, ) = 4>, that is snM = sinM, cnM = eosM : the elliptic function becomes the ordinary circular function. At the other extreme, when the particle just goes completely round the circle, k=l, and --/;^-i°^™-8--^-'(-<^)' (^) and so, for <;()=7r/2, t is infinite. The integrals K and F can be calculated easily by expanding (l-z&^sin^c^)"^ by the binomial theorem, and integrating term by term. This proceeding is legitimate (since ^ < 1 and the series are convergent), and yields for t the equation The first two terms of this series form an approximation sufficient for many purposes. This approxi mation can be arrived at directly by assuming that l/\/l -Psin^4'=\/T+WSi^ and integrating. From the expression on the right of (5) the multiplier K of x's/^r in (9) of § 91 can be calculated. There are more convenient methods of -W|{>-©"*'+(H)"^-GTtl)''--} <^) 164 A TREATISE ON DYNAMICS. [CH. III. calculating this integral ; but it is unnecessary to discuss the subject here. The values of K for different values of k were tabulated by Legendre ; for values of \Qq proceeding by successive steps of 1° from 0° to 90°, a table is given to 4 places of decimals in the Smithsonian Physical, Tables drawn up by the late Professor Thomas Gray. When |^o = 90°j ^o = 180°, so that the particle in the guide tube (not the pendulum) just goes completely round from rest to rest in half a period. The time required for this is infinite ; for a range, however, of 5^Q, from a deflection of 89° on one side to 89° on the other, the time required is 3"3 times that required for a very small swing from one side of the vertical to the other. Values for other amplitudes can be obtained from the following short table : i^o K ¥o K k% K ■\o. K 0° 1-5708 5° 1-5738 15° 1-5981 50° 1-9356 1 5709 6 5751 20 6200 60 2-1565 2 5713 8 5785 25 6490 70 2-5046 3 5719 10 5828 30 6858 80 3-1534 4 5727 12 5882 40 7868 90 CX) It is worth noticing that (7) of § 91 shows that the period always lies between the limits 27r/»j and 2?r sec Q^ln. For the angular speed of § about O in Fig. 36 is 2^, and the equation shows that 2to>2^>29icos|^„, and the same inequality holds for the mean angular speed of §. Hence 27r/M < Period < 2jr sec Bt^jn. 93. Revolution of Particle in Vertical Circle. Now let the particle in the circular guiding tube be making complete revolutions under gravity. Here we may have given the time of revolution and be required to find the speed of the particle at any position and the time of describing any part of the circle, or we may have given the speed at top or bottom of the given circle and be required to find the period of revolution, the speed at any point, and the time of describing any part of the circle. If the speed at top of the circle is known, that at the bottom is also known, and mce versa. For since the speed v along the circle when the thread makes an angle 6 with the vertical is W, we have v^^=-glsme-^ (1) or v^ = 2glcose+C, (2) where C is a constant. Thus if Vi be the speed at the lowest point (6=0), we have C=v\-'i,gl. Hence v^=v\-'igl{\-cose), (3) ^92, 93, 94] EXAMPLES ON MOTION IN VERTICAL CIRCLE. 165 and thus at the highest point, where v = V2 say, <=<-^9l, (4) so that we must have, for the motion to be possible, v\ > igl. Now 1 - cos ^ = 2 sin^<^, if <^ = ^9, and therefore v'^=v\- 4Lgl sm^(j> (5) Thus, since v=ds/dt = ld6/dt, we get dt= , ^^'^'i' =g-^_^ (6) V, Now it is here known that igl/v^Kl, and thus if we write k^ = igllv\ we get for the time from the lowest point to the inclination 6, '=^iov/l-Fsin^^=^^^^"^> (^) The time for any arc, from <^=<^i to <^ = (^2i say, is thus 97 t2-h=~{nh 4>2)-F{lc, <^l)}, :..(8) and, by (11) of §91, stands in the constant ratio Z-Jgl/vi to the time of describing the corresponding arc in the oscillatory motion in a vertical circle of the same radius. Again, from the lowest point to the highest, the time t is given by 2Z r d4> =^JfU^) (9) The table in §92 may be used to obtain numerical values in par- ticular cases. If the speed at the highest point be zero, Vi—-J4gl and ^ = 1 ; the value of T is infinite, as we have already seen. Since here V 1 - P sin^ <^ = cos (f), the integral I /rf<^/cos(^> can be found for' any limits and a., if a. < n-/2, by ordinary integration. 94. Examples on Motion in a Vertical Circle. Ex. 1. P is a, point on a vertical circle of which AB (reading downward) is the vertical diameter and is the centre. From a point C taken above A on this diameter produced upward, lines CE, CF, each equal in length to a tangent drawn from C to the circle, are laid oif respectively downward and upward along the diameter, and a circle is described on EO as diameter. A line PEQ is drawn from P intersecting the latter circle in Q. Prove that if h, K be the vertical distances of P and § below C and F respectively, and P move 166 A TREATISE ON DYNAMICS. [CH. III. under gravity g in the first circle with speed due to h, then § will move in the second circle as would a particle under gr3.\iiy g . EO^I% A 0" with speed due to h'. Ex. 2. Two vertical circles touch one another at their lowest points. B denotes the point of contact and ^S, A'B are the coincident diameters, of which A'B has the greater length. A line perpendicular to the vertical diameter cuts the circles in P, P'. Show that if P move under gravity g with speed due to its vertical distance from A', then P' oscillates in the larger circle with .the speed due to its vertical distance from A and gravity g. A'E^jAB'. Ex. 3. Prove that if two particles be projected from the same point with the same speed and in the same direction, but at different times, along a narrow circular tube in which they move without friction, and which has its plane vertical, the line joining them always touches a fixed circle. Ex. 4. To find the condition that a carriage may " loop the loop," that is pass round a vertical circle or curve, and to find the reaction on the guide. Eegarding the carriage as a particle, we see from § 92, that a particle attached to a cord fixed to the centre of the circle will exert outward pull on the cord at the highest point of the circle, if the value of v^Jr exceed g, where v is the speed at the highest point, and r is the radius of the circle. But if the speed is ac- quired by the descent of the particle from a starting platform, as is customary in " looping the loop " apparatus, v must be the speed acquired by a particle in falling from the level of the platform to that of the top of the circle, that is if the difference of levels be h, we must have v'^='igh, so that the least possible value of the "head" h is r/2. The head must be greater than r/2 to a sufficient extent to allow for loss of head caused by friction and the resistance of the air. If the curve traversed by the carriage is not a circle, then r is the radius of the circle of curvature at the highest point. Properly, in the case of a carriage, we ought to take the curve for which the head is reckoned at different points as that in which the centroid of the carriage moves, and in so doing we should still neglect the rotation of the wheels. The equation of energy of the body moving in the gravitational field is 1 •> / , N where c is the height of the starting level above the level of the origin from which the vertical distance y is measured downward. Thus, for two distances y, y\ the energy equation becomes ^wiv^ - \mv'^=mg(jj — y'). The reaction against the path, R say, is given by the equation mv^ „ , =n+ mg cos y, §94] EXAMPLES ON MOTION IN VERTICAL CIRCLE. 167 where i^ is the inclination of the normal to the curve at P to the vertical, as in Fig. 37. Thus R is zero if v^=grcos'\jr, and the carriage ■will leave the path in its upward journey at the point where this condition is fulfilled. "We can write the last equation in the form R _ '2i/ — r cos \lr mg r ' where y is the head HP required for the speed v. We then have PC=r, MP=r co%i^, and R vanishes when PM='2iHP Fig. 37. Fig. 38. When the track is a circle, we see again that for the highest point to be reached 2y must be greater than r. At the lowest point of the circle cosi/r=-l, and so Rlmg={2^+r)/r. But if at the highest point 2i/>r, at the lowest point ^y+r>r + 4r+r, that is (2y +?•)/?■> 6. The reaction of the track on the carriage is thus greater than 6 times its weight. If the carriage be running on the convex side of a curve in a vertical plane, as in Fig. 38, the equation of normal force is — - = mg cos ^ — R, or with the same notation as before R _rcosi/r — 2y mg ■ r Thus R will vanish if ?'cosi/r = 2y, and will become negative if r cos i/r < 2y, and the carriage can then only be kept on the track by a guard-rail. The figure gives r=PC, MP=r coayp-, and if MN=^y, Rjmg = NPjPC=APIPM. Thus R will change sign if MN becomes greater than MP, and the carriage will leave the track unless prevented by a guard-rail. 168 A TREATISE ON DYNAMICS. [CH. III. K°RcoiL Ex. 5. At the crown of an arched bridge the curve in which a motor-car passes over it has a radius of 50 feet : at what speed will the wheels of the car just cease to press on the road ? By the last example we have as the condition to be fulfilled v^—gr, that is «^=32x50, or v=iQ, that is the limiting speed is 40 ft./sec, or 27'27 m./h. The apparent failure of the steering gear may no doubt be sometimes explained in this way. The car running at a high rate of speed passes over a convex part of the road of con- siderable curvature, and the wheels lose their grip of the surface. 95. EoLuilibrium of a Plummet under Gravity. Apparent and Real Gravity. A plummet P is hung by a cord of length I from a point fixed relatively to the earth : to find the effect of the earth's rotation. We suppose the earth (Fig. 39) to be a sphere of radius R attracting the plummet at P in the direction to- wards the centre G with a force G per unit mass. The plummet is in rela- tive equilibrium, and is therefore carried round with the angular speed of the earth in a circle of radius R cos L, where L is the geocentric latitude of P, that is the angle PCE. The plummet is under acceleration n^RcosL to- wards the centre of the circle in which it moves, and for this a force Tnn^RcosL is required. This is supplied by the force of gravity mG, which acts towards C; so that we must conceive mG as resolved into two components, one mn^R cos L towards M, and another mg in a direction PD, to be determined, and of such amount that with the first component the resultant is mg. PD is clearly the direction of the cord which supports the plummet. The sides of the triangle PCD are in the directions of the three forces, and are therefore of lengths proportional to the numerical values of the forces. We thus have §§94,95] TRUE AND APPARENT GRAVITY. 169 sin LGPD/sinLPCD = {n^R COS L)/g. But lPGD=L, and therefore 27? sin^0Pi) = -i-^^sin2i (1) If, in Fig. 39, n^R be represented by CQ, then DQ represents n^R sin L, and DR represents in}R sin L cos L or ^v?R sin 2Z. If the figure were drawn to scale and CP were taken to represent G, CQ would, as we shall see presently, be only 1/17 of GP. The direction PD is that of apparent gravity g, and is the line of the plummet-cord. The angle GPB is the deviation of the plumb-line from the true direction of gravity, the direction of G, or, what is the same thing, the excess of the geographical latitude PDE as shown by the inclination of the plumb-line (or the normal to a horizontal mercury surface) to the plane of the equator, over the geocentric latitude L. The acceleration of a particle moving freely under gravity is G; the excess of this over g, and the difference of direction, are however so small that for many purposes, for example an elementary discussion of the flight of projectiles, they may be neglected. If the earth were not rotating, the plummet at P-would be held at rest by a force Tnn^R cos L, applied outwards at P in the direction MP (Fig. 39), without alteration of the direction of the plumb-line, or of the two forces, 7nG towards G and the pull of the cord vng outwards in the direction DP actually applied to the bob. It is often convenient to put aside the rotation in this way, and consider equilibrium as produced by the introduction of a force acting outward, which is then called the centrifugal force. The value oi n^R cos L is greatest at the equator, and we have in f./s.^ units. This is about 1/289 part of the value of at the equator, and therefore, since 289 = 17^, the speed of the earth's rotation would have to be increased to 17 times its present amount in order that the force of gravity might be all 170 A TREATISE ON DYNAMICS. [CH. III. employed in giving the necessary centre-ward acceleration to bodies at the equator carried round by the earth. Gravity would then be apparently zero. In the latitude of Glasgow, n^RcosL is only slightly more than 062 (f./s.^), and CPD is about 5 J minutes of angle. 96. Pltiininet in Railway Carriage. Apparent Gravity. A plummet is hung in a railway carriage which is subjected to acceleration. The position of equilibrium of the plummet-cord is not along the real vertical, but is inclined to it at an angle depending on the acceleration. If the carriage were running uniformly the equilibrium direction of the plummet would be vertical; but in the case of acceleration there is inclination of the cord, so that the deflection of the plummet is in the opposite direction to that of the acceleration (see Fig. 40). Backward deflection (X)() TTOU Fig. 40. accompanies forward acceleration in the line of motion, forward deflection accompanies retardation, outward de- flection accompanies motion of the carriage round a curve. The position of equilibrium is that in which the pull exerted by the cord on the bob is just that required to give it the acceleration of the carriage; and if disturbed from that position the plummet will oscillate as a pendulum about it, under a directive force of apparent gravity, differing from the real force of gravity in a manner similar to that in which g was found in the last section to differ from G. Let a. be the equilibrium inclination of the pendulum to the vertical when the acceleration of the carriage is a, and let P be the pull exerted by the cord on the plummet- bob. Then we have ma = P sin a, mg = Pcoaa., so that tanot. = -, P^mjg^ + a? (1) §§95,96] PLUMB-LINE IN A CARRIAGE. 171 If the plummet is displaced from this position through an additional angle 6 — a. in the plane of a, so that the inclination to the vertical is now 6, its motion relatively to the carriage (that is the motion apparent to an observer in the carriage who takes no cognisance of objects external to it) can be found in the following manner. The hori- zontal acceleration of the carriage in the direction taken as positive (the line of motion forward say) is a; if the acceleration of the plummet in that direction be x, its acceleration relatively to the carriage is x — a. If P be the pull applied to the bob by the cord and the deflection be 6, we have mx = P sin 6, and therefore m{x — a) = F sin 6 — Tna. The upward vertical acceleration y is given by mij = P cos 6 — mg. Thus by Fig. 40, taking the angle the cord there makes with the vertical as 0, we have 7rblO= — (Psin 6 — ma) cos + (P cos 6 — mg) sin 6 or 16= —{gsinO — acosd) (2) But we have seen that if a is the equilibrium inclination of the thread, we have a = gta,n(x. Hence (1) may be Thus, if — 0. be small, the motion relative to the carriage is one of simple-harmonic oscillation of the plummet about the inclination a. The period is given by r=27rV-r^. (4) that is, the period is that of a pendulum of length I oscillating under the effective gravity glcoscL = >Jg'^ + d^, or, as it may be otherwise put, it is the period of a pendulum of length I cos a., under gravity g. If the plummet were deflected slightly sideways from the plane of CL, it would oscillate about that plane in the same period 2'7r\/l cos a./g. If the carriage be going round a curve so that the bob moves in a circle of radius R at uniform speed v, the 172 A TREATISE ON DYNAMICS. [CH. III. plummet is in equilibrium at each instant in the vertical plane through the centre of the curve and the point of suspension, when inclined outward at the angle For the acceleration a. is in this case toward the centre of the curve and is v^/R, so that a/g = v^/gR. The pendulum if disturbed in this plane oscillates about the inclination (X in the period . — T=^^^-d:z7:> (5) ' gf/cos a' which, as can easily be seen, is also the period of a small transverse oscillation. Gravity in the carriage thus seems to have altered in direction by the angle oc, and to be increased in intensity in the ratio Jl-\-a?lg'^ to 1. 97. Cycloidal Motion. Oycloidal Pendulum. Consider now motion in a cycloid which has its plane vertical and the tangent at its vertex horizontal. The curve is that traced out by a point P (Fig. 41) of a circle (radius a say) as the circle is rolled without sliding — that is so that the centre of the circle advances a distance aB when the circle is turned through an angle Q in its own plane — along a horizontal line AB. If the circle make more than one turn the point traces out successive cycloids which meet in cusps, for example at A and B, for the move- ment of the point P is there away from or towards the base AB, and the successive cycloids have common vertical tangents. The point / of the circle is at the instant at rest, and therefore the extremity P of IP, regarded as turning about /, is tracing out an element of the curve. It is easy to show that the radius of curvature of the cycloid at P is 21 P = QP = ia sin [see Gibson's Calculus, § 146]. Now let a guiding tube in the form and position of the cycloid in Fig. 38 be provided, and let a particle be placed on it at rest in any position P^. If, as we suppose, there be no friction, the force of gravity along the tube is rngcos^, and acts towards the lowest point. We have therefore ^,^ ^=-^cos^=-^s. : (1) The particle will therefore swing from the position Pq to another at the same distance from the vertex on the other side, in time W4a/gf. The whole period will be 2Tr\/4sa/g ; that is the period is equal to that of a simple pendulum of length 4a ( = KV) vibrating through an infinitesimal arc. If we differentiate s = 4acos^, we get for the speed, s = — 4a sin 0.0; and for the acceleration, s— —4a cos .0^ — 4asin0 . !p (2) Hence we see by (1) that the motion under gravitj' along the cycloidal guide from cusp to cusp is one in which has the constant value \/g/4a be made, and be hung from the cusp of contact of the two halves of the evolute (supposed made as material cycloidal cheeks, with the end of the string clamped between them at K (Fig. 41)), and be made to vibrate, the string will at first be wound upon the cheek on one side, will then unwind itself from that side, next wind itself on the other cheek, then unwind itself, and so on, while the bob moves in an equal cycloid. The motion of the bob, as the student will easily see, is exactly that of the particle in the cycloidal tube discussed above, and the period, whatever the amplitude may be, is ^TrJ^ajg. In the construction of such a pendulum the cycloidal cheeks should be made with exactness. Their form is often quite incorrect, especially near the cusp. The length of the pendulum is adjusted by making the string longer than 4a, and pulling it through between the clamp- ing surfaces, until the bob just reaches to the extremity of one of the cheeks on which the string is wound. It is instructive to hang a circular pendulum of the same length alongside the other, and vibrate them together ; when it will be seen that the circular pendulum lags behind the cycloidal for large arcs of vibration. 98. Tautochronous Motion. The discussion, just given, of motion along a cycloid, and the theory of simple-harmonic motion, set forth in §§32-34 above, illustrate the fact that the condition s= —n\ is sufficient to ensure that the same time will be taken by a particle to move along any path from rest in any initial position s = Sq, to a point of arrival s = [s is measured along the path]. The motion is said to be tautochronous, and the path for a given field of force is called a tautochrone. We can prove that the condition is necessary for the case §§97,98] TAUTOCHRONOUS MOTION. 175 in which the force depends on the position of the particle. The force must clearly be towards the point of arrival : let it be /(s). We have dv =f(s) dt ; and therefore vdv= —f(s)ds. Thus, vl=-2\y(s)ds = 2{F{s,)-F(8,)}, (1) if -^1 be the speed at distance Sj and I f(s)ds = F{s). Jo ds ^2^Fis,)-F(8)' '^ and if t be the time of motion of the particle from s = s^ to s = 0, weget 1 p ds ,„- ^~ j2]o J F(s,)-F(s-j Now let s = Sf,u, and the last equation becomes T =-ir. s.du °o (4) '^i,^F{s,)-F(s^u) which for tautochronism must be independent of s^. We have dT^l_ {' 2{F(s„) - F(s,u)} - s,{F\s,) - uF'(s,u)} ^^ . dso J2}, 2{F(s,)-F(s,u)}^ and in order that dT/ds^ may vanish, we get the condition 2F{s,)-s,F'(s,) = 2F(s,u) - s,uF'{s,u), that is 2F(s) = sF'(s) throughout the motion. Thus we get, by integration, F{s)=Gs^ that is F'(s) = 2Cs=f(s). The force /(s) is thus proportional to s. Ex. 1. If the motion of the particle is resisted by a force proportional to the speed, the motion is still tautochronous. The equation of motion is s + B+nh=0. If in this we write « = «<«"**', we get for the transformed equation 176 A TREATISE ON DYNAMICS. [CH. III. This, when ra2>jF, is the equation of a tautochronous motion of displacement u. The time required for the passage of the particle to the point s = from rest at any initial distance «o is the smallest positive root of the equation tan {-Jn^-lk^. t)= - 2s/n'-ikyt Ex. 2. If a particle move along a catenary [equation, under a force at each point proportional to the ordinate y, and in the direction of y decreasing, the motion is tautochronous for the point of ordinate c, as point of arrival. Let the force be n^y. Then the component toward the point of arrival is n^y dylds = n^ysly=n^s, where s is the distance of the particle at the instant considered from the point of arrival. The proposition is therefore proved. By the last example, it also holds when a resistance proportional to the speed also acts on the particle, and the time of passage is given by an equation similar to that at the end of Ex. 1. Ex. 3. A particle is constrained to move in an equiangular spiral while acted on by a central force towards the pole of the spiral of amount rfir, where r is the length of the radius- vector. The component of force along the spiral is wV cos <^, if <^ be the constant angle which the radius-vector makes with the tangent. But the distance «, along the curve from the pole, of the point to which the radius-vector is r is r/cos(t, A, B), and the motions are tautochronous. Ex. 6. The differential equation of tautochronous motion stated in Ex. 5 may be written /;\ -^^©- For sjs we write u/f(u), and the equation becomes The last two terms, it is to be noticed, form a homogeneous function of the first degree in u and/(M). The motion begins when u=0, and ends when f(u)=0. For n=0 when s=0, and since s=us/f{u), we see that, as i is not zero when g=0, sjf{u) cannot vanish when s=0. Thus, when «=0, we have f{v,)=0, that is at the point of arrival /(m)=0. It will be observed that the equation just found may be written in the form y.,(^) • (.^2 x 5. / ^ I /W /(m) •" ' \.f{v)) For instead of f{v) we may write f(u)IC, where C is any constant ; and it is obvious that C cannot appear in the first and last terms. The theorem stated in this example is due to Lagrange {M^moires de Berlin, 1765, 1770). The proof here given is a version of that due to Bertrand {Liouvill^s Journal, xii. 1847). It is shown by Bertrand that the equation states a sufficient but not necessary condition of tautochronous motion. Ex. 7. If the equation of motion is where p and q are given functions of s, prove that the condition of tautochronism is /ig' + 2c?g'/cfe = const. We have here dv^jds= —2s=pv^ + q. Thus, by the second form of Lagrange's equation given in Ex. 6, we get, putting u=s, But p and q are functions of s only. We can write f(s)J^{slf(s)} in the form As+Bf(s), and in the present case A=0. Thus we get for the second of the equations just written and its derivative, Bf(s)=-iq, £/'(^)=-iJ. Substitution from these in the first equation gives pq + 2-^=i const., G.D. M 178 A TREATISE ON DYNAMICS. [CH. III. as stated above. Here, as generally above, s is measured from the point of arrival. If s is measured in the opposite direction, the condition becomes ^ pq - %j- = const. It may be proved that this constant is positive. Ex. 8. If the particle be constrained to move along a given path and be subject to resistance 2Ky+KV : to find the force P along the path at each point which will make the motion tautochronous. We here suppose v to be the speed along the path towards the point of arrival, so that if s be measured from that point v= -dsldt. The equation of motion is ^ ^+2ct + kV=P. at Now putting dsjdt for v and multiplying by e-"'*, we get — e - «'«« "+ k'« - k'ss^ _ 2x6 - K'»i = Pe - «'», that is if u= -e-«'«s, w + 'i.Kit,-Pe-<''=0. The particle will therefore arrive at the point m = in the same time, whatever the initial value of u may be, if -Pe -«'» = «%, where n^ is real and positive. But ■■ and so if u and s begin together, we have G= — 1/k'. Thus M = -;(e-'=s -1) K and P=^-,(e<'«-\). K This theorem is due to Euler. The reader will notice that P has the value nh if k.' be vanishingly small, the result obtained above for zero resistance. We infer from this that the term %kv in the resistance does not affect the tautochronism under the law of force found for zero resistance. It is supposed here that m^ > ^i^ ^taA the time from the initial values of u and s to the point of arrival is the smallest positive root of the equation i—^ » tan(N/w^-K^O=- ^'^~'' , in which it is to be noticed that k' does not appear. On the other hand the coefficient k has no influence on the value of the force P. The first of these curious results is noticed by Laplace [M^canique Celeste, t. i. p. 35], who also remarks that the value of the time of passage would not be altered if terms k"v^ + k"'v'^+ ... were added to the resistance. The discussion here given is a version of Eouth's modification of Laplace's process for the case of the resisted motion of a particle under gravity. [Laplace, loc. cit., Routh, El. Rigid Dynamics, § 492.] §§ 98, 99] BRACHISTOCHKONES. 17& Ex. 9. If in the last example the impressed force P be due to gravity we get, measuring z downwards, dz Tfi , , ,, ^df -9z=^^(e'^''-K's) + a If we suppose that k'=0, this equation becomes the equation of a cycloid. The curve in which the particle is con- strained to move must therefore be a cycloid if the impressed force be that of gravity, and there be no resistance depending on the second or higher powers of the speed. This may be compared with the result of § 97 for the cycloidal pendulum. 99. Brachistochrones. The problem of the line of quickest descent, or, to put it more generally, of the path of quickest passage, in a given field of force from one given point in the field to another, is of great interest. It was proposed in 1696 (in the Acta Erud. Lipsi.) by John Bernoulli for a particle moving under gravity, and a solution was pub- lished by his brother James Bernoulli, in the same journal, in 1697. It seems to have been solved also by John Ber- noulli himself and by Leibniz. The following is a short version of James Bernoulli's solution. In the first place, as the student may easily satisfy himself, the path must lie in the vertical plane con- taining the two points. Let OGD (Fig. 42) be the curve, and let a small portion of it, CB, be divided into two parts at (?; if we assume that the time for each element of the path is a minimum as well as the time for the whole path, then the time along a near element OLD terminated at G and D must be indefinitely nearly equal to that along CGD. For if we pass gradually to the path of minimum time from paths nearly coinciding with it, there must, from 180 A TREATISE ON DYNAMICS. [CH. III. the fact that the time is a minimum, be only a very slight variation from one to the other. The rate of variation of a continuous quantity in the immediate neighbourhood of a maximum or minimum is extremely small — it is absolutely zero at the maximum or minimum itself. Now, in Fig. 42, we may regard CG, OD as straight, the first coinciding with the tangent to the curve at G, the other with the tangent at G; and similarly for CL, LD. Draw LM at right angles to the element CG, and GN at right angles to LD. Then if t, t' be the times of passage along GG and GB, and i,, t[ those for GL and LB, we have t + lf = ti + t[, and therefore t — t^ = t[ — t'. Also if y and y' be the vertical distances of G and G below 0, the starting point, the speeds in the curve at C and G are \/2gy, J'igy'. The latter is also the speed in the adjoining path at L, since LG is horizontal. Thus taking, as we may, the speed along GG and GL as that at C, and the speed along GB and LB as that at G, we have t-t^^MG/^/^gy, t[-t' = LN/>j2g^. But MG = LG sin GGE, LN= LG sin BGH. Calling the first angle , and the second 0', we get finally sin 0/i! = sin t^'jv' or sin (p/'j2gy = sin fp'/xj2gy'. The curve therefore has the property that the speeds along it at successive elements are proportional to the sines of the angles which the tangents to the elements make with the vertical. This, as we have seen (§ 97), is a characteristic property of the cycloid. Since the particle starts from rest at the highest point 0, the cycloid has there a cusp. This fact, together with the condition that the final point lies on the curve, determines the cycloid. The cycloidal path is of course not a free path. A f rictionless guide must be provided. 100. Brachistoclirone in Conservative Field of Force. Euler's Theorem. The result just obtained holds for the motion of a particle under any conservative system of coplanar forces. For precisely similar reasoning shows that if 6 be the angle which the resultant force F due to the field (that is the resultant of the forces applied to the particle, exclusive of the reaction of the guide) makes with the normal to the path at the element ds, then v = CcosO, where C is a constant. ^99,100] BRACHISTOCHRONBS. 181 We can now find the reaction of the guide on the particle. Since dv/dt = vdv/ds, mv-j- =Fsin6, (1) for the reaction gives no component along the path. But ^ = Ocos0 gives i^^_sjne vde COS0 ^ '' Dividing the former equation by the latter, we obtain -5-= -Fcose, (3) where R is the radius of curvature of the path at ds. On the left is the force toward the centre of curvature which is supplied by part of the reaction of the guide on the particle. It is important to remark that it is equal and opposite to the normal force with which the particle is pressed against the guide by the field, and which is also balanced by the reaction of the guide. Hence the total reaction is 2J''cos Q toward the centre of curvature, or twice that which would exist if the particle were at rest. This theorem was first given by Euler. In the cycloid therefore we have — ^= —7ngcosB= —rngsuKj). In a free path, from one point to the other, we should have „ -^-=Fcos9, (4) that is the field would supply exactly the force on the particle towards the centre of curvature that is required. The concavity of the path would therefore be turned the other way. The brachistochrone would thus be the free path for a system of forces which left the tangential com- ponent everywhere unaltered, but reversed the normal component without altering its amount. If the forces of the actual field were replaced by forces represented by 182 A TREATISE ON DYNAMICS. [CH. III. the reflection of the former in a mirror containing each element of the path, and perpendicular to the plane of the path at every point, the motion would not be changed, but the guide would be rendered unnecessary. Conversely, any ordinary free path can be changed into a brachistochrone for a field of force composed of the same tangential component and the normal component reversed. For example (see § 126), a particle moves freely in an ellipse under a force directed towards, and varying inversely as the square of the distance of the particle from one of the foci. If this attraction were replaced by a repulsion of the same amount, but directed from the other focus, the path would become a brachistochrone for the new field. [See a paper by Tait, Trans. R.S.E., 24, 1865.] 101. Variational Method for Brachistoclirone under Gravity, It is fairly evident that the motion along the cycloidal guide from one point to the other must be one of least time ; but the elementary method adopted above, though instructive in several respects, is defective in that it leads to no general process by which such problems of maxima and minima of integrals as occur in geometry and physics can be solved, and gives no criterion by which to judge whether the result is a maximum or a minimum, or only one of a succession of stationary values. It will be noticed that the problem of the line of quickest descent from one point to anotber differs from the ordinary questions of maxima and minima dealt with in the differential calculus, for there it is only expressions of known form that are discussed, while in the former we have to find what the expression itself must be, in order that its integral may have a maximum or a minimum value under the circumstances stated. Thus, in the case of the brachistochrone, it is specified that the integral / dsjv, taken along some path joining the two given points, is to have a minimum value, and we must first discover the form of the path, and so find the manner in which v varies along it, before we can find the least value of the time of passage. As we may have in what follows to employ the method of variations in the discussion of certain problefhs, we shall give here its solution of the brachistochrone question, in order that the student may understand the notation and form some conception of the general process, which has many applica- tions in higher dynamics. First, we notice that if we take as rectangular coordinates of an element ds, x horizontal and y vertically downwards from the starting point, we have ds={\ + {dyjdxf ]^dx, and v='J'igy. Hence, if t be the time of passage, and a>=a be the §§ 100, 101] BRACHISTOCHRONES. 183 abscissa of the point of arrival, we have for t the equation -fV^- ■^::^ « where p is written for dy\dx. If we denote ■J{\+p'^)ligy by V, we have t= I Udx (2) Here ?7 is a function of y and p, which are both functions of x ; we heme to find what functions they must he in order that t may he a minimum. We therefore impose on y, and consequently also on p, a small variation of value, while x is kept unchanged. It is clear that this will bring about a small change in the course of the curve, which would not be made any more general by also varying x. By equating the eft'ect of this variation (taken to the first order of small quantities) on t to zero, we obtain a condition fulfilled by a curve of the shape desired, and a curve of this shape can then be fitted to the given data, and, if it is desired, t calculated. We can then, by carrying the, effect of the variation to the second, or, if need be, to a higher order of small quantities, determine whether the condition obtained leads to a maxi- mum or to a minimum, or to neither. The student will observe the similarity of this pi'ocess to that adopted for ordinary maxima and minima. Denoting the variation of Z7 by S(J, oi t by St, and of y and p by Sy and Sp, we get St= rWdx (3) Jo But since U contains only y and p, «^=-W«^+-f«^' (^) in which the differential coefficients are partial. Now ^='fx=i'y^ (^) for by the variation y becomes y + Sy, and therefore dyjdx becomes p + Sp = d(^+ Sy)/dx =p + d(Sy)/dx. Ti^- ^'=f f i^^-'^^+rf ^^'^ («) If we integrate the first term on the right by parts, we obtain ff^*-*=[f*]:-/*if-. ; <') where the symbol [ ]j means that the quantity enclosed is to be evaluated for x=0 and x=a, and the former value subtracted from the latter. But at each limit y is fixed, and therefore Sy=0, so that the integrated term vanishes. 184 A TREATISE ON DYNAMICS. [CH. III. We get then, 5.=f ( -^,f +f) 8y^- (8) and for the condition sought, -Pw^W-' ' (9) ax op oy Now, differentiating totally, we obtain du^-dudp -du^^df -din (10) dx "dp dx 3y dx dx\ 2p/ by the last equation. Hence u=p'^+C, (11) where (7 is a constant. Thus U is determined. This equation may be written VI +p _ V =5r+C or ^gy{l+p^) = igc, (12) where c is another constant. This is the differential equation of the curve. Since v = \l^gy and \/l+p^ = l/sin <^, if (^ = tan-'(l/jo), we have v = '2,>Jgcsm <^, (13) which is the result obtained above by James Bernoulli's elementary process. It is obvious from this result that the curve has a cusp at the starting point. We can also obtain easily the integral equation of the curve. By (12), 2„_^ P = -, y and therefore dx=—iJ=J- — (14) Integrated, this gives x= -V2cy-y2 + ccos-i(l-|j + 6, (15) where 6 is another constant of integration. This is the equation of a cycloid. By carrying Sf/ to terms of the second order, we should find the effect of these terms to be positive, and should therefore infer that the condition obtained above renders t a minimum. 102. Variational Method for Brachistochrone in Conservative Field of Coplanar Forces. In the more general case of any system of coplanar forces, we have , = t=r^i±idx, (1) §§101,102,103] BRACHISTOCHRONES. 185 and by precisely the same process as before we get as the condition of least time ^ _^a.^^i±Z ^f-n m rf^wr+p"^ v^ ^j ' ^' where it is to be remembered that the first differentiation is total, the second partial. This may be written, if a. denote the inclination of the element of path to the horizontal, that is (x. = tan~*jc>, in the form d^ sma. 1_ 3_ 1_ , dx V cos a.'dy V Hence by reduction, putting djdx='d/'dx+p'd/'di/, we get cos n. da. 1 / . "dv 'dv\ 1-= —„\ sin (x.= — cos a.;^— I ; V ax ir\ ox ay J or, since cos a. =dx/ds, sin a.=dy/ds, mv^ Cdv dy 'dv dx\ ,„. -R='^AWxd^-^Ts) (^) If now the forces, other than those due to the reaction of the guiding curve, be conservative, that is be derivable from a function V of the coordinates as explained in § 50 above, so that the equation of energy T+V=h (4) holds for the motion, we have „ clV 'dv „ "dV 3w ,-, Jl= — ~— =?nt»75-, Y= ~T^- = mv7K- (5) ox ox oy oy Hence (3) becomes ^f = zf^- F^= -Fees ^, (6) R ds ds where, as in § 100, B is the angle between the direction of the resultant force F of the field, at the element ds of the path, and the normal to the element. Hence we have again the theorem stated in § 100, regarding the reaction and the system of forces which would give the same motion unguided. We can now verify the cosine law of velocity assumed in (2) § 100. From (6) and the relation mvdv/ds = F am 6, we obtain (R dvlds)/v = - sin 6/cos 6 — = — tan t? . do, V which gives by integration v = Ceos Q, (7) where C is a constant. 103. Brachistochrone in any Field of Force. If the path is in space of three dimensions and the components of force are X, T, Z, we have, as before, for the time along the path as prescribed, t- -!^-' « 186 A TREATISE ON DYNAMICS. [CH. III. and therefore &= f^Sds-dsSv (2^ with vSds = xSdx+ySdi/+zSdz, (3) mds8v=(X8a;+rS2/+ZSz\dt (4) The latter equation follows from the equation of energy on the supposition that the forces are conservative. Hence, since the opera- tions Sd may be taken in the order dS, as the student may easily convince himself, fvSds_ f xdSx+ydS)/+idSz J "^-j p =l^^y[{s4f,^s4i,...)d. (^) Again, mj -^ = j ^ dt. Hence, since the integrated terms enclosed in [ ] in (5) vanish when evaluated at the limits — the starting and final points — we obtain by (2), as the condition of a minimum value of t, or, since Sx, Sy, Sz are arbitrary, and this relation must hold whatever values are assigned to them, d X , X -^ d ii , Y „ d z , Z ^ ,„. dt v^ v' dt «^ v' dt ir v' If we write vdlds for dldt in these, we obtain d l\dx\X ^ \ K^^S-4S)+^^=0' ••■'••■•/ Since the direction-cosines of the tangent to the path at an element ds are dx/'ds, dyjds, dzjds, and- those of the normal in the osculating plane are proportional to d^x/ds% ... (see § 17), we see that if I, m, n are the direction-cosines of a normal to the osculating plane — the binormal — j 7 ? jdx dy , dz l-j- + m-f -f- n-j- =0, ds ds as ,d^x d'^y dH . ^^+'"^ + '^58^ = °' and the three equations last obtained give lX + m7+nZ=0, (9) so that the resultant of the applied forces lies in the osculating plane. §103] BRACHISTOCHRONES. 187 If we multiply the first of the equations (8) by d'^xjds^, the second by d^i/lj, 0^2 . ^2 respectively, we can write (1) in the form X-^= CljCCj + 0ii^2'l /-|\ where tCj, x^ have the same meanings as before. [It will be noticed that here a^ = \. These are the equations of motion of the system of two spiral springs shown in Fig. 45.] Let now x^ = A^e^''\ X:^ = Aj''^^ (where i = sl —\), which will give simple-harmonic motion if n be real. The real part and the imaginary part of e"' will satisfy the differential equations separately, as will be found on trial, so that we can easily "realise" the solution. Substituting in (1), we obtain {^^ - a^x^-^\x^ = ^\ These give for the determinatioil of w^ the equation (w^ — ai)('n^ — 62) — 0^2^! = or n^ — (a-^->rh^n^-\-a^^ — a^.^^='^) (3) The roots of this quadratic in v^ are real if (aj-|-&2)^>4(a.j^&2~*2^i)' that is if {a-^^ — h^^^—^a^-^, which is always the case since a^ and h^ are positive. Mdreover, the roots are both positive. For the expression on the left of (3) is positive when -n,^ = -f 00 , and also when n^ = 0, since a-p^'^a,^^; it is negative when n^ = a.^ and when n^ = b„. One root, therefore, lies between + 00 and the :::}<^' 19Z A TKaATiSK UJN Ui^JNAMlUS. IVil. 111. greater of a^, b^ (that is a^), and the other root lies between the lesser of these (that is 63) and 0. Equation (3) also gives, as the student will easily per- ceive, the two modes of steady vibration of the system as a double conical pendulum. We can show that when the two connected pendulums are vibrating according to either one of the two modes, the vibrations are in the same phase. For, realising the values, of ajj, X2 assumed above, we write x-^ = K cos nt + L sin nt, X2=M cos nt + N sin nt. Substitution of these values in (2) gives {{n^-a^)K+b^M}cosnt+{(n^-a^)L + b^N}amnt = 0, {('n?-b2)M+a2K}cosnt + {{'n?-b2)N+a2L}8mnt and these must hold for all values of t. They will so hold if (ii^ — ai)(TO^ — 62) — ^■2^1 = (which is the equation already found f^r the determination of n^), and we obtain K _ _b^__ _b^ — 7i^ L_ b^ _ ^2~'^^ (K\ M~ a.^ — n^~ ctg ' N~ a-^ — 'n? a^ Thus we have \x2={a.^ — 'nP')x.^, or, which is the same, C'^\ = (^i~'^^)^2- Either of these shows that x^ and x^ have the same phase in the same mode of vibration. Taking the latter, we get ^2 _ "•f^i ~ ^iPl _ 9 ^2~'^l /gX since b2 = a2=g/l2- Thus, for the period, we have T=27rJ-2-^^ (7) which verifies the result otherwise obtained above, as to the length of the equivalent simple pendulum. 107. Double Pendulum. Discussion of Cases. Now it has been proved (§ 106) above that nn^, for one mode of vibration, is greater than a-^ (the greater of a^ and 62)) ^-^d for the other mode is less than h^- The period is 1-K\n, and we see from (7) that in the former case, that of the smaller §§ 106, 107] DOUBLE PENDULUM. 193 period, the ratio x-^jx^ is negative, that is the pendulums are at the same moment deflected in opposite directions, and that in the latter case, when the period has the greater value, the ratio x-^^jx^ is positive, that is the pendulums are at the same moment deflected in the same direction. It is obvious from the action of the forces applied by the cords to the bobs that such a difference of period must exist in the two cases. If mj be very great in comparison with mj, two cases arise, namely, l^ small and l.^ large in comparison with l^. In the former we have, approximately, ai=g/li, and, exactly, a^^h^^gjl^, while 6j is very small. Equation (5) of §106 gives approximately nn? = a^ + aJ3j{a^ — h^, and ■n^ = 62 — ct2^i/(«i — ^2)- Thus the period of the first mode of vibration — which is nearly lir-Jl^g and is that for which xjx^ is negative — is diminished in the ratio of l/\/l+«2^i/'*i(^i~"^2)' ^^^ ^^ period of the second mode — which is nearly 1-Ks]Wg and is that for which x^lx^ is positive — is increased in the ratio of l/Vl — ap^\h^(a-^ — b^). In the case of l-^ large in comparison with l^, we have still, if m^ be so great as to make b^ sufficiently small, both these approximations. In all cases regard must be had to the genesis of the motion, and this will be fully considered later in the dis- cussion of the compound pendulum [Chap. VII.]. But in the case of l-^ large in comparison with l^, if the large upper pendulum be set into oscillation, and so made to drive the lower pendulum, the period of the former will be little affected, but a steady oscillation of the lower will be set up, and the deflections from the original vertical will be on the same side for both — that is the oscillations of the driven pendulum will be, as we say, direct — and the actual period of the driver will be somewhat increased. But if the natural period of the lower pendulum be greater than that of the other, the effect of driving the lower by the upper will be to produce in course of time inverse oscillation of the former, that is the deflections will be on opposite sides of the vertical at each instant. The actual period of the driver will be slightly diminished. 194 A TREATISE ON DYNAMICS. [CH. Ill 108. Physical Analogues of Double Pendulum. We have a physical example in the fact that oceanic tides, produced by the rotation of the earth relatively to the tide-producing bodies, the sun and the moon, are, speaking generally, in- verted in low latitudes and direct in high latitudes. The rotating earth is here the driving pendulum. According to what is called the " canal theory " of the tides, the natural period (for an endless canal parallel to the equator) of oscillation of the water in low latitudes (where the canal is longer) is longer and in high latitudes is shorter than that of the tide-producing force, and hence the result stated. If the free period of the driven pendulum be equal or nearly equal to that of the driver, oscillations of the former of great amplitude will quickly arise. In the tidal case violent oscillations are not found at the latitude of transition from direct to inverted tides : the results of the theory illustrated by the complex pendulum are so modified by friction as to prevent such disturbances (see § 225). A good example, however, is that of two similar pendulums tuned to the same period, and hung opposite one another on the two sides of a plank of wood. When one is set in motion, the other is gradually started by the slight disturb- ances communicated to the common support. As the motion of the second pendulum increases, that of the first diminishes to almost zero. Then the motion of the second diminishes and that of the first increases, and so on con- tinually until the whole energy has been dissipated in overcoming friction in the air and in the support by which the motion is transferred. • The energy is continually exchanged from one pendulum to the other. A similar case of " sympathy of vibrations " was discussed by Euler in his papers " De Sympathieis Pendulis " {Nova Comment. Petrop. xix.). The transference of oscillatory motion from a beam to the scales suspended from its ends and back again was observed by Daniel Bernoulli, and described by him in the Petersburg Memoirs. Its theory was given by Euler in the papers referred to : his treatment of the problem and the discussions of later mathematicians have done much to further the theory of the oscillatory motion of connected systems. §§ 108, 109] DOUBLE PENDULUM. 195 Lord Kelvin has applied the theory of the complex pen- dulum to the investigation of the influence of the mode of suspension of a clock or chronometer on the rate of the time-keeper. His paper contains many instructive observa- tions ; we can only notice the following. The practice of hanging a watch on a nail (often followed by watch- makers), or in a bag or " watch-holder " hung on a nail, is objectionable, as causing a serious change in the rate of the watch : it is much better to lay it face up on a moderately hard cushion or under a pillow. A marine chronometer should be " firmly attached to the middle of a two feet long plank, with heavy weights near its ends," and this plank should be strapped down on cushions to avoid damage from the tremors of the ship. [See also the worked examples on double pendulums in Chapter VII.] 109. Two Connected Spiral Springs in Same Vertical. The theory of the double pendulum applies also to the arrange- ment shown in Fig. 45. A mass m^ is hung by a spring Sj from a fixed support, and from ■TOj is hung a mass m,^ by a spring Sj. If the system be displaced from the equilibrium position along the vertical, vibrations ensue which are given by equa- tions perfectly analogous to those which hold for the double pendulum. For let ajj, iCg be the down- ward displacements of rrij, ttv^ from the equilibrium position, Cj, Cg the forces, per unit elongation in each case, which these springs apply to the fastenings at their ends; then if, as we suppose, the masses of the springs be negligible, the spring Sj pulls up- wards on wij with a force CjCCj and the spring Sg pulls downwards on ■wij and upwards on m.^ with a force c^(x^ — x^. The equations of motion are m,X,= -C,{x,-X,), j ^^Fia. 45. or, if we write a^, fej, a^, \ for (Cj-|-C2)/'mi, cj'm-i^, cjm^, cjm^ (so that, as before, a^ = h^, which are precisely equations (1) of § 106. 196 A TREATISE ON DYNAMICS. [CH. III. The solution is in every respect precisely as before, with a quadratic for n^ identical with (3) of § 106, the roots of which are real and positive. The vibrations are along the vertical, and there are two modes, as above described, one in which the two masses move at each instant in the same direction, the other of shorter period, in which the masses are moving at each instant in opposite directions. The most general motion is compounded of these two motions superimposed. 110. Three or More Connected Springs with Attached Masses. If a third mass be hung from m^ by a spring Sj, the equations of motion are easily obtained. They are left as an exercise for the student, who may verify that a cubic is now obtained for n^, the roots of which are real and positive. Thus there are three modes of vibration in general, one in which the three masses are all moving at each instant in the same direction, one in which the two lower or the two upper masses move in one direction while the third moves in the opposite direction, and one in which the first and third masses move in the same direction while the second moves in the opposite direction. Similarly, the case of four or more springs with attached masses might be discussed. If there be p springs with p attached masses, an equation of the p*^ degree in n^ gives p distinct modes of vibration. Lord Kelvin has applied the theory of an arrangement of this kind to the dynamical explanation of the phosphor- escence of bodies. To the series of masses thus connected by springs is attached a terminal spring carrying a handle, by means of which a forced vibration of any desired period can be applied. For a description of the apparatus see Popular Lectures and Add/resses, vol. ii., or The Baltimore Lectures, passim. EXERCISES III. 1. A particle of mass m moves in a spherical bowl without friction. Axes are taken at the centre of the surface, z downwards. Show that the equation of energy can be written |m (i? + ^2 + i^) = mgz + h, where A is a constant. §§ 109, no] EXERCISES. 197 If r be the radius of the bowl, show that the normal force applied by the surface to the particle is given by N=-{2h + Zmgz). 2. Prove that if C denote the constant double rate of description of area by the projection of the radius drawn from the centre of the bowl to the particle, on a horizontal plane, Hence, show that the equation of energy can be written 2i2 = 2(^^+^2)(r2-22)-C2 3. The time * for any part of the motion of the particle between the planes z=Zq and 2=2:1 is given by _ r^i rdz where <^(a) = 2 {^+gz)) {r^-z^)-C\ Prove that <^(2)=0 has three real roots, one a between - 00 and -»•, another h between —r and z^, and a third c between ^q and +r, and that 6 + c is positive, so that c is always positive. [We have a6 + 6o + cai= -r^.] 4. Show that if 2=20 initially, where 6<% ^^- ^'X'"' ' "^^ 7 Now, as the reader may verify, TT >/(?• - b)(r -o) + \/(r + b)(r + c) _Tr ^%r^ + 2bc + 2\/(^^^b^)(r^ - (?) so that dao lies between two limits which are both greater than ir/2. 8. Show that as 6 and c approach equality, dao approaches the value Trrjslr^+Zc^. [The value of 9bc cannot exceed it (Halphen, Fond. EUip. t. ii., and de Saint-Germain, Bull, de Sci. Math., 1901.] 9. Prove that if the particle is projected horizontally on the surface, in the plane 0=0, with speed v^, the values of 6 and t from the starting point are connected by the relation 2 r sl'igif-z'') III.] EXERCISES. 199 Here C^ = vlr^, 2hl'm,=vl, and therefore the values of t and 6 (Exs. 3 and 7) become _ r rdz r v^r^ds From these the result stated can be deduced. It is to be noticed that the upper limit of Vf,^lzjs|%g{r'^-z''') is 1, and that therefore z cannot exceed the value given by 'igz^ + v^z='igr'^. The value of z is given as an elliptic function of t, and this with the relation stated above enables x, y, and z to be expressed in terms of t. [This theorem is due to Sir George GreenhilL] 10. Reckoning t from the instant at which the particle is at the lowest level it can reach, so that z must be negative, prove that, if = c-(c-6)m^, F=(c-6)/(c-a), and k=»J'iig{c-a)l'i/r, z is an elliptic function G-{c-b)sn^Xt, which has the real period 2 p du 11. Write down the equations of motion for a, position of the particle very close to the bottom of the bowl, and hence show that the X and y equations are x+^x = 0, y+^y=0. Hence prove that if when t=Q, x=x^, y=0, j6=0, ^ = «'o) the path of the particle is an ellipse of semi-axes Xq, VQ\/r/g. 12. A particle moves on a concave surface of revolution, the axis of which is vertical. The origin of coordinates z, p is taken on the axis ; z is measured downward, p horizontally, and z=f(p) is the equation of the surface. Prove that the energy equation is im{p%l+n + pW} = mgf{p) + h, where /' stands for f'{p). Prove also that the description of areas by the horizontal radius p leads to the equations f:''Jr^r^T^ VJfi'/W+^l-c^ where t is the time of passage from the distance Pq from the axis to the distance p, and O-dgis the angle turned through by the horizontal projection of the radius- vector in the same time. 200 A TREATISE ON DYNAMICS. [CH. The square of the resultant speed at time t is p2 + p2^ + i2 = ^2(l+/'2) + p2) he negative. Thus, we get , jt^-^ where p is the radius of the parallel and v the speed with which it is traversed. 15. Show that, if c^ = C^ml2h, the equations of Ex. 12 become when the particle is under no force, except the reaction of the surface. c2" 16. Prove that the particle in this case moves along a geodesic curve on the surface. The value of v is in this constant and equal to Vg. Also h—^v^, and therefore G^mj2h=G^/vl. Hence (Ex. 13) psma.=c. This relation is a characteristic property of a geodesic. [A geodesic is a curve drawn on a surface so that its osculating plane at each point contains the normal to the surface at that point.] III.] EXERCISES. 201 17. Find the path of the particle on a right circular cylinder when no force except the reaction of the surface acts on the particle. Here jO is constant, =a say, and therefore sina = c/ci!. The path is therefore a helix on the surface. This is otherwise evident. 18. An india-rubber tire of cross-sectional area a is shrunk on the wheel of a motor car, and the tensile force per unit area in the tire is T. If r be the radius of the wheel and P the normal force per unit length of the rim exerted upon the wheel by the tire, show that when the car is at rest P=aTjr. The density of india-rubber is 112 lbs. per cubic foot. If the tensile force in the tire when the car is at rest is 224 pds. per square inch, show that the maximum possible rim-velocity is nearly 65 miles per hour. 19. A particle suspended from a fixed point by a string of le ngth a hangs vertically ; it is projected horizontally with speed sJ'Jaq/Z ; show that the string will become slack when the particle has risen to a height 3a/2. 20. A particle is projected from the lowest point of a vertical section of a smooth hollow circular cylinder', of radius r, whose axis is horizontal, so as to move round the inside of the section. Prove that if the velocity of projection is ^'Jgr the particle will leave the circle when the radius through it is inclined to the vertical at cos~\2/3). Prove also that the particle will rise to a total height of 50r/27 above the point of projection. 21. A particle moves under gi'avity in a smooth groove in a vertical plane. Write down the equations from which the velocity, and the reaction of the groove on the particle in any position, can be obtained. If the groove have the form of the parabola 3c^='2,Ku^yjg, with axis vertical and vertex upward, and a particle of unit mass is projected horizontally from the vertex along the groove with- speed ii, show that at a point where p is the radius of curvature, the reaction of the groove on the particle is v?{K-\)lp. 22. A particle starts from rest at the highest point of an ellipse of eccentricity e placed with its major axis vertical. Show that if there be no friction the particle will leave the curve at a point for which the cosine, z, of the eccentric angle fulfils the equation eV-32-t-2 = 0. 23. A heavy particle moves on the inside of a smooth paraboloid of revolution, axis vertical, vertex downwards, latus rectum 4a. Prove that when it describes a horizontal circle its angular velocity about the axis is slgj^a. If Xi , x^ are its greatest and least heig hts a bove the vertex, show that the corresponding speeds are ij^gx^^ slzgx-^ , respectively. Prove also that when it is at a height x its angular velocity about the axis is {'Jgx^x^l'2.a)lx. 202 A TREATISE ON DYNAMICS. [CH. 24. A train is running smoothly along a curve at the rate of 60 miles an hour, and a pendulum which would ordinarily oscillate seconds is observed to oscillate 121 times in 2 minutes. Show that the radius of the curve in which the train is running is very nearly a quarter of a mile. 25. A heavy particle of mass m, moves within a smooth circular tube (radius I) in a vertical plane. It starts with speed V from the lowest point ; show that when the line joining the particle to the centre of the tube makes an angle Q with the vertical, the force applied by the particle to the tube is Zmg cos Q — 'img+Tn V^/l. A carriage of mass 30 pounds moves round the inside of a vertical circular track of radius 8 feet. Its speed when at the lowest point is 40 feet per second. Find the speed at the highest point, and the reaction of the carriage against the track. 26. The bob of a simple conical pendulum of length I, suspended from a point 0, is constrained to describe a horizontal circle of radius ^l on the inner surface of a smooth sphere of radius a, of which is the hig;hest point. If the angular speed of the pendulum be "J^g/l, determine the stretching force in the pendulum thread, and the thrust on«the surface of the sphere. 27. The bob of a simple pendulum is drawn aside through a right angle and let go. Prove that when the th read make s an angle 6 with the vertical, the resultant acceleration is \/l + 3 cos^ft 28. Two spiral springs are connected in a vertical series. The two supported masses are 5O0 grammes each, and each spring is of such strength that 100 grammes produces an extension of 3 cms. Find the period-equation and solve it. Give also the integral equation when the initial displacements are (1) +1 cm., +1 cm., (2) +1 cm., -1 cm., and the initial speeds are zero in each case. 29. A rocket is fired off and rises vertically. If m is the mass burnt, and E the mechanical energy generated per unit time by the burning, prove that the speed of the burnt products relative to the rest of the rocket is 'JzEjm. 30. PQ is a focal chord of a parabola lying in a vertical plane. If PQ is vertical, and TP and TQ are tangents drawn to the parabola at P and Q respectively, show that heavy particles started simul- taneously from rest at P and T, and falling along the lines PQ, TQ, will reach Q at the same instant. 31. A number of heavy particles start from rest from a point and slide down straight lines inclined at various angles to the horizontal. Show that the locus of the points reached by them with a given speed is a horizontal plane ; show also that the locus of the points reached by them in a given time is a sphere whose highest point is the starting point. III.] EXERCISES. 203 32. A particle falling freely from rest in vacuum acquires a speed L in j8 seconds. The same particle falls from rest in a medium in which the resistance varies as the speed ; its limiting speed is L. Show that the speed after time -/^jS seconds from the start of the motion is nearly \L^ and that after j^J/S seconds it is nearly f i. 33. A particle of mass m moves in a straight line under a constant force F in the direction of motion and a resistance ct^, where c is a constant and v the speed. Show that if V be the speed acquired in traversing a distance & from rest, -5= 57 log 2c ^F-cV^' 34. A train moves on a level at Ffeet per second under a resistance of R pounds per ton given by ^=6 + '009F^. Its mass is 100 tons, and a speed of 30 miles per hour is acquired in travelling 1 mile from rest under a constant tractive force F. Show that i^is 1'31 Tons and that the limiting speed is nearly 36 miles per hour. 35. A particle is projected vertically upwards with an initial speed V in a medium whose resistance varies as the square of the speed. If L be the speed for which the resistance offered by the medium is equal to the weight of the particle, show that the time of ascent is i(tan-' VIL)lg, and the distance ascended is Z2{log(l + V^/L^)]j2g. If the speed of projection be small in comparison with k, show that the particle returns to the point of projection with speed V(l - V^/2Z^). 36. An engine capable of exerting a maximum pull of P Tons can draw a train weighing M tons with speed V on the level, against resistances which vary as the square of the speed. Prove that the limiting speed of the train when running without steam down a hill inclined at an angle a. to the horizontal is V\/Msina./P, and that the maximum speed with which the train can ascend the incline is V^l-Msma-IP. 37. A ship of 1000 tons displacement is towed at a uniform speed of 15 miles per hour, the pull required being 25 Tons. If the towing rope be slipped, prove that the speed of the ship will fall in five minutes to about j^ of its initial value. [Assume the resistance to vary as the square of the speed.] 38. The engines of a steamer going at full speed are reversed and the steamer is brought to rest in a distance d. Prove that d=iMV^I2F)logA where M is the mass of the steamer, V is full speed, and F is the propelling force (supposed the same for motion ahead and astern). The resistance to the motion is supposed to vary as the square of the speed. 204 A TREATISE ON DYNAMICS. [CH. 39. On the experimental law that the resistance of similar steamers is proportional to the wetted surface and to the square of the speed, prove that if a 6 ft. model run at a speed of 2 knots in an experimental tank experiences a resistance of 0'2 pound, a similar steamer 600 ft. long and having a displacement of 10000 tons would experience when run at 20 knots a resistance equivalent to an incline of 1/112, and require over 12000 effective horse-power. 40. A ship is steaming at a speed of x knots (relatively to the water) against a tide the speed of which is a knots. If the resistance to motion varies as the ti"" power of the ship's speed through the water, show that for maximum economy in fuel consumption »+l x= a. n 41. The motion of the bob of a simple pendulum, of length I, is resisted by a force proportional to the speed. The force is equal to the weight of the bob when its speed is nsfgl, where m is a large number. The pendulum is performing small oscillations. Prove that if 1/ji^ is negligible, the period is unaffected by the resistance, while the amplitude of the oscillations diminishes in n periods to about 1/20 of its original value. 42. A thin uniform spherical shell of mass m. is filled with a frictionless liquid of the same density. The system descends a rough inclined plane from rest in time *j, and a solid sphere of the same density and radius makes the same descent in time t^. li M be the whole mass in each case, show that t^J(l=2lM/{15(M- m)+ l(hn}. 43. A system which has one degree of freedom has kinetic energy T=^fjiffi, and potential energy V=f{ff). Prove that the motion is tautochronous if r=c(f[idoY. [Appell, C.E., 1892.] Put ij,d=S : then V=^CsK Hence the theorem by Ex. 8, § 98. 44. Under what condition may the system for which T=i(A^+2Be4,+ G^), V=f(e,) where A, B, G are functions of 6 and <^, be tautochronous? [Appell, loa. cit.'] Put (l> = F{ff), and use the last example. 45. Prove that if a particle move under gravity from rest on one curve in a vertical plane, to another curve in the same plane, in the shortest time, the path is a cycloid which meets the lower curve at right angles and has a cusp on the upper, and that the tangents where the path meets the curve are parallel. It follows from § 101 that the path is a cycloid, with a cusp as stated. The variation of the time of passage due to displacements of the ends along the curve is to be found. The values of this when first III.] EXERCISES. 205 one end, then the other, is fixed must vanish separately, and the results stated are verified. 46. Prove that a given plane curve will be a brachistochrone for a central force F=fj.jr^, and a free path for a central force fJ,'lr"', if n + n' = 2 and the speed in each case varies as a power of the distance r. [See the last paragraph of § 103.] 47. Show that the lemniscate of Bernoulli is a brachistochrone in a field of potential fir^, where r is measured from the node. Find the necessary speed. 48. Show that a given plane curve is a brachistochrone for a particle under a central force varying as pdpjdr when the speed vanishes with p. CHAPTER IV. RESISTED MOTION OF A PARTICLE IN A UNIFORM FIELD OF FORCE. 111. Uniform Field. Resistance kv. We have considered in § 43 the motion of a particle under a force proportional to its distance along its path from a fixed point in the path, and resisted by a force proportional to its speed at each instant, and in §§ 84 ... 86 the rectilineal motion of a particle under a constant force in the line of motion, and resistance varying according to different powers of the speed. We now take the more general case of a projectile in a uniform field of applied force (such as that to which the field of gravity is an approximation), in which a particle is acted on by a force proportional to its mass, and shall suppose that the particle is subject to a resistance according to some power of the speed. For the sake of brevity of reference, we call any direction perpendicular to the field a horizontal direction, and refer to any line of applied force as a vertical, and speak of the downward or upward vertical according as the direction is with or against the applied force. We have then, for the horizontal motion of a particle of mass TTi under a resistance /cS proportional to the speed s, the equation / ^v m{x + ks -j-j = m{x + kx) = (1) It will be noticed that the component resistance in the direction of x is in this case proportional to x. The same thing holds of course for y. For the motion along the. up ward vertical, we have m{y + ky)= -mg, (2) § 111] RESISTED MOTION IN UNIFORM FIELD. 207 where g denotes the uniform force per unit mass on the moving partial^. Thus, we have the two differential equations m+kx = 0, y + ky=-g (3) It is clear, in the first place, that ij is zero when y is such that ky+g = 0, that is when y=—gjlc. Thus, when the particle has a downward speed =glk, the resistance just balances the downward force of gravity, and there is no acceleration. The particle then falls with uniform speed We denote this limit of speed by L. Integrating (3), and putting Fcos a, F'sin a for the initial values of x, y, we obtain «+fec=F'cosa, y-'rlcy^ —gt+V sm a. (4) We multiply by e*', and again integrate, determining the constants by the conditions that a; = 0, y = when ^ = 0. The results are hx= Fcos 0.(1-6-*'). ky=-gt + {L+ Fsina)(l-e-*')- (5) From the first of these, we get , _ , Fcos a. ° Vcosa. — kx' and therefore the second becomes y=-I^°g FcosT-fa + KFcL^+^^^4 (^> which is the equation of the path (the trajectory) of a particle projected from the point x = y = 0, with speed F in a direction inclined at the angle a to a plane drawn through the point of projection perpendicular to the field. The first of (5) shows that when t is very great, X = Fcos a/k, and that then the horizontal velocity is zero. The second of (5) shows that the speed is then L, and that the motion is then vertically downward. Hence a vertical line at distance Fcos a./k from the point of projection is a tangent to the path at a great distance from the origin, measured along the path ; that is it is an asymptote to the path. If a tangent to the path be drawn from the point of projection 0, to intersect this vertical asymptote in T^, then. 208 A TREATISE ON DYNAMICS. [CH. IV. clearly, V=lc. OT^. But any point P, at which the speed is V, may be taken as the point of projectioi^. Hence, if the tangent at P (Fig. 46) intersect the vertical asymptote in T, we have v = k. PT. O O' Fio. 46. [The points 0, 0' are not on the same level.] If in (5) we put t= —cc , we find an asymptote at a point far anterior to the point of projection. The inclination of the trajectory there to the axis of x is tan-i(y/i;)j^_„ =tan-i{(i+ Fsin a)/ Fcos a} ; and a line touching the trajectory at the point thus suggested is another asymptote. We shall write j8 for tan-^{(i+ Fsinot.)/Fcosa.}. This asymptote is of much use in the construction of the trajectory. The equation of the trajectory, with reference to horizontal and vertical axes drawn from any point of the path as origin, can be written y=xta,n^—Lt, (7) §111] RESISTED CURVILINEAR MOTION. 209 which shows that if a line be drawn from any point, P say, at inclination j8 to the axis of x, a point on the curve whose abscissa with reference to P is x, is, after an interval t, vertically below the corresponding point on the straight line at a distance equal to that which a particle would travel in time t at the speed L. Moreover, the perpendicular distance of the point x, y on the curve from the line y = x tan B is Lt cos /3. The component velocity at right angles to this line has thus the constant value Zcos/3. It follows that at the point M (Fig. 46), where the direction of the tangent is perpendicular to the line PZ7, i/ = a;tan/3, the speed is L cos j3, and this is the minimum speed in the path, as we shall see presently. Differentiating (7) with respect to the time, we obtain y=d)ian^ — L, (8) or, if we put f, rj for x, ij, ;,=rftan/3-X, (8') the equation of the hodograph, which is thus a straight line op (Fig. 47) inclined at the angle ^ to the axis of x. Fl8. 47. and passing through the point ^= 0, >j = — Z. The velocity at any point P of the path is represented in magnitude and direction by the line op drawn from the hodographic origin in the direction of the tangent to the path at P to intersect the line (8), that is wp, in f (see Fig. 47). The shortest line 210 A TREATISE ON DYNAMICS. [CH. IV. which can be drawn from o to meet the line (8) is the line om, which meets the hodograph at right angles. Hence, as stated above, L cos ^ is the minimum speed. 112. Resistance kv. Trajectory. For the purpose of giving a graphical representation of the path, we calculate the co- ordinates (1) of the vertex, (2) of the point at which the speed has the least value. By (4) of § 111 , «= Fcos ol — Jcx, hence y = {Vco3a. — kx)ta,n^ — L, and therefore at the vertex, where ^ = 0, Jcx= Fcoso.— r — o= Fsinacot/3, (1) since tan ^ = (L+ Fsin a.)/ Fcos a.. The horizontal distance of the vertex from the vertical asymptote is thus L/k tan /8. Again, by (4) of §111, when y = 0, y= —Lt+Vama./k. But by the first of (5) we can eliminate t from the ex- pression for y and obtain with the value of a; in (1), , J. 1 Fcos a. tan B „ . Icy==^ —L log J 1- -I- Fsin a. If Xq, y^ denote the initial horizontal and vertical speeds, and v^ the minimum speed, we obtain for the vertex kx = " sin/3' •(2) Again, at the point M of minimum speed Vm, i:= X cos /3 sin /3, y= —L cos^/3. Hence, for this point, since «= Fcos «.—/«, y+ky=—gt+Vsm.a., kx^x^ — VrnSai^, I % = ..cos/3+3/o-i^log^^.| (^> If for any point (for example, the point P) on the trajectory, a line be drawn at inclination /3 to the hori- zontal, the intersections of this line, and the tangent at P §§112,113] RESISTED CURVILINEAR MOTION. 211 with the vertical asymptote, will be at a distance Ljk apart, which is the same wherever P may be on the curve. For we, find for the distance in question the value y(cos a. tan /3 — sin cl) = j i^) by the value of tan;8. By this means we can draw a tangent to the curve at any point P, the position of which is known. We have only to draw a line from P at the angle j8 to the horizontal intersecting the vertical asymptote in V, then measure down a distance Ljk to a point T on the asymptote. PT is the direction of the tangent at P. In comparing for a given initial speed the trajectories in different media, each of which resists directly as the speed, we take in each case the vertex V (Fig. 46) as the point of projection, so that the initial speed is hori- zontal. We have then tan^ = -^ = -|^ or lc = ^coip (5) Now the distance of the vertical asymptote from the vertex is Vjk. Thus if D denote this distance {O'L in Fig. 46), tan^=T^ or k = jyCot/3 (6) For simplicity we take k as equal to cot/3, that is we choose the scale of the diagram so that g/V=l ; (6) shows that this amounts to making in the drawing D = L. 113. Construction of Trajectory. We shall now draw the trajectory in this way for /3 = 30°, and therefore make k = /J3. We lay down first the vertical asymptote, and choosing a point L upon it lay off" the arbitrary length LK. From L we draw the line LQ inclined at 30° to the vertical, and through K draw the line KMO' at right angles to LQ, meeting' ZQ in M and the horizontal line through L in 0'. Then, if we suppose O'L to represent the speed L, the line LQ is the hodograph turned through 90° in the plane of the diagram. Hence, any line O'Q represents the velocity at a point P in the path where the tangent is at right angles to (JQ. As the projectile moves in the path, the 212 A TREATISE ON DYNAMICS. [CH. IV. velocity of Q along the hodograph is the acceleration in the path ; and Q moves as a particle in a medium resisting as the first power of the speed. With the value of k taken, nve have for the coordinates of the vertex referred to any origin on the trajectory, ^ = *otan/3-^. 2/ = 2/otan^-Ztan^logfL!ilL^ ...(1> "^ C0S;8 Vm (where iig, y^ refer also to the point 0), and for the co- ordinates of the point of minimum speed, sm lm \ " ^ ""cos^ . I (2) 1/ = -y^ sin /3 + 2^0 tan /3 - i tan ;8 log ^^-^-g- j Thus, since the distance of the asymptote from the origin is dijk, the distance of the asymptote from the vertex is Vm/cos8. Hence, if we take M as the point of minimum speed, O'M is. v^ and O'L is Vm/cos^, and the vertex is on the ordinate through 0', as already shown above. The coordinates of 0' relatively to are a;'=i»gtan/3- «o«^ ^ \ (3) 2/' = yotan^-Xtan^log^^-^.j Subtracting y' from the value of y for the vertex, we find that the distance of the vertex from 0' is ZtanjSlog(l/sin^|8) or Z tan /3 X logc4, for (8 = 30°. Multiplying the distance LK of the diagram by loge4, and laying oft' a length equal to the result from 0' along the vertical, we reach the vertex. 114. Besisted Motion. Tangential and Normal Resolution. We now consider the motion of a projectile under a resistance in its line of motion proportional to a higher power of the speed than the first. It is convenient to use here the tangential and normal resolu- tion of accelerations that has been explained in § 10 above. There it has been shown (1) that the acceleration along the normal to its path at each instant is v towards the centre of curvature, where (j> is the angular speed (taken positive) with whiqh the tangent is turning at the §§113,114,115] RESISTED CURVILINEAR MOTION. 213 point, and (2) that the acceleration in the direction of motion is simply V. It has also been shown (§§ 10 and 19) that "=-" rfs-2' ^•^=p' ^^> where p denotes the length of the radius of curvature. If the resistance per unit mass be some function /(i;) of v, we have i=-i^-^i=-f{'>')-gsmir, (2) where t/' is the angle (Fig. 46) which the forward drawn horizontal line PH at the point makes with the forward drawn tangent PT to the path. But now = —xp, and therefore, if u=x=vcos'\jr, — v\j/=g cos yjr, - v^\p=gv cos iff =gib (3) Also, since the force in the horizontal direction is the resolved part of the resistance in the line of motion, -jf= -/Wcosi/r (4) The first of equations (3) gives, since m = «cosi^, — uip=gcos^\jf (5) Let tani/r ( = di//dx) be denoted by p ; then, if u be known as a function of p, the following relations derived from (5) are useful. Divide (5) by m^cos^^ ; then, since ^/u=dyj/-ldx and {d-^ j dx)l cos^^ = dp jdx, we have — =- = — (6) dp g Divided by a, this is _.^=!^ (7) dp g Again, multiplying (6) by p, we get -pdxldp=pu^jg. But since p = dyldx, pdxldp=dyjdp. Hence -f=p''~ (8) dp ^ g Also VI +p^ . dxjdp = ds/dp, where ds is an element of the path. Hence _^=Vh:^2^ (9) dp ^^ g Thus the last four equations enable x, t, y and s to be found by direct integration if u is known as a function of p. 115. Resistance = kv". "We now suppose that /(«) = fe'', where n is some positive integer. If the particle were to fall vertically in the resisting medium, it would finally attain a velocity X, at which 214 A TREATISE ON DYNAMICS. [CH. IV. the downward acceleration would be zero. Then ]clJ'=g, and therefore L={g/ky. Now, from (4) and (5) of § 114, we obtain, with »^ = *,,"« = *(^) , (1) since {duld£)l{d^ld()=duld^. Thus we have du _ d^ ,„. ■M»+^~(cosi/')"+.i V ; Integrating from ' (^) Here, if n be even, so that m — 1 is odd, the positive square root of (l+jo^)""' is to be taken, since the subject of integration is to be positive throughout the possible range of integration, that is from P=P^ top= —00. If the starting point of the integration be the vertex of the path, where p=0, and Mq now refer to that point, we get, putting F(p)= rii+p^f''-''dp, Jo the equation ^^'UIlY -nF(p)Y (5) f=vi{(|)"-"M* <«> which is the polar equation of the hodograph in terms of v and i^. 116. Particular Cases. Hodograph. Intrinsic Equation of Path. If M=l, F{p)—p, and we obtain, still taking the vertex as starting point, L(l/ug-l/u)=p, an equation which can also be deduced from (8) of § HI. We have also vttF' ^^^ that is —veos'\jf — vamTfr= — ^ — r] = L, (2) if ^, 77=»(cos yjr, sin T/r\ The hodograph is thus a straight line inclined at the angle tan~'(Z/MQ) to the horizontal, and passing through the or V \U(, ^ I §§115,116,117] RESISTED CURVILINEAR MOTION. 215 points whose horizontal and vertical coordinates are «q, 0, and .0, -L. This result has already been obtained in §111 above. (See Fig. 46.) If n=2, the law of resistance moBt nearly fulfilled in a large number of cases in practice, 2i?'(p) =p\/r+p + log(^ +\/lTp^). Therefore ^=;7j= {(|)'-p^/^TP-log(p + ^/^+?)} , (3) the equation of the hodograph. Expressed in terms of coordinates ^,17, it is .r\2 , ri 1 tj ^^-'/VF+?-^^log||(^+Vr +')')} = ^ (4) If «=3, ZF{'p) = Zf-^'p^, and so the equation of the hodograph is or i^i^-Z^\-r?==U (5) From the nature of the case, ^ cannot have a negative value ; its smallest value is zero. It will be seen that each hodograph gives for 1 = 0, 1)= —L. Hence we infer that in each case the path has a vertical asymptote. If the expression on the right of (5), § 115, be denoted by Q, we have, by equations (6), (8), (9) of § 114 for the equation of the path, gh Q" ^ 9 Jo Q" ff h Q' ^' ^' the last of which is .the "intrinsic" equation. For the time of passage, we have, by (7) of § 114, 4/:f <" 117. Intrinsic Equation of Path for Resistance kvK We can find the intrinsic equation of the path for the case of resistance pro- portional to v% as follows. By (4) of § 114, we have — = — fc^C0Sl/r= —kUj, (1) dt Oct Hence, dividing both sides by u and integrating, we find M = «o«"*', (2) if s start from the point at which m=Mo. Now, from (3) of § 116, we have ^2 7-2 ■^=^2-P^^^-'^'>S(P+'J^^i¥) (3) 216 A TREATISE ON DYNAMICS. [CH. IV. and therefore the value of u just found gives |-2(l-e^)=px/i+p + log(jtj + v/T+P), (4) which is the equation sought. The point for which w = Mo is here that for which ^ = 0. The same equation can be obtained at once from the last of (6) (§ 116). For here q^^I^jul- 2F(p), and since F(p)= ["(l+p'rdp, Jo F\p)=iJ\+p'^. Thus, integrating the equation just referred to, we get, since L^^gjk, ^ ^2 ~^^¥k^°^ L^ -9.ulF(p)' which agrees with (4). It is clear from the relation 'U,=u„e'^ that as s increases towards + x, M diminishes towards zero. Equation (4) shows that then p increases numerically towards - ao , that is the motion approaches more and more nearly without limit to the vertical. The path has, like that for the case of resistance simply proportional tospeed, a vertical asymptote at a finite distance from the vertex on the right, as shown in Fig. 46. To see that this distance is finite, consider the integral [see (6), § 115] ., x= - \ —dp j-i g taken from p=—q (where g' is a small finite positive quantity) to p = - 00 . Then x is the horizontal distance between the points for which p has these values. Now, if we take from (5) of § 115 (with the value of F{p) for m=2), L^lu^=j^ throughout this integration, we shall take l^jv? too small, and therefore u^ too great. Thus we have r'l^ dp J-i 9 f' that is x<\lkq. Thus x is finite taken between these limits, and must also be finite taken from ^=0to^=-<30. 118. Flat Trajectory when Resistance = kv^. If the trajectory be so flat that we may identify s with x, equations (6) of § 114 and (2) of § 117 give for the case of resistance =fo)*, the relation d/p_g ^,e^=k ©^^ (^) dx u^ \U(,/ Hence, integrating and determining the constant by the condition that, when x=0, p—q, we obtain ^.-U^ p = q i\i |)V-1) (2) §§117,118,119] RESISTED CURVILINEAR MOTION. 217 Integrating again, and putting «/=0 when x=0, we find y=9'''Tk{^y^''^-^^''-'^) • (3) "■0' for the equation of the path. For the time of flight we get, by (2) of § 117, dt 1 1 da; II Uq Hence, integrating and determining the constant by the condition that when t = 0, ,» = 0, we obtain '=^(«"-^> w If the resistance to the motion were to cease at the point x=y=0, the particle would thereafter move in a parabola of semi-latus rectum l=uyg (see § 21). Hence we may write (4) in the form *-ikr-'^ (^) Equations (3) and (5) are formulae sometimes used by artillerists. Expanding y from (3) in powers of kx, we get '=^--S(?-^4+-) y=^''-i-^4r-^ (6) which shows, by the third term on the right, how, in the case of slight resistance, the trajectory deviates from the parabolic form given by y=i''-i w 119. Experimental Laws of Resistance to Shot. Ballistic Tables are given m the Text-Book of Ounnery used at the Ordnance College, Woolwich,- and contain the results of very elaborate experi- ments made by the Rev. F. Bashforth, B.D., in 1865-1867 and in 1878-1879, by means of a chronograph which enabled the speed at different points in the path of the projectile to be ascertained. We have no space in which to pursue the subject in its more technical aspects, but the reader will find full information in the text-book referred to as to resistances for different speeds and diflerent projectiles, times of flight and distances traversed between different speeds, altitudes attained and so forth, with examples of the solution of practical problems by the tables. One point may however be referred to. No simple law of resistance is found to fit the experimental results. For very low speeds the curves there given show resistance 218 A TREATISE ON DYNAMICS. [CH. IV. at first nearly proportional to the speed, then resistance increasing more rapidly, a sudden further increase at a little over 1000 feet per second, and then a range from about 1100 ft./sec. to 2200 ft./sec, which the tables show to be one of resistance nearly proportional to the cube of the speed. About 2300 ft./sec. there is a sudden lowering of the upward slope of the curve of resistance, that is at the speed at which air rushes into a vacuous space such as presumably exists at the base of a verj' quickly moving projectile. When the projectile moves at a higher speed than that of sound — about 1100 ft./sec. — waves produced by its progress cannot outstrip it, and therefore the projectile constantly moves forward into undisturbed air. Observations were made by Newton in 1687, of the time taken by balls of different diameters and weights (glass shells filled with different materials) to fall a distance of 220 feet from the dome of St. Paul's Cathedral ; and it was then found that the resistance at a given speed was proportional to the square of the diameter. This result was confirmed by Bashforth for projectiles of dififerent shapes. Whatever the shape of the shot used — ogival,* hemispheroidal, spherical, or flat headed — the resistance for each shape was propor- tional to the square of the diameter. The relative resistances may be taken as 2 for flat headed shot, 1-7 for spherical cannon balls, ■95 for modern pointed projectiles, and -8 for the magazine rifle bullet. Mr. Bashforth constructed a table of values of a coefficient K, which, used in the equation f v \^ ^=nioooJ' gave the resistance p in pounds on an ogival headed shot of 1 inch diameter moving at a speed of v it.lsea. The following short extract gives some of the numbers : •0 K V K 100 578-1 1100 106-9 150 385-4 1200 109-6 200 289 1400 104-7 300 192-7 1500 97-9 500 121-9 2000 ■68-8 1000 75 2800 52 EXERCISES IV. 1. If r denote the retardation produced by the resistance of the air (a given function of the speed) and i/f denote the angle defined in 8 114, prove that at a point of minimum speed r-|-^sini/r=0, that where the curvature of the path is greatest (that is, where i^/J is *Shot having a cylindrical body and a pointed head the longitudinal section of which is formed of two arcs of equal circles. § 119] EXERCISES. 219 numerically a maximum), r + ^g sin -^ = 0, and that where tf/ is numerically a maximum, r + 25fsini^=0. 2. Let the resistance vary as the speed, and draw through an origin on the path a line parallel to the oblique asymptote meeting the vertical through a point P on the path in F. Show that if ^, 7) denote OF, PF, Vf, be the speed at 0, a the elevation at 0, and t be the time from to F, ^ = {vl + I^ + 2Lv„sma)^(l- e-")/k, ri=Lt. 3. In a medium in which the retardation is kv^, the length of the arc, measured from the point of projection to any point at which tan'\/r=p, is s. If the medium had been non-resisting the length of this arc would haye been S. Prove that ,S=^(«--1). 4. Show that if the semi-latus rectum of the unresisted trajectory be I, and a. be the angle of elevation, tana.-^ = 2j^(e2»»-l) for a flat trajectory. 5. A projectile under gravity is resisted by force fo". The speeds at the two points where the inclinations of the direction of motion to the horizontal are i/r and n- — ^ are Vj, v^, and v is the speed at the vertex. Prove that 1 1 _ 2cos"T/r < <~ «" Prove also that Jl _ 1 = ?^ eos'h/r [* sec"+ii/f d^r. vl d" 9 Jo 6. If the equation of the trajectory be cosi/r=/(pcosi/r), find the law of resistance. [Here, by (2) and (4), § 114, «= — j-cosi/r, v= — r-^sini/r, where r is the retardation, so. that rvd(cos'\lr)/dv= —gsm'[jrd(-vcos-<{r)/dv; and, since v'^/p =g cos i/r, cos ■\jf = f{v^lg), so that, since the function / is given, r is found.] 7. If p cos 1^ is constant, the path is the catenary of equal strength. Let the concavity be downward. Show that v is constant, and that ?•= -g'sim/r, so that r is a positive acceleration on the upward slope and a positive retardation on the downward. 8. Apply the result of Ex. 6 to the parabola p cos^=2fli. 9. If y8 have the meaning assigned to it in § 111, prove that if the retardation be kifl the speed is a minimum at the point given by the negative root of p^ — tan /3(tan'''y8 -H 3)^ -1=0. CHAPTER V. FREE MOTION OF A PARTICLE UNDER A FORCE DIRECTED TO A FIXED POINT. 120. Path lies in a Plane. Differential Equation. For a particle moving under the action of a force continually directed towards a fixed point or "centre of force," the equation of motion has been found in §§31 and 50. It has been seen that the path lies in a plane, and that the full determination of the motion is theoretically possible when the law of force is given, and the differential equation can be integrated in accordance with the specified initial conditions. Before proceeding to the discussion of some important particular cases, we give another proof of the fundamental differential equation, including the case in which a force acts on the particle in the line of motion ; so that we may have before us all that is necessary to deal with the motion of the particle in a resisting medium, or against such a resisting force as there seems reason to believe may be experienced by a planet absorbing the sun's radiant heat and light, and kept at equilibrium of temperature by its own radiation. [See Nature, Aug. 4, 1910.] Let 1/tt be the distance of the moving particle P (Fig. 48) at time t from the centre of force — or, as we call it, the length of the radius-vector — and Q the angle which the line OP makes with a fixed line in the plane of motion. The momentum in the outward direction along the radius- vector is md{l/u)/dt= —(mdu/dt)ju^. The time-rate of change of this is _rn^ mdu §§ 120, 121] ORBITAL MOTION. 221 The momentum in the forward direction at right angles to the radius-vector is mQju ; and because of this, and the turning round of the radius-vector and lines connected with it at angular speed 6, momentum in the outward direction along the radius-vector is growing up at time- rate — m0^lu (see §9). Hence the whole rate of growth of momentum along the radius-vector in the position which it occupies at time t is cZ/1 du\ 02 -TO-r, -o TT -m— . ' dt\u^ dtJ u Now let h = 7^d=6/u\ so that at the instant the angular momentum about the centre of force is mh = 7nd/u^; then mO^/u = mh^u\ Also we have dt = dd/hv,^, so that the rate of growth of momentum along OP is Hence, if mF is the inwa/rd force towards the centre, we ^(^S) + ^^ = ^2 ^^^ as the radial equation of motion. 121. Effect of Force transverse to Radius-vector. If, as is here supposed, h is not constant, a force transverse to OP in the plane of motion must account for the variation of h. If S be the force, per unit mass of the particle, reckoned positive when in the forward direction, we have dhldt = Sju, or, taking as before the moving particle as timekeeper, that is putting dt = dd/hu^, we dbtain ^^fe='- ^1) This enables us, when 8 is known, to write, (1) of § 120 in the form -» n o j am _ Ji IS du (2) 222 A TREATISE ON DYNAMICS. [CH. V. Hence, eliminating h, we obtain F _SdM d u^ u^de 2S do d^u , V? .(3) In the cases in which the particle is acted on by a force resisting or accelerating in the line of its motion, as well as by a force towards a fixed point 0, it is convenient to consider the particle as subject to two component • ac- celerations, one in the forward direction of motion, the other towards 0, and to write two corresponding equations of motion. We shall show first that these accelerations can be wiitten in the forms h dh , h^ r —R -J , and —5 - p'' ds p^ p or h^ dp p^ dr where p is the radius of curvature of the path and p the length of the perpendicular let fall from on the tangent to the path at the position P of the particle at time t. For, denoting the tangential and radial accelerations by at, ar, the angle between the radius-vector OP and the §121] ACCELERATION IN ORBITAL MOTION. 223 tangent at P by ^ (as shown in Fig. 48), and the radius of curvature at P by p, we have by §10, v-j- = at — ttr cos (j>, — = ffl,.sm0, (4) since vdvjds and v^jp are the rectangular components of acceleration along the tangent in the direction of motion and towards the centre of curvature respectively. The second of these gives, since sin

= dsfp, which, since cos (p = drlds, gives dp/dr = rlp. Hence also _h^ dp _h^ r /K-. p^ dr p^ p We have now, by (4), _ dv h^ dp dr_ dv h^ dp *~ ds p^ dr ds~ ds p^ ds But since v = h/p, vdv/ds = {hdh/ds)/p^ — (h^dplds)/p^, and therefore , ,, fi an /n\ at = ^-j- (o) p'' ds Thus if F be the force toward the point and 8 the tangential force in the forward direction, each taken per unit mass of the particle, we have the two equations of motion ,„ , , ,, AJ^_ 4^=S (7) p^ dr p'' ds If h is constant, 8 is zero ; but the first of these equations still holds, and may be used as an alternative for (1) of § 120. As an example, let (Sf be a resistance proportional to the speed V, that is, let 8= —lev. Then (hdh/ds)/p^= —lev. 224 A TREATISE ON DYNAMICS. [CH. V. Multiplying both sides hjv( = ds/dt) and substituting h^/v^ for p^, we obtain , „ ^ 1 dh_ , hdi~~ ' sothat h=Ge-'*, (8) where C is the value of h when ^ = 0. Again, let S= —kv^; then, after reduction, we get 1 dh_ , hds''"' sothat h = Ce-^\ (9) where C is the value of h at the point from which s is measured. Thus h diminishes exponentially as the time increases in the former case, and as the distance travelled increases, in the latter. The resistance which a planet experiences in its orbit, according to the modern theory of light pressure, is directly proportional to the speed, and therefore h diminishes in that case exponentially as the time increases, according to (8). The coefficient k of the resistance is, however, inversely proportional to the radius of the planet, so that except for a planet of exceedingly small size the effect here calculated is quite insensible. When S is zero h is constant, and the single differential equation , 2 j |l = ^' ^1«) or its equivalent d^u _ F ,,^, ^"•"^"AV' ^^^> determines the motion. It is useful to remember that, since t)2 = s^ = r^02 + f^, and h = 7^6=pv, we have «2 \dd) \2 (12) §§ 121, 122] SPEED FROM INFINITY. 225 122. Speed from Infinity. Exhaustion of Potential Energy. Let us suppose that F^/x/i^, where /it is a constant (called the intensity of the centre) and r is the length of the radius-vector to the position of the particle; and let the particle describe any path in the field of force from a position at distance r^ from the centre 0, to a position at a final distance r^. If V be the speed of the particle at the instant under consideration, the distance ds = vdt is described along the path at P in time dt. If (p be the angle which the direction of motion makes with the line drawn from P to the point 0, the increase of speed dv is ficos^dt/ir^K But dt = ds/v, and therefore v dv = /j. ds cos (jt/r'^^ By Fig. 48 it will be seen that ds cos (p= —dr, and so vdv= —/j-dr/if": Thus we obtain, if 7i.>-l or ^l, 1 a 1 2 rd/r_ fx ( 1 1 \ , This is the equation of energy. On the left is the increase of kinetic energy per unit mass of the particle, and on the right the work done by the force of the field in the displacement. The latter it will be seen is independent of the path of transference. If the particle start from rest at rg= oo , and if to>-1, *^-^.f' • (^> and Vj is the speed acquired in the transference of the particle under the action of the field from infinity to the distance r^. -It is called the speed from infinity at distance rj. In the very important case of n = 2, we have v\ = fxlr.^. The quantity on the right of (2), multiplied by in, is the amount of potential energy transformed into kinetic energy in the passage of the body of mass m from infinity to the point at distance r^ from the centre of force. We shall refer to this as the "exhaustion of potential energy from infinity." The quantity on the right of (1) is the exhaustion of potential energy from distance r^ to distance rj, per unit mass. G.D. P 226 A TREATISE ON DYNAMICS. [CH. V. As an example, we may find the speed from infinity to the surface of the earth, acquired under the influence of the earth's attraction. Here n = 2. The speed acquired in time dt at distance r is // dt/r'^, if we suppose the path to be a straight line towards the earth's centre. But at distance r^, the earth's radius, the speed acquired in time dt is gdt^ndtjrl, and therefore n=g'r[. Thus we obtain lv\ = gT^ or v\=^2gr^ (3) The speed acquired is therefore that which would be acquired by the particle in falling through a distance equal to the earth's radius under constant acceleration g, equal to that of a falling body at the surface. Taking, as rough values of g and r^, 32 ft./sec.^ and 21 x 10^ ft. respectively, we obtain ,,2 _ 9 ^ q.t ^01^1 <« t*-, 2 ^2= 2 X 32 X 21 X 106 ft.2/sec. 2 or ■!; = 36rOO ft./sec (4) nearly. When TO>-1 the contribution made by the term l/(')i — l)r""' in (1) is zero for r^ = 'x>\ but when tKil, this contribution is infinite. We take in that case the speed % acquired by the particle in passing to 0, the centre of force, from rest at Pj, distant r^ from 0, and have then ^< = l^<"' (5) If 71 = 1 , both the speed from infinity and the speed from P^ to are infinite. We shall see presently how the determination of the orbit in particular cases depends on the speed from infinity. 123. Concavity or Convexity of Orbit towards Centre of Force. It is evident that, if the central force is an attraction, the path, or orbit as we shall call it, is concave towards the centre of force. For the attraction is con- tinually causing the direction of motion to deviate from the tangent in the direction towards the centre 0, that is to bend round 0. If the central force is a repulsion the bending is the other way, that is the orbit is convex towards the centre of force. It is easy to arrive at these §§ 122, 123, 124] LAW OF DIRECT DISTANCE. 227 results analytically. We have already, § 121, established the equation h^r^h^dp^j,^ p^ p p^ d/r It follows that if F is positive, . that is, is an attraction, dpjdr is positive (for p is always taken positive), that is p increases with r and diminishes with r, in other words the orbit is concave toward 0. If F is negative the force is a repulsion, and p increases or diminishes as r diminishes or increases, and the path is convex toward 0. 124. Force varying directly as Distance. Consider a particle moving under the influence of a force which acts along the line joining the position of the particle to a fixed point 0, and is proportional to the length r of this line. If fxr, where fx is positive, be the magnitude of the force per unit mass of the particle, the equations of motion with reference to axes of x and y with origin at are x + fj.x = 0, y + fiy = 0, (1) if the force is an attraction, and x-fj.x = 0, y-fiy = 0, (2) if it is a repulsion. The axes need not be at right angles to one another, and we may choose their directions so that Ox is in the direction of the initial displacement and Oy in that of the initial motion. Thus initially we have x = a, d) = 0, y = 0,y = Vf), since the complete solution of either difFerential equation gives for x ov y a, value of the form A cos J/xt + B sin •J]j.t. For the case of attraction then, we have x=a cos ijjit, y^bsinj/it, (3) where b = vj'jjj.. For these values of x and y satisfy the difFerential equations and the chosen initial conditions. Eliminating t by the relation cos^\//«^ + sin^\//ii5=l, we get |+P=1' W 228 A TREATISE ON DYNAMICS. [OH. V. the equation of an ellipse of which the axes of coordinates are a pair of conjugate axes and the centre is the centre of force. In the case of repulsion the complete solution of either differential equation gives for x ov y a, value of the form Ae'^* + Be-'^^\ and, in accordance with this, values of X and y which satisfy the same initial conditions are a; = ^a(e^*+e"^i^O. i/ = J6(e^'-e-^i^0. (5) where b = vj\/ji. From these we get %-t, = l, (6) a^ b^ the equation of a hyperbola of which the axes of coordinates are conjugate axes, and of which the centre is the centre of force. The construction of the path in either case is simply the construction of a conic of which a pair of conjugate axes are given in position and magnitude. The reader may verify that the criterion of concavity or convexity stated in last section is satisfied. It is clear that the period of description of the path is 2Tr/V)u in the case of the ellipse, that is, since twice the area is 2Trab sin a, where a. is the angle between the axes, the double rate of description of area is J fiab sin a., or -Jfiab if a, b be the principal semi-axes. In the case of the hyperbola, if we calculate (xy — ysb) sin a. from the values of X and y given above, we obtain h = (xy — yd)) sin = J /nab sin a.. It will be observed that by (3) the rate of description of area leads in the case of the ellipse to the value J fit for the eccentric angle described in any time t, and that the period of revolution depends only on /i, and is therefore the same for all ellipses described about the same centre of force. 125. Examples of Force in Different Cases. The equations are very useful for finding the force when the orbit is given. i 124, 125] EXAMPLES. 229 Ex. 1. To find F when the orbit is an ellipse with the centre of force at a focus 0. Let r be the distance of a point P on the orbit from 0, p, p' the lengths of the perpendiculars from and the other focus on the tangent at P, and 2a the sum of the focal distances of P. Then, for the ellipse p'lp = {2a — r)/r, or, since pp' = W (where 6 is the length of the semi- axis minor), W]p'^=(^-r)lr. Thus differentiating, we obtain {dpldr)jp^ =ajb'r\ and therefore 62 ^2- ' I r-2' where I is the length of the semi-latus rectum. Thus the force varies as the inverse square of the distance from 0. Ex. 2. Prove that at P (Ex. 1) the curvature, lip, of the ellipse has the value aam^jb^, where (f> is the angle between OP and the tangent at P. By §121 we have l/p=(dp/dr)/r=ap^/b^i^. But we have similarly from the other focal distance 2a — ?' and perpendicular p', llp = ap'^/b\2a-rf. since rsin<^=^, (2a-?-)sin^=p'. Hence 1 a . , , -=psin3<^. Ex. 3. To find F when the orbit is an equiangular spiral with the centre of force at the pole of the spiral. The equation of the spiral is (see Ex. 3, at end of Chapter I.) ?' = ae*™t*, where <^ is the constant inclination of the tangent to the radius-vector from the origin to the point of contact. We have then Hence p=rsin<^ and dpldr=sm. A2^^ A2 1 p^ dr sin^iji 'fi' Since ^ is constant, F varies as the inverse cube of the distance' r. Since l/p=smjr,_ F also varies as the cube of the curvature. Ex. 4. To find F when the orbit is the lemniscate of Bernoulli, and the centre of force is the node of the curve (Eig. 49). The equation of the curve is r'=a^coa26. Hence if, as before, (j) denote the inclination of the tangent to the radius-vector, we have Fig. 49. 230 A TREATISE ON DYNAMICS. [CH. V. (since sin <^=rde Ids), l/p^=(dr^+r'de')lr*de^ = a*lr^ We get by differentiation for the value of F, h'dp hV The force varies as the inverse seventh power of the distance r. The curvature is 3r/a^. In this case the orbit passes through the centre of force 0. At that point the speed is infinite, for it will be seen that the force when the particle is near the origin is along the path and is ver3' great. It will be observed that the acceleration is very great when the particle is approaching the centre, and that the retardation is correspondingly great after the particle has passed the centre, so that at an infinite distance the speed is finite. Ex. 5. To find the force when the orbit is the curve of which the equation is r"=a"coan6. This curve is the lemniscate when n=2. By the same process as in Ex. 4, we get \lp^=a^lr^^K Hence 126. Solution of Differential Equation in Various Cases. Energy Relations. We can use the differential equation either to find the force when the orbit is given or to find the orbit when the force is given. In the latter case the differential equation must be solved, and this is not always possible except under special conditions, for example equality of the speed of projection to the so-called speed from infinity. We now consider first the motion of a particle under gravitational attraction directed towards a fixed point 0. In this case n = 2, so that F=fivP; and the equation of motion is ^2^ 50^+^=f- -v (i> The complete solution of this equation is tt = ^+ilcos(0-a), (2) where A and a. are constants. (See § 28 above for another method of obtaining this result.) §§ 125, 126] LAW OF INVERSE SQUARE. 231 For 0-oc = O, u = iui/h^ + A, r=l/(fi/h^ + A), and for — (X=7r, u = filh^ — A, r=ll{fij}i? — A). If the first value of r be denoted by a(l— e) and the second by a{l+e), or the first be denoted by a(e — 1) and the second by — a(l + e), according as fx/h^'^ or << J., we obtain in the first case 4 — ^ /^ — ^ /Q^ a{l-e^y h'~a{\-e^y ^ ' and in the second (x(e2-l)' /t2~a(e2-l) ^ '' In the first case e-l. Equation (2) becomes ^ ail-e') u l + ecos(0 — oc.) in the former case, and r=l= ^("'-^> (6) u l + ecos(0 — a) ^ in the latter. In each case the equation is that of a conic section of which the centre of force is a focus The curve is an ellipse in the former case and a hyperbola in the latter, the length of the major axis is 2ct and the eccentricity e in both cases; while 2a(l— e^) is the so-called parameter of the ellipse, and 2a(e^— 1) that of the hyperbola, that is twice the length of the radius-vector drawn from at right angles to the major axis. Now v-=h?[[^) +u^\ = h- —a-jpi^ -• -(7) But by what precedes h^/a{l — e^) = iui or h^/a(e^ — l) = fi, according as e < or >■ 1 , that is according as the curve is an ellipse or a hyperbola. Thus we have „ r2{H-ecos(0-a)}_ 1-e^ 1 /2_1\ ,_, L ±a(l— e^; a(l— e-'^J ' \r a/ according as the orbit is an ellipse or a hyperbola. When e = l, the orbit is a parabola, and the equation for v^ is 9 ,, ^^ = f (9) 232 A TREATISE ON DYNAMICS. [CH. V. We shall deal with this case specially when it arises. It is to be noticed that the speed at each point is that from infinity to the point in question. Hence J^fifr is called the parabolic speed. It will be shown later [see Ex. 15, §131] that m,m/2a is the time-average of the kinetic energy in either orbit, so that (8) asserts that the kinetic energy ^nnv'^ of the par- ticle at distance r from the centre of force in the hyper- bolic orbit exceeds, and in the elliptic orbit falls short of, the exhaustion of potential energy (§122) Tnfi/r, from in- finity to the distance r, by the time-average of the kinetic energy in the orbit. 127. Discrimination of Orbit. The speed from infinity to the distance r is, as we have seen (§ 122), 2fi/r. Hence we have, by (8) of § 126 (putting now v^ for the speed from infinity), , , ^ h' " a a\l-e?)' ^^ in the case of the ellipse, and V' -V•'=:-^t= y (2) a a^{e^ — \) ' in the case of the hyperbola. Thus the speed from infinity is greater or less than v, according as e is less or greater than 1, that is according as the orbit is an ellipse or a hyperbola, and conversely. Thus a comparison of the speed at any distance with the speed from infinity enables us to discriminate between the two forms of orbit. The reader may satisfy himself that when the force is a repulsion the only possible orbit is a hyperbola, and that the motion is in the branch which does not contain the focus at which the centre of force is situated. When the force is an attraction the particle moves along the branch within which lies the centre of force. Now let (1) ro = a(l-e), (2) r^ = a{e-\)\ then by (5) and (6) of §126, dulde^Q, and vl = h^la\\-ef in both cases. But M = /ia{l—e^) in the first case and h^ = fia{e^—l) in the second; therefore vl = /n{l + e)/a{l—e) in case (1) and vl = /n(e + l)/a(e — l) in case (2). This agrees with (8) §§ 126, 127, 128] PERIOD IN ORBIT. 233 of §126, which gives in the former case |w^ = /i(l/r— l/2a) and in the latter Ji)^ = /i(l/r+l/2a). Equation (8) of §126 may be taken as that of energy! For we may write it imi)^ — m-= +m^ (3) On the left we have the kinetic energy ^ttov^ of the particle, and a term, —vi/u/r, depending on the position of the particle with reference to the centre of force, which may be taken as the potential energy of the system, while on the right we have a constant, —m/n/2a or m.yu/2a, the constant sum of the kinetic and potential energies. 128. Period of Particle in Orbit. The period of revolution, or time of description, in an elliptic orbit is finite, in a parabolic or hyperbolic orbit it is infinite. If a be the semi-axis major of an elliptic orbit of eccentricity e, the length of the semi-axis minor is ajl — &, so that the area of the orbit is iro^'Jl — e^. But twice the rate of description of area is h = i>/fjia(l—e^y, so that if T denote the period, 3 2'=27ry^, (1) the period of a simple pendulum of length a under acceleration /u/a^. It is to be observed that the squares of the periods in different orbits about the same centre of force are by this equation proportional to the cubes of the corresponding values of a. This, as we shall see, is one of the observations made by Kepler regarding the planetary motions. It is important to remark that by equation (8) of § 1 26 the length a of the semi-axis major depends only on the speed and the distance of the point of projection from the focus. Thus, if a number of particles be projected with the same speed, greater than that from infinity, in different directions from the same point under the attraction of the same centre of force, the major axes of the orbits of the different particles will all be of the same length, and the periods will be the same. The particles will therefore all return after the lapse of a period to the same point. 234 A TREATISE ON DYNAMICS. [CH. V. 129. Determination of Orbit from Distance and Velocity, etc. When the direction and speed of projection, the centre of force and its constant, (jl, are given, the orbit can be at once constructed. We join the centre of force (Fig. 50) with the point P of projection, and calculate the value of 2/R— V^lfi, where R is the distance and V the speed of projection. If the result is positive, the orbit is an ellipse, and the value found is the reciprocal l/o. of the length of the semi-axis major. If the result is negative, the orbit is a hyperbola, and the reciprocal 1/a of the semi-axis is V^lfi-2IR. Now draw the direc- tion of projection through P and produce it both ways to T and U (Fig. 50). The line so drawn is a tangent to the path. In the case of the ellipse draw from P a line PC/, making the angle OPU equal to the angle O'PT, and take the length P0' = 2a — r. Then 0' is the second focus, and the orbit is determined completely. It can be drawn in the usual way. The eccentricity e is the ratio of the length 00' to 2a. It is to be observed that any point of an orbit and the velocity there may be taken as the point and velocity of projection. The relation v^ = 2jui/r— fji/a enables the orbit to be drawn if v, r, and fi are given, and the line OP is given in position, provided one other datum is supplied. That may be the direction of the tangent at P, or it may be the condition that the particle shall pass through another given point Q (Fig. 51). In the latter case we find a as described above, and with radius 2a — r describe a circle from P as centre in the plane of 0, P, Q. Then from Q as centre with radius 2a — r' (where r' = OQ), we draw another circle which, if it meet the former circle at all, will Fig. 50. §129] CONSTRUCTION OF ORBIT. 235 Fig. 51. generally intersect it in two points, 0', 0". Either of these points is the second focus of an orbit which passes through and 0'. It will be seen that as Fig. 51 is drawn, if the second focus be 0' the two points P, Q are on the same side of the major axis, and that if the second focus 0" be taken, P, Q are on opposite sides of the major axis. If the circles meet in one point 0' only, that is if P, 0', Q be in line, one orbit only can be drawn; if the two circles neither intersect nor touch, a solu- tion of the problem does not exist. To construct a hyperbolic orbit, we draw (Fig. 52) PO' so that lO'PU= l. UPO, and take the distance P0' = 2a+0P as the second focus, and again the orbit is determined completely. It will be seen that both for the ellipse and for the hyperbola, if the plane of motion is fixed, but not the direction of motion, the 7^ locus of the second focus is a circle with centre at P. The student may prove that the locus of the centres of the pos- sible orbits is a circle with centre at the middle point M of OP (Fig. 50) and radius = ^PO'. If the plane of motion is not fixed, these loci are spheres with centres and radii as stated. [See Ex. 18 below.] If V^lfi = 21 R the orbit is a parabola, .since a is infinite. The semi-latus rectum, which in the ellipse has length Fig. 52. 236 A TREATISE ON DYNAMICS. [CH. V. a(l— e^), however remains finite. If it be denoted by I, we have for a parabolic orbit, as the student may verify for himself, h=\J fiL. If for a parabolic orbit the focus and a tangent with point of contact P be given, we join OP (Fig. 63) and draw another line 0T= OP to meet the tangent in T. This line is in the direction of the axis of the curve. A perpendicular ON let fall on the tangent and a perpendicular NA on OT, determine the vertex A of the path, and a distance OD = 20 A laid off along 07 gives the directrix, to be drawn through D perpen- dicular to the axis. Further points on the path can be found from the condition that every such point is equidistant from the directrix and focus. 130. Newton's Eevolving Orbit. If an orbit for a central force f{u) is known, an orbit for the central force f(u) + /ijV? can be found. When the force is f(u) the differential equation of the ,2' '^) Fig. 53. dff' - + U=' h?v? and if the force is tusAq f{u) + fi-{u?, the equation is •(2) If in this we substitute dff/s/l-fjijh^ (or dd'/k) for d9, and put h'^ for h? — ^■^, we obtain dHh , fOw) dS"' h'^u^' .(3) a differential equation of the same form as that first written above. Hence if u = (j){Q) be the equation of an orbit for the force f{u), u = (p{B') is the equation of an orbit for the force f{u) + fx-^u^. We imagine, thus, the particle to describe the first orbit, and that orbit to revolve at the same time with angular §§ 129, 130, 131] EXAMPLES. 237 speed (k — 1)0 in its own plane about the origin. The path of the particle is then the second orbit required. This is the theorem of Newton's revolving orbit. Or we may suppose the orbit u = (p(9) constructed, and then con- struct another in which the u for any value 6 is the u for an angle Jc9 in the former orbit. [Frincipia, Lib. I. Sect, ix.] 131. Examples. Ex. 1. li f{u) = iiu\ prove that the orbit for the force inu^ + fi-^v? has the equation ^ +a(l-e^) '^~\-\-e cos {key The orbit for the force iijr^ has the equation ±a(l-e2) y. — ^ -^ 1+ecos d where the positive sign is to be taken for an ellipse and the negative for a hyperbola, for which, of course, e^> 1. The orbit for the force fi/r^ + /j^jr'' is, by the theorem, ±a(l-e2) T^ ^ • l + ecos^' But 6' is the actual vectorial angle in the new orbit, correspond- ing to the radius- vector r, multiplied by h', and so we have r — >^ ^ "l-l-ecos(*^) Ex. 2. Prove that if the force [j,r towards or from the centre of a conic be increased by iiji^, the orbit is changed from cos2^/a2±sin2 (9/62 = 1/^2 to cos2(/i;6l)/a2±sin2(i^)/62=i/^2. Ex. 3. To find the law of force towards the same centre, by which the inverse of a given orbit with respect to a circle, with its centre at the centre of force, may be described by a particle. For the known orbit, we have F=(A^dp/dr)/p^, and if F', k', p', r' be corresponding quantities for the inverse, we can, by the relations rlp=r'lp' = flpt',p=f{r) and h/r^=h'/r'^, find F'={h'Mp'idr')/p'\ Or we may derive from the equation p =f(r) of the known orbit the relation cy/r'^=f(c^/r'), and since F= -idv^/dr, write F'= -ih'^d{l/p'^)/dr'. Between these two last equations p' is eliminated and F' obtained as a function of r' alone. 238 A TREATISE ON DYNAMICS. [CH. V. Ex. 4. Prove that the attractive force F under which a particle de- scribes the inverse of an ellipse, a focus of which is the centre of force, is given by ^ , 3aA^c^ 1 2AV 1 ■^ 62 r* 62 r*' where the accents used in Ex. 3 are dropped. Ex. 5. If the centre of the ellipse is the centre of force, prove that „_c^ / 2(a2 + 62)A2 3c*An Ex. 6. When the orbit is the reciprocal of a known orbit, and the centre of force is the same for both : to find the law of force. In this case rp'—r'p = (?, and if the equation of the known orbit be )-=/(p), then (!V=/(«V»^)- As before, also, F' = -\K^d{\lTp"^ldi', and from these relations P' can be determined. Ex. 7. Show that if the orbit be the reciprocal of a conic, a focus of which is at the centre of force. F=d^- "("-if and that if the centre be the centre of force, F=^'^Br. Ex. 8. Find F for the pedal of a given orbit, and show that, according as the central pedal or the focal pedal of an ellipse, with centre of force at the focus, is takea. ^=^<^^-^-^), -(62 + ^2)3 Ex. 9. Find F when the orbit is the pedal of a circle (radius a) with the centre of force at an eccentric point in the plane of the circle distant c from the centre, Ex. 10. Show that for the cardioid derived from this circle, Ex. 11. Verify the second result of Ex. 8 from the fact that the focal pedal of a conic is the circle described on the major axis as diameter, by finding the force required to give a c ircular orbit of radius a, when the centre of force is at a distance n/o^— 6^ from the centre. [Most of the foregoing examples are taken from a paper by Curtis, Mess. Math. xi. 1881.] §131] EXAMPLES. 239 Ex. 12. Prove that if e be the eccentricity of an orbit describee^ under a force /x/r^, and v be the speed at a point where <^ is the angle between the tangent and the radius-vector, /J. ^ . (G. W. Hill.) For brevity, we shall work out for an ellipse. The modifications of the proof for a hyperbola are obvious. Clearly h^ = v^r^aixi'^ e cos(d-a.)— ^-1 by the former result. Now produce the tangent and major-axis to meet, and let i/r be the angle of intersection (see Fig. 54). Let fall perpendiculars (lengths ^p') from the foci on the tangent. The distance between the foci is 2a«, and by the figure 2oe cos \^ = (2a — ?•) cos <^ -f- r cos (^ = 2a cos <^. Hence ecosi/r = cos<^, which is a very useful relation. FiQ. 54. Again, p' -p — (2a - r) sin ^ - r sin =2(a — r) sin ^. But also p' — p=2ae sin ■\jr. Hence, we have ae sin yjr = (a — r) simp- But in the figure LAOP=d-a-, so that sin(5 — o(.) = sin(<^-)-i^), and therefore e sin(^ — a.)=esin ^oos i/f-t-e cos <^ sin •^=sin <^ cos i^-l sin <^ cos 1^, 240 A TREATISE ON DYNAMICS. [CH. V. by the relations just established. Thus esin(9 — a.) = ri 1 sin esini 0-0.) = -■ /J. Squaring this relation and the corresponding one found above for acos(^-a.), and adding, we obtain „ , „ ?;Vsin^rf> . vV^sin^A e'=l-2 ^-\ n— ^. Ex. 13. A particle describes an el liptic orbit about a centre of force at a focus, in the period T=2irija^lix, where a is the mean distance : to infer the time required by the particle to reach the centre of force if placed at rest at any distance. It is to be noticed first that the period depends only on the mean distance ; hence if we keep a unchanged and make e approach unity, we shall, without altering the period, cause the orbit to approximate to a very narrow ellipse with the centre of force, 0, close to one extremity. The motion from the other extremity to very near is then a path diiTering little from a straight line towards 0. The transverse speed of the particle at the remote extremity of the path is annulled by the small transverse component of the central force, and the generation of motion toward takes place as if the particle had been given at rest at the remote extremity. Thus we have for the time T, from a distance 2a, ^ T = 7r V — - If the particle were placed at rest at any other distance 2a', we should have for the time t' required for it to reach the centre of force For instance, if 2a' = a, the mean distance in the orbit, we get "2n'2 V/*~W2' [Thus, if the earth were deprived of its orbital motion at any instant, it would begin to fall into the sun, which it would reach in about 65 days.] Ex. 14. Verify the results of the preceding example by direct integration from the differential equation § 131] EXAMPLES., 241 Ex. 15. To find the value of 1 1>^ dt for a complete period of a particle describing an elliptic orbit under attraction to a focus ac- cording to the Newtonian law, and to show that it is independent of the eccentricity of the orbit. r Since vdt=ds, this integral may be written ivds, and is then called the "action" (per unit mass) for the path along which it is taken. We are to take it once round the orbit. Now we have „ „ /I 1 \ 2a-?- /xr' r2 = 2ul --5- )=M = - -> where / is the distance of the particle from the " empty " focus. But if p, p' be the lengths of the perpendiculars from the centre of force and the other focus to the position of the particle at the instant considered, we have r'/r—p'jp, and therefore, since pp' = b^, r'jr=p'^/b'^. Hence \vds='\i- r\p' ds. Biit clearly \p' ds taken round the orbit is twice its area, that is 27ra6, and therefore \vds = 27r\/i lia. The action for one revolution is thus 27rv/ia, and is the same for all orbits, whatever their eccentricities, for which a has the same value. If we denote it by A, then A = SWjLia = 27r-/=== i-Tr' -^ ■ A is the value of the integral Iv'^dt taken over the period T ; therefore the time-average of the kinetic energy (per unit mass) of the particle is.4/27', and we have 2^ r' 2 /^ ^"" [Thus while the area described about the centre of force by the radius vector is proportional to the time, the area described by the radius vector to the particle from the empty focus is proportional to the action.* It is shown in §100 that the actual motion of the particle is brachistochronic for a centre of force in the empty focus. Thus the mode of representing the time in the free motion about that focus has become the representation of the action. Now we can write the energy equation (13), of § 127, in the form , „ , mu u. •^ 2a r * Since this Example was written we have found that this fact had been noticed by Tait : Proc. R.8.E., 5, 1865 and Trans. R.S.E., 24, 1865. G.D. Q 242 A TREATISE ON DYNAMICS. [CH. V. where the upper sign applies to the elliptic and the lower to the hyperbolic orbit. We thus get the curious theorem that the kinetic energy of the particle, at distance r from the same centre of force, in a hyperbolic orbit, of semi-transverse axis a, exceeds, and in an elliptic orbit, of major semi-axis a, falls short of the exhaustion m/i/r of potential energy from infinity to the distance r (see § 122), by the time-average of the kinetic energy of the elliptic motion. We infer that the ultimate constant velocity in the hyperbolic orbit at a very great distance from the centre of force — that is along the asymptote — is the square root, \//I7a, of twice the tinie-average of the kinetic energy in the elliptic orbit. Furthei', since the time along the asymptote at the constant speed V/i/a is infinite, the mean kinetic energy in the hyperbola is »i/i/2a, the same as that in the ellipse. This can be seen to be the case by a consideration of areas in the hyperbola. It is worth noticing that for one revolution lvds=>/ij^/abjd,t/p, so that fds Ex. 16. To find the angular speed of a planet about the " empty " focus. If r, r' be the lengths of the radii from the centre of force and the empty focus to any position of the particle, and 6, 6' be the corresponding angular speeds, then rS—r'Q'. For if ^ denote, as before, the inclination of the tangent to either radius-vector, we have sin cj>=rd9/ds=r'dd'/ds, and therefore the relation stated. Hence 0'=rdlr' = h/rr', since d=hjr\ Now, by the properties of the ellipse, r = a — ex, r' = a + ex, if x be the distance, parallel to the axis, of P from the centre. Thus we get Thus if powers of e above the first can be neglected, 6' may be taken as constant. [See Exercise 4, p. 301, below.] Ex. 17. To integrate the equation of central orbits for the case of a central force jU,(a-t-6cos2^)/r^. The equation is -^ ™ -|-M = |^(a-t-6cos2^), which may be written --a <-^f) A particular integral is easily found by substitution to be u.a ah „ „ § 131] EXAMPLES. 243 and this plus the complementary function (which satisfies the difierential equation when 6 = 0) is the complete integral. Thus we get „ By putting x = rco%6, y=rsva.6 (r^l/u), we can show that this is an algebraic curve of the fourth degree, unless 6 = 0, when it reduces to a conic. Let it be given that at the point where u = c, 6 = 0, the velocity is at right angles to the radius vector, that is, that the value of dujdd is there zero. Then we get c = 4+/x(3a-6)/3A2 or ^={3A%-/i(3a-6)}/3A2, and by difi'erentiation and the condition (110/(10=0, we find B=0. The equation is then u=:^^l{Sh^c- iJi.{Sa-b)\cos e+tJ.(3a - b cos 2^)], . and the orbit is completely determined. Ex. 18. To find the locus of the centres of all the orbits that can be drawn for a given centre of force and given speed Fof projection from a fixed point P. Let R be the distance OP, and suppose for the present that the orbits are ellipses. The value of a is found from the equation ^F2=/i(l//J-l/2a), and is ju,^/(2/t- V^R). The second focus lies on the circle described from P (see Fig. 51), with radius = 2a — ^. If the origin of coordinates be taken at 0, and a., /3 be the coordinates of the point of projection, the equation of the circle is (^ - a.)2 + (y - /8)2 = (2a - Rf. The coordinates of the centre of the orbit are ^=i^', ■'?=|y, and therefore the equation just written can be put in the form (f-t)V(-f)'=(»-f)' which shows that the point ^, i; lies on a circle with centre at the point a./2, /3/2, that is the middle point M of OP [Fig. 50]. The radius is a — Rj2, that is ^PO', if 0' be a second focus. We have ae=^00', so that a^e^=^^-{-r)^ = (a-Rl2f-i(a? + ^) + ^a.+T]P=a^-aR+$a. + r]f3. Thus e^ = i-^+lBL}gL§, which gives e^ as depending on ^, ij. The modification of this process for hyperbolic orbits may be written out by the student. [The present discussion affords another proof of the expression for e^ given in Ex. 12. Taking the axis of ^ along OP, we make /S=0, and the equation of the locus of centres found above may be written i^ + ri^=a'-aR+iR, 244 A TREATISE ON DYNAMICS. [CH. V. since now a.= R. But Fig. 50 shows that Hence p + 7f = a?-aR{\-cosZ4.)+B?siti^4> = a^-^aR sin2 <^ + RHvo? 4>. But ^^ + rf = ah\ and l/a = (2jii- V^R)ffiR, and so we get „ , ^iJsin^d) , iJ^sin^ii 6^=1-2 —^H — »^^ a a' , „ F2^sin2 ^ /^ , as before. It will be noticed that, when R=^a, this gives del'dfi=0, and also de/dR = (for then V^a = ij,).] 132. Acceleration in terms of Tangential and Radial Forces. Returning now to equations (4)) and (7) of § 121, we have dv_h^dh h^dp_„ „dr q^ ds~p^ ds p^ ds" ds Hence we obtain by integration, |'y2= \'sds- (Fdr (2) where r^, s^, r, s are corresponding values of r and of distance travelled , along the path ;f rom some chosen point. If F is some function /(r) of r, iv^ = fsds-{f(r)d/r (3) Thus, if 5^ = 0, V is a function of r, and is the same at the same distance from the centre of force. It follows therefore that when >Si = 0, du/dO is the same at the same distance, that is the radius-vector makes the same angle with the --"'• - .=.'{©%«'). and, since h is constant and v has the same value for the same value of u, du/dd must also return to the same value when u does. Again, by (4), § 121, v^/p = F sin 0, that is v'^ = 2Fxl chord of curvature at position of particle. . . .(4) §§ 131, 132, 133] HODOGRAPH OF PLANET. 245 The speed v at any position of tlie particle in the orbit is thus equal to the speed which the particle would acquire if it traversed from rest, under constant acceleration F, a distance equal to I of the length of the chord of curvature. It will be noticed that this theorem holds whether a tangential force S acts or not : the radius of curvature p, for a given speed v, however, is affected by such a force. Since, when 8 = 0, the speed is the same at the same distance, and also the angle which the tangent makes with the radius-vector, the chord of curvature and the radius of curvature of the orbit are the same at the same distance from the centre of force. 133. Hodograph of Particle describing Orbit. The relation v = hlp shows that any polar reciprocal of the path, turned through 90°, represents the hodograph of the particle's motion, whatever the orbit may be. When the path is a conic section with the centre of force at a focus, the circle described on the axis of length 2a. as diameter may be taken as the hodograph, pro- vided the hodographic origin be taken at the "empty" focus, and the direction of motion be taken turned back through ' 90°. For let 0, 0' be the foci, of which the former is the centre of force, and a tangent be drawn to the path at any position P. If the tangent meet the circle referred to in M, M', as shown in Fig. 55, the lines OM, O'M' are, by a property of the ellipse, perpendiculars to the tangent, and if p, p' be their lengths, the product pp' is equal t o h^, w here h is the length of the minor semi-axis CB, db-Jl—e^, for the ellipse, and the length of the conjugate semi-axis, aje^^l, for the hyperbola. Thus we have h h , 1 / « , /IN Fig. 55. p'a'il-e'y ~a^a{\-e^) 246 A TREATISE ON DYNAMICS. [CH. V. for the ellipse, and ^h^ h for the hyperbola, p (2) 1)"' ctVa(e2-l) Thus 0' is the hodographic origin, and the velocity is represented on the scale indicated in (2) by the line M'O', which is the direction of motion turned forward 90°. A circle of radius 2a described from the focus as centre serves still more conveniently as hodograph. For it will be seen that if the perpendicular O'M' from 0' (Fig. 56) be continued to meet the circle in Q, then 0'P = PQ, and 0'Q = 2p'. Thus, since V X p', we may take O'Q as representing the velocity at P along the tangent to the ' path turned back through 90°. The true direction is O'S. Fig. 56, which we shall use in what follows, represehts the path and the hodograph thus constructed. 134. Velocity resoluble into Two Components of Constant Amounts. It will be noticed that the velocity represented by O'Q can be resolved into two components O'O and OQ. Of these O'O is fixed both in amount and in direction, the other, OQ, is fixed in amount but changes in direction with OP. The lines O'R, RS, perpendicular to OQ and O'O respectively, represent the true directions of these components. We can find their magnitudes very simply from the consideration that their sum must be the speed with which the particle moves at right angles to the line joining it to the centre of force, when the length of that line has its least value. For the ellipse the least length of the line is a{l—e), and the speed is therefore h/a(l — e). Fig. 56. §§ 133, 134, 135] DEDUCTION OP LAW OF FORCE. 247 This is made up of two components proportional to the lengths of 00' and OQ, that is to 2ae and 2a. Thus the components are eh/a(l—e^) = e\/iui./a{l — e^) = ejfjia/b, and the same divided by e. For the components in the case of the hyperbola we have in the same way, eh/ a (e^ — 1 ) = ej/x/a (e^ — 1 ) = &>J/j.a/b, and the same divided by e. In the latter case, since e>-l, the constant speed at right angles to the axis ip greater than the component at right angles to the radius-vector, and we see from another point of view why the orbit is closed when e--l. As an example of motion with such components as exist for the particle moving round a centre of force, we may take a steamer rounding a buoy moored in a tidal stream which flows past the buoy with constant speed. If the steamer have, besides the motion of the water, always a constant speed at right angles to the direction of the buoy, it will describe a conic section relatively to the land, with the buoy in a focus, just as if it were a satellite moving round a stationary primary which attracts with a force inversely proportional to the square of the distance. If the speed of the stream be greater- than that of the steamer, the path will be a hyperbola, and in the contrary case an ellipse. If the two speeds are the same, the orbit is a parabola. 135. Deduction of Law of Force from Form of Orbit and Uniform Description of Area. If we assume that the orbit is an ellipse with the centre of force in a focus 0, and that the radius-vector to the particle describes equal areas on the plane of the orbit in equal times, we can prove that the particle is acted on by a force which varies inversely as the square of the distance from the focus. For join the centre of the circle to Q, then, as the particle moves in time dt along the path from P to an adjacent point P', Q moves along the circle to an adjacent point Q'. The lines O'Q, O'Q' (Fig. 56) represent on a certain scale the velocities at the beginning and end of the interval dt. Thus QQ' represents on the same scale the change of 248 A TREATISE ON DYNAMICS. [CH. V. velocity which the particle has sustained in the interval. But QQ' = 2aedt and h = r^e, so that Q(/ = 2ahdt/r\ The acceleration is therefore, on the scale of the diagram, 2ah/7^, and, since the hodograph, with the lines of construction, is turned back through 90°, is directed towards 0. Its absolute value is obtained by multiplying 2ah/r^ by the factor 'J/u/a{l — e^)/2a for the ellipse, and is therefore fi/r^. In the same way the hyperbolic orbit might be dealt with and the same result obtained. 136. Kepler's Laws. Verification. The manner in which the plaiiets move about the sun was inferred by Kepler from a large number of observations, especially those made of the planet Mars by Tycho Brahe, who preceded him as astronomer at Prague. The results are contained in his Astronomia Nova, which appeared in 1629, and though the ideas on dynamics set forth in it are in great part erroneous, this work led to the establishment of the physical theory of gravitation which accounts for the motions of the planets by a consistent dynamical theory. As Newton showed, the planets move in obedience to mutual forces between the different bodies along the lines joining them, and tending to bring them together, a tendency prevented by the relative motions from having the apparently direct and simple effect which the ordinary undynamical intelli- gence expects. Kepler in vain endeavoured to fit the observations of positions and times into the hypothesis that each planet moved in a circle with uniform angular speed about an eccentric point, midway between which and the sun the centre of the circle was supposed to lie [see Ex. 16, § 131]. Observing the motion of the earth in the manner indicated in the next paragraph, he noticed that at the points of greatest and least distance from the sun, the earth had speeds inversely as these distances, and (of course with deviation from the hypothesis of uniform angular speed about the eccentric point) concluded that the speed of the earth in the imagined circular orbit was at every point inversely proportional to its distance from the sun. He noticed, moreover, that at the greatest and least distances §§ 135, 136] KEPLER'S LAWS. 249 there was the same rate of description of areas by the radius-vector drawn from the sun to the earth, and was led finally to adopt the law of uniform description of areas for the whole motion. This conclusion, however, he found to be utterly irreconcileable with the hypothesis of a circular orbit when applied to the planet Mars; and so finally he abandoned that hypothesis in favour of the true notion of an elliptic orbit with the sun in a focus. He thus formulated the two laws of the planetary motions : I. The radius-vector from the sun to each planet sweeps over equal areas on the plane of the orbit in equal times. II. Planets move round the sun in ellipses which have a com/mon focus at which the sun is situated- The observations of Kej)ler on the motion of the earth can be verified by anyone who cares to examine the tabulated values of the sun's apparent diameter from day to day throughout the year, as they are set forth in the Nautical AlTnanac, and to compare these with the longitudes of the earth's position at different times. The longitude from the perihelion position (the position nearest the sun) is the angle for a planet moving in the plane of the ecliptic denoted by d — CL in equation (6) of §126. Now the apparent diameter of the sun is the angle which the sun's diameter subtends at the earth, and in radians is djr, where d is the actual diameter and r the earth's distance from the sun. This apparent diameter is measured in various ways, e.g. by observing the time taken by the sun's disk to pass over the cross-wires of a telescope ; while the advance of the earth in longitude is obtained for successive equal intervals of time by observing with an equatorial telescope the corresponding changes, of the sun's Right Ascension (that is of the angle between a meridian containing the sun's centre and a certain zero meridian — that of the "first point of Aries"). The advance in Right Ascension (say in an hour) is not exactly the same as the advance in longitude, but enables the latter to, be calculated, and is, roughly proportional to the square of the sun's apparent diameter, as anyone may verify by means of the Nautical Almanac. 250 A TREATISE ON DYNAMICS. [CH. V. Taking the tabulated values, which, of course, are derived from and checked by observations, we find djr varying with the longitude from perihelion according to the equation - = D{H-ecos(0-a)},' (1) where D is a mean value of the apparent diameter for the different positions, that is we have, if i be a constant, (2) I l+ecos(0 — fx.) which, if e-<], is the equation of an ellipse (see § 126). The value of e can be calculated with great ease. Take the apparent diameter of the sun when the earth is at perihelion, that is about Dec. 21, and again at Midsummer, June 21, when the earth is at aphelion — these are the greatest and least values. Call them D^ and B^. Then we have by (1), D^ = D{l + e), D^^D{\-e), and so , = A:iA, (3) which we find to be 8/481 = 1/60, nearly. Thus the orbit is an ellipse of small eccentricity, that is an ellipse differing perceptibly but not greatly from a circle. Taking the sun's mean distance as 92,600,000 miles, this eccentricity gives as the distance of the focus at which the sun is situated from the centre of the ellipse about 1,540,000 miles, which is rather less than 1"8 times the sun's diameter. We see then how the law of the elliptic orbit is established for the earth at least ; that of the equable description of areas is proved by the fact, referred to above, that the daily or hourly advance in longitiide varies directly as the square of the sun's apparent diameter, that is, by what precedes, inversely as the square of the length of the radius- vector. Thus it is verified that r^Q is constant and r^Q is twice the rate of description of areas, as has been shown in § 25 above. By the mean distance of a planet from the sun is meant the length of the major semi-axis. We have already seen that the period of a particle about a centre of force, of §§ 136, 137] NEWTON'S DYNAMICAL DEDUCTIONS. 251 constant /x, is ^Trjo^ffi.. Hence, if /x is the same for different particles revolving about centres of force at different distances a^, a^, cig, ..., and T^, T^, T^,... be the periods of revolution, /TT2 7^2 712 ii-ii-ii^...., (4) a\ a\ a\ and conversely. By comparing the mean distances of the different planets from the sun, measured in terms of the earth's distance, with their periods, Kepler found that this relation of periodic times to mean distances held good, and he enunciated a third law of the planetary motions : III. 'Hie squares of the periodic times of the different planets are proportional to the cubes of their mean distances from the sun. Kepler's third law, dynamically interpreted, thus shows that fx is the same for the forces between the sun and the different planets of the solar system. The law, however, as we shall see presently, requires a correction which could only be foreseen and applied when the dynamical theory had been worked out, and the agreement of which with observation affords a strong confirmation of the truth of the theory. 137. Newton's Dynamical Deductions from Kepler's Laws. From these laws, which, so far as they go, merely state the observational facts of the motions of the planets, Newton made certain dynamical deductions [Principia, Lib. I. Props. II. XI. XV.]. (1) From the law of areas : that the force, if any, between the sun and a planet is along th§ line joining the planet with the sun. For the product, mr^d, of the double rate r^6, of descrip- tion of areas by the mass m of the planet, is the angular momentum of the planet about the centre, and this cannot remain constant under the action of force on the planet in the plane of the orbit unless the force have no moment changing mr^O, that is, tlie force must be in a line through the sun's centre. A component of force perpendicular to 252 A TREATISE ON DYNAMICS. [CH, V. the orbit would of course alter the plane of mdtiorii a change which is here supposed not to take place. (2) From Kepler's law of the elliptic orbit: that the planet is acted on by a force toward the sun which varies, as the planet moves in its orbit, inversely as the square of the distance. The proof of this deduction is contained in § 125, Ex. 1, and again in a simple form in § 135. (3) From Kepler's third law: that the forces towards the sun on the different planets at any given instant of time are inversely proportional to the squares of the distances of the planets at the instant. This deduction was proved above, when it was shown that if the squares of the periodic times are proportional to the cubes of the mean distances, fi, the so-called " force of the centre " is the same for all the bodies. The correction of this law, referred to above, is necessary to take account of the acceleration of the sun towards the planet, which is sensible when the mass of the planet is comparable with the mass of the sun. For if P and S be the masses of the planet and sun, we have, since the force F on the planet towards the sun is equal to the force on the sun towards the planet, acceleration of sun towards planet _ F/S _P acceleration of planet towards sun F/P 8' so that if P be very small in comparison with M the sun may be taken as being at rest. In the cases of the large planets, such as Jupiter and Saturn, the masses are so great that they must be taken into account. We shall now show how this may be done. 138. Effect of Mass of Planet. Since the observed motion of a planet is taken with reference to the sun's centre, regarded as at rest, the foregoing theory must be corrected by substituting for the actual acceleration, /x/r^, of the planet the acceleration with reference to that point. We have seen that, on the supposition that the sun is at rest, the accelerations of the planets along the lines joining them to the sun would be the same at the same distance : let us suppose this to be true in the actual case and compare the result §§ 137, 138] EFFECT OF MASS OF PLANET. 253 with observation. The forces on the different planets are proportional to their masses. Hence we take kS/r^, where k is a constant and 8 the mass of the sun, as the force per unit mass on each planet, or its real acceleration, so that fi = kS. The. whole force on a planet of mass P is kSP/r^ at distance r, and this must be equal to the opposite force on the sun. The acceleration of the sun in the opposite direction is therefore kP/r^. These two oppositely directed accelerations may be taken as relative to the centroid of sun and planet, the position of which cannot be affected by their mutual action. To enable the theory set forth above to give the motion of a planet relatively to the sun, we must apply to both planet and sun an acceleration kP/r^ in the direction from planet to sun. This does not alter the relative motion, but .cancels the planetward acceleration of the sun and gives k(S-\-P)/r^ for the sunward acceleration of the planet. Hence, in the application of the foregoing theory, we take fx, for a planet of mass P, equal to k(S+P). Thus we have different values of /z for the different planets in the field of solar attraction. The differences, however, are but slight, since S is great in comparison with every P: for example, 5f/P = 332000 for the earth and =1047 for Jupiter. If the student does not perceive why the process here described is followed, the following discussion may serve to explain it. The position of the centroid G of the two bodies is not affected by their mutual action, and is con- venient, therefore, as a point of reference from which to measure the distances of the sun and planet. We denote these distances by r^, r^, and the distance of the planet from the sun is r-^ + r2 = r, say. The two bodies remain in line with their centroid, and so the lines joining their centres to G, remaining as they do pai-ts of one straight line, are turning with the same angular speed .6 in the same direction at each instant. Thus, since the forces per unit mass oh the sun and planet are respectively kP/r^, kSjr^, we have p „ ri-'r-^eF = k-,, r,-r,^ = k^^ (1) 254 A TREATISE ON DYNAMICS. [CH. V. Adding, we get, since ri+r2 = r, ^_re^ = k^, (2) which is the differential equation of time-rate of variation of momentum along r for either body. Again, for the angular momentum (per unit mass) of the planet about the centroid, we have rie = h, (3) or, since r2 = 8r/(S+P), r'e = h,{^)"^h (4) These lead, in the manner already explained, to the differ- ential equation (Fu ^ , 8+P (n\ ^ + u = /.-^, (5) where k{S+P) takes the place of fn. For the period of revolution §128 above gives with this value of fjL the equation ^=2-V^' <'> where a is the mean distance. For another planet of mass P' and mean distance a', the period 2" is given by ^'^^''^MS+P') Thus we obtain (7^ 'k(S+P') if k be the same as in the former case. Thus we obtain T^_a^ S+P' T"-~a'^ S+P 139. Correction of Kepler's Third Law by Theory of Gravita- tion. Now Kepler's third law asserted, as we have seen, that T^/T'^ = a?ja'^. Equation (6) shows that, according to the theory just explained, this statement is not quite correct. The following table, taken mainly from Maxwell's Matter and Motion, shows that observation confirms equa- tion (7). The values of a are the mean distances of the §§ 138, 139] CORRECTION OF KEPLER'S THIRD LAW. 255 planets expressed in terms of that of the earth, taken as unity; in the same way the periods T are taken. It will be seen that for the planets of smaller mass than that of the earth, a?—T^ is very small and negative, while for the much larger planets, Jupiter, Saturn, Uranus and Neptune, it is positive. In proportion to a^ or to T'^ this difference is greatest for Jupiter, the planet of greatest mass. We have here referred to the sun and a planet as the two bodies, but of course the same theory is applicable to any primary and a satellite of that primary. /', Mureury. Venue, Earth. Mars. 0-476 0-82 1 0-1073 a, 0-387098 0-72333 1 1-52369 T. 0-24084 0-61518 1 1-88082 a\ 0-0580046 0-378451 1 3-53746 ^2 _ 0-0580049 0-378453 1 3-53747 a? - T\ - -0-0000003 -0-000002 -0-00001 Jupiter. Saturn. Ui'anus. Neptune, P, 317 94-8 14-6 17 a. 5-2028 9-5388 19-1824 30-037 T, 11-8618 29-4560 84-0123 164-616 a?, - 140-832 867-914 7058-44 27100-0 T\- - 140-701 867-658 7058-07 27098-4 a?-T\- + 0-131 + 0-256 + 0-37 + 1-6 The third law of Kepler is thus corrected by the gravita- tional theory of the motion of a planet about the sun. This is an important result of the theory of The Motion of Two Bodies, as it is called ; but it is to be remembered that both the sun and the planet considered are acted on by all the other planets, to say nothing of more distant bodies. While the problem of two bodies is thus comparatively simple, the solution of that of three bodies has so far only been obtained by successive' approximations, and the same method has enabled the various perturbations due to the other planets to be evaluated in each case of motion, and tables of the approximate positions of all the planets for future time to be constructed. 256 A TREATISE ON DYNAMICS. [CH. V. 140. Weighing the Planets. The determination of the mass of a planet can be effected by this theory if the planet has a satellite the period and distance of which are known. Let the mass of the satellite be m, the period T^, and the semi-axis major of its orbit a^. Assuming what observation shows to be the case, that the same constants applies to the attraction between a planet and its satellite as to the solar attraction, we get ^.=2'a/k^ <" On the other hand, for the period of the planet, we have ^-^-aGs?; <^> 'HS+P) Therefore Tl_oi S+P (3) T^ a? P+m and if m be neglected, _T^^S__ (4) Ex. 1. Take the case of the earth and the moon. We have 7'j/7'=l/13'369, and the mean values of the angles subtended by the earth's radius at the sun's centre and moon's centre are 8-8" and 57' 2" respectively, so that a?/a^=(57g^)5 x eO^/S'S^ approximately. Thus we get, neglecting m, S 1 (57^)^x60^ l-32<»00 P-T3^3692 8^P 1-329000. The accepted value of this ratio is slightly greater, 332000. Ex. 2. A satellite of Saturn makes one revolution about the primary in 16 days (true period 15 d. 22 h. 41m. 23"2s.), -while the Saturnian year is (see Table, § 139) 10760 days nearly. The radius of the orbit of the satellite subtends at the sun an angle of 176^ seconds, so that the ratio aja of the distance of the satellite from the primary to the distance of the latter from the sun is 176"25/206265. Thus we obtain g _ ^ 16 V 20626.5^ P-Il0760y 176^25^"^— ^•^*'' nearly. The sun's mass thus comes out 3543 times the mass of Saturn. The value accepted is (Young's Aitronomy) 3502. In this way the masses of the superior planets have been measured. The miniature solar system which we have in §§ 140, 141] UNIVERSAL GRAVITATION. 257 Jupiter and his family of moons affords in itself examples of Kepler's third law with its Newtonian correction. The observed distances of the moons enable their accelerations to be calculated, and a comparison of these with the ac- celeration of the planet towards the sun confirms the supposition that it is the same constant h which enters into the value of all the attractions between different planets. This constant, commonly called the constant of gravitation, is the force of attraction between two units of mass, say two grammes of matter, concentrated at two points at unit distance, say a centimetre, apart. An experimental comparison of the gravity pull .on a body at the earth's surface, with the pull between that body and a sphere of lead,* has enabled the earth's mass to be deter- mined and the value of h to be calculated. For the units just specified it is about 67 x 10"^ dynes. If we alter the units of length, mass, and time to, say L cms., M grammes, T seconds, the value of this constant will be altered in the ratio of 1 to UM-'^-T'^. For the force F, between two masses m, m', at distance r apart, is krrmi'lr^, so that k = F7^lmnn' ; and so the multiplier would be as stated. If we take Z = l, M=l, and 1^=10^16-7, k will become 1. Thus the new unit of time would be 10*/\/6T = 3862, in seconds, 262 seconds more than an hour. This has been called by M. Lippmann " I'heure naturelle " (G.R. 1899), but it would hardly be a convenient interval of time to adopt in practice. 141. Newton's Theory of Universal Gravitation. It occurred to Newton to compare the acceleration of the moon in its orbit relative to the earth with the acceleration of a body falling at the earth's surface. The former can be calculated if the moon's distance from the earth is known, for the orbit is nearly circular, and the average period of revolution has been very exactly determined. In Newton's time, the ratio of the moon's mean distance to the earth's radius was fairly accurately known ; the radius of the earth, however, had been very inaccurately estimated, and the moon's distance deduced from it was in error, of course, to the * See Gray's Treatise on Physics, Chap. XIII. G.D. R 258 A TREATISE ON DYNAMICS. [CH. V. same extent. Thus his calculation of the acceleration of the moon towards the earth was in error, and when multi- plied by the proper ratio failed to give as the corresponding acceleration at the surface of the earth a value sufficiently nearly equal to the observed acceleration of a falling body. The comparison was effected on the assumption that the force towards the earth on a particle at or near the surface was the same as it would have been if the matter of the earth had been collected at the earth's centre. Newton laid the calculation aside until, in 1682, he learned at a meeting of the Royal Society that a new measurement of an arc of the meridian had been carried out by M. Picard in France, which increased the former estimate of the earth's radius in the ratio of 7 to 6. Resuming the calcula- tion, he now found very fair agreement between the calculated and observed values of the acceleration of a falling body. He found, in fact, that the acceleration of the moon towards the earth was to the acceleration of a falling body in the inverse ratio of the moon's distance and the earth's radius. It was not, however, until three years later that he published his conclusion that it was the same gravitation that kept the moon in its orbit and caused the fall of a stone at the earth's surface. In the interval he had overcome the difficulty of obtaining a satisfactory proof of the assumption above referred to, on the basis of the theory of universal gravitation to which his in- vestigations had led him. According to this theory there existed a force between every pair of particles of matter, urging each toward the other, which was directly pro- portional to the product of the masses of the particles and inversely proportional to the square of the distance between them ; that, in fact, if m, m! be the masses of two particles (that is portions of matter of dimensions so small in com- parison with the distance between any point in one and any point in the other that they might be regarded as concentrated at points) and r the distance just referred to, the mutual force F between them was given by the equation „ , min' § 141] UNIVERSAL GRAVITATION. 259 where kiaa, constant — the "constant of gravitation" already referred to — which is the same for every pair of particles. Now, on this principle, Newton at length succeeded in proving that the whole force exerted on a particle of matter in consequence of the presence of a sphere of matter, either uniform in density or made up of concentric shells which were each of uniform density, but differed in density from one another, was the same as if the whole mass were collected at the common centre. The agreement of the two earthward accelerations — that of the moon and a stone at the earth's surface — was strong presumptive proof of the truth of Newton's theory, and later investigations, in which the theory has been applied in an immense number of ways, with in all cases results which agree with observation, have confirmed it in the most complete and triumphant manner. That the force between two particles is referred to as an attraction is sometimes made a ground of criticism of this theory : for it may be, it is urged, that each body is pushed toward the other. It is true that this might be a perfectly correct way of describing what takes place ; but when we say that two bodies A and B mutually attract one another we mean no more than that A is urged toward B and B is urged towards A, with equal forces, in con- sequence of the presence of the two bodies in the field. The cause of gravitational action is in no way prejudged by this mode of referring to the phenomenon. The comparison made by Newton may be restated as follows, taking the earth as a sphere and neglecting its rotation. If the moon's mean distance from the .earth be 383000 kilometres, and its time of revolution be 2732 mean solar days, its acceleration towards the earth is, in cm./sec.^ units, (2T32lkooy^«^«««^l«««00 = -27l. The acceleration of a falling .body at the surface of the earth, taken as a sphere of radius 6365 kilometres, ought therefore to be, in the same units, /383000Y 27] 1^6365"; "" ^^^' 260 A TREATISE ON DYNAMICS. [CH. V. if it is the same gravitational attraction that keeps the moon in its orbit and causes a stone to fall to the earth. Now we have /qssooON^ (25^) X -271 = 982-4, \ 6365 / which is nearly the (uncorrected) value in cm./sec.^ units of the acceleration of a body falling freely under gravity at the earth's surface. Laplace (M&anique Celeste, P" Partie, Lib. II.) calculates from accelerations estimated from the distances the value of the moon's horizontal parallax, and compares the calculated value with the observed value. 142. Does Newtonian Gravitation extendi to the Fixed Stars ? The question of the extension of the theory of gravitational attraction to the fixed stars is not one that can be settled by means of observation alone. For though it is seen that the components of a binary star revolve round one another, so that it is clear that each component is acted on by a force toward the other, observation cannot decide what the position of the primary of such a pair is with respect to the relative orbit described about it by the secondary. For the apparent orbit is only seen projected on a tangent plane to the celestial sphere at the point, and it is the projected position of the primary that is observed, not the real position. Now the orbits as seen are always ellipses, and the real paths are no doubt also ellipses ; they are certainly ellipses if they are plane curves. But the position of the primary is neither at the centre nor at a focus of the ellipse observed. It may, however, be situated at a focus of the real ellipse, for when an ellipse is projected on a plane the foci do not project into foci of the curve obtained by projection, though the centre projects into the centre. The relative movements of the components of a large number of double stars are known, and these motions are very different in different systems; and we are led to assume that the central force is such that each component describes an elliptic orbit about the other, which depends only upon the position and velocity of the body at the initial instant. If then we take axes Ox, Oy, drawn from the centre of the primary in the plane of the real orbit, we §§ 141, 142, 143] UNIVERSAL GRAVITATION. 261 may write for the relative motion of a secondary, the coordinates of which are x, y, the equations mx= —Fx/r, iny— —Fy/r, where r = \/x^ + y^, and inquire what function F is of x, y in order that the orbit may be a conic whatever are the initial values of x, y and of ob, y. This problem was proposed, by Bertrand in Gom/ptes Rendus, 84, and. the same volume of that journal contains solutions by Halphen and Darboux, who have shown by very different methods that two laws, equivalent to those stated in § 168 below, are the only laws of force which give a conic as the orbit for any initial conditions. If the force is assumed to be independent of the vectorial angle, = tan"^2//a;, that is, to be a function of the distance r only, there are only two laws which give always a conic, namely, F=mfjir and F=mfji.jr^. The first of these cannot be the law of force for the components of binary stars, since the primary would then be at the centre of the projected orbit, which is not found to be the case; there remains therefore only the other law, which is that of the Newtonian gravitation. 143. Experimental Illustration of Gravitational Attraction. The motion of a satellite round a primary can be illustrated experi- mentally for different initial conditions by the following arrangement, in which electrical forces varying inversely as the square of the distance from a fixed point are made to play the part of gravitational forces. Two Leyden jars are arranged on a table with their knobs on the Same level, and from two to three feet apart. Between them is hung by a thin fibre of silk a pith ball, or, better, a small silvered bead made of thin glass, so as to be as light as possible. The silk fibre should be at least fifteen or twenty feet long, and the ' point of support should be adjusted so that the ball may hang about the level of the centres of the knobs, and midway between them. The jars are noW removed and charged, one positively, the other negatively, and replaced in Ae same positions. The siuall ball will be attracted towards one of the knobs, will touch it; and then be repelled. As.it is driven away from that knob it acquires speed under the continual repulsion, and if it moves, as it probably will, towards the other knob, the' repulsion lis aided by the attraction which the ball also -experiences towards the 262 A TREATISE ON DYNAMICS. [CH. V. latter centre. Thus the ball arrives in the vicinity of the second knob with a considerable speed in a direction which depends on initial conditions, which are different in different experiments. When it has thus arrived, with what we may call a "speed of projection," at a point near the second knob, it is acted on by the attraction due to the charge on that, and hardly at all by the repulsion due to the charge on the first. The orbit round the second will be clearly seen by the persistence of impressions on the retina, and will take different forms according to the speed and direction of " projection." Sometimes the ball will be seen to pass round the second knob like a comet in its perihelion passage round the sun, passing off in what appears at first to be a long ellipse till it comes again under the influence of the first knob, to be thrown back again, perhaps, to describe a second orbit round the other. Or, on coming into the lield of the second knob, the ball may be moving with just the velocity necessary to enable it to describe a circular orbit round the centre of force, which it will do two or three times in quick succession before the adjustment of force and velocity necessary for the circular path has broken down, and the ball falls in on the centre. It will be seen that, except in so far as the charge on the adjacent knob is disturbed by that on the ball, the arrangement, when the ball is near the first knob, is such as to give force varying inversely as the square of the distance from the centres of the knobs. The action of gravity is well-nigh annulled, and this is essential, by making the fibre very long. The spectators should stand some little way off, to prevent disturbances from air- currents, which should be otherwise avoided as far as possible. 144. Elements of an Orbit. The orbit of a planet lies in a plane coinciding more or less nearly with the plane of the ecliptic or path of the earth round the sun. The line of intersection of the plane of the orbit with the ecliptic is called the line of nodes: the nodes are the points on the orbit which lie in the ecliptic. To an eye placed away beyond the north pole of the earth a planet will appear at one node to come from the south to the north side of the ecliptic, and at the other to pass from the north side to the south. The former is called the ascending, the latter the descending node. Let a line be drawn from the sun's centre to the ascending node and another from the same point to the vernal equinox : the angle between these lines is called the heliocentric longitude of the ascending node. Let a line be drawn from the sun's centre to the perihelion position of the planet in its orbit, and another to the vernal equinox, and let these lines be projected on the plane of the ecliptic. The angle between the projections is the i 143, 144, 145] TIME IN AN ORBIT. 263 heliocentric longitude of the planet's perihelion. The longitude of the planet at a specified instant is defined in the same way by the projections of a line drawn from the sun's centre to the planet and the line to the vernal equinox. These angles are reckoned positive only when measured one way round, so as to avoid confusion. Thus for the complete determination of an orbit six elements are required : 1. The major semi-axis, a. 2. The eccentricity, e. 3. The inclination of the plane of the orbit to the plane of the ecliptic. 4. The longitude of the perihelion, ot.. 5. The longitude of the ascending node. 6. The longitude of the planet at a given instant, Q. 145. Time in an Elliptic Orbit. In Fig. 57 let APA' be the orbit, with foci 0, 0' and centre C, NP the ordinate perpendicular to the major axis AA', to the position P of the particle, meeting when produced the circle described on A A' as dia- meter in Q. P is joined to and Q to G and 0. As P moves, let the ordinate NPQ accompany it: the point Q is called the eccentric follower of P. The angle AOP is called the true anomaly of P and the angle ACQ the eccentric anomaly. We denote the former by Q and the latter by u. [This is the usual notation, though u is also generally used for 1/r.] The mean angular speed n with which the radius-vector turns as the particle moves is I-kIT. But T=2Tr-Ja^//UL = 2Trab/h, so that n = \filja? = hlah. The quantity nt, where t is the time in which the particle 264 A TREATISE ON DYNAMICS. [CH. V. moves from A to P, is called the mean anomaly. We shall now find relations connecting 6, u and nt. In the first place we may regard the ellipse as derived from the circle in Fig. 57 by shortening every ordinate (as NQ to NP) in the ratio b/a. Hence area AOP = - area 40Q = -(area ^CQ-area OCQ) = - ( ^ua^ — \a^e sin u). But if t be the time in which the particle moves from A to P, area AOP=\ht = \abnt. Equating this to the result just found, we get nt = u — esmu (1) To connect Q with u, we have NO = a{e — caBu) and also 0N= — OP cos Q, with OP = a{\ — e cos u), so that 0N= a(e cos tt— 1) cos 6. Thus we obtain „ e — cosu e + cosO cost^ = ^, cosu=. ecosw — 1 1 + ecost^ This equation may also be written 1 — COS0 1+e 1— cosw or 1+COS0 1 — e l+costi tan ^0 = j^Y — tan ^u. •(2) Also we have 8intt = \/l — e^=— ■ ;; (3) 1+e cos Finally, by (1), (2) and (3), we obtain nt = 2i3.n-^(J\^ia.-a\e)-eJT^^ ^ ^'""^ . . ...(4) \V]+c "'/ 1 + e cos From this last equation the time can be reckoned from perihelion when the true anomaly 6 is known. 146. Time of Describing any Arc. Lambert's and Euler's Theorems. The time t^ — t^ of describing any arc from a point P-^, §§ U5, 146] TIME IN AN ORBIT. 265 where the radius-vector is ri, to a point J\, where the radius-vector is j'2, can be found for any central orbit by the equation tn — t-. 1 /"*2 where 0^ and 9^ are the angles corresponding to ri, r^. The in- tegration can be carried out by the relation connecting r and 6. For an elliptic orbit the following theorem has been given by Lambert for this case. Let, besides r^, r,^, the length c of the chord Pi 7*2 be known ; then, if 0, <^' be angles defined by the equations sin ^(^ = \/{ri + »-3 -I- c)/4ffi, sin|<^' = N/(?-i-|-r2-c)/4a, n{t2-ti) = ') (2) To prove it we note first that if %, u^ be the eccentric anomalies for the positions P^, P.^, we have n(t2 — ti) = U2 — Uj^ — e{smii2~ sin Ui) (3) Taking and 0' first as undetermined, putting ^ — ^' = U2-Ui, and choosing (^-)-<^' so that co&\{<^ + 4'')=eQ,oa\{ui+u^, we get J! (<2 - ^i) = ^ - ^' - (sin <^ - sin c^'), the form (2) given to m(*2- l/i be the coordinates of P^, P^, (;2 = (^2 -^,)2 + (2/2-^1)^ = a!^(cos 1*2 - COS M])^ -I- 6^(sin u^ - sin Mi)^ = 4a2 sin^C-^ - <^') sin^K^ -t- <^'). Thus c = 2a sin i{rj) - <^') sin J(<^ -t- 4>'), where the positive 'value of the square root has of course been taken. Again, i\ + r2 = a(1 -ecos«,)4-a(l -ecos«2)=2a{l -cos^(<^-<^')cos^(<^-l-<^')}- Hence ?'i-|-?'2-l-c = 2a(l -cos^) =4asin^-|i^, ^i +r2 - c = 2a(l - cos (^') = 4fl(sin^^<^', that is sin^<^ = -i^!i^±f, sini<^'=V^^i^^ (4) The theorem is therefore proved. The ambiguity resulting from the radicals in these equations is of no consequence if the positions of Pi, P2 on the ellipse are known ; the student may consider different possible cases for given values of r^ , ■/ 2 and c. Inserting the value of n, V/i/a'*, in (2), we get <2-«] = \P{<^-^'-(sin<^-sin.^')} (5) 266 A TREATISE ON DYNAMICS. [CH. V. Now let the eccentricity of the ellipse be increased towards unity, a will increase towards infinity, and the ellipse will approximate to a parabola in the part near the centre of force. When a is very great we may take <^-sin<^=J(^^ cf)' -sm' = l'^, since (f) and <^' are now very small. We have then a^(<^-sin<^)=J(n + »-2 + c)*, a*() }. Similarly, the area of a focal sector of a parabola for which the same quantities r^, r^, c are given is, since h in this case is v/iii, where I is the length of the semi-latus rectum, For a solution of Kepler's Problem — the expansion of the true anomaly 9 and the radius-vector r, in terms of the time t — the reader is referred to Routh's Dynamics of a Particle, § 476 et seq., or to Tait and Steele's Dynamics of a Partiile, § 163. The expansion of m in terms of t is given in Gray and Mathews' Bessel Functions, Chap. I. p. 4 ; see also the other books cited. 147. Disturbed Orbits. (1) Tangential Impulse. We now find the effect of a small impulse on the particle in changing the elliptic orbit which it describes about the given centre of force. The impulse may produce an increase of the speed of the particle without changing the direction of motion, or generate a small speed in the direction at right angles to that of motion, or some combination of these. Let first the impulse be tangential, and change the speed from v to v + Sv without changing the direction. The distance r of the particle from the centre is thus given, and the problem is to find (1) what change in the eccentricity of the orbit is produced, and (2) the amount of turning of the major axis. The most convenient method ! 146, 147] DISTURBED ORBITS. 267 of dealing with such problems consists in finding first what change is produced in the position of the empty focus. From that the change of eccentricity and the new position of the major axis can be found at once. Let P (Fig. 58) be the position of the particle when the impulse is applied, PT a tangent meeting the axis produced in 2\ the centre of force and 0' the empty focus. The point A, at which the major axis intersects the orbit, is an apse (§ 152), and the Fie. 58. alteration of direction of the major axis is often referred to as the change of the position of the apse A. The effect of Sv is to change the focus 0' to 0", where O'O" is t^ice the distance Sa obtained by differentiating the equation that is '^\r a/ 2v Sa ■ a^Sv. •(1) Now we have 0P + P0" = 2(a+Sa), and 0" must lie on PO' produced, since the equal angles OPT, SPO' are unaltered. The distance 00' is 2ae, and 00" is 2(a+Sa)(e+Se), so that 00"-00' = 2(aSe+eSa). Now, by Fig. 58, 00" -00'= O'O" cos L 00' P. But if ^=lOPT=cSPO' and yp-=LPT0, then LOO'P=,p-\lr and L0'0P =

-\l^)-2ae}Sv (2) by the value of Sa, found in (1) above. Now, by Fig. 55, 2(ie=(2a— r)cos(^-\/r)+rcos(0+\/r), and therefore 2a cos(^ — ^) — 2ae = 2r sin ^ sin \^ (3) Also, by the equation for v^ already used, v//j. = V(2a — r)//Mr. Hence we get Se = 2\- J^ sin-\b- Sv = 2-i=.sin\^ Sv, (4) since if p, p' be the lengths of the perpendiculars let fall from and 0' on the tangent at P, r sin <}> =p, {2a— r)sin

/»• = 2 (a — r) sin 0, so that sin (j> = ae sin ■»/r/(a — r). Substituting in the last equation, we get, since a—r = ex, eSylr = —-F= sm-yb-Sv, (4) It will be observed that by (4) and (6) of §147 a tangential impulse with the motion (see Fig. 57) increases or diminishes the eccentricity according as the particle, in its motion in the direction ABA'B'A, is between B' and B, or between B and B', while the apse advances or recedes according as the particle is between A and A' or between A' and.^. On the other hand, for a normal impulse applied inwards, the eccentricity is diminished or increased according as the particle is between A and A' or between A' and A (see Fig. 57) ; the apse, on the other hand, advances when the particle is between K' and K, and recedes when the par- ticle is between K and K'. From the last result we get at once Callandreau's theorem,* that if the orbit of a comet lie within the orbit of Jupiter, so that the comet finds itself near the planet only in the vicinity of aphelion, the disturbing action of Jupiter's attraction is to turn the major axis of the orbit round in the direction of the comet's motion. Here the normal impulse acts outwards, and of course when this is the case, and the tangential impulse retards the motion, the values of de and e S\/r found above must be reversed. If the impulse is in neither the taiigential nor the normal direction, it must be resolved into its tangential and normal components, and the effects, found as above, added together. The effect of continuous action, the law of variation of which is known, can be formed by integration from the results obtained above, in which §v and Su are then to be regarded as the changes of velocity in the tangential and normal directions produced in time dt, over and above those which arise from the displacement in the orbit. 149. Disturbed Orbit. (3) Change of Intensity of Central Force. So far the constant fi. of the centre has been supposed * Ammles de I'OhservcUovre de Paris, 1892, t. 20. §§148,149,150] DISTURBED ORBITS. 271 unaltered. If it be suddenly changed to fx', we must have v^ = IJ.(2/r-l/a) = fj.'(2/r-l/a'). so that , _ fi'ar ~ fir+2{fi' — fi)a which gives the length of the new major semi-axis. Since the direction of the motion and the position of the centre of force are not altered, the angle remains the same, and thus the new position of the empty focus can be found at once from the value of a. Fig. 58 illustrates this case. If the change fx' — fi is a finite one, we get from Fig. 58, since L0P0'==Tr — 2 — \fr)+r cos { — \fr) = {2a — r)cos(l e'^ - 1. The alteration of peiiod is from iirsla?!)!, to 27r\/a^(l — ^aJf/ij,, that is to 27r(l - Sa.)>Ja^lfi.. The period is thus diminished in the ratio of 1 - So. to 1. Ex. 8. A particle describes an elliptic orbit about a centre of force at a focus, and the centre of force is suddenly shifted a small distance a.a towards the centre of the orbit : to find the change of eccentricity and the turning of the apsidal line. The distance r is altered by -^(x.acos((/>+T/r), where i/r is the angle between the tangent and the major axis, as shown in Fig. 58. Hence a is shortened by a.a'cos(<^ + i/r)/r2„ and therefore 2a -r is shortened by |(x.a(4a2 — y?)cos(+^)/*'^. Both radii-vectores are turned towards the centre through the angle aoLam( + ylr)/r. Reference to a figure shows that the second focus is carried to the other side of the major axis from the particle through a perpendicular distance (2a-r) — sin(<^+V')cos(<^-i/r)- Ja.a(4a2-r2)-2Cos(^ + ^)sin(<^-^), 276 A TREATISE ON DYNAMICS. [CH. V. and therefore the turning di/r of the apsidal line is this quantity divided by 2ae. Again, the distance between the foci in their new positions is less than the former distance 2a!e by a.a{l + — g-j— cos(^+i/f)cos(^-i/r)H sin((^ + i/r)sin(^-i/r)}, and this is -(2aSe+e8a)= -2{aSe-2e— 2-cos(<^ + ^)}, from which we obtain Se. •Applying these results to the particular case when the particle is at an extremity of the latus rectum through the focus, we notice that to the first order of small quantities 8a =0, and obtain 8\jr=-p! Se=— a, where I is the length of the semi-latus rectum. To the second order of small quantities we have in the same case &r=^(Sf=la.^ayi, and therefore Sa = 2Sra^lr^=a?a*/P. Thus (a + Saf=a^(l +Sa.VjP). The period ^Tr^/a^/iJ, is thus increased by the fraction ^a.V/P of its former value. 151. Orbit slightly disturbed from Circular Form. In the preceding §§147-150, we have considered the effect of a small disturbance of the motion of a particle about a centre of force attracting according to the law /lu'^, and have seen that it is to cause the particle to describe a new orbit, the deviation of which from the original orbit is specified by the alterations produced in a, e, and \fr. We now suppose a particle which moves in a circular orbit about a centre of force situated at the centre of the circle, and attracting according to the law fiu", to be slightly disturbed from its path in such a way that the value of h is not altered. In the first place, if 1/c be the radius of the circular orbit and v the speed of the particle in it, v^c^fid^, so that h} = v''lc'- = fid^'^. Further, the equation of motion is de''^'^~h^u^~ d^-^ ^ ' Now let ^ = 0(1+ a;), where x is small. We obtain ^02= -(l+a;)+{H-(w-2)a;+...} (2) §§ 150, 151] NEARLY CIRCULAR ORBIT. 277 or J+(3-%)a; = 0, (3) if we neglect higher powers of x than the first. If S^-n., this equation has the solution ir = ^cos(\/3 — 710 — oc), (4) where A and a are constants. Thus we have ii = c{l+^cos(s/3-7i.0-a)}, (5) and as the radius vector turns through the angle 27r/\/3— %, the value of u oscillates from the maximum value c{\-\-A) to the minimum c(l— J.) and back again. The value of A is to be found from the conditions of the disturbance to which the orbit is subjected. To carry the solution to a higher degree of approximation, we write the differential equation as ^2 v. ||=(»i-3){^ + i(»»-2M (6) from which all terms above ^^ have been excluded. Substituting the approximate value of x, A cos(\/3 - n d - a.), just found in the term in x% we get for the differential equation to be solved |^=(?i-3)[*- + i(?i-2)^2{l+cos2(v/3^e-(x.)}] (7) For the solution of this form of equation the student may consult Gibson's Calcvlus, § 170. But he will find by substitution that it is satisfied by x=Acos{'JZ^e-a.) + A'^{G+Dcos{^.\IZ^6-a.)}, (8) and will at the same time determine the two additional constants C, D. In the same way the approximation may be pushed still further, but it will be found that the coefficient of Q is no longer a multiple simply of sj'd-n. If »i > 3 the solution of (7) is of the form so that unless A is zero x will increase indefinitely with 6. We must therefore regard the circular motion in this case, as unstable, and when TO < 3 as stable, inasmuch as whatever the constants A and tx. may be the radial deviation can never exceed the values correspond- ing to the maximum and minimum values of u, c{l + A), c(l -A). 278 A TREATISE ON DYNAMICS. [CH. V. It will be observed that if w = 2, the differential equation is d'u d^x +^=0,, and X is under no restriction to be small. Thus we have x=Aco%{B — a^, and M=c{l + 4cos(^-a.)}, .(9) (10) (11) the equation of a conic of eccentricity A and semi- latus rectum 1/c. Thus we fet.again the solution fully iscussed above. But from the present point of view, we regard it (at least when A<.\) as an oscillatory deviation from the circular orbit, described from the centre of force (a focus of the conic) as centre, with the senii-latus rectum OX = 1/c as radius, as shown in Fig. 63. The period of oscilla- tion is that of revolution. PiQ g3_ For the ellipse which we have when A ( = e) <1, the radial deviation at the point nearest the centre is l/c-l/c(l + J) = e/c(l-|-e), since A = e, and at the point furthest from the centre is l/c-l/c(I-^)=-e/c(l-e). The double rate of description of area retains the value in the elliptic orbit which it had in the circular orbit, but the period in Uie ellipse is 27r\/a3/7I=27r/\//«^(l -e2)3, while in the circle it was 2?r/N//j^. By this we can reckon easily the alteration of period produced by a slight disturbance of a circular orbit. Thus, if e be very small, the period is changed from 27r/>/;^ to (2ir/\/;ic3)(l-l-3e2). If »i = l, the differential equation of the approximately circular orbit is ,2 ^f+2a;=0, (12) so that for x we have the value x=Aq.o%{'J^O-(x), (13) §§ 151, 152] THEORY OF APSIDES. 279 where A is small in comparison with 1, and the equation of the path is _ w=e(l+x) = c{l+Acos(\f2e-a.)} (14) The variation of u thus passes through one complete period while the radius- vector revolves through the angle 'JZir. The student may prove that the area swept over by the radius- vector and the period remain unaltered to the second order of small quantities. 152. Theory of Apsides. An apse is a point on the orbit at which it ia met at right angles by the radius-vector from the centre of force. The condition fulfilled at an apse is therefore du/d6 = 0, that is u is a maximum or a minimum. A planet is at an apse when in perihelion or aphelion, but not elsewhere in the orbit. At perihelion u is a maximum and r a minimum, at aphelion the reverse is the case. The radius-vector from the centre of force to an apse is called an apsidal distance. We can show that, whatever may be the number of apsides in an orbit or a branch of an orbit, there cannot be more than two apsidal distances, if the central force is a function of the distance alone. For an apse may be taken as the point of projection, and the velocity there as tlie velocity of projection. If two particles be projected in the plane of the orbit from an apse in the two opposite directions at right angles to the apsidal radius- vector, under a central force which has always the same value at the same distance, the paths of the particles will lie symmetrically on the two sides of that line. Thus every radius-vector on one side will be repeated on the other at an equal angular distance from the apsidal radius. The curve on one side will, in fact, coincide with the image of the curve on the other in a mirror at right angles to the plane of motion and coinciding with the apsidal radius. Now the radius of curvature at every point of an orbit must be the same for both directions of motion along the tangent at an apse, since v^/ p = F sin (p; and therefore a particle which has reached an apse in its orbital motion will, if its motion were there suddenly reversed, simply return along the path by which it arrived. Thus an apsidal radius-vector divides an orbit into two parts, which 280 A TREATISE ON DYNAMICS. [CH. V. lie symmetrically on its two sides. It is clear then, that if we take any three apsidal radii to successive apsides A,B,G, the radius to G is the same as that to A, and the radius to B is repeated again at the next apse in order, D say, and so on. Thus there cannot be more than two apsidal dis- tances in any distinct branch of the orbit, and these are reached alternatively as the particle describes it. We can prove a somewhat more general proposition analytically as follows. We have If F=/nu'^, where n is an integer, ?;2 = 2Lu»-2dtt = ;^^«,"-i-|-C (1) Hence, for an apse, since there v^ = h^u^, where h is the angular momentum about the centre of force, we get w»-i-^^A%2+^^ C=0, (2) where (7 is a constant ; in fact, it is twice the constant value of the energy of the motion. This is an equation in descending powers of u, if n'^2, but whatever n may be the equation can always be so arranged, and as there are only three terms there cannot be more than two mutations of sign of the coefficients. Thus, by Descartes' rule, the equation has no more than two positive roots. Thus there are, for this law of force, in all the branches of the curve (if there be more than one branch) not more than two apsidal distances. The case of n a fraction p/q (in its lowest terms), can be dealt with by writing u = u'^, taking care that the sign of F is properly settled when g is even. Ex. 1. Let the central force be /«.«", where n>3. The value of C is zero when v^=2fiu"-^/(n-l), that is when the speed is the speed from infinity. To make C positive, therefore, we take l!2>2/iM"-V(n-l). Then we notice that a superior limit of the positive roots of (2) is a value of tt ( > 0), which makes the expression on the left of (2) positive. §§ 152, 153] LAW OF INVERSE CUBE. 281 Such a value of u is one which satisfies the equation ."-3 = (.-l)|. Hence 1/m cannot be less than a positive root of the equation (l/M)"-' = 2/i/(ra-l)/i2. Again, transforming (2) by substituting 1/m for u, we get and a value of \ju which makes the expression on the left positive is one which satisfies the equation (lluf=Clh\ Thus l/u cannot be greater than the positive root of this equation. Ex. 2. To find the apsidal angle. By what has been stated above as to the symmetry of the orbit about each apsidal radius- vector, it is clear that there is only one apsidal angle, that is the angle between two apsidal radii. It can be determined at once when the equation of the curve is known, by differentiating v, with respect to and putting duld6=0. Thus, for the ellipse we have dujd6= — esin(^ — a.)/a(l — e^), and this vanishes for 6 — cl=0, it, 27r, Stt, Thus the apsidal angle is 77 and the apsidal distances are a(l —e), a{l+e). In the case of the approximately circular orbit, discussed in § 151, it will be seen that the apsidal angle for both the approximations there given is 7r/\/3 — n. For a higher approximation, in which it is found that x = Acosp(9-a.) + A^{C+I)co32p{9-a.)} + A^Ecos3p(e-a.), the values of C, D, E, and p are to be found by substituting in the differential equation, and equating coefficients on the two sides of the result. It is found that p^ = {Z-n){\-i.2{n-%){n + \)A^}. The apsidal angle is then irfp. [For fuller information regarding the Theory of Apsides and the Classification of Orbits, the student is referred to Eouth, Dynandcs of a Particle.'] 153. Centre attracting according to Inverse Cube of Distance. A discussion of the motion of a particle attracted according to tlie inverse cube of the distance (F=/nu^) is very- instructive from the point of view of the effect of initial circumstances on the form of the orbit. The differential equation is gi^ +(i-|)^=o (1) 282 A TREATISE ON DYNAMICS. [CH. V. We may have 1>, =, or < iui./h^. In the first and last eases we have, if k = Jl — fjilh^, u = Acosk(e-a.), (2) «=4ie*» + ^2e-" (3) respectively, and in the transition case of iui/h^=l, u==G{d-a.) or r(0-a)C=l, (4) where the constants A, A-^, B^, and Care assigned according to initial conditions. The speed from infinity to distance R is -J/jl/R = V, say. If the particle be projected with speed V at distance R, in a direction inclined at an angle cj> to the radius-vector, then h=VRmn(p, and thus iu/h? = fM/Vm^sin^(l>. The cases enumerated above are therefore those in which V^ > , = , or , = , or < the speed in the equidistant circle, we have the three cases enumerated above. Fig. 64. (1) ysin0>F', or \> fijh?. Differentiating the ex- pression for u above, we get du/d6= —kA sin k(6 — o.), which vanishes when 6 — oi = n7r/k, where n is any integer, included. Measuring 6 from the radius-vector for which 71 = 0, we get ^ = ^cos/<;0 or rcoskd = a (5) §153] LAW OF INVERSE CUBE. 283 for the equation of the path. Each value of a. gives a branch of the curve, and these branches occur at successive intervals of tt/Zc. They are all precisely alike in the sense that each in succession is the one before it, turned forward through an angle tt/Jc in its own plane about the centre. The curves are represented in Fig. 64. (2) Vsm^=V', or m/h^ = l. The equation is then re=C, (6) and the curve is known as the reciprocal spiral. It is shown in Fig. 65. Fig. 65. (3) Fsin9!.V'. In the first of these, dujdd = ku, and Fig. 66. therefore .(11) TfM = e*^ is the equation of the curve, which is an equiangular spiral (see Fig. 23, §43). In the remaining case V^^V'^, and so we have (du/d9f = {ju/h^ - 1 )%« + ( r^ - V'^)/h^. As we have seen, there is no apse. If we go back to the solution of the differential equation, we get ^ = k{A,e'''-A,e-^% .(12) and, by squaring, i^)' = Bv?-WA,A^ (13) Thus the positive quantity {V'^-V"^)lh?= -4sk^A-^A^, so that in this case A.^, A^ have opposite signs, and are such that A^A^=-{r^- V'^)/4>kW. Thus, writing ¥ for we have tt = ^Je*«_ e-**] (14) §§ 153, 154] INVERSE n'* POWER. 285 Both u and du/dO increase without limit as 6 increases, and therefore r diminishes. When = 0, u = A.^ — b^, and if initially A^ = b^, then r=oo. Of course du/dd maybe either positive or negative, and so there are two branches, as in Fig. 67. They have a common asymptote, as shown in the diagram, and the curves are described as shown by the arrows. Fig. 67. By putting Ae^'- = Ai, ^6-*° = 67^1 "w^e get A = b and A;a = log j4,— log6, so that we can write the equation for u in the form u = ^{e*(8+«)_e-*(9+«)} (15) or, changing the initial value of 6, in the form % = ^(e*»-e-*»), (16) which differs from that for the case of F' > F only in the sign of the second term. The curves for the motions treated in this section are known as Cotes' Spirals {Harmonia Mensurarum, 1722). 154. Force varying as Inverse n*^ Power of Distance. The differential equation for the case of F= ixu'^{n >■ 1 ), namely de'-^ h'^ ' '' ' can (except in the case, just treated, of n = S) be integrated by aid of the energy equation when the speed of the particle at every point of its path is the speed from infinity. If this is the case at one point — the point of projection — it will be the case at all. The energy equation can then be written 286 A TREATISE ON DYNAMICS. [CH. V. m, -r=de, (3) This gives, after a little reduction, du where a = 2iu/(n — l)h^. By the substitution au'^-^ = l/z^ (which is only applicable if w / 3), this transforms into dz n — S 4\-z^ 2 -dQ, (4) which gives cos~is = — ^ — (0 — «.), (5) where ot is a constant. Thus the equation of the orbit is -^ >cos— g— (9 — a) = c ^ cos „ (0 — a.). ...(6) \(n-l)hy The result of the integration is unaffected by the ambiguity of sign introduced by the radical, which is to be interpreted according to the sign of the initial dujdQ. This equation shows that when %]>3 the path consists of one, two, or more loops, according to the value of n, with a common node at the centre of force, and a maximum radius vector equal to c, which recurs at the angular interval 47r/('K, — 3). When tc-<3 the orbit has infinite branches; for example, when n=2, it is a parabola, and c is the minimum radius-vector. As another example we take the case of ri = 5. We have then r = c cos(0-o(.), (7) the equation of a circle with the centre of force on the circumference. The maximum radius-vector is c, the diameter of the circle. If 71, = 7 the equation of the curve is r2 = c2cos2(0-a) (8) the equation of the lemniscate of Bernoulli (see above, Ex. 4, §125). In all these cases the condition is imposed that the speed at each point is that from infinity. But the integrations can be effected under other conditions in these and other §§ 154, 155] DIFFERENT CENTRES. 287 cases. (Reference may be made to Routh, Dynamics of a Pa/rticle, and Greenhill, Elliptic Functions.) Ex. 1. Carry out the integration -when the force is repulsive, force varying as the inverse w"" power of the distance, and the speed at each point is that from the centre of force to the point. Ex. 2. Integrate the equation of energy when the force fs repulsive and varies directly as the n*-^ power of the distance, and the speed at each point is as in Ex. 1. Ex. 3. If the speed at each point in a central orbit bear a constant ratio to that in an equidistant circle, find the orbit and the law of force (see § 153.) Ex. 4. The speed and direction of motion at any point and the centi-e and law of force : find the radius of curvature of the orbit at the point. Ex. 5. Find the equation of the orbit when F= fiM" + fj,iu^, where n>l and ^3, when the speed in the orbit for F=fm" is equal to the speed from infinity (see § 130.) 155. Different Centres for Same Orbit. Newton's Theorem. Newton proved (Principia, Lib. I. Prop. VII. Cor. 3) that if the force towards a centre 0, by which an orbit can be described, is known, ^ the force towards a new centre Oj, by which the same orbit is described, can be found. Clearly, the orbit will be described if the force toward the centre of curvature have always the proper value. Let P (Fig. 68) be any point of the orbit, p the radius of curvature there, r, rj the distances, OP, 0-^P, and j9,Pi the lengths of the perpendiculars let fall from 0, Oj on the tangent drawn to the curve at P. Then, if v be the speed at P and F, F^ the forces which give the same value of v^/p, Fig. 68. Ft= F, ^, so that #=^. 288 A TREATISE ON DYNAMICS. [OH. V. The double rate of description of area is h =pv, under the force F towards : if the force F-^ towards Oj is taken for a new speed v', we have h^=pjv', so that v'^/v^ = hlp'^/h'^pl, and in this ratio we must increase F^. Thus, we obtain for the forces towards and 0^, either of which will enable the orbit to be described, F^Jlp\ (1) F h'^^r If we draw from 0^ a line O^H parallel to OP to meet the tangent in H, then pjp = Oj^H/r, and therefore pS/pl = 7^/0^H^ Thus t\_Mjr^ (2) F-h?O^IP As an example, take as the focus of an elliptic orbit at which the centre of force is situated, and let 0^ be the centre of the ellipse. Then F= fijr^, and from the geometry of the ellipse we have O^H=a, so that ^=^M (^) Thus the force toward the centre of the ellipse under which the orbit can be described varies directly as the radius- vector rj, drawn from the centre. Writing f^■^r■^ for F^^, we f^==fm? ^^^ But when the orbit is described by a particle under a force toward the focus, h^ = fia{l—e'^), so that we obtain or h^=sj]ji^.ab, (5) where a, b are the principal semi-axes. 156. Hamilton's Theorem. The force toward any centre Oj (Fig. 69), under which a particle will describe an orbit, is, as we have seen, given by the equation j^^^r^lAV (^) p p P p^ §§ 155, 156] HAMILTON'S THEOREM. 289 where p is the radius of curvature at a point P of the orbit to . which the radius- vector is r, and f is the length of the perpendicular let fall from Oj on the tangent at P. Now, if the orbit be a conic of semi-axes ah, and the polar LM of the point Oj with respect to the conic be drawn, and cr, vs' denote the lengths of the perpendiculars let fall from P and from the centre of the conic respectively to this polar, it can be proved that p'a'l^JS^P- •■■• • '''^> Fig. 69. In proving this proposition we shall assume that the curve is an ellipse, but the proof may be easily modified to suit any other conic. Let /, g be the coordinates of Oj with reference to the principal axes; the equation of the polar is f^,mL_, If X, y be the coordinates of P, the perpendicular from P to the polar has length / _£ ,[P7f 'Jph'+g^a'^ while that from the centre to the polar has length a?W vf = T 290 A TREATISE ON DYNAMICS. [CH. V. The length of the perpendicular from 0^ to the tangent at F, of which the equation is is found in the same way to be _ a^b^ — W'fx — a?gy Hence we get, by (2), 1 CT'3 „ a*¥ .„v 1,3= {6) But if we calculate 1/p by the usual formula, d?y 1 dx^ P {>-©T we find (disregarding sign, since Ijp is to be taken positive here) precisely this value. Hence by (1) we have The values of r and cj vary from point to point on the curve, the other quantities remain constant. Thus we have the theorem that if 0, be any centre, a force varying as the distance of O^ from P, and inversely as the cube of the length of the perpendicular from P on the polar of Oj and directed towards Oj, will enable the conic to be described. This theorem is due to Sir W. R. Hamilton {Proc. Royal Irish Academy, vol. 3). As an example, let 0^ be the focus of the ellipse. Then the polar of 0^ is the directrix, and rs=r/e, r^ = a/e. „ IP h^ /®V ua{l—e^)a^ u Hence ^='~a:2\^l '''=471 2\ Z^'''=2' a^o^K-GjJ a*{l — e^) 7^ r^ ■ the known law of force. 156, 157] HAMILTON'S THEOREM 291 157. Second Statement of Hamilton's Theorem. Hamilton's theorem can be put in another form. Still considering a given orbit, let, if possible, two tangents O^A, O^B (Fig. 70) be drawn from the proposed centre of force Oj to the orbit, and from any point P of the orbit let fall perpendiculars to the lines O^A, O^B, AB: let the lengths of the first two be de- noted by a., j8; the length of the third is cr since AB is the polar of Oj. Then, by the properties of conies, we have a.l3 = kTS^, where k is a constant for writing k for /iW%^/a^6^ we get P=«^^ (1) Fig. 70. the conic. Hence, (0.137 The equivalence of these two statements of Hamilton's theorem may be seen as follows. The general equation of the second degree referred to 0^ as origin may be written ax^ + 2hxy + by^ + 2gx +2fy+c = 0. The polar of the origin has equation gx+fy + c = 0, and therefore if P be any point on the curve of coordinates x, y, the perpendicular from P to this polar has length If ^, rj, be the coordinates of the centre of the conic, we obtain by transforming the equation of the conic to the centre as origin, ai+hri+g = 0, h^+hr,+f=0 (2) as the conditions that the new / and g should vanish. By these we write the equation of the curve in the form ax^+2hxy+hy'^+g^+fri-\-c = a£^+2hxy+hy^+^l^—j^^=0, where A = abc+2fgh — af^ — bg^—ch^, the result, multiplied 292 A TREATISE ON DYNAMICS. [CH. V. f by ah — h?, of substituting the values of ^, tj derived from (2) in the expression gi+fy + c. But since ^ , rj are the coordinates of the centre, (gi+fv + (^)/'^P+9^ is *^® length c/ of the perpendicular let fall from the centre on the line gx+fy+c = 0, the polar of Oj with respect to the conic. Thus . -'^^?^=^ ^'^ Again, if a^, h^ be the lengths of the principal semi-axes of the conic, we get, by turning the axes so as to make the new h vanish. We obtain, therefore, a^6j\CT/ ^ ^{gx+fy + cf (gx+fy + cf But X, y, the coordinates of P, satisfy the equation ax^ + 2hxy + by^ + 29a; + 2fy + c = 0, which can evidently be written in the form c{ax^+2hxy + by^)-(gx+fyf=-{gx+fy + cf; that is, if we put A=g^ — ac, B=f^ — bc, C=fg — ch, Ax^+2Cxy+By^ = (gx+fy+cf. We get then, by (1), with the meaning of h in (4), § 156, F^k- TA , (5) (Ax^+2Gxy+By^y Now the equation Ax^ + 2Gxy+By^ = represents two straight lines O^A, O^B meeting the curve at the points ^, B in which it is met by the polar of the point Oj. These lines are therefore, if real, tangents to the curve at A, B. If then X, y be, as here, the coordinates of a point P on the curve which does not lie on either line, Ax^+2Gxy+By'^ is, to a factor which is the same for all points on the curve, equal to the products of the lengths of the perpendiculars a, /3 let fall from F on the lines. The second form of Hamilton's theorem is thvis established. §§157,158,159] MULTIPLE CENTRES. 293 158. Orbit a Conic touching Two Straight Lines drawn from C.F. The very important result follows that if Oj be any centre of force and O^^A, O^B be any two lines drawn from 0^, any conic touching these two lines is an orbit for the centre of force, and the law of force is given by (5). Again, if a particle move under the action of a force directed to a fixed point 0^, and varying directly as the distance of the particle from the fixed point and inversely as the cube of its distance from a fixed straight line, the orbit is a conic with respect to which the given straight line is the polar of Oj. For if a point P in the plane of the two lines 0-^^A, O^B, or in the plane of the centre of force Oj and the fixed straight line, be specified and a velocity at that point be also specified in direction and magnitude, a conic passing through P and touching 0-^A, OJi, or with respect to which the given straight line is the polar of 0^, can be determined, which is an orbit described about the centre of force 0^ under the influence of a force as specified in (4) of § 156 or (1) of § 157, with velocity at P and direction of curvature as indicated. And every other possible orbit so described will coincide at P with the conic in regard to speed, direction of motion, and curvature, and the variation from P in direction of motioii and curvature will be the same iii both — that is the two orbits are solutions of the same differential equation which fulfil the same initial conditions. There is only one such solution, and the orbits are identical. Analytical proofs of propositions equivalent to the state- ment that the two laws (or rather two versions of th« same law) stated above are the only laws which always give conies as orbits have been given by MM. Halphen and Darboux (Gomptes Bendus, t. 84). 159. Particle acted on by Forces from Several Centres. Bonnet's Theorem. Provided a certain condition is satisfied, a particle can descpibe a given path under the combined action of any specified system of forces F^^, F^, ... directed to any given fixed points. Let N and T denote the normal and tangential components of force on the particle, due to these forces. Then N-=-ZF,f^, T=--ZF,'^, (1)^ „ 1 rf, 294 A TREATISE ON DYNAMICS. [CH. V. where «» denotes the length of the perpendicular let fall from the centre for the force Fk to the tangent at P, the position of the particle at the instant considered, and rt is the distance of F from that centre. Now if p be the radius of curvature of the path at F, we have 2 T= d{v^)lds = d{Np)lds. Hence, inserting the values of N and T given above, we get ^^^4*^^l(^'?)=« <^) But we have since llp = {dp^jdr^lrt. Thus the equation just found can be written ^IK')=«' (^) which is the required condition. Of course, the speed Fof projection must be such that V^/p = JV. Let /*!, P2, ... be central forces, each of which if it acts alone causes a particle to describe a central orbit ; then we can prove that any such system of forces acting together will enable the particle to describe the orbit, provided the speed v at any chosen point is given by »^=«j+«2+ ... , where v^, V2, ... are the speeds at that point for the separate forces Pj, F2, ... . Thus, for the combined forces, the energy is the sum of the energies for the separate forces. Since each of the forces F^, F^, ... enables the particle to describe the orbit, we have i'j/jo = P,jBi/ri, V^J p = F.^pjr2, ..., Vjdvilds= -F^drjds, v^vjds= — F^dr^jds, Now, with the value of v stated above, we have r^s^^^^-- ^£=-(^^S+^^'S+-) (^) and the normal and tangential forces required are just furnished by the combined system. This is Bonnet's theorem. [Liouville's Journal, t. ix.]. 160. Theorem of Curtis. The following is another general theorem with regard to the description of an orbit under combined forces. If a given path is described by a particle under the separate action of forces F^, P^, ... , which act from fixed centres, it can be described also under the combined action of forces F-^, F^, ... acting from the same centres, provided 2c*P, mh". <■) where c* is the chord of curvature of the path in the direction of P*. [Curtis, Mess. Math. x. 1880]. «.3 i 159, 160, 161] MULTIPLE CENTRES. 295 We have, as before. If, then, F^, F^, ... , acting from the same centres as /"j, Pj, ... , enable the particle to describe the path, they must satisfy the simultaneous equations -V dvlds= Fidrjds+F^rjds+ ... , v^lp = Fi^+F/-^ + (3) and these give the condition or, with insertion of the values of Pilr^, p^fr^, ... from the first of (2), - dvdfFi ^\ that is by the second of (2) -^^t-^^^^'^^^'^xi-y^ (^) from which, noticing that v\=Pjfi^, by § 132, and that the two first terms cancel one another by the first of (3), we get finally equation (I) as the condition to be satisfied [A. H. Curtis, Mess. Math. x. 1880]. 161. Examples of Multiple Centres of Force. Ex. 1. Deduce from this theorem the relation to be fulfilled "for two forces F^, F^ towards the foci of an ellipse under which acting together a particle can describe the ellipse. [Note that /i/r^, /a/?^ directed towards the foci are forces under which the ellipse can be described.] Show that the general solution of the equation 1 d „ ,, 1 c? / „ o\ ?,dir,^^^<^=rid;jM)' which holds for elliptic motion, is rlF,=frl{f^(r,)+y,{2a-r,)}dr, + C„ rX=f<{f {rd+M2a-r,)}dr,+ G„ where f^, /j are arbitrary functions and Cj, C, are constants, which may be included in the integrals if it is understood that different constants may be used for Fi and F^. 296 A TREATISE ON DYNAMICS. [CH. V. Ex. 2. Show that by properly choosing the functions/, we obtain as forces towards the foci under which a particle can describe the ellipse. Ex. 3. A particle moves in an ellipse with the speed a\lK {a - r)/r(2a - r), where 2a is the length of tjie major axis and k is a constant : to show that its acceleration consists of two components, one towards the near focus and the other from the farther focus, both varying as the inveise square of the distance. s If r'=2a-r, we have v^=^a^{K/r-K/r'), and v^lp=^a^b(Klr-Klt'')/(rry^, since l/p=ab/(rr')^. Now substitute for '2,aj{r¥y the value {p-\-p')l^pp', and obtain v'/p=ia^{Kp'/rh^ - Kp/r'^r), and therefore ^_^fK P K p'\ the first equation, which is consistent with the statement to be proved. This, with dv_ 1 j/k dr K dr'\ which is at once obtained, establishes the theorem. It will be noticed that the motion here specified cannot exist except in the half of the ellipse on the same side of the minor axis as the attractive focus. Outside these limits the speed is imaginary. Ex. 4. Show that a particle will describe an ellipse if its speed along the tangent at any point P is given by and it is acted on by forces towards the foci. Ex. 5. Particles of difierent masses mij, nij, ... , which have speeds Vj, Wg) ■•• -at the same point, describe the same path under the action of given forces F-^, F^, Show that a particle of any mass M, which has kinetic energy at the same point equal to the sum of the kinetic energies of the particles, will describe the path under the combined ' action of F^ , F^, Ex. 6. Prove that if a conic section is described under the action of either of two forces directed to the foci, each varying as the inverse square of the distance, or of a force towards the centre and varying directly as the distance, it will describe the curve under §§ 161, 162] EARTH-MOON SYSTEM. 297 the combined action of the same three forces, provided the particle is projected with speed given by ''=-a{^^-r^>'?yi''''' where r, r' are the focal distances at the point of projection and /u, fi!, ij." the intensities of the centres [Lagrange, J/^c. Anal. t. ii. sect. vii. § 83]. 162. Earth-Moon System disturbed by Action of Sun. We give here an interesting application of the equations of motion, with reference to revolving axes, to the ap{)roximate determination of the influence of the sun's attraction in disturbing the motion of the moon relative to the earth. Let S be the position of the sun's centre, supposed to be fixed, and E that of the earth's centre. We take a to denote the distance of the earth from the sun, and x, y, z for the coordinates of the moon's centre M relative to that y Fig. 71. of the earth as origin, choosing the direction of x in the prolongation of the line SE, that of y at right angles to 8E in the plane of the ecliptic, and taking z as the distance of the moon's centre from that plane. The coordinates of the moon's centre relative to the sun's centre as origin are therefore a-\-x, y,z. The equations of motion of the moon are therefore (see (2), § 14), ii X, Y Z be the component applied forces, X — 2ny — yn— n^(a +x) = X,-\ y + 2nib+{a+x)n—'n^y =Y,V (1) z=Z.) The acceleration of the earth toward the sun is at any given instant n^a, which is the force per unit mass toward 298 A TREATISE ON DYNAMICS. [CH. V. the sun at distance a. Hence the force per unit mass toward the sun on a particle at M is The components of this in the direction of x, y, z, increasing, are —nn?a^{a-\-x)IB?, —n^ahjIR^, —n^a^z/BK Since x, y, z are small in comparison with a, these components are approximately —n^a+2n^x, —n^y, —n^z. Besides these there are the component forces of attraction exerted on the moon by the earth, which, if r denote Jx^ + y^+z^, are — fix/7^, —fiy/r^, — fizjif^ per unit mass. Thus, if we suppose 'n. = 0, we have the equations of motion x—2ny — \^n''—^\x = 0, y + 2nx+^y =0, z+n^z+^,z =0. ■(2) Multiplying these equations by x, y, z respectively, integrating and adding, we get v2-3%2a;2+wV-2^ + (7=0, (3) r which is known as Jacobi's equation of the relative energy of the moon's motion. 163. Stability of Earth-Moon System. Hill's Theorem. This result affords an example of a very useful method of assigning limits to the possible relative displacements of the parts of a system. For a given value of v, the body here considered — the moon — must have its centre some- where on the surface given by (3) of § 162, which intersects the plane of x, y in the curve 3w2a;2-f2^-(7=0 (1) or, if 03= r cos 0, 37iVcos20-Cr-|-2yU = O (2) If this cubic equation for r has at least one real positive i 162, 163] EARTH-MOON SYSTEM. 299 -e- root for every value of d, the curve possesses a branch closed round the centre, within which the body must always remain. It may be verified by the student that the roots of (2) are all real if cos^ d < G^HlnY- K will be found that when 6=± 7r/2, so that cos 6 — 0, the equation has one finite root r=2iuL/C, and two infinite roots, one positive, the other negative. For = or x, so that cos20 = l, one root lies between 2/J./G and S/jl/C, and between these values of 6, r alters continuously. Besides the closed branch, which is oval in shape, the curve has, as shown in Fig. 72, two infinite branches, which have the two lines Sm W = G as asymptotes. In the space between these infinite branches and the closed branch, v'' is nega- tive, and V, therefore, a pure imaginary. The body must therefore either be inside the closed branch or outside the two infinite branches, and, if once within the closed branch, can never escape to the space beyond. The closed branch and the infinite branches are shown in Fig. 72(a). A A', BB' are the lines ZnV = G: In diagrams a, b, c, the trace of the surface on the plane of xy is shown for the three cases (7^>-, =, Za^Sji^, where M is the mass of the satellite, S that of the primary, a the radius of the satellite and r the distance of the centres of the two bodies apart. Show that the same condition is necessary for the retention of a particle on the line of centres, but on the side turned from the primary. 17. A particle of mass 1 gramme is hung by a very fine quartz fibre 2 metres lone, and is at rest with the fibre vertical. A sphere of lead 30 eras, in diameter is suddenly placed with its centre 20 cms. from the particle on the same level. Find the equation of motion of the particle and how far the particle moves towards the sphere in a second. [Density of lead 11 '47 grammes per cub. cm.] 18. Two homogeneous spheres of matter of the same density (22 grammes per cub. cm.), one 30 cms., the other 60 cms. in diameter, have their centres 300 cms. apart and revolve round their common centroid as a double star. Show that the period of revolution is 20 h. 58 m. 22 s. 19. If two homogeneous spheres, of masses E and M, move under their mutual gravitation and that of a fixed homogeneous sphere of mass so that ZS/dx, ... are proportional to the direction-cosines of a line perpendicular to the displacement dx, dy, dz.] Hence we must have -Fi+Br,-D^=4, -Ei-Br,+ Ci;=Kr,, ..(2) where k is a constant. §§166,167,168] PRODUCTS OF INERTIA. 313 By elimination of ^, »?, f from equations (2), a cubic equation, -{ABC-2DEF-AD'-BE^-CF^) = 0, (3) called the discriininating cubic, for the determination of K is obtained, the three roots of which can easily be proved to be all real ; so that there are three axes which can be drawn to the ellipsoid to meet it at right angles, and ecich •pair of these are at right angles to one another. They are called the principal axes of the ellipsoid. It will be observed that according to (4), § 167, if one of the axes of reference, say that of f, be a principal axis, the products D, E of inertia must, from the third of these equations, be zero. Special cases are (1) that in which two of these axes are of equal length, when also all the axes in the plane of these two are equal, that is, the ellipsoid is one of revolu- tion, and (2) that in which all three axes are equal in length, when the ellipsoid is a sphere. The roots of (3) substituted successively in (2) enable a set of values of the cosines ^jp, ri/p, ^/p to be found for each root, which, when used in (4), § 166, enable the length of the axis corresponding to the root to be calculated for an assumed k. The length of this axis is thus found as a function of k. Hence the roots of the discriminating cubic are independent of the choice of axes, provided the origin is fixed, and therefore the coefiicients of the powers of K in the cubic have the property of invariance. These coeflBcients, the values of which are invariant, that is, inde- pendent of the choice of axes, are, as will be seen, A + B+C, AB+BG+CA-B^-E^-F\ ABC - 2DEF- AD^-BE^- GF^. 168. Meaning of a Product of Inertia. It is important to gain a clear idea of the meaning and eflfect of a product of inertia. Consider a body of any form revolving about a shaft, fixed in position and so strong that it is not sensibly disturbed by the action of the body upon it. The body 314 A TREATISE ON DYNAMICS. [CH. VI. may be of any form whatever; for example, one of the arms of a Watt's steam-engine governor, or the crank-axle of a locomotive, may serve to fix the ideas. The body is attached to the shaft by bearings, the reaction on which we shall also consider. Let aaiy origin on the central line of the shaft be chosen, and the central line taken as axis of z, while the other axes are taken in a plane at right angles to Oz through 0. The coordinates x, y, z of each element of mass are taken for the configuration of the system at a given instant. A particle of mass m at the point P{x, y, z) is moving about Oz in a circle of radius Jx^+y^. H ence, it is under acceleration towards Oz of amount (o^-Jx^+y^. ~{viw x-irmwyj' 77ttiy^+mtijy Fig. 77. This is applied to it through the action of the rest of the body. The particle reacts on the system with an equal and opposite force. This reaction, being outward from Oz, has no moment about Oz, and can be resolved into two components, mw^a: parallel to Ox and nnoo^y parallel to Oy. These act as shown in Fig. 77. Again, if the angular speed is varying there is a force on the particle at P, of amount mdujx^ + y^, in the direction of motion, and, as before, a reaction of the same amount in the §§ 168, 169] REACTIONS ON BEARINGS. 315 opposite direction on the rest of the system. This reaction has components may, —thwx in the direction of Ox and Oy respectively, as shown in Fig. 77. We shall consider the aggregate of these reactions and their effect, which is to exert certain forces upon the supporting shaft or axle. For this purpose we introduce at the origin forces ■mM^tc + mwj/, mco^y — 7nwx along Ox, Oy with two equal and opposite forces to balance them. The force mw^x+mwy at P and the force —(mw^x+mwy) at give a couple of moment 7nw^xz + m(hyz about an axis in the direction of Oy, and similarly, the forces mco^y — vmx at P and —nuo^y + mwx at give a couple of moment — mw^yz + mwxz about an axis in the direction of Ox. The forces at P in their action on the body, and ultimately on the axis of support, are equivalent to these two couples, and the two forces at 0, Tnco^x + mcby acting along Ox and mu?y—nwx acting along Oy. This process, applied to all the particles of the system, reduces the reactions to two resultant couples of moments ai^1,{mxz) + oD^(myz\ —w^'2(myz)+w^(mxz) [or as we may write them, Ew^ + Dw, —Du?-\-Ew, where B, E are the products of inertia 1,{myz), ^(mxz)], about axes parallel to Oy, Ox and two Jorces w^^mx + w^my, w^^my — dj^Tnx along Ox, Oy. 1^ x, y be the coordinates of the centroid, these forces become w''Mx-\-wMy, w^My — wMx, where M denotes the whole mass of the body. If the products of inertia D, E are zero, the moments of the couples are zero, and there is no couple of reaction given by the resolution with respect to the chosen axes : only the forces w^Mx + wM y, co^My — ioMx remain applied at 0. 169. Reactions of an Unsymmetrical Rotating Body on its Bearings. Free Axis of Rotation. So far only the reactions have been inchided, and they appear as the reversed mass-accelerations. But the applied forces X, Y on the particle of mass m at P, give couples Xz, - Yz about axes parallel to Oy, Ox respectively, with forces X, Y applied at 0. Thus we obtain resultant couples SZz, — SF« about these axes, and resultant forces 2X, 2 Fat 0. Now let the body be held by two bearings on the axis Oz at distance «!, a^ from 0, and let X,,Fi, X^, Y^, denote the components of forces exerted by these bearings respectively on the body. The latter forces and the aggregate of reactions must form a system in equilibrium. 316 A TREATISE ON DYNAMICS. [CH. VI. Taking moments about the bearings in succession, we obtain „ { M(o>^ x + mi) + SXfg; - (go)^ + Zid) + 2Xz) ' JL-t — — ■ ~) X,. 1^1 = \M{ta^-uix)-\-l,Y}a^-{D^-iDx) + l,Y}ai-{D^-Ei, + l.Yz 2a ' Dio^-Eih+'EYz la .(3) If E denote the resultant of il/((o%' + (i^) + 2A' in the direction of Ox and if((o^y-uj^) + 2F in the direction of Oy, and R the resultant of (^ft)2 + 2>w + 2Xz)/a in the direction of Ox, and {BuP-- Eii>-Vl.Yz)la in the direction of Oy, these equations show that at one bearing, forces -\E, \R, and at the other bearing, forces —\E, —\R, are applied to the body. The forces applied by the body to the bearings are equal to these forces reversed. The two equal but oppositely directed forces \R, —\R are entirely due to the products D, E of inertia if there are no applied forces, and in that case vanish when these products are zero. They tend to turn the supporting shaft or axle round in their plane of action. We have thus an example of the effects of products of inertia on the supporting axis of a rotating body. We may take the driving axle of a bicycle with its pedals, which give products of inertia for an axis through the centroid, or the driving axle of a locomotive with its cranks and attachments as practical examples. It will be observed that the §§ 169, 170] EQUATIONS OF ROTATIONAL MOTION. 317 closer together all attachments, such as pedals or cranks, are placed, the smaller are the products of inertia and the resulting couples. Undue spreading out of the parts along the axis of rotation increases the products of inertia, and augments the couple, causing unsteadiness of running. If we choose the axes Ox, Oy through so that ^{mxy) is zero, and if at the same time '^{rm/z), '2{mzx) be each zero, then the three axes of coordinates Ox, Oy, Oz are principal axes of moment of inertia of the body. It will be observed that if be taken at the centroid Mx—My = Q, and that therefore if the axes Ox, Oy,Oz be principal axes through the centroid, and there be no applied forces, there is no action whatever exerted on the axis of support or exerted by that axis on the body. Hence, in the absence of other forces, the body, if rigid, will, when set rotating about Oz, continue to do so without support. Oz is then what is called a free axis. It will be clear that any principal axis of moment of inertia through the centroid is a free axis. 170. A.M. about any Axis. Equations of Rotational Motion. Let now the axis OA in the direction I, m, n be one about which the whole system is turning with angular speed to. The angular momentum H about the axis is given by H=w(Al^+Bm^ + Cn^-2Dmn-2Enl-2Flm), ...(1) or, as we may write it, H=oo{l{Al-Fm-En)+m{-n+Bm-Dn) +n(-El-Dm+Cn)} (2) But the angular momentum is also IH-^+m.H^+nH^, where H^, H^, H^ (the F, G, H of § 71) are the components of angular momentum about the axes of x, y, z. Hence, since (Oj, oo^, w,, the angular speeds about the same axes, are Ice, mw, nee respectively, if^ = J.«i — Fw^ — Ewg , H^=— Fco^ + Boo^ — Doos, \ /g\ H^=-Ew^-Boi>,+ Gw^. ] '"^ ^ If L, M, N be the moments of forces about the axes, the time-rates of variation of these components give three equations of the form Aui^ — Fa)^ — Ew^+A(ii)^—Fw^—Ew^ = L; (4) for it will be noticed that as the body is in motion relatively to the axes, the quantities A, B, C, D, E, F cannot be taken as constants. 318 A TREATISE ON DYNAMICS. [CH. VI. The equation of the ellipsoid referred to its principal axes may be written A'i^+BV+C'C^ = l, (5) where A', B', C are the moments of inertia about the principal axes. If I, m, n be the direction-cosines, with reference to the principal axes, of the axis about which the angular speed is w, the angular momenta about the prin- cipal axes are A'lw, B'mw, C'nu> = A'u).^, -B'ajg) G'''>s\ ^^^ it is only when the axes of reference and the principal axes are coincident that these simple values of the components of angular momentum are obtained. If at the instant under consideration the principal axes coincide with the axes of reference, A' = A, B' = B, C' = G and D = E = F=0. The equations of motion are now Aw^ + Awj^-Fw2-Ewg = L, (6) and two others of similar form. It will be noticed that though D, E, F are zero, their time-rates of variation are not in general zero, for the body is changing in position with reference to the axes, and the coincidence of axes existing at time t no longer exists at time t+dt. We now suppose the system to be a rigid body, so that the values of A, F, E are to be calculated subject to this condition. Since A = 'L{iJ.{y^-\-z^)}, E== I,(fizx), 'F= I,{fixy), A = 21{iui(yy + zz)}, F='2{fi(xy + yx)}, F=-E{iu{zx + xz)}. But since the body is rigid and is turning with angular speeds w^, w^, w^ about the axes, x = w^z-w^y, y = w^ — w^z, z = m^y^m^x, (7) so that y 1/ -I- i2i = (o^xy — w^xz. Hence A ='2{iii{yy + zz)} —0, since i)== E= F== at the instant. Again, xy + yx= a)^{x^ + z^) — cog(y^ + z^) + w^yz — co^xz, and therefore F=w^{B — A). Similarly E=w^{A — G). The equation of motion becomes therefore Au>,-{B-G)w,w,^L (8) Similar results hold for the other two cases. § 170] EQUATIONS OF ROTATIONAL MOTION. 319 Here A, B, G are the moments of inertia about the principal axes, which we have supposed to coincide at the . instant with the fixed axes of reference. The angular speeds coj, Wg. '^s ^^^ those about the fixed ax^s of x, y, z; but since, obviously, there is no difference between the angillar speed about a moving axis and that about a fixed axis with which the moving axis coincides, we may regard Wj, Wg, (n^ as the angular speeds about the principal axes of the momental ellipsoid, which moves with the body. It requires examination, however, to decide whether w^, the time-rate of variation of the angular speed w^ about the fixed axis of x, may be identified with the time-rate of variation of the angular speed about the moving axis ; for after the lapse of time dt the moving axis has separated from the fixed axis. To decide this point, we find, what the angular speed about the moving axis is at time t + dt. In time dt the principal axis, which coincided with Ox at time t, has turned ^ round in the plane xOz through the angle w^dt by the rotation about Oy, and in the plane xOy through the angle w^dt by the rotation about Oz (Fig. 78). In other words, a line Oa of unit length, which coincided with Ox, has been turned about so that its outer extremity a " fio. 78. has now coordinates 1, w^dt, — tiifit, and these quantities may be taken as its direction- cosines. Hence the cosine of the angle between the new position of this line and the fixed axis, now of direction cosines l-\-dl, m+dm, n+dn (about which the angular speed is now w + wdt), is l+(mwg — nu)2)dt, and the angular speed about it is (w+cbdt){l + (moi}g — nw2)dt}=(oj+wdt)l+(a}2Ws — (OgW2)dt, to the first order of small quantities. Hence wldt is the change in the angular speed about the moving axis; which is precisely the change which takes place in time dt 320 A TREATISE ON DYNAMICS. [CH. VI. in the angular speed about the fixed axis with which the moving axis of Ox coincided at the beginning of that interval of time. We have thus obtained for a rigid body, moving about the fixed point 0, the very important result that the equations of motion with respect to principal axes of moment of inertia passing through the point and tnoving with the body A(i)i — {B—G) W2'«'3 ~ -^' ] Bd,2-iC-A)ivs(o^ = M,V (9) Cwg -(A-B) 0)^0)2 = N,j where A, B, G are the principal moments of inertia for the fixed point 0. These equations were first given by Euler and are of continual application in the theory of rotational motion. Another proof is given in § 251. [See also Ex. 17, p. 336.] 171. Moments of Inertia in DiflFerent Cases. In the previous sections the part which moments of inertia play in the dynamics of a rigid body has been illustrated. We now consider a little in detail the practical subject of the calculation of moments of inertia in different cases. In the first place we make some deductions from the theorem of the momental ellipsoid. First, we see that, if the principal moments of inertia A, B, G are known, the moment of inertia about an axis, the direction-cosines of which with respect to the principal axes are I, in, n, is Al^ + Bw?+Gn\ Hence, if A=B=C, the moment of inertia about the given axis is A. We have such a case in a uniform cube ; clearly by symmetry principal moments of inertia for axes passing through the centre are those about the three axes at right angles to the three pairs of opposite sides, and are equal. Thus the moment of -inertia about an axis joining two opposite corners of the cube has the same value as that for any one of these axes; in fact the moment of inertia is the same for every axis through the centre of the cube. In this and other such cases consideration of the momental ellipsoid, with the theorem of § 165, enables the moments of inertia about different axes to be found with great ease. §§170-173] M.I. OF A LAMINA. 321 172. M.I. of a Lamina. In the case of a plane lamina, or plate, it is clear that one principal axis is at right angles to the plate, through whatever point as origin axes in different directions are taken. For if ZOZ' be an axis at right angles to the plate, and AOA' an axis inclined at an angle Q to ZOZ', the distance d of any element m from ZOZ' is greater than its distance d' from AOA', since d' = d cos 0, where ^ is some angle between and 6. The axis ZOZ' is therefore one of maximum moment of inertia, and the theorem of the momenta! ellipsoid shows that it meets the ellipsoid at right angles. The other two prin- cipal axes therefore lie in the plane of the plate. If now we take the plane of the plate as that of xy, and any chosen point as the origin through which axes of reference are taken, we get, since 2 = for every particle, A = Y.{fiy% B=I,(/JLX^), G=I,{fi{x^ + y^)}, whether the axes in the plane of xy are principal axes or not. Thus we have G=A+B for a plate; and however the axes may be taken, provided only that of z be perpendicular to the plate, the products of inertia I) and U vanish, and F also vanishes if the axes of x and y are principal axes. The moment of inertia of the plate about an axis through the origin is thus AP+Bm,^+{A+B)n^ or AP+Bm^+2Fmn+{A+B)n^ and the equation of the momental ellipsoid is A'f+£V+(^'+S')r=i (1) or Ai'+Br,^+2Fir, + (A+B)^^ = ¥, (2) according as the axes of x and y are or are not principal axes. As explained above, any constant value can be assigned to k in (2). 173. M.I. of Triangular Plate. As a first example we find the moment of inertia of a uniform triangular plate about any axis in the plane of the plate. The theorem just proved will then enable the momental ellipsoid to be found for any point through which axes are taken in different directions. Let ABC (Fig. 79) be the triangle, OK the axis in its plane, and D, E, F the feet of perpendiculars, of o.n. X 322 A TREATISE ON DYNAMICS. [CH. VI. lengths pi, 592, pg, let fall on the axis from the vertices A, B, 0. The line AH drawn parallel to OK divides the triangle into two, ABH, AHC, of which we can very easily find the moments of inertia. Taking first ABH, con- sider a strip LM of breadth du, the length of which is parallel to the axis, and let u be its distance from the axis. The length of the strip is AH{u-p^)/{Pi-Pz), and its moment of in- ertia about the axis is the product of this by (j-to^du, where a- is the mass of the plate per unit area. Thus the moment of inertia I^ of the triangle is , cr. AH (vi Fi F2 J Pi = a.AH {\{p,+p^){p\+V';}-\pM+p{p^+fd)--{^) For the other triangle the length of a strip parallel to the axis and at distance u from it is AH{pg — u)/{pg—pi), and its moment of inertia is Fig. 79. .AH rPs I (Ps~u)u^du Pa Pi jpi or I, = 'r.AH{ip,(pl+p,p,+pl)-l(p,+p,){pl+p';)}. (4) The whole moment of inertia is the sum of these^results, or if A denote the area of the triangle, that is ^AH(pg—p2), I=i^Mpl+Pl+pi+P2P3+PsPi+PiP2) (5) I=.^^l(Pi+B]\(P,+P3)\(Ps+Pi)'\ (6) or ^=4.A{(S±B)%(&+B)%(Pa+ay}. If OK intersect the triangle, the 33 or ^Js on one side have positive numerical values, on the other side negative. §173] M.I. OF A LAMINA. 323 Clearly, (6) gives the moment of inertia, about the axis, of three particles each of mass ^crA, placed at the middle points of the sides. Here we have an example of an equiTnomental systenn. In every way, as regards moment of inertia, this system of particles is equivalent to the triangular plate. If we consider two perpendicular axes through and in the plane of the triangle, and call one of these the axis of X and the other the axis of y, tlie ordinates of the vertices will be i/j, 1/2, 2/3, and their abscissae ajj, x^,x^, and we shall have for the moment of inertia about Ox, /.=i.A{(^^)%(^3)V(^.y|, (7) and for the moment of inertia about Oy, 7, = i.A {(^4^-f +(^^y+(^^0^} (8) If now we take an axis through in the plane of the triangle and inclined at an angle Q to the axis Ox, the distances of the vertices from that axis are y^ cos — 33^ sin Q, ... , and the moment of inertia caa be written down by substituting these values for p^, p.^, Pg in the expression found above. It will be found to have the form Acos^e + Bsm^d-^FsmOcose, (9) where ^ = tVA [{(2/1 + 2/2)P +...],. B=^-,^ while that about the axis of y at Oi or Og is BA-M{A-B')IM=A. Thus the moment of inertia is the same for every axis at 0| or O2 in the plane of xy, that is the momental pje §0. ellipsoids with centres at 0, or O2 are ellipsoids of revolution. The points 0-^, 0^ are called foci of inertia. If we take any point P in the plane of two of the principal axes at G, it can be proved that one of the principal axes at that point is perpendicular to the plane. For let the plane be that of x,y, and let /i, h be the co- ordinates of P, and X, y, z the coordinates, with reference to 330 A TREATISE ON DYNAMICS. [CH. VI. the principal axes through the centroid, of an element of mass /I. We have then for an axis through P parallel to Gz, the products of inertia 2{/z(aj-/0^}, 2{/x(2/-/i)4, which are both zero since 1.{fi.xz) = Jl(fiyz) = 0, and 'Lifiz) = 0, for the axes at G are principal axes and G is the centroid. Hence the proposition stated. Again, taking the plane of x, y at G, let Ox (Fig. 80) be the axis about which the moment of inertia is A{1;> B), and Oj, Og be the foci of inertia. Then, if we take any point P in the plane and join it with 0^, 0^, the moments of inertia about O^P and Og-^ ^^^ ^'^ same in amount, and one principal axis at P has been shown to be at right angles to the plane O-^PO^. The other two must be the internal and external bisectors of the angle O^PO^- For if a momental ellipsoid be described from P as centre it will meet the plane O^PO^^ (the plane xGy) in an ellipse, and the radii- vectores from P through 0^, 0^ will be equal in length. The principal axes of the ellipse are the bisectors referred to. Thus if an ellipse or hyberbola be described through P with Oj and 0^ as foci, the two principal axes at P required are the tangent and normal to the curve at P as shown in the figure. 178. Ellipsoid of Gyration. Finally may be noted here some theorems regarding other ellipsoids which also repre- sent conveniently the moments of inertia of a material system about different axes through a specified point. For example, if, M being the total mass, we write Mk\, Mk\, Mk\ for the moments of inertia A, B, G about principal axes through the specified point, we may use the equation of the momental ellipsoid in the form Ke+]4^+ki^'=i ....(1) The quantities k^, k^, \ are called radii of gyration of the system about the principal axes. It is convenient sometimes to use the ellipsoid t+l + ^l=L, (2) §§ 177, 178] ELLIPSOID OF GYRATION. 331 which is said to' be reciprocal to the momental ellipsoid Ai'' + Bn'' + Gt^=l, (3) and is called the ellipsoid of gyration. With the constant term on the right chosen as l/M, where M is the whole mass, the ellipsoid represents moments of inertia about different axes through its centre in the following manner. For every such axis two parallel planes can be. drawn to touch the ellipsoid and be perpendicular to the axis. These planes are at the same perpendicular distance p from the centre, and the moment of inertia about the axis is Mp\ For the direction-cosines of the normal I, m,, n, say, are given by the equations 1, 1, 1 l,m,n= 7 (4) Hence, if ^, >?, f be the coordinates of a point in which a tangent plane touches the surface, the length of the per- pendicular is l^+m>] + nt=p = T i^) But since l/(^/A) = 7n/{t]IB) = n/(^/G), we get by squaring the fractions, multiplying the numerator and denominator of the first squared fraction by A, oi the second by E, and of the third by G, and then adding numerators and denomi- nators, PA+mW+nW _ 1 '1/M ~ ^jA 2 -I- tf'lE^ + f 702 ~ ^ '' ^ ^^' by (5). Hence we have V'A+m?B+n^G=Mp'', (7) which is the proposition stated above. Thus the perpen- dicular p on the tangent plane is exactly the radius of gyration of the body about the axis with which the perpen- dicular coincides. In this lies the convenience of this mode of representation. 332 A TREATISE ON DYNAMICS. [CH. VI. It will be seen that if /cj, Ic^, k^ be the principal radii of gyration, we can write the ellipsoid of gyration in the more compact form, ^2 2 P2 Wk-' <^> This ellipsoid is said to be reciprocal to the momental ellipsoid, (1) above, for the following geometrical reason. Let the momental ellipsoid be constructed, and concentric with it a sphere of such radius that it lies wholly within the ellipsoid. Then taking any point P on the momental ellipsoid, draw the polar plane of P with reference to the sphere, that is the plane which contains the points of contact of all tangent planes to the sphere which pass through P. Then if we cause the point P to travel over the momental ellipsoid, we get a succession of polar planes which all touch a second ellipsoid — envelope it, as it is said. This second ellipsoid is coaxial with the first and reciprocal to it. The ellipsoid of gyration, (2) or (8), may be regarded as thus produced with a suitable choice of the radius of the sphere of reference. 179. Ectuimomental Cone. Theorem of Binet. At any point P, the principal moments of inertia at which are A, B, C, the axes for which the moment of inertia has the same value / form a cone of which the principal diameters are the principal axes at P. For if I, m, n be the direction-cosines of an axis about which the moment is /, we have V^A +m^B+rfiC=I or P{A-I) + m.^B-I) + n^(C- 1) = 0. Multiplying by p=^jl=ri/m = ^/n, we get (A-I)i'' + {B-I)r,^ + C-I)C'=0, (1) which is the equation of a cone, called an equinioniental cone, on which lie the axes in question through P. The principal axes of this surface are coincident with the principal axes of the momental ellipsoid at P, since there are no terras involving the products of coordinates ^rj, t;^, fA Different equimomeutal cones are obtained for different values of /, but it is to be carefully remarked that all have this property. To further determine the principal axes of moment of inertia at any point P, we consider the surfaces which pass through P and are confocal with the ellipsoid of gyration which has its centre at G (the central ellipsoid of gyration), which for brevity we shall refer to as the ellipsoid E. The equation of a surface through P confocal with E is 1 178, 179] BINET'S THEOREM. 333 Its pi'incipal sections have the same foci as those of the ellipsoid E. Equation (2) may be regarded as a cubic equation for the determination of X, for any given iixed real values of x, y, z. Thei'e is no difficulty in proving that all three roots are real, and that one is less than the least of ^j, h.^, kl and negative or positive accoiding as F is within or without E, while the other two roots aie both negative, and have numerical values which, taken positive, lie one between the greatest and next greatest and the other between the least and next least of ^j, k7,, yfcg. Thus, if ^j, /4 kl be in order of magnitude, the greatest first, ^j + A, Aj + A, ^]^+A are all positive for the first root, ^^ + A is positive and kl + X, Aj + A are negative for the second, and kl+\, ^g+A are positive and ^3 + A is negative for the third. The surface represented by the equation is an ellipsoid when the first root is used as the value of A, a hyperboloid of two sheets when the second root is taken, and a hyperboloid of one sheet in the third case. Thus thiough any point whatever can be drawn these three surfaces confocal and therefore coaxial with the ellipsoid E. It can easily be proved that at the point of intersection the normals to the three surfaces are mutually perpen- dicular. Moreover, the normals to the hyperboloids at F are tangents to the lines of curvature of the confocal ellipsoid at the same point, that is, the intersections of the surface by the two planes at right angles to one another which contain the radii of greatest and least curvature of the ellipsoid at the point. Now, from the specified point P, at which it is required to I find the principal axes, draw a ' tangent cone to a«y surface confocal with E. To fix his ideas the student may take the confocal ellipsoid. This cone is the locus of the intersections of planes which all pass through P and touch the surface. Take any one of these planes and calculate its distance from G. If I, m, n be the direction-cosines of a normal to the plane from O, the square of this distance is, by (2), l2(^kl + X)+m\kl + \) + nHi:l + )^) = l'kl + ^'i^l+n'l:l+>^ (3) Thus if (Fig. 81) a perpendicular from O meet the tangent plane at Eund a parallel tangent plane to E in ffo, and we write p, po for GH and Gffo, we get X=p^-pl This interprets A, and we see that it is the same for all tangent planes drawn from P to the same surface confocal with E. Fig. 81. 334 A TREATISE ON DYNAMICS. [CH. VI. But if we write aP'^=r\ we have PH'^=r^-p\ Hence, by § 165, if we take the moment of inertia of the body about GH, namely iip\, the moment of inertia about a parallel axis through P is M'^„ + M{r^-p^)=M{r^-\). (4) Thus r^-X is the square of the radius of gyration about the axis parallel to GH through P, and hence the axes drawn through P parallel to all the axes GH, corresponding to planes through P enveloping the surface characterised by A, form an equimomental cone with /-" as vertex, and the principal axes at P are those of this cone. Such an equimomental cone can be drawn by enveloping any one of the confocal surfaces by planes through P, and we have seen that the principal axes of all such cones coincide. We may therefore use any one of the three confocal surfaces which intersect in P. In this case the equimomental cone has one axis at P perpendicular to the surface which it envelopes ; and thus, by drawing an equimomental cone to envelope each surface, we see that the three principal axes at P are normals to the three confocal surfaces which these intersect. This is Binet's theorem. EXERCISES VI. 1. Prove that if the mass of a system is symmetrically distributed on the two sides of a plane, a pair of principal axes lie in that plane for every point of it. 2. A uniform plate is in the form of a regular polygon. Prove that its M.I. about an axis through its centre at right angles to its plane is ^M{r'^ + 'ir'^), where r, r' are the radii of the inscribed and circumscribed circles. 3. Find the m.i. of a uniform elliptic plate, of semi-axes a, 6, and mass m, round an axis at right angles to its plane and passing through an end of a diameter inclined &,t an angle 6 to the major axis. Find also the m.i. about a tangent to the elliptic boundary at the extremity of the diameter specified [Ex. 4, § 175]. 4. Prove that the m.i. of a homogeneous right circular cone about an axis through the centroid perpendicular to the axis of figure is ^M{h^ + iaP), where a is the radius of the base and h the height of the cone from base to vertex. Prove also that the m.i. about the axis of figure is ^Ma^. Hence find the m.i. about an axis I, m, n through the centroid, and the equation of the central m.e. Find also the m.fs. for the vertex. 5. Prove that in a momental ellipsoid the length of the shortest of the three axes is not less than that of the perpendicular let fall from the centre on the line joining the extremities of the other two axes. § 179] EXERCISES. 335 6. A uniform elliptic lamina of mass M is loaded with two particles each of mass m placed at the extremities of the minor axis. Find what condition must be fulfilled in order that the principal axes for any point on the boundary of the lamina may be the tangent and normal at the point. 7. Prove that a body of given density and of mass M will have minimum m.i. about an axis through a given point if it is a sphere with its centre at the point. 8. Prove that the m.i. of a uniform hemispherical shell is the same for every axis through the centre. Hence show that the same thing is true for axes drawn through the vertex of the surface. 9. A uniform plate is bounded by a parabola {y^ = iax) and a straight line perpendicular to the axis at distance o from the vertex : prove that the m.i. of the plate about the axis of symmetry is ^Mca, and about the tangent at the vertex is ^Mc\ Hence find the m.e. for the vertex. 10. A solid ellipsoid may be regarded as made up of infinitely thin similar and similarly situated ellipsoidal shells, each of uniform density. This density varies as the distance from the centre along the axis a. Show that the m.i. about that axis is ^M(b^ + c^), where M is the mass of the ellipsoid. [Equation (24), § 175. Put b=pa, e = qa, p=ka, where jB, q, k are constants,] 11. Show that the foci of inertia for a u,niform elliptic lamina lie on the minor axis at distances from the centre each equal to \ae. 12. A rigid body has its mass M symmetrically distributed on the two sides of a vertical plane and revolves under gravity about an axis at right angles to the plane of symmetry. At the instant considered the angular speed is w : show that the resultant forces on the axis are M{to^h+g cos 9) outwards along the perpendicular let fall from the centroid on the axis and Mg am 6. k^/{h^+k^) at right angles to this perpendicular, where h is the distance of the centroid from the axis, % the radius of gyration about a parallel axis through the centroid, and 6 the inclination of the perpendicular to the vertical. 13. If the perpendicular from the centroid to the axis of rotation have an initial inclination a. to the vertical, show that the forces are respectively j^^, { (3A2 + k^) cos e - 2A%os ol}, Ma J^, sin 6. Hence show that the resultant stress is a minimum when 6 = 0.. 14. A planet of mass M has a satellite of mass m which revolves about it at a constant distance r. The planet has moment of inertia / about a diameter and rotates with angular speed n about an axis per- pendicular to the orbit. Show that the a.m. of the system about its centroid is 2n + 2^KMm(M+m)~^r^, where k is the force of attraction between two particles of unit mass at unit distance apart. 336 A TREATISE ON DYNAMICS. [CH. 15. Show that the total energy E kinetic and potential of the planet and satellite in Ex. 14 is given by where C is a constant. 16. A body turns about a fixed point with angular speeds (o,, ojj, Wj about axes fixed in space. Show that the kinetic energy T is given by 7"= ^ ( J a>f + Sa>| + Cwg - 2 DwjWj - 2 E'coaWi - 2 /^'wjUj), and verify that the equations of motion are dt 3a)i ' dt 3(1)2 ' dt 9(03 [See § 170.] 17. By means of equation (1), § 9, and the values of jSj, £^21 ^z (F, O, H oi § 9) given in § 170, find the general equations of motion with respect to moving axes for a rotating body. Also by (1), § 9, taking for P, G, H the angular speeds oj,, m^, 0)3 about the moving axes, and for L, M, N the angulai- speeds about fixed axes with which the moving axes at the instant coincide, prove that (i)i = (i)j, a)„=a)2, iii=bs^. [See also § 170.] 18. If the axes of x and z are the axes of greatest and least m.i. for the origin, in the case of any material system, prove that the equi- momental cones intersect the m.e. in curves which, projected on the planes of yz and xy, give ellipses, and projected on the plane of xz give hyperbolas. 19. Find the locus of P (§ 179) so that the moment of inertia with respect to the principal axis \Pz, say) at the point may have a constant value Mk^. By (4), § 179, the m.i. about a principal axis of the surface is i!/(r^-A), where A. is a root of equation (2) of that section, and r'^=x^-Vy^ + z\ Now here Mk'^=M{r^-X\ so that X=r^-k\ Sub- stituting in (2), we get for the required locus r^-h^+k\ fi-k-^+kl 'fi-k^+K. a surface of the fourth degree. If for (1) on the right we substitute (x2+y2+^2)/r^ and put a' = B-t^, h^=W-k\, &=¥-k\, the equation takes the form aV V^y^ cV _ a^ — r^ h^ — r'^ c^ — r^ ' the well-known equation of the wave-surface. VI.] EXERCISES. 337 20. Find the condition that a given straight line {x-a.)ll = (y-P)lm = {z-y)ln may be a principal axis at some point, and find the coordinates x, y, z of the point. The line must be a normal to the quadric surface, (2) § 179, which passes through the point and is confocal with the central ellipsoid, and therefore, if we put /i=.j;/;(Aj + A)=y/m(^+A)=«/w(^3+A), and substitute the values of x, y, z which these give in the equation of the line, we get fJi-=(.»-ll- l3/m)/{k\-l!^={p/m-y/n)/{kl-kl), which is the condition required. The value of ft, is thus known, and so x, y, z can be found. The value of A is assigned by substituting the 'values of x, y, z in the equation of the quadric. This gives k + Ph\Jrirfik\+rfihl==\l)i?. 21. Prove that in the plane cnx+fiy + yz — l^Q) there is a point for which it is a principal plane of the m.b. (with centre at the point) of a given material system. Prove that if A, B, C be the principal moments for the centroid, the coordinates of the point are those of the intersection of the plane with the straight line xla.-A=ylfi-B=zly-C. 22. Prove that a straight line drawn on a uniform thin plate is a principal axis of m.i. for some point on the line. Find the point if the straight line pass through the centroid. 23. A principal axis at a point P meets a principal axis for a point § in a point B. Two planes are drawn through P and Q respectively perpendicular to these principal axes. Show that their line of inter- section is a principal axis for the point in which it meets the plane PQR. 24. Ox, Oy and Ox', Oi/ are two pairs of rectangular axes in the same plane, and d is the angle xOx'. The moments about the first pair are A, B, and the product of inertia "Eiimxy) is zero. Prove that 2l(m:^y')=^{A-B)sin2d, where 6=LxOx'. CHAPTEE VII. APPLICATIONS OF DYNAMICAL PEINCIPLES. 180. Practical Applications. In the present chapter we shall deal with a considerable number of illustrations of dynamical principles, drawn as far as possible from practical affairs, such as mechanical traction, workshop appliances, and various contrivances made use of in the industries or in daily life. By practical examples drawn from ordinary experience and from engineering and the arts generally, the relations of the different fundamental ideas, and indeed their precise significance, are mad^ clear, and by a careful study of these the student can obtain a hold of the subject which no mere study of abstract formulae can provide. We shall in this chapter use gravitationaL or practical units in many examples, and shall distinguish between a force equal to that of gravity on a pound or a ton — or briefly a force of one pound, or one ton — and the mass or weight of a pound or a ton by the use of an initial capital for Pound or Ton in the former case. There can be no question that the use of the word weight in the Act of Parliament defining the standard pound, and the process of comparing masses by weighing, renders difficult and inconvenient the restriction of the word weight to forces. • The combination foot-ton and the like are in no danger of being misapprehended. 181. Acceleration in the Direction of Motion. We take first the simple case in which the acceleration of a body along the path in which it moves is alone considered. And here we consider a body moving without rotation as that is exnlained in 8 45 above, or onlv the motion of the §§ 180, 181] UNIFORMLY ACCELERATED MOTION. 339 centroid of a body which does not fulfil that condition. Acceleration is rate of growth of velocity of a body. When the component in the line of motion is of uniform amount a, the speed which grows up in t units of time is at. If this is in addition to the speed u which the body had at the beginning of the interval of time, the speed at the end is ^, where v = u + at (1) As has been already explained, speeds are measured in centimetres per. second (cm/s), in feet per second (//s), in miles per hour (m/h), or in knots (k). An acceleration of amount a may be a rate of growth of af/s per second (written af/s^, or of am/A per second (written amjhs) or of a knots per hour (ak/h), or of a knots per minute {ak/m). It is to be clearly understood, that acceleration is not velocity, but rate of growth of velocity, and has in every case its own direction which does not depend on the direction of motion, but on the action of other bodies which produces it. The speed in a given direction which grows up in time t depends on the average value for that time of the acceleration in that direction, and on the magnitude of the interval of time t. If that average value be a, or if the acceleration in the specified direction is uniform and of amount a, the speed in that direction, produced in the interval t, is at; and if we desire to specify the units, it will be written at . f/s, at . m/h, or at . k as the case may be. This will be done when necessary : it is undesirable to encumber our equations by always inserting the units specification. When the acceleration of a body in the direction of motion is uniform in amount, the average speed during' time t is ^(u + v) = u + ^at, and the distance travelled in the time is given by s=h(u + v)t = ut + iat^ (2) If in this we substitute (v — u)/a for t, we get as=^v^-iu^ (3) Of course u may denote an initial speed which is negative and a may at the same time denote a positive acceleration — as when an engine exerts a forward pull on a train moving backward ; or the initial speed may 340 A TREATISE ON DYNAMICS. [CH. VII. be positive while a denotes a negative acceleration (a retardation) — as when a train approaching a station is being slowed down by the action of the brakes. The direction taken positive is a mere matter of convenience ; the formulae hold in all cases, with proper interpretation of course when numerical valuesof the quantities symbolised are inserted. 182. Motion of a Railway Train. Time lost in Stoppages. If now F denote the force applied to the body and R the resistance to motion due to gravity, friction, etc., and these be constant, the whole work done by F in time t is Fs, and the part of this spent in increasing the kinetic energy is {F— R)s. But if W be the weight of the body in pounds or tons, F— R = Wa, and so {F-R)s^Was = ^Wv^-\Wu^ (1) In the specification of units here employed, the unit of force is that which gives unit of acceleration to the unit of weight, a pound, or a gramme, or a ton, and so the unit of work or energy, maybe expressed as lb. {ffsf or ton {fjsf, as the case may be. Let, however, the force of gravity on a weight of 1 lb. be taken, as it often is, as the unit of force, then, since this force gives to the 1 lb. weight let fall under gravity a downward acceleration of which the numerical reckoning is g, the values of F and R are the former values divided by g. Roughly, g = S2f/s^. The unit of work when F and g are reckoned in the units now specified is 1 ft. lb., the work done in overcoming a force equal to the gravity of 1 lb. through a space of 1 foot. We have then in ft. Iba. 1 W 1 W (F-R)s = ^—v^-^—u\ (2) ^ 2 g 2 g ^ ' where it is to be clearly understood that W is the number of lbs. which the body weighs. As an example of what precedes, we take the case of an express train fitted with continuous brakes, which is brought to rest before entering on a block, and started again after the block has been declared clear. If the speed of the train was 60 miles an hour, and the brakes, and other resistances, produced a retardation of Sm/hs, if §§ 181, 182, 183] DYNAMICS OF RAILWAY TRAIN. 341 the train remained at rest 2 minutes, and was then started and regained its full speed- under uniform acceleration in travelling 1 mile, it is required to find the running time lost. The student may construct the speed diagram for such a case. The ordinates of the curve are speeds, and the abscissae are times measured from a convenient zero. The distance travelled in any time is numerically equal to the area contained between the curve, the line of abscissae, and the terminal ordinates for the interval of time. The graph for a journey, including stoppages between which the maximum speed is attained, is a succession of curves rising from and falling again to the line of abscissae, with gaps between. The length of a gap along the line of abscissae is the duration of the stop. The distance lost in the running of the train is the area of the gap between the straight line of maximum speed, the line of abscissae, and the curve. If this latter area is measured in any way, the running time lost is got by division by the full uniform speed of the train. The time required to bring the train to rest is 20 seconds, and the distance traversed in stopping is, since the average speed is 30 m/A, or 44//«, 880/ or 1/6 of a mile. The time taken to start is 120 seconds, and so the whole time from the instant of application of the brakes to that of regaining full speed is (20 + 120+120)sec. or 260 sec. In this time, at full speed, the train would have travelled 260 x 88/= 4-^l;)C-+S(i.S) « If the vehicle is only slowed down from speed v to speed u, the values of i5 ands are to be found for the slowing 0) 348 A TREATISE ON DYNAMICS. [CH. VII down and the speeding up again from the equations r, \ f-i , ^2\f^ h\fv-u v^-u^ \ '■■■■ respectively. In a motor-car the height h is not very great, though it varies wiih the number of people in it ; thus the change of load on the wheels is small. On a motor-bus, however, the height h is considerable, and varies a good deal according to the distribution of the passengers between the inside and on the top. In the case of a horse-drawn vehicle of any kind, the value of TFj for forward pull is the weight of the horses. For stopping the case is different according as the hind wheels, or the front wheels, or both, are braked, and as the horses exert or not a backward push. In the last case, of course, only the shaft-horses can be taken account of. Ex. A motor-bus weighs 9 tons with passengers ; 6 tons are on the hind wheels and 3 tons on the front wheels, without driving or retarding force applied. It is driven and braked from the hind wheels and is limited to a speed of 10 miles per hour. Taking the adhesion coefficient as J, and the height of the c.g. and the length of the wheel- base as 5 feet and 15 feet, iind the time and distance for starting and stopping, and the time occupied in a journey of 300 yards between two stopping stations. Here we have i»=44/3 (ft./sec), and Hence, for starting, t = 3^^ (sec), s = 23'5 (ft.), and for stopping t = m (sec), s = 26-9 (ft.). Thus 6J seconds are occupied with starting and stopping and 50'4 feet are traversed in the time. There remain 8496 feet to be travelled at 10 miles per hour, that is 44/3 feet per second, and for this 57"9 seconds are required, making in all 64'8 seconds as the time spent in travelling the 300 yards. In this way, with an allowance for the duration of each stop, the time required for any specified journey can be found. 188. Dynamics of Vehicle on a Curve. A vehicle, such as a bicycle or a motor-car, moving round a curve on the level, or on a track inclined inward towards the centre of §§ 187, 188] VEHICLE ON CURVE. 349 the curve, aiFords an example of the relative equilibrium referred to in § 95. We take first a level track on which the vehicle, a motor-car say, of weight If, moves so that its centre of gravity describes a circle of radius r with speed v. The force toward the centre of the curve required to give the acceleration v^/R towards that point is Wv^/R. That is supplied by the action of the ground transverse to the vehicle brought into play by the continual change of direc- tion of the motion. The force thus applied is equivalent to a force Wv^jR towards the centre of the curve, applied at the C.G., and a couple of moment Wv^d/R, where a is the height of the C.G., tending to capsize the car outwards. This capsizing couple is balanced by the action of the weight of the car and the vertical forces applied by the ground to the wheels on the two sides ; for these vertical forces are, as we shall see presently, unequal. If b be the breadth of the wheel-base, and P, Q the upward forces on the inner and outer pairs of wheels, we have, taking moments about the line of contacts of (1) the inner wheels, (2) the outer wheels, Wv^^+lwbg = Qb, -Wv^^+~Wbg = Pb (1) Hence the forces applied to the wheels are P=F^*^=|^, Q=f2^^^ (2) We see then that the inner wheels will just cease to bear on the ground, that is, the car will be in imminent danger of upsetting, when 2v^d=gbR ; that is, when ^^ = J> (^) Ex. Take the case of a motor-car for which d is 30 inches and b 5 feet, turning a corner of a street 30 feet wide on the level (that is in a curve of l5 feet radius). We get in feet per second "4 ,g32xl6.21«. or slightly less than 15 miles per hour, as the limiting speed at which the wheels on the inner side just cease to press, and the car is on the point of upsetting. 350 A TREATISE ON DYNAMICS. [CH. VII. The force Wv^jR is, if the weight of the car W be taken as 2 tons, 143360 in lb. ft./sec^ units, or the gravity of 4480 lbs., that is, as it happens in this case, just the force of gravity on the weight of the car. It is the force towards the centre of the curve applied by the ground to the tires, and therefore also the force applied by the tires to the ground — the force tending to produce skidding. If the ground is too slippery to allow this force to be produced, the car will skid. 189. Bicycle on Curve on Banked Track. We now consider the more general case of a vehicle on a track canted over towards the centre of the curve to obviate risk of upsetting. For a bicycle on such a track, the condition of greatest safety is adjustment of the speed so that the plane of the bicycle shall be normal to the slope of the track, and there- fore no transverse force along the slope be applied to the tires. The reaction of the track on the machine is then upwards in the plane of the frame, and balances the resultant of the outward centrifugal force Wv^/R, and the downward force of gravity Wg, which acts in the same plane. Then tana = v^/gR. If a. be given, we can find v from v'^ = gR/tana.. Ex. 1. If the radius of the curve be 120 feet, find the cant of the track for a racing speed of 40 miles per hour. [o(.=48|°.] A bicycle is said to have run from a steeply canted track to a vertical bounding wall on the outside of the slope, described a curve on the wall and returned to the track without losing contact with the wall or the rider losing his balance. The path described on the wall was no doubt convex upward as well as outward. Ex. 2. If R be the radius of the circle which the bicycle describes, and )>v^> gR tan (a. - o 2\< W t 1/1 1\ v^+v-i, \+vJcs "" 2s As remarked in § 119, the results of Mr. Bashforth's experiments show that for speeds from 1100 //s to 2200 //s, the resistance is very nearly proportional to v^. 192. Uffect of Small Periodic Variation of Uniform Speed : (1) Time-Periodic, (2) Space- Periodic. The effect of a simple harmonic variation of an otherwise uniform speed is of considerable practical importance. The motion of a boat is periodically disturbed by the action of the oars, the motion of a steamship by the variation of the action of the screw which occurs with every revolution of the engines. These variations are too complicatedly periodic §§ 191, 192] PERIODIC VARIATION OF SPEED. 355 to be completely represented by a single harmonic term, but such a term gives a first approximation to the effect. It is to be observed that they are time-periodic, that is the speed v is capable of being represented by the equation v = v,+v,mnnt, (!) where in general v^ is small in comparison with v^. The interval of time — the period — in which the variation goes through a cycle of changes is 27r/n. In this case the distance s travelled in the interval 2-7r/n is simply V(,2Tr/n, that is the distance which the body would have travelled if there had been no periodic disturbance. For we have pv™ 27r (f D + f J sin nt)dt = — v^ (2) Jo n since the integral of the harmonic term vanishes. Thus Vg is the mean speed of advance for any time containing a whole number of time-intervals each equal to 2-7r/n. Just as much time is gained when sinw^ is positive as is lost when sin nt is negative, and this is obvious without calculation. This, however, is not the case when the variation of speed is space-periodic, as when a ship passes over a succession of waves of equal length, or better (since the wave motion varies with time as well as with distance), when a bicycle or motor-car traverses a series of regular up and down undulations of the road, so that the speed is given by the equation v = Va-\-v^smmx (3) where x is distance travelled forward from some chosen point in the path. The length of an undulation is 27r/m. The distance travelled in a period is now given by \{v^-\- v-^ sin mx)dt taken over the interval of time occupied in traversing an undulation. It is easy to see without calculation that the time gained when sin ma; is positive is less than that lost when sin ma; is negative. A boat is driven at speed v through water 356 A TREATISE ON DYNAMICS. [CH VII. which is flowing at speed v'. The speed of the boat relative to the land is v — v against the stream and v-\-v' with the stream. To travel a distance s with the stream takes time s/{v + v'), and if the boat then travels the same distance against the stream, the time taken is s/(v — v'). The time required for the distance 2s, half travelled with the stream and half against the stream, is thus s/{v + v') + s/{v — v') = 28/v{l—v'''/v^), which is greater than 2slv, the time required when v' is zero. When v' = v the time is infinite, for in the second half of the journey the boat then makes no headway against the stream. If v' be small in comparison with v, this expression is approxi- mately {2s/v)(l + v'^/v'^). We shall show that in the case of the periodic variation the factor 1 +v'^/v^ is replaced by The time required to traverse an undulation is t, given by the equation ^ ^ p-'M dx _ p'^/' " dx Jo V Jo Vf^+VySinmx ^ ' where Wj ■< -y,,. For the calculation of this integral, the transformation u = tan J-wia; may be employed (see Gibson's Calculus, § 117), but care must be taken in evaluating for the limits. It wil l be found that the integral has the value 2irl'm'Jvl — v\, or provided vjv^ is small, t-- Mtl « The time in the undisturbed motion is lirlrnv^, and the increase caused by the term ■y^sinmcc is thus 50v\lvl per cent. 193. Small Periodic Variations of Speed of Ship. Effect of Relative Motion of Parts of Ship. It will be seen that it conduces to uniformity of speed in the first case, that of time-periodic variation of speed, to have everything in the boat or ship made quite fast, so that the periodic variation of the driving force which gives rise to the term v-^ sin nt may have as little effect as possible. For if any part of the boat or cargo be loose and move relatively to §§ 192, 193, 194] PERIODIC VARIATION OF SPEED. 357 the boat, its inertia is in whole or in part withdrawn from the reaction against acceleration or retardation, and the change is greater in consequence, that is, the amplitude v^ of the harmonic term is increased.* Moreover, though the time-periodic change does not affect the average speed it makes the boat, as we shall see, more difficult to drive, so that for the same mean speed the rowers or the engines must work at a greater rate. The swing of the oars forward and backward which produces a corresponding backward and forward swing of the boat is more or less nearly counteracted by the swinging of the bodies of the rowers. On the other hand, in the case of space -periodic variations of speed, due to waves or undulations in the path pursued, the value of v^ is fixed, and consequently the smaller the mass which takes up the periodic change completely, the more nearly does the momentum as a whole remain uniform. Thus everything should in such a case be as loose as possible, even to the masts.t 194. Activity with periodically Varying Speed. The rate of working, or activity, of the propulsive force is easily calculated in each of these cases. We shall suppose, what is approximately true for a ship, that the resistance R to motion is proportional to the square of the speed, that is, that R = jj.Wv^, where Wis the weight of the vessel and fi a coefficient. If, then, F be the propulsive force applied at any instant, we have Wv=F-R or F=W{v + ixv^) (1) *Sir George Greenhill (Notes on Dynamics) quotes Joseph Pitts' Account of Mohammetans (1704) regarding the pirates of Algiers whose galleys then infested that part of the Mediterranean, "... so careful are they that nothing may hinder their speed, that they will scarce suffer any Person in the Ship to stir, but all must sit stock-still, unless Necessity otherwise require. And all things that are capable of any motion must be fasten'd or unhang'd (even the smallest weight), lest the Pursuit should be something retarded thereby." + " Pipe the hammocks down and each man place shot in them, slack the stays, knock up the wedges and give the masts play. " — Sir Edward Berry's orders on board the "Foudroyant" when in chase of the " Le G^nereux," Feb. 18, 1800 (Mahon's Life of Nelson, ii. p. 25). 358 A TREATISE ON DYNAMICS. [CH. VII. t The rate of working is thus Fv or //Ft;^+ Wvv. To find the average activity, we integrate this over one complete period of a variation and divide by the time occupied. For the time-periodic variation, we have Fv = fiW{Va+v^saintf+ W{Va+v^&mnt)nv.^cosnt. ...(2) When this is expanded, we get Fv = fiWvl+^nWvy^s\n^nt+ (3) The terms which are not written down on the right are all such as, when integrated over the time 27r/'n., give a zero result. Thus \'yvdt^f.Wvf-^^-lt.Wv,v\''^ -(4) Dividing this by l-wln, we get for the mean value A^ of the activity the equation A^ = fiWvl{l + \'^^ (5) The first factor is the activity for uniform speed i;^ ; and it appears, therefore, that the periodic variation brings about an increase of activity amounting to \bQv\lv\ per cent, of the uniform activity. The propulsive force F is given by F=ix Wv^ +Wv = ix W{vl + IVfiVy sin nt + v\ sm nt) + Wvincosnt (6) Neglecting the term in v\ and writing F=Fa[l+ksm{nt+e)}, we obtain Fo = fi Wvl, FJc cos e = 2/ii WvaVi , FJc sin e = WviTi, . . .(7) so that /c cose = 2—, A;sme = — ^-, tane= „" \°) In the other case considered, that of space-periodic variation of speed, we shall calculate the work done in traversing the distance 27r/m — the length of an undulation f2iT/m /■2ir/m Fdx §§ 194, 195] PASSAGE OP CARRIAGE OVER OBSTACLE. 359 — and then, dividing by the time 2Tr{l + vyvl)/mVo, which we have seen is occupied in traversing that distance, we shall obtain the mean activity. Thus /•27r/ni = (nWv^+Wv)dx (9) But v^ = vl+ 2'i;(,Wi sin mx + vl sinmx, and v = Vim cos Tnx . x = mVi cos trx (v^+Vi sin mx). Inserting these values in the integral to be found, we note that no term makes any contribution to the integral except the group of two iuW(vl + vlsm^mx). Thus we get \ydx=,w^^{i+l^^ (10) Dividing this by the time 2x(l + |'yJ/t;o)/'niv„, we get simply fj.Wvo for the mean activity, that is, the mean activity is not affected by the space-rate of variation of v. Never- theless, as the time of traversing any given distance across such undulations is, if the distance is great in comparison with 2'!r/m, increased by 50v\lvl per cent., the energy expended is increased in the same proportion; so that besides the delay there results an increased cost of propulsion. Denoting in this case the force F by FQ{l-\-lcsm{nt + e)}, we have, neglecting as before terms in v\, F„ = iuiWvl, FJccos e—2fxWvgVi, FJc sin 6= TnWvgVi, ...(11) so that A;cose = 2— , A;sine = — rr^, tane- „„ ■ ••■U^) Vo -to ■'-Po 195. Work done in the Passage of a Carriage over an Obstacle. Extra Work on a Causeway. Effect of Springs. We now consider the passage of the wheel of a carriage over an obstacle in its track. We suppose that the radius of the wheel is r, and the height of the obstacle h, and that the centre of the wheel, before impact occurs, is moving with speed v horizontally. If W be the weight of the 360 A TREATISE ON DYNAMICS. [CH. VII. wheel and the part of the carriage and its load which rests upon it, the kinetic energy of the wheel and its load is ^ Wv^ in absolute units. When impact with the obstacle occurs the direction of motion is changed, for the wheel at that instant begins to turn about the point of contact B with the obstacle (see Fig. 83); but the radius to this point is inclined, at an angle ot. = cos"^(l —hjr), to the radius which at the same instant has its outer extremity in contact with the horizontal plane along which the wheel was rolling. Just before impact the momentum associated with the wheel was Wv in the horizontal direction ; just after the impact with the obstacle the momentum is in the direction CD and is of amount Wv cos a. For the impulse applied to the wheel cannot alter the angular momentum about the point of contact B, and this just before impact was proportional to Wv cos a. The centre of the wheel now gradually ascends through a height h, and the speed changes from v cos a. to u, say. Let now the average forward force in the direction of motion, on the wheel and its associated load, be F, and the resistance be R. During the ascent the centre of the wheel travels a distance roc, and the forward displacement of each part of the load will also be ra., if, as the wheel mounts the obstacle, the load is displaced so that its centroid remains vertically above the axle. We suppose this to be the case, and that the forward speed u when the obstacle has just been surmounted is v. This involves the supposition that the energy of rotation of the wheel is not altered. Then the principle of energy gives (F-R)ra.= Wgh + ^Wv^sin^a. (1) On the left we have the w^ork done by external forces in making the wheel surmount the obstacle, and on the right that work is seen to be made up of two parts, Wgh the gain of potential energy and the part ^Wv^sin^oi due to the Fig. 83. § 195] EFFECT OF CARRIAGE-SPRINGS. 361 sudden change of direction of motion — the jerh. Eut sin2(x=2A,/r-/i.7r^ and so {F-R)ra.= Wgh+Wv^(^-~^ (2) In general the second term on the right is the more important. For example, if a wheel 4 feet in diameter carrying a load of 10 cwt. pass over a pebble an inch high, at a speed of 10 miles an hour (44/3//s), we get, taking 1 cwt. as the unit of mass, Tfg'A. = 10x 32 XTV=26f (or in foot-pounds, 93^) and Ift;%/r=10x447l2x 2x 9 = 90, nearly (or in foot-pounds, 313). Let now the wheel descend again to its former level. The force F in the direction of motion has not the same value as before : we suppose it to be such that the speed of the centre of the wheel, when contact with the horizontal plane has been resumed, is still the same. By the same process as before, we get (F-r)r--- be the resulting angular accelerations of the wheels in the train geared in succession, we have ^ , „ , „ L = wicih + wjc\w^ + ftj/c^Wj + . . . = {wk^ + w;ic\n\ + wjclnl+...)w, (3) where w, Wi, w^,... are the weights on the main and successive axles, k, k^, k^,... the radii of gyration, and %i, n^, ... the speed-ratios of the successive wheels to the first. The effective moment of inertia / of the train is thus given by I=wk^+wJclnl + oDjclnl+ (4) that is the moment of inertia of a single wheel and axle in which L would generate the angular acceleration to has this value. 199. Motion of a Wheeled Vehicle on an Inclined Plane. The motion of a vehicle on an inclined plane — for example of a railway carriage on a gradient — may be determined from (1), § 197, which enables account to be taken of the inertia of the wheels. The gradient is measured by the rise h of the track in a distance I travelled along it. When h and I are measured in the same units, the inclination a §§ 198, 199] VEHICLE ON INCLINED PLANE. 367 of the track to the horizontal is given by sin ex. = hjl. Thus, for a gradient of 1 in 20, we have sin a = 1/20. The value of P in (1), § 197, is then {W+nw)jB\na.-R, where Wg + nwg is the force of gravity in absolute units on the whole weight, and R is the resistance to motion, apart from the backward pull applied to the carriage at the contacts of the wheels with the rails, which has been taken account of in forming the equation referred to. Thus, if the only applied force be that due to gravity, [W+nw — 2 — jv = {W +nw)g smcL — R (1) {w+nw^^^ye = ]^{{W+nw)g&ma.-^R), ...(2) where d is the rate of increase of 6, the angular speed of turning of the wheels, supposed to be rolling without slipping. If these equations be multiplied by v ( = ds/dt, where s is distance measured along the slope in the direction of motion) and respectively, and B be supposed con- stant, they become directly integrable, and we obtain or ^( W+nw — ^- jv'^={(W+n'w)gsma.—R}s+C, (3) (4) where is a constant. If v, s, 6, Ohe Vg, Sg, dg, 6^ initially, we have = {{W +nw)g sin a.— R}(s — So), ^{Wr'+nw (h' + r«)(0^ - Sl) = {{W+nw)gsma.-R}r{e-eo); The first of these equations may be written also in the form \{W + nw){v'-vl) + lnwk\&'-el) \ = {{W +nw)g&ma.-R]{s-So). j ^ ' 368 A TREATISE ON DYNAMICS. [CH. VII. These are equations of energy ; on the left-hand side is the kinetic energy gained in descending the distance s — s^ along the incline from initial speed Vq ; on the right is the loss of potential energy and the work done against the resistance R. In the last form of the equation the kinetic energy is separated into two parts — the translational kinetic energy of the whole moving system and the energy of rotation of the wheels about their axes. 200. Boiling of a Solid of Revolution on an Inclined Plane under Gravity. For the motion of a solid of revolution or "wheel" rolling down the incline under gravity, without resistance, we have ^wv^ + \wl^{i-+;^z)('"^-'vl) = wg(s-s^)sma., (2) where s — s^ is the distance traversed by the centroid while the speed increases from Vg to v. These results may of course be obtained directly. At the centroid a force wg sin a. acts down the plane, and at the point of contact a force F acts up the plane. Hence for the motion of the centroid, we have wv = wg sin a. — F. (3) Again for turning about the centroid, we get wJS^Fr; (4) or if there be pure rolling, since then 6 = v/r ivm = Fr^ (5) w §§199,200,201] SLIDING ON INCLINED PLANE. 369 With the value of F given by (4) inserted (3) becomes Vl + ^j^ = '«^9'sina (6) from which multiplying by v and integrating we derive 4^ (^1 + ^ j (^^ - ''^o) = '>^9 (s - So) sin a which is (2). Also (3) and (6) give F. By (6) the value of v is uniform, and so the time of motion can be obtained as in uniformly accelerated motion. The " wheel " may be any solid of revolution witli matter symmetrically distributed about its centre. For example, it may be a uniform sphere of radius r, in which case /c2 = -|r2, and we have \w\v^ = ■^■^imP' = wgs sin a. . , (7) The effective inertia of the sphere is thus increased by rolling in the ratio of 7/5. For a thin hoop the equation is, ' ^w2i^ = wv^ = wgssva.CL, (8) so that the effective inertia is 2w. If the wheel is a uniform disk, ^{h^-\-v^)lr^ = \, so that the effective inertia is increased in the ratio of 3/2. This is an approximate estimate for the wheels of railway carriages. Ex. 1. Two spheres have the same external diameter and the same weight. One is a gilded sphere of brass and the other a hollow shell of gold. Compare the speeds acquired in the same time from rest by the two spheres in rolling down an inclined plane through the same distance. Thus one sphere may be distinguished from the other. Ex. 2. A rough homogeneous sphere of radius r is placed within a hollow cylindrical garden roller of radius R and comes to rest at the lowest point. The roller is suddenly set rolling on a level track so that its eentroid moves with speed F. Show that the sphere will roll completely round the interior of the roller if V'^>^ g{B — r). [If the line from the centre of the sphere to the point of contact with the roller make an angle 9 with the vertical at time t, the angular speed of the sphere is {R — r)6/r— V/r. Hence find the equations of motion]. 201. Sliding Motion of a Body along an Inclined Plane with Friction. The approximate laws of friction between dry G. D. 2 A 370 A TREATISE ON DYNAMICS. [CH. VII. solids are no doubt known to the student, but they may be here recapitulated. (1) Friction acts always tangentially to the surface of contact of two bodies in the direction to annul or prevent relative motion. Thus a body resting on an inclined plane is prevented from sliding down if the inclination is less than a certain limiting value. Just sufficient resisting force is developed to prevent motion, and this force, which must equal wg sin a., if w be the weight of the body and a. the inclination of the plane to the horizontal, increases until wg sin a. = fiwg cos oc, where yu is a coefficient depending on the nature of the surfaces in contact. Thus /n = ta.na., and OL is called the limiting angle of friction. When pure rolling is possible just enough friction is developed to make rd = v. (See § 204). (2) Friction is independent of the extent of the surfaces in contact. (3) Friction is independent of the relative speed of the surfaces in contact. (4) Friction at each part of the surface of contact is proportional to the normal force with which the two surfaces there press against one another, that is, if F be the friction and N the normal force, F=/j.N, where /j. is the "coefficient of friction." [Thus in the case of the in- clined plane and body resting on it referred to in (1), the normal force N is ivgcosxx. and /U = tana. The angle a., at which the body begins to slide, thus enables the co- efficient of friction to be determined for any two substances if the inclined plane is made of one, and the body placed upon it is made of the other.] The value of /i thus obtained is, however, rather greater than that which an experiment with an inclined plane gives, if there is sensible relative motion. For it is found that if the plane is held at the limiting angle, and the body is started down the plane with a small speed, it will have a slight acceleration, showing that now the value of F is somewhat under wg sin a. Experiment has shown that the value of the coefficient of friction — nearly constant for finite speeds — increases for a small range of values of t; as V is diminished towards zero. § 201] FRICTION. 371 Let now an inclined plane make an angle /3 >■ a. with the horizontal, and let a body of weight w be placed upon it. For simplicity we shall suppose for the present that the body is a particle. The particle will slide down along the line of greatest slope of the plane through the initial position, that is along a line at right angles to the inter- section of the inclined plane with a horizontal plane at the place. The equation of motion is wv = wg sin /3 — fiwg cos /3 (1) or i;=gi(sin;8 — yUC08j8) = g^secasin((8 — a.), (2) since fi = ia.ncL. Thus the acceleration is the same as if there were no friction, the inclination were /3 — (x., and gravity were increased to gseca.. If the motion start from rest, V =gt sec a. am(^—(x.), (3) and the distance s traversed in time t is given by s = ^gt^ sec a sin (|8 — ot.) (4) But the equation for v gives t = v/gsec(xam(l3 — • 1^%, the angular speed will have changed sign before rolling has been set up. The distance s of the centroid from the point of projection at the same instant is given by Thus s is positive or negative according as VaQc^ + 2r^)> or (Va+r9o)kVfi{k^+r^)g. On the other hand, if /(;V0o>- t;g(/c^ + 2r^), s is negative, that is the body has had the speed of its centroid reversed, and been brought back beyond the starting point before the setting up of pure rolling. If then k^O^ ]> v^r, the body will be rotating in the same direction as at first, and will roll still further away from the starting point. The (8) 378 A TREATISE ON DYNAMICS. [CH. VH. condition /<;V0q> Dg(A;^ + 2r^) precludes the possibility of the case k^Og-i^v^r, which therefore cannot be associated with a negative value of s. If the solid returns the time t' occupied in retraver.sing s is given by s/6r, by (8), that is t' = s(k^ + r^)/Ar. Thus ^,^ Vo+rdo P Vg{k^+i^)-Ar ^g^ 2 fig k^+r^ At To find the whole time from projection to return, we have to add to this the value of t. Thus if A=W-Q^ — v^, t + t'= \ ^o+r0o 1 or t. Find also the whole time from the instant of projection to that of return to the starting point. Ex. 3. The solid, still with initial underhand spin, is projected down the inclined plane of last example : find the equations of motion and show that, if /x cos yS > sin /3, and pure rolling has not begun, v=0 §§204,205] COMPOUND PENDULUM. 379 when t — VQlg(fjLC0sj3-smP). Prove that at that instant the solid is rotating in the same direction as at first, and will therefore turn back, provided jU, > tan;8/(l -v^rjBd^. Prove also that pure rolling will never begin unless /n > i^tan j8/(F+»-2). Ex. 4. A uniform cylinder of radius r, revolving with angular velocity w about its axis, is gently laid with its axis horizontal on a horizontal table. If the coefficient of friction between the cylinder and table be ;«,, show that the cylinder will slip for a^ time r(a/3[j.ff, and then roll with angular velocity (o/3. Ex. 5. A shaft on loosely fitting bearings, radius a feet, carries a weight of W lbs. If /A=tan<^, show that the resisting couple is Wa sin <^, in pound foot units. 205. Compound Pendulum. Any rigid body movable about a horizontal axis may be used as a compound pendulum, provided the centroid does not coincide with the axis about which the body is pivoted. A wheel mounted with its axis horizontal and put out of balance by a weight attached to the rim may serve as an example. The position of stable equilibrium of the body is that in which the centroid is in / Fig. 92. the lowest possible position. For when the body is deflected a little way from that position and left to itself at rest, the forces upon it have a moment causing it to return to that position, that is the forces tend to bring it into the position of minimum potential energy consistent with rotation about the fixed axis when the moment vanishes. 380 A TREATISE ON DYNAMICS. [CH. VII. GenerfiUy a compound pendulum constructed as such is a body supported on a horizontal line of knife-edges, and so shaped that its parts are symmetrically distributed about a plane through the centroid at right angles to the knife- edges which are also symmetrically placed with regard to that plane. Fig. 92 shows one form — a massive ring of rectangular- section supported on an upturned knife-edge which touches it along one of the generating lines of its inner cylindrical surface. Fig. 93 shows a more usual form with attached knife-edges and sliding weight. 206. Theory of Compound Pendulum (C.P.). Equivalent Simple Pendulum. We shall suppose first that the body is provided with a cylindrical bearing of radius c (Fig. 94) supported on a horizontal surface, so that at any instant of its motion the body is rolling on that surface and turning therefore at the instant about a generating line of the cylinder. We denote by h the distance of the centroid of the body from the axis of the cylinder and by WM' the moment of inertia of the body about an axis through the centroid parallel to that of the cylinder.' Then, if the deflection of the line through the centroid perpendicular to the axis from the vertical be B, the moment of inertia about the instantaneous axis is Tr(/c2 + /i,2 + c2_2/iccos0). The forces on the body are (1) the resultant force of gravity acting vertically downward through the centroid, and (2) the reaction R of the axis. The latter has no moment about the axis, and thus Wg is the only force concerned in altering a deflection from the vertical position, for we assume that there is no friction at the axis which exerts a moment on the pendulum, for example no couple due to what is called " rolling friction " (see § 201). Wgh sin is then the moment of Wg about the instantaneous axis of , turning, and tends to produce angular momentum in the §§205,206] COMPOUND PENDULUM. 381 direction of diminishing 6, and the angular momentum at the moment is W{k^ + h^ + c^ — 2hc cos 6)6. We have there- fore W{k^+h^+c^- 2hc cos 6)6 +2 Whc sin 6. 6^=- Wgh sin 6, and therefore {h^+J (-5) so that l ^^f+kJ) + w,(hl+Jct) + ... (6) The compound pendulum of Fig. 92 consists of a ring of iron of rectangular section, made of quarter-inch boiler or ship plate, well hardened after construction. The inner and outer surfaces are truly coaxial cylinders, and the ring, as already stated, oscillates on an upturned knife-edge touching it in a generating line of the inner surface. If r-j, r^ be the inner and outer radii, the moment of inertia about the axis of figure is where m is the weight of the plate per unit of area, that is, the M.I. is ^W{rl+rl), where W is the whole weight, and the moment of inertia about a generating line of the inner cylinder is ^Wir^ + Srl). The period is therefore given by V r,g '^i9 207. Suspension Axes and Oscillation Axes. Interchange- ability. Suppose the pendulum to be hanging in stable equilibrium, while capable of turning about an axis A (Fig. 93) at distance h from the centroid 0, and let a circle be imagined to be described from the centroid as centre with h as radius, in the plane in which the centroid moves. Now let the pendulum as it hangs be imagined connected, if necessary, by a framework of negligible mass, rigidly attached to the pendulum with an axis through B parallel to A, and then to be loosed from the axis at A so as to be free to turn about that at B. The period of unresisted oscillation about B for any range of deflection will be the same as the period for the same range about A. If another circle be described from G as centre in the same plane as before with l — h (or l — h + c) as radius, a §§ 206, 207] COMPOUND PENDULUM, 383 perpendicular let fall from Q to any of the equivalent axes A, B,..., and produced backward, will meet the second circle in a point A', B',.... Then l = AA' = BB'= .... The points A, B, ... have been called centres of suspension for the pendulum, and A', B',... centres of oscillation. The pendulum has, as will be shown presently, the same period of oscillation about a parallel axis through A', B', ... that it has about the axis through A, or any of the equivalent axes through B,G, This is the principle of " convertibility of the centre of oscillation and suspension " ; but the principle is often so expressed as to suggest that for a certain period there is only one centre (or axis) of suspension and a corresponding centre (or axis) of oscillation. As generally made with fixed knife-edges compound pendulums admit of only one axis of suspension being used, and the problem is then that of finding the centre of oscillation which corresponds, and some range of variation of that is provided for by a sliding weight which can be fixed in different positions on the pendulum. But as a matter of dynamics there are an infinite number of equivalent axes and corresponding oscillation axes, or, as they would be more properly called, conjugate axes. It is convenient to have an arrangement to illustrate this, and one has been made as follows. A sheet of steel- plate, thick enough to remain rigidly plane, is loaded with a diametral bar across the centroid formed by two strips of steel riveted to its two sides. Holes of equal size, large enough to admit an upturned knife-edge projecting from a fixed support, are bored with their centres in a circle round the centroid, and another concentric row of similar holes is made nearer the centroid, so that the radii of the circles touched by the outer edges of the holes in the two series have the radii h and l — h for the arrangement when used as a compound pendulum. The outer series of holes is cut first, and then the position of the second series is fixed with allowance for the material to be cut away. The cross-bar is made a little too long at first, and the arrangement is finally "tuned" to agreement of period by filing a little away from each end. The same period of 384 A TREATISE ON DyNAMICS. [CH. VII. oscillation is given' whatever hole is used for the suspension of the body on the knife-edge. The theorem of convertibihty of axes referred to above is proved as follows. The period T is given, by -=W^-W^ 0. where l = (h^ + k^)/h. This last equatijn can be written in either of the forms h?-}il + lc^ = 0, (l-hf-{l-h)l + ¥ = 0, (2) so that, if h is one root of the equation, l — h is the other. The sum of the roots is I and their product is F, as aiBrmed by the quadratic equation. We infer that if I be the length of the equivalent simple pendulum for the distance h from the centroid to the axis, it is also the length of the equiva- lent simple pendulum for the distance l — h. Thus for the infinite series of parallel axes, for which I has a given value, there is a conjugate series at distance l — h from the centroid, for which I has the same value. If h be chosen very great l — h will be correspondingly small, and the periods will be the same; and when h is infinite l — h will be zero, and the periods in both cases will be infinitely long. Hence, if we diminish h from infinity l — h will be increased from zero, and the periods will be diminished; and clearly the two distances coincide when the period is a minimum. We have then I — l — h, or l = 2h and h = k, so that l = 2k is the smallest length of the equivalent simple pendulum. If the matter of the pendulum be concentrated in two particles, one of weight Wk^/{h^ + k^) at the centre of suspension and the other of weight Wh^/{h^ + k^) at a point L at distance l = (h^ + k^)/h from the suspension, the period will be the same. This arrangement, as was pointed out by Maxwell (Matter and Motion), is kinetically equi- valent to the compound pendulum. For if the compound pendulum have its suspension at and its centre of oscillation at L, the two centroids coincide, the moments of gravity forces, and the moments of inertia about 0, are the same. The moments of inertia about an axis through.: §§ 207, 208] COMPOUND PENDULUM. 385 the common centroid are the same, and therefore the moments of inertia about any axis whatever are the same. 208. Experimental C.P. A compound pendulum used for experiment is generally furnished with two pairs of knife- edges, one pair fixed relatively to the pendulum and a movable pair, and also with a sliding weight to enable the distribution of matter in the pendulum to be altered, and the experimenter is required to arrange the apparatus so that the pendulum swings about the two pairs of knife- edges in the same period. In the Repsold pendulum the sliding weight is within a containing tube, which keeps the external form the same for all distributions of the mass, in order to avoid inequalities of air-resistance. This resistance may be regarded as made up of two parts, a true frictional resistance and a dragging of air with the pendulum, by which its inertia is virtually increased. Further, the pendulum being immersed in the air has its gravity virtually diminished by the buoyancy of the air displaced. We shall show presently how the virtual increase of mass and the effect of buoyancy may be estimated. The adjustment of the period of swing about the two axes to equality is made easier by hanging a simple pendulum alongside the compoimd one (when the latter is made to oscillate about the fixed knife-edges) and altering its length until the two just keep pace. The position of the second pair of knife-edges should then be shifted to a distance from the first equal to the length of the simple pendulum thus found; and this, with a slight correction for the change produced by the sliding piece carrying the knife-edges, will give the required arrangement. The distance of one line of knife-edges from the other is then I, and, if the period T of small oscillations be determined, g can be calculated from the equation 9^^^- (1) This method of determining g was carried out by Captain Kater (Phil. Trans. R.S. 1818). In preference to a simple pendulum he carried a compound pendulum from place to 6.D. 2 b 386 A TREATISE ON DYNAMICS. [CH. VII. place, and so made a gravity survey over a considerable part of the country. It will be clear from Fig. 93 that besides A and A' there are in the same line AA' two other points, A-^^, B^, at which the second knife edge can be placed to give the same period. The student is not likely to place the second knife-edge at Aj^, but occasionally he hits on B^ as the position. This, it will be seen, gives 2h, not I, as the dis- tance between the two lines of knife-edges. Thus, twice one root of (2), § 207, generally the greater root, is obtained and taken as I; the student ought to be advised of his error by the absurdly large value of g then given by (1). 209. Buoyancy and Air-Drag of C.P. The buoyancy and air-drag modify the equations as follows. Let w be the weight of air displaced by the pendulum, that is the weight of the air at the density of the surrounding atmosphere, which fills a volume equal to that of the pendulum : the assumption is made that the air dragged with the pendulum is proportional to w. This assumption is derived from the fact that, for example, an infinitely long cylinder moving with uniform speed u in a, direction at right angles to its length in an infinite incompressible frictionless fluid, has an apparent kinetic energy greater than that corre- sponding to the mass of the cylinder by ^^lnl? : in the case of a sphere moving in any direction on such a fluid, the excess of energy is \wu''; for an ellipsoid it is Kvyu?, where /f is a coefficient depending on the direction of motion relatively to the principal axes. We take then k' such a length that wl^f^ is the increase of moment of inertia of the pendulum, supposed of sym- metrical outward shape, and situated about positions of the knife-edges, symmetrical about the centre of figure, and giving nearly equal periods Tj, T,^. The buoyancy of the air is a force wg acting upwards through a point which is called the centre of buoyancy of the immersed body, and this for a pendulum in which the knife-edges are symmetrically placed, as here supposed, is at a distance (k^-\-h^l2 from either axis, if h^, h^ now denote the dis- tances of the centroid of the pendulum from the two (1) §§ 208, 209] COMPOUND PENDULUM. 387 knife-edges. Thus the moment of inertia is increased to W{hl+k'^)+w}e'\ and the moment of forces is diminished to Wghj^ — ^wgihj^+h^), about the first knife-edge. Similar expressions hold for the other knife-edge. The lengths li, l.^ of the equivalent simple pendulum for the two knife- edges are given by _ W{h\ + B) + Wh'^ _ 9 rpA _mK+if)±wi^_j_ , By means of these equations, we can eliminate WW + wk'^, and so find an equation for g containing a small correction term, depending on the value of w, which can be approxi- mated to more or less nearly in various particular cases; and this, with or without the term depending on wf W may be used to find g, when the distances /ij, h^ giving periods Tj, T^ are measured. We find g^ 87r^(/t,-H/t,) (-2) This divided by 47r^ is the length of the equivalent simple pendulum which would have a period equal to unity, and divided by tt^ it gives the length of the equiva- lent simple pendulum which would beat seconds. The length is thus expressed in terms of \, h^, T-^, T^ and the ratio wjW. The latter furnishes a small correction term, which can be estimated more or less nearly in different cases, according to the shape of the pendulum. For a clock-pendulum the air carried with the bob is the only thing regarded, and then it is suiEcient to take it as a particle added at the centroid of the bob. The weight added is some fraction fi of the weight w of air displaced, and so the equation for the Corrected value I' of the length of the equivalent simple pendulum is f- - — (jr:^, — -T^wV'^^vi' ^^^ 388 A TREATISE ON DYNAMICS. [CH. VII. where l = {h^+k^)/h, the uncorrected length. If the bob is a cylinder moving at right angles to its length, ju may be taken as 1, though this supposition is rendered in- accurate by the fact that the cylinder is of finite length, and the atmosphere in which it moves is limited by the clock-case. For a spherical bob jul may be taken as J. The factor of w/W is the ratio of the specific gravity of air to that of the material of the bob, and if the bob be of lead the ratio is about 8000. Thus, taking h/l=l and // = J for a spherical bob (wlW)(l + fih/l) = 3/16000, so that a clock regulated by such a pendulum, of the length I to beat seconds, would, in consequence of buoyancy and air- drag combined, lose about 8 seconds in 24 hours. Very accurate sidereal clocks at observatories are now enclosed in partly exhausted air-tight cases, so that this correction may not fluctuate as it would otherwise do with the barometric pressure, owing to alterations produced in the ratio w/ W. 210. Examples on the Compound Pendulum. Ex. I. A compound pendulum is formed of a uniform rod of length 21 and mass m loaded with a mass m' at each extremity. Find the length of the equivalent simple pendulum for vibrations about a horizontal axis at right angles to its length. Find also the position of this axis when the period has its least value. Let the distance of the axis from the centroid be k. The length L of the equivalent simple pendulum is then given by j._ {^ + 2m')P+{m + 2m')h^ {m,+2m')h This equation may be written m + 2m The roots of this quadratic in h cannot be imaginary, and therefore the least possible value of L is to be found from the relation (w -t- 2»i') i2 = 4Qra -f- 2m') P. If m = m', this gives 91^ = 28^^, or 3Z = 2^J^l. We have then A = Ja/T^. If the axis of suspension intersects the rod, it divides the rod into segments in the ratio of 3 - >/? to 3 -1-n/7. Ex. 2. A homogeneous sphere rolls within a hollow right circular cylinder which is fixed. Find the time of a small oscillation of the sphere about the lowest position. §§ 209, 210] COMPOUND PENDULUM. 389 The centre of the sphere moves in a circle of radius R-r, where R is the radius of the cylinder and r that of the sphere. Let i/r be the inclination to the vertical of the perpendicular from the centre of the sphere to the axis of the cylinder, and d the angular speed of the sphere at any instant. Then we have (Fig. 95), noticing that 6 and ^p are in opposite directions r6={R-r)^. The kinetic energy of the sphere (weight w) is by the relation between 6 and \p. The potential energy in the position indicated is wg{R - »•)(! - cos i/f). Hence if a. is the extreme value of t/^, /jw {R-rf\p^=wg{R-r) (cos ^ - cos a.) ; and we get, by differentiation, l{R-r)\i'+gsm-\lr=0. The period of a small oscillation is therefore 2^^7.^E-r)/g. The problem may also be solved as follows. The sphere is turning at the instant about the point of contact A with the cylinder. Hence, if 6 be the angular acceleration, ^_—'wgrsinf_ bgainyjf ^wr^ + wr^ 7 r But, by the relation between 6 aiid xp, we have 0=(R-r)if'/r. Substituting in the equation just found for 6, we get the same equation as before. Ex. 3. A uniform plank, of length I, balances on a fixed horizontal cylinder of radius R : the length of the plank is at right angles to the axis of the cylinder. If the plank is set rocking without slipping, show (neglecting the thickness of the plank) that the period is that of a simple pendulum of length -^P/R. Derive also the energy equation. The plank rolls without slipping : at time t let it be inclined at an angle 6 to the equilibrium position : the radius to the point of contact now makes an angle 6 with the vertical, and if O be the centroid, and D the point of contact (Fig. 96), DO=Rd. Hence, jf F be the weight of the plank, W{^li+B?ffi)e= - WgRd cos d, or, when 6 is small, 0+ 3-— 0=0, that is, the length of the equivalent simple pendulum is ^^P/R. 390 A TREATISE ON DYNAMICS. [CH. VII. The kinetic energy of the rod is J W(^l^+RW)e^; and at time t the point of contact of the rod is at a distance ^(1 - cos ^) above the centre of the cylinder. Hence, the centroid is at a distance /?(9sin6l + i?(l-cos6') above C. The potential energy may therefore be taken as Wg{Resm d+Il{l-cos 9)}. Let a. be the extreme value of 6, for 6 ' which, of course, 9=0. The equation Fio. 96. of energy is i W(^P+IPe^)e'+ WgRHesin e-a.amoC)-(cos6-cosa.)\=0. Ex. 4. A uniform plank 13 feet long is balanced on a horizontal log of circular section, 4 feet in diameter, and two boys of equal mass seated at its ends use it as a " see-saw." Taking the mass of the plank as 40 lbs. and that of each boy as 84 lbs., and regarding the plank as a thin rod ahd the boys as particles each at a distance of 6 feet from the middle of the plank, show that the period of a small oscilla- tion is 4'43 seconds. Ex. 5. Show that if the internal and external radii of the ring pendulum described in § 205 be j\ and r2, the length of the equivalent simple pendulum I is given by Irl S 2 /-J 2 '■ If the mechanic in cutting the ring has made r^jr^ have nearly an assigned ratio (so that only the outside diameter is measured), show that a small error in this ratio has zero efifect on the calculated length of the equivalent simple pendulum when the ratio has the value >/3/l. Ex. 6. At a point P. in the line joining the centre of suspension and the centre of oscillation of a compound pendulum a mass v) is attached. It is required to find the change in the length I of the equivalent simple pendulum, and to show that if the mass w be small it produces the greatest change in I when attached at a point halfway between the two centres. Without the additional mass l=(h^ + k^)/h. When w is attached the moment of forces for a deflection 6 becomes (Wgh+tag3;)am 6, where X is the distance of F from 0. The moment of inertia becomes Hence iil+^ denote the new value of I, and p the ratio w/W, we have h+px ~ h+px §§210,211] . COMPOUND PENDULUM. 391 Thus y vanishes when ^ = 0, and when x=l^ and is a maximum when x=ll%, if p be small. Clocks are sometimes regulated by varying a small mass placed on a shelf carried by the pendulum. It is here shown that the shelf should be midway between the two centres. Ex. 7. If y denote the excess of the length of the equivalent seconds pendulum when the mass w is at distance x from the centre of suspension over the length when the mass w is at distance a, show that the graph formed with values of y as ordinates and values of so as abscissae, is a hyperbola, of which the asymptotes are the lines ^=0, !t:=y. We have here _hl + px^ hl + pa^ _hl + px^ j ^~ h+px h + pa ~ /i + px ' which leads to the equation px^ — pxy — pLx -hy — h{L-r)=0, the graph of which is clearly a hyperbola. The terms of the second degree pa? - pxy equated to zero give the asymptotes, which are therefore the lines ^^^^ x-y=0. This relation can be used to graduate a metronome, an instrument for beating time in music. The period is altei'ed by changing the position of a sliding weight, which is comparable with the whole vibrating mass. Ex. 8. A compound pendulum is formed of a sphere as bob, consisting of a uniform shell of iron filled with water, and suspended on knife-edges attached to a rod rigidly connected with the spherical shell. If "we neglect the friction of the shell on the water we must take the water as a mass every particle of which has at each instant the same velocity as the centroid. Hence if W be the weight of the water, TFj that of the shell and rod, A the distance of the centroid of the whole and K the distance of the centre of the sphere from the knife- edges, k the radius of gyration of the solid part ahout the knife-edges, ( W+ W^)h ' 211. Beactions due to Accelerations. Case of C.F. It is important to find the reactions due to the accelerations impressed on the different parts of a body or system moving in any manner, as these give the forces which are exerted by each different part on the rest of the system, and when properly summed lead to expressions for the forces exerted on the supports of the system. As an 39-2 A TftEATISB ON DYNAMICS. [CK. VII. example we take here the compound penduhim ; but the same process may easily be applied to any swinging body, such for example as a ship in a seaway. Any small part of weight Wj, say, at a point P^ at distance Tj from the axis of suspension has, in the motion of the pendulum at any instant, an acceleration r^Ol in the direction towards the axis, and an acceleration rSj in the direction of 6 in- creasing, if 0j be the inclination to the vertical of the perpendicular let fall from the point Pj to the axis about which the pendulum turns. For a positive value of 6^ that of 6i is negative, and vice versa, so that the accelera- tion rdi is really always in the direction of diminishing 0j. Fig. 97. The corresponding forces are w^r^Oi and tu^r^Oj. These are forces applied to the part by the rest of the system. Now we hav e WjTjdi = —wgsmd + P-^, tWirjOj = —wgcos6 + Ri, where Pj is the force tangential to the circular motion in the direction of increasing 0^, and P^ is the inward radial force, both applied to w^ by the body itself. Hence w^r^dj^ + w^g sin 6-i^ = P^, WjTi&l+Wjg cos 6i = Ej^. ...(1) The addition can be carried out graphically, by drawing vectors from the point Pj to represent the mass-accelera- §211] COMPOUND PENDULUM. 393 tions of w-^ reversed, and then combining with these the vertically downward vector w^g. The vector obtained is equilibrated by that given by the P, R forces in their actual directions. The latter vector represents the result- ant of the forces applied to the body by the axis, since the resultant of the internal forces is zero. Thus, in Fig. 97, the symbols w^a, W{n, indicate the two reversed vectors so drawn for u\ at the point Pj. The result- ant is the vector F inclined at an angle tan"'a/% to the line AP^ produced. Now, since the angular acceleration 0, and the angular speed Q, are the same for every part of the oscillating body at any instant, this angle is the same for every radius-vector AP drawn from the axis to the position of an element. Thus the resultant of the reversed mass- accelerations is, at every point, along the forward tangent to the equiangular spiral of equation re'*"''"/", {j' = AF), drawn through the point from a pole at A. This result is due to Sir George Greenhill {Notes on Dynamics). With this is to be combined the uniformly directed vector wg. This can be done by constructing another spiral of the same angle = tan''(a/'n.) from a new pole A' found by the following process. We draw AB inclined to the vertical at the angle 0, which represents the de- flection of the pendulum at the instant (so that AB is a line which would be vertical in the equilibrium position of the body), and tnake AB = l, the length of the equivalent simple pendulum, and on the upward vertical through A take a point G, such that AG=g/d^. Then through B we draw a line at right angles to AB, to meet a horizontal line through A in E, shown in Fig. 97. C is then joined to E and a perpendicular A A' let fall to CE. Then, as can easily be shown, lAEG= LA'AC= = w^g(AA'IAE)/{AA'/AE) = w^g (3) Hence, if with this be compounded the vector Wggi down- wards, the resultant is zero. Since the resultants of a and n for the two points P^ and A' are by (2) proportional to the lengths of the lines AP^ and AA' (at A' the resultant is g upwards), and are inclined to one another at the same angle as are these two lines, the result of compounding the downward acceleration g with a and n at P■^ is the same as that of compounding a vector represented by AP.^, with a vector equal and opposite to one represented by A A', and is a vector represented in the same way by A'P^. Thus the resultant at Pj is inclined to A'P^ at the angle , and is proportional to A'P.^, and so for other points. A' is thus the pole of a new spiral by which the resultants at other points may be found graphically. It is to be observed that, since a and n vary as the pendulum swings, the positions of E, C, and A' also change. For example, in the middle of a swing 6^ has its maximum value, and AG its least, while, since a = 0,

= irl% so that A' is at E, the position of which is that given by the construction when Q is the whole amplitude of deflection. By resolving the forces P^, R^ horizontally and vertically, and summing for the whole body, we find the components in these directions of the whole force exerted on the body in consequence of its internal connections and the action of its support. These "are partly purely internal forces, and partly forces transmitted through the body from the axis §§ 211, 212] IMPULSIVE FORCES. 395 of support. Since the purely internal forces equilibrate one another, we shall obtain by a summation of the forces described above the equilibrant of the whole system of forces . applied to the body by the axis of support, and therefore also the reaction of the body on the axis. Thus, for the total force in the direction of x, taken positive when taken along AE in Fig. 97, applied to the body by the axis, we have X = SP^ cos 0j — 2i2i sin 0j = ^w^r^dj^ cos 6^ — St«irj9^sin 0) = Whcoad.e-Wh8mP ^ '^'^^ (6) and for the total force in the vertically downward direction - Wh cos d.&'-Wg,]"" which give Jr= Wg 12 I 12 ^i° ^ '^o^ ^ ~ hd^ain 6, Y=-Wg- Wh&'cos O+Wg p^ sin^e. Besides the forces here calculated the axis may apply to the pendulum, a couple depending on the manner in which it is attached to the axis. An overhanging pendulum, for example, turning about an axle carried in a bearing before or behind the plane in which the centroid -moves, applies a couple to the bearing, and a reaction couple must be applied by the bearing to the axis to keep it horizontal. 212. Theory of Impulsive Forces. Impulse. An impulsive force is a force of very great amount F applied to a body during a very short interval of time t: a blow from a hammer, the impact of a common-shot on an armour plate, are examples. The smallness of t makes the change of momentum produced in that time by ordinary forces applied to the body, such as the force of gravity, vanishingly small in comparison with the time integral {^Fdt of the impulsive force F. The change of momentum 396 A TREATISE ON DYNAMICS. [CH. VII. produced by the impulsive force is equal to this integral, which is called the impulse; and though, as a rule, the manner of variation of F within the interval t is unknown, the value of the integral can be calculated, or observed experimentally, in many practical cases. We shall take some examples, and then consider how the equations of motion of a system are to be modified in the case of impulsive forces. 213. CoUision of Inelastic Bodies. Theory of File-Driver. We consider here collision between inelastic bodies. These are bodies which after impact remain in contact, so that two rigid bodies which impinge on one another move afterwards as one rigid body, or, if there be any relative motion, there is no elastic deformation of either to be extinguished by frictional resistance to vibrations. As an example, consider the impact of a wooden mallet on a chisel used to cut wood or soft stone. There is little or no elastic deformation, or "rebound," of the mallet, the momentum which it possessed before impact is distributed between the mallet and the chisel just after impact, and the kinetic energy of the system is then used up in the cut made. If the mallet rebounded from the chisel there would be a greater forward momentum of the chisel just after the blow, and that would be useful in making a slight cut in hard and resistent material like steel and granite, and for this reason it is found advantageous to substitute a steel hammer for the mallet. We shall return to the discussion of this example. Anything like a complete discussion of the impact of elastic bodies in- volves the theory of vibrations of an elastic solid, and therefore we do not enter on the subject in the present treatise. Consider now an inelastic "pile" — a beam of wood pointed at the lower end — which is being driven into the ground by successive blows of a hammer, or pile-driver, of weight W let fall from a height h above the head of the pile. The speed of the hammer at the beginning of the impact is \/2gh, if the friction of the guide and of the air be disregarded. Hence the momentum is then Wj2gh. § 212, 213] THEORY OF PILE-DRIVER. 397 Just after the impact the hammer and pile have the common speed v given by the equation W v = - w+w-^^ah (1) For the momentum of the pile has been increased from zero to wv, and that of the hammer has been diminished from Wjlgh to Wv. Hence we have 'W{'j2gh — v)= wv, which gives the equation just written down. We have in this case ~r -rn- \m^wv==-^.m (2) Now let the pile descend a short distance d in consequence of the blow. The energy expended is l{W+w)v^+{W+w)gd.. But if R be the space-average of the resistance of the material to the downward progress of the pile, the work done is Rd. Hence, inserting the value of v just found, we get 1 W2 or {R-(^w+w)g)d = .^gh (3) The average upward thrust R' on the hammer during the penetration rf is different from R. For that we have in the same way {R-Wg)d=w{J^"gh. (4) The quantity {W+w)gd is small in comparison with the right-hand side of (3), and therefore this quantity in (3), and Wgd in (4), may be omitted without seriously impairing the accuracy of the values of R and R'. From (3) it follows that the weight which must be applied to the top of the pile, so that it may descend by dead weight alone, is ^\W+wd^V- 398 A TREATISE ON DYNAMICS. [CH. VII. The method of dead weight is adopted for the sinking of caissons, which for several reasons cannot be driven down by blows from a hammer. It is to be observed that if, after a little penetration, a smaller resistance than R were offered to the pile, the dead weight would be too great and the pile would go down " with a run." The discussion here given for a pile is applicable to the driving of nails by a hammer, though here the action between the nail and the hammer is elastic. A wooden hammer does not drive nails well, even if it is made of very hard wood, for the head of the nail deforms the hammer face, and energy is lost thereby, which in the case of the steel hammer is utilised in causing the nail to penetrate the wood. 214. Energy-Change in Inelastic Impact. Advantage of Heavy or Light Hammer. The error, which is not uncommon, of equating the kinetic energy of the hammer just before impact to the work done in causing the pile or nail to penetrate the ground or wood is to be avoided. The first action is one of redistribution of momentum, and this has been effected before penetration to any sensible distance has taken place, so that the resistance of the material has not been sensibly brought into play. In the impact itself a loss of kinetic energy of amount takes place, and it is noteworthy that for a hammer and pile or nail, of given weights, it is always the same fraction of the whole kinetic energy just before impact. This is expended in deforming the head of the pile and, to a much less extent, the face of the hammer. It is important to observe that the energy lost bears to the whole kinetic energy available the ratio wl{W+w), which is nearly equal to unity, or to zero, according as W is small or great in comparison with w. Thus the hammer should have a weight W, great in comparison with that of the pile, or nail, for then very nearly the whole available kinetic energy will be utilised in causing the penetration desired. On the §§213,214,215] THEORY OF PILE-DRIVEU. 399 other hand, if w be great in comparison with W, or of the same order of magnitude, nearly the whole or a large part of the kinetic energy available will be expended in deforming the head of the body struck. Hence if the object is to indent, or fashion the surface of a body according to any pattern, by hammer blows, the hammer should have mass W stnall in comparison with the mass w of the body struck, as then the available kinetic energy is expended in producing the result j-equired. Thus a blacksmith uses a small hammer to give detail of form to the surface of a piece of iron, and employs a sledge hammer to drive a steel chisel through the material, softened by heat, when he cuts a thick bar in two, and also a sledge hammer when a bar is to have its cross- section much reduced or altered. 215. Duration of Impact. How far a Pile should be Driven. From (1), § 214, and (4), § 213, we can obtain the distance, a say, by which the pile is shortened in consequence of the blow. For the energy Wwghl{W+w) lost from the hammer in the impact must be equal to the work R'a done by the force R' exerted by the hammer on the pile. Hence _ Wwgh _w(W+'w) _, , , "'~(W+w)R'~ w^~^ ' ^ ' by the approximate value of R' from (4), §213. From this result we see that, according as W is great or small in comparison with w, the distance a is small or great in com- parison with d. The times t^, t^, t of traversing the spaces h, d, a can be compared. First we have t-^ = 'j2h/g. Again, the time- average of the speed during the interval t^ is ^v = y2ghW/iW+w), so that t^ = 2d(W + w)/W' V' ^^^ the angles which perpendiculars let fall from any point, xyz, to the axes of x, y, z, make with the axes of y, z, x respectively. The lengths of these perpendiculars are r^, yy, r^, and (8x, Sy, Sz) = {(R cos ^ - Q sin ^)rx, "j (P COS x-R sin x)ry, [ (6) (Q cos •\/r — P sin i/r)rz}.J The values of 0, X' ^ ^^® different for perpendiculars drawn from different points, but if the body be rigid, (p, X, yp- must be the same for all. The perpendiculars may therefore be supposed drawn from the centroid to the given axes. 217. Impulse applied to Compound Pendulum. Consider now a compound pendulum hanging vertically on its axle, or on knife-edges, and let it receive a horizontal impulse in the vertical plane through the centroid at right angles to the axis of turning. We can prove that, if the line of application of the impulse pass through the centre of oscillation, no shock will be applied to me axle or bearing by which the body is supported. We suppose that the duration t of the impulse is so short that whatever change of motion of a body is produced by an impulse applied to it is brought about before the body has moved through any perceptible distance, or turned through any sensible angle. Now the change of motion of the centroid of a body, whether rigid or not, effected in any time, is the same (§ 60) as it would have been if all the forces on the body had been applied to a free particle of mass equal to the mass of the body ; and so if the horizontal component of the reacting impulse at the axis be I' (measured as before by the time integral of an impulsive force), and v be the change of speed of the centroid in the direction of the im- pulse, we have Wv==I-I' (1) §§216,217,218] THEORY OF IMPULSIVE MOTION. 403 Again, the moment of the impulse 'about the axis must be equal to the angular momentum generated about the axis, so that if a be the distance of the line of application of I from the axis, ,, Ia=Wvh+Wk?i-' h, since vjh is the angular speed generated. Thus we have by (1). Thus we get /a= Wv^^^^= Wvl = (I-r)l, (2) / '=-i. Thus a a = l, I' = 0, and there is no shock to the axis. There is no vertical component of impulse called into play. The production of angular speed vjh about the axis calls into play a force towards the axis, of amount Whv^/h^= Wv^/h, on the body, and a reaction of the same amount on the support; but this being a force of finite amount, depending as it does on v/h, cannot in any very short time produce a sensible change of momentum. 218. Impulse applied to Bod on Smooth Table. Impulse on Pivot. Consider a rod of length 2h resting on a horizontal table: If the rod be of uniform weight per unit length and friction be absent, it will, if struck at one end A by an impulse along the table at right angles to the length of thi rod, begin to turn about a point B, at a distance of | from A. B is the centre of oscillation of the rod as a compound pendulum hung from the point A, and if the rod is struck by an impulse at G, directed as before, it will begin to turn about A. In neither case will there be any impulse or shock applied to an axle at the point of turning. The proof is similar to that given above for the compound pendulum. Let / be the impulse and v the speed of the centroid produced. Then if W be the weight of the rod, we have Wv = I. (1) If the rod turn about a point A at distance x from the end 404 A TREATISE ON DYNAMICS. [CH. VII. struck, the angular speed is v/(x—h). Hence, taking moments about the point A, we get Ix=Wv{x-h)+W^-^^> (2) or, substituting the value Wv of / on the left, and rejecting the common factor Wv, xh = ^h? (3) Thus we obtain x = %h, and the statement made above as to the point of turning is verified. Moreover, this result has been obtained on the supposition that no impulse, except that at the end, has been applied to the rod, and we infer that if there had been a pivot at the point B, it would have experienced no shock. But let a pivot be provided at a distance x^ from the end, so that the rod is constrained to turn about it, and let the impulse / be applied at the end as before. If /' be the impulse now applied to the rod by the pivot, in the direction parallel to that of /, we have Wv = I+r (4) and Ix.= Wv(x,-h)+W^^-^ (5) ^ '- 3 ajj — ft These equations, since cc^ is now fixed, enable us to deter- mine /' and V. We see first that if x^ = h, v must be zero ; this is obvious without calculation, for then the centroid is fixed and its speed v is zero. It will be seen that the equation reduces to /'=F^|^i=^, (6) which vanishes when x^ = ^h. For v we get the equation Wv= p^'(^'-^^) _/. (7) 3x\-6}ix^ + 4sh- Ex. 1. Verify that when .ri = |A the kinetic energy is a minimum for given speed u of the end struck. The angular speed is Jt/.r,, and the moment of inertia about the pivot is W{lh^ + (x\-/i)^}. Hence the kinetic energy is and this, by the ordinary criterion, is a minimum when a,\ = ^h. §§218,2)9] DOUBLE COMPOUND PENDULUM. 405 Ex. 2. Show that if the impulse / applied to the end of the rod be given, the kinetic energy is a maximum when Xi = ^h.'- . It will be observed that in this case u is not given but depends on the value of /. But the kinetic energy is still \ Wu^{l h^ + {x^ - hf}jx\, and since u=XjV/{x^^h), we have Wu^= Wv^x\l{x^-h'f. Thus, by (7), we get for the value of the kinetic energy, 1 (■__£?__ yi^ 2ljA2 + (.ri-A)2/ W We have seen that the reciprocal of the fraction in the brackets is a minimum when x-^ = ^h, hence the fraction is a maximum for that value of «i. But PjW is given, and so the kinetic energy is a maximum. These examples illustrate by particular cases a general theorem of maximum and minimum energy of a system set in motion by impulses, which we shall explain later. Ex. 3. An impulse /is applied at the distance x=^h from one end of the rod ; prove that turning will begin about that end. We have again for the motion of the centroid /= Wv, and if the rod begins to turn about a point in itself, or in line with it on the table, at distance X from the point struck, we have, taking moments about that point, Ix= Wv{x-\h)+ W\h^vl{x-^h). Substituting Wv for /, we obtain x{x-lh) = (x-lhf + lh\ or lh{x-lh) = lh\ that is fl;=*A. The rod therefore begins to turn about the farther extremity from the point struck. The student may easily experiment on this subject in any smith's shop or engineering laboratory. Let him take a uniform bar of iron, measure off two-thirds of its length from one end, and mark the point. Then, gripping it by the farther extremity from the mark, let him strike the bar forcibly against the edge of an anvil, at a point a little beyond the mark to allow for his grip of the bar. He will feel little or no jar from the blow. But if the point of the bar which strikes the anvil be much farther off, or much nearer the hand, the jar will be very unpleasant. Again, if the bar is held at a distance of twcs-thirds of its length from one end, and is then made to strike the anvil, there will be a very perceptible jar, unless the point of collision is the farther extremity of the bar. Ex. 4. Show that a uniform sphere, oscillating about a point on its surface under gravity, has a period equal to that of a simple pendulum of length equal to '7 of the diameter of the sphere. Hence explain why the cushion of a billiard table is at a height above the table equal to '7 of the diameter of a billiard ball. 219. Double Compound Pendulum. We take next the problem of two compound pendulums, one hinged to the other, like a bell and its clapper. To deal with this we take axes as in the last problem. 406 A TREATISE ON DYNAMICS. [CH. VII. and denote by .r, y the coordinates of the point of attachment of the second pendulum to the first (so that x^-^-y'-^V; where I is the distance of the hiuge from the fixed point 0), by hxll, liyjl the coordinates of the centroid of the first pendulum (distant h from 0) and by ^, -i) the coordinates of the centroid of the second pendulum. We suppose that the hinge is on the line from through the first centroid. We have x = lsm6, y=l cos 6, kvll=h sin 6, hy /I = h cos 6, $■=1 sin 6+a sin , where 6, 4> ^^re the angles which the line in each pendulum through the centroid and the hinge makes with the vertical, and a is the distance of the centroid of the second pendulum from the hinge. These lines are supposed to remain in one plane — the plane of vibration. We have then j±=/icos6. 6, -.ij= -hsin.6 .6, $=lcos 6. d + acos4>- 4'> V= -^sin 6. d-asin(t>. . ) + (lsin 9. 6' + asin <^. ^)(Zsin ^+asin <^) + fe^}, that is W.Aiv-^^)+ W^U=W,{Pe + {a' + k^i> + alcos{ct>- 6)0 + 4)]. (2) The total angular momentum of the system about is thus Fi(A2 + *i)(9+ W2{lW+(a^ + kl)4,+alcos(^ - e)0 + )\. The time-rate of change of this equated to the total moment of the forces of gravity gives an equation of motion W,{h^+kl)e+ W.,{Pd+{a^+kl)4, + alcosi - 6)0+^) \ (3) -al sin {?(^ -■»)}> and in taking the time-rate of change of this, we must regard x, y as constant. We thus obtain, as the reader may verify, Tf2{(a2 + ^|)^ + a?cos( 408 A TREATISK ON DYNAMICS. [CH. VII. 220. Small Vibrations of Double Pendulum. If now in the double pendulum we suppose 6 and to be so small that we may write 0, t/> for sin (9, sine)!), 1 for co8(i^-V'), and neglect terms in e^ \ we obtain (§ 219) (h^+k'^+:!^^i-^)e+-^^ai^+[k+^^i)ffe=o.j ^'^ Writing h, d, e for the coefficients in the first of (1), and o, pb,fiov these in the second, as they stand in each case, we get bd+d4> + e^ = 0,'\ (2) ce+pb4,+fd=o,j where p= Tiy W,. If now we put 6 = Amn(nt + a.), (j> = Biiin(nt + a.), and substitute in these last equations, we obtain bAn^+ dBn^-eB=^0, cArfi+pbBn^-fA which by elimination of A and B give {cd- pW)n* -{ce + fd)n^ + ef=0, (4) a quadratic equation for n? which, since cd> pb'^, has two positive roots. Thus, as before, we get two modes of vibration of different frequencies, one, the mode of greater period 2ir/%, in which both pendulums are deflected at the same instant in the same direction from the vertical, the other, of smaller period 2?r/»i2, in which they are deflected in opposite directions. Solving (4), we get ^^ = 2(crf-pfe^) {oe+fd + ^ice-fdf+ipem] which, since cd> pb'^, proves that n^ and m^ are real and positive. The complete solutions of the differential equations (2) are thus 0=Aiam(nit+OLi) + A2sm(n2t + cL2),\ /»» (f> = Bisin{nit + a.i} + BiSin (n^t + a^).) The constants A, B for each frequency are connected by a fixed relation, namely, Bi = k,Ai, B^ = k2A^. Multiplying the first of (3) by pb and the second by d, and subtracting, we obtain the ratio BIA={fd-{od- pb^)n''}lbpe, so that ^P'^ \ (7) §§ 220, 221] COMPOUND PENDULUM. 409 ^^"' C^ = 2^,{-(««-/«) + V(c«-/<^?+46W}-] W Thus K, is essentially positive and k^ essentially negative. Multiplying these values together, we obtain «lK2=-/- : (9) ep It is important to notice that if ce=fd, the values of kj and kj are eaual in numerical value but have opposite signs, and further that, when the relation ce=fd is not fulfilled, the imposition of the value 1 on either of the coefficients /o> ^0) 4>a- We obtain easily from equations (6) of § 220, 6o = Ai sin a.i + .4asin aj, q = KiAi sin a.] + K3.:42sin 0-2, 6ij=7iiAiCos a-i+n^A^coa 0L2, <^(,= WiKi^Ij cos (Xi+ra^Kg-^ 2*^0® "-zi from which to find /lisinaj, .4.2sino(.2, ^icoso-i Thus we have K2-K1 I. Ml ' M2 I (]) - (c/),^ - Ka^o) cos Wi« + ((^0 - " i^o) cos n^t y,\ with for (j> the same expression, modified by multiplying the first and third terms by Kj, and the second and fourth by Kj. Hence (2) +(kj - l)|4zJll^ sin «2i; + (<)bo - Ki^o) cos W2AI If we impose the condition that" ^o=(/)o=0, we get K.2 - Kj I. », ' 112 J with the same expression for , modified by multiplying the first term by Kj and the second by k^. Also in this case, K2 - Kj I. ?i, n^ J 410 A TREATISE ON DYKAMICS. [CH. VII. From these last results we draw at once the conclusion that if both pendulums be started with the same angular speed 6'o = <^o> (/'-^will remain zero, if k^ = 1. The coefficient k^ will then have the value -f/ep (see (9) § 220). These are therefore the conditions that the two pendulums should continue to vibiate as one, if started together as one pendulum. Such an arrangement of a bell and its clapper would, if started as here supposed, fail to give any relative motion of the parts, and the bell would not ring. We shall, however, consider the failure of a bell more fully presently. If in (2) we assume that K] = 1, 6o = ^> '"^'' ^° ""'' suppose that (f>Q is zero, we get ; a (^ _ ^ = 22 gj J, ,j^^ ^ ^^ gQg ^^^^ ^4^ ^2 of which the maximum value is v(^„-6'(,) +«i!o/i2- Thus, if the arrangement is such that Ki = 1, then, although the second pendulum may have an initial deflection <^q, the relative deflection cannot exceed the maxiniupi value here stated, which it will be seen approximates to Q if 742 b^ great, that is, if the period of the second pendulum vibrating alone is short. Thus the relative deflection remains very nearly ^q, and the two pendulums still practically vibrate as one. This result still holds when 4>a=0. Ex. 2. Prove that if the condition ce=fd is fulfilled, the values of K,, Kg are i^fjpe, -'Jfjpe, and of jij, n^ are 'Jfd-'XpefWl^cd-plfi, \lfd+\l pefb^l*Jcd— ph'^, respectively. Find the finite equations of motion for this case. Ex. 3. 11 p(= WJWi) be small, prove that to a first approximation ni = 's/flc, n^=^ejd, and to a second approximation n^=-J\fd{ce -fd) - pefb^/ice -fd^cd-pb^), n^ = 'J{ce{ce^fd') + pefV'}l{ce -fd){cd- pS'). Ex. 4. A rod of length 2a is suspended by a string of length I attached at one end of the i-od, from a fixed point 0. Show that if the inclinations of the string and the I'od to tlie vertical at time the 6 and (j), the equations of motion are ld+acos( -6)4) -a sin(<^ - 0)4>^+g ain 6=0, ia4, + lcos(-e)e + lsin(-e)e^+gsin4> = 0. Hence find the equations for small motions, and show that the equation of frequencies is (an^ - 3g)(ln^ - 4g) - 9g^ = 0. Show that this equation has four real roots, and write down the complete integral equations. §§221,222,223] THEORY OF SEISMOGRAPHS. 411 222. Driving and Driven Pendulums. Forced Vibrations. The corresponding values of K], n'2 ^^ *^^ ^^- ^ '^^^ approxi- mately Ky = hfj{oe-fd), K,^= - (ce -fd)/bpe. The value 'Jf/c of ^lis very neai'ly \hg/{k^ + ^f), that is, >/g/L, where L is the length of the equivalent simple pendulum for the fii'st pendulum oscillating alone, and \/eld is the same thing for the second pendulum. Thus the two fundamental periods of the system are simply those of the two pendulums, each hung up alone. If, then, p= WJVI\ be small, and the double pendulum be started from rest with da = o=0, and <^q=0, by imparting an initial angular speed 6^(1 to the first pendulum, we have, by (1) above and the approxi- mate values of Kj, Kj, ^=V^^osinVJ^ + (S^V!«inVI^ 0) Since p is here supposed small, and it is assumed that there is no approach to fulfilment of the condition ce—fd=0, the second term makes onlj' a small addition to the first term, which is the main value of 6; but so far as the second term goes it varies in the period 27r\/d/e. The value of cf> is obtained from that of by multiplying the first term by K; and the second by kji that is, ^=AAf«WJ-^V!^W^' (^) Of course, if the effect on the period of oscillation is required with exactness, the more closely approximate values of n^, n^ must be used. If the second pendulum has a very short equivalent simple pen- dulum, vc// may be so great in comparison with iJdje, that the main part of <^ is represented by the first term, and this varies in the period of the first pendulum. This, of course, is what we should anticipate without calculation ; the massive upper pendulum acts as driver, and the small attached pendulum is driven in the period of the other, and acquires a steady amplitude of vibration of considerable amount, while the vibration of the driving pendulum is but little affected. The term, however, in the value of ^, which has the period of the second pendulum, is great in comparison with the corresponding term in the value of Q- It is to be observed that here fd or ce = Ki6, along with which will be freejvibrations of the seismograph itself, in its much longer period ^-n-Jd/e. All these will be registered by the apparatus, but there is no difficulty in distinguishing those due to the earth or building from those of the instrument. We have thus <^ = /, d=4>f and a — M, al-'rl?^ = liy. The length of the equivalent simple pendulum is then for the first (the bell) vibrating alone, (A" + ^j)//i = a + /;^/a = a + (aZ + /i;2)/a = Z + a + F/a = ; + Z, §§223,224,225] FORCED VIBRATIONS. 413 where L is the length of the equivalent simple pendulum for the second (the clapper) vibrating alone. The two pendulums if started together with B=^, and 0=<^, will thus vibrate so that Q remains equal to <\>, if the distance of the centre of oscillation of the second from the point of suspension of the system, when the centroids are in line, is equal to the distance of the centre of oscillation of the first pendulum from the same point. Thus, if the fii'st pendulum is a bell and the second its clapper, and the conditions of starting are as stated, the bell will not ring. One way of curing a bell from behaving in this way would be to lengthen its clapper considerably. This is said to have been done for a bell in the Cathedral of Cologne. 225. Forced Vibrations. The subject of forced vibrations referred to above is of great importance. Examples of it are found in the phenomena of the tides, which are oscilla- tions of the water on the earth's surface and of the earth's substance, produced by the periodic action of forces which are not to any appreciable extent controlled by the earth itself, in such a way as to enable tidal vibrations to have any of the free periods of such disturbances. A ship is made to vibrate by the revolution of the more or less unbalanced parts of the engines, and it is made to pitch in the period of the waves it passes over, and to roll in the period of the waves that pass under it transversely. In a great number of such cases it will be seen that the control of the driven body by the driving oscillator is absolute : the energy of the latter is practically unlimited, or, at least, the part abstracted by the driven body is so small a fraction of the whole that no modification of the driving oscillations is noticeable. It is otherwise, however, in such cases as a pendulum driven by another pendulum of energy of motion comparable with that which the former possesses when in full swing. Take, for example, a beam and scales, in which the beam oscillates about its knife-edges (when the scales hardly swing about their suspensions), in nearly the same period as that in which the scales swing alone. When the beam is set oscillating the scales gradually increase their pendulum swing about the extremities of the beam, which in its turn comes to rest, to begin oscillating again as the scales in their turn become the driver, and so on. Thus, if the system is left to itself, a continual backward and 414 A TREATISE ON DYNAMICS. [CH. VII. forward transfer of energy takes place, from the beam to the scales, from the scales to the beam, and so on, until all the energy has been transformed into heat by the friction which retards the motion throughout. This is the problem referred to in § 108 as having been discussed by Euler. It is obviously an example of the double pendulum of which the theory is given above. The reader may now work it out for himself and trace the energy changes, leaving friction out of account. 226. Simple Pendulum with Vibrating Support. As an example of forced vibrations, we take first Q__$_P_C the case of a simple pendulum hung from a point P, which is constrained to vibrate in a horizontal direction about a mean position (Fig. 98), so that its distance x from that point is given by the equation x = asm'pt. Let I be the length of the cord, m the mass 1 of the bob, the angle which the thread \ makes with the vertical at any instant, and ^_^ P the pull which the cord exerts on the bob. Fio 98 "^^^ horizontal distance of the bob from at time t\fix + l sin (p, and the vertical distance I cos (j). Hence the equations of horizontal and vertical motion are m m-Tj^{x-\-lHm 0)= — Psin ^)= —Pcoatp+nng.) Multiplying the first of these by cos , the second by sin 0, and subtracting the second product from the first, we eliminate P and obtain l^ + g sin (p= — ,TCOS0, (2) which, if is always small, may be written ^ + J<^=-1 (3) §§225,226] FORCED VIBRATIONS. 415 But .(.: = — p2fl3, and hence this equation of motion is really ^+^^=p^jsmpt (4) We now assume that ^ = A sin pt is a particular solution of this differential equation. Substituting, we obtain A(g/l-p^)=p^a/l, so that ^=(pV)/(^/^-p')■ But if Tj be the period of the forced oscillation of the point of suspension, and T^ the natural period of the pendulum, p' = 4>Try Tl, gjl^'^ir^lTl, arid we get A ^^'^ ^^ (5) g{T\~'ry T\-T\l Thus, adding the complementary function to the particu- lar solution, with the value of A just found, we obtain 4>= - ,j,J^j,2. asmpt + BiSmy^'j-t + B^cosyJ^t ...(6) or ^=- -,y2--y2Tasm^^+C'sm(^^^-aj (7) where C and a. are constants. This, as the reader should notice, agrees with (5) of § 223, for the quantity Id {I has there a different signification) which occurs in the first term of (p in that equation is clearly the present a ainpt. It is important to observe that, if T^'^T-^, the forced vibration term in the solution is opposite in phase to the exciting vibration, and that the amplitude of the latter is altered in the ratio of 47r^ to g{T\ — T''^, a ratio which is greater the more nearly the two periods coincide. Examples are a plank of wood, which when floating in water has a very short free period, and follows at once the motions of the waves in a seaway, and a vessel, the period of rolling of which is greater than the half period of the waves in a seaway, and which therefore oscillates in the opposite phase to what must be regarded as the exciting oscillation in this case. Other examples are the driven pendulums discussed in § 107. 416 A TREATISE ON DYNAMICS. [CH. VII. 227. Agreement of Natural Period with Forced Period: Resonance. When, however,, the two periods — the natural period T^ of the pendulum and the impressed period T^ given by the motion of the point of support — coincide, the particular integral which we have assumed is not applicable. We now assume that (j> = At cos pt is a solution. Substi- tuting in the differential equation, ^+g^/l = Msinpt (M^p^a/l), we find that A= — M/2p, and therefore we have 0= —a^tcospt + B-^siny^.yt+B.^cos^J^t, with p = \/g/l,, or, by the values of T-^, Tg given above, which are now equal {=T, say), ^ 47H'a, . /27r, 7r\ „ . /27r, \ ,.,. ^=-^-tsm[-^-t-^) + Cam\^j,-t-a.j (1) Thus the "forced" part of the oscillation is a vibration of frequency ^/2Tr, a quarter of a period behind the exciting vibration in phase, and of amplitude increasing uniformly with the time. The exciting vibration is given by x = asmpt, and therefore at t = we have x = 0. The forced part of

, and so the equation of motion is «'+J=(^+'^)=Mi+w>=«- (^> Thus the period is changed from •l-'Jl^FjE to •2-s'^-i'jE{\-\-wk?l WK^), and the frequency f rom / to /(I -t- «'F/ WK'^)-. Thus the watch goes more quickly. A pocket chronometer belonging to Ai-chibald Smith of Joi"danhill (presented by the Admii-alty for his work on the Deviations of the Compass in Iron Ships) when thus suspended was found to gain 1 second in 1299. In this case the i-atio of fi-equencies was 1300/1299, and therefore ^wl^/WK^ was appi-oxiniately 1/1299, that is, the moment of inertia of the watch was about 649 times that of the balance-wheel. [Lord Kelvin, "On the Rate of a Clock or Chronometer fis Influenced by the Mode of Suspension," Popular Lectures and Addresses, vol. ii.] 231. Ex. 2. Watch hong by Bifilar Suspension- Theory of Bifilar. A watch is hung with its face horizontal by two thi-eads each of length I attached at their upper ends to two points on the same level at a distance 2a apart. The lower ends ai'e sym- metrically attached to the watch at a distance 2r apart. It is required to find the effect of the vibi-ations of the support on the rate of the watch. It will be clear that when the watch is hanging in equilibrium each thread is inclined to the vertical at the angle sin~' {{a - r)jl\. When the watch is turned thixmgh an angle <^ about the vertical p,Q iqq^ thiough its centre, the inclination of each thi-ead to the vertical is sin"'^.Bj^ Hence if P be the pull applied by each thread the two thi-eads together apply a couple to diminish 422 A TREATISE ON DYNAMICS. [CH. VII. e, of which the moment is P. ABjl .asmLOAB (Fig. 100). But sin L OAB/sia =r/AB, and the moment of the couple is 2Par sin (ft/L Now, if, as we shall suppose to be the case, I be great in comparison with r and a, we have 2P=( W+w)g, since the vertical acceleration will then be small. Here W is the weight of the watch (without the balance) and the sling, and w is the weight of the balance. Hence if K be the radius of gyration of the watch and sling, and Ic that of the balance, the equation of motion, if the watch is not going, is {WK^+wF)4>+{W+w)^g s.m4>=0 (3) Thus, for small motions, the frequency is n/( W+ w){argll)j{ WK^ + wk^j^w. We shall denote this by F, so that 4^^F^WK^+wk') = {W+w)arg/l (4) If now the balance be vibrating, and its deflection from the position which it occupies when everything is at rest be 6, the equation of motion of the watch, etc., without the balance, is WK^ + (W+w)^g4,+i:(^-&) = 0, (5) and that of the balance alone is whW-E{-6) = 0. .'(6) If the balance were vibrating alone the equation of motion would be wk^d+E6=0, and the frequency of vibration would be 'J'EjwWl'2,Tr=f, say. Thus E^Air^fwkK The two equations of motion (5) and (6) are satisfied by d=Asm'2i-Knt, (^=5sin27r?i< (7) (so that n is the frequency), provided the equations -iTr^ii^WK'^^+{W+w)'^g4> + E{4>-e)=o\ (gs - iir'^n^whW - E{4> -e)=0,] hold simultaneously. The necessary condition for this is B^ E ^E-iTT^wlV ,g. ^ -4Tr^WEV+{W+w)^g+E ^ Substituting for E and ( W+w)argll, the values 47ry2w^2 and 4Tr^F^WE^+wk^), found above, we have B^P-n^_ ^^^2^ A f -WKhl^^-F^{WK'^^wl■^)■^wlt?■p^ ^ ' §231] FORCED VIBRATIONS. 423 If 1+e be put for l + iok^j \rK% equation (10) may be written in the form {n^-0- + e)F-^}{n^-(l + e)f^}-eF^P(l + e) = O. (11) This equation in n^ has two positive roots, one between + co and the greater of (H-e)i^ and (l+e)/^, and another between the smaller of these and 0. For when n^=+a:i, the left-hand side is positive, when n^={l + e)F^, or n^ = {l+e)f-, it is negative, and when n^ = 0, it is again positive. We see therefore that if n be greater than F'Jl + e, or greater than/\/l-f-e, it must be also greater than the other, and the watch gains. But (10) shows that then B/A is negative, that is, the watch and balance ai'e then deflected in opposi te di rections at each instant. On the other hand, if n be less than F-JY+e, or less than f\/l + e, it must also be less than the other, and the watch loses. Then AjB is positive, and the watch and balance swing in the same direction at each instant. It is important to consider what will happen if the bifilar suspension is held at rest while the watch goes, and is then left to itself. The discussion of the analogous case in §107 above answers this question. The watch becomes the driving pendulum, and we see that if the natural period of the bifilar be greater than that of the watch balance, the two vibrations, that set up in the bifilar arrange- ment and the vibration of the balance, wUl be in opposite phases and the watch will gain. This case can be ari'anged for by placing the upper points of attachment of the threads sufficiently close together, so as to make the value of F', which varies as a, small enough. The mode of vibration is shown in Fig. 44, where the upper pendulum represents bj"^ analogy the watch balance, and the lower pendulum the bifilar pendulum. On the other hand, if the natural period of the bifilar arrangement be smaller than that of the watch balance— and this can be arranged for by placing the upper ends of the threads sufficiently far apart — the vibration set up by the going of the watch, and that of the balance will be in the same phase, and the watch will lose. The mode of vibration is represented by the diagram of two pendulums in Fig. 44, where, as before, the upper pendulum corresponds to the watch balance, and the lower to the bifilar pendulum. This last result is of great importance in its bearing on the proper mode of supporting a clock which is intended to keep accurate time. Very frequently the supporting fi'amework from which the pendulum is suspended is not sufficiently massive, while it is rigid enough to have a short period of free vibration. The result is that the going of the clock is influenced by the mode of suspension just as that of the watch is in the second case just considered. Ex. 3. Prove that if the watch be hung by the ring from a nail (as a watch under adjustment sometimes is in a watchmaker's shop) so that it oscillates like a compound pendulum under the influence of 424 A TREATISE ON DYNAMICS. [CH. VIl. the vibrations of the balance, the equations of motion are W(K'' + h^)4,+{ W+w)gh4> + E{4> - e) = 0, where h is the distance of the common centroid of watch and balance from the point of suspension. Hence show that the frequencies (values of n)ot vibration are given by the equations rf/^ _ f-n^ ^4> where F^=(W+ w)gh/{ fVK" +wk'' + { W+ w) W\ ^ifi, and p= E/ATrhvk^ are the squares of the natural frequencies of free vibrations of the watch hanging on the nail with the works stopped, and of the balance vibrating with the watch at rest. Approximate values of n^ are p + (wkyWK^)P/(P-F^), and F^-(wP/WK^)PI{f-F^). The frequency of oscillation and beat of the watch is n in the two cases. It will be noticed that iu all these problems no account has been taken of the effect of the motions of other parts of the watch than the balance, e.c/. of the escapement, etc. These motions must, of course, affect the results to some extent. Ex. 4. A carriage of weight W, mounted on side springs at a distance b apart, oscillates about a longitudinal axis mid-way between • the springs and on a level with their top : if the c.o. of the oscillating body be at a height h above the springs, and its radius of gyration about a parallel axis through the centroid be k, and the springs be compressed a distance c when the load upon them is in equilibrium, find the period of small oscillations. If the compression be x at any instant during an oscillation, the return force of the springs on one side will be J Wx/c. One spring will be under compression, the other under stretch, and therefore the carriage will be acted on by a " righting couple," due to the springs, of moment ^ Whxjc. The angle of inclination is then 2a:/6, and if we call this 6, the moment of the couple is ^Wh'^Ojc. Besides this a couple is applied in consequence of displacement of the c.o. This has moment Wgh sin 6, and tends to increase 6. Thus the equation of motion for small oscillations is The period of oscillation is 'iir'Jic(h?+k''')l{b'^-Ach)g, and the length of the equivalent simple pendulum is {k^+k^)l(b^l4:C-/i). The period is thus greater the greater h and the greater c ; but b^ must be greater than 4c/i to ensure return to the equilibrium position. 232. Dependence of Steadiness of a Vehicle on Period of Vibration. For steadiness, a long period of free vibration is essential : a carriage mounted on stiff springs (that is, §§231,232,233] FORCED VIBRATIONS. 425 for which c is small) with its c.G. low, will vibrate in a short period, and its motion will be unpleasant. Thus, if the carriage is hung low on the springs, these must not be stiff or the motion will be very uneasy. Care, however, must be taken not to turn corners quickly if the C.G. is high, otherwise the carriage may capsize. [See § 190.] The C.G. of a locomotive, or of a railway carriage, is made fairly high to ensure easy running; the righting moment of a ship, for small angles of heel to one side or the other, is usually made so small, that in a moderate sea the period of rolling is long. But the ship is so con- structed that, as the angle of heel increases, the righting moment increases more rapidly, so that there may be no risk of capsizing in a heavy sea, or, when a succession of waves passes transversely under the ship, the period of which may nearly coincide with the free period of rolling of the ship. Lord Kelvin's compass card is made exceedingly light, and most of the weight is distributed round the rim, which is kept in shape by radial silk threads under tension. The card has thus a large moment of inertia, and there- fore a long period of free vibration, thus ensuring great steadine.ss. 233. Pendulum with Point of Support in Vertical Vibration. If the point of support is subjected to a vertical vibration instead of a horizontal one, we can write down tlie equations of motion in a similar way to that used in § 226. We have in this case y = asmpt, and the equations of motion become then 7n(y — lsia^. ^ — icos0. ^^"°^-^. -(2) v(%^— j3^)^+/cy u^— p^ + zcptanoc For the equation found by substitution gives on the left terms in sin(pi+;8), cos(j3i + y8), the aggregate of which are equal to 'p^asa\{jpt-\-cC)\ and since the equation must hold for all values of t, we are entitled to equate the coefficients of sin^f, cos^i on the two sides. This process gives the values of tan^S and A written above, as the * Equation (3) belongs to the class of linear differential equations in which the coefBcients of the terms are harmonic functions of the inde- pendent variable. This class of equations is discussed in various treatises. §§233,234,235] EFFECT OP FRICTION ON RESONANCE. 427 student should verify. We have then only to add the complementary function to complete the solution, which is therefore, if ni? > \k^. i= p^a ■- sin (pt+^) +e-i'"(Bi sin n't + ^^ cos n't), (3) where n' = \/n^—{K^. The condition n^'^lK^ must hold if the body is to be capable of vibrating about the position of equilibrium when left entirely to itself after displacement. If Ik^ > n^, the body will, when left in the displaced state, Fig. 101. gradually lose its displacement according to the exponential law ^-KKW^-'-in'')t^ thus losing it all in an infinite time, but the greater part of it in a moderate interval of time. As it is, the free oscillations once started are gradually wiped out by the exponential multiplier e"*""* in the manner shown by Fig. 101. 235. Resonance modified by Friction. Tidal Example. It will now be seen more clearly what happens when the periods 2'7r/p and 2Tr/n, of the exciting vibrations and the free vibrations, coincide. The forced vibrations do not, as might at first sight appear from (6), § 226 above, become infinite in amplitude ; the amplitude approximates to pa/K, 428 A TREATISE ON DYNAMICS. [CH. VII. as p and n tend to coincidence. Thus, according to the canal theory of the tides, with friction left out of account, the excess of the natural period in a canal parallel to the equator, above that of the forced tidal wave at the place, causes the tides to be inverted in low latitudes, while in sufficiently high latitudes where this excess has become negative, the tides are direct, and at the latitude of transi- tion the tides are infinite. The effect of friction is to modify these results profoundly, and to bring them into something like agreement with actual fact. What takes place is this. It will be seen from the equations found above, that- tan (oc - /3) = ^^^^ . By comparing the values of t for which sm{pt-\-a) and sm{pt + fi) are equal to unity, it will be found that at the equator, where n<^p, the phase of the tide is behind that of the tide-producing action by an angle oc — y8, lying between '7rf2 and tt ; in other words, high water occurs later at any place than the maximum of the tide-producing action, by the time-interval corresponding to this angle. But without friction the angle would have been tt, and therefore, relatively to the state of affairs, with no friction, the existence of friction has set the phase forward by an angle in value between and •7r/2. As we go to higher latitudes, where still n found above. This enables the pendulum to go on turning, until, at a deflection Q, the whole kinetic energy has been turned into potential energy, by the raising of the pendulum to a higher level on the whole. The centroid has been raised through the vertical distance /i(l— cos0), and the bullet through a height /i'(l — cos 0). The potential energy is thus §236] BALLISTIC PENDULUM. 431 in the same units. Hence we have the energy equation 1 w^vW^ \ 2 FFTii)P=(^^+^'^')^(l-^°^^) I (2) and so we get ,.2_ . ( Wh+wh'){ WB+wh'^)g sin^e fg) The value of 1^ has been, it is understood, previously found by allowing the pendulum to oscillate through a small angle about its knife-edges, and determining its period of oscillation, with and without a massive cylinder of weight TTj, which can be attached below the bob with its axis of figure parallel to the knife-edges in the plane through these and the centroid. This cylinder can be made fairly long and thin, so that its diameter may be neglected. If T, 2\ be the periods of the pendulum alone and with the cylinder attached, and A^, the distance of the axis of the cyhnder from the line of knife-edges, we have, by the theory of the compound pendulum, ¥lh = ^T VV, (/<;2 + W^h\l W)/(h + W^hJ W) = gT'y 4>Tr^. Thus we have two equations from which to find k^ and h. When h is found we can use the value k^=ghT^/4!Tr^, in the equations for v\ If I be the length of tape drawn out, we have 2L sin ^9 = 1 or ain^d = l/2L. Making these substitutions in (3), we obtain for the calculation of v. For the sake of the example we have worked out the exact value of v^, with neglect, of course, of the resistance of the air, which it is difficult to estimate. In view of this neglect, and of the smallness of the ratio of w to W, we may leave out of account the effect of the retention of the bullet in the pendulum, and obtain, ^ = 2^^^ be the coefficient of friction. Prove that if the flywheel is left to itself it will come to rest after making Tr^W^/lSOOog' sin ^ turns in Trk^JV/QOOagsmip minutes, where k is the radius of gyration of the wheel about the axle. [Ex. 5, § 204.] 5. A truck consists of a framework with the wheels and springs, which carries a box above it hinged along the front of the truck. The box is filled with material so that it may be regarded as a uniform rectangular block of length 2a, height 26, and mass M, with its centroid G* at a distance k from the line of hinges. If the truck be suddenly stopped, find the speed so that the box may just turn over. Prove that if the box thus turns over, the horizontal and vertical components of force on the hinges vanish when the plane through the line of hinges and G is inclined at the angles sin~i(2/3) and sin~'(l/3) respectively, and that the total force- on the hinges has a minimum value J\/7/ll . Mg when the angle is sin~'(20/33). 6. A pulley (weight M) has a fine cord wrapped round a groove (radius a) in its edge and its middle plane is coincident with that of a fixed vertical pulley over which the cord is passed in a groove (radius b). The free end of the cord carries a weight, M', and the parts of the cord depending from the fixed pulley are vertical. Find the motion. G.D. 2 E 434 A TREATISE ON DYNAMICS. [CH. Let i//f 2 be the m.i. of the first pulley about its axis «iF that of the fixed pulley, 7, T' the forces applied to the latter, by the cord, a., al the downward linear accelerations of the movable pulley and the weight respectively. Let also (u be the angular acceleration of the first pulley. The equations of motion are MK'^ih=Ta, Ma. = Mg-T, M'a; = M'g-T', mFa.' = (?"- 7)62. The reader may verify that since u)a — a.'=a., these equations give «.= hi ' * ■ {M'a.^-(M-M')K^)g {M+M')K'^ + M'a^ + 'n£{K'^ + a?) yand T are determined by T=M{g- 2t«/ Jo vinr§' with a final angular speed >J%afl{a^ +}!'■), where 2a is the breadth of the door, and h the radius of gyration about a vertical axis through the centroid. 13. A uniform rod is turning (without friction) about one extremity on a horizontal table and drives before it a particle of mass equal to its own, which starts from rest indefinitely near to the fixed extremity of the rod : show that when the particle has described a distance r along the rod, its direction of motion makes with the rod the angle tan-»e=>fc/\/r2TP. Why does the particle move outward along the rod ? 14. A uniform solid spherical ball of mass m and radius a is at rest in a cylindrical garden roller of radius b, when the roller is seized and made to roll along the level with uniform speed V. Find the motion of the ball, supposing that it does not slip on the roller. Prove that the inclination 6 of the line of centres to the vertical varies as does the inclination to the vertical of the thread of a simple pendulum of length |(6-a), and the ball will lose contact with the roller when d=^{lO-1V^lih-a)g}. Hence find Fso that the ball may just go completely round. 15. The wheel of an Atwood's machine is supported by placing the two ends of its axle in the well-known manner on two pairs of over- lapping friction wheels or rollers : to work out the theory of the machine and explain the action of the rollers. 436 A TREATISE ON DYNAMICS. [CH. Let L be the moment of frictional forces applied to the axle (radius a, moment of inertia MK^) of the wheel, T, T (T> T') the pulls applied to the rim of the wheel by the cord on the two sides. Hence if (2 = angular speed of wheel at time t, MKm={T-T')R-L. Let the sum of the frictional couples on the axles of the rollers be L', and the sum of the moments of the rollers about their axes 4/t^2. The angular speed of each roller is Q,Rlr, and therefore Elimination of L between these two equations gives (mK^+4^¥ ^) il^{T- T')R- L ~. But if a( = /Jf2) be the linear acceleration of m downwards and m' upwards, {m + m')a=(m-m')g-{T-T'). (m — m') a — L'—T-. Hence r- - ''^ If the radius a of the axle be small in comparison with both ■/■ and B, the eifeot of the frictional couple L' becomes negligible. The rollers therefore prevent friction from causing any sensible dissipation of energy. In many pieces of mechanism ball-bearings are used for this purpose. The theory of friction rollers here given requires modification for such bearings, since the balls are displaced bodily as they roll in the ball-races ; but the action is similar. The couple L' is avoided by having no bearings for the balls, but on the other hand the motion of the balls is retarded by friction in the " races." In Atwood's machines, as usually made, the ends of the axle are enormously too thick. A short length at each end should be turned down to the thickness of a darning needle. 16. Find the length of the shortest equivalent simple pendulum for a uniform solid hemisphere oscillating about an axis parallel to the base in a plane through the oentroid perpendicular to the base. 17. A uniform rod of mass m and length 2a is hung from a fixed point by a fine cord of length I attached to one end, and the system moves in a vertical plane through the fixed point. Find the exact equations of motion, and prove that the equation of frequencies (m/Sjr) for small oscillations is aM -(/(4a + 3l)n'+Sg^=0. 18. Solve Ex. 17 when the string is replaced by a uniform rod, of mass fji, and length 21, to which is freely jointed at the outer extremity the rod of mass m and length 2a. vn.] EXERCISES. 437 19. Two uniform and equal rods AB, EG, freely jointed at B, are moving forward in line with speed v, the direction of motion being perpendicular to the lengths of the rods. If A is suddenly fixed, show that the speed of the middle point of AB is immediately reduced to -^v, the angular speeds of AB, EC are made 94>/14a, - ZvllAa, respec- tively, and the kinetic energy is reduced by \ of its original value. 20. Determine the speed acquired by a block of wood, weighing W lb., free to move in a straight line, when struck directly by a bullet weighing w lb. moving with a speed of v feet per second ; and prove that if the bullet is imbedded a feet, the resistance of the wood to the bullet, supposed uniform, is in Pounds Wwv^l{W + w)'iga. Prove also that the time of penetration is %ajv seconds, during which time the block travels through a distance of waj{ W+w) feet. 21. A railway carriage of weight W and moving with speed v impinges on a carriage of weight W at rest. The force necessary to compress a buffer to the full extent I is equal to the force of gravity on a weight w. Assuming that the compression is proportional to the force, prove that the buffers will not be fully compressed if v'^<'2.wgl{\IW+\IW'). If the yielding of the backing against which the buffers are driven be neglected, prove that when v exceeds the limit stated the ratio of the final speeds is {Wv- n/2w W'gll{\ + W'l W)l{ Wv + \/2w%?/(l+ F/Tf' }. Let the speed v be just sufficient for driving the buffers home, and let the common speed of the carriages when this is done be v'. Then by the principle of energy '{W+W')v'^=Wv^ — 2mgl. But ( W+ W')v'= Wv, and so v'='2wgl{l/W+l/W'). A smaller value of v would not give complete compression of the buffers. If this value of v be exceeded, let the final speeds be v^, % Then finally the kinetic energy is what it was at first, and so Wvl+ W'vl= Wv\ Also Wvi+W'v2=Wv. Hence if p=i;i/'Bi we get W{ Wp^+ W')l(Wp+ F')2 = 1, or p = (W- W')/2W, which is zero when W= W. But at the instant of complete compression of the buffers we have ( W+ W')v'^= HV - mgl, ( W+ W')v^= Wv, and therefore Wv^^Zwglil + W/ W'), W'v^=2wgl{l + W'l W), and the ratio can be written as stated. 22. A thin lamina moves, without rotation and unimpeded by friction, in contact with a horizontal plane, when a point A at distance X from the centroid is suddenly fixed. Prove that the speed of the centroid is changed to vx^ sin 0/(k^ +3:^), where v denotes the original speed of the lamina, k its radius of gyration about a vertical axis through the centroid, and the an^le between the original direction of motion and the line from the centroid to the point A. [The angular momentum about the point of space with which A coincides at the instant cannot be changed by the fixing.] CHAPTER VIII. ROTATIONAL MOTION. 237. Motion of a Rigid Body about a Fixed Point. When a rigid body turns round an axis every point of the body receives a displacement in a circle, the centre of which is on, and the plane of which is at right angles to the axis : these displacements are all in the same direction round the axis, and are proportional to the distances of the points from it. An example is the turning of a wheel about a stationary axis. The particles of the body retain the same configuration relative to one another, since clearly the particles in any plane whatever containing the axis retain the same con- figuration relative to one another, and the particles in any plane perpendicular to the axis remain also in the same relative positions as the plane turns. At any instant while such a displacement is taking place, all the particles have speeds, in the coaxial circles which are their paths, proportional to the radii of these circles; that is, the perpendiculars to the axis from the different points are all turning with the same angular speed in the same direction. We can prove that any displacement of a rigid body, one point of which is fixed, can be effected by a rotation of the body about a definite axis passing through the point and fixed in the body. Describe a spherical surface in the body with the fixed point as centre, and let the displacement be one (however effected) in which points A, B oi a, spherical sheet of the body (centre 0), are i ! §§237,238] MOTION OF A RIGID BODY. 439 carried to A'B'. [The student should construct a suit- able Figure.] The different points of the sheet do not alter their relative positions. Join A and A', B and B', by arcs of great circles, and through the middle points G, D of these arcs draw great circles on the spherical sheet meeting AA', BB' at right angles. These will meet in two diametrically opposite points /, /' on the spherical surface. Join AI, BI, A' I, B'l. The body might have been carried from the initial to the final position by a rotational displacement about the line //'. For by this turning A is carried to A' and B to B', and it is clear that the particles which lay on the part of the spherical surface bounded by the spherical triangle ABI are in the same relative positions on the part bounded by A' B'l; and all the particles in the spherical surface are in the same relative positions. 238. Every Rigid Body Displacement parallel to Fixed Plane is equivalent to a Rotation. A rigid body is displaced in such a way that a plane A fixed in the body, initially and finally coincides with a plane B fixed in space, and three points (not in line) in A come from P, Q, R to P', Q', R' : it is clear that the displacement, whatever it may be, could be effected by first displacing the body so that every point of the plane A receives a displacement equal and parallel to PP', and then turning the body round an axis through P' at right angles to the plane B, until Q, R coincide with Q', R'. [Here again the reader should draw the necessary Figure.] This displacement may also be effected by a turning, of the same amount and in the same direction, about an axis at right angles to the plane B. For, except in an extreme case, the lines joining P, P' and Q, Q' will not be parts of the same straight line. Excluding that case (in which no change of direction of lines in the body is involved), let these lines be drawn, and their middle points 0, D be found, and lines perpendicular to PP', QQ' drawn through C and D. These meet in a point /. A turning about an axis through / at right angles to the plane B would evidently bring the three points P, Q, R to P', Q', R', and likewise all 440 A TREATISE ON DYNAMICS. [CH. VIII. other points of the body from their initial to their final positions. If s denote the displacement PP', and 6 the angle of turning, which is clearly the same in both the modes of effecting the displacement described above, the co- ordinates of / in the plane B are evidently PC—s and (7/=^s/tan^0, to be measured from C so that the angle PIG{ = ^6) is in the direction of turning. If the displacement s be the small displacement effected in time dt with speed S, we have PP' = s dt, and if dO be the angle of turning, we have CI=sdt/dd. The displacement might be effected in time dt by a turning about the axis at /, with angular speed 9, such that 6dt = d6. We have then CI=s/d. The axis through / is then the instantaneous axis about which the body may be regarded as turning. 239. Any Bigid Body Displacement is equivalent to that of a Nut on a Certain Screw. We can prove that any displace- ment whatever of a rigid body can be effected by a displacement of the body without rotation (a translation) parallel to a certain direction; and a rotation about an axis parallel to that direction. For the displacement can be effected by displacing the body without rotation, so that a point in it initially at P is transferred to its final position P', and then rotating the body about some axis through P'. The different points of the body in the latter displacement move in planes at right angles to the axis, and the direction of the axis and the angle of turning are independent of the choice of the point P- The former displacement, the translation PP", can be resolved into two components, PM and MP', at right angles to one another, of which MP' is at right angles to the axis of the rotational displacement. But, as we saw in §238, the displacement MP', and the rotation about an axis through P' at right angles to MP', may be replaced by a turning about a parallel axis. Hence the displacement can be effected as specified in the proposition. The two displacements may be supposed effected together, in such a manner that the amount of turning effected is always proportional to the translation; that is, the body §§238,239,240] MOTION OF A RIGID BODY. 441 may be regarded as having the motion of a nut along a screw, so that each point of the body moves in a helix. If s be the distance which measures the translation, and 6 the angle which measures the turning, the ratio s/0 is the advance of the body per radian turned through, the "pitch " of the screw, while 27rs/0 is the advance per complete turn, called the " step " (often also the pitch) of the screw. Thus all the helices in which the points of the body move have the same pitch. If the motion is a pure rotation s = 0, and the pitch and step are zero; if the motion is a pure translation 6 = 0, and the pitch and step are infinite. The motion of a rigid body has been discussed very fully from this point of view by Sir Robert Ball in his Theory of Screws, which the reader may consult for further particulars. We shall find this mode of regarding the subject illustrated later by the theorem of the central axis [see § 247 below. In Chap. XI. below the central axis of a system of forces is considered.. The system is reduced to a " wrench," a single force along the central axis, and a couple about a line parallel to the central axis]. That the central axis is a single determinate line will be proved in § 247. 240. Motion of a Bigid Body parallel to a Given Plane. Space and Body Centrodes. We have seen (§238) that any small displacement of a rigid body, which is moving parallel to a fixed plane, may be produced by turning the body through an angle dO about an instantaneous axis which meets the fixed plane in the point /. To find /, we take two points P and Q in a plane in the body coinciding with the fixed plane, and apply the construction described in §238. Since PP', QQf are very short lines, the con- struction can be carried out by drawing, two lines from P and Q, perpendicular respectively to the direction of motion at P and the direction of motion at Q. These meet at /. The points P and Q are obviously turning about I : the reader may prove that any other point in the body is turning about the axis drawn through I at right angles to the fixed plane. As the body moves continuously the positions of the 442 A TREATISE ON DYNAMICS. [CH. VIII. instantaneous axis change in space and in the body. Thus, taking their intersections with the fixed plane, and with the plane in the body which moves in coincidence with it, we get two curves which are called respectively the space-centrode and the hody- centrode ((7„ Cb). We can find their equations in the following manner. Take two axes Oxy in the fixed plane, as shown in Fig. 103. Let / be the inter- section (coordinates x, y) of the plane by the instantaneous axis, and P any point (co- ordinates a, h). Then if I be the distance IP, and u, v the components of velocity of P, 6 the inclination of IP to Ox, we have — u = l6 sin 6 = d(b — y), v = 16 cos 6 = d(a—x). a P / A X 1/ 1 Fio. 103. Hence x = a — -> y = o + -- •(1) If a, b, u, V are known functions of the time, we can eliminate the time between these two equations, and thereby obtain the equation of C,. To find the equation of Oj, take two axes of f, ^ fixed in the plane of the body which moves in coincidence with the fixed plane, and let ^, tj be the coordinates of / with reference to these axes. Let 6 have the same meaning as before, and 6' denote the angle at the instant between the fixed axis Ox and the axis of ^. The coordinates of P are now ^^^ cos (0-6'), »? + isin(0-0'), and therefore, since leoad = a—x = v/d, lain6 = b — y= —u/d, they are ^-1- {v cos d'-u sin eye, ti - (u cos 6' +v sin 6')/ 6. §§240,241] MOTION OF A RIGID BODY. 443 If now P be the origin of the ^, j; coordinates, we get i=-.(iism.e'-v cos ff), r, = -(u cos e'+v sin 6'), . . .(2) 6 6 which give the equation of Cj by elimination of the time. The two curves C^, Oj (Fig. 104) are two series of points such that, as the body moves, the points of the second series come in succession into coincidence with corresponding points of the first series, and each in doing so comes to rest at the instant, though, as the body moves con- tinuously, it does not remain at rest for any interval of time, however short. At the instant of rest it coincides with the point / of the instantaneous axis. After an interval of time dt has elapsed, the instantaneous axis '' ^k 104 has passed to another position I', and another point F' of the body coincides with it. 241. Velocity and Acceleration of Body-Point. The velocity of the point of Cj at the instant of coincidence with / is zero ; its acceleration has in general a definite finite value. If s be the speed with which the point I moves along the curve Cg, it is plain that this is also the speed with which the position of the instantaneous axis moves along Cj,. For clearly the distance IF' is equal to //', if P', I' he the points in the two curves which coincide after the interval dt. The curve Cj, may be regarded as rolling without slipping along (7,, as a wheel rolls without slipping along a rail, and the distance which the instantaneous axis travels along the circle of contact of the wheel in any time is equal to the distance which it travels along the rail. Now, since the curve Cj, may be regarded as turning about I, the point F', at distance from I=sdt, has speed COS dt (where 00 = 6) at right angles to IP', and this speed is annulled in the interval of time dt, in which P' travels to /'. Thus the acceleration of the point P' infinitely near 444 A TREATISE ON DYNAMICS. [CH. VIII. P is w.s, in, the direction perpendicular to the curve G^, opposed to that along which P' approaches the curve just before arrival. For a point P' on Oj, but on the other side of P, the acceleration is also ws, in the same direction as before. It may be affirmed of course that P' has an acceleration u?s dt towards /, but this is infinitely small in comparison with COS. If P (Fig. 104) be any point in the body, in the plane for which the centrodes are drawn, and at distance r from P, the velocity of P is rw at right angles to IP. Hence, relatively to /, the acceleration of P consists of two com- ponents rw at right angles to IP, and rw^ in the direction from P towards /. But after dt, P is turning about /', and we must take account therefore of the acceleration of the point of the body coinciding with /. Thus we must add to the acceleration of P a component sd, in the direction of the normal IN drawn to Cb (Fig. 106). That direction is also indicated by the dotted line through P in Fig. 104. Or the acceleration may be found as follows (Fig. 105). From /, /' draw lines to the position of P for /, and lay off Pp, Pp' to represent rw, r'w ; the geometrical difference j.jg j^Qg p'p between these lines repre- sents the change in rd due to the displacement in dt from I to /'. The other changes produced are rwdt and roc^dt, the directions of which have been specified. But Pp, Pp' are proportional to IP, I'P, and the angle pPp' is equal to the angle IPF. Hence the triangles IPr,pPp' are similar, and, since IPp is a right angle, pp' is at right angles to //'. Thus we have pp' = ir..Pp/IP = sdt.rw/r = swdt. Thus the acceleration due to the motion of / along the space-centrode is sod, and is parallel to the direction IN. 242. Curvature of Path of Body-Point. We can apply these results to find the curvature of the path described §§241,242] MOTION OF A RIGID BODY. 445 by any point of the body. For, (Fig. 106), let the curve Gft roll on the curve 6'^ in the plane of the paper, and G, D be the centres of curvature of the two curves at the point of contact /. Then, if /' and P' are corresponding points, we have arc /P' = arc //', and if the arcs be those traversed in time dt, sdt^IC. LlGP' = ID.LlDr. UtIO=p, ID — p', and we have . TCP' +LlDI=sdt(-+ -). ^p py p p' But this is the angle turned through by the body in time dt, and therefore (i+i). V p' (1) Fig. 106. This is on the supposition that, as shown in Fig. 106, the curvatures are oppositely directed. If the curvature of C^ be in the same direction as that of C„ »=<---A (2) \p P' \p p> and, of course, p' >• p. We get then for the acceleration of the body-point at /, the value (3) wS = a? ^ , or COS = co^ , > P + P P-P according as the curvatures are opposed or in the same direction. Also s/a) = pp'/{p'+p). For the point P (Fig. 105) in the body the total accelera- tion in the direction PI is, by the results obtained above, ft)V — i'tocos ^, if denote lPIN. Thus wV — SO) cos ^ = wh^jR, where R is the radius of curvature of the path of P at the instant. Thus 1 _ft)V — s<»cos^_l 1 R~ rV ~r~^ijo' PP cos 0. .(4) 446 A TREATISE ON DYNAMICS. [OH. VIII. This expression for the curvature vanishes if r = pp COS /(p'±p), (5) that is for all points on a circle in the body-plane touching both curves at I, and of diameter pp'/(p'±p). All such body-points therefore pass points of inflexion in their paths at the same instant. 243. Signs of Angular Displacements. The direction of turning of a body about an axis is positive or negative according to the manner in which it is regarded. Thus the direction of turning of a flywheel may be taken as positive or negative according to the side from which it is viewed. Seen from one side the motion of the top (supposing the wheel vertical) is from right to left, seen from the other side it is from left to right. We usually take the former direction, or, to make the distinction applicable to all cases, the counter-clock direction, as posi- tive, the clock direction as negative. If we have to take account of FiQ. 107. turnings about a number of parallel axes, for example the turning of each of the wheels of a train of wheelwork, we may, viewing the arrangement from either side, reckon all those moving in the counter-clock direction as having a positive turning motion, and all those moving the other way as having a negative turning motion. Again, when we have a number of axes in different directions which all pass through one point or origin, it is convenient to settle in each case, according to convenience, a direction from the origin outward along the axis, which i» to be regarded as positive. Let O-^A, O^B, 0^6,... be these directions chosen as positive. Then, regarding the turning about each axis from the point A,B, C,...,ii is classed as positive or negative according as it is in the counter-clock direction or in the clock direction. Thus, in the figure the rotations about 0^.4 and O^B appear §§242,243,244] MOTION OF A RIGID BODY. 447 to an eye situated at A and B respectively, counter-clock rotations, and are to be reckoned positive, while that about OjO is in the clock direction, and is to be reckoned negative. As already noticed in § 76, the turning of a rigid body about an axis is one in which each point of the body moves at right angles to the perpendicular drawn from the point to the axis, in the same way round for each point, and through a distance to dt .p, where us dt is the (infinitely small) angle of turning, the same for all points, effected in time dt in consequence of the angular speed to, and p is the length of the perpendicular. The arrangement of the particles and their relative distances are evidently undisturbed by this motion. 244. Composition of Angular Displacements. The displace- ment of a point P of a rigid body, due to a turning of the body through any small angle wdt above any axis, say O^A, being equal to the product wdt.p of the angle turned through into the perpendicular distance of P from the axis O^A, is numerically equal to the moment of a force F= ft) dt, acting along the axis, about the point P, or about an axis at P at right angles to the plane of OjJ. and P. This leads to the conclusion that angular speeds about different axes (that is angular velocities) are to be compounded like forces of the same numerical amounts along the same axes. The theorems regarding the com- position of angular velocities might be inferred from those of composi- tion of forces, but for clearness we shall give a separate investigation. Let first P be a point on O^M, and consider the motion of P, due to the turnings in an element of time dt, in consequence of angular speeds (Oj^tca^, about O^A and O^B, in the directions shown by circular arrows in Fig. 108. Let PA, PB be perpendiculars from P on OA, OB, and denote the lengths of these perpendiculars by Pi, p^. By Fig. 108. 448 A TREATISE ON DYNAMICS. [CH. VIII. the turning w-^dt about O^A the point P is depressed below the plane of the paper a distance la^dt.pi, and by the turning te^dt about O^^B it is raised above the paper a- distance w^dt.p^. The point P will be undisturbed if tOjPj = £1)2^2 > that is if Wj/wg = sin ^/sin a, where a., j8 denote the angles AOP, BOP, and r the distance O^P. The same thing can be proved for any point lying in the line 0-^M, and for no other set of points. The line 0-^M is thus an axis about which the body may be regarded as turning: this will be seen more clearly from what follows. Let now P be any point in the plane AOB, which also contains OJi, and let PA, PB, PM, of lengths p-^, p^, p, be perpendiculars let fall from P on O^A, O^B, O-Ji (Fig. 108). Denoting the angle PO^M by 6, and, as before, OjP by r, and the angles AO-^M, MO^B by a, /3, we have for the displacement of P, due to the turnings about OA and OB, the expression dt{w^p.^ + w^^ = Tdt{w^sm{Q-\-a.) + w^w\{Q — ^)), which may be written r sin . dt{w^ cos cl + w^ cos P), since co^sina — (B2sin|8 = 0, as we have already seen. If then we write „ = »iCosa+t«2Cos^, (1) we have for the displacement the expression w dt . p. Now WiCosfX+WgCOSiS would, in the ease of forces Wj, w^ along the lines 0^-4, O^B, be recognised as the resultant of the forces, acting along OM, since the relation eo^ sin a = coj sin (3 would show that there was no component of force at right angles to 0-^M. Hence we take w^ cos a + co^ cos j3, the result as we say of resolving w^, w^ about OjM, as the resultant angular speed, and the turning at this angular speed is about O^M, the line which we have seen remains at rest when the two turnings specified about O^A, O^B are superimposed. Now let P be taken at a distance h from the plane AOB. Fig. 108 will serve for this case also. We have to show that the displacement of P, compounded of those due to §§ 244, 245] MOTION OF A RIGID BODY. 449 the two turnings oojdt, w^dt about the axes O^A, 0-^B, is equal to that produced by the single turning (wj^ cos a. + (Bg cos /3) dt about OM, which is so situated in the plane AO-^B that ft)j sin a = Wg sin jS. First consider the displacement Wjdt.p^: this is at right angles to the perpendicular from P on OA, and lies in a plane at right angles to the plane A OB and containing that perpendicular. Evidently it can be resolved into two components, one at right angles to the plane AOB, and one parallel to the latter plane. Let P^ be the projection of P on the plane AOB and r,, the distance OP^,. Then, if 6 is the angle Pfi-Ji, the two components just specified are r^dt . Wj sin (0 + «■) and <»i-Ji dt. Similarly we have for the components due to the turning w^dt about O^B the expressions r^c?^ . tOg sin (0 — ^8) and wji dt. It has already been proved that the displacement wdt.p-^^==rf^dt{w.^sm.{Q + a.) + w^sm{Q — P)}. There remain the components wji dt, w^hdt. These are parallel to the perpendiculars from P^ on O^A and O^B respectively, and since they are proportional to Wj, w^ have a resultant in the direction of the perpendicular from P^ on OM. The magnitude of that resultant is h dt(w.^ cos o(.+ Wg cos ^8), that is whdt. Hence it has been proved that the displacement at P, compounded of the independent displacements due to the two rotations specified, is identical with the displace- ment at the same point due to the resultant rotation, that compounded of the rotations w^, toj about O^A, O^B. The rotations about O^A, O^B may be each the result of compounding the rotations about a pair of axes, and so on, so that the rotation about O^M may be the resultant obtained by compounding the rotations about any number of axes given in position. 245. Turning about any Axis expressed by Component Turn- ings about Three Rectangular Axes. We shall now show that if a rigid body turns with angular speed to about an axis O-^M, passing through any point Oj, the same motion G.D. 2F 450 A TREATISE ON DYNAMICS. [CH. VIII. is produced by the displacement due to independent turn- ings at angular speeds Iw, mw, nw^p, q, r, about three fixed axes 0-^x-jj^z^ at right angles to one another, and making angles with O^M, the cosines of which are I, m, n. The distance nr of any point P (coordinates x, y, z with reference to axes O-^xyz) of the body from O^M is {a;z + jy2 ^ ^2 _ ^i^ +my + nzf)^. Hence the displacement effected in dt is this multiplied by w dt, that is wxsdt= dt{(7nz — nyy+(nx — lzy+(ly — 7nxy}- I , , = dt {{qz—ryf +{rx —pzf+{py — qxf}.^ The expression on the right can easily be seen to be equivalent to the three displacements of the point P, P'Jy'^+z^.dt, qjz^+x''.dt, rjx^+y^.dt, due to the turning through the angles pdt, qdt, rdt about the axes of the set OjX^y^Zj^ respectively, where each turning is supposed to be effected independently of the others, from the same initial position of the body. The three displacements result- ing from these turnings are not generally at right angles to one another, but they give the displacements parallel to the axes as follows: — ^zdt and py dt parallel to O^y^ and OiZi, the components of pjy^+z^.dt, —qxdt and qzdt parallel to OjS^j and O^ajj, the components of q-Jz^+x^.dt, and —ry and rx parallel to O^x.^ and 0^y■^, the components of rjx^+y^.dt. 246. Component Linear Velocities of Point in Turning Body. The rates of displacement of the point P parallel to the axes are therefore qz — ry, rx—pz, py — qx; and so we have the equations u — qz — ry, v = rx—pz, w=py — qx, (1) which are useful in many applications. These are component velocities of the point P due to the rotations of the body about the axes. From them we can obtain at once the expressions for the components of velocity for a set of mutually rectangular axes Oxyz which are turning with angular speeds p, q,r about Ox, Oy, Oz §§245,246] MOTION OF A RIGID BODY. 451 respectively. For we may suppose these axes fixed in the rigid body, and, in the course of their turning with it, to be at time t in coincidence with the fixed axes O^x-^y-^z.^. The motion of the particle P, if that particle is fixed in the body, has in consequence of the rotation of the bodj'' the components just written down. But if the particle at P be moving in the body, and we now regard x, y, z as the coordinates of P relative to the moving axes at the instant, which of course we may do, since at the instant the axes are coincident with the fixed axes, the components x, y, z, relative to the moving axes, give the rates of displacement of the particle in the body with respect to axes fixed in it and therefore moving with it. The components u, v, w of the velocity with respect to the fixed axes O^x^y-^z^ are therefore given by u = cb + qz — ry, v=y + rx—pz, w = z+py — qx. ...(2) Thus if we draw the vector OP, the components of the motion of the outer extremity, P, are given by these equations. The terms qz — ry, ... , are those which depend on the motion of the system of axes Oxyz, those, in fact, due to the motion of the rigid body here supposed to carry the axes, and with them also the point P. Thus qz — ry, ... are called by French writers the components of the velocity of entrain'me7it ; perhaps they may be termed the com- ponents of co-velocity. If the origin of the moving axes does not coincide with that of the fixed axes, the values of u, v, w require no modification, provided the two systems are parallel and the origin is not in motion. If, however, is in motion with components u^, V(,, w^, the equations become, as the reader may easily convince himself, u = x+u^ + qz-ry, v = y + Vf,+rx-pz,'\ w = z + Wg+py — qx. f ^ ' Then the components of the velocity of entrainTnent are u^+qz-ry, v^+rx-pz, w^+py-qx. The same results may be obtained in the following manner. Let the fixed axis O^Xj at time t make angles, 452 A TREATISE ON DYNAMICS. [CH. Vlll. the direction-cosines of which are a., ^, y, with the axes OX, OY, OZ, and let the projection of O-fl on the axis OjJTj be x^. Then if x-^ be the projection of O^P on 0-^X^, we obtain x^ = x^ + ajic + ^y + yz, (4) and therefore Xi = Xg + cdt + ^y + yz + dx+$y + yz (5) But if the directions of Oxyz coincide with those of O^x^y-^z^, then (x. = 0, j3 = y = Tr/2, and, since d(cos6)/dt= —smO.O, a. = 0, /3 = 0, fi=—r, y = 0, y = q, and therefore, writing u for Xj, and u^ for x^, we get u = x + 'u,f^+qz—Ty (6) Similar expressions are obtained in the same way for V and w. 247. Central Axis. Let, now, the point P be fixed in the body so that x = y = z = 0. Then u^, Vg, Wf, are the same for all points of the body, and are the components of a motion of translation of the body as a whole parallel to the fixed axes with which the moving axes at the instant coincide. We can find for each instant a line in the body which fulfils the condition that the motion at every point of it is in the direction of the axis of resultant angular velocity. The equations of the line are qz-ry + Ug ^ rx-pz + Vg ^ py-qx + Wg _^ ^j\ p q r It is called the central axis of the motion. Its equation depends on the components UgjVg, Wg of the translational motion, and the components p, q, r of angular velocity, just as the central axis defined in § 343 below for forces depends on the resultant of a set of forces and their moments about the axes. It may be observed that the central axis is the locus of points for which the resultant speed v={{qz-ry + Ug)^ + {rx—pz + Vgf + (py-qx+Wgf}i (2) is a minimum. For if this expression be difierentiated partially with respect to x, y, z, and the derivatives be equated to zero, we get exactly the equation of the line. (3) §§246,247,248] CENTRAL AXIS : EXAMPLES. 453 With respect to the fixed axes, to which Oxyz are parallel, the equations of the central axis are p p If we denote the common value of these ratios by Vjw, V/o) is the ratio of the rate of displacement of a point of the body parallel to the central axis to the rate of rotation around it, the pitch of the screw at the point (see § 239). If V= 0, the body has motion of rotation only, if o) = the motion is of translation only. 248. Examples on Central Axis and Botation. Ex. 1. Planes are drawn through any two points F, Q of a, rigid body at right angles to the trajectories of these points, and meet the central axis A in M, N. Show that M, N are the feet of the perpen- dioulais let fall from P, Q to the centi'al axis, and that MJSr= PQ cos {PQ, A ). [Chasles.] No generality is lost by taking the central axis A as the axis of x, the coordinates of P as x-^, 0, zi, and the coordinates of Q as x^, y^, z^. The point P has speeds i% parallel to the axis of x, — oizj parallel to the axis of y, and zero parallel to the axis of z. The corresponding speeds, of (^ are Mq, -0)22) ^Vi.- Hence the equation of a plane through P at right angles to the trajectory of P is Wo (^ -«■!)- 0)01 ?/ = 0, which meets the central axis in the point x = x-^. Again, the equation of a plane through § normal to the trajectory of that point is u,{x-x,)-^z^{y-yi,^.y!),{z-z,)=Q, that is, u^{x- x^ - m^ + vty^ = 0, which meets the central;axis in the point x=X2- Hence the first part of the proposition is proved. The second is self-evident, since MN is the projection of PQ on the axis A. Ex. 2. From an arbitrary origin, say the origin of coordinates, are drawn three vectors OA, OB, OC, representing in magnitude and direction the velocities at three points P, Q, R of the body : to show that the central axis is at right angles to the plane of the three points A, B, C, and to find the intersection of the central axis with that plane. X, y. n. 1 SI' Vu C^, 1 S2; %. C., 1 ^3, V3> L 1 454 A TREATISE ON DYNAMICS. [CH. VIII. If the coordinates of the three points be xi,yi,Zi, •»2)2'2>%i ■^3j.%)^3> the coordinates of A, B, C are '^o+ih-'>'2/u Vo+rx^-pzi, w^+py^-qx^ for A, with similar expressions for B and C, to be obtained by changing the suffixes to 2 for B and to 3 for G. If we denote these three sets of coordinates by ^j, rjj, fi, ^g, tjj, fj, ^3> %. 4> "^^ get for the equation of the plane ABG, =0. The coefficient of x in this is •>7i(f2-fi) + »?2(f3.""fi)+'73(fi~4)) ^"d the coefficients of y and z can be written down by symmetry. Sub- stituting the values of the coordinates, we find that these three coefficients are respectively p, q, r multiplied by a common factor. Hence the direction-cosines of the normal to the plane are pro- portional to jo, q, r, and the proposition is proved. The completion of the example is left to the student. Ex. 3. A rigid body is turning about a fixed point ; show that if the instantaneous axis of rotation, 01, is fixed in the body it is also fixed in space, and conversely. Let the direction-cosines of a fixed axis Ox^ , referred to the moving axes, be I, m, n. Then the angle between 01 and Ox^ has cosines a.={lp+imq + nr)j(a, Hence 6.=-{lp + mq+nr+lp-\-mq + rvr) — ^(Ip + mq + nr). If the axis 01 is fixed in the body, we must have, however p, q, r may change, p/=k2, rju> = k,, where k^, k^, k^ are constants Hence pjio =p/a), q/i) = q/m, rj6)= rjoy. Thus (a(lp+mq+nr)=(lp+mq + nr)i}), and therefore a.=-(ip + mq + nr). Now for the motion of a point of coordinates x, y, », we have u = qz-ry, v=rx-pz, w=py-qx. Take a point on the axis Ox^, at unit distance from the origin ; its coordinates with respect to the moving axes are I, m, n. The values of u, v, w therefore give l—nq-mr, m,=lr — np, n=mp-lq, and so ip + mq + nr=0. Thus i=0, that is the angle which 01 makes with the axis Oxi, is constant, if (p, q, r)lm remain constant ; and Ox^ may be any fixed axis. The direct proposition is therefore proved. §248] ROTATIONAL MOTION: EXAMPLES. 455 To prove the converse, we begin with a=0 and note that we have also lp+mq+nr=0. Hence we obtain (o{lp + m^ + nr) = =p/ii), ... , the conditions that the axis should be fixed in space. Kx. 4. Prove that if a body be turning about a fixed point, the angular acceleration about an axis, the cosines of which with respect to the moving axes are I, m, n, is lp+m^-\-nr. If coj be the angular speed about the axis in question, we have u>^=lp + mq+nr. Hence ii>i = lp+m([-\-nr-\-lp + mq+mr. But in last example it is proved that lp-\-mq + nr=Q, and therefore we have 6)\ = lp-'r'm(l + 'nf. Ex. 5. A body moves along a curve in space in the following manner. A point fixed in the body describes the curve, and three lines fixed in the body and intersecting in that point are always directed so that' the lirst is along the tangent to the curve in the direction of motion, the second along the principal normal, and the third at right angles to the osculating plane. These lines are at right angles to one another and in the order named form a system of moving axes Omyz, in which the turning is from y to z, z to x and X to y. It is required to find the angular speeds about the axes. Let the radius of curvature at any position of the body be p, and the radius of torsion — that is the reciprocal of the rate at which the o.sculating plane is turning round the tangent per unit distance travelled along the curve— be t, and let s, the distance travelled along the curve from <=0 to the instant considered, be/(<)- According to the sign usually given to the analytical expression denoted by t, the rate ot turning of the body about Ox is -«/t= —f'{t)JT. The rate of turning about the radius oi curvature is zero, since the changes in the direction of the tangent lie wholly in the osculating plane for the instant. Finally, the rate of turning in the osculating plane is that about the binomial as an axis,. and is s/p=f'(t)/p. Hence we have P=-mir, q = 0, r=f{t)lp. To find the equations of the central axis, we note that the axis of x here taken has, with reference to axes Oj.«,?y]«, , direction-cosines rf^/rfs, drilds, difdz, wliere ^, ?/, ( are the coordinates of the position at time t of the point which moves along the curve. The cosines of the y-axis are p{d^^lds\ d^r)lds\ d^/ds^), and of the z-axis, p {drilds . d^/ds^ - dC/ds . dhi/d^) Calling these three sets of cosines a, b, c, a', 6', e', a", b", c", and putting a?j, ^i, 2] for the coordinates of a point P of the body, we get 3!=a{Xi-^)+..., y=a'(xi-^) + ..., z==a"{xi-^) + ..., 456 A TREATISE ON DYNAMICS. [CH. VIII. and therefore the equations of the central axis are V q{a"(iVi-$)+...\-r{a'(,a;i-0 + ...}=p~-s, where p, q, r have the values found for them above, and F is « cos (w, s). Ex. 6. Prove that if the ratio p/t= const., the curve in the last example is a helix traced on a cylinder. • If we draw axes from a fixed point parallel to the moving axes O.vi/z, then, since p/r is constant, the line through parallel to the central axis is fixed relatively to the system of axes. It is therefore fixed in space (see Example 3), and therefore the axis Ox makes a constant angle with the instantaneous axis, which remains fixed in direction as the body moves. 249. Acceleratibns of Point in Botating Body. Equations of Motion. In precisely the same way if we set up a vector from 0, the components of which along the axes Oxyz, that is, parallel to Ox, Oy, Oz, are u, v, w, we obtain the components of acceleration in terms of x, y, z, p, q, r, X, y, z, Wq, v^, Wq, and their rates of variation. If the components of acceleration along the fixed axes be a-g, ay, a^, we have aa; = u + qw — rv =x + U() + 2qz—2ry + qz — ry +p (qy+rz) —(q^-\- r^) x, ay = v+ru —pw = y + VQ+2rd; — 2fpz+rx—pz + q (rz +px) — (r^ +p^)y, an =w+pv —qu=z +'WQ+2py — 2qx+py — qx +r{px + qy)-{p^+q^)z.. The first two terms on the right in these expressions are the components of relative acceleration, the next two terms, 2{qz — ry), 2{rx—pz), 2{py — qx), form the com- ponents of what is called the complementary acceleration, and the three groups of four terms remaining form the components of the acceleration of entrainment or co- acceleration. Thus, denoting qz — ry +p(qy + rz) — (q^ + r^)x ..(1) §§248,249,250] ANGULAR MOMENTA. 457 by a^c, 2qz — 2ry by a'^^, and so on, we have where u^, Vq, ly^ are supposed included in x, y, z. Thus, if X, Y, Z be forces on a particle at P, the equations of motion are These equations may be written m* = Z+Z<,, my=Y-\-Y^, mz = Z+Zc, (3) where Xc= — 'm{axc + o-Lc); • • ■ ■ Thus the equations of motion are now of the form for axes at rest, but the force in each 'case is the applied force, with a force added sufficient to produce an acceleration equal and opposite to axc+a'„, ... . It will be noticed that the components a'^, a'yc, a^ represent a vector at right angles to the plane of the vector {p, q, r) (the instantaneous axis about which the body is turning with angular speed ft) = s/p^ + g^ + r^) and the vector Vr = y/d!^ + y^ + z^ (where x, y, z include x^, y^, z^ if these exist), that is, the vector representing the relative velocity. Its magnitude is ItoVy sin {w, v,.), where (to, Vr) is the angle between the positive direction of the instantaneous axis and that of Vr, and its sense is that in which the turning w tends to carry the outer end of. the vector Vr- 250. Angular Momenta. By (2), § 71, the components of angular momentum about the fixed axes O-^x-^y^Zj^ are given by the equations Hj^ = 'E{m(zy-yz)}, H^ = 'Z{m{xz-zx)},\ ^^^ Hg = I,{m(yx-xy)}, f and therefore by (1), § 246, we have Hj =pX{'m{y^+z^)} — q'E{mxy) — r^{7nxz), (2) with similar expressions for H^, H^. But 2{m(3/^ + 3^)}, ... are the moments of inertia about the axes and are denoted hy A,B, G, while l!,(myz), ... are products of inertia and are denoted by D, E, F. Hence H,^Ap-Fq-Er, H, = Bq-I)r-Fp,] .^. H^^Gq-Ep-Dq. j L..(5) 458 A TREATISE ON DYNAMICS. [CH. VIII. When the axes are principal axes of moment of inertia D=E = F=Oa.nd H, = Ap, H, = Bq, H, = Cr (4) The kinetic energy T is given by T=^1,[m{(qz — ryy+{rx—pzy+Qpy — qxf}'] — 2qr^{myz) - 2rp'Z{mzx) — 2pql,{mxy)],) and so by (2) ^=H„ ||=fi-„ ^=H, (6) Also clearly T=\{pH^ + qH^+rH^) (7) A proof in all respects similar to that given in § 244 above for the composition of angular velocities might be framed to show that the angular momentum about the axis, the direction-cosines of which are proportional to jffi, H^, H„ is (Hl+Hl+Hlf, and that, if this be called H, the angular momentum about any axis inclined at an angle 6 to that of H is jff cos 6 ; but the subject has already been discussed in § 71. It has been proved in §75 that the time-rate of change of angular momentum, about any axis, is equal to the sum of the moments of the impressed forces about the axis, or, as it is sometimes put, to the moment of the impressed couple about that axis. This holds whether or not the system is a rigid body. 251. Representation of A.M. as a Vector. Bates of Change of A.M. about Moving Axes. It is convenient to measure a distance from along any axis OA , OB, ... , in the positive direction, in length numerically equal to the angular momentum, or angular speed, or moment of forces, as the case may be, about the axis. The points A,B,... may be the outer terminal points of these distances or vectors, and will show by their displacements as time passes, how the direction or the numerical value of the directed quantity represented by the vector is varying. §§ 250, 251] EULER'S EQUATIONS. 459 Now, the time-rates of change of angular momentum for the fixed axes OiX^y^Zi have been found in § 72 above, and if these be denoted by H^, H^, H^, the time-rate of change of angular momentum about any axis 0-JiJ^, the direction cosines of which with reference to O-^x-^y^z^^ are I, m, n, is IH-^ + mH^+nH^. Also, in § 170 have been derived the rates of growth of angular momentum for a rigid body, about the principal axes of moment of inertia supposed fixed in the body, and therefore turning with angular speedy? about O^x^, q about Oji/i, and r about O^z^ («i> (»2, Ws' ill § I'^O). For the case of angular momentum referred to a system of axes Oxyz turning with angular speed Q-^, 6^, Q^ about fixed axes, with which the moving axes at the instant coincide, we can find the equations of motion by an applica- tion of the method explained in § 9 above. We shall denote the components of angular momentum referred to the system of rotating a,xes by \, \, \, and identify these with the directed quantities L, M, N referred to in § 9. The symbols L, M, N, thus set free, we shall use to represent the moments of impressed forces — the impressed couples — about the axes. Thus we get, if the origin be at rest, K-dA+esK=M, I (1) If the axes are fixed in space 0^, 62, 6^ are zero, and the expressions on the left reduce to their first terms. If the axes are fixed in the body, then 0j, 6^, 63 are the angular speeds, p, q, r say, of the body at the instant about these axes fixed in itself. If moreover the axes fixed in the body coincide with the principal axes of moment of inertia of the body, we have h^ = Ap, h2 = Bq, h^=Cr, so that the equations of motion become Ap-(B-C)qr=L, Bq-(G-A)rp = M, Cr-(A-B)pq = N, ) (2) 460 A TREATISE ON DYNAMICS. [CH. VIII. which are the Euler's equations, found in § 170 by another method. These give only the rates of change of the angular momenta, so that the body is either turning about a fixed point, for which the principal axes are taken, or the translational motion is ignored. 252. Body with One Point fixed. Deductions from Euler's Eciuations. From these equations we can obtain some important results. Multiply the first equation by p, the second by q, and the third by r, and add. The sum on the left is App+Bqq + Grr, so that App + Bqq + Crr==Lp + Mq+Nr (1) The kinetic energy T is given by the equation T=^^(Ap^ + Bq^+Cr^), (2) so that the equation just obtained can be written rIT ^^Lp + Mq+Nr (3) The quantity on the right is the time-rate at which work is being done by the impressed couples as the body moves, and this is the time-rate of growth of the kinetic energy. Hence, if L = M=N=0, the system moves subject to the condition that the kinetic energy is constant. If we multiply tlje first Euler's equation by Ap, the second by Bq, and the third by CV, we get A^pp + B^qq + C^rr = LAp+AtBq + NCr (4) The square, h^, of the resultant angular momentum is given ^y- h^ = AY + BY + C^r^-, (5) and therefore the equation just found can be written ^ = l(LAp+MBq+NCr), (6) that is, the time-rate of growth of resultant angular momentum is equal to the resultant moment of the im- pressed couples about the axis of resultant angular momen- tum. If L = M = N=0, the resultant angular momentum (iT below) remains constant §§251,252,253] ROTATIONAL MOTION. 461 in amount, and its axis {OH say) unchanged in direction as the body moves. The axis of resultant angular velocity 01, the instantaneous axig, is not, however, stationary, and the value of w {=\lp^-^q^+r''') also varies. 253. Body with One Point fixed. Eolation of Axis of Resultant A.M. and Instantaneous Axis, The cosine of the angle between the axes OH and 01 (angle POH in Fig. 109, below) is {Ap'^ + Bq^ + Crf/Hw, that is, if T be the kinetic energy, 2T/Hw. Now, if i = if = iV= 0, both T and H are constant, and we have then w cos I0H=2T/H, a constant. The direction-cosines of a normal to the plane of lOH are {{B-G)qr, (O-A)rp, (A -B)pq}/MH sin lOH, and so the resultant couple, or rate of growth of angular momentum, represented by {(£- C')Yr2+(C^-4)Vy + (^ --B)2pV}* lies in the plane lOH, and its magnitude is ooH sin lOH. This is sometimes called the centrifugal couple, or couple due to centrifugal forces. For we can write Euler's equations intheform Ap = L+(B-G)qr, ..., (1) and then (B—G)qr, ..., appear as moments of centri- fugal couples. On the other hand, we have interpreted — {B—G)qr, ... , above as the rates of growth of angular momentum about the instantaneous positions of the principal axes, due to the motion of the body. In fact, the angular momentum H about the axis OH resolves into two com- ponents wHcosd along 01 and ft)fi"sin0 at right angles to 01, and the turning with angular speed w about 01 gives a rate of growth of angular momentum wHsm IOH= {{B - Gfq^r^+iG-AyrY+i^ - B)Yq^^ (2) about an axis at right angles to the plane lOH. The reader may consider for the sake of the analogy the equation of radial acceleration in the motion of a particle in a plane, 'm(r-rd^) = F, (3) where F is the outward impressed force along the radius- vector from the origin. We may consider the term rtird'^ either as appearing in the rate of growth of momentum 462 A TREATISE ON DYNAMICS. [CH. vin. —irwd^ along the radius- vector, due to the turning with angular speed G, and consequent momentum mrQ transverse to the radius-vector, or as a centrifugal force 7nr^ added to F, which gives the rate of growth of momentum due to the rate at which invr is changing, according to the equation 'mr=F-\-m/rQ'^. 254. Motion of a Eigid Body under No Forces. We now go on to consider very shortly the motion of a rigid body under no forces, by means of the momental ellipsoid, according to the method of Poinsot. We may suppose one point of the body to be fixed in space ; but the conclusions will be applicable in other cases, for example to the oscilla- tions and rotations, with respect to the centroid, of a quoit, or of a stick thrown into the air, since these relative motions are not affected by the action of gravity when the body is free in the air. Fig. 109. First, we prove that the angular speed about 01 is pro- portional to the length of the radius-vector of the momental ellipsoid (m.e.) (described for the body about as centre), with which 01 coincides. For let 01 intersect the M.E. in P, and x, y, z be the coordinates of P with respect to the principal axes of the ellipsoid. Then, since p, q,r are the angular speeds about the principal axes, we have p/x = q/y = r/z = wjOP, §§253,254,255] INVARIABLE PLANE. 463 The square of each of these ratios can be written But the numerator of this is 2T, where T is the kinetic energy, and by the equation of the ellipsoid we can put ' Ax-'+By^+Cz^ = l. Thus we get ^2 = ^^, (1) a constant. Again, the value of w . cos lOH is constant. For coa I0H=2TlwH, since the direction-cosines of 01 are proportional to p, q, r and those of OH to Ap, Bq, Gr, and therefore weoB IOH=~, (2) and T and H are both constant from the condition i = if=i\r=o. 255. Invariable Plane and Invariable Line. Boiling of M.E. on Invariable Plane. The perpendicular from the centre on the tangent plane at F has direction-cosines proportional to Ax, By, Cz, that is to Ap, Bq, Gr. The perpendicular therefore coincides with OH. Its length is IJiA'^x'+E^y^+G^z^f, that is w/{OP.H), which, by (1), § 254, has the constant value >J21'/H. The m. e. thus always touches a plane perpendicular to the axis of resultant angular momentum H at the distance wKOP .H), or >J2T/H ( = 57), from the centre of the ellipsoid. The plane through at right angles to OH is frequently called the invariable plane. In this plane an impulsive couple, of moment H, which instituted the motion from rest would have to be laid. The fixed line OH is called the invariable line. The M.E. is turning about the instantaneous axis 01, which is coincident with OP. At P the m.e. is in contact with a fixed plane parallel to the invariable plane, and so rolls on that plane. The angular velocity about OP may be resolved into two, an angular velocity of amount wcoa I0H^2T/H 464 A TREATISE ON DYNAMICS. [CH. VIII. about OH and an angular velocity of amount w sin lOH about an axis OG, the intersection of the plane HOI with the invariable plane. If then we suppose the invariable plane to turn with the M.E. about OH with speed 2T/H, the motion of the ellipsoid relative to that plane will then be simply that of rolling about OG with angular speed ft) sin lOH. In the course of the motion, OG describes in the body a cone, and in space a portion of the invariable plane about 0. The angle turned through by the invariable plane will give the time. Ex. 1. Prove that the ellipsoid which is confocal with the m.k., touches a plane parallel to the invariable plane in a point Q, the coordinates of which are x = R{l + Ah)p, y = R{l+Bh)q, z=R{l + Ch)r, where R=ll(ZT+hH^). Also prove that the line OH intersects this plane of contact in a point Z, at a distance ■J%T-\-hH''jH, a constant. Ex. 2. Calculate the speed of the point Q, that is w.OQ sin QOI for any instant, and hence the angular speed (o sin QOIjsm QOL, of Q round OL at the same instant, and show that this angular speed reduces to hff. [This gives Sylvester's measure of the time required by the bodj' to perform any part of the motion ; namely the angle turned through by Q about the line OL, divided by hff.] Ex. 3. Prove that the line 00 describes a cone in the body. Let O be the projection of P on the invariable plane. Then 0F=7S. Also if the coordinates of G with reference to the principal axes be ^, 17, Ct we have, since OP is parallel to Off, Also, since 00 is perpendicular to OH, Ax^-\-Byrj + Cz^=0. The first set of relations give ^=x+fLAx, rj=y+u,By, ^=z+^Cz. Multiplica- tion of these by Asc, By, Cz gives 2la;2+By^+&2+^j,(^2^2+2j2^2+C^2^)=0, or /;!= -ra2 (since Ax^^-By'^-\-Gz^ = \, AV+my^+Ch^ = \lrS^). Thus x = ^l{\--!SU), y=-ql{l-VflB), z=il{l-WiC), which, substituted in Axi+Byt) + Cif=0, gives A& , Bf CC' W^A - 1 ^VPB- 1 "^CT2(7- 1 ~"' the equation of a cone. j 255, 256] POLHODE AND HERPOLHODE. 465 256. Polhode and Herpolhode. As the M.G. rolls on the fixed plane referred to above, the successive points of contact trace out two loci, one on the ellipsoid, the other on the plane. The former is called the polhode, the latter the herpolhode. The former is the locus of points on the ellipsoid, the tangent planes at which are at a constant distance fi"om 0. If ct as before be the constant length J2T/H of this perpendicular, we have the equations Ax^+By^+Gz^-=1, Ah;^+B^y^+C's'-=\. (1) which give the equation the equation of a cone fixed in the body. This is called the body-cone. It rolls on a cone fixed in space, the space- cone, the intersection of which with the fixed plane of contact is the herpolhode. The cone is imaginary unless H^/'2T, that is l/cT*, lies between the greatest and the least of A,B,C. c Fig. lia If C be the greatest moment of inertia and A the least, then 1/uJ- = A or l/nr* = C converts the equation of the cone into B{A-B)y^-\-C(A -C)z^ = (3) or A(A-C).v--[-B(B-C)i,-' = (4) each of which represents a pair of imaginary planes, in the G.D. 2g 466 A TREATISE ON DYNAMICS. [CH. VIII. former case meeting in the axis of x, in the latter case in the axis of z. The cone degenerates into two real planes if 1/ts^ = B, where B is the intermediate moment. We have then A{A-B)x^-O{B-C)z^ = (5) These two planes intersect on the axis of intermediate moment, and they separate the polhodes which are closed curves surrounding the axes of greatest and least moment, as shown roughly in Fig. 110. Their intersections with the M.E. are therefore called the separating polhodes. 257. Stability of Motion of Rigid Body under No Forces. It has been shown in § 169 that an axis of principal moment of inertia is an axis of free rotation for a body under no impressed forces. The figure shows that if the body be set rotating about the axis of greatest or least moment, any slight deviation of the axis of rotation from the principal axis will not result in any further large divergence of the axes; the instantaneous axis moves in the body so that its intersection with the m.e. describes the small closed curve of points at the same distance cr from the centre. But if the body be set rotating about an axis nearly coinciding with the axis of intermediate moment, the axis of rotation will wander off in the body along the polhode, which it will be seen passes nearly to the opposite side of the ellipsoid before returning to the original position. The motion is therefore stable in either of the former cases and unstable in the latter. If the body rotates exactly about either the axis of greatest or the axis of least moment, the polhode is a mere point. 258. Projections of the Polhodes. If we eliminate z be- tween the two equations (1) and (5) of § 256, we obtain A{A-G)x-'+B{B-C)y^=^(l-^) (1) Whether G be the greatest or the least moment the locus, which is the projection of the polhode on a plane at right angles to the axis of z, is an ellipse. The ratio of the cc-axis of the ellipse to the y-axis is JB{B — C)/A(A — C), 1 256-259] POLHODE AND HERPOLHODE. 467 and is therefore more nearly a cii'cle the nearer A and B are to equality. But \i B—G and 4 — be very unequal, the axes of the ellipse differ widely, and in one direction there will be a comparatively large displacement of the instantaneous axis in the body. For the highest degree of stability, therefore, we should have in this case A=B. Eliminating y between the equations (1) and (2) of § 256 to project the polhode on the plane of xz, we get A(^A-B)3?-G{B-G)z^ = \{).-Brs\ (2) the equation of a hyperbola. For 1 >• Bis^, we get one hyperbola, and for 1 ■< Brs- the conjugate. The asymp- totes are the two lines A{A-B):^-G(B-G)z'' = ^, (3) that is the lines >JA{A-B)x^^QkB-(J)z-- „,, )3 = 0,| )0 = O,J ^A{A-B)x - ^fC{B-G)z which are tlie projections of the separating polhodes. 259. Form of the HerpoUiode. With regard to the herpol- hode we have not space to go into detail. It is a curve consisting of different parts, which correspond to the successive repetitious of the polhode, and from the manner of its description, bv the rolling of the ellipsoid, it must always have its concavity turned towards the foot of the perpendicular from the centre of the ellipsoid to the plane of contact, and therefore cannot have a point of inflexion. The distance of the point of contact at any instant from the foot H of the perpendicular from the centre is -JOP^—vy-, and it is evident from the form of the polhode as displayed by its projections just indicated, that OP varies between a maximimi and a minimum value, in each fourth part of its description. Thus, the distance ffP = sjOF^ — cj* similarly varies, and so the herpolhode is a curve lying between two circles which have the pro- jection of the centre of the m.e. as their common centre, and touching the outer circle internally and the inner 468 A TREATISE ON DYNAMICS. [CH. VIII. externally, as shown in Fig. 109. The herpolhode is not in general, however, a closed or re-entering curve ; unless the angle turned through by HP, fronj contact of the herpolhode with one circle to contact with the other, be commensurable with 27r, the curve will not be repeated. When w^ = l/5, the intermediate moment, the herpolhode has an interesting form shown in Fig. 110. The polhode then passes through the extremity of the principal axis OB of the M.E., and is therefore one of the ellipses which form the separating polhodes. When the extremity of the axis OB is in contact with the fixed plane, HP = 0, and so the radius of the inner limiting circle is zero. Let the motion of the m.e. begin at any point of the polhode distant from the extremity of the axis OB, say at the maximum value of OP, then the motion consists, as we have seen (§ 25.5), of a spin about an axis through the point of contact, of angular speed 2T/H, and a rolling motion about the line HP with angular speed oo sin lOH. As the ellipsoid moves and the point of contact approaches B, the motion becomes more and more one of spin merely, and so the herpolhode consists of a succession of constantly diminishing arcs of a spiral closing down on a pole P. The spiral is a double one, but only one half of it is de- scribed by the point of contact. (See Ex. 4, § 260 below.) 260. Examples on Motion of Bigid Body. Ex. 1. If p be the distance HP, then from the equations x^+f+z^^p^+Zi^, Ax"+By'^+Cz^=l, CT2(^V-|-B2/ + CV)=1, fulfilled by the coordinates of the point of contact of the m.e. with the plane on which it rolls, prove that BC{C-B){p^-,t.) ,_ CA{A-C){p-^-IB) ,^ ^ ~{A-B){B-C)(C-Ay -^ (A-B)(B-C)(V-Ay "■' where o.= J-^^^^^g^, /^=... , y^... . Ex. 2. By means of the relations p/,v=q/i/=r/z=(o/OP^^fM' (§ 254), prove that Euler's equations may be written for the case of no forces in the form 4*--y27*(fi-C)y2=0, B'g-»j2f(C- Ay. T=0,.... §§ 259, 260] ROTATIONAL MOTION : EXAMPLES. • 469 Hence show that 06-- ^M^xvz i^-B){B-C){C-A) PR— -^ ^ixyz ABG and by the last example that from which p^ can be found in teims of t. Ex. 3. By calculating the rates of description of area by the projections of the radius-vector OP on the coordinate planes (that is, yz-zy, zA-xz, r^-yx), and taking the sum of the projections of these on the plane of contact, show that if <^ be the rate of increase of the vectorial angle 4> corresponding to the radius- vector p = HP of a point of the herpolhode, and that this, by Example 1 above, reduces to f?^=TSJrr(p^+E), where E={rsU-\){VflB-\){rs^G-\)lT;s^ABG= -sf-a.f3y/m. Hence show that the differential equation of the herpolhode is d4>^ ZS(p'+E) <^P P'J-{p^-o.){p'- fiW-y) Ex. 4. In the case (§259 above) in which r:;'^=\IB, show that this differential equation reduces to d4>_ and that therefore dp 'jBpsf^^' P ^'Jfi so that the curve has the form shown in Fig. 66. (See equation (10), §153.) [These examples are mainly due to a Note by M. Darboux in Despeyrous' Traiti de la Micanique.'] Ex. 5. Erom Euler's equations of motion of a rigid body turning about a fixed point under the action of no forces, deduce the stability of the motion, when the axis of greatest or the axis of least moment of inertia is the instantaneous axis. Let the axis of rotation coincide with OA, then the equations of motion are Ap = 0, Bq = 0, Cr = 0. If, however, the axis deviate slightly from OA, and the angular speeds be Po+p', q, r, where p', q, r are small, then we can show that in certain circumstances q, r can never become large. The equations of motion are now, if products of small quantities be neglected, .4^' = 0, ^g'-(C-J^)r^o=0' Or-{A-B)p^q^O. Diflfer- entiating the second equation and eliminating f between the result 470 . A TREATISE ON DYNAMICS. [CH. VIII. and the third equation, we obtain Now, if {A-C)(A-B) be positive, that is, if either A>B>C or C>B>A, this equation can be written q + n^q=0, (2) where n'={A - C){A - B)pyBC is real and positive. If initially q = qo and q = qo, we get . q — qo cos nt+^ sin nt (3) But initially qQ = (C-A)rQpJB, and therefore C—A q = qaCOsnt+^;^ roposinnt (4) Thus if, as we suppose, q^ and qg be small initially, q can never acquire more than the small value given by the last equation, and a similar result can be found for r. The instantaneous axis therefore remains in proximity to OA. This will be seen more clearly if we find the position of the instantaneous axis. By the relation B^ = {C- A)rp^, we get for r the equation r = Bql(C- A)pg, or Bn go, ■Gpo^ Hence, for the angle AOI which the instantaneous axis makes with OA , we have /-r- — s tanJO/=^V±r! (6) Po For the angle ^05^ which the axis of resultant angular momentum Off makes with OA, we have UnAOffy^y^^' (7) ■^Po Thus, if A be the greatest moment, the fixed cone lies within the moving cone. Ex. 6. Show that if A be the greatest or least moment, and B=G, that the instantaneous axis describes in the body a right cone round OA, the axis of figure, and that this cone rolls on a right cone fixed in space. We have here 7fi = (A- Bfp^jB^ or n = {A- B)pJB. The equations for q and r become q = qo cos nt - j-q sin nt, r=rg cos nt + ^o sin nt, or q=Rcoa{nt + e), r=Ra'm{nt+e), where R=>Jq^^ + r\ and tane=>o/?o- The resultant of q and r is there- fore an angular speed about an axis which lies in the plane of B and C, and makes an angle nt+e with the axis OB. That angle increases at rate n, and the resultant axis moves round from OB r=rnB, and in the contrary direction ii AB, n is positive, and the axis 01 moves round OA in the body in the direc- ^ — ^^ tion of rotation. The /•-' A. WT^i /'''^•.'^ ^^N^.'A angle which 01 makes with OA is tan-i(\/?f+^/po). that is, tan~^(fl/po), that which OH makes with OA is tan-i(55/4p„), which is less or greater than the former according as A>B or i,'Pi sin Q along OP-^ and ^\P\ sin ^ — ii>. But a, 6 are the sides of a spherical triangle the vertices of which are the intersections of OL, OA, 01 with a sphere of unit radius described about as centre, and <^ is the angle between these sides at P. If f (see Ex. 5) be the third side, sinasin Scos<^=cos{'-cosacosft Hence ^i sin % = w cos f — u cos o. cos B. But co cos ^ is the turning about OL, that is ^TjE, and w cos d = b>i. Hence ^j sin-'a = -gr — 0) J cos 0'—^~-t cos'a, since Wi = ffcosa./A. This gives the rate at which the plane AOL is turning away from a plane intersecting it in OL and fixed in space. Similar insults hold for the planes BOL, COL, inclined at the angles b, c to the fixed plane. 7. Show that if the invariable line OL make angles with the principal axes the cosines of which are a., /3, y, Euler's equations, for a body with one point fixed and under no' forces, may be written in the form BGa. — H{B-- G)/3y = 0, with two similar equations. Taking along with the equations of last Example, written in the *°™ 6L~B{\IC-\IB)l3y=0,..., the equations of Ex. 6, ,, „i 2r Ha? show that if two bodies of principal moments A, B, C, A', B, C, be initially placed with their principal axes parallel and be set in 474 A TREATISE ON DYNAMICS. [CH. VIII. motion by parallel impulsive couples H, H' which fulfil the relation H{\IC- ljB) = n'{llC'-llB'), ... , prove that after any time t the prin- cipal axes will still be equally inclined to the axes of the couples. 8. Prove that if the kinetic energies of the two bodies in Ex. 7 be T, r, B_W_H_B^_H_H;^(T_r\ A A'~B B'~ C C'~ \H H'P and hence that the angles between the corresponding planes AOL, A'OL', ... will increase at constant rate ^{TjH- T'jH'). 9. A body free to turn about a fixed point is impulsively set into motion : to find the equations of motion about principal axes fixed in the body. [Let L, M, N denote the time-integrals of the impulsive couples L, M, N, and use Euler's equations.] 10. To find when an impulse applied in a given straight line to a body movable about a fixed point produces no rotation of the body about a perpendicular from the fixed point to the line. Take the direction of the impulse as Ox and the perpendicular as axis Oz. Then, impulse about 02=0, and (03=0, if D=E=0. Oz is then a principal axis for the sections xOz, yOz. 11. Find the direction of the axis of the impulse of moment H in order that the initial kinetic energy of the body may be a maximum. Here 2'=^(LV4-1-M 2/5-1- N^/C). Let A>B>C, then clearly T will have its greatest value if N be made equal to H, and L = M=0. The impulse should be applied about the axis of least m.i. 12. To find the values of L, M, N for fixed axes. These are given by (3), § 170. They are Ao)i-Fiy- io^x. "Rut z=OP, x=y=0, and therefore V=w^, a),=0. Also N=0, that is, G//-. We have ^=6„.|oZ_| (6) By the value of -o~^i j'^st obtained we can write this in the form ^ ;^a V-c l-r, (7) ^ bnz^-z^l-o- The quantities z^—c, -q — ^i are both small: their ratio depends on the value of c. The value, however, of the ratio (1 — c^)/(l — S-) is (^since o can differ but little from Sj) verj' nearly 1. We see from {2) that c„>-z,, that is, -:^) is the value of s for the upjw limiting circle, and Cj is that for the lower. We notice that when c = So, that is at the upper circle, 1^ = 0, or the curve of intersection of the axis with the spherical surface is that shown in Fig. 115, for it is clear from (2). (.3) and (7) that yjf^{l — :-) is small in com- parison with $ when z approximates to z^, though both 6 and \jr appi-oach zero, and the path meets the upper circle in a series of cusps. Again, when o — z^, we have yj/- = a'bn = '2Wgh,Cn. We shall see that the average value of \^ is Wgh Cn in this case. 266. Approximate Solution for Bapidly Rotating Top. In the present case, Sj— 0^ i^ small, and 6 lies between 0^ and 0j. Let 6 = 6Q+th where 9 is a small quantity. We get cos(? = cos0,— j;sin9a, that is, c = Cg — i;sin0,. Now, substituting in (4), g 265, we get, neglecting terms affected by the factor »//«'-, and remembering that 1 — c^ = sin*0g, ij* = a »; si n 0g — 6=^ H -ij- (1) From this we obtain bndt= ^ J(t-sin^^o / I rtsinOo VP I Wn* V' 2 bh^- ) ] 484 A TREATISE ON DYNAMICS. [CH. IX. and so j; = a sin 6(,{l~ cos bnt)/2b^n^. Hence we have 6 = d,+'^^(l-coBbnt), (2) and therefore 0i = 0o + 5£isin0o (3) It only remains to determine -/(tx. - a cos 0) sin'''6' -{/S-bn cos 0f' But, initially, 0=9o, and 0=\p=0 ; and therefore a. = acos^Q, fi = bncosda- Substituting in the last equation, reducing, and writing p for \bVja, ^« g«* d±^ sing dO J\ - 2p cos 0Q +p^ -{p-cos 0y Now, at the other limit 0i of 6 we have cos g=p-\/l--2pcosgo+;t)^,. and therefore we get, by integration, cos^o+j» 488 A TREATISE ON DYNAMICS. [CH. IX. and for, /? the value of this integral between the limits 6= do and 6 = cos~'{^ - \/p^ - 2p cos Oq + i\. Hence s/p^-2pcos6f, + l Vp'-'-Sjocos^o + l O 1 • -1 P- cos On or j8 = i7r-sin , ^ " - ■ \/jo2-2)OCOS^„+l This is, of course, also the measure of the angle turned through by the axis of the top on the surface traced out by the axis. The expression found for /3 gives Vjo^-Spcos ^0+1 and it follows that sinj8= / " V p^ - 2jB cos ^0+1 Hence, we have (p - cos 6o) tan /3 = sin do, the relation to be proved. 270. Steady Motion of Top Rapidly Rotating about a Fixed Point. Stability of Steady Motion. It is proved in § 266 above, that if a top be set spinning about its axis of figure at a high speed and then be left to itself, with one point fixed, it will perform small oscillations about a state of steady motion between narrow limits of 6, the smaller of which is the initial inclination of the axis of figure to the vertical. But as an example of a method which is of frequent application, another discussion is here given. We have, § 261, the equations Ae + (Gn- A\p- cos e)\ir sin e= WghsmO,] (j) Cn COS d+A\lram^e = G. / We notice first that if 6 be changed slightly by action which has no moment about the vertical, d(GncoHe+Axiysm^e) = (2) The peculiarity of steady motion is that 6 is permanently constant, so that = 0. Hence \^ must be constant also: let its value be /jl. We get therefore by the first of (1) for steady motion, the equation (On - A /JL cos 6) fi=Wgh (3) The factor sin 6 is dropped as we do not suppose that = 0. §§269,270,271] STABILITY OF TOP. 489 This is a quadratic equation in fi. The condition that its roots should be real is that CV > 4J. Wgh cos ft Unless this condition is fulfilled, steady motion is not possible. For example, a top cannot spin upright in steady motion unless (7 W > 4 J. iy(//i. We shall return to this question presently. Now let the steady motion be deviated from, so that the inclination becomes 6 + a., where 6 is the steady value, and the azimuthal motion, or precession as we shall call it (see § 275), becomes fi+t}. Substi- tuting in the iirst of (1) and in (2) multiplied by sin 6, combining the results and using (3), we obtain, as the reader may verify, A^lJ?!i.+(AY-'^WghAti?(iose+ Wy/j2)(x.=0 (4) The quantity in brackets can be written as the sum of two squares, and is therefore positive. Hence the deviation from steady motion is simple harmonic. The period is qr^ 'JttAix (4 V - 2 WghAii? cos 6+ Wyh^)^ If the motion had been unstable, the period would have been imaginary. The result shows that if a top is in steady motion, and is slightly disturbed without violation of (2), the motion is then one of oscillation about the state of steady motion, in a period which is shorter the greater the spin. The period here obtained is a more exact value than that, 2TrA/Cn, found in § 266, to which, however, it reduces if the terms in /a*, /i? be neglected. The two values of /x given by (3) are Cn 2A cos ,(.W.-i^o»4 Either of these values of /i is possible and may be realised by stai'ting the top properly. The smaller root, which approximates to Wg/i/Cn, or, more exactly, to Wgk{l+(A Wgh cos d)IC'^n']ICn, when n is great, is that which applies when the top is held with its axis inclined at some angle 6 to the vertical, set into rapid rotation by the unwinding of a string, and then left to itself. The motion is not then strictly steady, but is one of oscillation through_a small range of 6, and a range of ji, from twice the initial value of ^ to zero. For truly steady motion, the top must, besides being set into rapid rotation, have given to it at starting the proper amount of azimuthal motion fi. 271. Graphical Representation of Condition of Stability of Steady Motion. The dynamical stability can be illustrated by a very elegant geometrical construction due to Sir George Greenhill. In "Fig. 117 OC is the vertical, OC the axis of the top. OC and OC are made of lengths to represent respectively the angular momentum 490 A TREATISE ON DYNAMICS. [CH. IX. G about the vertical, and the angular momentum Cn about the axis of figure. These are components of an angular momentum OK in the plane COZ, obtained by drawing lines in that plane at right angles to OG and OC, and the line OK to their point of meeting K. KM is also drawn vertically, and KN is drawn parallel to OC' to meet OG in N. [From K the line KII at right angles to GK is drawn to represent A6 ; and so 05" represents the resultant angular momentum.] OK is equivalent either to the two components OC', G'K or to the two OG, GK. Now G'K is A^ain 0, so that CK= OC ame-C'Kcosd=(Gn-A4^ cos 6*) sin 6. Also MK=ON= CK/sin e=A4>, and NG= Cn cos 0-A\p cos^ 6 =(Cn -Atp cos 6) cos 0. Thus NK^Cn-A^ cos 6. Now, if the motion is steady, 6=0, and H coincides with K. Let the steady value of tp be /u. The point K then moves round OC, keeping 6 constant, at such a speed that angular momentum in the §271] STABILITY OF TOP. 491 difection of the motion of K, measured by the speed of K in that direction, grows at rate CK . ij,= Wgh sin 6- But CK={Cn-Aiico&e)sme, and therefore we have {On — Afi cos 0)[J.= Wgh, the quadratic equation foi' fjt found in § 270 above. By the diagram, Wgh . . Wgh , ON MK ,. = -^sme=^|. and ^=^ = ^- Thus we get MK . NK=A Wgh, and thus, for OC with the given length and inclination d to the vertical, K lies on a hyperbola of which OC and OC are the asymptotes. If E be the middle point of C'F, we get C'E'' - KE'^=C'K. KF=KM. NK. sin tan ^= Wgh sin 6 tan 6. If the line C'A' intersect the hyperbola again in K', another value ;u,' of the azimuthal angular speed exists for K', and is the larger root of the equation. When the roots are equal the line C'K touches the hyperbola. Then KM = \OF, KN = \OC', and therefore A Wgh = KM. NK=iOF. OC'=iCVsec 6. Hence Cn — 'i.J A Wgh cos 6, tJi. = iWgh/Cn. It will be seen that the hyperbola depends only on the angle 6, so that if OC be too short Ck will fall below the vertex of the branch of the curve shown dotted in the diagram, and steady motion will be impossible. The roots of the quadratic are then imaginary. What happens, when the top is started with the given a.m. Cn, at a given inclination 6 with 6 and \p zero, is, first (since the term JCm^ of the kinetic energy remains unaltered, while terms iA6^ a,nd ^ Aij/^ sin^ 6 are called into existence at the expense of the potential energy) a sinking of the axis below the inclination 0. This sinking continues while 6 increases, and 6, at first a maximum, diminishes until when is zero S is a maximum. At that instant tp has the steady motion value /x, as appears from (1) of § 270. Fig. 117 shows that at the starting of the top K lies within the hyperbola, and that when 6=0 the value of ip is the smaller root of the steady motion equation just referred to. It cannot possibly be the greater root unless a sufficiently large initial value of \// is given to make (On -A\p cos 9)4' > Wgh, when, by (1), § 270, AS must be negative, and the axis will rise toward the point K. After 6 has thus become a maximum, and K has reached the hyperbola, the axis continues to sink, and becomes negative. We 492 A TREATISE ON DYNAMICS. [CH. IX. have then {Cn - Atp cos 0)\p >■ Wgh, and xp increases, until it attains a maximum value just when the absolute value of 9 is greatest, as we see from (3) of § 261, for then 6 = 0, and therefore ^ = 0. Then the absolute value of 6 diminishes, a negative value of 6 grows up and the axis rises. Unless the initial position is such that the line C'K intersects the hyperbola, there does not exist a value of xp, with which if the top were started it would continue in steady motion. 272. Additional Couple about OD. Effect of forcing Pre- cession above Free Value. Now let an additional couple N about OB be applied to the top, say by the action of a ring similar to that which constrains the model in Fig. 121, so that the whole moment about OD is WghwaQ + N, and let the top be in steady motion in these circumstances. We have then the equation {On — AfjLCOsO)//. sin 9 = Wgh sind + N or ^^= -A fjL^ cos d +071/1 -Wgh (1) If yUj, yUg (yUj >■ fi^) be the roots of the equation Afi^ cos 6 - Gnfi + Wgh = 0, we can write (1) in the form N A nr \/ X KM.NK ,„ , .„, ^^ = Acos6(Hi-fi){fi-H.,) = 2 ^3fi •••(2) by § 271 above. According as N is positive or negative JUL does or does not lie between fi^ and /Zg, that is, the point K in Fig. 117 does or does not lie within the hyperbola which gives the values of fi for free steady motion. But if N be positive, it must arise from the exertion of a force by the ring on the axis tending to increase 6, so that the axis presses upwards against the ring, that is, the outer end of the axis tends to rise. On the other hand, if If be negative, the point K in Fig. 117 lies outside the hyper- bola, and the axis tends to fall from the ring. In free steady motion no ring is required, but it is now clear that any increase of the precessional speed /i from the value /n^ will, in the absence of constraint such as that §§271,272] EFFECT OF PRECESSION. 493 given by the ring, cause the outer end of the axis to rise, and that any decrease of fj. will cause the axis to fall. It is shown in § 261 that a.m. is produced by the motion about the axes 00, OE at rate (On — A\jr cos 6)^^-810. 6, and so we have the equation of free steady motion (On — Ajj. cos 6)iu sin = Wgh sin 9, which can be written, without change of signs, Acose{fJ.i-fx)(fi-fi2) = (1) If this equation is fulfilled because in = /J.2> the smaller root of the quadratic in fi, any sudden increase in jul, without change in 6, must give the quantity on the left a positive value, that is make {On — A\fr coa 6)\j/- sin 6 exceed Wgh sin 6, and so by (1), § 261, A6 must be negative, that is begins to acquire a negative value, and the top rises. On the other hand, if the equation is fulfilled because /« = /«i, the greater root, any increase of jul beyond that value will make (Cn — A^JA cos d)\fr sin — Wgh sin 6 acquire a negative value, that is AO must be positive ; in other words, 6 begins to ac- quire a positive value, and the top falls. Similarly dim- inution of fi from the values jUL^, jn^ causes the top to fall and rise respectively. It is important to notice that the common rule " hurry- ing the precession causes the top to rise, delaying the pre- cession causes the top to fall " is not, as it is usually given, correct. The efl^ect of either depends on whether the smaller or the larger of the two possible values of fi is that of the steady motion. In the majority of cases which occur in experiments with tops, it is the smaller value of fj. which characterises the motion, Fig. 118. 494 A TREATISE ON DYNAMICS. [CH. IX. and so the rule in its ordinary form gives results in accordance with experiment. Fig. 118 shows the effect of imposing precession about a vertical axis in a balanced gyrostat. Precession about a horizontal axis is produced. 273. Reaction of Bing-Guide or Space-Cone on Top. If, as in the model of Fig. 121 and in the toy shown in Fig. 119, the top be furnished with a material cone or axle, fixed round the axis of figure, which rolls on a cone fixed in space represented by the ring in Fig. 121 or the curved wire of Fig. 119, and the point of support be at the centroid, the couple on the' top must be applied by the pressure of the fixed against the moving cone. The circle of the points of contact on the moving cone is the polhode on the top, and the fixed ring or curved wire is the herpolhode. (See Chap. VIII.) The pressure of the axle on the ring-guide, that is of one polhode on the other, is to be found from the calculation of the rate of growth of a.m. given in §261. This is the rate of displacement of the extremity H of the vector OH representing the a.m., and is clearly about an axis at right angles at once to the axis of figure and to the vertical, an axis, therefore, which may be represented by the axis OD of Fig. 112. For OH is always in the plane ZOG of Fig. 112, which is perpendicular to the path of the point / of the instantaneous axis along the guide. But the A.M. grows in the direction OB at rate Ae+{Gn-Aylr cos 0)i/r sin 0, and therefore, if N be the couple, Ad + {Gn-AyjrCOB^)ylrs\ne = N, (1) or if the motion is steady, (On -Aficoad)/!ism6 = K (2) This equation is sometimes written in this connection* in the form {Cw-(A-G)fxcose}fismd = N, (3) *See Klein and Sommerfeld, Theorie des Kreisels, p. 173, where, however, a different mode of derivation is used. §§272,273,274] EFFECT OF PRECESSION. 495 where w is the rate of turning of the top relatively to the plane ZOO (§261). If ^ = 0, we have the steady precessional motion, under couple N, of a spherical top, that is, the equation is C(n — fi COS 6) ft sin 6 = N, (4) as in §272 above. We shall see below that the term introduced by the inertia of the case of a gyrostat enables a similar equation of steady motion to be obtained for that form of top (§281). The pressure on the ring is N/l if I denote the distance of the point of contact of the axle with the ring from the point of support. If a slight push or blow be given to the axis of the top, an impulsive couple is applied which produces an increase of the component .4i/rsin0 of a.m. about the axis OE, that is, changes -yf/- to \ir + S\fr, if 6 is kept unchanged by the guide. This increase in i/r makes the rate of growth of A.M. about OD more rapid than is acc^Snted for by the couple N, and so the top endeavours to turn about OD in the direction to keep the rate of change of a.m. the same as before, that is so as to press with so much greater force against the guide, that the enhanced value of N is that required for the greater precession. [The reader should as an exercise verify this by the consideration of an actual case, drawing the momentum axes, and determining the sense of the couple iV.] 274. Explanation of Clinging of Axle of Top to Curved Ghiide. The action of the top shown in Fig. 119 is very curious, but its explanation may be made out easily from the above discussion. The axle rolls round the curved guide following all the convolutions, however sharply curved, and on coming to the end of the guide in one direction turns rapidly round the end of the wire and rolls back on the other side. The axle has been described as clinging to the wire like a piece of iron to a magnet. For simplicity we have supposed that there is no gravi- tational couple on the top. The action of the guide may be analysed as follows. Consider a right circular cone 496 A TREATISE ON DYNAMICS. [CH. IX. with the vertical through as axis and the hne 01 as a generator; a short element of the guide at the point of contact is at the intersection of the guide and a circular section of the cone. Such a cone may be made to pass through any element of the guide, and 9 is now the semi- vertical angle of that cone. The element will in general give a component of action on the axis of the top in the plane through the axis of the cone. We have for the couple applied to the axle in the plane through 01, the equation Ad + (On -Ayjr cos 6)^^31116 = ]^. (5) Fig. 119. Besides this couple N, a reactional couple in the tangent plane to the cone through 01 is applied to the top. For clearly a component F of reaction of the guide acts on it at / with or against the direction of motion along the circular section, according to the angle between the section and the guide, and F and —F inserted at the point of support give a couple of moment iV^', the axis of which is at right angles to 01, in the plane COI. This can be resolved into two components N'sina., N'cosa. {a. = IOC) about OG and OE (at right angles to OC) in the plane COI. The former couple of comparatively small moment alters the speed of rotation, the latter gives change of \Ja at numerical rate ^sin0 = iV"'cosa, §§274,275] ASTRONOMICAL PRECESSION. 497 where 6 is negligible. The axle therefore presses more or less on the guide from this cause. There is also a frictional couple which in general splits into two components, one with or against N, and the other helping or retarding ■^, according to the direction of the guide. Now let the axle come to a discontinuity in the guide, for example one of the ends. The couple N may be regarded as there suddenly annulled, and therefore (since any thing like steady motion ceases) .40 as taking at the same time a value — {Cn — Ay^r cos 0)'— -"" ; ""'--^^ applied by gravity to the top. ^ IN eboo '^- '^^^ result is the same; just as YE^RS ; the top does not fall down, but has an azimuthal motion in virtue of the couple, so that the ; /' axis of rotation, if the motion i IN 13000 ^^ steady, moves in a right cone, '^°^ \^ 1 y^/(rs so the earth's axis does not \ I / approach perpendicularity to \ ; /' the ecliptic, but, relative to the *';'' earth's centre regarded as a fixed Fig. 120. point, has a conical motion in space about a line drawn from the earth's centre to the pole of the ecliptic, which answers to the vertical OZ in the case of the top (Fig. 120). The angular speed of a point on the earth's axis about the axis of the cone is M\Cn sin Q, where J/ is a certain mean value of the moment of the couple referred to above as applied by the sun's attraction. This is exactly analogous to the value Wgh sin Q/Cti sin 0, which the theory of the top gives for the precessional motion of angular speed i/r^ about the vertical. The conical motion of the earth's axis has a period of 26,000 years, and causes the astronomical phenomenon of precession of the equinoxes, that is the continual revolution of the line of equinoxes in the plane of the ecliptic. This is illustrated by Fig. 121, which shows a terrestrial globe with the lower half cut away, and the upper part §275] ASTRONOMICAL PRECESSION. 499 loaded so that it can turn about a point of support at the centre, with the pin P in contact with the inside of the horizontal ring RP at the top. The pin P is the upper end of a cone iixed on the body, having its vertex at the centre of the globe ; this cone rolls on a cone fixed in space. The latter cone is represented by the ring RP, which is enough to guide the moving cone : all the rest is cut away, but it is understood that the vertex in this case is also at the centre. As then the globe turns about the axis of figure the cone P rolls on the fixed cone, and travelling round the axis of figure describes a cone in space, in the model a cone of 23° 27' semi-vertical angle. The equator of the globe is shown by the dark line intersecting a meridian through P in N. The upper surface of the rim, to which the supports of the ring R are attached, represents the plane of the ecliptic, and the point N represents the intersection of the equator with that plane. N therefore represents an equinox. As the globe revolves in the counter-clock direction (as seen from outside P) the pin P rolls round the ring in the clock direction, and so the point N moves from right to left along the ecliptic, in the direction to meet the rotation, that is to make the equinox occur earlier in time. This is the precession of the equinoxes, which is thus completely illustrated by the model. Ex. Supposing the model enlarged to the size of the earth and to spin with tne same speed as the eaith, find the diameter at the north pole of the cone fixed in the earth with vertex at the centre, which, rolling on the internal surface of a cone of semi- vertical angle 23° 27' FiQ. 121. 500 A TREATISE ON DYNAMICS. [CH. IX. with its vertex also at the centre of the earth, gives precessional motion of 26,000 years' period. The rolling of a cone iixed in the body on a cone fixed in space represents exactly the steady motion of a top. The body as it rotates about the axis of figure with speed n has each point of that axis carried round the vertical OZ with angular speed i^^- The point I in Fig. 112 is there- fore, in consequence of the rotation about OC, being carried in the direction from the reader, while, in consequence of the turning about OZ, it is being carried towards the reader. Let the position of I be so chosen that the one motion just counteracts the other. Then, as we shall show, the body is turning about the line 01, which is the instantaneous axis. 276. EoUing of Body-Cone on Space-Cone. As shown in the figure, I lies on two circles described about OZ and OC as axes. Denote the angle IOC by a., then Z0I=6 — (X. The radii of the two circles are O7sin(0 — a) and 0/ since But / has speed at right angles to the paper, of amount n . 0/sin oc, due to the rotation about OC and speed ■xjrsmO . Oleosa., due to the rotation i/r sin about OE. Thus we have tana. = t^^ (1) n The resultant angular speed is thus about 01, and is 01 always lies in the vertical plane ZOC, which turns round OZ with angular speed 1/^9. Hence, if 6 does not vary neither does a., and 01 moves round OZ in the cone of semi-vertical angle IOZ=Q — a., the cone fixed in space. It will be noticed that the moving cone rolls in this case on the convex surface of the cone fixed in space, and that therefore precessional, or azimuthal, motion is in the same direction as the rotation. In the case of the earth, the moving cone rolls on the concave surface of the fixed cone, that is inside the latter, If this be called positive pre- cession, that of the top is negative. §§ 275, 276] BODY -CONE AND SPACE-CONE. 501 We can now analyse the motion in the following manner, which gives a geometrical picture of what takes place. Consider two axes OA and OB fixed in the body, at right angles to one another and to OC, and therefore principal axes about which the moment of inertia is A, to coincide with OD and OE, and let a short interval of time t elapse. The moving cone has rolled forward on the fixed cone, and the instantaneous axis is now 01'. The change of direction lOI' on the surface of the cone is towards the position which OA occupied at the beginning of t, that is towards the position then of OD. The angle lOI' is clearly i/fT sin (6 — a.). By the turning of 01 towards OB in this way the angular speed about the position of OA at the initial instant of t has (as we see by the principle already frequently applied) been increased by \/n^ + \j/'^am^6.cos{7r/2 — \irTsin{d — a.)}, that is by Jn^ + \fr sin^0 . \jrT sin (0 — a.). [The semi- vertical angle of the cone has in the. time t been increased by Or, but this has only moved the instantaneous axis parallel to the plane ZOO, and therefore can have produced no effect on the angular speed about OA.] Now the figure shows that sin (0 — a)/sin Q = {n — yp- cos 6)1 J n^ + ^j/^wa^Q, so that the change of angular speed just calculated is ■\jrr{n — \}r cos 6) sin 6. [For n—\}^cos9 is the angular speed about OC relative to the plane ZOC, and therefore ('n—\fr cos 6) sin a. is balanced by i/rsin(0 — ex.), so that sin(0 — a)/sin a. = (n — i/r cos 6)/-\l/: But 8in0/tana = 'n,/i/r, and therefore sin (0 — a)/sin Q = (n—\j/-cos 6)/(n/cos a.) = {n-ypr cos e)lJn^+yp-Hin^e.'\ 502 A TREATISE ON DYNAMICS. [CH. IX. To this change of angular speed falls to be added any change Or which has grown up in the angular speed 6. The total rate of growth of angular speed about the instantaneous position of OA is therefore 6 + \i/-{n — \fr cos 6) ain 6 ; and this is the rate of change of the angular speed about OA in its position at the instant. We have proved (§ 170) that this is also the rate of change of the angular speed about OA as it moves with the body. The angular acceleration about the axis OD, the instantaneous position of which was taken as coinciding with that of OA, is uninfluenced by the rotation of the body with angular speed n — yp- cos 6 relative to the plane ZOO, and is therefore simply 6. The reader may in like manner find the position of the axis OH of resultant a.m., and find the equations of motion from a consideration of its motion. 277. Motion of a Top deduced from Euler's Eauations. The equations of motion of a top, with reference to the special axes 00, OD, 0^ which have been used above, are often obtained by means of Euler's equations, and to complete the discussion we indicate how that is done. We have to use axes fixed in the body: one of these is OC, the others are OA, OB, which are in the plane of OB and OE (see Fig. 122). Since OE moves with the plane ZOO, we may take EOB as the angle through which the body in its turning about OG has outstripped the plane ZOO. Denoting this angle by 0, we have

sin d, \ Aq -{C - A)rp =-Wgh sin ^ am d, \ (2) Hence r =

sin 6 = Wgh sin 0, (3) which is (1) of §261. Multiplying the first equation obtained by the substitu- tions by sin is zero, and OA and OB therefore coincide with OD and OE, p = (^+\j/-(n — \l/- cos 6) sin 6, q = \fram d—6(n — 2\j/^co8 6). (5) The first of these agrees with the value obtained otherwise in § 276, and the second can be obtained in a similar manner. The reader should also obtain them by the method of § 9, proceeding as shown in § 261. The reader should also carefully note the fact here illustrated, that p, q, r, the angular accelerations with respect to fixed axes with which the moving axes OA, OB, OG coincide at the instant, are also (see § 170 above) the angular accelerations with respect to the moving axes 504 A TREATISE ON DYNAMICS. [CH. IX. OA, OB, 00. In other words, the change in dt of angular speed about OA, OB or 00 in their new positions is the same as for the fiKed axes, with which at the initial instant of dt they coincided. On the other hand, while the angular accelerations about the fixed axes, with which OD and OE coincide at the instant, are the values stated above in the equations for p, q, the accelerations about the moving axes OD, OE are simply Q, and d{yj/^ sin Q^jdi. The former is less than p hy\ff{n — \jy cos 6) sin 6, and the latter greater than q by (n — xj^ cos 0)6. 278. Gyrostats. Motion of a Gjrrostat. The theory of a top given above applies with some slight modifications to _the-^otion of a gyrostat, that is, a fly-wheel mounted in a case or on a framework, and set into rapid rotation about its axis. Figs. 123, 124, and 125 show different gyrostats made for ditferent purposes. The first shows a fly-wheel with hea^V^y rim, mounted on an axis the ends of which are car^lly rounded points held in cup bearings, adjustable Fig. 123. by screws, and secured by locking nuts which prevent any possibility of loosening of the bearings as the wheel revolves. The bearings are attached to a case shaped to enclose the wheel and its axis; so that the central part §§277,278] GYROSTATS. 505 is a wide and shallow cylinder with at each side a longer, narrower cylinder surrounding the axle. Round the wide cylindrical box is a projecting edge, on which in the diagram the gyrostat is shown resting. ■ A part of the case surrounding the axis is cut away to allow the thread by which the spin is generated to be passed round the axle. A strong fine cord about 6 or Fio. 124. Fig. 125. 7 yards long has one end passed round the axle, and the two ends are then knotted together. The cord is then passed over the over-hanging pulley of a small electric motor, so that the plane of the now endless string is at right angles to the axis, and the string is crossed to give it a better grip of the axle. The motor is now started while the gyrostat is held by the operator, who pulls only slightly at first, so as not to stop the motor. After a time the fly-wheel will have been got into motion, and the string is cut by a blow from a sharp knife near where it is running to the axle, and runs off". A simpler form is that familiar to nearly everybody as a scientific toy, in which the case is reduced to a ring carrying the fly-wheel bearings, and provided with a stand on which the gyrostat can be placed in different positions. 506 A TREATISE ON DYNAMICS. [gh. IX. The gyrostatic action of a bicycle wheel is familiar to every one. The angular momentum of such a wheel is great though its speed of rotation be small. A simple form of gyrostat (or rather top) may be constructed, as suggested by Sir George Greenhill, by mounting a bicycle wheel at one end of a straight rod as axle, and hanging it from a fixed point by a universal joint at the other end. The wheel can then be spun by a stick placed between the spokes, and the phenomena of precession, reactions, etc., studied.. The gyrostatic action of the wheels of a vehicle (a rapidly moving motor-car or railway carriage, for example), moving round the curve, gives a couple aiding centrifugal force to upset the vehicle, which must be balanced by the reaction of the ground or rails. The reader may calculate this couple by the methods explained below. 279. Gyrostatic Stability. Two positions of a gyrostat which experiment and theory show are stable are indicated Fig. 126. Fig. 127. in Figs. 126, and 127. In the first, the gyrostat is supported on two stilts, one rigidly attached to the case and parallel to the plane of the wheel, the other merely a stiflf wire with rounded points, the upper of which rests loosely in §§278,270] GYROSTATIO STABILITY. 507 a hollow in the projecting arm seen in the diagram. The lower ends ol" the stilts rest on a metal plate. If the gyrostat is I'roo to oscillato in azimuth, it will be stable when thus supported. in the second case, the gyrostat is supported on gimbals, with its axis nearly vertical. It can thus turn its axis nway i'rom oi- towards the vertical in any direction. It has ill fact two freedoms to turn from the vertical, one about the axis of each giinbal ring. The upright position is thoroughly stable when the fly-wheel is spinning. The roinarkabm fact will be proved below that the gyrostat must be unstable for both freedoms when the fly-wheel is not rotating, otherwise it cannot be made completely stable by rotation. In point of fact only an even number of freedoms can ho rendered stable by the angular momentum. Fig. 128. Ill Kig. 128 a gyrostat is shown supported on a bifilar sling, arranged in different ways. In the third and fourth (liiigrains oi' this figure the two threads are crossed by putting one througli a ring placed in the other. Here azimutiial oscillations are possible. It is clear that the inclinatioual equilibrium in 1 and 3 is stable without rotation ; in 2 and 4 it is rendered stable by rotation of the fly-wheel. The azimuthal equilibrium in 3 and 4 is only rendeixvl stable by rotation. These arrangements are due to Lord Kelvin. [See Thomson and Tait's Natural Pli ilomph I/, §345x.] One oi the most striking experiments which can be made with a gyrostat is that shown, carried out in slightly 508 A TREATISE ON DYNAMICS. [CH. IX. different ways, in Figs. 125, and 12!J. In Fig. 129 the cased gyrostat is shown hung by its rim, while a weight is hung from one end of the part of the ease surrounding the axis. The gyrostat thus supported is pulled by the weight, so that it is acted on by two equal and vertical forces at a considerable distance apart, and would, if the wheel were not rotating, turn round so as to bring the centre of gravity of the whole under the supporting thread. But if the wheel is in rapid rotation, the axis of rotation remains approxim- ately horizontal while the whole revolves about a vertical axis. The axis of rota- tion of the fly-wheel turns round in a horizontal plane, that is to say, turning is produced about an axis perpendicular at once to the axis of rotation, and to the axis about which the vertical forces tend to turn the gyrostat. One almost naturally expects (though any other behaviour of the gyrostat than that which actually takes place would be , really unnatural), the axis to be tilted Q 2 down. This does not happen; the axis *" '" moves round sideways. The result is not, however, more wonderful than the azimuthal motion of an ordinary top under the action of gravity. The same thing is shown in Fig. 125, and perhaps in the latter case more strikingly. The whole gyrostat is hung by a cord attached outside the containing ring, and by its weight pulls the centre of gravity down. As before, the axis, if free to do so, turns round in azimuth. It is to be noticed that the direction of this azimuthal turning of the whole gyrostat is (like that of the top under gravity) always towards making the fly-wheel face in the direction in which it would face if the rotational motion of the wheel were produced by the turning moment, or torque, due to the weight of the gyrostat and the pull in the supporting cord. As the vertical line of action of the weight moves round with the gyrostat, the turning in Fig. 129. §279] EXPLANATION OF PRECESSION. 509 asiimuth goes on continuously, and is always towards giving to the system angular momentum about the axis round which gravity tends to generate such momentum. We shall refer to this azimuthal motion as the precession of the gyrostat, according to the analogy between it and the precession of the equinoxes, explained in § 275 above. The precession may be explained in an elementary way as follows. Consider a ring of balls contained in a circular tube as shown in Fig. 130. Let the balls move round in the tube in the direction shown by the arrows, while a couple acts tending to turn the whole system round the axis AB, so that C comes forward towards the reader. A ball when at B has no A.M. about AB, but as it rises above .4J5 it will, if the ring have any turning about that line, be made to take up such A.M. The ball will therefore press against the tube in the direction from the reader. Fig. 131. Similarly, a ball below, the level of B losing its A.M. as it rises presses against the tube in the same direction. The right-hand half of the tube is thus pressed away from the reader. It will be seen in the same way that the balls in the left-hand half press on the tube towards the reader. Thus the tube is made to take a processional motion about CD. The directions of the motions are shown by the circles in Fig. 131. 510 A TREATISE ON DYNAMICS. [CH. IX. 280. Experiments with Gyrostats. Bising and Falling of Ordinary Top. If the gyrostat is held in the hand with the axle in the line of the outstretched arm, and an attempt is made to strike a sudden downward blow with it, as with a mallet, the gyrostat gives a violent sideways wrench. The explanation of this is obvious. The downward turning of the gyrostat gives a rapid rate of production of angular momentum about a vertical axis, while the action of the operator has a moment, not about a vertical, but about a horizontal axis. The gyrostat as a whole, therefore, moves round sideways about a vertical axis in the proper direction to annul the production of angular momentum about that axis. When the gyrostat is supported by a cord, or on a glass plate or stone slab, so that a couple is applied to it by gravity tending to change the direction of the axis of rotation, it will be noticed that when the precessional motion is impeded by applying a couple round the vertical axis, the gyro- stat at once begins to fall down, and that if a couple is applied in the opposite direction, that is so as to hurry up the precession, the axis actually rises. It ^!on is thus, as was long ago Fio 132 " pointed out by Jellett in his Theory of Friction, that a top is made to rise in the first part of its spin and fall in the latter part. In the first part of the spin the rotation is so rapid that the point of contact of the peg with the surface of the stone slab is moving relatively to that surface in the direction opposite to that indicated for the precession in Fig. 132, so that the friction applied to the top gives a couple about its axis hurrying up the precession; in the latter part the spin is so slow that the point of contact is moving the other way, so that the couple due to friction delays the precession, and the §280] EXPERIMENTS WITH GYROSTATS. 511 top falls. [It is very instructive to experiment with two identical tops, one with a peg ground sharp, the other with a well-rounded peg. The former, if supported on a glass or marble slab, does not rise up from its initial inclined position — the latter does.] A dynamical explanation of all this will be found later; and the phenomena here described, though apparently not directly connected with the subject, will help to make clear the dynamical dis- cussion. Another experiment, which it is convenient to describe here, is made with the gyrostat (Fig. 133, § 283) spun as before. It is provided with a pair of trunnions, attached at extremities of a diameter to the edge surrounding the case in the plane of the fly-wheel. These rest in bearings on the two sides of this rectangular frame of wood ; and the gyrostat when thus supported, and the frame held level, has its axis nearly vertical. Moreover, the centre of gravity of the gyrostat (wheel and case) is almost exactly in the plane through the trunnions at right angles to the axis of rotation, so that there is little or no stability due to gravity with either end of the axis uppermost. The direction of rotation of the fly-wheel is shown by the arrow-head marked on the case. If then, holding the tray in his hands, the operator carries it with the gyrostat I'ound in azimuth in the direction in which the wheel is rotating, the gyrostat remains at rest so long as the azi- muthal motion imposed on the whole system coincides with the rotation ; but if the azimuthal motion is reversed, the gyrostat at once capsizes so as to bring its rotational motion into coincidence with the azimuthal motion. This will also aflbrd an illustration of the theory of the in- strument. Finally, consider the arrangement in Fig. 137, (like that of Fig. 129 without the attached weight). A gyrostat has the centre of gravity of the fly-wheel and the case (which is supposed to be symmetrical on the two sides of the fly- wheel) at the centre of the fly-wheel. The fly-wheel is spun rapidly, and the gyrostat is hung at the lower end of a long vertical steel wire, so that the axis of rotation is very nearly, if not quite, horizontal. If the gyrostat 512 A TREATISE ON DYNAMICS. [CH. IX. is turned round in azimuth, so that the wire is twisted, and is then left to itself, it swings in azimuth about the vertical in consequence of the torsional elasticity of the wire, performing also inclinational oscillations in the same period, and the period of this torsional vibration is much greater than that of the vibrations which the same system would execute if the fly-wheel had no rotation. The moment of inertia of the gyrostat round the vertical axis is virtually enormously increased. This arrangement is analogous to that of a large and very rapidly rotating fly-wheel supported in a certain way on board ship, with its axis across the horizontal line about which the ship rolls. If this wheel were of great enough moment of inertia and rotated sufficiently rapidly, it would virtually increase the moment of inertia of the rolling vessel and lengthen the period of rolling. The virtual increase of moment of inertia is proportional to the square of the angular momentum of the fly-wheel. This arrange- ment will be referred to again later. 281. Equations of Motion of Gyrostat. The equations given in § 261 above for the motion of a top require modification for a gyrostat to take account of the fact that only part of the instrument — the fly-wheel — has the angular speed n about the axis of figure. We suppose, however, that the distribution of matter is symmetrical about the axis of the fly-wheel, that the wheel has moment of inertia G about its axis, and that the rest of the arrangement, which we shall call the case, has moment of inertia G' about the same axis. Frequently a point on the axis of the gyrostat may be taken as fixed ; we shall denote then by A the moment of inertia of the whole about an axis through that point at right angles to the axis of figure. We refer to Fig. 112. First, then, we suppose that the angular speed n is only taken by the fly-wheel, while the case turns with the angular speed ■\fr about the vertical. The angular speed of the case is thus yfrCoaO about the axis of figure and \/rsin0 about OE, and the whole system turns about OD §§280,281,282] STEADY MOTION OF GYROSTAT. 513 with angular speed 6. The a.m. about OD is Ad, about OE it is Axjr sin 0, and about OC it is Gn+G'\j/- oos 6. Thus the rate of growth of a.m. about OB is (0%+ C'yj/- cos 0)\/r sin 6 due to the turning about OE, and — Ay}/^ sin cos due to the turning with angular speed i/rcos^ about 00. a.m. therefore grows about the instantaneous position of OD at total rate Ad + {On - (A - C')\p- cos 6} yjr sin 6 =Wgh sine. ....{!) In a similar way the reader may calculate the total rate ,of growth of A.M. about the instantaneous position of OE, and, since there is no moment of forces about OE, verify the equation A-\^ sin e+{2(A-C')-x}y cos 6 -On} 6 = (2) As before, we notice that this equation of motion is derivable from that of constancy of a.m. about the vertical through the fixed point, which is now (On +C'\i^ cos 6) cos d+Ayp-sin^e = (3) Equations (1) and (2) are exactly the same as those obtained in §261, with A—C substituted for A in the terms within brackets on the left, but not in the first term in each case. 282. Steady Motion of Gyrostat. Period of Oscillation about Steady Motion. We may find, in precisely the same manner as for the ordinary top, the condition of steady motion at a constant inclination 6 of the axis to the vertical, and the period of a small oscillation of the gryostat about steady motion. The equation of steady motion is {On - (A -C')^ cos 6} fi^Wgh (1) The period of oscillation is y_ 2Tfi{(Asin^e+G'cos^e)A}^ ] {Wyh^ -2WghfjL\A-G') cos d'y (2) + A{A-G')im*}^sine) If G' = 0, this reduces to the period obtained for the ordinary top. 6.D. 2 K 514 A TREATISE ON DYNAMICS. [CH. IX. It is interesting to notice that the top may be so con- structed that G' = A. In that case, the equation of steady- motion is Cnfi=Wgh, (3) and there is only one possible value of jul. The period becomes j,^_^zM_^^jrA_ (4) Wgh sin d Cn sin 6 283. Gyrostat with Axis Vertical, Stable or Unstable accord- ing to Direction of Azimutbal Motion. We now take some cases of gyrostatic motion. First, let the gyrostat be sup- ported (as shown in Fig. 133) by two trunnions screwed to the projecting edge in the plane of the fly-wheel on a wooden tray as shown. The axis of the fly-wheel is very nearly vertical, and the wheel is spinning rapidly in the direction of the arrow shown on the upper side of the case. The centre of gravity of the whole instrument is nearly on the level of the trunnions, so that there is no stability due to gravity. Fig. 133. If now the tray be carried round horizontally with constant angular speed fi in the direction of spin, the gyrostat remains quite stable. If, however, it be carried round in the opposite direction, the gyrostat immediately turns on its trunnions and capsizes so that the other end of the axis is uppermost, and if the azimuthal motion is continued in the same direction, the gyrostat is now stable. It will be observed that the fly-wheel is now spinning in the direction of the azimuthal motion. Hence the gyrostat is in stable equilibrium when the azimuthal motion is in the same direction as the rotational motion. §§2S2,283,284] GYROSTATIC PENDULUM. 515 This result follows from equation (1) adapted to fit this particular case. It will be seen that the terms mgh sin 6 and Aji^sinQcosB are small in comparison with Cnfjiwa.6, the former because /(- is practically zero and the latter because fx is small in comparison with n. Hence the equation is Ae + Gnne = Q (]) The solution of this differential equation, if n and /jl be in the same direction so that nfx is positive, is oscillatory motion of period ^Tr-jAjGiifi about the vertical position, so that this position is stable. On the other hand, if n and fi have opposite signs the solution of the differential equation is of another form, curiously connected with the former, but representing a different state of things. It shows that if the gyrostat is disturbed from the vertical position of its axis it tends to pass further away from it ; the instrument capsizes. These results are indeed indicated by the differential equation. The moment CnfiQ, producing rate of change AQ oi A.M., is in the first case in the direction to check motion away from the vertical position and to bring the gyrostat back to that position, while in the other case CnfxO, having tlie opposite sign, produces a.m. in the direction away from the vertical. It will be seen that in this arrangement of the gyrostat it has only one freedom of motion as regards inclination of the axis to the vertical ; it can turn about the trunnions but not about a horizontal axis at right angles to the line of the trunnions. Hence, as we shall now show, it cannot have complete dynamical stability. [See § 284.] 284. Gyrostat on Gimbals. Gyrostatic Pendulum : Analogy of Motion of Electron in Magnetic Field. Consider the arrange- ment shown in Fig. 127 of a gyrostat on gimbals. One end of the part of the case which surrounds the axis carries knife-edges in a line at right angles to the axis and intersecting it. These knife-edges are pivoted on a ring, which itself carries knife-edges at right angles to the bearings on which the former rest, and these in their turn 516 A TREATISE ON DYNAMICS. [CH. IX. rest on bearings carried by a second but fixed ring, or on a fixed support as shown in the figure. The gyrostat is shown with its axis vertical, and the two sets of knife- edges enable it to turn about one horizontal axis or about the other, or about both at the same time, so as to be inclined to the vertical in any desired azimuth. The two pairs of knife-edges are not exactly, but nearly, on the same level. The part of the case surrounding the axis may be supposed prolonged so as to give any required " preponderance " Wgh to the gyrostat above either axis. Let the total mass which turns about the axes formed by the knife-edges be W and W, the heights of the centroids above (or distances from) the axes be h, h', the moments of inertia about the axes be A, A', the respective angular deflections (supposed small) be \fr, cp, and the moment of inertia and angular speed of the fiy-wheel be C, n. We get then, by the process so often employed for the rates of growth of a.m. about the axes, fixed in the present case, A^r + Gn^ = Wgh^rA A'^-Gny^=W'gh'(l>] ^^^ or, if we write B= Wgh, B' = Wgh', Air + Cn -B-^ = Q,\ A'^-Cnyir-B'4> = 0.\ ^^^ Now let xfr = ae*"', = be'''K Then, by substitution, we get -i^i/M a-^-ivCnb- Ba = 0, l iVA'b-ivGna-B'b = 0, } ^^^ and therefore, since i^= —1, (v^A + B){vW+B') - v^C^n^ = (4) The quantities A, B, A', F are all positive according to the supposition made above, and the roots of the quadratic in v^ are real and positive if the inequality {Chi:'-AB'~A'B)->'\^AA'BB' is satisfied. This is the condition of complete dynamical stability, for, if it be fulfilled, \}r and represent simple §284] GYROSTATIC PENDULUM. 517 harmonic deflections from the position in which the axis of the gyrostat is at right angles to both lines of knife- edges — in the diagram the upright position. Each de- flection may have either of the two periods given by the two real roots v^, v\ of (4). The motion is therefore stable, and there are two modes of vibration which the gyrostat may take either separately or in combination. It is important to notice that if (contrary to the figure of course) B, B' have opposite signs and the product B, B' therefore be negative, one of the roots of the quadratic in )/^ is positive, the other negative ; and consequently there is only one possible mode of stable motion, for the negative root of the quadratic in v^ gives an imaginary period. Let now h = h', W= W, A=A', and let n-^, —n^, n^, —n^ be the four roots of the determinantal equation in v ; then, since the real and imaginary parts of x = ae'"'; y = be'"' must separately satisfy the difl'erential equations, and since the expressions for the ratio a/b exhibited above give a = ib, we get \lr = Li cos Tilt + L\ sin n^ + L^ cos nj, -\- L'^ sin nji,,^ / p. n (j> = L^ sin Thit — L\ cos n^t + L^, sin n-jj — L'^ cos n^tj whei'e Lj, Z'l, L^, L'^ are arbitrary constants. We see that the first terms on the right give a circular motion of a point on the axis in the period 2'7r/n-^, that the second terms give a circular motion of the same period in the opposite direction, and that the third and fourth terms give circular motions in opposite directions in the period 2%ln^. The radii of the circular paths are the values of i,, L\, etc. If we combine two of these circular motions, say those given by the first two terms, or the last two terms, on the right of equations (5), we get Fig. 134 as the path of a point of the axis of the gyrostat. [The rays are not drawn in to the centre.] For here the radii are equal, the periods unequal, and the motions of the circular components oppo- sitely directed. If we take the motions given by the first and third, or by the second and fourth terms in each equation. 518 A TREATISE ON DYNAMICS. [CH. IX. the path is as shown in Fig. 135. Here everything is as before, but the circular components are in the same direction. The resultant radius bisects the angle between Fig. 134. Fig. 135. the component radii, and the resultant angular speed is the mean of the components. When the two motions exist together, we have the path shown in Fig. 136. There the present arrangement inverted is represented by a pendulum with a fly-wheel rotating about the axis of figure contained in the bob, so that there is gravitational stability for both displacements apart from rotation. The theory is essentially the same in both cases. In the pendulum, however, the universal gimbal joint is replaced by a short piece of steel wire which bends easily but resists torsion very greatly. Without serious error, h may be taken as equal to h', and so the motions are circular, as we have seen. The period of describing the circle in one motion is 2'7r/n-^ and in the other ^-wln^. The student may verify that in the case of the gyrostatic pendulum shown in Fig. 136, where the Fig. 136. §284] GYROSTATIC STABILITY. 619 whole mass may be regarded as concentrated at the centroid, provided the fly-wheel have moment of inertia C, the periods are 4!Tr/{2p + k) and 4 ([>=d=0 is given ■' ^=^iSin(x.i< + ^2sino(.2<, t/) = fficos(x.,< + ^2cosoi.2<, (5) where, since when t=0, (^ = <^q, and 6=0, we must have a.i^i + a2i2 = 0, Ki + K2 = 4>o (6) Now, by (2), we have in any case, k^ ia.jCn i «.j.4' + t Ki ir'a^^A + Wga ia.-fin k.^ ia.,fin i^a^A'+T ,.(7) K.^ i a^A + Wga ia..^Cn I so that, again, in any case whatever, K^ _ (i^a^A' + T)(iV^A + Wga) k^ K^ i^a-^a^C^n^ k^ In the present case ^ia.i= -k^^, and so putting - 1 for i% we get K,_ (A'<4-rXAa^- Wga) _ K~ Wr?^ ^^^ We might have supposed the wire at rest without torsion and the 524 A TBEATISE ON DYNAMICS. [CH. IX. gyrostat at rest initially, but tilted through an angle d^- Then we should have had / j^ ■ t , ir ■ , \ 4> = Ki siu u-it + K.^ sin (x.jj«, I 6 = ki cos o-i* + ^2 cos (Xji, J with the condition Kia.i + K.p^=0, ki + k^=9o (10) Then we should have found also Ic^ (A'a.l-r){Aa.\-Wga) ^^^^ It will be noticed that if Aa.^ be small in comparison with Wga (which in § 286 was supposed to be the case), the two frequencies of vibration have approximately the common value I /"" Wgar 2ir y'GV + A'Wgar so that the period is T=2^J^^+fJS^ , (12) the result obtained above [3, § 286]. If the angular momentum Cn of the fly-wheel is zero, (3) becomes (A'a?-T)(Aa?- Wga)=0 (13) The first factor gives the period Ztts/A'/t of the free oscillations of the wire and the attached gyrostat, when the fly-wheel is at rest and the gyrostat is moving only in azimuth with the lower end of the wire : the second factor gives the period Ztts/AI Wga of the free pendulum oscillations which the gyrostat can perform about the point of attach- ment to the wire, when the wire is held at rest. By means of these periods, or the corresponding frequencies, the quantities t, Wga can be eliminated from the equations set forth above. 288. Gsnrostatic Controller of Boiling of Ship: Schlick's Apparatus. The solution here given is applicable to the oscillations of a ship in which is fixed a gyrostat G, as shown in Fig. 138. When the ship is upright and the gyrostat in equilibrium, the axis of the fly-wheel is vertical. The wheel is pivoted in a frame F, as shown. The frame turns on the bearings 6,6, and a weight W gives the arrangement gravitational stability. In an arrangement of this kind, devised by Herr Otto Schlick to diminish the rolling of a ship, a brake pulley B surrounds the axis 66, about which the frame turns, and friction of a graded amount is applied by a special device. The brake damps out the free oscillations of the system and also serves to reduce the forced oscillations. But the action of the brake must not be so violent as to prevent the swinging of the gyrostat, as that would annul the inertia effect, which is of the greatest importance for the forced oscillations, according to the principle illustrated in § 287. If the ship is set rolling in still water, the theory of the motion §§287,288] GYROSTATIC CONTROLLER. 525 is precisely that set forth above. For r/A', WgajA we write iiz^Ii"^, 4;ry^, where F, f are the frequencies of the free oscillations of the ship and gyrostat, the first oscillating with the gyrostat rigidly fixed within it, the second when the ship is at rest, in both cases without rotation of the fly-wheel. A', A are the moments of inertia of the ship and fly-wheel for the axes about which the ship rolls and the gyrostat frame turns. The equation of periods, (3) of § 287, is thus {a?-4.TT'^F%a?-Aw'^p) '2\ -JA''- = 0. .(1) Similarly the other equations may be modified. Fig. 138. When the ship rolls in a sea-way, the main oscillations of the ship are forced oscillations of the period of the waves, and the natural period of the ship is so increased by the gyrostat that any resonance effect, due to near agreement between the period of the waves and that of the ship, which might exist without rotation of the fly-wheel, is rendered impossible. The differential equations of small oscillations are, as we see at once from what has been stated above, A'^-\- N'j,- Cnd+M<^ = C' cos pt,\ (2) A9+Ne+Cn4,+ Wgad =0, I where A, A' are the moments of inertia of the gyrostat for the axis 6, h and the ship for the longitudinal axis about which she rolls, N6 is the frictional couple applied to the gyrostat frame by the brake B and otherwise, N^ is the frictional couple applied by the water to the ship as she rolls, M is the righting moment per unit of the angle of heel, Wga is the "preponderance" of the weight W, C is the amplitude, and pl^TT is the frequency of the forced rolling produced by the waves. The forced oscillations are given by supposing /,, tan ^=7, : — . , „,' (4) C»u)SinA+ Wgli ' The components of o) about the vertical and about the horizontal in the meridian are u sin A, and to cos A. The latter has a component (u cos A cos (^ about a horizontal axis towards the east of north at right angles to the line of bearings. This, in its turn, gives an angular speed about an axis perpendicular at once to the horizontal axis just specified and to the axis of rotation, of amount to cos A cos <^ cos ^. The component w sin A, about the vertical, gives a component, (u sin A sin 6, about the axis last mentioned. The precessional angular speed about that axis is therefore a)(cos Acos^cos^-sin Asin S). Hence, since the couple about that axis has moment Wgh sin 6, we ^^® Cnta (cos A cos <^ cos 6 - sin A sin ^) = Wgh sin 6, and therefore tan ^=7= -. — . , ,„, (4) t7Bo)sm A+ Wgh Here it is supposed that n and w are the same way round. If they are not, the denominator has the value Cnm sin A — Wgh, and the upper end of the axis is turned towards the south, instead of to the north as in the former case. 290. The Brennan Monorail Oar. In this invention gyro- static action is used to keep a carriage in stable equilibrium on a single rail, and the apparatus is entirely self-acting. It forms at once the nerve-system which detects the need for the application of a righting couple to the carriage, and mechanism by which the couple is applied. Two gyrostats are placed in the carriage with their axes in line, and trans- verse to the rail, as shown in Fig. 140. The wheels W, W are driven by motors and revolve about the axes AA, A' A', at the same speed in opposite directions, as indicated by the arrows. The wheels are enclosed in cases G, C, from which the air has been exhausted, and which turn about the axes BB, B'B'. The system can turn as a whole about the axis which is parallel to the rail. By means of two segments, B, B', above the apparatus, the gyrostats are made to take equal and opposite precessions, when any •precession occurs; then, of course, the axes cease to be in line. When the car is upright and in equilibrium, the gyrostats are upright, with their axes in line transverse to the rail. §§289,290] MONORAIL OAR. 529 Suppose, now, a couple to be applied to the car, say by a gust of wind, or the displacement of part of the load, so as to tilt the car over on the rail, to the right, say. In con- sequence of the rotation the axes of the wheels retain their directions, and the carriage turns relatively to the gyrostats. This brings the shelf D, which is fixed to the car, into contact with the spinning axis R of the left-hand gyrostat, Fig. 140. and the axes begin to be tilted. Each gyrostat therefore begins to produce by its motion a.m. about a vertical axis, and the gyrostats therefore precess in opposite direc- tions. This precession is assisted by the couple exerted by the force of friction on R, enhanced by slipping of the rapidly revolving spindle R bn the shelf D, which is in the direction of forward motion of R, that is, in the direction 530 A TREATISE ON DYNAMICS. [CH. IX. away from the reader, with the result that a restoring couple is applied to the shelf D, and therefore to the car. This couple, which is due to the acceleration of the pre- cession, is sufficient to arrest the tilting and turn the car in the opposite direction. The shelf D extends away from the reader, and on the right there is a corresponding shelf i)' extending towards the reader, as shown by the plan, on which the end R of the spindle acts in the case of a deflection to the left, as explained above for R. There are two other shelves E, E' which are arranged to come into contact with rollers 8, S', mounted on sleeves turning loosely on the spindles. The shelf E extends inwards towards the reader, the shelf -£" outwards. It will be clear that in consequence of the precession of the gyrostats brought about by the pressing of the shelf D on the end R of the rotating spindle, tihe roller >Si' has been brought over the shelf E'. Consequently, as the car swings over to the left in consequence of the couple applied by the gyrostats, the roller S' comes into contact with E'. Precession in the opposite direction to the former pre- cession is caused, but there is not now any accelerating couple, but really a retarding one, since the roller sleeve turning round on and supporting the spindle applies a friction couple to the gyrostats resisting the precession, which, it is to be remembered, is now back towards the mid-position. The gyrostatic axes do not, however, greatly alter their inclination to the horizontal while precession occurs in obedience to the couple applied by the pressure of the shelf on S'. As precession goes on, the axes of the gyrostats are brought once more into the line RR', with R lowered. They go beyond the mid-position and R' begins to roll on the shelf D', and so applies a frictional couple to the gyrostat, just as R did before, with the result that the gyrostats now begin to turn over and apply a couple to the car from left to right. The car tilts over, and the roller 8 comes into contact with the shelf E, the axles are brought once more into line, R presses on D and rolls along it as before, and a couple to the left is applied to §§290,291] GYROSTATIC ACTION OF TURBINES. 531 the car, and so on. Thus the car vibrates about an equili- brium position under the deflecting couple, that is a position in which it is heeled over to meet the couple (supposed still existing) through angles which rapidly diminish in amount. Finally, . the vibration has been wiped out, and the car stands in the new position of equilibrium. Thus the car is held over against the deflecting couple, if that is maintained constant. When the car runs on a curve the two gyrostats exert equal and opposite gyrostatic actions, and the car takes the curve without the gyrostatic resistance which a single gyrostat would have applied, and which would have been very inconvenient. The mode of action of the gyrostats on the car has been modified in various ways by Mr. Brennan in later models ; but the principle is perhaps sufficiently explained in the description here given of the arrangement which he ex- hibited to the Royal Society in May 1907. [See the article by Professor Perry, in Nature for March 12, 1908.] 291. Gyrostatic Action of Turbines in Steamers. Interest in the gyrostatic action in steamers in which the main propelling engines are of the steam turbine type was excited at the time of the Cobra disaster, and a series of letters from engineers and others appeared in the technical journals. These letters were informing in very varying degree, but the general conclusion come to was no doubt correct, that the gyrostatic action could not produce any breaking moment so great as to affect a ship's safety. For example, to break the ship, as the Cobra apparently was broken, by a breaking moment applied to it in a vertical plane, the ship's head would have had to turn round at an impossible rate. Rolling could bring no gyrostatic action into play, the axes of the turbines being fore and aft; pitching would produce a moment no doubt, as will be seen, much greater than the former, but tending to bend the vessel in a horizontal plane, that is, about vertical lines. The following discussions are based on authoritative estimates of the data necessary for the calculation of the 532 A TREATISE ON DYNAMICS. [CH. IX. gyrosfcatic moments applied in possible circumstances to the hull (1) of a large Atlantic liner (the Carmania), (2) of a torpedo-boat destroyer, and (3) of a cross-channel steamer. The mode of calculation will be clear from the preceding discussion. When, for example, the ship's head turns round, the direction of the axis of the rapidly revolving turbines is changed at a rate /m, the fji of the equations above; that is a precession of speed jul about a vertical axis is imposed. But to correct the generation of a.m. about an athwartship axis, which this produces, the turbines make an effort to turn about that axis, and so a couple is applied to the ship, and an equal and opposite couple to the turbines. Hence the turbines may be regarded as having a precession of angular speed /x in azimuth pro- duced by the couple just referred to, which, therefore, has the moment Cn/n, if On be the A.M. of the turbines. If the turbine rotors be equal in all respects, and run at the same speed, but in opposite directions, the total couple exerted on the ship, as a whole, will be zero. But each turbine will exert a couple on the ship at the bearings, and an opposite couple will give the precession ju to the turbine. Internal stresses will be exerted on the ship in consequence of the opposite couples, and the stresses will be a self-balancing system within the ship. A corresponding action of course takes place when the ship is pitching with angular speed fx. For the Carmania* the total weight of the rotors, three in number, may be taken as 200 tons, and the radius of gyration as 4 feet, so that in ton-foot units, the moment of inertia of the rotor on each wing-shaft is 1280, on the supposition that the weight of each rotor is f of the whole, and the moment of inertia of the rotor on the centre shaft is therefore 640. The number of revolutions is 200 per minute, and therefore the value of fx is 20^/3, in radians per second. The ship's head can be turned through |- of a degree, or about ^l of a radian in a second. Hence the gyrostatic couple of moment Cn/x which must be * For these data we are indebted to Mr. W. J. Luke, of Messrs. John Brown & Co., Limited, Shipbuilders, Clydebank, who built the Carmania. §291] GYROSTATIC ACTION OF TURBINES. 533 applied by the ship to each wing-rotor to give it the precession which the turning of the ship involves, and therefore also the moment of the equal and opposite couple exerted on the ship, is 1280x20TrXi X yVXT5\=ll'2, in ton-foot units ; that is, the moment is that which would be produced by a force of 11 2 Tons acting at an arm of 1 foot, or a couple of ^28 ton acting at an arm of 40 feet. Such a couple cannot have any perceptible effect in straining the ship. If we take 12° as the range of pitching, and the period as 6 seconds, the maximum angular speed is 27rx 6/(6x57-3)= 1/9, in radians per second, and this is to be substituted for the 1/75 in the above calculation. The couple is thus 8'3 times the former couple, or 2-3 Tons at an arm of 40 feet : still quite a small couple when regarded from the point of view of breaking the ship, even if relatively as lightly built as was the Cobra. The engines of the Cobra were, of course, very small as compared with those here considered. The gyrostatic couple due to pitching is, however, reversed twice in each (double) period of pitching. For a range of pitching half as much again, and a period of 9 seconds, the gyrostatic action would just be the same. If there were only two shafts, one right-handed, the other left-handed, the moments applied to the ship would be equal and in opposite directions. Of course, internal stresses of a kind easily analysed would be set up in the structure. These would tend to produce alternately com- pression and extension at the bow, and extension and compression at the stern, athwartships in each case; but they would be quite negligible. For three shafts, if two turn one way, and the third the other way, and the weight of the turbines be supposed distributed among them in the ratio of two parts to each wing-shaft and one part to the centre shaft, the resultant gyrostatic couple is much less than ^ of that calculated above, inasmuch as the radius of gyration of the centre rotor is only 3 feet. The couple may be taken as 9/32 of that due to each wing-shaft. The couples due to the 534 A TREATISE ON DYNAMICS. [CH. IX. wing-rotors being oppositely directed at each instant, will produce internal stresses, which can only be of importance in the event of their coinciding in period with a free oscillation of the ship as an elastic structure, an event which seems very unlikely. If, however, one wing-shaft be driven ahead, the other astern at full speed, so that the direction of rotation is the same in both, and the centre shaft be stopped, the gyrostatic couple (due to pitching) applied to the ship will be twice that due to each wing-shaft, or 186 Tons at an arm of 1 foot. If the centre shaft be at the same time driven full speed ahead, the couple will be that just stated, with 9/32 of its amount added or subtracted, according as the centre shaft runs in the same direction as the wing-shafts, or in the contrary direction. If the centre shaft is run at diminished speed, the latter couple must be diminished in proportion. For a destroyer the weight of each rotor may be taken as 6 tons, the radius of gyration as 2 feet, and the revolutions 900. This gives moment of inertia, in ton-foot units, 24 for each rotor on wing-shafts. The angular velocity is SQtt iu radians per second, and tlie angular velocity with which the ship can be turned round is 3° per second or yV oi a radian per second. The gyrostatic couple for the two rotors running in the same direction would be 48x307rx iVx-A- = 7"4, that is, 7-4 Tons at an arm of 1 foot. With the same period and range of pitching the gyrostatic couple for the destroyer would be about twice the couple just calculated. Here, again, to get the true values of the resultant couple, we must take one -half, or, if the vessel has triple screws, some other fraction of the values just found. For a cross -channel steamer, the following data have been furnished by the Hon. C. A. Parsons : weight of each L.P. rotor 7 tons, radius of gyration 21 inches, speed 700 revolutions. The moment of inertia of each rotor is thus 7x175^ or 21-4 in ton-foot units, and the speed is 707r/3, in radians per second. The maximum gyrostatic couple of each rotor, for the same amplitudes and periods §§'291,292] GYROSTATIC ACTION OF MACHINERY. 535 of pitching as those supposed above, is thus above 1 "8 Tons acting at an arm of 1 foot. If the turbine on the centre shaft lias, as Mr. Parsons states it has in this class of vessel, less than half the mass of the others, the resultant couple on the ship will be less than one-half of that just calculated. ' The stresses seem quite insignificant. Their only im- portance, if they have any, must be in their rapid reversal and the consequent forced vibration of the structure. Danger is not likely to arise from near agreement of the period of this forced vibration with that of some natural free period of the structure, but this is a question for naval architects. Nor are natural vibrations in the rotor itself likely to correspond in period with that of the gyrostatic couple. [See a paper by Dr. Henderson, Transactions of the Inst, of Engineers and Shipbuilders in Scotland, 1905.] 292. Gryrostatic Couple on a Locomotive or Carriage. Gyrostatic couples of practically insignificant amount have been found for a new locomotive recently built in Glasgow, part of which consists of a rapidly rotating steam-turbine and dynamo mounted with their common axes in the " fore-and-aft " direction. Numerical par- ticulars cannot be given here, but the couples due to passing round curves, or over parts of the track where the gradient is changing, can have but little efiect on the running of the engine. Ex. 1. A carriage, which has wheels of total moment of inertia G and radius a, runs on a curve of radius R with speed v : find the gyrostatic couple on the train. The angular speed of a wheel is vja, and the a.m. of the wheels is Cvja. Hence a.m. is being generated by each wheel of amount Cvjna . vlR-=Cv^lnaR per second, if n be the number of wheels, about an axis drawn from the wheel in the direction backwards along the track. In order to counteract this, the carriage will tend to turn about this axis in the direction outwards from the centre, until the couple required to produce a.m. at the rate due to the turning is applied to the carriage by the excess of pressure on the outer rail. Thus the gyrostatic action provides a couple of moment WjaR, which tends to upset the carriage in the same direction as the couple due to cetitrif ugal force, and is balanced with the latter by the action of the rails. 536 A TREATISE ON DYNAMICS. [CH. TX. If the track is on the level, and b be the gauge, the excess of thrust exerted on the wheels by the outer rail over that exerted by the inner rail, is Cv'^/baR. The ratio of this to the centrifugal force Mv^i'R, where M is the weight of the carriage, is C/Mab, and is obviously very small. Ex. 2. "Work out the action of the steam-turbine referred to above as mounted on a carriage with its axis in the fore and-aft direction. The A.M. of the tui"bine may be denoted by Cn, where n is the angular speed of rotation. As the carriage moves forward on a curve, there is a rate of production Gnvjlt of a.m. about an axis in the direction of the radius of the curve at the position of the turbine at the instant. This throws more weight on the front wheels and less on the back, or vice versa, according to the direction of rotation and of turning in the curve, until the reaction couple applied to the carriage by the rails has moment CnvjR. If d be the distance between the front and back sets of wheels, the difference of weights borne is Ciiv/Rd, which is the fraction CnvjMgRd of the weight M of the carriage. If the locomotive, with the "fore-and-aft" turbine, referred to above, is not on a curve but on a convex part of the track, of radius of curvature R, there will be a rate of production of a.m. of amount CnvjR about a normal to the track at each instant. If the rotation is in the counter-clock direction, as seen by an observer standing behind the carriage, the rate of growth of a.m. is about the outward normal, and so the rear of the carriage tends to slew round towards the observer's left, and the front towards his right. The reverse is the case with reversed rotation, or with concavity of the track. 293. Drift of a Projectile. The turbine thus moving forward while rotating, may be compared to a projectile fired from a rifled gun. The rotation of the projectile is right handed in that case as looked at by an observer at the firing point, and the shot drifts in its trajectory, which is convex upwards, towards the right. But with this direction of rotation of the turbine, the front of the carriage would turn towards the left ; so that the idea of the projectile as a gyrostat moving forward on a convex track with its axis in the direction of motion throws no light on the drift of the projectile. The cause of this drift is not yet fully understood, but it is con- nected with the rotation, as its direction is reversed with that of the rotation. It amounts to 25, 1-1, 4-4, 11-5 metres on ranges of 500, 1000, 2000, 3000 metres respectively. Since the rapidly spinning pro- jectile tends to keep the direction of its axis unchanged, it is presently moving forward on the convex trajectory with its axis in the plane of the trajectory, but pointing a little upward relatively to the path. Thus it has a motion in the direction of the axis together with a lateral component. Hence, by §80, a couple is applied by the air tending to increase this obliquity of the axis of spin to the direction of motion ; but, as the projectile spins rapidly about its axis, it ^•292,293,294] STABILITY OF ROTATING PROJECTILE. 537 precesses about the instantaneous position of the axis of resultant momentum, as explained in § 294, with of course modification of the resistance in consequence. As a result the projectile moves forward in air, and relatively to the path its point is directed slightly upward and to the right, and the shot is continually deflected towards the right by a side thrust applied by the air. 294. Stability of Rotating Projectile in Air. We now consider the stability of a rotating projectile in an unlimited frictionless liquid.* Let the projectile rotate about its axis of figure with angular speed n, so that its a.m. about that axis is Cn. By § 80 the projectile will experience a couple depending on its motion with speed v in the axial direction, and in a direction perpendicular to the axis with speed u. The moment of the couple is (C2-Ci)ttf, where Cj, Cj are the effective inertias in the directions of v and u respectively, what are denoted by M^, M^ in § 80 above. Now let the shot have processional angular speed /i about an axis parallel to the direction of the resultant momentum, that is, the resultant of c-(o and CjM. This is the direction of the impulse which would be required to produce these components of momentum. If 6 be the angle which this direction makes with the axis of figure, tan Q=c.^\c^. We suppose the motion to be steady. The shot now " precesses " as if it were an ordinary top (Fig. 112) spinning about a fixed point with the line of resultant momentum vertical, and endowed with a.m. Cn about the axis of figure, and an effective a.m. Afi.%vad, about an axis OE, at right angles to the axis of figure OC, and in the vertical plane containing OC. The couple iV^acts about an axis represented in the case of the top by OD. For steady motion, § 272 above gives the equation (Cra- J/i.cos 0)/isin Q=N. (1) Now, since iV=(c2-Ci)MW and tan d—c^ujc^v, we have and therefore (1) becomes (C«-4/i,cos0)/x= J(c2 — Ci)i;^sec0, (2) ^2 for we do not suppose that d=0, which would mean that the shot did not swerve from the axial direction of motion given to it by the gun. * The discussion here given is a version of that by Sir George Greenhill (see "Gyroscope and Gyrostat," Encyc. Brit., 10th edition, vol. xxix.), to whom most of recent investigation in this subject is due. 538 A TREATISE ON DYNAMICS. [CH. IX. The roots of the last equation are real if S^^^t^^^-'^i)' (^^ which gives the least value of n compatible with stability. In what follows, we assume that n has the value given by converting this inequality into an equation. Now we can put Ci = M+M'a., C2=M+M'fi, (5) where M' is the weight of the displaced fluid and a., ^ are coefHcients depending on the shape of the body. Thus .■2-Ci = if'(/8-a.). If y8 be the angle of rifling and d the diameter of the bore at the muzzle of the gun, '' /: ■ (5) ^^^WdHH-a.)] If /fcj, ^3 be the radii of gyration of the body about the axis of figure, and the other axis about which the body revolves with angular speed f), sin d, we have, writing k for M'jM, C=Mk\, A=Mkl+M'k'l and tani'/3=(^^ + OpY^/<(/8-a) (6) Now we may apply this theory to a shot in air, and in that case we may neglect M'^l/M ia comparison with k^, and write tan2;8=rf2-^«((S- sin 9+(2Axfr cos e-Cn)0 = F'x (2) The motion of the axes produces no change of a.m. about GC, and therefore Cn=-F'y (3) Now let u, V be the speeds of the centroid parallel to DM arid perpendicular to the plane GOG respectively. Then, since is supposed to be at rest, u = MG.6 = (xcos9 + ysin6)d, v = yn — -\frxaind. ...(4) Here v is taken in the direction DG. The rates of change of momentum in these directions and along MG are given by M{u+v^p-)=F, M{v-u\ir) = F', Ml=R-Mg....{^) These equations give the whole motion. A relation between x and y is given of course by the form of the surface. For steady motion = 0, = 0, ^ = 0, z = 0, ^^ = 0, so that ^ = fJ; a constant, F' = 0, —F=MiJ.{/nxsinO — ny), R = Mg. Hence (1) becomes {{G+My^)n-{A +Mx^)fi cos 0}/£sin + Mxy{nficosQ — fi^sin^Q) = Mg{xsinQ-y cosQ). ...(6) Since in steady motion 'w = v = 0, we have by the first of (5), and the value of F for steady motion, v = ny — fjLX sin 6, that is V is constant. The direction of v turns round with uniform angular speed /z, and therefore G moves in a circle. §§ 295, 296] TOP ON ROUNDED PEG. 541 and M in a, parallel circle in the horizontal plane. But fxv is the acceleration of M towards the centre of the circle, that is if r be the radius jjlv = v^/r and r = v/fi. Thus r = ^ — a!sin0 (7) The azimuthal motion /x is in the counter-clock direction to an observer looking from above on the solid, and there- fore the circle has the position shown in Fig. 132. 296. Rising and Falling of Top Spinning on Rounded Peg. We see that a top supported on a rounded peg rises under certain circumstances. Initially the top is spun in various ways, generally by throwing it from the hand so that it alights on the ground on its peg. The speeds u, v, \fr are small as a rule, the speed of rotation n is large. The result is that the friction F' is for the turning indicated in Fig. 141 in the direction there shown, and is as great as the force R can make it: for the point of contact of the solid, owing to the rapid rotation, slips back on the plane, and the friction is not limited to that required for pure rolling. The total couple given by friction resolves into the two components on the right of equations (1) and (2) ; one accelerates the precession, the other reduces the spin about the axis of figure. The axes OC, GD, GE show, by the directions in which they are drawn from G, the sense of the angular momentum about each. Now the acceleration of the precession produces a.m. about the axis GB towards which the precession is carrying the axis GO, with its a.m. of amount Gn, so fast that the rate of growth (Cn — Axj/- cos 6)ylr sin Q of A.M. is greater than the applied couple producing a.m. about GD; and therefore AQ is negative, that is the cen-; troid of the solid, if Q before was zero, is now rising. If 6 is still positive, it is now diminishing, and the action is towards raising the centroid of the solid. We can study this quantitatively by means of the equations. ]V[ultiplying both sides of (2), § 295, by sin Q 542 A TREATISE ON DYNAMICS. [CH. IX. and substituting the value of F' given by (3), we obtain j^iAyjr sin^e + Gn cos d) + On -^^^^^- sin e^O (1) Now let the solid roll on a part of its surface, including the extremity of the axis of figure, which may be taken as spherical of radius a, and let c be the distance of G from the centre of curvature. We have then (x — y cot 6) j(y /sin. 6) = c/a, so that we get from (1), by integration, the result Ay}rsiii'e + Gn(cose + ^ = Cno(cose^ + ~^, (.2) where the constant expression on the right is the initial value of the quantity on the left, for initially 6 = do, n = no, rfr — O. Now n is being continually diminished by friction, and if we suppose n large and therefore i/r small, there will be a value of n = no{cos6f, + c/a)/(l + c/a). But, as the reader may verify from (2), for this value of n, 6 = 0, that is the body is spinning with its axis vertical. If this value of n satisfy the condition (§ 270 above) G^n^ >■ 4iA Wgh, steady motion in the upright position is possible ; the top will rise up and " sleep " in the vertical position. 297. Disk or Hoop on Horizontal Plane, Oscillations about Steady Motion. For a circular disk or hoop rolling without slipping on a horizontal plane, equations (1) and (2), § 295, become, since x=0 and y = a, the radius of the circular edge A9+(0n- Aif' cos e)4'sme=-a{Rcose-'rFsme),'\ ^(4^isin26()-C«(9 sin 61=0, i (1) Cn=-F'a, 1 with M{i+v4')=F, M(i-u^)=F', M'(=R-Mg, where (§ 295) M = a^sin 6, v = an, f=asin 6. Thus Ma{esme + ff'cose + nrP) = F, Ma(n-e>p sin d)=F' (2) §§ 296, 297] OSCILLATIONS OF ROLLING DISK. 543 Hence we obtain for the first of (1), {A + Ma^)e+{{C+Ma^)n- A^' cos e}\f' sm 6 + Mga cos 6 =0,-\ ~(A^ sin" 6) -CnO sin 0=0, ..(3) dt {C+Ma")n - MaW4' sin ^=0j Now, let the motion be steady. Then ^=0, 0=0, 0=0, m=0, M = 0, 11=0, M = 0, and we get {{C-ifMa")n-AiJ.cos0}ti.sm0-\-Mgacos0=O (4) with v = na. For a uniform circular disk, G=\Mcfi, A=^Ma", and for a hoop, C=Ma\A=^Ma\ Now, let n', ff, fj. be steady values of n, 0, xj/ and n' + v, ff + CL, ;u, + ;S be the values at any instant for a slight deviation from steady motion. Then equations (1) become (A+Ma")^ + {{C+Ma")v-A0cose' + Aa.iism0'}ij.sm0'' + {{C + Ma^yn- A fi, cos 0'}($ sin 0' + iJ.a. cos 0") — MgoLasin 0=0,^ ...(5) A'$sinff + (2AiJLCos0-Cn)6i=O, (C+i/'a2)v-JI/a2i/Asin 0'=O.. From the last equation, we get (C + Ma") v = Ma'oLIJ. sin ff, (6) since there can be no constant of integration. Substituting in the first equation, we obtain {A + Ma")a.+{Ma"a.ij.sin&-APcosff + Aa.t^sinff)fisinff\ + {(O+Ma")n-Aij.cos0'i{$sinff+ij.a.cosff) I ...(7) -Mga.asin 0=0.) Putting now a.=r sin (pt-f), fi =s cos (pt-f), and substituting in the last equation, we obtain {(C + Ma")n - A ij. cos ff} 11. cos ff ] s _ +(A + Ma")(,fj?sin"ff-p")-Mffasin ff \ (8) ,.- {(C + Ma")n -2AiJ. cos ff}p sin ff ' Again, substituting in the second of (5), we get s_2i4/Acos ff -On r Ap sin ff The value of f? can now be obtained by equating the two values of sjr. It is given by the equation A {{K+ Ma"n)fi cos ff + Lij?sin"0' — Mga sin ff } ' a _ +(K-Aii.cosff ) {K+ Mahi - Ajl cos ff) \ (9) P ~ AL i} 544 A TREATISE ON DYNAMICS. [CH. IX. where K=Cn- Afiaosff, L = A + Ma?: Here d „, „ •%, .^ (2) For a hoop this is n>'d-^, (.3) '4a! and for a uniform disk it is n> A/^ (4) The hoop is therefore more stable than the disk, requiring for the same radius less speed of rotation, in the ratio of v/3 to 2, in order to remain upright. §§ 297, 298] EXERCISES. 545 EXERCISES IX. 1. A heavy flywheel is pivoted at the exti'eniities of a horizontal diameter AOB, and this diameter is carried round a vertical axis tlirough its centre with uniform angular speed )j.. At a point P at distance a from the centre on a diameter at right angles to yl^ an additional weight w is attached. Find the equation of motion. Take as axes OA, O/'and the axis OC of the wheel drawn from Oon the other side of the vertical from OP. If 9 be the Inclination of OP to the downward vertical, the angular speeds about OP and OC are fxcosO and /xsin^, while the angular momenta are Aacosd and (C+wa^)jLiain 6. The total rate of growth of a.m. about OA is there- fore, by §9, -{A+wa^)B-\-{C+wa^)t)?smdmsd-Aii?cos9sm6, and the moment of applied forces is wgasmO. The equation of motion '^ {A +wa'^)6~{G+wa^-A)iJ?co% Ssin 6= -wga sin 6. 2. A top is set in rapid rotation and is placed on a frictionless horizontal plane with its axis inclined at an angle On to the vertical, and is constrained by two smooth planes parallel to the angle 6, so that its axis must remain in that plane. Prove that the top must fall. No action of the constraining planes can alter the energy of rotation, or the angular speed 0. If ^ = when 6= da, and A be the m.i. of the top about a horizontal axis through its centroid, we have J { /J (92 + wifi an\W.e^)=wgl (cos ^„ - cos 6). The left-hand side is positive, so 9 must be greater than 9q. [It is important to remember that any constraints may impair the stability of a top or gyrostat. Conclusions, for instance, derived from the behaviour of a top mounted on a tray as in § 283, where a certain diameter of the flywheel is constrained to lemain horizontal, cannot be regarded as necessarily holding for a top perfectly free to precess — e.g. a planet rotating in free space.] 3. A gyrostat is suspended from a fixed point by a string of length a fastened to a point P in the axis of rotation, and is in steady motion with the axis horizontal. Prove that if a. be the angle which the string makes with the vertical, n the angular speed of the flywheel, h the distance of the point from the centi'e of gravity of the gyrostat, M the mass of the gyrostat, and C the moment of inertia of the ■wheel about its axis, MVi^ tan a. =5-^^^ (A + a sin (x). [The string applies horizontal force Mfi,\h+asm a), and the gravity couple is Mgli. Thus (j?=M^gVi^lCV, and so horizontal force = My/i^h + a sin a.)ICV.. By equating this to the horizontal component of the pull exerted by the string and the gravity of the top to the vertical component, the reader will obtain the required result.] on. 2m 546 A TREATISE ON DYNAMICS. [CH. 4. A simple qonical pendulum is inclined at an angle a. to the vertical, and its length is I. Find the period of a small oscillation about the steady motion. 5. A ring of wire, of radius c, rests on the top of a smooth fixed sphere of radius a, and is set rotating about the vertical diameter of the sphere with an angular speed n. Prove that the motion is unstable if wV* < 2g{2a^ - c^)'Ja:' - cK [Math. Tripos, 1885.] Since the ring moves on the surface of the sphere it may be regarded as a top turning about the centre of the sphere. 6. Show that a gyrostat, balanced and free to turn about an axis AB through the c.o. at right angles to the axis of rotation, is in stable or unstable equilibrium with the axis of rotation vertical according as the rotation of the wheel is with or against the earth's rotation. Show also that if the gyrostat be placed v\rith its axis of rotation horizontal in the meridian, and the axis AB vertical, it will be in stable or unstable equilibrium according as the direction of rotation of the wheel is from west to east or from east to west. Show that the periods of oscillation about the p ositions of stable equilibrium are 27r\/j4/Cnft) sin A, 'i.ir'J A jCmo cos X, where n is the angular speed of the flywheel, w that of the earth about its axis, and A the latitude of the place. 7. Two intersecting rods are at right angles to one another. One is placed vertical, the other can turn in a horizontal plane about the lower end of the first. The ends of the axis of a gyrostat slide freely on these rods, and the axis (of length Sa) is initially inclined at an angle 6^ to the vertical, when also the horizontal rod is turning with angular speed xp^. If at time t the inclination of the axis to the vertical is 6, and the azimuthal speed \p, prove that (i/aH ^ )(\^ sin26l - ;^„sin2 ^o) + Cw (cos e - cos e„) =0, {Ma? + A) e + {On - {Ma^ +A)\jy cos 6} i^siae + Mgasm 6 = 0, where M is the mass of the gyrostat and C and A are its principal moments of inertia. [The M.i. of the ease about the axis of symmetry is neglected. Take axes at the centroid and apply the method of § 26.] 8. Four rods, each of length 2a, are freely jointed together so as to form a rhombus. At the centre of each rod is a gyrostat of mass M, the axis of which is along the rod. The rhombus is hung with one diagonal vertical, and the hinges at top and bottom are attached to rods which swivel in hooks, so that the frame can turn freely in azimuth, while a weight that does not turn is hung at the lowest point. The gyrostats are all equal and are set spinning with the same angular speed n, in the same direction in each case to an eye looking downward along the rod. A weight W is hung at the lowest point : prove that if the angle at the lowest point be 2a, the arrangement will IX.] EXERCISES. 547 turn with steady azimuthal angular speed /x given by {Cn-(A+Ma^)lJLC0sa.}iJ,+(2M+ W)ga=0. [Discuss each of the two gyrostats on either side, as in Ex. 7.] Find the period of oscillation about this state of steady motion. [It was stated by Loi'd Kelvin in his lecture on "A Kinetic Theory of Matter " [Popidar Lectures and Addresses, vol. i. p. 238] that this arrangement forms a spring balance which is drawn out a vertical distance proportional to any addition of weight made to W, and vibrates vertically when disturbed just as a spring balance does, that in fact, if the rotating and precessing masses were enclosed in a case leaving only the hooks accessible, it could not be distinguished from a spring balance. The reader may endeavour to verify these statements.] 9. Considering the earth as a rotating body with its centroid at rest in space, find the equations of motion of a particle with reference to axes Ox, Oy, Oz drawn from the centroid parallel to the horizontally southward, the eastward, and the vertically downward directions at a point /"q on the surface. The coordinates of P^ are thus 0, 0, a, where a is the vertical distance of Pfl from 0, approximately the earth's radius at the point. Let the direction of the gravitational force G on unit mass at F^ make, as in Fig. 39, a small angle 6 with the vertical. The components of gravity at Pq are Ocos{^+0) along Ox, zero along Oy, and Ccos 6 along Oz. The angular speeds (clockwise) are wcosA, 0, and TisinA, if n be the angular speed of the earth's rotation and A the geographical latitude (see Fig. 39). The coordinates of any other point P with reference to these axes are x, y, z, and the components of force there are Z+G'cos(j7r+6l), Y, Z+0 cose due to gravity, with components X', Y', Z' due to any other applied forces. The equations of motion are therefore ic — 2ymsin A + m^sin A(2Cos A-i»sinX)=X4-Jjr'-6'sin 6, y + 2xft sin A — 22m cos A - r^y = Y+ Y', 2 + 2^wcos A — Ti^cos A(0cos A-^sin A)=.^+^'+G'cos 0. If, as is generally convenient, the axes be taken in the directions specified, but from Pq as origin, it is only necessary to substitute z+a for z in these equations, and to add, on the right, force-components equal and opposite to those required to give the acceleration n^a cos A, of a particle at Pq towards the earth's axis of rotation. 10. Apply the equations of last example to a simple pendulum suspended from P„ and executing small vibrations under gravity. Change the origin to Pq, as explained in Ex. 9, and neglect terms in n^x, n^y, nh after this is done. If F be the pull per unit mass applied by the thread to the bob, X=-Fx/l, Y=--Fyll, Z=-Fzil=-F. These are the only applied forces besides those due to gravity. Verify that the third equation gives F=0 cos d=g, nearly, where g is the apparent force of gravity on unit mass. 548 A TREATISE ON DYNAMICS. [CH, Verify also that the first two equations of motion are, if to be written for n sin A, and O sin 6 be neglected, and that these equations are satisfied to terms involving w^ by x= a cos mt cos (at, i/= —acosmtsimot, where m^=g/l. Hence when t=0, x=a,i/=0, and at time t, r = \/x^ + T/^ = a cos mt, ta,a~^(i/lx)= - Vv %' ^^2' 2/2' 2^2. ••• . ^n, Vn, ^n, t) = 0,^ Jii^V Uv ^1' ^2' 2/21 ^2' ••• > *»' Vnt ^n> t) = 0. ..(1) Jm(Pit Vl) %) ^2> 2/2> ^2' •■• ' "^nt Vni ^nt t) — "•• These are the equations of constraint or simply the constraints. According as t appears or not in these equations, the constraints are said to be variable or in- variable. Fixed guides along which some of the particles move are an example of invariable constraints; if the \ \ §299] DYNAMICS OF CONNECTED SYSTEM. 551 guides are themselves in motion, the motion will be recognised by the explicit appearance of t in the equations of constraint, which are then variable. If the components of active force (§63) on a specimeij particle of mass m. be X, Y, Z, and 8x, Sy, Sz be any variations of the coordinates of the particle which are possible acccording to the conditions which exist at time t, we have, summing for all the particles, l{m{xSx+ijSy+zSz)} = 'E{XSx+YSy + ZSz) (2) We cannot, however, equate coefficients of Sx, Sy, Sz on each side, since the forces X, Y, Z are not necessarily the only forces which act on the specimen particle ; a sum "ZiX Sx+YSy + ZSz), due to inactive forces, which is zero, is left out on the right-hand side. But if we replace (1) by ^Sx, + ^Sx^ + ...+ ^1 Sy,+ ...=0, Sy,+ ... = 0, &1+. = 0, (3) we have a set of equations connecting Sx, Sy, Sz for each particle which coexist with (2). Now let the first of (3) be multiplied by \, the second by \^, and so on, and let the sum of the products be added to (2). We get I,{m(xSx + ySy + zSz)} = -E(XSx+YSy + ZSz)+(x,'^+X,'^^+...)sx^ ...(4) It is possible, as we shall see, to choose Xj, X^, ... , X^ of such magnitudes that the coefficient of each Sx, Sy, or Sz shall be zero in (4), and of course the multipliers can be taken of such dimensions that every product of the form 552 A TREATISE ON DYNAMICS. [CH. X. Xdfjdx . Sx shall have the dimensions of work. Thus we get the equations, 3% in number, m,x, = X, + \^+\^ + ... + X ^3a;„ 2 °^2 .(5) That Xj, \, ... can be thus chosen is clear from the fact that we have Su coordinates and m multipliers \,\^, ..., X,„, Zn + rn in all, and that the 3w equations of (5) and the m equations of (1) give Zn + in equations wherewith to determine them. The equation (4) of virtual work (so called because Sx, Sy, ... are virtual displacements, that is any arbitrary small displacements possible under the conditions of the system, as they exist at the instant t) only holds for the conditions of the system at time t. If we consider Actual displacements dx, dy, ..., effected in an interval of time dt, we have to replace (3) by 3A^^ -lM dXi 'dx, dx, + :^dx,+ . .+^dy^ + ...+^}dt ¥2 2t 0, M. •+S^2/.+-+tdi=o, ix,'^'"''^- =0. ,.(0) If we multiply these respectively by Xj, X^.-.-X^ and add, we get (7) §§ 299, 300] INDEPENDENT COORDINATES. 553 and so the whole work done in an actual displacement of the system in consequence of the fulfilment of the m equations of condition (1) is If we multiply (4), the first by x^, the second by x^, ... and add, we get by the result just obtained, since dTldt = 'L{'m{xx+yy+zz)], The interpretation of this result is that the time-rate of increase of the kinetic energy is equal to the rate at which work io done by the forces X^, Y^, Zj^, ... plus the activity due to the forces brought into play by the varying of the kinematical conditions (1), § 299. Hence when dfjdt, dfjdt,... are all zero, the fulfilment of the kinematical relations has no effect on the energy of the system. Thus if the forces X, Y, Z,... are conservative, that is, are derived from a function V of the coordinates only, the sum of the kinetic and potential energies remains constant during the motion, provided t does not appear explicitly in (1). 300. Reduction to Independent Coordinates. From (1) of § 299 any in of the Sti coordinates can be determined in terms of the other Sn — m, or k, coordinates. Thus, by elimination by means of (1) of any chosen 7n coordinates, say the first m, the discussion of any problem may be reduced to one regarding a system characterised by k independent coordinates. Instead of using the k coordi- nates left, we may substitute k parameters ^j, q^,---, qk> which are known functions of the coordinates. These are connected with the x, y, z coordinates by definite relations 554 A TREATISE ON DYNAMICS. [CH. X. such that the cc, y, z for each particle can be expressed in terms of the parameters, either by finite equations « = ^(g'i. 92. •••.?»), y = xi^v92'---'9k), z = y}r{q^,q^,...,qu){V) or by differential relations Sx = a^Sq.^ + a^Sq^ +...+ a^Sq^, \ Sy = b,Sq,+b^Sq^ + ...+hSq,A (2) Sz = Ci^gi + c^Sq^ +■■■+ c„Sq^. j Of these equations there must be as many sets as there are particles, and they take full account of course of the connections or constraints of the system, as expressed in (1). The displacements typified by Sx, Sy, Sz are arbitrary, but must be such as can take place under the conditions of the system (1), as they exist at the instant t. If t appear explicitly in (1), the actual displacements which take place in dt in pursuance of the motion are given by dx = a^dq-^ + a^dq^ + . . . + a^dq,^ + adt^ dy = \dqi+\dq2+ . . . + hdq^+ bdt,\ (3) dz = Cjdg^ + c^dq^ + . . . + c^dg'^ + cdtj where dq^^, dq^, ..., dq,c are the actual variations of the parameters in dt. The coefficients a, b, c are zero if t does not appear explicitly in the conditional equations (1), § 299, when the displacement specified by Sq^, Sq^, ... , Sq^ is one that is consistent with the conditions of constraint as they exist both at time t and at time t + dt. We have, in the general case, ^ = o^iS'i + «29'2 + • • • + %9'* + (^'l y = biq^+b^q^ + ... + b^q^ + b,\ W 301. Generalised Coordinates. The parameters q-y,q^,...,qk are called the generalised coordinates : they are supposed to be such as sufiice to express the configuration or the system at any instant. It may be remarked here that there are cases in which the motion can be expressed in terms of velocities (for example in the motion of a rigid ^300,301,302] HOLONOMOUS SYSTEMS. 555 body, the angular speeds o£ a body about its principal axes which are fixed in the body), unrelated to coordinates, which fulfil this condition. It is important to observe that the differential relations (3) may or may not be equivalent to a set of finite equations like (1). If they are, aj.a^, ... ,6^, 63, ... , c^Cg, ... must be partial differential coefficients of equations (1), § 301, and one set of conditions for this is 3a, _ 302 ^*i _ ^% 3^~3^i' 3^3-3^/ ■■ 36i_362 36,_363 \ (^^ The finite equations (1) and the differential relations (2) of § 299, then, express exactly the same thing — one can be derived from the other. But if (1) and all the similar equations are not fulfilled by the coefficients a^, a2, ... , b^, b^, ... , a, complete set of finite* equations does not exist, and the conditions (2) of § 299 are not integrable, as a whole at least. 302. Holonomous and Not Holonomous Systems. Derivation of Lagrange's Equations. Lagrange's equations were given for the case of finite equations of condition, and that these exist has been tacitly assumed in most of the expositions of the subject since his time. That they "fail" for the case of non-integrable relations has been pointed out by several writers, and systems are now called holonomous or not holonomous, according as the constraints are or are not defined by finite equations. We shall now derive the equations of Lagrange from the equations of motion of a system of free particles ; as this mode of derivation shows very clearly where the assumption that the system is holonomous is introduced, and where, therefore, the process should be corrected if the system is not holonomous. 556 A TREATISE ON DYNAMICS. [CH. X. The equations of motion of a system of particles are of thetype m,x = X, my=Y, m!i = Z, (1) and, of' course, these .are the equations of a particle of a connected system, when the forces due to the connections are included in X, Y, Z. Now, from (4), § 299, find the values of x, y, z. They are * = "i^i + "^2^2 +•••+<**?*+ *i?i + '*2?2 +•••+ <**!Z* + «-.-•• (2) with similar equations for y, z. Thus we get E {m{a^x + h^j + c{z)} = 'L{a^X+ h^ F+ c^Z)\ 2 [mia^x + h^y +c^z)}= ^(a^X + 1^ Y+ c^Z), t (3) The quantities on the right-hand side are called the generalised forces of the Lagrangian system, and will be denoted in what follows by Q-^, Q2, — It will be observed that, by (3), § 300, I,{X6x+YSy + ZSz) = I,{a^X + b^Y+c-^Z)Sq^ + 'Z{a^X + b^Y+c^Z)Sq. + ... + ^{a^X + b,Y+c^Z)Sq^ = Qi<5?i + Q2^?2+--- + QA- Thus any Q is the coeflScient of Sq in the expression Q Sq for the work done in a possible variation of the parameter q, and is not necessarily a force in the true dynamical sense ; e.g. if Sq is an angular displacement Q is a moment of dynamical force or a couple. Q does not depend on any of the inactive forces, that is, forces such as those due to guides and constraints which are invariable. Thus the results obtained from a system of free particles hold for a constrained system. Now, by (2), aiX + h^y + c^z = {a\ + 6? + Cj )(^i + {a^a^ + 6162 + CiCj) qi+-..\ + (x,(d,(?i+(X2^2 +...)+ 6, (6,^1 +62^2+...) \ (5) + ...-|-aid-|-6,6-|-CiC. J 303. K.E. in terms of Generalised Coordinates. The ex- pression of the right-hand side of this equation, by means of the kinetic energy transformed to generalised coordinates, (4) §§302,303,304] GENERALISED MOMENTA. 557 is the cliaracteristic feature of the Lagrangian equations. To effect this transformation we substitute on the right- hand side of the equation T=h'E{m(x^ + y^+z^)} ; (1) the values of x, y, z given by (4), § 299. Thus we obtain T= \ {^„j?-|- 2il 12^1^2+ 2^ 13M3+ ■ • • + ^22^1+ 2^23^3+ ■•• +-4 l^l+^2^2+---+^ft^* + ^o}. •■••(2) where ^n, A-^^, ...,A^, A^^, ... are functions of the co- ordinates ^1, ^2. •■• . ft- The expression thus consists of a liomogeneous quadratic function of the speeds q^,qi, ... , %, a Hnear part, 4j2i + -^222+---+-4k2)i> '^''^^ ^ term j„ which is a function of the coordinates only. These will be referred to below as T.^, T^, T^. 304. Generalised Components of Momentnm. the first place that from (2), § 303, we have We notice in v-=4„gi 4-^,222+. ..+^li?ft + ^l, Hi = 4,221 + ^2222+ ••■+^2& + ^2. 9r = 4ii,2i -t- A^^q^ + . . . + A^i,+A^. ■(,1) The expressions on the right-hand side are called the generalised components of momentum, and will be denoted in what follows by p^, p^, ..., p^. Equations (1) enable the speeds q.^, q^ ft *« be expressed in terms of Pv P2' ■■■' Pk ^^^ ^^^ coefficients J-n, A.^^,... (which are functions of the coordinates q-^, q^, ...), and therefore also the kinetic energy to be expressed in terms oip^,p^, ... ,pt and functions of the coordinates. It is to be observed that the determinant {A^^, A^, ... , A^) of equations (1) cannot vanish, since if it did the values of q^, q^, ■■■, it given by these equations would be zero, and T would be zero. 558 A TREATISE ON DYNAMICS. [CH. X. 305. Equations for Holonomous System. Modified EoLuations for Not Holonomous System. Now we have, taking first the coefficients a^, b^, c^, a-^x + h-^i) + c^z = j^ («!« + \y + c^z) - {a^x + \y +.c^z), and therefore d 1 2 {m(ai* + %■ + Ci^)} = ^ [{Em(aj« + 6ijf + CiZ)}] I (-J) — S { m(a.ia! + 6i^ + Ciz)} . J But if we form JS{m(«^ + 2/^+2^)}, or T, from the values of X, y, z given in (4), § 300, we get 2(m(aia!+6i2/+Ci2)} = gj (2) and therefore obtain ^{'m{a^x+\y + Ciz)) = j^ ^-^{m{a^x-\-\y+6iz)]. (3) Again, by the value of T thus formed ~ V3* di 3^1 32/ dt ■dq^'dz dt dqj' provided the relations (1), § 300, are derivable from a set of k finite equations, for then ctj, 6], c, are partial differential coefficients dx/dq^, ... of aij, y^, z^ expressed as there shown. Now d'dx_'ddx,'d'dx. 3 3a;.33a; ,., d^ 3^~3^ 3g ^1 3^2 3^ ^^^'"^"dq^ dq ^"'^dt dq""^^ Hence, since 3a;/3g' is supposed to be the partial derivative with respect to q of an explicit function of g^, g'j, ... , q,„ we S^*' ^^_3_3^ _L^_^^ 3^^__3_^ 3?i 39'~3g' dq^' dq^ dq~dq dq^''"' dt dq~dq dt' and therefore d dx 3 /3a3 . , 3a; . , , dx . , dx\ .c\ 3-<3^ = 3^V3^^i + 3^,*'- + -+3^^*+3n-T; {'r^d), so that we get the same equations of motion as before. 309. Hamilton's Transformation of Lagrange's Equations. Lagrange's equations admit of a remarkable transforma- tion due to Sir William Rowan Hamilton. Let equations (1), §304, be solved for q^, q^'---' ?*> ^^ terms of the components of generalised momentum, and the values be substituted in the expression for- T, which then becomes a function of the pa, made up, if the untransformed ex- pression was, of a homogeneous quadratic part and a linear part in terms of the p s and a function of the coordinates. The transformation of the equations of motion can be investigated by the following method, given by Jacobi (Vorlesuiigen iiber Dynamik). Consider the function K defined by the equation K=^qp-T,, (1) where T, is T supposed expressed in terms of the ^s. [For clearness we shall denote T when expressed in terms of the speeds, the ^s, by T,, and when expressed in terms of the p s, the momenta, by Tm-] Now, let the coordinates be subjected to slight variations Sqi, Sq^, ..., g^, and the speeds to slight variations Sq^, Sq^, ..., Sq^, which are all consistent with the conditions of the system as they exist at time t. Then, if K become in consequence K+SK, and T become T+6T, we have SK='EipSq+qSp)-ST„ (2) where Sp is the variation of p due to the changes Sq and Sq ST. = E(f Sq+^^ Sq) = E(i.^^) + Z(f Sq), §§308,309] HAMILTON'S EQUATIONS. 569 and therefore, by (2), 6K = nqSp)-^(^^Sq) (3) So far, each (5p has been taken as depending on the related 8q and 8q, regarded as independent variations. But if we please, we may take the q& and ps as independent variables, that is the Sqs and Sfs may be arbitrarily assigned, and the change Sq deduced from them. The expression for SK holds in this sense. But now, if K be expressed in. terms of the variables p and q, we have SK^^'^-^Sp + ^'^-^^Sq (4) and, since in each case the variations Sp and Sq are arbitrary, we may identify (4) with (3), and obtain the typical equations ■dK_ dK^_ dT, (^s dp oq dq which are of great importance. They enable the typical Lagrangian equation j ^y gy di'dq~dq~^ to be transformed to Hamilton's form, which, written along with its companion equation, gives dp dK ^ dK . , , where it is to be remembered that ^ is a function of the q s and the p s. It will be noticed that if T^ is a homogeneous quadratic function of the qs,, that is, if the equations which express X, y, 2 in terms of the q s do not involve the time explicitly, K=2T, and that then (5) become ■dp ^' dq dq' and Hamilton's equations are of the form dp dTr^_f^ 3^m_. /ox 570 A TREATISE ON DYNAMICS. [CH. X. If we now suppose that each Q is derivable from a function V of the coordinates (and it may be also explicitly of t) by the relation —dV/dq=Q, and write H iov K+ V, where K is supposed expressed in terms of the ps, then, since V does not contain any p, we can write the two equations (8) in the form _ dp ?)H dq _dH /^x dt dq' dt~ "dp which typify Hamilton's so-called canonical equations of motion of a connected system. There are as many pairs of such equations as there are variables. It will be observed that the system is supposed to be holonomous. The function H is called Hamilton's reciprocal function. Ex. 1. Verify the relations "dq 'dq' " dp by direct differentiation. On the left K is supposed to be a function of the ps and the qs. If the ps were replaced by their values given in (1), § 304, so that K is expressed in terms of the qs and the qs, that is, becomes Kg, we should have M'=2 f^^'^^UH "dq \3p "dqj "dq' But we have also by the definition of K, dK, ^-.f .dp\ dT so that dq X^dql 'dq' dK 'd^^_dT dp' dq dq' ?=%:' Ex. 2. A particle moves in a plane and its positions are referred to axes 0.V, Oy which turn with constant angular speed n about 0. If the forces on the particle are derivable from a function V oi x and y, it is required to find the equations of motion in the canonical form. The equations of motion are, by § 1 4, dV dV m(.v-2m)-n^x)= -^, m(i/+2nx-nh/)= -■=^. From these we can deduce, by multiplying the first equation by x, the second by ij, and integrating, the equation f (*2+f )+ F-m|'(a;2+y2)=const., which is the equation of energy. §§ 309, 310] EXAMPLES. 571 Now write qi = x, '*-Kf»'),-^(i*»).- "' SS = ^(pSq\-^{pSq)o, (2) to which must be added 'dSfdt .St, if t is not taken as independent variable, and is therefore made to vary. The function S was called by Hamilton the Principal Function. This expression is very important in connection with the integration or the equations of motion (9) of § 309 above. Such integration means the expression of L, or H, as a function of t, and of 2k constants of integration (there are k independent coordinates), so that 5 is obtained by integration in terms of these 2^ + 1 quantities, which may be the time t and the 2k initial values of the ^s and qs, typified, we suppose, by (p^,, g'o)- Thus, in (pq, g'o) and (p, q) with * we have 4^ + 1 quantities connected by the k equations of motion or their integrals, and the & equations q = dq/dt. Thus any Zk of these can be found in terms of the other 2^+1. If the 2k chosen to be found in terms of the remaining 2^ + 1 be (p, p^), and these be substituted in )S' already expressed as above, we obtain ^ as a function of i and (q, q^. The variation of S when thus expressed is S^=2(|8.)^^-S(|84. (3) If we compare this with (2), we see that -dS dS ... ^=P^ 3^0="-^" ^^ R, dS J dS,^/dS.\ ,.. ^"* W=^=Wt+^d^V' ('> since in the most general case t is explicitly contained in S, and implicitly in the qa. But by (4), this is or '^+^(pq)-.T+V=0, 576 A TREATISE ON DYNAMICS. [CH. X. that is with 2(p j) - T replaced by its value in terms of i^Sfdq, q), OS. ■dt' •■'■"W: 37,' ••■' Tqi' ^" ^^' •••' ^- 0=° ^^> This is the Hamiltonian differential equation. A second differential equation was given by Hamilton, but we shall not discuss it here. 314. Jacobi's Theorem. It was shown by Jacobi that if a complete integral of (6) is known, that is an integral which contains k constants, aj, a2> ■■■> aj-,.. besides the additive constant, the canonical equations can be integrated. For let the integral S be expressed in terms of §■,, q,^, ..., q^, and a^, a.^, ..., at, then if ij, 621 ■•■> ^* be Mother constants such that 3,5 , 35 , 35 , 3^=*-> "d^r ■'' •••' 3^=**' (1) these equations, together with dS dS -ds ,„. 3^=^!' 37r^^' ■••' wr^" ^^^ are the integrals of the canonical equations of the type dq_'dH dp _ JdH ,„, dt^'dp' irr 3? ^> This proposition is proved indirectly as follows. If a known integral of the equation (6) is substituted in that equation, the left side identically vanishes. Differentiate then any equation 'hSI'da=bi with respect to t, which can be done, since S is supposed known in terms of q-^, q^, ..., q^, t. We get 325 , 3^5 ■ , , 325 . „ ,^, 3<3^,+3^i3^?+-+3^a>-" ^^^ But by the first set of canonical equations this becomes 3^5 ^ 3^5 3g^ ^ ?t'S dH^Q ,g.. 'dt'dat 'dqi'dcii'dpi '" dq^dui'dp^ But the differential equation (6) of § 313, differentiated with respect to a,-, gives -^ -dH-dp, ?^^*^0 • (6) 'daidt 3p, 3a( "" 'dp^'doi ' for, since S in (6) is a f\niction of q^, q^, ..., q^ and the constants a,, ag, ..., a,., these constants must be contained in thepj, p.^,, ■■■,p,, of the function ff, from which as it stands in (6) the constants are supposed to have been eliminated. §§313,314,315] JACOBI'S THEOREM. 577 Now substitute in the last equation obtained the values indicated by pj=dSldqj, and we get dpJdai-di^S/daidqj. Thus equation (6) becomes 32^ ^ d^S m ?^S dH^^ which agrees with (5). The first set of canonical equations are thus verified. The second set can be verified in a similar way. Begin by differentiating p.^'dS/dq^ with respect to t, and substitute for the qs and for p, from the canonical equations. This gives, since p^=3ff/3g',., ■dqJ-dfdqJ-dq,dq,^pJ-^-dq;dq,^p-'^ ^^^ But (6), § 313, gives 9^^ ?)E -dH ?^S _Ml_?!^-n rQ^ dqfir-dq+'^;dS dq;dqJ-+^^S_ 3^*" ' ^' and if the values oi p^jp^, ■■■ ,p^ be inserted in this from the equations (2), we obtain (8). Thus again there is verification. 315. Case in wMcli H does not contain t. If the function does not depend on t as an explicit variable, (6) of § 312 becomes f + ^-«' W where h is put for the constant value which, as we have seen, H now possesses. Integrating from •■•> (pX for give the "path" by the first ^-1, and the time of passage is given by the last. G.D. 2 O 578 A TREATISE ON DYNAMICS. [CH. X. 316. Examples on Jacobi's Theorem. Ex. 1. Prove that the path of a particle the position of which is defined by three coordinates g',, q.^, q^ cuts the surfaces W=C at right angles. Let X, y, z be the component speeds of the particle at x, y, z and hx, Sy, hz the components of a step perpendicular to the direction of motion. Then the condition of orthogonality is a; 8a;+y 8j/+3 S«=0. Now, from the equations ^=<^(?o ?2> g's). y=x(9'i.?2. ^X 2='f (?1. ?2. ?3)> find i, ij, i, S.V, &!/, Sz and T. Then it will be found that the condition of orthogonality is But since pi = c)T/dqi = clW/dqi, ... , this is which proves the proposition. Ex. 2. If any coordinate, q^ say, be absent from IT, so that V7has the form H(pi,p2, ...Pj, Ja, qs, ■•• , q,,, t), show that, with olj a constant, 8=a.^qi+U{t, q^, q^, ...q^). Ex. 3. Prove that if coordinates q^, q^, ..., q^h& absent from H, "S = a-i?i + a-2?2 + ...+a.tqi+U{t, g' .^, , q^). Ex. 4. Eind the function W for the motion of a particle under the action of a central force which is a function -'dVfor of the distance of the particle from a force centre. By Ex. 3, § 309, we get and therefore the differential equation (1), § 315, is into which does not enter explicitly. Hence, we can write W=a.e + R, where /J is a function of r only. Hence, the differential equation becomes -^-Mm^hnr)^- 5 316] JACOBI'S THEOREM: EXAMPLES. 579 Thus R=jdr\l'im{h-V)-% so that W= a.e + jdr^J2m(h -V)-~ The finite equations are therefore J r^yj^ "dW f mdr The former of these last equations gives the path, the latter the time of passage from any position to any other. [The student should work out for P'=-^fji,r% K=-ju,/n] Ex. 5. Discuss the elliptic motion of a planet referred to three coordinates. [Jacobi, Varies, u. DT/namik.] Let the mass of the planet be 1, and adopt, to begin with, Cartesian coordinates. Then, Now, at time t = tQ, let the planet be at {,Vq, i/q, Zq), and at time t be at (.V, y, z). Then, if p be the distance between these two points, show that i^4-{ dWV p^+i^-rl^WdW /3iry 3r^ / rp "dp 'dr \c)p / Next, putting a-=r + rQ+p, o-' = »• + Jq — p, show that d W_d W 'dW dW^dW dW "dr "dcT 'da'' 3p dtr 'dcr' so that and the differential equation is =/a(o- - o-') - ^ (0-2 - o-'2) + ^(o- - (r')»-o. 580 A TREATISE ON DYNAMICS. [CH. X. Show that this can be integrated by splitting it into the two, fdw\\ ,fdwy /It, ,, ,, , c)W ,-^/42r^ dW ^ /-^I4a-, show that / d, so that W'=\/a/x(sin2(^ + 2^-sin 2^'-2^'). Ex. 6. By giving tlie same sign to "dW/da and BTT/So"', show that a second ellipse is obtained through the same initial point and having the same centre of force, and the same length of major axis. Ex. 7. Show that in Ex. 5, 31f/3A = (2a2//i)3TF/3a, and hence find t-ta, proving Lambert's theorem [§ 146 above]. Ex. 8. By calculating dW/drg, find the equation of the path. Write s=a-rQ, s' = a}^ ' M 2 ' a),. ' w+r,, Hence noticing that 3« _j'q — j'cos ^ 3*'_ r^ — rcosO calculate 'dW/drg, and show that the path has the equation /^ ^rp -rco sgN / 4a-a- A »o - >• cos 6* Wia - g-' ^ ^ where C is a constant. 317. Lagrange's Equations for Impulsive Forces. The equations of Lagrange may be modified in the following manner for the case of impulsive forces. We have only to integrate each over the infinitesimal time t, during which the impulsive forces act. The integral of any finite quantity, such as dL/dq, over that interval is zero. Thus §§316-319] IMPULSIVE FORCES. 581 we get, putting / for the time integral of the impulsive force Q,p~P(, = I. The equations are therefore Pi-{fi) = h> p%-{v^) = h, •••. P -(P*) = ^„ ■■■(1) where the brackets denote values of the quantities en- closed for the beginning of the interval r. If T^ be a homogeneous quadratic function of the speeds q^, q^, ... , we have ^ T,-(T,) = mpi-(pq)}, that IS, T,-{T,) = i:E[{p-(p)}{q + {q)}-p(q) + ipm- -(2) But if the terms in '^{p{q) — ip)q} be written out in full, it will be found that the sum is identically zero, so that (2) becomes T.-{T,)=hmp-(p)}{q+{m=mi{q+iqm (3) Thus if W be the work done by the impulses in time dt we have, since there is practically no displacement of the system effected in time r, and therefore no work done except that represented in the change of kinetic energy, w=hmp-(p)}{q+iq)U-mi{q+{m (4) 318. Reciprocal Relation between Two States of Motion. The theorem 2(^9)} =2{(p)Pi be the momenta which measure the impulses applied. We have for the kinetic energy 2T=2(p^) + 2(p'g') (2) For another system of impulses p + ^p, p'+Sp', for which the velocities are q-\-Sq, q'-^Sq', we have 2{T+ST) = mp + Sp){q + 8q)) + ^{{p' + 8p'){q' + Sq')],...{^) and therefore 2ST=-L{qSp +pSq + SpSq) + I,(q'Sp'+p'Sq' + Sp'Sq'). . . .(4) Now let the impulses p'^,p'^, ... be all zero, so that the two systems of impulses the effects of which are to be compared are ^^,^2, ... ,p^, and p^, p.^, ..., p., Sp[,Sp'^, ...,Sp], and take the first set as those applied at the specified points ; then by the conditions of Bertrand's theorem, we ^^^^ ^{Sp'{q' + Sq')}=Q (5) But since ^jp = 0. 'i^T=i:{pSq-lrq'Sp'+Sp'Sq') (6) §319] THEOREMS OF BERTRAND AND KELVIN. 583 Now (p, ^,0, q') and (p, q + Sq, Sp', q' + Sq') define two states of motion of the same system, and therefore, by the theorem of (2), § 318, we get ^{p(Hm = ^(pq+q'Sp), that is, 2(29 Sq) = I,(q'Sp') = - j:(Sp'. Sq'), by (5). Hence 2§T= -1.{8p'Sq') (7) Thus the motion produced by the second set of impulses has less kinetic energy than that produced by the first set by the amount 'EiSp'Sq'). For Lord Kelvin's theorem we have, since the speeds of the first set of points are the same for both sets of impulses, Sq^, 8q^, ... , Sq^ all zero, with, as before. Then 28T=^{qSp + q'8p' + S-p'§q'), (7) and in the same manner as before the theorem of (2), §318, ^""^^ ^(qSp + q'Sp')==0. Thus we have 2ST= I,(Sp'Sq'), or the kinetic energy of the first motion is smaller than that of any other motion with the same velocities q-^, q^, ■■•, ii by 'L{Sp'8q'), that is, the kinetic energy of the motion which combined with the first motion would give the second. As Lord Rayleigh has remarked {Theory of Sound, vol. i. § 79), both theorems are consequences of the fact that the imposition of any constraint on the system increases the effective inertia of the system. [Thus, if there were only a single coordinate, the inertia might be regarded as measured by the ratio pjq^ If then the ^s be fixed, any constraint, for example that required to make the velocities of q-^, q^, ... , q^ the same as before for the second set of impulses, will increase the value of p and therefore the kinetic energy ; and on the other hand, if the ps be fixed, the constraint will diminish the ^s, and diminish the kinetic energy. Of these theorems, §218 and the examples there given are illustrations. 584 A TREATISE ON DYNAMICS. [CH. EXERCISES X. 1. A heavy particle moves without friction along a vertical guiding circle which turns with uniform angular speed about a vertical axis in its own plane. Write down expressions for the kinetic and potential energies of the particle, and find the equation of motion relatively to the circle. Find the positions in which the particle can rest in equilibrium on the circle. 2. A particle is constrained to remain on a plane which turns about a horizontal axis in the plane with uniform angular speed w. The particle is otherwise free, and moves without friction. Show' that if r be given by the equations y=rsin(o<, z=rcos, integrate this equation. 4. A vertical shaft is hollow and has a hollow projecting horizontal arm in which slides, without friction, a particle of mass m. To that particle is attached a string which passes inwards along the arm, over a pulley at the junction of the arm and the shaft, and then down along the inside of the vertical shaft to a second particle of mass vi', which it carries at the lower end. The masses of the string and pulley are negligible. The vertical shaft is set rotating about its axis with angular speed w by external forces. Find the energy of the system, and the equations of motion. If T be the kinetic energy of revolution of the particle m, and dW the work done in time dt in driving the system of two particles, and r be the distance of m from the vertical axis, prove that ^=§^{log(^ Fj, and so T «2l9l + «22?2+ ••• +«2J?t+^21?l +*229'2+ ••• + *2J?* = 0> 586 A TREATISE ON DYNAMICS. [CH. X. Show that if we write qi = Aje*"', q2 = A^, ... where i=\P^, we get the determinantal equation for n, If we write X for n^ in this equation, we get an equation of the A''' degree in A. The student may prove that all the roots of the equation in X are real, and may also show that if the function F- V„ of Ex. 7 can be reduced to a sura of k squares, multiplied by positive coefficients, these values of A are all positive, and therefore the periods of vibration all real. This condition amounts to the statement that the potential energy Fq is a minimum for the equilibrium configura- tion. {See Routh, Elementary Bynamici, chap, ix.] 9. Prove that when the expression of F- F„ thus obtained is written in the form where Aj, A2, ..., Aj are the roots of the determinantal equation, we can write als6 The coordinates ^,, ^^i ■■■ > ^si which have thus taken the place of g'l, ^2, ... , ^j, are called ^^ •principal coordinates. The equations for the x, y, z of any particle are now, by Ex. 7 and equations (1) of § 300, where «.i,/3i,yi, a2, ..., are constants. Thus the motion of any particle in the mode determined by the coordinate ^, say, in period 2jr/\/Ai is determined by the values of ixj, ;8i, yi for that particle, with all the other coordinates put equal to zero. The existence of equal roots of the determinantal equation does not involve instability. The equations obtained in Ex. 8 by substituting ^( = j4je*"', ... enable n-\ of the coefficients jd,, A^, ..., At to be obtained in terms of the assumed value of any one. If two roots are equal, then two of the coefficients are to be arbitrarily assumed and the remaining n — 2 determined, and so on. [See Routh, loc. cit.'\ 10. By means of Bertrand's theorem (§ 582) prove that if a circular disk, radius a, receive an impulse in its plane along a line distant p from the centre, it will begin to turn about a point on the diameter perpendicular to the impulse and distant (a^+2p^)l2p from it. CHAPTEE XL STATICS. 320. Equilibrium of a Particle. A particle is in equi- librium under the action of a system of forces, when it is at rest or in uniform motion : hence the forces must have a zero resultant. If there are two forces acting, they must be equal in magnitude and opposed to one another ; if the number of forces is three, any one of them must be equal and opposite to the resultant of the other two; and if there are n forces acting, any one of them must be equal and opposite to the resultant of the remaining n — 1 forces. In other words, the graphical representation of the forces must result in a closed polygon. Analytically, we proceed as follows. The forces are referred to rectangular axes passing through the particle. Each force is resolved into components in the directions of the axes. Denoting the components of a specimen force by X, Y, Z, we see that the resultant force R acting on the particle is given by R = J(2Xf+(^Yf+(IZf. (1) Its direction-cosines are I,X/R, 'EY/R, EZ/R. If the particle is in equilibrium, R = 0, and consequently SX = 0, ZF=0, andE^=0. 321. Particle in Ec[uilibrium in a Smooth Tube. As an example, we may consider the case of a particle maintained in equilibrium in a smooth tube. We refer the particle to axes of reference Oxyz. Let the forces acting on the particle (leaving out of account the action of the tube) in the directions of the axes be JT, F, Z. If the length 588 A TREATISE ON DYNAMICS. [CH. XI. of the tube from a fixed point up to the point occupied by the particle be s, the direction-cosines of the tube at the point {x, y, z) are dx/ds, dy/ds, dz/ds. The force acting on tiie particle along the tube is X dx/ds + Ydy/ds+Z dz/ds, and there is no component of force along the tube due to the action between it and the particle. Consequently, the condition that there should be no acceleration of the particle along the tube is X^+Y^+Z^ = 0. ds ds ds FlO. 142. 322. Flexible String in Ectuilibrium. The consideration that three forces meeting at a point are in equilibrium, provided that they can be represented graphi- cally by the sides of a triangle taken in order, suffices to solve the problem of a string sus- pended from two fixed points, and subjected to forces applied at points distributed along its length. We suppose the string perfectly flexible, inex- tensible, and of negligible mass. In Fig. 142 it is shown attached to fixed points S^, S^: at points A, B, C, D forces are applied to the string in the directions in- dicated by the arrows. The relation that must exist among the weights may be determined graphically aa follows. Selecting any point 0, draw a line 01 parallel to S-^A, and from any point 1 in it, draw 12 parallel to the direction of i'\ ; from draw 02 parallel to AB. Now, the three forces 'I\, F^, and T^ are in equilibrium, and hence are proportional to the sides 01, 12, 20 of the triangle 012. Again, from 2 draw 23 parallel to F^, and from draw 03 parallel to BC: the three forces T^, F^, and 2 3 are evidently proportional to the sides 02, 23, 30 of the triangle 023. Proceeding similarly for the points G and §§321,322] EQUILIBRIUM OF STRING. 589 D, we obtain the diagram 0123450. Such a diagram is called a force-diagram or force-polygon : the polygon S^ABGDS^ assumed by the string is called a funicular polygon. The relation which must hold among the forces is evident. Provided that they are proportional to the sides 12, 23, 34, 45 of the force-polygon, the funicular polygon will be that of the figure. The student will see that if the forces are given in magni- , tude and direction, and likewise the / stretching force in, and the direction jj of, any one side of the funicular / polygon, the stretching forces in, and the directions of, the remaining sides can be determined. We now consider the case where the forces are due to weights attached ^^ to points in the string. The applied forces are all vertical, and if the force-polygon is constructed, the lines j.^^ ^^3 12, 23, etc., will lie in a vertical straight line. To solve the problem analytically, let P (Fig. 148) be one of the vertices of the funicular polygon. The three forces T, T', and Wn meet in a point, and are in equilibrium. Resolving vertically and horizontally, we obtain rsina-rsin^=W„, (1) T COB a. = T' cos ^. (2) Equation (2) shows that the horizontal component of the stretching force is constant throughout the string. Denoting it by H, we have rip tana-tan /3 = -J (3) Equation (3) shows that when all the weights, together with the inclination of anyone side and the stretching force in that side, are given, the inclinations and stretching forces for the other sides can be determined. This is evident from consideration of the force-polygon, as has already been pointed out. »«-i 590 A TREATISE ON DYNAMICS. [CH. XI. Now suppose that the suspended weights are equal. Further, let the lowest portion of the string be horizontal, as shown in Fig. 144. Denoting the inclinations of the Fig. 144. succeeding portions of the string to the horizontal by 6^, 6.2, etc., we obtain ta.nd^ = W/H; ba.ne^ = taxi6^+W/H=2W/H; tana3 = tan0,+ W/H = BW/H; ... ; t&nd^ = tan0^.i+ W/H=nW/H. (4) 323. Horizontal Projections of Sides of Funicular Polygon equal. If the horizontal projections of the sides of the funicular polygon are equal, it is easy to prove that the vertices lie on a parabola. Taking horizontal and vertical axes in the plane of the string and passing through the mid-point of the lowest side, we obtain |a, as the co- ordinates of 4, a being the length of the lowest side ; those of B are fa, c, where c is the vertical distance apart of B and A; those of G are -fa, c + 2c; those of D are |a, c + 2c + Sg; and if x and y are the coordinates of the %"" vertex counted from A, we have 2n + l x = — j5 — a, y =^<^+^)c (1) Eliminating n between these two equations, we obtain .2 , 2a^ , a'' x^= — V + -T c ^ 4 ■(2) as the equation to the curve passing through the vertices. Hence the vertices lie on a parabola whose axis is Oy and whose vertex is at a distance c/8 below the origin. j 322, 323, 324] SUSPENSION BRIDGE. 591 324. Chain of Suspension Bridge. If the number of vertices be very great and the suspended weights all equal, the parabola on which the vertices lie coincides with the string. An example is furnished by the chain of a suspension bridge (Fig. 145). The vertical bars carry the weight of the flooring, and are equally spaced throughout the span. The vertices of the funicular polygon formed by the chain thus lie on a parabola, and if the number of bars be great the polygon will coincide with the curve. Fig. 14.5. We refer the chain to axes of reference in its plane, with the origin at the lowest point. The curve in which the chain lies is represented by the equation 0? = 4iay, where a has not the same meaning as in § 323. Differenti- ating, we obtain dy/dx = x/2a = 2y/x. Hence, if 2s is the span of the bridge, and h is the height, the tangent of the angle a, made by the chain at the highest point with the horizontal, is 2h/s. If T is the stretching force in the chain at the point of attachment, and W the total weight carried, we have Tsma.= W/2, or (1) and if H denote the stretching force in the chain at the lowest' point, -i „ 4 h (2) These results may be obtained more directly as follows. Let OPB (Fig. 146) represent a portion of the chain, and T denote the stretching force in the chain at the point 592 A TREATISE ON DYNAMICS. [CH. xr. (x, y). Then, taking axes of reference as shown, denoting by w the load per unit of the span, and resolving hori- zontally and vertically, we have T^^ = H, T^~- = wx. as as „ dy wx Hence ^- =-==-, ax H which gives on integration y= 1 w the constant of integration being zero, since y = when x = Q. Again, we may suppose the load carried by the portion OPB of the chain to act in the vertical line PA, where A is the mid-point of OG. We have at B dy/dx = ta,n lBAC. Now the forces H, wx, and T are parallel to the sides of the triangle BAG. Consequently T AB H AG wx BC wx~BG' from which the expressions for the stretching force at the highest point may easily be determined in terms of the total load, the height, and the span. 325. Catenary. The form of the curve (called the catenary), as- sumed by a perfectly flexible, homogeneous, inextensible cord when suspended from two fixed points, and acted on solely by its weight, and the forces applied by the supports, can be found as follows. Since the cord is perfectly flexible, the action of one part of the cord on a neighbouring part will be everywhere along the cord. Let the weight of unit length of the cord be §§ 324, 325] CATENARY. 593 w, and take axes as shown in Fig. 147, with the origin at the lowest point. If T be the stretching force at a point P, at a distance s from measured along the cord we have, resolving vertically and horizontally, ^S=-' ^S=^' (1) where H is the stretching force in the cord at the lowest point. Hence , ^ ay _ws _s /2\ dx ~ H~ c where c = Hjw. Now, if ds denote an element of the cord at P, we have ds^^dx^ + dy^- From (2), we obtain dy^ _ s^ dx^ _ (9) the equation of the catenary in terms of x and s. The X, y equation may easily be obtained from it by (5). Or we may proceed thus : we have dy/dx = sjc, and therefore t-W-'-") <-) Therefore y^-^le' + e M + const. The value of the constant is —c, and hence 2/+c = |(e°+e'') (11) If we transfer the origin to a point at a distance c vertically below the lowest point of the cord, equation (11) takes the simpler form y = ^-{f+e-') (12) Equation (5) becomes y=»/c^+s^, (13) so that (3) may be written in the form dy^s__ dx^c /j4^ ds y' ds y The stretching force at any point in the cord may easily be obtained. We have' T dy/ds = ws, a,nd substituting the value of dy/ds given by the first equation of (14), we get T=wy (15) The stretching force at any point in the cord thus equals the weight of a part of the cord whose length is equal to the ordinate of the point. It is to be remembered that the origin is now at a distance c below the lowest point of the cord. §§ 325, 326] CATENARY. 595 326. Geometrical Properties of Catenary. We may here establish some geometrical properties of the curve. If 6 denote the angle made with the horizontal by the tangent at a point P, we have tan 6 = s/c, and therefore sec^6dd/ds = l/c, that is, (,l+s^/c^)de/ds = 1/c. Hence, if R is the radius of curvature at P, (13) gives R = ds dd' X c (1) In Fig. 148 a point P is taken on the curve, and a perpen- dicular is let fall from P upon the axis of x, meeting the axis in N. On PN as diameter a circle, centre 0, is con- structed; N'T is a chord of the circle of length c; TM is a perpendicular let fall from T upon PJV^. Joining T to P, we have the angle MTP equal to the angle PNT. Again, cos LTNP=c/y, by construction: hence cos LMTP=c/y. But by equations (15) above, dxjds = cjy. Hence PT is a tangent to the curve at P. A line drawn through P perpendicular to TP is the normal at the point P If G'P = y^/c, then G' is the centre of curvature of the curve at the point P. Now, let the normal PC be produced backwards to meet the axis of x in L. It is easy to see that the triangles PNL and PTN are similar, and hence PN/NT=PL/PN, that is, PL = y^/c. Hence PL is the length of the radius of curvature. This suggests a geometrical method of con- structing the curve if the lowest point A, and the value of O N Fig. 148. 596 A TREATISE ON DYNAMICS. [CH. XI. O B, Fig. 149. c are given.' From the value of c we find the origin 0. From 0, Fig. 149, we draw OA and produce it to ^' making AA' = OA; with A' as centre, and A'A as radius, we draw a short circular arc AB; A' ia now joined to B, and A'B produced to meet the axis of cc in Bj. BB^ is the radius of curvature of the curve at the point B. We now pro- duce BA' backwards to B' making BB' = BB^; B' is the new centre of curva- E, ture. With B' as centre and HB as radius, we draw a short circular arc BD. Repeating this process, the complete curve may be built up. 327. Flexible Chain under Great Stretching Force. The radius of curvature for the lowest point of the curve is c. Consequently, if the curve is flat, the value of c is great. This will be the case if the sag is small in comparison with the distance between the points of attachment. If the span 2x and the sag d are given, we have y = c-\-d, and hence {c + d)^ = c^-\-s^, or d^+2dc = ^. If the cord is very tightly stretched, c is great, and we get as an approximation from (9) of § 325, by expansion, s = x+lx^/c^, which leads to c = x^/^d. 328. Transmission of Power by Belt. Power is often transmitted by means of a belt passing over two wheels or pulleys, and tightly stretched to prevent slipping. In Fig. 150, let W.^ be the driving wheel and W^ the driven wheel. When the motion is uniform, let the stretching forces in the parts AB, CD of the belt be T-^ and 1\ respectively. To find the relation which holds between T^ and T^ when slipping is about to occur, let PQ (Fig. 151) represent a small portion of the belt. The forces acting on PQ are (1) the force T at P, (2) the force T+dT at Q, (3) the reaction dR of the pulley. Since these three §§326,327,328] DRIVING BELT. 597 forces are in equilibrium, tiiey must meet in a point, and since slipping is on the point of taking place, dR must make with the radius O^P of the wheel an angle (j>, where _ C vT D Q B^ A T+- Hence, for the equilibrium of the portion PQ of the string, we have dT d / r7r\ clr with two similar equations. If o- be the mass per unit length of the string at P, fi = a- 3s, and the equations become ■(1) If the forces X, Y, Z are derivable from a potential, they take the form 7 / v . ^rr i(^l)-w-».«'« (^' If the string is not in equilibrium we have to equate the §§ 328, 329, 330] CATENARY FROM GENERAL EQUATIONS. 599 forces on the left of (1) to cr«, a-y, crz respectively, and the equations of motion - = I(^S) + -^' «tc (3) are obtained. Resolving along the string we get, since {dxjdsf + {dyjdsf + {dzjdsf = 1, /..dx , ..dy , ..dz\ dT , ( .^dx , „dy , „dz\ ... ''V'rs+yi+'ds)=d^+<^d^+^i+^ds)- (^) On the left is the acceleration of the element along the string, on the right is the rate of variation dTjds of the stretching force, and the tangential component of applied force along the string. If now as in Ex. 5, p. 95, the only sensible forces applied to the string be due to the normal action of the peg, the second term on the right is zero. Thus we get for an element of length ds, , ( ..dx , ..dy , ..dz\ dT , ,^. '"^'v'ds+y-^+''dj=-ds'^' ^^^ The integral of this is small if the part ,s integrated over is small. This is the justification of the assumption of the equality of T^ and T^ made in the Example referred to. 330. Application of General Ectuations to Catenary. As a first example, we may apply equations (1) of §329 to the case of a uniform flexible string suspended from two points and hanging under the action of gravity. For axes of reference in the plane of the string, the equations become, since X = 0, Y= —g, Integrating the first of these equations we obtain ^S = ^' ^') where iT is a constant. This equation shows that the 600 A TREATISE ON DYNAMICS. [cH. XI. horizontal component of- the stretching force is constant throughout the string. Integrating (2), we obtain where c' is a constant. If the origin be taken at the lowest point of the curve and the weight of unit length of the string be denoted by w, the last equation becomes ^| = --- (3) Equations (2) and (3) agree with (1) of § 325, from which the equations of the catenary were derived. 331. Equation of Catenary of Uniform Strength. Again we may apply the equations to find the form assumed by a flexible string hanging under gravity, when its cross-section at any point is proportional to the stretching force there existing. Here T varies as o-, so that we may write r=X(7, where X is a constant. We have Introducing this value of T in the second of (1), § 330, we obtain „ , jrO'V dx _ dx^ ds~ '^' and since H= Tdxjds and T=\a, the equation just obtained may be written d^y /drV 1 dx'Kds/ c where c is written for X/g. Writing l/{l+{dy/dxf} for (dx/dsf and integrating, we get tan-i T^=--|- const (2) If the origin is taken at the lowest point of the curve, the constant is zero, and we have dy/dx = ta,Ti(x/c), which gives y = c log sec - (3) §§330-333] GRAPHICAL STATICS. 601 For the reason that the stretching force per unit area is constant throughout the string, the curve determined by (8) is called the catenary of equal strength. 332. Rigid Body acted upon by Forces. The equilibrium of a rigid body is best regarded as the limiting case of the conditions set forth in Chaps. II. and IV., in which the accelerations are zero. But it is sometimes useful to con- sider the subject separately, and therefore the following outline of the statics of a rigid system is given. The effect produced by a given force upon a body depends on (1) the magnitude of the force, (2) its direction, (3) its line of action. It is easy to see that the force may be supposed to act at any point in its line of action. Thus, let F act at the point A in the line BA (Fig. 152). At B apply two -F F F t 1 ^^ 1 V B A Fig. 152. equal and opposite forces of magnitude F, one along BA and the other along AB, Provided that the point A is rigidly connected to the point B, it is evident that the three forces specified are together equivalent to the force F at A. But the force F at A and the force —Fa,tB are in the same line, and hence have a zero resultant. Thus the force ^ at ^ is equivalent to the force F at B. Hence we may suppose a force applied to a body to act at any point in its line of action, provided that the point be rigidly connected to the body. If a rigid body is acted upon by a system of forces which are concurrent, the conditions of equilibrium are easily established. Each force may be supposed to act at the point of intersection of the forces, and the conditions of equilibrium are identical with those found above for the case of a particle. 333. Resultant of Two Parallel Forces. Before dealing with the general case of a rigid body in equilibrium under the action of forces, it is necessary that we should discuss the properties of parallel forces. A force P 602 A TREATISE ON DYNAMICS. [CH. XI. Fig. 153. (P'ig. 153) is applied at A and a force Q in the .same direction at B. Join AB and apply at J. a force F in the direction BA, and at 5 a force i^in the direction AB. Evidently these two forces together produce no effect upon the equilibrium of the body. The forces P and F acting at A are equivalent •^' ^ -F to a force R acting in the line OA. Similarly, the forces Q and F acting at B are equivalent to a force 8 acting in the line OB. At 0, where the lines of action of R and 8 intersect, we resolve R into the components F parallel to BA and P along OG, which is parallel to the directions of the forces P and Q. Treating the force /S in a similar manner, we obtain a force F a,i parallel to AB, and a force Q along 00. The two equal and opposite forces at may be removed, and we are left with a force of amount P+Q acting in the line OG. Thus the two parallel forces P and Q acting at the points A and B are equivalent to a force of amount P + Q acting in the line OG. Now the sides OG, GA, / ' \ \ \ \p-Q OA of the triangle OGA are parallel to the forces F B/ P, F and R. Hence \7\o A F G OG/GA=P/F. FiG.1.54. Similarly, from the triangle 0GB, we obtain OGIGB=Q/F. These two equations give P . GA = Q . GB, which de- termines the -position of G. The case in which the parallel forces are in opposite directions is dealt with in an exactly similar manner. The procedure is illustrated in Fig. 154. The student will §§ 333, 334] GRAPHICAL STATICS. 603 have no difficulty in proving that the tviro forces P and Q applied at the points A and B are equivalent to a single force of amount P-Q acting in the line OG, which is parallel to the lines of action of P and Q. The position of is again given hy P .AG=Q. BO. 334. Centre of System of Parallel Forces. Obviously vvrhere the number of parallel forces is greater than two, the magnitude and line of action of the resultant may be found by repeated application of this method. Let the forces be F^, F^, F^, ..., Fn, and let them be applied at points whose coordinates are i^^V Vv 2^l). (^2- 3/2. ^i)' (^3' Vi' ^3). ••• . i^n, Vn, Zn). If we denote the coordinates of the point of application of the resultant of F^^ and F^ by (x', y', z'), those of the point of application of the resultant of F^^, F^, and Fg by (x", y", z"), etc., we have at once, by the previous paragraphs, F^(x'-x{) = F^{x^-x') or iFi + F^)x=F^Xj^ + F^x^, (1) with similar equations for y' and z'. Proceeding a step further, we obtain, (F^ + F^)(x" - x') = F^ix^ - x") or {F^+F^+Fg)x" = F^x^+F^^+FgX^ (2) with similar equations for y" and sf'. Dealing with all the forces in turn, we obtain finally for the coordinates (x, y, i) of the point of application of the resultant, {F^+F^+Fg+... + F„)x = F^x, + F^x^ + /^gCCg + . . . + FnX„, (3) with similar equations for y and i. Hence we have _ I,Fx _ "LFy - XFz /4^ ''=xF' y^YF' ^=xr ^ ^ It is to be noted that the expressions for x, y, and z do not depend on the direction of the parallel forces. It follows that the position of this point is not changed by turning all the forces about their points of application, provided that they remain parallel. For this reason the point (x, y, z) is called the centre of the parallel forces. 604 A TREATISE ON DYNAMICS. [CH. XI. 335. Centre of Gravity of Body. A body situated at the surface of the earth is acted on by a system of very nearly parallel gravity forces, since the body may be supposed built up of a system of particles rigidly connected together. Supposing the body divided up into such particles of masses in^, m^, m^, etc., we have F.^ = m.^, F^ = 'm^, etc. Hence __2mgra; -_1,mgy -_ 'Emgz , ~ 1,'mg ' ^ ~ H/mg ' "Limg ' that is x = Swi y=- 2m Swi- (1) The point {x, y, z) is called the centre of gravity of the body. It coincides with the c.i. as found in § 59. In strictness a C.G. does not exist except for bodies belonging to a limited class called centrobaric bodies. But the dis- cussion of centrobaric conditions belongs to the subject of Attractions, which is not dealt with in this book. 336. Graphical Method for Parallel Forces. The line of action of the resultant of a system of parallel forces applied to a rigid body may be found by a graphical process. We take as an example the case of a bridge carrying a series of loads as shown in Fig. 155. The load W.^ may be supposed Wa to act at any point A' in its line of action, the line being =^ supposed rigidly connected to A. The force W^ at A' may now be resolved into two components, one (arbitrary) in the line I, and the other in line II, it being of course understood that the lines I and II are rigidly connected to A'. The line II is pro- duced backwards to meet the vertical through B in B'. Let now a force equal and opposite to that acting at ^' in the line II act at B'. Combining this with the force TTj acting in the line BB', we obtain a force in the line III. j 335, 336, 337] GRAPHICAL STATICS. 605 A force equal to this reversed must now be supposed to act at the point G\ the point of intersection of the line III and the vertical through G. Combining it with TTg acting in the line GO', we obtain a force acting in the line IV. This force together with the force acting in the line I are equi- valent to the forces W-^, W^, and Wg actiug at the points A, B and G. Producing the lines I and IV until they meet, we obtain a point G in the line of action of the resultant. 337. Application to Loaded Bridge. The method of carry- ing out the graphical construction is shown in Fig. 156 for a bridge carrying loads W^, W^, W^, W^, and W^. On a vertical line set off parts 12, 23, 34, 45, and 56 to represent Fig. 156. the loads. Then, selecting any point as pole, join 01,^ 02, 03, 04, 05, and 06 as shown. Starting at any point S'l, in the vertical through the left-hand point of support of the bridge, draw a line 8[A' parallel to 01, meeting the vertical through A in A'; from A' draw A'B' parallel to 02, meeting the vertical through B in B' ; and from B' draw B'C parallel to 03, meeting the vertical through C in C". _ The process is continued until finally we arrive at the point 82 606 A TREATISE ON DYNAMICS. [CH. XI. in the vertical through the right-hand point of support of the bridge. The diagram on the right is called the force- polygon ; the polygon S'.A'B'G'D'H'S;, is called the funicular polygon. From the force-polygon we see that the force represented by 12 is equivalent to the forces represented by 10 and 02 ; the force represented by 23 is equivalent to the forces represented by 20 and 03 ; and similarly for the remaining vertical forces. If, now, we suppose the weights to act at the points A', B', etc., we see that we may replace Ifj by the forces represented by 10, 02 acting in the lines 8'iA', A'B' ; similarly we may replace 1^^ by the forces tepresented by 20, 03 acting in the lines A'B' and B'C ; and similarly for the other weights. We observe that we have two equal and opposite forces acting in each of the lines A'B', B'C, CD', D'E'. Consequently the forces Ifi, W^, Fg, W^, and W^, acting at the points A, B, G, D, E, are equivalent to the forces represented by 10, 06 acting in the lines *Si^-4', S'^E' ; producing these lines until they meet, we obtain a point in the line of action of the resultant. The vertical thrusts exerted at the points of support Si and 82 are readily deduced from the diagram. If from we draw 07 parallel to the line S'lS'^, it is easy to see that 17 represents the force applied to the left-hand support and 76 the force applied to the right-hand support. The force represented by 10 is equivalent to the forces repre- sented by 17 and 70, and the force represented by 06 to the forces represented by 07 and 76. Consequently, if the force along S'^A' be resolved into a vertical component and a component along S'lS'^, and likewise the force along S'^E'i into a vertical component and a component along 8'.^'i, the two component forces in the line S'^S'^ are equal and opposite. Consequently the vertical thrusts applied to the supports are represented by 07 and 76. 338. Theory of Couples. The methods described above for the finding of the resultant of a pair of parallel forces break down in the case where the two forces are equal in amount, parallel, and opposite in sign. Such a system of ! 337, 338] GRAPHICAL STATICS. 607 forces is termed a couple. The subject of couples has already been touched upon in §79. We give here some further explanation. Referring to Fig. 153, we have for the point 0, P. OA = Q . OB. If we put P = Q in this equation, we have QA = OB, which is true only when is at infinity. The resultant force is P — Q, and hence in the case of a couple the resultant force is zero, and its line of action is a line parallel to the forces, and at an infinite distance from them. The perpendicular distance apart of the lines of action of the two forces is called the arm of the couple. In Fig. 157 let the arm be represented by AB. The product of either force into the arm is called the moment of the couple. Thus, in the case of the couple shown in the diagram, the moment is FxAB. This moment evidently measures the moment of the forces about any point in their plane. Thus, if we produce AB to and take moments about 0, we have Moment of forces about = F.OA-F.OB^ F.AB. ♦F -ri ■--Ji. F B' ■F Fig. 157. /0~--. Fig. 1D8. F A' The same result is obtained if lies between A and B ; in this case the moments of the forces are of the same sign. Certain tlieorems hold for couples acting on a rigid body. In the first place, we shall prove that the effect of a couple is not changed by translating it in its own plane or to any parallel plane. In Fig. 158 let ABhe the arm of the couple in its initial position, and A'B' the arm of the couple after the translation. Let the magnitude of each of the forces of the couple be F. Now introduce at A' and B' two equal and opposite forces, each equal and parallel to the forces at A and B ; obviously the system is 608 A TREATISE ON DYNAMICS. [CH. XI. in no way altered. Now the force +F a,t A and the force + F a,t B' combine to give a force +2F a.t 0; likewise the force —FaiiB and the force — i'^at J.' combine to give a force — 2F at 0. These two resultant forces being equal and opposite have a zero effect upon the system. We are left with the force +F a^t A' and the force — i'' at B', which proves the proposition. The effect of a couple is not changed if it is rotated in its own plane. To prove this proposition, let AB be the arm of the couple in its initial position, and A'B' the arm turned about through an angle. At A' and at B' let two equal and opposite forces each of amount F be introduced, each force being at right angles to A'B'. The force — ^ at A' and the force +F a,i A combine to give a resultant along ED; and the force -F at B and the force +F at B' combine to give a resultant along DE. These two resultants are equal and opposite, and we are left with the force +F a,t A' and the force — i^ at B', which constitute a couple equal to the original couple in all respects. The effect of a couple is not changed if the magnitude of each of its forces and its arm are changed, provided that the moment of the couple remains unaltered. Let AB=p be the original arm of the couple, and let A'B'—p' be the new arm. At A' and B' introduce two equal, and opposite forces, each of amount F' = Fp/p', in directions parallel to the original forces. The force —F' at A' and the force —F at B combine to give a resultant —{F+F') at 0; likewise the force F a,t A and the force F' at B' combine to give a resultant (F+F') at 0. We are left with the force F'- at A' and a force —F'atB'. It thus appears that the effect produced by a couple upon the equilibrium of a^ rigid body depends on (a) the Fig. 1.59. §§ 338, 339, 340] COUPLES. 609 moment of the couple, (b) the direction in which the couple tends to produce rotation, (c) the normal to the plane in which it is situated. 339. Graphical Eepresentation of a Couple. It follows from the preceding section that a couple may be repre- sented completely by a straight line. The line is drawn at right angles to the plane of the couple; its length represents the moment of the couple, and the direction in which it is drawn indicates the direction in which the couple tends to produce rotation. The con- vention adopted in drawing the line is as follows: if the couple, as viewed from one side, tends to produce counter- clockwise rotation, the line is drawn towards the observer ; if it tends to produce clockwise rotation, the line is drawn away from the observer. In Fig. 160 the couple shown in the plane abed tends to pro- duce counter-clockwise rotation as viewed from above; we therefore represent it by a line OA drawn / ^ upwards at right angles to the / p / plane. The student will see that // if the couple is viewed from below / ^~'^ /■ it will tend to produce clockwise « b \ rotation, and hence the line OA fig. 160. must be drawn upwards as before. Since the effect of a couple is not altered by translating it in its own plane or to a parallel plane, it is immaterial where the initial point of the line OA is taken. The line OA is called the axis of the couple. 340. Composition and Resolution of Couples. Now let two couples in planes inclined to one another act on a rigid body. It is easy to show that the two couples are equivalent to a single couple, the axis of which is obtained by compounding the axes of the two couples according to the parallelogram law. Let the couples act in planes perpendicular to the paper (Fig. 161) ; let OA be the trace of one plane, and OS the trace of the other. The two planes intersect in a line, which is represented in plan by in the figure. We may represent the couple in the plane OA G.D. 2q 610 A TREATISE ON DYNAMICS. [CS. XI. by its axis Oa, drawn for convenience from 0, and the couple in the plane OB by its axis Ob, as shown. In the figure the couples are both supposed to be counter- clockwise, as seen by an eye placed at G. The axis of the resultant couple is obtained by completing the parallelogram and taking the diagonal passing through 0. For let the arm of each couple be so changed that each of the forces become unity ; the magnitudes of the couples will then be represented by their arms. Now let the couple in the plane OA be translated until one of its forces passes through towards the reader; and let the couple in the plane OB be translated until that one of its forces which is from the reader passes through 0. The two forces at being equal and opposite, we are left with a force of unit amount at A at right angles to the paper and away from the reader, and a force of unit amount at B at right angles to the paper and towards the reader; that is, we have a couple in the plane of which AB is the trace, whose magnitude is represented by AB. The student will have no difficulty in proving that the triangle oaO is equal to the triangle A OB, and that the line OG is perpendicular to the line AB; that is, that the two couples in the planes OA and OB are equivalent to the couple whose axis is OG. When a number of couples act on a rigid body, their resultant is found by adding their axes geometrically. We resolve each axis into components along three rect- angular lines of reference Ox, Oy, Oz. The axes which lie along Ox are added, and likewise those along Oy and Oz. If L, M, N are the sums of the axes in these direc- tions, we have for the magnitude G of the resultant couple, and its direction-cosines I, m, n, G=JIJ+W+N'^; ...(1) , L M N (2\ §§340,341] FORCE AND COUPLE. 611 It is easy to see that a couple and a force F in the same plane are equivalent to a force of equal amount, and in the same direction, acting in a line at a distance OjF from the line of action of F. To prove that this is the case, we merely have to rotate and translate the couple in its plane until that one of its forces which is opposite in sign to the force F lies in the same line. Keeping the line of action of this force fixed, we trans- form the couple so that each of its forces is of magnitude F. The two equal and opposite forces annul one another, and we are left with a single force of amount F acting in the line specified. Conversely, a single force F applied at a point P in a rigid body can be replaced by an equal and parallel force F applied at any other point Q of the same body, together with a couple formed by ^ at P and — ^ at Q. 341. deduction of System of Forces to Force and Couple. Let forces F-y, F^,..., Fn, having components X■^, Y^, Z^, X^, Y^, Z^, ..., Xn, Y^, Zn, be applied to a rigid body at points {x^,y^, %), {x^, ■2/2, %), ...,{xn,yn, ^n)- Let X, Y, Z be the components of a representative force F applied at the point P(x, y, z). We drop a perpendicular from P (Fig. 162) upon the plane yx meeting it in m, and from m we draw a line 7nn parallel to Oy meeting the plane xz in n. 1 ^^ The force Z may be supposed 2 applied at the point n. At each of the points n and we introduce two equal and opposite forces, each of amount Z. The force Z at P and the force — Z ai n form a couple +Zy with axis Ox; likewise the Pig. 162. force Z at n and the force — Za,t form a couple -Zx with axis Oy. Hence the force .^ at P is equivalent to the force .Z' at together with the two couples specified. Dealing with the forces X and F in like manner, we arrive at the result that G.D. 2q2 *z y o 612 A TREATISE ON DYNAMICS. [CH. XI. the force F' at (x, y, z) is equivalent to the force F a,i together with a couple Zy — Yz with axis Ox, a couple Xz — Zx with axis Oy, and a couple Yx — Xy with axis Oz. We thus see that the system of forces F.^, F^, etc., applied at the points (ajj, y^ z-^), etc., are equivalent to a system of equal and parallel forces applied at the origin together with a couple 'E{Zy— Yz) with axis Ox, a couple l!,(Xz — Zx) with axis Oy, and a couple 'E(Yx — Xy) with axis Oz. If F is the resultant force, we have F^ JCEXf + (2 Yf + (EZf ; (1) its direction-cosines are (2X)/i2, (SF)/B, C2Z)/R (2) If G is the axis of the resultant couple, and L, M, N are its components, we have L = ^{Zy-Yz), M=I,(Xz-Zx), N=^{Yx-Xy); ...(3) 0=JT?+M^Tn'^; (4) the direction-cosines of are LjQ, M/Q, N/Q. 342. Conditions of Equilibriiun. To obtain the accelera- tion of the centroid of a body, we suppose all the forces transferred to the centroid without change. Hence the centroid will be without acceleration if 2X = 0, 2F=0, I,Z=0 (1) Further, the rate of change of moment of momentum of the body about all axes will be zero if X(Zy-Yz) = 0, i:(Xz-Zx) = 0, ^{Yx-Zy) = 0. ...(2) When a rigid body is without linear acceleration of its centroid and angular acceleration about any axis, it is said to be in equilibrium. The equations (1) and (2) are called the equations of equilibrium. (See § 75 above.) 343. Poinsot's Central Axis, Wrench. We have seen that any system of forces acting on a rigid body is equivalent to a single resultant force F, acting at an arbitrary origin 0, and a resultant couple 0. In Fig. 163 let be the origin, F the resultant force, and the axis of the couple, drawn J 341, 342, 343] CENTRAL AXIS. 613 for convenience from 0. We resolve Q into two com- ponents Om and On along and perpendicular to F. The component Om represents a couple in any plane perpen- dicular to F, and On represents a couple in any plane perpendicular to On. This latter couple and the force F are equivalent to a force of amount F acting in a line O'T, whose dis- tance from is OnjF, that is, G sin QjF, where Q is the angle between F and G. The line 00', it will be observed, is per- pendicular to the plane containing F and G. The force i^ at and the couple G are thus equivalent to a force F in O'T together with a couple in a plane perpendicular to O'T. The line O'T is called Poinsot's central axis. The combination of a force acting in a straight line and a couple whose axis co- jiq_ igs. incides with the line is termed a wrench. The ratio GjF, which evidently represents a length, is termed the pitch of the wrench. Let X, Y, Z be the components of the force F at 0, and L, M, N those of the couple G. The component couples about any point of coordinates x, y, z are L-Zy+Yz. M-Xz + Zx, N-Yx+Xy. Now the central axis is a line in the direction of the force F, such that the force system reduces to the parallel force F along that line, and a couple about that line. Hence the axis of the couple must have direction-cosines proportional to X, Y, Z. They are also proportional to the component couples written above. The equations of the central axis are therefore L-Zy+Yz M-Xz-Zx N-Yx+Xy /gx X ~ Y ~ Z This as the reader may verify can be transformed to x — a_y — b_z — c /^\ ~^~^~^~~z~' ^ ^ where a,b,c = {NY- MZ, LZ- NX, MX -LY)/F, (5) 614 A TREATISE ON DYNAMICS. [CH. SO that a, b, c are the coordinates of a point through which the central axis passes. [Compare the discussion of the Central Axis of the Motion of a Body, § 247.] We conclude the chapter with some examples, worked and unworked. EXEBCISES XI. 1. A man walking at the rate of 5'5 feet per second drags behind him 19 feet of flexible rope weighing five pounds per foot. If he holds the end of the rope 5 feet above the ground, show that he works at the rate of '12 h.p. in dragging the rope (coefficient of kinetic friction between rope and ground = 0'2). Let I feet be the length of the rope dragged along the ground ; the remaining 19 — Z feet will hang in a catenary. Denoting the weight of one foot of the rope by w and the coefficient of friction by jj., we see that the stretching force in the catenary at the lowest point is wwl. Hence c=fil. The value of y at the highest point is 5 + c, and the length of the catenary is 19 I, so that Introducing the value of fi, and reducing, we get Z2- 40^+336 = 0, which gives Z = 20 ± 8. Hence the length of rope dragged along the ground is 12 feet. The horizontal force applied is therefore 12 Pounds, and the rate of working in horse-power is 12 x 5'5/550=0'12. 2. A heavy uniform chain 110 feet long is stretched between two points in the same level 108 feet apart. Find the stretching force in the chain at either of the points of attachment. If I denotes the length of the chain and d the span, we have Expanding the right-hand side of this equation and remembering that is great, we obtain c2=rf3/24(Z-